U.S. patent application number 17/163219 was filed with the patent office on 2021-07-29 for fusion based quantum computing.
This patent application is currently assigned to Psiquantum, Corp.. The applicant listed for this patent is Psiquantum, Corp.. Invention is credited to Mercedes Gimeno-Segovia, Naomi Nickerson, Terence Rudolph.
Application Number | 20210232963 17/163219 |
Document ID | / |
Family ID | 1000005475096 |
Filed Date | 2021-07-29 |
United States Patent
Application |
20210232963 |
Kind Code |
A1 |
Gimeno-Segovia; Mercedes ;
et al. |
July 29, 2021 |
FUSION BASED QUANTUM COMPUTING
Abstract
A method includes receiving a plurality of quantum systems,
wherein each quantum system of the plurality of quantum system
includes a plurality of quantum sub-systems in an entangled state,
and wherein respective quantum systems of the plurality of quantum
systems are independent quantum systems that are not entangled with
one another. The method further includes performing a plurality of
joint measurements on different quantum sub-systems from respective
ones of the plurality of quantum systems, wherein the joint
measurements generate joint measurement outcome data and
determining, by a decoder, a plurality of syndrome graph values
based on the joint measurement outcome data.
Inventors: |
Gimeno-Segovia; Mercedes;
(San Jose, CA) ; Rudolph; Terence; (San Francisco,
CA) ; Nickerson; Naomi; (San Francisco, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Psiquantum, Corp. |
Palo Alto |
CA |
US |
|
|
Assignee: |
Psiquantum, Corp.
Palo Alto
CA
|
Family ID: |
1000005475096 |
Appl. No.: |
17/163219 |
Filed: |
January 29, 2021 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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62967513 |
Jan 29, 2020 |
|
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63140210 |
Jan 21, 2021 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06N 10/00 20190101 |
International
Class: |
G06N 10/00 20060101
G06N010/00 |
Claims
1. A method comprising: receiving, by a qubit fusion system, a
plurality of quantum systems, wherein each quantum system of the
plurality of quantum system includes a plurality of quantum
sub-systems in an entangled state, and wherein respective quantum
systems of the plurality of quantum systems are independent quantum
systems that are not entangled with one another; performing, by the
qubit fusion system, a plurality of joint measurements on different
quantum sub-systems from respective ones of the plurality of
quantum systems, wherein the joint measurements generate joint
measurement outcome data; and determining, by a decoder, a
plurality of syndrome graph values based on the joint measurement
outcome data.
2. The method of claim 1, wherein performing the joint measurements
includes performing fusion operations.
3. The method of claim 1, wherein performing the joint measurements
include performing a destructive joint measurement via a Type II
fusion operation.
4. The method of claim 1, wherein performing the plurality of joint
measurements on different quantum sub-systems from respective ones
of the plurality of quantum systems includes performing the
plurality of joint measurements on only a subset of the plurality
of quantum sub-systems that are received by the qubit fusion system
thereby resulting in a subset of unmeasured quantum
sub-systems.
5. The method of claim 4, further comprising, receiving, by the
qubit fusion system, a second plurality of quantum systems, wherein
each quantum system of the second plurality of quantum system
includes a second plurality of quantum sub-systems in an entangled
state, and wherein respective quantum systems of the second
plurality of quantum systems are independent quantum systems that
are not entangled with one another; receiving the subset of
unmeasured quantum sub-systems; and performing, by the qubit fusion
system, a second plurality of joint measurements between i) second
quantum sub-systems from respective ones of the plurality of second
quantum systems and ii) respective quantum sub-systems from the
subset of unmeasured quantum sub-systems, wherein the second
plurality of joint measurements generate second joint measurement
outcome data.
6. A system comprising: a qubit fusion system comprising a
plurality of fusion gates, wherein the qubit fusion system is
configured to receive a plurality of quantum systems, wherein each
quantum system of the plurality of quantum system includes a
plurality of quantum sub-systems in an entangled state, and wherein
respective quantum systems of the plurality of quantum systems are
independent quantum systems that are not entangled with one
another; wherein the plurality of fusion gates are each configured
to perform a joint measurement on different quantum sub-systems
from respective ones of the plurality of quantum systems, wherein
the joint measurements generate joint measurement outcome data; and
a decoder communicatively coupled to the qubit fusion system and
configured to receive the joint measurement outcome data and to
determine a plurality of syndrome graph values based on the joint
measurement outcome data.
7. The system of claim 6, wherein the fusion gates include a
photonic circuit and the plurality of quantum systems comprise
photons as the quantum sub-systems, wherein the photonic circuit
comprises a Type II fusion gate.
8. The system of claim 6, wherein the joint measurement comprises a
two-particle projective measurement onto a Bell basis.
9. The system of claim 6, further comprising a quantum memory,
coupled to at least one the qubit fusion system and that receive
and store a subset of the plurality of quantum sub-systems.
10. The system of claim 9, wherein the quantum memory is an optical
fiber.
11. The system of claim 9, wherein the quantum memory is coupled to
the qubit fusion system such that the joint measurement is
performed between i) quantum sub-systems from respective ones of
the plurality of quantum systems and ii) respective quantum
sub-systems from the subset of the plurality of quantum sub-systems
that are stored in the quantum memory.
12. The system of claim 6 further comprising a qubit entangling
system that is configured to generate the plurality of quantum
systems.
13. The system of claim 12, wherein the qubit entangling system
includes a quantum gate array.
14. The system of claim 13, wherein qubit entangling system
includes a photon source system that is optically connected to an
entangled state generator.
15. The system of claim 14, wherein the entangled state generator
is configured to receive output photons from the photon source
system and convert the output photons to an entangled photonic
state.
16. The system of claim 15, wherein the qubit entangling system
includes a plurality of output waveguides that are optically
coupled to the qubit fusion system and are configured to provide
the entangled photonic state to inputs of the fusion gates.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This patent application claims the benefit of U.S.
Provisional Patent Application No. 62/967,513, filed Jan. 29, 2020,
and also claims the benefit of U.S. Provisional patent Application
No. 63/140,210, filed Jan. 21, 2021. The disclosures of both
applications are hereby incorporated by reference in their entirety
for all purposes.
TECHNICAL FIELD
[0002] One or more embodiments of the present disclosure relate
generally to quantum computing devices and methods and, more
specifically, to fault tolerant quantum computing devices and
methods.
BACKGROUND
[0003] In fault tolerant quantum computing, quantum error
correction is required to avoid an accumulation of qubit errors
that then leads to erroneous computational outcomes. One method of
achieving fault tolerance is to employ error correcting codes
(e.g., topological codes) for quantum error correction. More
specifically, a collection of physical qubits can be generated in
an entangled state (also referred to herein as an error correcting
code) that encodes for a single logical qubit that is protected
from errors.
[0004] In some quantum computing systems, cluster states of
multiple qubits, or, more generally, graph states can be used as
the error correcting code. A graph state is a highly entangled
multi-qubit state that can be represented visually as a graph with
nodes representing qubits and edges representing entanglement
between the qubits. However, various problems that either inhibit
the generation of entangled states or destroy the entanglement once
created have frustrated advancements in quantum technologies that
rely on the use of highly entangled quantum states.
[0005] Furthermore, in some qubit architectures, e.g., photonic
architectures, the generation of entangled states of multiple
qubits is an inherently probabilistic process that may have a low
probability of success.
[0006] Accordingly, there remains a need for improved systems and
methods for quantum computing that do not necessary rely on large
cluster states of qubits.
SUMMARY
[0007] Described herein are embodiments of fault-tolerant systems
and methods for quantum computing that do not necessary rely on
large cluster states of qubits.
[0008] According to some embodiments, a method can comprise:
receiving a plurality of quantum systems, wherein each quantum
system of the plurality of quantum system includes a plurality of
quantum sub-systems in an entangled state, and wherein respective
quantum systems of the plurality of quantum systems are independent
quantum systems that are not entangled with one another; performing
a plurality of destructive joint measurements (such as fusion
operations) on different quantum sub-systems from respective ones
of the plurality of quantum systems, wherein the destructive joint
measurements destroy the different quantum sub-systems and generate
joint measurement outcome data and transfer quantum state
information from the different quantum sub-systems to other
unmeasured quantum sub-systems from the plurality of quantum
systems; and determining a logical qubit state based on the joint
measurement outcome data. The logical qubit state can be determined
in a fault tolerant manner.
[0009] According to some embodiments, a method can comprise:
receiving a plurality of quantum systems, wherein each quantum
system of the plurality of quantum system includes a plurality of
quantum sub-systems in an entangled state, and wherein respective
quantum systems of the plurality of quantum systems are independent
quantum systems that are not entangled with one another; performing
a logical qubit gate by performing a plurality of destructive joint
measurements (such as fusion operations) on different quantum
sub-systems from respective ones of the plurality of quantum
systems, wherein the destructive joint measurements destroy the
different quantum sub-systems and generate joint measurement
outcome data and transfer quantum state information from the
different quantum sub-systems to other unmeasured quantum
sub-systems from the plurality of quantum systems; and determining
a result of the logical qubit gate based on the joint measurement
outcome data. The result of the logical qubit gate can be
determined in a fault tolerant manner.
[0010] According to some embodiments, a quantum computing apparatus
can comprise: a qubit entangling system to generate a plurality of
quantum systems, wherein each quantum system of the plurality of
quantum systems includes a plurality of quantum sub-systems in an
entangled state, and wherein respective quantum systems of the
plurality of quantum systems are independent quantum systems that
are not entangled with one another; a qubit fusion system to
perform a plurality of destructive joint measurements on different
quantum sub-systems from respective ones of the plurality of
quantum systems, wherein the destructive joint measurements destroy
the different quantum sub-systems and generate joint measurement
outcome data and transfer quantum state information from the
different quantum sub-systems to other unmeasured quantum
sub-systems from the plurality of quantum systems; and a classical
computing system to determine a logical qubit state based on the
joint measurement outcome data.
[0011] According to some embodiments, a quantum computing apparatus
can comprise: a qubit entangling system to generate a plurality of
quantum systems, wherein each quantum system of the plurality of
quantum systems includes a plurality of quantum sub-systems in an
entangled state, and wherein respective quantum systems of the
plurality of quantum systems are independent quantum systems that
are not entangled with one another; a qubit fusion system to
perform a logical qubit gate by performing a plurality of
destructive joint measurements on different quantum sub-systems
from respective ones of the plurality of quantum systems, wherein
the destructive joint measurements destroy the different quantum
sub-systems and generate joint measurement outcome data and
transfer quantum state information from the different quantum
sub-systems to other unmeasured quantum sub-systems from the
plurality of quantum systems; and a classical computing system to
determine a result of the logical qubit gate based on the joint
measurement outcome data.
[0012] According to some embodiments a method includes receiving,
by a qubit fusion system, a plurality of quantum systems, wherein
each quantum system of the plurality of quantum system includes a
plurality of quantum sub-systems in an entangled state. Respective
quantum systems of the plurality of quantum systems are independent
quantum systems that are not entangled with one another. The method
further includes performing, by the qubit fusion system, a
plurality of joint measurements on different quantum sub-systems
from respective ones of the plurality of quantum systems. The joint
measurements generate joint measurement outcome data. The method
further includes determining, by a decoder, a plurality of syndrome
graph values based on the joint measurement outcome data.
[0013] According to some embodiments performing the joint
measurements includes performing fusion operations.
[0014] According to some embodiments performing the joint
measurements include performing a destructive joint measurement via
a Type II fusion operation.
[0015] According to some embodiments performing the plurality of
joint measurements on different quantum sub-systems from respective
ones of the plurality of quantum systems includes performing the
plurality of joint measurements on only a subset of the plurality
of quantum sub-systems that are received by the qubit fusion system
thereby resulting in a subset of unmeasured quantum
sub-systems.
[0016] According to some embodiments, the method further includes,
receiving, by the qubit fusion system, a second plurality of
quantum systems, wherein each quantum system of the second
plurality of quantum system includes a second plurality of quantum
sub-systems in an entangled state, and wherein respective quantum
systems of the second plurality of quantum systems are independent
quantum systems that are not entangled with one another. The method
further includes receiving the subset of unmeasured quantum
sub-systems and performing, by the qubit fusion system, a second
plurality of joint measurements between i) second quantum
sub-systems from respective ones of the plurality of second quantum
systems and ii) respective quantum sub-systems from the subset of
unmeasured quantum sub-systems. The second plurality of joint
measurements generates second joint measurement outcome data.
[0017] According to some embodiments a system includes a qubit
fusion system comprising a plurality of fusion gates. The qubit
fusion system is configured to receive a plurality of quantum
systems, wherein each quantum system of the plurality of quantum
system includes a plurality of quantum sub-systems in an entangled
state, and wherein respective quantum systems of the plurality of
quantum systems are independent quantum systems that are not
entangled with one another.
[0018] According to some embodiments the plurality of fusion gates
are each configured to perform a joint measurement on different
quantum sub-systems from respective ones of the plurality of
quantum systems, wherein the joint measurements generate joint
measurement outcome data.
[0019] The system further includes a decoder communicatively
coupled to the qubit fusion system and configured to receive the
joint measurement outcome data and to determine a plurality of
syndrome graph values based on the joint measurement outcome
data.
[0020] According to some embodiments the fusion gates include a
photonic circuit and the plurality of quantum systems comprise
photons as the quantum sub-systems, wherein the photonic circuit
comprises a Type II fusion gate.
[0021] According to some embodiments the joint measurement
comprises a two-particle projective measurement onto a Bell
basis.
[0022] According to some embodiments the system further includes a
quantum memory, coupled to at least one the qubit fusion system and
that receive and store a subset of the plurality of quantum
sub-systems.
[0023] According to some embodiments the quantum memory is an
optical fiber.
[0024] According to some embodiments the quantum memory is coupled
to the qubit fusion system such that the joint measurement is
performed between i) quantum sub-systems from respective ones of
the plurality of quantum systems and ii) respective quantum
sub-systems from the subset of the plurality of quantum sub-systems
that are stored in the quantum memory.
[0025] According to some embodiments the system of further includes
a qubit entangling system that is configured to generate the
plurality of quantum systems.
[0026] According to some embodiments the qubit entangling system
includes a quantum gate array.
[0027] According to some embodiments the qubit entangling system
includes a photon source system that is optically connected to an
entangled state generator.
[0028] According to some embodiments the entangled state generator
is configured to receive output photons from the photon source
system and convert the output photons to an entangled photonic
state.
[0029] According to some embodiments the qubit entangling system
includes a plurality of output waveguides that are optically
coupled to the qubit fusion system and are configured to provide
the entangled photonic state to inputs of the fusion gates.
[0030] The following detailed description, together with the
accompanying drawings, will provide a better understanding of the
nature and advantages of the claimed invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0031] Aspects of the present disclosure are illustrated by way of
example. Non-limiting and non-exhaustive aspects are described with
reference to the following figures, wherein like reference numerals
refer to like parts throughout the various figures unless otherwise
specified.
[0032] FIGS. 1A-1C are diagrams illustrating a cluster state and
corresponding syndrome graph for an entangled state of physical
qubits in accordance with some embodiments.
[0033] FIG. 2 shows a quantum computing system in accordance with
one or more embodiments.
[0034] FIG. 3 shows a quantum computing system in accordance with
some embodiments.
[0035] FIG. 4 illustrates an example of a qubit entangling system
in accordance with some embodiments.
[0036] FIG. 5 shows one example of qubit fusion system in
accordance with some embodiments.
[0037] FIG. 6 shows one possible example of a fusion site as
configured to operate with a fusion controller to provide
measurement outcomes to a decoder for fault tolerant quantum
computation in accordance with some embodiments.
[0038] FIGS. 7A-7C illustrates a fusion based quantum computing
scheme for fault tolerant quantum computation in accordance with
one or more embodiments.
[0039] FIGS. 8A-8C show one example of a lattice preparation
protocol for fusion based quantum computing in accordance with some
embodiments.
[0040] FIGS. 9A-9B show one example of a lattice preparation
protocol for fusion based quantum computing in accordance with some
embodiments.
[0041] FIGS. 10A-10E shows a flow chart and example lattice
preparation protocol for illustrating a method for fusion based
quantum computing in accordance with one or more embodiments.
[0042] FIGS. 11A-11E show representations of dual-rail-encoded
photonic qubits and photonic circuits for performing unitary
operation on photonic qubits in accordance with some
embodiments.
[0043] FIGS. 12A-12B show representations of dual-rail-encoded
photonic qubits and photonic circuits for performing unitary
operation on photonic qubits in accordance with some
embodiments.
[0044] FIG. 13 shows photonic implementations of beam splitters
that may be used to implement one or more spreaders, e.g., Hadamard
gates, according to some embodiments.
[0045] FIG. 14 shows photonic implementations of beam splitters
that may be used to implement one or more spreaders, e.g., Hadamard
gates, according to some embodiments.
[0046] FIG. 15 shows one example of a Bell state generator circuit
that can be used in some dual-rail-encoded photonic
embodiments.
[0047] FIG. 16 shows an example of a Type II fusion circuit for a
polarization encoding according to some embodiments.
[0048] FIG. 17 shows an example of a Type II fusion circuit for a
path encoding according to some embodiments.
[0049] FIGS. 18A-18D show effects of fusion in the generation of a
cluster state according to some embodiments.
[0050] FIG. 19 shows examples of Type II fusion gates boosted once
in polarization and path encodings according to some
embodiments.
[0051] FIG. 20 shows a table with variations of the Type II fusion
gate for different measurement basis in a polarization
encoding.
[0052] FIG. 21 shows examples of photonic circuit variations of the
Type II fusion gate for different choice of measurement basis in a
path encoding according to some embodiments.
DETAILED DESCRIPTION
[0053] Reference will now be made in detail to embodiments,
examples of which are illustrated in the accompanying drawings. In
the following detailed description, numerous specific details are
set forth in order to provide a thorough understanding of the
various described embodiments. However, it will be apparent to one
of ordinary skill in the art that the various described embodiments
may be practiced without these specific details. In other
instances, well-known methods, procedures, components, circuits,
and networks have not been described in detail so as not to
unnecessarily obscure aspects of the embodiments.
1. INTRODUCTION TO QUANTUM COMPUTING
[0054] Quantum computation is often considered in the framework of
`Circuit Based Quantum Computation` (CBQC) in which operations (or
gates) are performed on physical qubits. Gates can be either single
qubit unitary operations (rotations), two qubit entangling
operations such as the CNOT gate, or other multi-qubit gates such
as the Toffoli gate.
[0055] Measurement Based Quantum Computation (MBQC) is another
approach to implementing quantum computation. In the MBQC approach,
computation proceeds by first preparing a particular entangled
state of many qubits, commonly referred to as a cluster state, and
then carrying out a series of single qubit measurements on the
cluster state to enact the quantum computation. In this approach,
the choice of single qubit measurements is dictated by the quantum
algorithm being run on the quantum computer. In the MBQC approach,
fault tolerance can be achieved by careful design of the cluster
state and using the topology of this cluster state to encode
logical qubits that is protected against any logical errors that
may be caused by errors on any of the physical qubits that make up
the cluster state. In practice, the value of the logical qubit can
be determined, i.e., read out, based on the results (also referred
to herein as measurement outcomes) of the single-particle
measurements that are made on the cluster state's physical qubits
as the computation proceeds.
[0056] However, the generation and maintenance of long-range
entanglement across the cluster state and subsequent storage of
large cluster states can be a challenge. For example, for any
physical implementation of the MBQC approach, a cluster state
containing many thousands, or more, of mutually entangled qubits
must be prepared and then stored for some period of time before the
single-qubit measurements are performed. For example, to generate a
cluster state representing a single logical error corrected qubit,
each of the collection of underlying physical qubits can be
prepared in the |+ state and a controlled-phase gate (CZ) state can
be applied between each physical qubit pair to generate the overall
cluster state. More explicitly, a cluster state of highly entangled
qubits can be described by the undirected graph G=(V, E) with V and
E denoting the sets of vertices and edges, respectively and can be
generated as follows: 1) initialize all the physical qubits to be
in the |+ state, where
| + = ( | 0 + | 1 ) 2 ; ##EQU00001##
and 2) apply the controlled-phase gate (CZ) to each pair i,j of
qubits. Accordingly, any cluster state, which physically
corresponds to a large entangled state of physical qubits, can be
described as
|.PSI..sub.graph=.PI..sub.(i,j).di-elect cons.ECZ.sub.i,j|+.sup.|V|
(1)
where the CZ.sub.i,j is the controlled phase gate operator and with
V and E as defined above. Graphically, the cluster states defined
by Eq. (1) can also be represented by a graph with vertices V that
represent the physical qubits (initialized in the |+ state) and
edges E that represent entanglement between them (i.e., the
application of the various CZ gates). In some cases, e.g., cases
involving a fault tolerant MBQC scheme, |.PSI..sub.graph graph can
take the form of graph in 3 dimensions. Like the examples shown in
FIG. 1A and FIG. 7C, such a graph can have a regular structure
formed from repeating unit cells and is therefore often referred to
as a "lattice." When represented as a 3-dimensional lattice,
2-dimensional boundaries of this lattice can be identified. Qubits
belonging to those boundaries are referred to as "boundary qubits"
while all other qubits are referred to as "bulk qubits".
[0057] After |.PSI..sub.graph is generated, this large state of
mutually entangled qubits must be preserved long enough for a
stabilizer measurement to be performed, e.g., by making X
measurements on all physical qubits in the bulk of the lattice and
Z measurements on the boundary qubits.
[0058] FIG. 1A shows one example of a fault tolerant cluster state
that can be used in MBQC, the topological cluster state introduced
by Raussendorf et al., and commonly referred to as the Raussendorf
Lattice as described in further detail in Robert Raussendorf, Jim
Harrington, and Kovid Goyal. A., Fault-Tolerant One-Way Quantum
Computer, Annals of Physics, 321(9):2242-2270, 2006. The cluster
state is in the form of repeating lattice cells (e.g., cell 120)
with physical qubits (e.g., physical qubit 116) arranged on the
faces and edges of the cells. Entanglement between the physical
qubits is represented by edges that connect the physical qubits
(e.g., edge 118), with each edge representing the application of
the CZ gate, as described above in reference to Eq. (1). The
cluster state shown here is merely one example among many and other
topological error correcting codes can be used without departing
from the scope of the present disclosure. For example, volume codes
such as those disclosed within International Patent Application
Publication No. WO/2019/173651, the contents of which is hereby
incorporated by reference in its entirety for all purposes, can be
used. Also the codes based on non-cubical unit cells described in
International Patent Application Publication No. WO/2019/178009,
the contents of which is hereby incorporated by reference in its
entirety for all purposes, can be used without departing from the
scope of the present disclosure. Furthermore, while the example
shown here is represented in three spatial dimensions, the same
structure may also be obtained from other implementations of codes
that are not based on a purely spatial entangled cluster state, but
rather can include both entanglement in 2D space and entanglement
in time, e.g., a 2+1D surface code implementation can be used or
any other foliated code. For cluster state implementation of such
codes, all of the quantum gates needed for fault tolerant quantum
computation can be constructed by making a series of single
particle measurements to the physical qubits that make up the
lattice.
[0059] Returning to FIG. 1A, a chunk of a Raussendorf lattice is
shown. Such an entangled state can be used to encode one or more
logical qubits (i.e., one or more error corrected qubits) using
many entangled physical qubits. The collection of single particle
measurement results of the multiple physical qubits (e.g., physical
qubit 116) can be used for correcting errors and for performing
fault tolerant computations on the logical qubits through the use
of a decoder. Many decoders are available with one example being
the Union-Find decoder as described in International Patent
Application Publication No. WO2019/002934A1, the disclosure of
which is hereby incorporated by reference in its entirety for all
purposes. One of ordinary skill will appreciate that the number of
physical qubits required to encode a single logical qubit can vary
depending on the precise nature of the physical errors, noise,
etc., that are experienced by the physical qubits, but to achieve
fault tolerance, all proposals to date require entangled states of
thousands of physical qubits to encode a single logical qubit.
Generating and maintaining such a large entangled state remains a
key challenge for any practical implementation of the MBQC
approach.
[0060] FIGS. 1B-1C illustrate how the decoding of a logical qubit
can proceed for a cluster state based on the Raussendorf lattice.
As can be seen in FIG. 1A, the geometry of the cluster state is
related to the geometry of a cubic lattice (lattice cell 120) shown
superimposed on the clusters state in FIG. 1A. FIG. 1B shows the
single particle measurement results (also superimposed on the cubic
lattice) after the state of each physical qubit of the cluster
state has been measured, with the measurement results being placed
in the former position of the physical qubit that was measured (for
clarity only measurement results that result from measurements of
the surface qubits are shown).
[0061] In some embodiments, a measured qubit state can be
represented by a numerical bit value of either 1 or 0 after all
qubits have been measured, e.g., in the x basis, with the 1 bit
value corresponding to the +x measurement outcome and the 0 but
value corresponding to -x measurement outcomes (or vice versa).
There are two types of qubits, those that are located on the edges
of a unit cell (e.g. at edge qubit 122), and those that are located
on the faces of a unit cell (e.g., face qubit 124). In some cases,
a measurement of the qubit may not be obtained, or the result of
the qubit measurement may be invalid. In these cases, there is no
bit value assigned to the location of the corresponding measured
qubit, but instead the outcome is referred to herein as an erasure,
illustrated here as thick line 126, for example. These measurement
outcomes that are known to be missing can be reconstructed during
the decoding procedure.
[0062] To identify errors in the physical qubits, a syndrome graph
can be generated from the collection of measurement outcomes
resulting from the measurements of the physical qubits. For
example, the bit values associated with each edge qubit can be
combined to create a syndrome value associated with the vertex the
results from the intersection of the respective edges, e.g., vertex
128 as shown in FIG. 1B. A set of syndrome values, also referred to
herein as parity checks, are associated with each vertex of the
syndrome graph, as shown in FIG. 1C. More specifically, in FIG. 1C,
the computed values of some of the vertex parity checks of the
syndrome graph are shown. In some embodiments, a parity computation
entails determining whether the sum of the edge values incident on
a given vertex is an even or odd integer, with the parity result
for that vertex being defined to be the result of the sum mod 2. If
no errors occurred in the quantum state, or in the qubit
measurement then all syndrome values should be even (or 0). On the
contrary, if an error occurs, it will result in some odd (or 1)
syndrome values. Only half of the bit values from qubit measurement
are associated with the syndrome graph shown (the bits aligned with
the edges of the syndrome graph). There is another syndrome graph
that contains all the bit values associated with the faces of the
lattice shown. This leads to an equivalent decoding problem on
these bits.
[0063] As mentioned above, the generation and subsequent storage of
large cluster states of qubits can be a challenge. However, some
embodiments, methods and systems described herein provide for the
generation of a set of classical measurement data (e.g., a set of
classical data corresponding to syndrome graph values of a syndrome
graph) that includes the necessary correlations for performing
quantum error correction, without the need to first generate a
large entangled state of qubits in an error correcting code. For
example, embodiments disclosed herein described systems and methods
whereby two-qubit (i.e., joint) measurements, also referred to
herein as "fusion measurements" or "fusion gates" can be performed
on a collection of much smaller entangled states to generate a set
of classical data that includes the long-range correlations
necessary to generate and decode the syndrome graph for a
particular chosen cluster state, without the need to actually
generate the cluster state. In other words, in some systems and
methods described herein, there is only ever generated a collection
of relatively small entangled states (referred to herein as
resource states) and then joint measurements are performed on these
resource states directly to generate the syndrome graph data
without the need to first generate (and then measure) a large
cluster state that forms a quantum error correcting code (e.g., a
topological code such as the Raussendorf lattice).
[0064] For example, as will be described in further detail below,
in the case of linear optical quantum computing using a Raussendorf
lattice code structure, to generate the syndrome graph data, a
fusion gate can be applied to a collection small entangled states
(e.g., 4-GHZ states) that are themselves not entangled with each
other and thus are never part of a larger Raussendorf lattice
cluster state. Despite the fact that qubits from the individual
resource states were not mutually entangled prior to the fusion
measurement, the measurement outcomes that result from the fusion
measurements generate a syndrome graph that includes all the
necessary correlations to perform quantum error correction. Such
systems and methods are referred to herein as Fusion Based Quantum
Computing (FBQC). Advantageously, the resource states have a size
that is independent of the computation being performed or code
distance used, which is in stark contrast to the cluster states of
MBQC. This allows the resource states used for FBQC to be generated
by a constant number of sequential operations. As a result, in
FBQC, errors in the resource state are bounded, which is important
for fault-tolerance.
2. A SYSTEM FOR FBQC
[0065] FIG. 2 shows a quantum computing system in accordance with
one or more embodiments. The quantum computing system 201 includes
a user interface device 204 that is communicatively coupled to a
quantum computing (QC) sub-system 206, described in more detail
below in FIG. 3. The user interface device 204 can be any type of
user interface device, e.g., a terminal including a display,
keyboard, mouse, touchscreen and the like. In addition, the user
interface device can itself be a computer such as a personal
computer (PC), laptop, tablet computer and the like. In some
embodiments, the user interface device 204 provides an interface
with which a user can interact with the QC subsystem 206 directly
or via a local area network, wide area network, or via the
internet. For example, the user interface device 204 may run
software, such as a text editor, an interactive development
environment (IDE), command prompt, graphical user interface, and
the like so that a user can program, or otherwise interact with,
the QC subsystem to run one or more quantum algorithms. In other
embodiments, the QC subsystem 206 may be pre-programmed and the
user interface device 204 may simply be an interface where a user
can initiate a quantum computation, monitor the progress, and
receive results from the QC subsystem 206. QC subsystem 206 can
further include a classical computing system 208 coupled to one or
more quantum computing chips 210. In some examples, the classical
computing system 208 and the quantum computing chip 210 can be
coupled to other electronic components 212, e.g., pulsed pump
lasers, microwave oscillators, power supplies, networking hardware,
etc. In some embodiments that require cryogenic operation, the
quantum computing system 201 can be housed within a cryostat, e.g.,
cryostat 214. In some embodiments, the quantum computing chip 210
can include one or more constituent chips, e.g., an integration
(direct or heterogeneous) of electronic chip 216 and integrated
photonics chip 218. Signals can be routed on- and off-chip any
number of ways, e.g., via optical interconnects 220 and via other
electronic interconnects 222. In addition, the computing system 201
may employ a quantum computing process, e.g., a fusion-based
quantum computing process as described in further detail below.
[0066] FIG. 3 shows a block diagram of a QC system 301 in
accordance with some embodiments. Such a system can be associated
with the computing system 201 introduced above in reference to FIG.
2. In FIG. 3, solid lines represent quantum information channels
and double-solid lines represent classical information channels.
The QC system 301 includes a qubit entangling system 303, qubit
fusion system 305, and classical computing system 307. In some
embodiments, the qubit entangling system 303 can take as input a
collection of N physical qubits (also referred to herein as
"quantum sub-systems"), e.g., physical qubits 309 (also represented
schematically as inputs 311a, 311b, 311c, . . . , 311N) and can
generate quantum entanglement between two or more of them to
generate entangled resource states 315 (also referred to herein as
"quantum systems" which are themselves made up of entangled states
of quantum sub-systems). For example, in the case of photonic
qubits, the qubit entangling system 303 can be a linear optical
system such as an integrated photonic circuit that includes
waveguides, beam splitters, photon detectors, delay lines, and the
like. In some examples, the entangled resource states 315 can be
relatively small entangled states of qubits (e.g., qubit entangled
states having between 3 and 30 qubits). In some embodiments, the
resource states can be chosen such that the fusion operations
applied to certain qubits of these states results in syndrome graph
data that includes the required correlations for quantum error
correction. Advantageously, the system shown in FIG. 3 provides for
fault tolerant quantum computation using relatively small resource
states, without requiring that the resource states become mutually
entangled with each other to form the typical lattice cluster state
required for MBQC.
[0067] In some embodiments, the input qubits 309 can be a
collection of quantum systems (also referred to herein as
quantum-subsystems) and/or particles and can be formed using any
qubit architecture. For example, the quantum systems can be
particles such as atoms, ions, nuclei, and/or photons. In other
examples, the quantum systems can be other engineered quantum
systems such as flux qubits, phase qubits, or charge qubits (e.g.,
formed from a superconducting Josephson junction), topological
qubits (e.g., Majorana fermions), spin qubits formed from vacancy
centers (e.g., nitrogen vacancies in diamond), or qubits otherwise
encoded in multiple quantum systems, e.g.,
Gottesman-Kitaev-Preskill (GKP) encoded qubits and the like.
Furthermore, for the sake of clarity of description, the term
"qubit" is used herein although the system can also employ quantum
information carriers that encode information in a manner that is
not necessarily associated with a binary bit. For example, qudits
(i.e., quantum systems that can encode information in more than two
quantum states) can be used in accordance with some
embodiments.
[0068] In accordance with some embodiments, the QC system 301 can
be a fusion-based quantum computer that can run one or more quantum
algorithms or software programs. For example, a software program
(e.g., a set of machine-readable instructions) that represents the
quantum algorithm to be run on the QC system 301 can be passed to a
classical computing system 307 (e.g., corresponding to system 208
in FIG. 2 above). The classical computing system 307 can be any
type of computing device such as a PC, one or more blade servers,
and the like, or even a high-performance computing system such as a
supercomputer, server farm, and the like. Such a system can include
one or more processors (not shown) coupled to one or more computer
memories, e.g., memory 306. Such a computing system will be
referred to herein as a "classical computer." In some examples, the
software program can be received by a classical computing module,
referred to herein as a fusion pattern generator 313. One function
of the fusion pattern generator 313 is to generate a set of
machine-level instructions from the input software program (which
may originate as high-level code that can be more easily written by
a user to program the quantum computer).
[0069] In some embodiments, the fusion pattern generator 313 can
operate as a compiler for software programs to be run on the
quantum computer. Fusion pattern generator 313 can be implemented
as pure hardware, pure software, or any combination of one or more
hardware or software components or modules. In various embodiments,
fusion pattern generator 313 can operate at runtime or in advance;
in either case, machine-level instructions generated by fusion
pattern generator 313 can be stored (e.g., in memory 306). In some
examples, the compiled machine-level instructions take the form of
one or more data frames that instruct the qubit fusion system 305
to make, at a given clock cycle of the quantum computer, one or
more fusions between certain qubits from the separate, i.e.,
unentangled, resource states 315. For example, fusion pattern data
frame 317 is one example of the set of fusion measurements (e.g.,
Type II fusion measurements, described in more detail below in
reference to FIGS. 18-21) that should be applied between certain
pairs of qubits from different entangled resource states 315 during
a certain clock cycle as the program is executed. In some
embodiments, several fusion pattern data frames 317 can be stored
in memory 306 as classical data. In some embodiments, the fusion
pattern data frames 317 can dictate whether or not XX Type II
Fusion is to be applied (or whether any other type of fusion, or
not, is to be applied) for a particular fusion gate within the
fusion array 321 of the qubit fusion system 305. In addition, the
fusion pattern data frames 317 can indicate that the Type II fusion
is to be performed in a different basis, e.g., XX, XY, ZZ, etc. As
used herein, the term XX Type II Fusion, YY Type II Fusion, XY Type
II Fusion, ZZ Type II Fusion etc. refer to a fusion operation that
applies a particular a two-particle projective measurement, e.g., a
Bell projection which, depending on the Bell basis chosen, can
project the two qubits onto one of the 4 Bell states. Such
projective measurements produce two measurement outcomes that
correspond to the eigenvalues of the corresponding pair of
observables that are measured in the chosen basis. For example, XX
Fusion is a Bell projection that measures the XX and ZZ observables
(each of which could have a +1 or -1 eigenvalue--or 0 or 1
depending on the convention used), and XZ Fusions is a Bell
projection that measures the observable XZ and ZX observables, and
the like. FIGS. 18-21 below show example circuits for performing
Type II fusions for various choices of basis in a linear optical
system but other Bell projective measurements are possible in other
qubit architectures without departing from the scope of the present
disclosure. One of ordinary skill will appreciate that in a linear
optical system, Type II Fusions perform probabilistic Bell
measurements. FIGS. 18-21 discuss the probabilistic nature of
linear optical fusion in the context of fusion "success" and
"failure" outcomes and will not be repeated here for the sake of
clarity.
[0070] A fusion controller circuit 319 of the qubit fusion system
205 can receive data that encodes the fusion pattern data frames
317 and, based on this data, can generate configuration signals,
e.g., analog and/or digital electronic signals, that drive the
hardware within the fusion array 321. For example, for the case of
photonic qubits, the fusion gates can include photon detectors
coupled to one or more waveguides, beam splitters, interferometers,
switches, polarizers, polarization rotators and the like. More
generally, the detectors can be any detector that can detect the
quantum states of one or more of the qubits in the resource states
315. One of ordinary skill will appreciate that many types of
detectors may be used depending on the particular qubit
architecture being employed.
[0071] In some embodiments, the result of applying the fusion
pattern data frames 317 to the fusion array 321 is the generation
of classical data (generated by the fusion gates' detectors) that
is read out, and optionally pre-processed, and sent to decoder 333.
More specifically, the fusion array 321 can include a collection of
measuring devices that implement the joint measurements between
certain qubits from two different resource states and generate a
collection of measurement outcomes associated with the joint
measurement. These measurement outcomes can be stored in a
measurement outcome data frame, e.g., data frame 322 and passed
back to the classical computing system for further processing.
[0072] In some embodiments, any of the submodules in the QC system
301, e.g., controller 323, quantum gate array 325, fusion array
321, fusion controller 319, fusion pattern generator 313, decoder
323, and logical processor 308 can include any number of classical
computing components such as processors (CPUs, GPUs, TPUs) memory
(any form of RAM, ROM), hard coded logic components (classical
logic gates such as AND, OR, XOR, etc.) and/or programmable logic
components such as field programmable gate arrays (FPGAs and the
like). These modules can also include any number of application
specific integrated circuits (ASICs), microcontrollers (MCUs),
systems on a chip (SOCs), and other similar microelectronics.
[0073] In some embodiments, the entangled resource states 315 can
be any type of entangled resource state, that, when the fusion
operations are performed, produces measurement outcome data frames
that include the necessary correlations for performing fault
tolerant quantum computation. While FIG. 3 shows an example of a
collection of identical resource states, a system can be employed
that generates many different types of resource states and can even
dynamically change the type of resource state being generated based
on the demands of the quantum algorithm being run. As described
herein, the logical qubit measurement outcomes 327 can be fault
tolerantly recovered, e.g., via decoder 333, from the measurement
outcomes 322 of the physical qubits. Logical processor 308 can then
process the logical outcomes as part of the running of the program.
As shown, the logical processor can feed back information to the
fusion pattern generator 313 for affect downstream gates and/or
measurements to ensure that the computation proceeds fault
tolerantly.
[0074] FIG. 4 illustrates an example of a qubit entangling system
401 in accordance with some embodiments. Such a system can be used
to generate qubits (e.g., photons) in an entangled state (e.g., the
resource state used in the illustrative examples shown in FIGS. 7-9
below), in accordance with some embodiments. Qubit entangling
system 401 is an example of a system that can be employed in an
FBQC system, such as qubit entangling system 303 shown in FIG. 3
above. One of ordinary skill will appreciate that any qubit
entangling system could be used without departing from the scope of
the present disclosure. Examples of qubit entangling systems can be
found in U.S. patent application Ser. No. 16/621,994 (published as
US Pat App Pub No 20200287631) titled, Generation of entangled
qubit states, U.S. patent application Ser. No. 16/691,459
(published as U.S. Pat. No. ______), titled, GENERATION OF
ENTANGLED PHOTONIC STATES, and U.S. patent application Ser. No.
16/691,450 (published as U.S. Pat. No. ______), titled, GENERATION
OF AN ENTANGLED PHOTONIC STATE FROM PRIMITIVE RESOURCES, the
disclosures of which are hereby incorporated by reference in their
entireties for all purposes. For example, in some embodiments,
rather than generating single photons, the photon sources may
generate entangled resource states directly, or may even generate
smaller entangled states that can undergo additional entangling
operations at the Entangled State Generator 400 to produce the
final resource states to be used for FBQC. As such, as used herein
the scope of the term "photon source" is intended to include at
least sources of single photons, sources of multiple photons in
entangled states, or more generally any source of photonic states.
One of ordinary skill will appreciate that the precise form of the
resource state generation hardware is not critical and any system
can be employed without departing from the scope of the present
disclosure.
[0075] In an illustrative photonic architecture, qubit entangling
system 401 can include a photon source system 405 that is optically
connected to entangled state generator 400. Both the photon source
system 405 and the entangled state generator 400 may be coupled to
a classical processing system 403 such that the classical
processing system 403 can communicate with and/or control (e.g.,
via the classical information channels 430a-b) the photon source
system 405 and/or the entangled state generator 400. Photon source
system 405 may include a collection of single-photon sources that
can provide output photonic states (e.g., single photons or other
photonic states such as bel states, GHZ states, and the like) to
entangled state generator 400 by way of interconnecting waveguides
402. Entangled state generator 400 may receive the output photonic
states and convert them to one or more entangled photonic states
(or larger photonic states in the case that the source itself
outputs an entangled photonic state) and then output these
entangled photonic states into output waveguides 440. In some
embodiments, output waveguides 440 can be coupled to some
downstream circuit that may use the entangled states for performing
a quantum computation. For example, the entangled states generated
by the entangled state generator 400 may be used as resources for a
downstream quantum optical circuit (not shown).
[0076] In some embodiments, the photon source system 405 and the
entangled state generator 400 may be used in conjunction with the
quantum computing system illustrated in FIG. 3. For example, the
qubit entangling system 303 illustrated in FIG. 3 may include the
photon source system 405 and the entangled state generator 400, and
the classical computer system 403 of FIG. 4 may include one or more
of the various classical computing components illustrated in FIG. 3
(e.g., the classical computing system 307). In this case, the
entangled photons that leave via output waveguides 440 can be fused
together by the qubit fusion system 305, i.e., they can be input to
a detection system that performs a collection of joint measurements
for use in a FBQC scheme.
[0077] In some embodiments, system 401 may include classical
channels 430 (e.g., classical channels 430-a through 430-d) for
interconnecting and providing classical information between
components. It should be noted that classical channels 430-a
through 430-d need not all be the same. For example, classical
channel 430-a through 430-c may comprise a bi-directional
communication bus carrying one or more reference signals, e.g., one
or more clock signals, one or more control signals, or any other
signal that carries classical information, e.g., heralding signals,
photon detector readout signals, and the like.
[0078] In some embodiments, qubit entangling system 401 includes
the classical computer system 403 that communicates with and/or
controls the photon source system 405 and/or the entangled state
generator 400. For example, in some embodiments, classical computer
system 403 can be used to configure one or more circuits, e.g.,
using a system clock that may be provided to photon sources 405 and
entangled state generator 400 as well as any downstream quantum
photonic circuits used for performing quantum computation. In some
embodiments, the quantum photonic circuits can include optical
circuits, electrical circuits, or any other types of circuits. In
some embodiments, classical computer system 403 includes memory
404, one or more processor(s) 402, a power supply, an input/output
(I/O) subsystem, and a communication bus or interconnecting these
components. The processor(s) 402 may execute software modules,
programs, and/or instructions stored in memory 404 and thereby
perform processing operations.
[0079] In some embodiments, memory 404 stores one or more programs
(e.g., sets of instructions) and/or data structures. For example,
in some embodiments, entangled state generator 400 can attempt to
produce an entangled state over successive stages and/or over
independent instances, any one of which may be successful in
producing an entangled state. In some embodiments, memory 404
stores one or more programs for determining whether a respective
stage was successful and configuring the entangled state generator
400 accordingly (e.g., by configuring entangled state generator 400
to switch the photons to an output if the stage was successful, or
pass the photons to the next stage of the entangled state generator
400 if the stage was not yet successful). To that end, in some
embodiments, memory 404 stores detection patterns from which the
classical computing system 403 may determine whether a stage was
successful. In addition, memory 404 can store settings that are
provided to the various configurable components (e.g., switches)
described herein that are configured by, e.g., setting one or more
phase shifts for the component.
[0080] In some embodiments, some or all of the above-described
functions may be implemented with hardware circuits on photon
source system 405 and/or entangled state generator 400. For
example, in some embodiments, photon source system 405 includes one
or more controllers 407-a (e.g., logic controllers) (e.g., which
may comprise field programmable gate arrays (FPGAs), application
specific integrated circuits (ASICS), a "system on a chip" that
includes classical processors and memory, or the like). In some
embodiments, controller 407-a determines whether photon source
system 405 was successful (e.g., for a given attempt on a given
clock cycle) and outputs a reference signal indicating whether
photon source system 405 was successful. For example, in some
embodiments, controller 407-a outputs a logical high value to
classical channel 430-a and/or classical channel 430-c when photon
source system 405 is successful and outputs a logical low value to
classical channel 430-a and/or classical channel 430-c when photon
source system 405 is not successful. In some embodiments, the
output of control 407-a may be used to configure hardware in
controller 107-b.
[0081] Similarly, in some embodiments, entangled state generator
400 includes one or more controllers 407-b (e.g., logical
controllers) (e.g., which may comprise field programmable gate
arrays (FPGAs), application specific integrated circuits (ASICS),
or the like) that determine whether a respective stage of entangled
state generator 400 has succeeded, perform the switching logic
described above, and output a reference signal to classical
channels 430-b and/or 430-d to inform other components as to
whether the entangled state generator 400 has succeeded.
[0082] In some embodiments, a system clock signal can be provided
to photon source system 405 and entangled state generator 400 via
an external source (not shown) or by classical computing system 403
via classical channels 430-a and/or 430-b. Examples of clock
generators that may be used are described in U.S. Pat. No.
10,379,420, the contents of which is hereby incorporated by
reference in its entirety for all purposes; but other clock
generators may also be used without departing from the scope of the
present disclosure. In some embodiments, the system clock signal
provided to photon source system 405 triggers photon source system
405 to attempt to output one photon per waveguide. In some
embodiments, the system clock signal provided to entangled state
generator 400 triggers, or gates, sets of detectors in entangled
state generator 400 to attempt to detect photons. For example, in
some embodiments, triggering a set of detectors in entangled state
generator 400 to attempt to detect photons includes gating the set
of detectors.
[0083] It should be noted that, in some embodiments, photon source
system 405 and entangled state generator 400 may have internal
clocks. For example, photon source system 405 may have an internal
clock generated and/or used by controller 407-a and entangled state
generator 400 has an internal clock generated and/or used by
controller 407-b. In some embodiments, the internal clock of photon
source system 405 and/or entangled state generator 400 is
synchronized to an external clock (e.g., the system clock provided
by classical computer system 403) (e.g., through a phase-locked
loop). In some embodiments, any of the internal clocks may
themselves be used as the system clock, e.g., an internal clock of
the photon source may be distributed to other components in the
system and used as the master/system clock.
[0084] In some embodiments, photon source system 405 includes a
plurality of probabilistic photon sources that may be spatially
and/or temporally multiplexed, i.e., a so-called multiplexed single
photon source. In one example of such a source, the source is
driven by a pump, e.g., a light pulse, that is coupled into an
optical resonator that, through some nonlinear process (e.g.,
spontaneous four wave mixing, second harmonic generation, and the
like) may generate zero, one, or more photons. As used herein, the
term "attempt" is used to refer to the act of driving a photon
source with some sort of driving signal, e.g., a pump pulse, that
may produce output photons non-deterministically (i.e., in response
to the driving signal, the probability that the photon source will
generate one or more photons may be less than 1). In some
embodiments, a respective photon source may be most likely to, on a
respective attempt, produce zero photons (e.g., there may be a 90%
probability of producing zero photons per attempt to produce a
single-photon). The second most likely result for an attempt may be
production of a single-photon (e.g., there may be a 9% probability
of producing a single-photon per attempt to produce a
single-photon). The third most likely result for an attempt may be
production of two photons (e.g., there may be an approximately 1%
probability of producing two photons per attempt to produce a
single photon). In some circumstances, there may be less than a 1%
probability of producing more than two photons.
[0085] In some embodiments, the apparent efficiency of the photon
sources may be increased by using a plurality of single-photon
sources and multiplexing the outputs of the plurality of photon
sources. In some embodiments, the photon source can also produce a
classical herald signal that announces (or heralds) the success of
the generation. In some embodiments, this classical signal is
obtained from the output of a detector, where the photon source
system always produces photon states in pairs (such as in SPDC),
and detection of one photon signal is used to herald the success of
the process. This herald signal can be provided to a multiplexer
and used to properly route a successful generation to a multiplexer
output port, as described in more detail below.
[0086] The precise type of photon source used is not critical and
any type of source can be used, employing any photon generating
process, such as spontaneous four wave mixing (SPFW), spontaneous
parametric down-conversion (SPDC), or any other process. Other
classes of sources that do not necessarily require a nonlinear
material can also be employed, such as those that employ atomic
and/or artificial atomic systems, e.g., quantum dot sources, color
centers in crystals, and the like. In some cases, sources may or
may be coupled to photonic cavities, e.g., as can be the case for
artificial atomic systems such as quantum dots coupled to cavities.
Other types of photon sources also exist for SPWM and SPDC, such as
optomechanical systems and the like. In some examples the photon
sources can emit multiple photons already in an entangled state in
which case the entangled state generator 400 may not be necessary,
or alternatively may take the entangled states as input and
generate even larger entangled states.
[0087] In some embodiments, spatial multiplexing of several
non-deterministic photon sources (also referred to as a MUX photon
source) can be employed. Many different spatial MUX architectures
are possible without departing from the scope of the present
disclosure. Temporal MUXing can also be implemented instead of or
in combination with spatial multiplexing. MUX schemes that employ
log-tree, generalized Mach-Zehnder interferometers, multimode
interferometers, chained sources, chained sources with
dump-the-pump schemes, asymmetric multi-crystal single photon
sources, or any other type of MUX architecture can be used. In some
embodiments, the photon source can employ a MUX scheme with quantum
feedback control and the like. One example of an n.times.m MUXed
source is disclosed in U.S. Pat. No. 10,677,985, the contents of
which is hereby incorporated by reference in its entirety for all
purposes.
[0088] FIG. 5 shows one example of qubit fusion system 501 in
accordance with some embodiments. In some embodiments, qubit fusion
system 501 can be employed within a larger FBQC system such as
qubit fusion system 305 shown in FIG. 3.
[0089] Qubit fusion system 501 includes a fusion controller 519
that is coupled to fusion array 521. Fusion controller 519 is
configured to operate as described above in reference to fusion
controller circuit 319 of FIG. 3 above. Fusion array 521 includes a
collection of fusion sites that each receive two or more qubits
from different resource states (not shown) and perform one or more
fusion operations (e.g., Type II fusion) on selected qubits from
the two or more resource states. The fusion operations performed on
the qubits can be controlled by the fusion controller 519 via
classical signals that are sent from the fusion controller 519 to
each of the fusion sites via control channels 503a, 503b, etc.
Based on the joint measurements performed at each fusion site,
classical measurement outcomes in the form of classical data are
output and then provided to a decoder system, as shown and
described above in reference to FIG. 3. Examples of photonic
circuits that can be employed as Type II fusion gates are described
in below in reference to FIG. 6 and also FIGS. 18-20.
[0090] FIG. 6 shows one possible example of a fusion site 601 as
configured to operate with a fusion controller 319 to provide
measurement outcomes to a decoder for fault tolerant quantum
computation in accordance with some embodiments. In this example,
fusion site 601 can be an element of fusion array 321 (shown in
FIG. 3), and although only one instance is shown for purposes of
illustration, fusion array 321 can include any number of instances
of fusion sites 601.
[0091] As described above, the qubit fusion system 305 can receive
two or more qubits (Qubit 1 and Qubit 2, shown here in a dual rail
encoding) that are to be fused. Qubit 1 is one qubit that is
entangled with one or more other qubits (not shown) as part of a
first resource state and Qubit 2 is another qubit that is entangled
with one or more other qubits (not shown) as part of a second
resource state. Advantageously, in contrast to MBQC, none of the
qubits from the first resource state need be entangled with any of
the qubits from the second (or any other) resource state in order
to facilitate a fault tolerant quantum computation. Also
advantageously, at the inputs of a fusion site 601, the collection
of resource states are not mutually entangled to form a cluster
state that takes the form of a quantum error correcting code and
thus there is no need to store and or maintain a large cluster
state with long-range entanglement across the entire cluster state.
Also advantageously, the fusion operations that take place at the
fusion sites can be fully destructive joint measurements on Qubit 1
and Qubit 2 such that all that is left after the measurement is
classical information representing the measurement outcomes on the
detectors, e.g., detectors 603, 605, 607, 609. At this point, the
classical information is all that is needed for the decoder 333 to
perform quantum error correction, and no further quantum
information is propagated through the system. This can be
contrasted with an MBQC system that might employ fusion sites to
fuse resource states into a cluster state that itself serves as the
topological code and only then generates the required classical
information via single particle measurements on each qubit in the
large cluster state. In such an MBQC system, not only does the
large cluster state need to be stored and maintained in the system
before the single particle measurements are made, but an extra
single particle measurement step needs to be applied (in addition
to the fusions used to generate the cluster state) to every qubit
of the cluster state in order to generate the classical information
required to compute the syndrome graph data required for the
decoder to perform quantum error correction.
[0092] FIG. 6 shows an illustrative example for one way to
implement a fusion site as part of a photonic quantum computer
architecture. In this example, qubit 1 and qubit 2 can be dual rail
encoded photonic qubits. A brief introduction to the dual-rail
encoding of photonic qubits is provided in Section 4 below, in
reference to FIGS. 11-14. Accordingly, qubit 1 and qubit 2 can
input on waveguides pair 621, 623 and on waveguide pair 625, 627,
respectively. Interferometers 624 and 628 can be placed in line
with each qubit, and within one arm of each interferometer 624, 628
a programmable phase shifter 630, 632 can be optionally applied to
affect the basis in which the fusion operation is applied, e.g., by
implementing the specific mode couplings shown in FIG. 21 to
implement what is referred to herein as XX, XY, YY, or ZZ fusions).
The programmable phase shifters 630, 632 can be coupled to the
fusion controller 319 via control line 629 and 631 such that
signals from the fusion controller 319 can be used to set the basis
in which the fusion operation is applied to the qubits. In some
embodiments the basis can be hard-coded within the fusion
controller 319, or in some embodiments the basis can be chosen
based upon external inputs, e.g., instructions provided by the
fusion pattern generator 313. Additional mode couplers, e.g., mode
couplers 633 and 632 can be applied after the interferometers
followed by single photon detectors 603, 605, 607, 609 to provide a
readout mechanism for performing the joint measurement.
[0093] In some embodiments, fusion can be probabilistic operation,
i.e., it implements a probabilistic Bell measurement, with the
measurement sometimes succeeding and sometime failing, as described
in FIG. 20 below. In some embodiments, the success probability of
such operation can be increased by using extra quantum systems in
addition to those onto which the operation is acting upon.
Embodiments using extra quantum systems are usually referred to as
"boosted" fusion. In the example shown in FIG. 6, the fusion site
implements an unboosted Type II fusion operation on the incoming
qubits. One of ordinary skill will appreciate that any type of
fusion operation can be applied (and may be boosted or unboosted)
without departing from the scope of the present disclosure.
Additional examples of Type II fusion circuits are shown and
described in Section 5 below for both polarization encoding and
dual rail path encoding. In some embodiments the fusion controller
319 can also provide a control signal to the detector 603, 605,
607, 609. A control signal can be used, e.g., for gating the
detectors or for otherwise controlling the operation of the
detectors. Each of the detectors 603, 605, 607, 609 provides photon
detection signal (representing the number of photons detected by
the detector, e.g., 0 photons detected, 1 photon detected, two
photons detected, etc.), and this photon detection signal can be
preprocessed at the fusion site 601 to determine a measurement
outcome (e.g., fusion success or not) or passed directly to the
decoder 333 for further processing.
3. AN EXAMPLE OF FBQC EMPLOYING GHZ RESOURCE STATES
[0094] FIGS. 7A-7B illustrate an FBQC scheme for fault tolerant
quantum computation in accordance with one or more embodiments. In
this example a topological code known as the Raussendorf lattice
(also known as the foliated surface code) is used but any other
error correcting code can be used without departing from the scope
of the present disclosure. For example, FBQC can be implemented for
various volume codes (such as the diamond code, triamond code,
etc.), various color codes, or other topological codes can be used
without departing from the scope of the present disclosure.
[0095] FIG. 7A illustrates one unit cell 702 of a Raussendorf
lattice. For the case of measurement based quantum computing, to
determine the value of the syndrome graph, referred to herein as
P.sub.cell, at the center of the unit cell, the qubits on the six
faces of the unit cell are measured in the x-basis resulting in a
set of 0 or 1 eigenvalues being determined for each of the six
M.sub.x measurements. These eigenvalues are then combined as
follows
P cell - M .times. B .times. Q .times. C = [ i = 1 6 .times. M x
.function. ( S i ) ] .times. mod .times. .times. 2. ( 2 )
##EQU00002##
where S.sub.1, S.sub.2, . . . , S.sub.6 correspond to the six sites
on the faces of the unit cell and M.sub.x(S.sub.i) corresponds to
the measurement outcomes (0 or 1) obtained by measuring the
corresponding face qubits in the x basis. (S.sub.1, S.sub.2, and
S.sub.3 are labeled in FIG. 7; S.sub.4, S.sub.5, and S.sub.6 are
located on the hidden faces of unit cell 702.)
[0096] In FBQC, the goal is to generate, through a series of joint
measurements (e.g., a positive-operator valued measure, also
referred to as a POVM) on two or more qubits, a set of classical
data that corresponds to the error syndrome of some quantum error
correcting code. For example, using the Raussendorf unit cell of
FIG. 7A as an illustrative example, the set of measurements that
can be used to generate the syndrome graph value in an FBQC
approach is shown in FIG. 7B. In this example, GHZ states are used
as the resource states, but one of ordinary skill having the
benefit of this disclosure will appreciate that any suitable
resource state can be used without departing from the scope of the
present disclosure. To get from the MBQC scheme shown in FIG. 7A to
the FBQC scheme shown in FIG. 7B, every face qubit of FIG. 7A is
replaced with individual qubits from distinctly separate (i.e., not
entangled) resource states. For instance, four resource states R1,
R2, and R3 (encircled by dotted ellipses), are each contributing at
least one qubit to what would be the face qubit S2 of the
Raussendorf cell are labeled in FIG. 7B. For example, the face
qubit S2 in FIG. 7A is replaced with 4 qubits, from three different
resource states: resource state R1 contributes two qubits; resource
state R2 contributes a third qubit; and resource state R3
contributes the fourth qubit. In operation, the system will perform
two fusions at each face (e.g., circles 721, 722 in FIG. 7B
represent fusions between the contributing qubits of resource state
R2 and R1 and R3 and R1, respectively). In an example where the
fusions are Type II fusions, all four face qubits are measured,
thereby generating four measurement results. The syndrome graph
value for the cell is obtained by Eq. (2) above, but now with
M.sub.x(S.sub.i)=[F.sub.1,XX(S.sub.i)+F.sub.2,XX(S.sub.i)]mod 2
(3)
where, for the ith face, F.sub.1,XX(S.sub.i) is the measurement
outcome obtained by performing the joint measurement on the qubits
associated with fusion 1 (e.g., as indicated by circle 721), with
the fusion 1 being a type II fusion performed in the XX basis and
where F.sub.2,XX (S.sub.i) is the measurement outcome obtained by
performing the joint measurement on the qubits associated with
fusion 2 (e.g., as indicated by circle 722), with the fusion 2 also
being a type II fusion performed in the XX basis. Like the
measurements associated with the X observable described above in
reference to Eqn. (2), the fusion measurements of the observable XX
(and ZZ) take the values of zero or 1 corresponding to the positive
or negative eigenvectors, respectively, of the measured operators
(XX and ZZ, in this example). In view of Eq. (3), to obtain each
measurement on a face M.sub.x(S.sub.i), correct fusion outcomes for
both the fusion measurements F.sub.1,XX (S.sub.i) and
F.sub.2,XX(S.sub.i) are desired. However, if due to some error
either fusions fails so that values for the operators cannot be
recovered, then in some embodiments, the measurement of the face is
considered failed and results in at least one erased edge in the
syndrome graph data. One of ordinary skill having the benefit of
this disclosure will appreciated that errors can be dealt with by
the decoder in a manner that is analogous to that described above
in reference to FIGS. 1A-1C. One of ordinary skill in the art will
also recognize that while our description of Eqn. (3) focused on
the XX observable, the fusion can also produce the measurement of
the ZZ observable and that those outcomes can also be combined as
per Eqn. (3) to produce an independent set of syndrome graph date.
In some embodiments these two sets of syndrome data are referred to
as the primal and dual syndrome graphs.
[0097] FIG. 7C shows an example of a cluster state made up on
several unit cells of the Raussendorf lattice. In an MBQC approach,
this entire cluster state would need to be generated, forming an
entangled state of many qubits with the entanglement of the state
extending across the lattice from one surface boundary to another.
In the MBQC approach it is this large entangled cluster state that
serves as the quantum error correcting code and thus can encode the
logical qubit. Computation proceeds by performing single qubit
measurements on each qubit of the entangled state to generate the
measurement outcomes that are used to generate the syndrome graph
that is fed to the decoder as described above in reference to FIGS.
1A-1C. As such, increasing the error tolerance of the computation
requires an increase to the size of the lattice and therefore an
increase to the size of the entangled state. In one or more
embodiments of the FBQC approach disclose herein, such a large
entangled cluster state is not necessary, but rather, smaller
resource states are generated, with the size of the resource states
being independent of the of the required error tolerance. As
described in detail above in reference to FIG. 7, the FBQC approach
can be constructed from any fault tolerant lattice by replacing
each node of the lattice with a set of fusions between two or more
adjacent resource states. This construction of replacing each node
of the lattice with a resource state/fusions is merely one example
of obtaining an FBQC scheme and one of ordinary skill having the
benefit of this disclosure will recognize that many different ways
of constructing an FBQC scheme from a fault tolerant lattice can be
employed without departing from the scope of the present
disclosure.
[0098] Furthermore, as described in more detail below, the process
can proceed by generating a layer of resource states in a given
clock cycle and performing fusions within each layer, as described
in FIG. 8-9 below. For example, in FIG. 7C, the horizontal
direction represents time in the sense that all or a subset of the
qubits in any given layer in the x-y plane can be
generated/initialized at the same clock cycle, e.g., qubits in
Layer 1 can be generated at clock cycle 1, qubits in layer 2 can be
generated at clock cycle 2, qubits in layer 3 can be generated at
clock cycle 3, etc. As will be described in more detail below, a
certain subset qubits in each layer can be stored/delayed such that
they are available to be fused with qubits from resource states in
a subsequent layer, if necessary to enable fault tolerance.
[0099] In some embodiments, in order to generate a desired error
syndrome, a lattice preparation protocol (LPP) can be designed that
generates the appropriate syndrome graph from the fusions of
multiple smaller entangled resource states. FIGS. 8-9 show an
example of a lattice preparation protocol according to some
embodiments. For the purposes of illustration, the resource states
are states such as resource state 800 shown in FIG. 8A; however
other resource states can be used without departing from the scope
of the present disclosure. The resource state 800 is equivalent to
a GHZ state up to the application of Hadamard gates to single
qubits. For example, the states used in the example disclosed
herein are equivalent to GHZ states up to the application of
Hadamard gates to the two terminating end qubits 800a-3 and 800a-4
in FIG. 8A. More specifically, a 4-GHZ state can be identified as a
stabilizer state having the following stabilizers: XXXX,
ZZII,ZIZI,ZIIZ. The resource state 800 shown in FIG. 8A is closely
related to this GHZ state, but the stabilizers of state 800 are
XXZZ, ZZII, ZIXI, ZIIX (with the ordering of the operators
corresponding to qubits 800a-1, 800a-2, 800a-3, and 800a-4,
respectively). One of ordinary skill will appreciate that 4-GHZ
state and the resource state 800 are equivalent under the
application of a Hadamard gate on qubits 800-a3 and 800-a4.
[0100] The time direction in FIGS. 8-9 is perpendicular to the page
such that a resource state having a shape such as resource state
810 represents a collection of qubits, qubits 1, 2, and 3 that are
mutually entangled within the same clock cycle and qubit 4 which is
entangled in the time dimension with, e.g., qubits 2 and 3. Such a
resource state can be created by, e.g., generating the full 4 qubit
resource state in a single clock cycle and then storing qubit 4 for
a fixed time period (e.g., one clock cycle) in a memory. As used
herein the term "memory" includes at any type of memory, e.g., a
quantum memory, a qubit delay line, a shift register for qubits, a
qubit itself, and the like. In the case of photonic resource
states, qubit memories such as these are equivalent to qubit delays
and can thus be implemented through the use of optical fiber. In
the example shown in FIG. 8C, the delay to qubit 4 is represented
schematically by a loop of additional optical path length (e.g.
provided by an optical fiber) placed inline with the existing
optical path of the qubit but that is not present in the optical
path of qubits 1-3. In this example the length of the fiber is such
that it implements a single clock cycle delay of duration T but
other delays are possible as well, e.g., 2T, 3T, etc. In terms of
physical delay times, such delays could be in the range of 500
ps-500 ns but any delay is possible without departing from the
scope of the present disclosure.
[0101] Returning to the FBQC process disclosed herein, FIGS. 8-9
show an example of how a lattice preparation and measurement
protocol for FBQC can proceed according to layers. FIG. 8A shows a
portion of the underlying layer of the Raussendorf lattice shown as
layer 810 (corresponding to a portion of Layer 1 shown in FIG. 7C).
In the example illustrated here, to process a layer like that shown
in FIG. 8A, first multiple resource states 800 are generated (e.g.,
in qubit entangling system 303 of FIG. 3). In this example, the
resource state 800 is an entangled state comprising 4 physical
qubits (also referred to herein as quantum sub-systems): qubits
800a-1, 800a-2, 800a-3, 800a-4. In some embodiments, the resource
state 800 can take the form of a 4-GHZ state where the two
terminating end qubits 800a-4 and 800a-3 have undergone a Hadamard
operation (e.g., for the case of a dual rail encoded qubit, by way
of applying a 50:50 beamsplitter between the two rails that form
the qubit). In some embodiments, not all qubits in the layer are
subject to fusions in this clock cycle, but rather some of the
qubits generated during this clock cycle from certain resource
states can be delayed, e.g., the measurement of qubit 820,
redundantly encoded qubit 805, or any other qubit can be delayed so
that the qubit will be available at the next clock cycle. Such
delayed qubits are then available to be fused with one or more
qubits from resource states that will only be available for fusions
at the next clock cycle.
[0102] In examples that employ a photonic implementation, the
qubits from the resource states can then be routed appropriately
(via integrated waveguides, optical fiber, or any other suitable
photonic routing technology) to the qubit fusion system (e.g.,
qubit fusion system 305 of FIG. 3) to enable a set of fusion
measurements that implement quantum error correction, i.e., that
will result in collecting the measurement outcomes that correspond
to the error syndrome of choice. While this example explicitly uses
a topological code based on the Raussendorf lattice, any code can
be used without departing from the scope of the present
disclosure.
[0103] FIG. 8B shows an example of a collection of GHZ resource
states arranged, i.e., that they have been pre-routed, such that
the qubits that are to be sent to a given fusion gate are
positioned graphically adjacent to each other. For qubits that are
adjacent to each other in this illustration, respective fusions can
be performed between pairs of qubits (also referred to herein as
respective quantum sub-systems, with each qubit from the pair of
qubits input on a fusion site belonging to a different respective
resource state). For example, at site 802, two Type II fusion
measurements can be applied, one between qubits 822 and 824 and one
between qubits 826 and 828. It should be noted that before the
fusions are performed qubits 822 and 824 (or qubits 826 and 828)
are not entangled with one another but instead are each part of a
distinct resource state. As such, the large entangled cluster state
known as the Raussendorf lattice is not present before the fusion
measurements are performed.
[0104] Referring to FIG. 9A, a portion of a second layer of the
underlying code structure is shown as layer 910 (corresponding to
layer 2 shown in FIG. 7C). In an FBQC system, to process a single
layer like that shown in FIG. 9B the FBQC method proceeds along the
same lines as described above in reference to FIGS. 8A-8B so the
details will not be repeated here.
[0105] FIGS. 10A-10E show in further detail a method for performing
FBQC in accordance with one or more embodiments. More specifically,
the method described here includes steps for performing the joint
measurements for a particular quantum error correcting code
according to some embodiments, where the different layers of the
code may be generated at different time steps (clock cycles) as
introduced above in reference to FIGS. 8-9 and entangled together
in a manner that provides for fusions measurements to extract the
necessary syndrome information for performing quantum error
correction. Like other examples provided herein, the Raussendorf
lattice is used for the sake of illustration but other codes can be
used without departing from the scope of the present
disclosure.
[0106] For example, FIGS. 10A and 10B shows portions of layers 1
and 3, and layers 2 and 4, respectively, from the Raussendorf
lattice of FIG. 7C (referred to here as the quantum error
correcting (QEC) code). FIGS. 10C and 10D illustrate a method for
processing these layers in an FBQC system, including example
resource states that could be used. For the sake of example, the
description is limited to vertices 1, 2, 3, and 4 of the QEC code
and the example focuses on how to perform the resource state
generation and measurements in an FBQC system.
[0107] Returning to FIG. 10A, in step 1001, a first set of the
resource states are provided during a first clock cycle. FIG. 10D
shows one example where, instead of single qubits being provided at
vertices 1, 2, 3, 4, 5, etc. with those single qubits being
mutually entangled across the lattice (as would be the case for a
MBQC system), two or more qubits are provided, each originating
from different, non-entangled, resource states, e.g., respective
resource states A, B, C, D, E, F, and G. As used herein, the
notation A.sup.i.sub.j is used to denote the j-th qubit from the
A-th resource state of the i-th layer. For example, the A-th
resource state of layer 1 in FIG. 10D is a GHZ state that includes
4 qubits, labeled A.sup.1.sub.1, A.sup.1.sub.2, A.sup.1.sub.3,
A.sup.1.sub.4 as shown. Likewise, the qubits comprising resource
state B that is provided as part of layer 1 can be labeled as
B.sup.1.sub.1, B.sup.1.sub.2, B.sup.1.sub.3, B.sup.1.sub.4 (but
this time with labels not explicitly shown in the figure to avoid
cluttering the diagram). Qubits that will be fused to generate the
syndrome information associated with vertices 1, 2, 3, 4, 5 are
also shown as enclosed by solid ellipses 1, 2, 3, and 4 in FIG.
10D. As used herein these vertices are each associated with
hardware for performing Type II fusions at fusion sites, as
described above in reference to FIGS. 3-6.
[0108] In some embodiments, the resource states for any given layer
can be generated/provided by a qubit entangling system such as that
described above in reference to FIGS. 3 and 4. However, one of
ordinary skill having the benefit of this disclosure will
understand that any qubit entangling system can be employed, and
that a given qubit entangling system can employ many different
types of resource state generators, even generating different types
of resource states. In this sense, an FBQC system is completely
agnostic to the choice of resource states and choice of
architecture for the qubit entangling system, or even the
architecture of the qubit itself, thereby leaving the system
designer a great deal of flexibility to implement a system that
results in the highest threshold for the given the prevailing
error/noise sources.
[0109] In Step 1003, fusion instructions in the form of classical
data (also referred to herein as a fusion pattern) are provided to
the fusion sites. Referring back to FIG. 3, for example, fusion
pattern data frame 317 is one example of the set of fusion
instructions (e.g., Type II fusion measurements in the XX basis)
that can be applied between pairs of qubits from different
entangled resource states at a fusion site during a certain clock
cycle as a quantum application is executed on the FBQC system. As
also described above, in some embodiments, several fusion pattern
data frames can be stored in memory as classical data. In some
embodiments, the fusion pattern data frames can dictate whether or
not XX Type II Fusion is to be applied (or whether any other type
of fusion, or not, is to be applied) for a particular fusion gate
within the fusion site. In addition, the fusion pattern data frames
can indicate that the Type II fusion is to be performed in a
different basis, e.g., XX, XY, ZZ, etc.
[0110] Returning to FIG. 10D, the fusion instructions for Layer 1
can include fusion parameters (qubit location and basis) to fuse
two or more qubits from different resource states (also referred to
herein as respective quantum sub-systems because the qubits reside
in, or are part of, respectively separate resource states). For
example, for fusion site 1 the fusion instructions can specify the
fusion parameters to indicate that XX Type II Fusions are to be
performed between qubits from resource states A.sup.1, B.sup.1, and
C.sup.1 (and similarly for site 3 between E.sup.1, F.sup.1, and
G.sup.1). More specifically, the two Type II Fusions to be
performed at fusion site 1 can be specified to be between
A.sup.1.sub.4 and B.sup.1.sub.2 and between C.sup.1.sub.1 and
B.sup.1.sub.3. Similar instructions are provided for the other
fusion sites in the layer. For example, for fusion site 2, the
fusion instructions can specify the fusion parameters to indicate
that XX Type II Fusions are to be performed between qubits from
resource states B.sup.1, D.sup.1, and F.sup.1. More specifically,
the two Type II Fusions to be performed at fusion site 2 can be
specified to be between B.sup.1.sub.4 and D.sup.1.sub.2 and between
D.sup.1.sub.3 and F.sup.1.sub.4. However, unlike the case for
fusion site 1, where all the qubits were measured, fusion site 2
includes a qubit that is to remain unmeasured until the second
clock cycle. This is because the underlying structure of the QEC
lattice requires that the quantum state of this qubit to be
preserved until it is to be fused to a qubit from a different layer
at a different clock cycle, i.e., if this were an MBQC scheme the
qubit associated with this vertex would be one that is entangled
with qubit in another layer, e.g., qubits 2 and 6 shown in FIGS.
10B and 10C, respectively.
[0111] Returning to the explicit example shown in FIG. 10D, the
fusion instructions can specify that D.sup.1.sub.4 will not be
measured until the next clock cycle, where it will be fused from
qubits in a later layer, e.g., layer 2 shown in FIG. 10E. In a
photonic implementation optical fiber can implement a qubit delay
for the above function, serving as a reliable quantum memory to
store qubits until they are needed for a future clock cycle. As
used herein, these unmeasured (delayed) qubits are referred to as
unmeasured quantum sub-systems.
[0112] Moving on to fusion site 4, this site is an example that
includes fusions between layers, i.e., fusion between qubits from
resource states that were generated in this clock cycle with qubits
from resource states that were generated in a prior clock cycle but
were not measured at that time but instead were delayed, or
equivalently, stored until the next clock cycle. For fusion site 4,
the fusion instructions can specify the fusion parameters to
indicate that XX Type II Fusions are to be performed between qubits
from resource states in three different layers C.sup.1, B.sup.0,
and B.sup.2. The fusion instructions can also include instructions
to delay (not measure) qubits C.sup.1.sub.2 and C.sup.1.sub.3 until
the next clock cycle. For example, in this case, the fusion
instructions can indicate that in the next time step, C.sup.1.sub.2
is to be fused with B.sup.0.sub.4 and C.sup.1.sub.3 is to be fused
with B.sup.2.sub.1.
[0113] In Step 1003, the fusion operations that are specified by
the fusion instructions are performed, thereby generating classical
data in the form of fusion measurement outcomes. As described above
in reference to FIGS. 3-6 and Eq. (2), this classical data is then
passed to the decoder and is used to construct the syndrome graph
to be used for quantum error correction.
[0114] These examples are illustrative. The choice of error
correcting code determines the set of qubit pairs that are fused
from certain resource states, such that the output of the qubit
fusion system is the classical data from which the syndrome graph
can be directly constructed. In some embodiments, the classical
error syndrome data is generated directly from the qubit fusion
system without the need to preform additional single particle
measurements on any remaining qubits. In some embodiments, the
joint measurements performed at the qubit fusion system are
destructive of the qubits upon which joint measurement is
performed.
4. INTRODUCTION TO QUBITS AND PATH ENCODING
[0115] The dynamics of quantum objects, e.g., photons, electrons,
atoms, ions, molecules, nanostructures, and the like, follow the
rules of quantum theory. More specifically, in quantum theory, the
quantum state of a quantum object, e.g., a photon, is described by
a set of physical properties, the complete set of which is referred
to as a mode. In some embodiments, a mode is defined by specifying
the value (or distribution of values) of one or more properties of
the quantum object. For example, again for photons, modes can be
defined by the frequency of the photon, the position in space of
the photon (e.g., which waveguide or superposition of waveguides
the photon is propagating within), the associated direction of
propagation (e.g., the k-vector for a photon in free space), the
polarization state of the photon (e.g., the direction (horizontal
or vertical) of the photon's electric and/or magnetic fields) and
the like.
[0116] For the case of photons propagating in a waveguide, it is
convenient to express the state of the photon as one of a set of
discrete spatio-temporal modes. For example, the spatial mode
k.sub.i of the photon is determined according to which one of a
finite set of discrete waveguides the photon can be propagating in.
Furthermore, the temporal mode t.sub.j is determined by which one
of a set of discrete time periods (referred to herein as "bins")
the photon can be present in. In some embodiments, the temporal
discretization of the system can be provided by the timing of a
pulsed laser which is responsible for generating the photons. In
the examples below, spatial modes will be used primarily to avoid
complication of the description. However, one of ordinary skill
will appreciate that the systems and methods can apply to any type
of mode, e.g., temporal modes, polarization modes, and any other
mode or set of modes that serves to specify the quantum state.
Furthermore, in the description that follows, embodiments will be
described that employ photonic waveguides to define the spatial
modes of the photon. However, one of ordinary skill having the
benefit of this disclosure will appreciate that any type of mode,
e.g., polarization modes, temporal modes, and the like, can be used
without departing from the scope of the present disclosure.
[0117] For quantum systems of multiple indistinguishable particles,
rather than describing the quantum state of each particle in the
system, it is useful to describe the quantum state of the entire
many-body system using the formalism of Fock states (sometimes
referred to as the occupation number representation). In the Fock
state description, the many-body quantum state is specified by how
many particles there are in each mode of the system. Because modes
are the complete set of properties, this description is sufficient.
For example, a multi-mode, two particle Fock state
|1001.sub.1,2,3,4 specifies a two-particle quantum state with one
photon in mode 1, zero photons in mode 2, zero photons in mode
three, and 1 photon in mode four. Again, as introduced above, a
mode can be any set of properties of the quantum object (and can
depend on the single particle basis states being used to define the
quantum state). For the case of the photon, any two modes of the
electromagnetic field can be used, e.g., one may design the system
to use modes that are related to a degree of freedom that can be
manipulated passively with linear optics. For example,
polarization, spatial degree of freedom, or angular momentum, could
be used. For example, the four-mode system represented by the two
particle Fock state |1001.sub.1,2,3,4 can be physically implemented
as four distinct waveguides with two of the four waveguides
(representing mode 1 and mode 4, respectively) having one photon
travelling within them. Other examples of a state of such a
many-body quantum system are the four photon Fock state
|1111.sub.1,2,3,4 that represents each waveguide containing one
photon and the four photon Fock state |2200.sub.1,2,3,4 that
represents waveguides one and two respectively housing two photons
and waveguides three and four housing zero photons. For modes
having zero photons present, the term "vacuum mode" is used. For
example, for the four photon Fock state |2200.sub.1,2,3,4 modes 3
and 4 are referred to herein as "vacuum modes" (also referred to as
"ancilla modes").
[0118] As used herein, a "qubit" (or quantum bit) is a physical
quantum system with an associated quantum state that can be used to
encode information. Qubits, in contrast to classical bits, can have
a state that is a superposition of logical values such as 0 and 1.
In some embodiments, a qubit is "dual-rail encoded" such that the
logical value of the qubit is encoded by occupation of one of two
modes by exactly one photon (a single photon). For example,
consider the two spatial modes of a photonic system associated with
two distinct waveguides. In some embodiments, the logical 0 and 1
values can be encoded as follows:
|0.sub.L=|10.sub.1,2 (1)
|1.sub.L=|01.sub.1,2 (2)
where the subscript "L" indicates that the ket represents a logical
value (e.g., a qubit value) and, as before, the notation
|ij.sub.1,2 on the right-hand side of the Equations (1)-(2) above
indicates that there are i photons in a first waveguide and j
photons in a second waveguide, respectively (e.g., where i and j
are integers). In this notation, a two qubit state having a logical
value |01.sub.L (representing a state of two qubits, the first
qubit being in a `0` logical state and the second qubit being in a
`1` logical state) may be represented using photon occupations
across four distinct waveguides by |1001.sub.1,2,3,4 (i.e., one
photon in a first waveguide, zero photons in a second waveguide,
zero photons in a third waveguide, and one photon in a fourth
waveguide). In some instances, throughout this disclosure, the
various subscripts are omitted to avoid unnecessary mathematical
clutter.
5. LOQC INTRODUCTION
[0119] 5.1. Dual Rail Photonic Qubits
[0120] Qubits (and operations on qubits) can be implemented using a
variety of physical systems. In some examples described herein,
qubits are provided in an integrated photonic system employing
waveguides, beam splitters (or directional couplers), photonic
switches, and single photon detectors, and the modes that can be
occupied by photons are spatiotemporal modes that correspond to
presence of a photon in a waveguide. Modes can be coupled using
mode couplers, e.g., optical beam splitters, to implement
transformation operations, and measurement operations can be
implemented by coupling single-photon detectors to specific
waveguides. One of ordinary skill in the art with access to this
disclosure will appreciate that modes defined by any appropriate
set of degrees of freedom, e.g., polarization modes, temporal
modes, and the like, can be used without departing from the scope
of the present disclosure. For instance, for modes that only differ
in polarization (e.g., horizontal (H) and vertical (V)), a mode
coupler can be any optical element that coherently rotates
polarization, e.g., a birefringent material such as a waveplate.
For other systems such as ion trap systems or neutral atom systems,
a mode coupler can be any physical mechanism that can couple two
modes, e.g., a pulsed electromagnetic field that is tuned to couple
two internal states of the atom/ion.
[0121] In some embodiments of a photonic quantum computing system
using dual-rail encoding, a qubit can be implemented using a pair
of waveguides. FIG. 11A shows two representations (1100, 1100') of
a portion of a pair of waveguides 1102, 1104 that can be used to
provide a dual-rail-encoded photonic qubit. At 1100, a photon 1106
is in waveguide 1102 and no photon is in waveguide 1104 (also
referred to as a vacuum mode); in some embodiments, this
corresponds to the |0 state of a photonic qubit. At 1100', a photon
1108 is in waveguide 1104, and no photon is in waveguide 1102; in
some embodiments this corresponds to the |1 state of the photonic
qubit. To prepare a photonic qubit in a known state, a photon
source (not shown) can be coupled to one end of one of the
waveguides. The photon source can be operated to emit a single
photon into the waveguide to which it is coupled, thereby preparing
a photonic qubit in a known state. Photons travel through the
waveguides, and by periodically operating the photon source, a
quantum system having qubits whose logical states map to different
temporal modes of the photonic system can be created in the same
pair of waveguides. In addition, by providing multiple pairs of
waveguides, a quantum system having qubits whose logical states
correspond to different spatiotemporal modes can be created. It
should be understood that the waveguides in such a system need not
have any particular spatial relationship to each other. For
instance, they can be but need not be arranged in parallel.
[0122] Occupied modes can be created by using a photon source to
generate a photon that then propagates in the desired waveguide. A
photon source can be, for instance, a resonator-based source that
emits photon pairs, also referred to as a heralded single photon
source. In one example of such a source, the source is driven by a
pump, e.g., a light pulse, that is coupled into a system of optical
resonators that, through a nonlinear optical process (e.g.,
spontaneous four wave mixing (SFWM), spontaneous parametric
down-conversion (SPDC), second harmonic generation, or the like),
can generate a pair of photons. Many different types of photon
sources can be employed. Examples of photon pair sources can
include a microring-based spontaneous four wave mixing (SPFW)
heralded photon source (HPS). However, the precise type of photon
source used is not critical and any type of source, employing any
process, such as SPFW, SPDC, or any other process can be used.
Other classes of sources that do not necessarily require a
nonlinear material can also be employed, such as those that employ
atomic and/or artificial atomic systems, e.g., quantum dot sources,
color centers in crystals, and the like. In some cases, sources may
or may not be coupled to photonic cavities, e.g., as can be the
case for artificial atomic systems such as quantum dots coupled to
cavities. Other types of photon sources also exist for SPWM and
SPDC, such as optomechanical systems and the like.
[0123] In such cases, operation of the photon source may be
deterministic or non-deterministic (also sometimes referred to as
"stochastic") such that a given pump pulse may or may not produce a
photon pair. In some embodiments, coherent spatial and/or temporal
multiplexing of several non-deterministic sources (referred to
herein as "active" multiplexing) can be used to allow the
probability of having one mode become occupied during a given cycle
to approach 1. One of ordinary skill will appreciate that many
different active multiplexing architectures that incorporate
spatial and/or temporal multiplexing are possible. For instance,
active multiplexing schemes that employ log-tree, generalized
Mach-Zehnder interferometers, multimode interferometers, chained
sources, chained sources with dump-the-pump schemes, asymmetric
multi-crystal single photon sources, or any other type of active
multiplexing architecture can be used. In some embodiments, the
photon source can employ an active multiplexing scheme with quantum
feedback control and the like.
[0124] Measurement operations can be implemented by coupling a
waveguide to a single-photon detector that generates a classical
signal (e.g., a digital logic signal) indicating that a photon has
been detected by the detector. Any type of photodetector that has
sensitivity to single photons can be used. In some embodiments,
detection of a photon (e.g., at the output end of a waveguide)
indicates an occupied mode while absence of a detected photon can
indicate an unoccupied mode. In some embodiments, a measurement
operation is performed in a particular basis (e.g., a basis defined
by one of the Pauli matrices and referred to as X, Y, or Z), and
mode coupling as described below can be applied to transform a
qubit to a particular basis.
[0125] Some embodiments described below relate to physical
implementations of unitary transform operations that couple modes
of a quantum system, which can be understood as transforming the
quantum state of the system. For instance, if the initial state of
the quantum system (prior to mode coupling) is one in which one
mode is occupied with probability 1 and another mode is unoccupied
with probability 1 (e.g., a state |10 in a Fock notation in which
the numbers indicate occupancy of each state), mode coupling can
result in a state in which both modes have a nonzero probability of
being occupied, e.g., a state a.sub.1|10+a.sub.2|01, where
|a.sub.1|.sup.2+|a.sub.2|.sup.2=1. In some embodiments, operations
of this kind can be implemented by using beam splitters to couple
modes together and variable phase shifters to apply phase shifts to
one or more modes. The amplitudes a.sub.1 and a.sub.2 depend on the
reflectivity (or transmissivity) of the beam splitters and on any
phase shifts that are introduced.
[0126] FIG. 11B shows a schematic diagram 1110 (also referred to as
a circuit diagram or circuit notation) for coupling of two modes.
The modes are drawn as horizontal lines 1112,1114, and the mode
coupler 1116 is indicated by a vertical line that is terminated
with nodes (solid dots) to identify the modes being coupled. In the
more specific language of linear quantum optics, the mode coupler
1116 shown in FIG. 11B represents a 50/50 beam splitter that
implements a transfer matrix:
T = 1 2 .times. ( 1 i i 1 ) , ( 4 ) ##EQU00003##
where T defines the linear map for the photon creation operators on
two modes. (In certain contexts, transfer matrix T can be
understood as implementing a first-order imaginary Hadamard
transform.) By convention the first column of the transfer matrix
corresponds to creation operators on the top mode (referred to
herein as mode 1, labeled as horizontal line 1112), and the second
column corresponds to creation operators on the second mode
(referred to herein as mode 2, labeled as horizontal line 1114),
and so on if the system includes more than two modes. More
explicitly, the mapping can be written as:
( a 1 .dagger. a 2 .dagger. ) input 1 2 .times. ( 1 - i - i 1 )
.times. ( a 1 .dagger. a 2 .dagger. ) output , ( 5 )
##EQU00004##
where subscripts on the creation operators indicate the mode that
is operated on, the subscripts input and output identify the form
of the creation operators before and after the beam splitter,
respectively and where:
a.sub.i|n.sub.i,n.sub.j= {square root over
(n.sub.i)}|n.sub.i-1,n.sub.j
a.sub.j|n.sub.i,n.sub.j= {square root over
(n.sub.j)}|n.sub.i,n.sub.j-1
a.sub.j.sup..dagger.|n.sub.i,n.sub.j= {square root over
(n.sub.j+1)}|n.sub.i,n.sub.j+1 (6)
[0127] For example, the application of the mode coupler shown in
FIG. 11B leads to the following mappings:
a 1 input .dagger. 1 2 .times. ( a 1 output .dagger. - i .times. a
2 output .dagger. ) .times. .times. a 2 input .dagger. 1 2 .times.
( - i .times. a 1 output .dagger. + a 2 output .dagger. ) ( 7 )
##EQU00005##
[0128] Thus, the action of the mode coupler described by Eq. (4) is
to take the input states |10,|01, and |11 to
| 1 .times. 0 | 1 .times. 0 - i | 0 .times. 1 2 .times. 01 - i | 1
.times. 0 + | 0 .times. 1 2 .times. | 11 - i 2 .times. ( | 2
.times. 0 + | 02 ) ( 8 ) ##EQU00006##
[0129] FIG. 11C shows a physical implementation of a mode coupling
that implements the transfer matrix T of Eq. (4) for two photonic
modes in accordance with some embodiments. In this example, the
mode coupling is implemented using a waveguide beam splitter 1120,
also sometimes referred to as a directional coupler or mode
coupler. Waveguide beam splitter 1120 can be realized by bringing
two waveguides 1122, 1124 into close enough proximity that the
evanescent field of one waveguide can couple into the other. By
adjusting the separation d between waveguides 1122, 1124 and/or the
length l of the coupling region, different couplings between modes
can be obtained. In this manner, a waveguide beam splitter 1120 can
be configured to have a desired transmissivity. For example, the
beam splitter can be engineered to have a transmissivity equal to
0.5 (i.e., a 50/50 beam splitter for implementing the specific form
of the transfer matrix T introduced above). If other transfer
matrices are desired, the reflectivity (or the transmissivity) can
be engineered to be greater than 0.6, greater than 0.7, greater
than 0.8, or greater than 0.9 without departing from the scope of
the present disclosure.
[0130] In addition to mode coupling, some unitary transforms may
involve phase shifts applied to one or more modes. In some photonic
implementations, variable phase-shifters can be implemented in
integrated circuits, providing control over the relative phases of
the state of a photon spread over multiple modes. Examples of
transfer matrices that define such a phase shifts are given by (for
applying a +i and -i phase shift to the second mode,
respectively):
s = ( 1 0 0 i ) .times. .times. s .dagger. = ( 1 0 0 - i ) ( 9 )
##EQU00007##
[0131] For silica-on-silicon materials some embodiments implement
variable phase-shifters using thermo-optical switches. The
thermo-optical switches use resistive elements fabricated on the
surface of the chip, that via the thermo-optical effect can provide
a change of the refractive index n by raising the temperature of
the waveguide by an amount of the order of 10-5 K. One of skill in
the art with access to the present disclosure will understand that
any effect that changes the refractive index of a portion of the
waveguide can be used to generate a variable, electrically tunable,
phase shift. For example, some embodiments use beam splitters based
on any material that supports an electro-optic effect, so-called x2
and x3 materials such as lithium niobite, BBO, KTP, BTO, PZT, and
the like and even doped semiconductors such as silicon, germanium,
and the like.
[0132] 5.2. Photonic Mode Coupler: Beam Splitters
[0133] Beam splitters with variable transmissivity and arbitrary
phase relationships between output modes can also be achieved by
combining directional couplers and variable phase-shifters in a
Mach-Zehnder Interferometer (MZI) configuration 1130, e.g., as
shown in FIG. 11D. Complete control over the relative phase and
amplitude of the two modes 1132a, 1132b in dual rail encoding can
be achieved by varying the phases imparted by phase shifters 1136a,
1136b, and 1136c and the length and proximity of coupling regions
1134a and 1134b. FIG. 11E shows a slightly simpler example of a MZI
1140 that allows for a variable transmissivity between modes 1132a,
1132b by varying the phase imparted by the phase shifter 1137.
FIGS. 11D and 11E are examples of how one could implement a mode
coupler in a physical device, but any type of mode coupler/beam
splitter can be used without departing from the scope of the
present disclosure.
[0134] In some embodiments, beam splitters and phase shifters can
be employed in combination to implement a variety of transfer
matrices. For example, FIG. 12A shows, in a schematic form similar
to that of FIG. 11A, a mode coupler 1200 implementing the following
transfer matrix:
T r = 1 2 .times. ( 1 1 1 - 1 ) . ( 10 ) ##EQU00008##
[0135] Thus, mode coupler 1200 applies the following mappings:
| 1 .times. 0 | 1 .times. 0 + | 0 .times. 1 2 .times. 01 | 1
.times. 0 - | 0 .times. 1 2 .times. | 11 1 2 .times. ( | 2 .times.
0 + | 02 ) . ( 11 ) ##EQU00009##
[0136] The transfer matrix T.sub.r of Eq. (10) is related to the
transfer matrix T of Eq. (4) by a phase shift on the second mode.
This is schematically illustrated in FIG. 12A by the closed node
1207 where mode coupler 1216 couples to the first mode (line 1212)
and open node 1208 where mode coupler 1216 couples to the second
mode (line 1214). More specifically, T.sub.r=sTs, and, as shown at
the right-hand side of FIG. 12A, mode coupler 1216 can be
implemented using mode coupler 1216 (as described above), with a
preceding and following phase shift (denoted by open squares 1218a,
1218b). Thus, the transfer matrix T.sub.r can be implemented by the
physical beam splitter shown in FIG. 12B, where the open triangles
represent +i phase shifters.
[0137] 5.3. Example Photonic Spreading Circuits
[0138] Networks of mode couplers and phase shifters can be used to
implement couplings among more than two modes. For example, FIG. 13
shows a four-mode coupling scheme that implements a "spreader," or
"mode-information erasure," transformation on four modes, i.e., it
takes a photon in any one of the input modes and delocalizes the
photon amongst each of the four output modes such that the photon
has equal probability of being detected in any one of the four
output modes. (The well-known Hadamard transformation is one
example of a spreader transformation.) As in FIG. 11A, the
horizontal lines 1312-1315 correspond to modes, and the mode
coupling is indicated by a vertical line 1316 with nodes (dots) to
identify the modes being coupled. In this case, four modes are
coupled. Circuit notation 1302 is an equivalent representation to
circuit diagram 1304, which is a network of first-order mode
couplings. More generally, where a higher-order mode coupling can
be implemented as a network of first-order mode couplings, a
circuit notation similar to notation 1302 (with an appropriate
number of modes) may be used.
[0139] FIG. 14 illustrates an example optical device 1400 that can
implement the four-mode mode-spreading transform shown
schematically in FIG. 13 in accordance with some embodiments.
Optical device 1400 includes a first set of optical waveguides
1401, 1403 formed in a first layer of material (represented by
solid lines in FIG. 14) and a second set of optical waveguides
1405, 1407 formed in a second layer of material that is distinct
and separate from the first layer of material (represented by
dashed lines in FIG. 14). The second layer of material and the
first layer of material are located at different heights on a
substrate. One of ordinary skill will appreciate that an
interferometer such as that shown in FIG. 14 could be implemented
in a single layer if appropriate low loss waveguide crossing were
employed.
[0140] At least one optical waveguide 1401, 1403 of the first set
of optical waveguides is coupled with an optical waveguide 1405,
1407 of the second set of optical waveguides with any type of
suitable optical coupler. For example, the optical device shown in
FIG. 14 includes four optical couplers 1418, 1420, 1422, and 1424.
Each optical coupler can have a coupling region in which two
waveguides propagate in parallel. Although the two waveguides are
illustrated in FIG. 14 as being offset from each other in the
coupling region, the two waveguides may be positioned directly
above and below each other in the coupling region without offset.
In some embodiments, one or more of the optical couplers 1418,
1420, 1422, and 1424 are configured to have a coupling efficiency
of approximately 50% between the two waveguides (e.g., a coupling
efficiency between 49% and 51%, a coupling efficiency between 49.9%
and 50.1%, a coupling efficiency between 49.99% and 50.01%, and a
coupling efficiency of 50%, etc.). For example, the length of the
two waveguides, the refractive indices of the two waveguides, the
widths and heights of the two waveguides, the refractive index of
the material located between two waveguides, and the distance
between the two waveguides are selected to provide the coupling
efficiency of 50% between the two waveguides. This allows the
optical coupler to operate like a 50/50 beam splitter.
[0141] In addition, the optical device shown in FIG. 14 can include
two inter-layer optical couplers 1414 and 1416. Optical coupler
1414 allows transfer of light propagating in a waveguide on the
first layer of material to a waveguide on the second layer of
material, and optical coupler 1416 allows transfer of light
propagating in a waveguide on the second layer of material to a
waveguide on the first layer of material. The optical couplers 1414
and 1416 allow optical waveguides located in at least two different
layers to be used in a multi-channel optical coupler, which, in
turn, enables a compact multi-channel optical coupler.
[0142] Furthermore, the optical device shown in FIG. 14 includes a
non-coupling waveguide crossing region 1426. In some
implementations, the two waveguides (1403 and 1405 in this example)
cross each other without having a parallel coupling region present
at the crossing in the non-coupling waveguide crossing region 1426
(e.g., the waveguides can be two straight waveguides that cross
each other at a nearly 90-degree angle).
[0143] Those skilled in the art will understand that the foregoing
examples are illustrative and that photonic circuits using beam
splitters and/or phase shifters can be used to implement many
different transfer matrices, including transfer matrices for real
and imaginary Hadamard transforms of any order, discrete Fourier
transforms, and the like. One class of photonic circuits, referred
to herein as "spreader" or "mode-information erasure (MIE)"
circuits, has the property that if the input is a single photon
localized in one input mode, the circuit delocalizes the photon
amongst each of a number of output modes such that the photon has
equal probability of being detected in any one of the output modes.
Examples of spreader or MIE circuits include circuits implementing
Hadamard transfer matrices. (It is to be understood that spreader
or MIE circuits may receive an input that is not a single photon
localized in one input mode, and the behavior of the circuit in
such cases depends on the particular transfer matrix implemented.)
In other instances, photonic circuits can implement other transfer
matrices, including transfer matrices that, for a single photon in
one input mode, provide unequal probability of detecting the photon
in different output modes.
[0144] 5.4. Example Photonic Bell State Generator Circuit
[0145] A Bell pair is a pair of qubits in any type of maximally
entangled state referred to as a Bell state. For dual rail encoded
qubits, examples of Bell states (also referred to as the Bell basis
states) include:
| .PHI. + = | 0 L | 0 L + | 1 L | 1 L 2 = | 1 .times. 0 .times. 1
.times. 0 + | 0 .times. 1 .times. 0 .times. 1 2 .times. | .PHI. - =
| 0 L | 0 L - | 1 L | 1 L 2 = | 1 .times. 0 .times. 1 .times. 0 - |
0 .times. 1 .times. 0 .times. 1 2 .times. | .PSI. + = | 0 L | 1 L +
| 1 L | 0 L 2 = | 1 .times. 0 .times. 0 .times. 1 + | 0 .times. 1
.times. 1 .times. 0 2 .times. | .PSI. - = | 0 L | 1 L - | 1 L | 0 L
2 = | 1 .times. 0 .times. 0 .times. 1 - | 0 .times. 1 .times. 1
.times. 0 2 ##EQU00010##
[0146] In a computational basis (e.g., logical basis) with two
states, a Greenberger-Horne-Zeilinger state is a quantum
superposition of all qubits being in a first state of the two
states superposed with all of qubits being in a second state. Using
logical basis described above, the general M-qubit GHZ state can be
written as:
| G .times. H .times. Z = | 0 M + | 1 M 2 ##EQU00011##
[0147] In some embodiments, entangled states of multiple photonic
qubits can be created by coupling modes of two (or more) qubits and
performing measurements on other modes. By way of example, FIG. 15
shows a circuit diagram for a Bell state generator 1500 that can be
used in some dual-rail-encoded photonic embodiments. In this
example, modes 1532(1)-1532(4) are initially each occupied by a
photon (indicated by a wavy line); modes 1532(5)-1532(8) are
initially vacuum modes. (Those skilled in the art will appreciate
that other combinations of occupied and unoccupied modes can be
used.)
[0148] A first-order mode coupling (e.g., implementing transfer
matrix T of Eq. (4)) is performed on pairs of occupied and
unoccupied modes as shown by mode couplers 1531(1)-1531(4).
Thereafter, a mode-information erasure coupling (e.g., implementing
a four-mode mode spreading transform as shown in FIG. 13) is
performed on four of the modes (modes 1532(5)-1532(8)), as shown by
mode coupler 1537. Modes 1532(5)-1532(8) act as "heralding" modes
that are measured and used to determine whether a Bell state was
successfully generated on the other four modes 1532(1)-1532(4). For
instance, detectors 1538(1)-1538(4) can be coupled to the modes
1532(5)-1532(8) after second-order mode coupler 1537. Each detector
1538(1)-1538(4) can output a classical data signal (e.g., a voltage
level on a conductor) indicating whether it detected a photon (or
the number of photons detected). These outputs can be coupled to
classical decision logic circuit 1540, which determines whether a
Bell state is present on the other four modes 1532(1)-1532(4) based
on the classical output data. For example, decision logic circuit
1540 can be configured such that a Bell state is confirmed (also
referred to as "success" of the Bell state generator) if and only
if a single photon was detected by each of exactly two of detectors
1538(1)-1538(4). Modes 1532(1)-1532(4) can be mapped to the logical
states of two qubits (Qubit 1 and Qubit 2), as indicated in FIG.
15. Specifically, in this example, the logical state of Qubit 1 is
based on occupancy of modes 1532(1) and 1532(2), and the logical
state of Qubit 2 is based on occupancy of modes 1532(3) and
1532(4). It should be noted that the operation of Bell state
generator 1500 can be non-deterministic; that is, inputting four
photons as shown does not guarantee that a Bell state will be
created on modes 1532(1)-1532(4). In one implementation, the
probability of success is 4/32.
[0149] In some embodiments, it is desirable to form resource states
of multiple entangled qubits (typically 3 or more qubits, although
the Bell state can be understood as a resource state of two
qubits). One technique for forming larger entangled systems is
through the use of a "fusion" gate. A fusion gate receives two
input qubits, each of which is typically part of an entangled
system. The fusion gate performs a "fusion" operation on the input
qubits that produces either one ("type I fusion") or zero ("type II
fusion") output qubits in a manner such that the initial two
entangled systems are fused into a single entangled system. Fusion
gates are specific examples of a general class of two-particle
projective measurements that can be employed to create entanglement
between qubits and are particularly suited for photonic
architectures. Examples of type I and type II fusion gates will now
be described.
6. EXAMPLES OF FUSION GATE PHOTONIC CIRCUITS
[0150] FIGS. 16-21 show some embodiments of photonic circuit
implementation of fusion gates, or fusion circuits, for photonic
qubits that can be used according to some embodiments using Type II
fusion. It should be understood that these example embodiments are
illustrative and not limiting. More generally, as used herein, the
term "fusion gate" refers a device that can implement a
two-particle projective measurement, e.g., a Bell projection which,
depending on the Bell basis chosen, can measure two operators e.g.,
the operators XX, ZZ, the operators XX, ZY, and the like. A Type II
fusion circuit (or gate), in the polarization encoding, takes two
input modes, mixes them at a polarization beam splitter (PBS) and
then rotates each of them by 45.quadrature. before measuring them
in the computational basis. FIG. 16 shows an example. In the path
encoding, a Type II fusion circuit takes four modes, swaps the
second and fourth, applies a 50:50 beamsplitter between the two
pairs of adjacent modes and then detects them all. FIG. 17 shows an
example.
[0151] Fusion gates can be used in the construction of larger
entangled states by making use of the so-called "redundant
encoding: of qubits. This consists in a single qubit being
represented by multiple photons, i.e.
.alpha.|0+.beta.|1.fwdarw..alpha.|0.sup.n+.beta.|0.sup.n,
[0152] so that the logical qubit is encoded in n individual qubits.
This is achieved by measuring adjacent qubits in the X basis.
[0153] This encoding, denoted graphically as n qubits with no edges
between them (as in diagram (b) of FIG. 18), has the advantage that
a Pauli measurement on the redundant qubits does not split the
cluster, but rather removes the photon measured from the redundant
encoding and combine the adjacent qubits into one single qubit that
inherits the bonds of the input qubits, maybe adding a phase. In
addition, another advantage of this type of fusion is that it is
loss tolerant. Both modes are measured, so there is no way to
obtain the detection patterns that herald success if one of the
photons is lost. Finally, Type II fusion does not require the
discrimination between different photon numbers, as two detectors
need to click for the heralding of successful fusion and this can
only happen if the photon count at each detector is 1.
[0154] The fusion succeeds with probability 50%, when a single
photon is detected at each detector in the polarization encoding.
In this case, it effectively performs a Bell state measurement on
the qubits that are sent through it, projecting the pair of logical
qubits into a maximally entangled state. When the gate fails (as
heralded by zero or two photons at one of the detectors), it
performs a measurement in the computational basis on each of the
photons, removing them from the redundant encoding, but not
destroying the logical qubit. The effect of the fusion in the
generation of the cluster is depicted in FIGS. 18A-18D, where FIGS.
18A and 18B show the measurement of a qubit in the linear cluster
in the X basis to join it with its neighbor into a single logical
qubit, and FIGS. 18C and 18D show the effect that success and
failure of the gate have on the structure of the cluster. It can be
seen that a successful fusion allows to build two-dimensional
clusters.
[0155] A correspondence can be retrieved between the detection
patterns and the Kraus operators implemented by the gate on the
state. In this case, since both qubits are detected, these are the
projectors:
h 1 .times. h 2 , v 1 .times. v 2 .fwdarw. h 1 .times. h 2 + v 1
.times. 2 2 ##EQU00012## h 1 .times. v 2 , v 1 .times. h 2 .fwdarw.
h 1 .times. h 2 - v 1 .times. v 2 2 ##EQU00012.2## h 1 2 , v 1 2
.fwdarw. .+-. h 1 .times. v 2 ##EQU00012.3## h 2 2 , v 2 2 .fwdarw.
.+-. v 1 .times. h 2 , ##EQU00012.4##
[0156] where the first two lines correspond to `success` outcomes,
projecting the two qubits into a Bell state, and the bottom two to
`failure` outcomes, in which case the two qubits are projected into
a product state.
[0157] In some embodiments, the success probability of Type II
fusion can be increased by using ancillary Bell pairs or pairs of
single photons. Employing a single ancilla Bell pair or two pairs
of single photons allows to boost the success probability to
75%.
[0158] One technique used to boost the fusion gate comes from the
realization that, when it succeeds, it is equivalent to a Bell
state measurement on the input qubits. Therefore, increasing the
success probability of the fusion gate corresponds to increasing
that of the Bell state measurement it implements. Two different
techniques to improve the probability of discriminating Bell states
have been developed by Grice (using a Bell pair) and Ewert &
van Loock (https://arxiv.org/pdf/1403.4841.pdf) (using single
photons).
[0159] The former showed that an ancillary Bell pair allows to
achieve a success probability of 75%, and the procedure can be
iterated, using increasingly complex interferometers and large
entangled states, to reach arbitrary success probability, in
theory. However, the complexity of the circuit and the size of the
entangled states necessary may make this impractical.
[0160] The second technique makes use of four single photons, input
in two modes in pairs with opposite polarization, to boost the
probability of success to 75%. It has also been shown numerically
that the procedure can be iterated a second time to obtain a
probability of 78.125%, but it has not been shown to be able to
increase the success rate arbitrarily as the other scheme.
[0161] FIG. 19 shows the Type II fusion gate boosted once using
these two techniques, both in polarization and path encoding. The
success probability of both circuits is 75%.
[0162] The detection patterns that herald success of the fusion are
described below for the two types of circuit.
[0163] When a Bell state is used to boost the fusion, the logic
behind the `success` detection patterns is best understood by
considering the detectors in two pairs: the group corresponding to
the input photon modes (modes 1 and 2 in polarization and the top 4
modes in path-encoding) and that corresponding to the Bell pair
input modes (modes 3 and 4 in polarization and the bottom 4 modes
in path-encoding). Call these the `main` and `ancilla` pairs
respectively. Then a successful fusion is heralded whenever: (a) 4
photons are detected in total; and (b) fewer than 4 photons are
detected in each group of detectors.
[0164] When 4 single photons are used as ancillary resources,
success of the gate is heralded whenever: (a) 6 photons are
detected overall; and (b) fewer than 4 photons are detected at each
detector.
[0165] When the gates succeeds, the two input qubits are projected
onto one of the four Bell pairs, as these can be all discriminated
from each other thanks to the use of the ancillary resources. The
specific projection depends on the detection pattern obtained, as
before.
[0166] Both the boosted Type II fusion circuits, designed to take
one Bell pair and four single photons as ancillae respectively, can
be used to perform Type II fusion with variable success
probabilities, if the ancillae are not present or if only some of
them are (in the case of the four single photon ancillae). This is
particularly useful because it allows to employ the same circuits
to perform fusion in a flexible way, depending on the resources
available. If the ancillae are present, they can be input in the
gates to boost the probability of success of the fusion. If they
are not, however, the gates can still be used to attempt fusion
with a lower but non-zero success probability.
[0167] As far as the fusion gate boosted using one Bell pair is
concerned, the only case to be considered is that of the ancilla
being absent. In this case, the logic of the detection patterns
heralding success can be understood by considering the detectors in
the pairs described above again. The fusion is still successful
when: (a) 2 photons are detected at different detectors; and (b) 1
photon is detected in the `principal` pair and 1 photon is detected
in the `ancilla` pair of detectors.
[0168] In the case of the circuit boosted using four single
photons, multiple modifications are possible, removing all or part
of the ancillae. This is analogous to the Boosted Bell State
Generator, which is based on the same principle.
[0169] First consider the case of no ancillae being present at all.
As expected, the fusion is successful with probability 50%, which
is the success rate of the non-boosted fusion. In this case, the
fusion is successful whenever 2 photons are detected at any two
distinct detectors.
[0170] As for the boosted BSG, the presence of an odd number of
ancillae turns out to be detrimental to the success probability of
the gate: if 1 photon is present, the gate only succeeds 32.5% of
the time, whereas if 3 photons are present, the success probability
is 50%, like the non-boosted case.
[0171] If only two of the four ancillae are present, two effects
are possible.
[0172] If they are input in different modes in the polarization
encoding, i.e. different adjacent pairs of ancillary modes in the
path encoding, the probability of success is lowered to 25%.
[0173] However, if the two ancillae are input in the same
polarization mode, i.e. in the same pair of adjacent modes in the
path encoding, the success probability is boosted up to 62.5%. In
this case, the patterns that herald success can be understood again
by grouping the detectors in two pairs: the pair in the branch of
the circuit where the ancillae are input (group 1) and the pair in
the other branch (group 2). This distinction is particularly clear
in the polarization-encoded diagram. Considering these groups, the
fusion if successful when: (a) 4 photons are detected overall; (b)
fewer than 4 photons are detected at each detector in group 1; and
(c) fewer than 2 photons are detected at each detector in group
2.
[0174] In these examples, the fusion gates work by projecting the
input qubits into a maximally entangled state when successful. The
basis such a state is encoded in can be changed by introducing
local rotations of the input qubits before they enter the gate,
i.e. before they are mixed at the PBS in the polarization encoding.
Changing the polarization rotation of the photons before they
interfere at the PBS yields different subspaces onto which the
state of the photons is projected, resulting in different fusion
operations on the cluster states. In the path encoding, this
corresponds to applying local beamsplitters or combinations of
beamsplitters and phase shifts corresponding to the desired
rotation between the pairs of modes that constitute a qubit
(neighboring pairs in the diagrams above).
[0175] This can be useful to implement different types of cluster
operations, both in the success and the failure cases, which can be
very useful to optimize the construction of a big cluster state
from small entangled states.
[0176] FIG. 20 shows a table with the effects of a few rotated
variations of the Type II fusion gate used to fuse two small
entangled states. The diagram of the gate in the polarization
encoding, the effective projection performed and the final effect
on the cluster state are shown.
[0177] Rotation to different basis states is further illustrated in
FIG. 21, which shows examples of photonic circuits for Type II
fusion gate implementations using a path encoding. Shown are fusion
gates for ZX fusion, XX fusion, ZZ fusion, and XZ fusion. In each
instance a combination of beam splitters and phase shifters (e.g.,
as described above) can be used.
7. ADDITIONAL EMBODIMENTS
[0178] Those skilled in the art with access to this disclosure will
appreciate that embodiments described herein are illustrative and
not limiting and that many modifications and variations are
possible. The measurements performed and the states on which they
act can be chosen such that the measurement outcomes have
redundancies that give rise to fault tolerance. For instance, a
code can be directly entered with the measurements, or correlations
can be generated in the measurements that directly deal with both
the destructiveness of the measurement and the entanglement
breaking nature of the measurement in a fault tolerant manner. This
can be handled as part of the classical decoding; for instance,
failed fusion operations can be dealt with as erasure by the
code.
[0179] With reference to the appended figures, components that can
include memory can include non-transitory machine-readable media.
The terms "machine-readable medium" and "computer-readable medium"
as used herein refer to any storage medium that participates in
providing data that causes a machine to operate in a specific
fashion. In embodiments provided hereinabove, various
machine-readable media might be involved in providing
instructions/code to processors and/or other device(s) for
execution. Additionally or alternatively, the machine-readable
media might be used to store and/or carry such instructions/code.
In many implementations, a computer-readable medium is a physical
and/or tangible storage medium. Such a medium may take many forms,
including, but not limited to, non-volatile media, volatile media,
and transmission media. Common forms of computer-readable media
include, for example, magnetic and/or optical media, punch cards,
paper tape, any other physical medium with patterns of holes, a
RAM, a programmable read-only memory (PROM), an erasable
programmable read-only memory (EPROM), a FLASH-EPROM, any other
memory chip or cartridge, a carrier wave as described hereinafter,
or any other medium from which a computer can read instructions
and/or code.
[0180] The methods, systems, and devices discussed herein are
examples. Various embodiments may omit, substitute, or add various
procedures or components as appropriate. For instance, features
described with respect to certain embodiments may be combined in
various other embodiments. Different aspects and elements of the
embodiments may be combined in a similar manner. The various
components of the figures provided herein can be embodied in
hardware and/or software. Also, technology evolves and, thus, many
of the elements are examples that do not limit the scope of the
disclosure to those specific examples.
[0181] It has proven convenient at times, principally for reasons
of common usage, to refer to such signals as bits, information,
values, elements, symbols, characters, variables, terms, numbers,
numerals, or the like. It should be understood, however, that all
of these or similar terms are to be associated with appropriate
physical quantities and are merely convenient labels. Unless
specifically stated otherwise, as is apparent from the discussion
above, it is appreciated that throughout this specification
discussions utilizing terms such as "processing," "computing,"
"calculating," "determining," "ascertaining," "identifying,"
"associating," "measuring," "performing," or the like refer to
actions or processes of a specific apparatus, such as a special
purpose computer or a similar special purpose electronic computing
device. In the context of this specification, therefore, a special
purpose computer or a similar special purpose electronic computing
device is capable of manipulating or transforming signals,
typically represented as physical electronic, electrical, or
magnetic quantities within memories, registers, or other
information storage devices, transmission devices, or display
devices of the special purpose computer or similar special purpose
electronic computing device.
[0182] Those of skill in the art will appreciate that information
and signals used to communicate the messages described herein may
be represented using any of a variety of different technologies and
techniques. For example, data, instructions, commands, information,
signals, bits, symbols, and chips that may be referenced throughout
the above description may be represented by voltages, currents,
electromagnetic waves, magnetic fields or particles, optical fields
or particles, or any combination thereof.
[0183] Terms "and," "or," and "an/or," as used herein, may include
a variety of meanings that also is expected to depend at least in
part upon the context in which such terms are used. Typically, "or"
if used to associate a list, such as A, B, or C, is intended to
mean A, B, and C, here used in the inclusive sense, as well as A,
B, or C, here used in the exclusive sense. In addition, the term
"one or more" as used herein may be used to describe any feature,
structure, or characteristic in the singular or may be used to
describe some combination of features, structures, or
characteristics. However, it should be noted that this is merely an
illustrative example and claimed subject matter is not limited to
this example. Furthermore, the term "at least one of" if used to
associate a list, such as A, B, or C, can be interpreted to mean
any combination of A, B, and/or C, such as A, B, C, AB, AC, BC, AA,
AAB, ABC, AABBCCC, etc.
[0184] Reference throughout this specification to "one example,"
"an example," "certain examples," or "exemplary implementation"
means that a particular feature, structure, or characteristic
described in connection with the feature and/or example may be
included in at least one feature and/or example of claimed subject
matter. Thus, the appearances of the phrase "in one example," "an
example," "in certain examples," "in certain implementations," or
other like phrases in various places throughout this specification
are not necessarily all referring to the same feature, example,
and/or limitation. Furthermore, the particular features,
structures, or characteristics may be combined in one or more
examples and/or features.
[0185] In some implementations, operations or processing may
involve physical manipulation of physical quantities. Typically,
although not necessarily, such quantities may take the form of
electrical or magnetic signals capable of being stored,
transferred, combined, compared, or otherwise manipulated. It has
proven convenient at times, principally for reasons of common
usage, to refer to such signals as bits, data, values, elements,
symbols, characters, terms, numbers, numerals, or the like. It
should be understood, however, that all of these or similar terms
are to be associated with appropriate physical quantities and are
merely convenient labels. Unless specifically stated otherwise, as
apparent from the discussion herein, it is appreciated that
throughout this specification discussions utilizing terms such as
"processing," "computing," "calculating," "determining," or the
like refer to actions or processes of a specific apparatus, such as
a special purpose computer, special purpose computing apparatus or
a similar special purpose electronic computing device. In the
context of this specification, therefore, a special purpose
computer or a similar special purpose electronic computing device
is capable of manipulating or transforming signals, typically
represented as physical electronic or magnetic quantities within
memories, registers, or other information storage devices,
transmission devices, or display devices of the special purpose
computer or similar special purpose electronic computing
device.
[0186] In the preceding detailed description, numerous specific
details have been set forth to provide a thorough understanding of
claimed subject matter. However, it will be understood by those
skilled in the art that claimed subject matter may be practiced
without these specific details. In other instances, methods and
apparatuses that would be known by one of ordinary skill have not
been described in detail so as not to obscure claimed subject
matter. Therefore, it is intended that claimed subject matter not
be limited to the particular examples disclosed, but that such
claimed subject matter may also include all aspects falling within
the scope of appended claims, and equivalents thereof.
[0187] For an implementation involving firmware and/or software,
the methodologies may be implemented with modules (e.g.,
procedures, functions, and so on) that perform the functions
described herein. Any machine-readable medium tangibly embodying
instructions may be used in implementing the methodologies
described herein. For example, software codes may be stored in a
memory and executed by a processor unit. Memory may be implemented
within the processor unit or external to the processor unit. As
used herein the term "memory" refers to any type of long term,
short term, volatile, nonvolatile, or other memory and is not to be
limited to any particular type of memory or number of memories, or
type of media upon which memory is stored.
[0188] If implemented in firmware and/or software, the functions
may be stored as one or more instructions or code on a
computer-readable storage medium. Examples include
computer-readable media encoded with a data structure and
computer-readable media encoded with a computer program.
Computer-readable media includes physical computer storage media. A
storage medium may be any available medium that can be accessed by
a computer. By way of example, and not limitation, such
computer-readable media can comprise RAM, ROM, EEPROM, compact disc
read-only memory (CD-ROM) or other optical disk storage, magnetic
disk storage, semiconductor storage, or other storage devices, or
any other medium that can be used to store desired program code in
the form of instructions or data structures and that can be
accessed by a computer; disk and disc, as used herein, includes
compact disc (CD), laser disc, optical disc, digital versatile disc
(DVD), floppy disk and Blu-ray disc where disks usually reproduce
data magnetically, while discs reproduce data optically with
lasers. Combinations of the above should also be included within
the scope of computer-readable media.
[0189] In addition to storage on computer-readable storage medium,
instructions and/or data may be provided as signals on transmission
media included in a communication apparatus. For example, a
communication apparatus may include a transceiver having signals
indicative of instructions and data. The instructions and data are
configured to cause one or more processors to implement the
functions outlined in the claims. That is, the communication
apparatus includes transmission media with signals indicative of
information to perform disclosed functions. At a first time, the
transmission media included in the communication apparatus may
include a first portion of the information to perform the disclosed
functions, while at a second time the transmission media included
in the communication apparatus may include a second portion of the
information to perform the disclosed functions.
[0190] All patent applications, patents, and printed publications
cited herein are incorporated herein by reference in their
entireties, except for any definitions, subject matter disclaimers
or disavowals, and except to the extent that the incorporated
material is inconsistent with the express disclosure herein, in
which case the language in this disclosure controls.
* * * * *
References