U.S. patent application number 13/111828 was filed with the patent office on 2012-02-02 for method for planar implementation of pi/8 gate in chiral topological superconductors.
This patent application is currently assigned to Microsoft Corporation. Invention is credited to Parsa Bonderson, Lukasz Fidkowski, Michael Freedman, Chetan Nayak, Kevin Walker.
Application Number | 20120028806 13/111828 |
Document ID | / |
Family ID | 44972958 |
Filed Date | 2012-02-02 |
United States Patent
Application |
20120028806 |
Kind Code |
A1 |
Bonderson; Parsa ; et
al. |
February 2, 2012 |
Method For Planar Implementation Of PI/8 Gate In Chiral Topological
Superconductors
Abstract
Disclosed herein is a topologically protected .pi./8-gate which
becomes universal when combined with the gates available through
quasi-particle braiding and planar quasi-particle interferometry. A
twisted interferometer, and a planar .pi./8-gate in CTS,
implemented with the help of the twisted interferometer, are
disclosed. Embodiments are described in the context of state X
(CTS) supported by an ISH, although the concept of a
twisted-interferometer is more general and has relevance to all
anionic, i.e. quasiparticle systems.
Inventors: |
Bonderson; Parsa; (Santa
Barbara, CA) ; Freedman; Michael; (Santa Barbara,
CA) ; Nayak; Chetan; (Santa Barbara, CA) ;
Walker; Kevin; (Santa Barbara, CA) ; Fidkowski;
Lukasz; (Santa Barbara, CA) |
Assignee: |
Microsoft Corporation
Redmond
WA
|
Family ID: |
44972958 |
Appl. No.: |
13/111828 |
Filed: |
May 19, 2011 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
61347022 |
May 21, 2010 |
|
|
|
Current U.S.
Class: |
505/191 ; 257/31;
356/450 |
Current CPC
Class: |
G06N 10/00 20190101;
B82Y 10/00 20130101 |
Class at
Publication: |
505/191 ;
356/450; 257/31 |
International
Class: |
H01L 39/02 20060101
H01L039/02; G01B 9/02 20060101 G01B009/02 |
Claims
1. A twisted interferometer, comprising: a source, a first drain,
and a second drain; a first tunneling path between a first vacuum
and a second vacuum; a second tunneling path between the first
vacuum and a third vacuum; and a third tunneling path between the
second vacuum and the third vacuum, wherein a voltage bias is
maintained between the source and the drains such that a current
flows around edges of the vacuums.
2. The interferometer of claim 1, wherein each of the vacuums is an
absence of X-state fluid or a transformed fluid without non-Abelian
properties, and the vacuum is created by electronic or magnetic top
gating.
3. The interferometer of claim 1, wherein a first distance through
the first and third tunneling paths between a first point on an
edge of the first vacuum and a third point on an edge of the third
vacuum is approximately 1/5 of a second distance through the second
tunneling path between the first point and the third point.
4. The interferometer of claim 3, wherein the first distance is
measured around an edge of the second vacuum, and the second
distance is measured around an edge of the first vacuum and an edge
of the third vacuum.
5. The interferometer of claim 1, wherein tunneling is allowed at
each of the tunneling paths for an interval of time that is
proportional to the time for a wave packet to propagate once around
the second vacuum at group velocity.
6. The interferometer of claim 1, wherein, while the first, second,
and third tunneling junctions are open, all current flows into the
first drain.
7. The interferometer of claim 1, wherein a wave packet of
.alpha.-particles with support small with respect to the first
distance is allowed to transit to the second vacuum with a first
amplitude.
8. The interferometer of claim 7, wherein the wave packet is
allowed to transit the first vacuum with a second amplitude that is
smaller than the first amplitude.
9. The interferometer of claim 8, wherein a second wave packet
arrives at the second tunneling junction such that the packet is
transmitted with a third amplitude.
10. The interferometer of claim 9, wherein the first wave packet
arrives at the third tunneling junction precisely when the second
wave packet also arrives at the third tunneling junction.
11. The interferometer of claim 10, wherein the first wave packet
arrives at the third tunneling junction after 2.5 trips around the
second vacuum.
12. The interferometer of claim 11, wherein the third tunneling
path is briefly made substantial such that the two wave packets
interfere.
13. The interferometer of claim 12, wherein the nature of the
interference is detected at the second drain.
14. A planar .pi./8-gate in a chiral topological superconductor,
comprising a low mobility electronic structure comprising a
spin-orbit coupled semiconductor, an ordinary superconductor, a
ferro-magnetic insulator or a fern-magnetic insulator.
15. The gate of claim 14, wherein the electronic structure is
capable of supporting universal topological, fault tolerant quantum
computation.
16. A topologically protected method for implementing a .pi./8-gate
in a chiral-topological-superconductor (CTS),
Ising-sandwich-heterostructure (ISH) device, the method comprising:
implementing universal quantum computation in a topologically
protected, fault-tolerant manner within the CTS-ISH device.
17. The method of claim 16, wherein the CTS-ISH device comprises a
low mobility electronic structure that is capable of supporting
universal topological, fault tolerant quantum computation.
18. The method of claim 17, wherein the low mobility electronic
structure comprises a spin-orbit coupled semiconductor.
19. The method of claim 17, wherein the low mobility electronic
structure comprises an ordinary superconductor.
20. The method of claim 17, wherein the low mobility electronic
structure comprises a ferro- or fern-magnetic insulator.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims benefit under 35 U.S.C. .sctn.119(e)
of provisional U.S. patent application No. 61/347,022, filed May
21, 2010, the disclosure of which is hereby incorporated herein by
reference.
TECHNICAL FIELD
[0002] The subject matter disclosed and claimed herein relates
generally to the field of quantum computing. Specifically, the
subject matter disclosed and claimed herein relates to methods for
planar implementation of the .pi./8 gate in chiral topological
superconductors.
BACKGROUND
[0003] The term chiral topological superconductor (CTS) may be used
to describe any 2D-system based on a spin-orbit coupled
semiconductor with superconductivity imported via proximity effect,
as well as any other Ising-like system with the topological
properties listed below. Examples include Sau et al.
(arxiv:0907,2239), Alicea (arxiv:0912.2115), and Qi et al. (arxiv:
1003.5448), the disclosures of which are incorporated herein by
reference. Such systems are topological superconductors and support
localized Majorana states. These CTS are not purely topological,
additionally supporting a classical order parameter .phi.. If the
CTS is not planar, but configured as a surface of genus >0, a
significant stiffness term .rho.|.gradient..phi.|.sup.2 in the
Lagrangian will prevent superposition of certain topological
states. For this reason it is desirable to devise a protocol for
executing a computationally universal set of gates in a strictly
planar context.
[0004] Previously, the term Ising sandwich heterostructure (ISH)
has been used for this concept. But, since it is hoped that an ISH
may be built without an effective order parameter .phi., the term
CTS is used herein to emphasize the presence of the order
parameter.
SUMMARY
[0005] Disclosed herein is a topologically protected
.pi. 8 - gate .ident. .pi. 8 0 0 - .pi. 8 ##EQU00001##
which becomes universal when combined with the gates available
through quasi-particle braiding and previously described planar
quasi-particle interferometry:
P = 1 0 0 i , H = 1 2 1 1 1 - 1 , and C N O T = 1 0 0 0 0 1 0 0 0 0
0 1 0 0 1 0 . ##EQU00002##
[0006] Key features of the Ising topological structure underlying
CTS quasi-particle excitation include:
[0007] Excitations I, .sigma., .psi. (trivial, twist, fermion),
[0008] Nontrivial fusions .sigma..sigma.=1+.psi. and
.psi..psi.=1,
[0009] S-matrix:
1 2 2 2 1 2 2 2 0 - 2 2 1 2 - 2 2 1 2 , ##EQU00003##
[0010] Twist-matrix:
1 0 0 0 2 .pi. 16 0 0 0 1 , ##EQU00004##
and
spin ( .sigma. ) = 2 .pi. 16 ##EQU00005##
[0011] To explicate the disclosed methods, the concept of quantum
mechanical measurement may be connected to topology change in
(2+1)-dimensional TQFTs. First, tensor contraction may be
illustrated, in Penrose notation [P], with a 3-tensor, as shown in
FIGS. 1A and 1B.
[0012] A measurement operator O with possible outcome vectors
v.sup.1, . . . , v.sup.n can be written as
O = l = 1 n v l .times. v l v _ l , ##EQU00006##
where v.sup.l is the detector state corresponding to outcome
v.sup.l. Measurement can be applied not just to a vector w but to a
tensor T (corresponding to a segregation of quantum information
into disjoint systems) in which case becomes
w w O = l = 1 n w v l v l v _ l ##EQU00007## T Tcontract O == l = 1
n ( k T ijk v k l ) v _ l . ##EQU00007.2##
[0013] Once the outcome of the measurement is observed, the system
is in a tensor product state. Thus, measuring v (i.e., observing
some v.sup.l) can be written as the right most alternative depicted
in FIG. 2. The final effect of an observed measurement is tensor
contraction with the observed state.
[0014] The situation for a (2+1)-dimensional TQFT, as shown in FIG.
3, is only slightly more complicated. The 3-manifold M plays the
role of the tensor T, but its valence is unspecified until the
boundary of M is dissected into "pieces". These pieces may be
closed or with boundary (and are not necessarily connected), and
serve as the index set for the tensor. The axioms for TQFTs
strongly restrict which tensors arise as the boundary decomposition
of M is varied. As an example, take M to be a solid torus
S.sup.1.times.D.sup.2 and the theory to be Ising. Decomposing
.differential.M into A.orgate.B in the following three ways yields
three different matrices (2-tensors) in the 1, .sigma., .psi.
basis, along the loops A.andgate.B.
[0015] These calculations may now be illustrated. Case (2) is
axiomatic: products correspond to identity morphisms. The identity
operator "glues up" to become the vector
v l .di-elect cons. v l ( T 2 ) , v l = 1 D ( d 1 , d .sigma. , d
.psi. ) = ( 1 2 , 2 2 , 1 2 ) ##EQU00008##
in the longitudinal basis. Now transforming by S to the meridial
basis we obtain v.sub.m=S(v.sub.l)=(1, 0, 0).epsilon.V.sub.m and
converting to an operator we should divide entries by
S.sub.l,i=d.sub.i/D to obtain case (1). Finally, to compute case
(3) note that if A=S.sup.-1T.sup.2S is the modular transformation
sending (1) to (3), then in this twisted basis (t), v becomes
v t = 1 + .omega. 2 0 1 - .omega. 2 0 1 0 1 - .omega. 2 0 1 +
.omega. 2 ( 1 0 0 ) = ( 1 + .omega. 2 0 1 - .omega. 2 ) , .omega. =
- 2 .pi. / 8 , ##EQU00009##
which converts to case (3).
[0016] We record also the vector and operator associated with a
case (3'), the same boundary data as case (3) but with the solid
torus containing a .psi.-charge Wilson loop running along its core.
In case (3')
v t = 1 + .omega. 2 0 1 - .omega. 2 0 1 0 1 - .omega. 2 0 1 +
.omega. 2 ( 0 0 1 ) = ( 1 - .omega. 2 0 1 + .omega. 2 ) ,
##EQU00010##
and the corresponding operator is
1 - 2 .pi. / 8 0 0 0 0 0 0 0 1 + 2 .pi. / 8 . ##EQU00011##
[0017] It is known that the operation of an interferometer within
an Ising system using .sigma. probe particles projects the
topological charge state of the interferometry loop to a charge
sector 1, .sigma., or .psi., along the interferometry loop .gamma.
up to an error exponentially small in the number of probes.
[0018] The TQFT analog to partial trace is to glue a 2-handle of
space-time topological fluid along the measured loop .gamma.. As
shown in FIG. 4, the .alpha. charge line at the core of the
2-handle is precisely the measurement outcome .alpha.=1, .sigma.,
.psi. (so if the trivial particle is measured, the 2-handle has no
Wilson line). Up to an overall scalar (=S.sub.1.alpha.) which has
no physical significance, if .gamma. is a loop on a torus boundary
component T, we may glue not just a 2-handle but also a 3-handle
(B.sup.3, .differential.B.sup.3) as well (containing a matching
charge line), so that measuring .alpha. along .gamma. is equivalent
to Dehn filling T with a solid torus S.sup.1.times.D.sup.2,
*.times..differential.D.sup.2 gluing to .gamma. and S.sup.1.times.*
being a Wilson loop of charge .alpha.. The analogy with the Penrose
TQFT: may now be completed. As shown in FIG. 5, a measurement Dehn
fills a solid torus along .gamma. with a Wilson loop of charge
.alpha. at its core--and in a disjoint system, we have a state |
.alpha. recording the fact that .alpha. was the measurement
outcome.
BRIEF DESCRIPTION OF THE DRAWINGS
[0019] FIGS. 1A and 1B illustrate tensor contraction in Penrose
notation.
[0020] FIG. 2 depicts a measurement using tensor contraction.
[0021] FIG. 3 depicts a (2+1)-dimensional TQFT.
[0022] FIG. 4 depicts gluing a 2-handle of space-time topological
fluid along a measured loop.
[0023] FIG. 5 depicts a measurement Dehn filling a solid torus.
[0024] FIG. 6A provides a sketch of a Beenakker-style
interferometer in the CTS-ISH context.
[0025] FIG. 6B depicts a twisted interferometer.
[0026] FIGS. 7A and 7B illustrate computation of operators in a
twisted annular basis.
[0027] FIG. 8 summarizes a protocol producing a topologically
protected .pi./8-gate using twisted interferometry.
[0028] FIGS. 9A and 9B illustrate how a .pi./8-gate may be
obtained.
[0029] FIGS. 10A and 10B show Wilson loops in relation to charge
lines corresponding to an original qubit in various states.
[0030] FIG. 11 shows an example computing environment in which
example embodiments and aspects may be implemented.
[0031] FIG. 12 depicts braids.
DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS
[0032] A CTS is any semiconductor/superconducting system resulting
in a "generic (no special symmetries) 2D topological
superconducting film with a single sheeted Dirac-like Fermi
surface." Our terminology sometimes identifies such a state (called
X) with the ISH which houses it, as in our co-pending U.S. patent
application Ser. Nos. 12/979,778 and 12/979,856, the disclosures of
which are incorporated herein by reference. As disclosed herein,
three tunneling amplitudes, t.sub.1, t.sub.2, and t.sub.3 may be
sharply regulated (essentially turned "on" and "off") on a
gigahertz time scale. This may be done either with 1) high speed
electronic and/or magnetic top gates, 2) optically using laser
pulses to disturb the ground state of X near a tunneling junction,
effectively reducing the bulk gap .DELTA. in this region and
increasing the amplitude to tunnel (note that
t.apprxeq.e.sup.-const {square root over (M.DELTA.L)}, where M is
an effective mass, .DELTA. the bulk gap, and L is the length of the
summary tunneling junction,) or 3) any other electronic, optical,
or magnetic intervention.
[0033] For reference and contrast, FIG. 6A provides a sketch of a
Beenakker (arxiv:0903.2196) style interferometer in the CTS-ISH
context. The dotted lines are tunneling junctions with amplitudes
t.sub.1 and t.sub.2 respectively. The single arrows represent
Majorana mode along edges, while the double arrow represents a
Dirac mode. FIG. 6B depicts a twisted interferometer. The
interferometry loop is labeled by l.
[0034] Note that, as used herein, "vacuum" is either an absence of
the X-state fluid or a transformed fluid without non-Abelian
properties. The "vacuum" is created by electronic and/or magnetic
top gating. In FIG. 6B, the distance L.sub.1 from a to b along the
island (through tunneling junctions 1 and 3) should be
approximately 1/5 the distance L.sub.2 from a to b detouring the
island and passing through tunneling junction 2. Tunneling is
allowed at t.sub.1, t.sub.2, t.sub.3 for an interval of
time.apprxeq.t.sub.0/10, where t.sub.0 is the time for a wave
packet to propagate once around the island at group velocity.
[0035] A voltage bias may be maintained between the source S and
the drains D and D'. While the tunneling junctions 1, 2, and 3 are
open, all current flows into D'. At time t=0, t.sub.1 is made
substantial for a brief time.apprxeq.t.sub.0/10 (order
10.sup.-10-10.sup.-9 seconds), allowing a wave packet of
.sigma.-particles with support small with respect to L.sub.1 to
transit to the island with amplitude t.sub.1 while, with amplitude
{square root over (1-t.sub.1.sup.2)}, the wave packet continues
along the left edge. Knowing the geometry and the velocity of the
edge mode, t.sub.2 is briefly--again for a
time.apprxeq.t.sub.0/10--increased from zero precisely as this
latter wave packet arrives at the second tunneling junction so that
the packet is transmitted with amplitude t.sub.2. Finally, the
first wave packet branch arrives, after 2.5 trips around the
island, precisely when the second wave packet branch also arrives
at the third tunneling junction.
[0036] At this moment, t.sub.3 is briefly made substantial (>0),
and the two branches interfere. The nature of this interference,
constructive or destructive, is detected at the drain D. The
condition L.sub.2/L.sub.1.apprxeq.5 allows the two branches to
arrive synchronously. In order to maintain superposition of trivial
topological charge and a topological charge along the edge of the
vacuum island, during the running of the twisted interferometer, a
magnetic flux should be threaded through the vacuum island which is
tuned to equalize the energies of these two states. This tuning
does not need to be exponentially precise; its purpose is not
precision of a computational state but rather maintenance of
superposition during the 2.5 laps around the interferometer. Error
in this tuning may reduce the twisted interferometer's visibility
algebraically.
[0037] Up to errors exponentially small in the number of Ising
.sigma.-particles admitted to the island, the twisted
interferometer acts on the topological charge enclosed within the
interferometry loop by either
1 + .omega. 0 0 1 - .omega. or 1 - .omega. 0 0 1 + .omega.
##EQU00012##
in the longitudinal {|1,|.psi.} basis according to whether |1 or
|.psi. is observed, where .omega.=e''.sup..pi.i/4.
[0038] In the untwisted (Beenakker) context, the measurement is in
the basis of topological charge {|1, |.psi.} enclosed in the
untwisted interferometry loop l (FIG. 6A). The operator O is the
sum of the two basis projectors tensored with the measurement
outcome
O = 1 0 0 0 1 + 0 0 0 1 .psi. , ##EQU00013##
or less formally,
P 1 = 1 0 0 0 if 1 ##EQU00014##
is observed and
P 2 = 0 0 0 1 if .psi. ##EQU00015##
is observed. One might expect in the twisted context to affect
conjugates of P.sub.1, P.sub.2--accounting for a change of basis
from an untwisted to a twisted interferometry loop. However, it is
immediate that if no charge lines enter or leave the twisted
interferometer (and we always assume there are no mobile charges)
that O.sub.t=O.sub.twisted must be diagonal in the basis
{|1,|.psi.} of topological charge.
[0039] As described above, making a charge measurement with trivial
outcome is equivalent to topologically Dehn filling the twisted
interferometry loop .gamma. with a solid torus
S.sup.1.times.D.sup.2 of pure topological ground state medium;
whereas outcome |.psi. is equivalent to Dehn filling along .gamma.
a solid torus with a .psi.-charge loop (Wilson line loop) along the
core circle S.sup.1.times.0.OR right.S.sup.1.times.D.sup.2.
[0040] In terms of operators in twisted annular {1,.psi.} basis, we
have:
O t 1 = 1 + .omega. 0 0 1 - .omega. = ( 1 + .omega. ) 1 1 + ( 1 -
.omega. ) .psi. .psi. ##EQU00016##
if |1 is measured, and
O t .psi. = 1 - .omega. 0 0 1 + .omega. = ( 1 - .omega. ) 1 1 + ( 1
+ .omega. ) .psi. .psi. ##EQU00017##
if |.psi. is measured.
[0041] In FIG. 7A, the two extra trips around the island mean that
measurement is affected along a topologically twisted "(1, -2)"
loop (using meridian, longitude basis) which is related to the
spatial perimeter of the interferometer l in FIGS. 6A and 6B, i.e.,
the usual interference loop, by a A:=S.sup.-1T.sup.2S change of
coordinates. Referring to FIGS. 7A and 7B, it can be seen that the
computation provided above computes O.sub.t.
[0042] Twisted interferometry employs a single burst of n
co-propagating probe particles (.sigma.'s in the Ising theory
(CTS)) which form a wave packet. A reasonable estimate for n to be
large enough both to converge the interferometer and to tunnel onto
the vacuum island in 10.sup.-10 seconds is 10.ltoreq.n.ltoreq.100.
The .sigma. probes follow trajectories mutually twisting -4.pi. and
linking each other (linking number=-2) as they make two (clockwise)
circuits around the vacuum island. Measurement of current into the
drain D can be compared with standard interferometric calculation
to yield a topological charge of either |1 or |.psi. along the (-2,
1) interferometric loop.
[0043] Ball park estimates for ISH Majorana edge mode velocities
are 10.sup.4 m/s. If L.sub.1=5 .mu.m and the wave packet is to have
most of its amplitude supported along a 1 .mu.m length with
exponentially decaying tails, then the tunneling junction should be
open for 10.sup.-10s. For reasonable tunneling currents, this would
permit between 10 and 100 .sigma.'s to tunnel, adequate to
effectively converge the interferometer.
[0044] FIG. 8 summarizes a protocol producing a topologically
protected .pi./8-gate using twisted interferometry. FIG. 8 depicts
a 1 or .psi.-qubit evolved in time. The first event is the creation
of a new dot of vacuum (the local minima). At the saddle point,
this dot of vacuum splits into two dots of vacuum, each with
topological charge .sigma.. Third, twisted interferometry is
performed along .gamma.. Note that .gamma. should not be read as a
Wilson loop in FIG. 8, but rather as our notation for twisted
interferometry.
[0045] The fourth event is a fusion of the .sigma.-charged dots of
the original qubit. The fifth and final event is charge measurement
along .alpha.. The charge .alpha. at the top can be measured to be
1 or .psi. by ordinary (previously described) Beenakker
quasi-particle interferometry. Twisted interferometry is used to
measure the charge around .gamma. (.ident.l' in earlier notation)
with the twists recorded by the two kinks in .gamma. corresponding
to the two loops around the island of vacuum in FIG. 6B. This
yields either states A(1) or A(.psi.) via projective measurement.
Using techniques of quantum topology, we will verify that the
initial .alpha.|+.beta.|.psi. is transformed by the matrix
following "outcomes" in the four measurement cases:
TABLE-US-00001 charge .gamma. charge .alpha. outcome correct by
A(1) A(1) 1 0 0 e 2 .pi. i 8 ##EQU00018## 1 0 0 1 ##EQU00019## A(1)
A(.psi.) e 2 .pi. i 8 0 0 1 ##EQU00020## 1 0 0 i ##EQU00021##
A(.psi.) A(1) - 1 0 0 e 2 .pi. i 8 ##EQU00022## 1 0 0 - 1
##EQU00023## A(.psi.) A(.psi.) - e 2 .pi. i 8 0 0 1 ##EQU00024## 1
0 0 - i ##EQU00025##
In all four cases, available transformations (as described above),
listed in the right-most column above, convert the gate executed in
FIG. 8 to the desired .pi./8-gate (up to an irrelevant overall
phase).
[0046] Freedman, Nayak, and Walker (arxiv: 0512.072 and 0512.066)
and our U.S. patent application Ser. Nos. 12/979,778 and
12/979,856, show that the .pi./8-gate may be obtained by cutting
along .beta. in FIG. 9A if .alpha.=1 (and its inverse if
.alpha.=.psi.). Thickening the surface in FIG. 9A results in FIG.
9B. Now the framed curve .gamma. in FIG. 9A is precisely the
surgery required to send .beta. to the meridian .mu. labeled in
FIG. 9B. Measuring 1 along .gamma. affects ordinary framed surgery,
while measuring .psi. affects an easily computed variant,
"defective surgery," which is correctable to ordinary surgery as
above, by the action of one or two braid generators. The matrices
in the table above give precise outcomes according to the two
measurements.
[0047] Since the original qubit has .sigma. charges on its internal
punctures, there will be a .sigma.-charge on .beta., but compared
to the original qubit at time t=0, the relative phase between the
two fusion channels 1 and .psi. is now changed by e.sup.2.pi.i/8.
The loop .beta.' in FIG. 9B is simply a copy of .beta. transported
across the product structure.
[0048] A (-1) Dehn twist on the loop .gamma.' throws .beta.' to the
meridian .mu.. Thus Dehn filling on a bulk parallel to .gamma.',
with a (-1) additional twist in its framing compared to the normal
framing .gamma.' inherits from the boundary of the bulk,
effectively endows the bulk with a new product structure in which
.beta. is connected by a cylinder to .mu.. .gamma., as drawn in
FIG. 8, is this (-1) framed bulk loop isotopic to .gamma.'. Thus
twisted interferometry with |1 as outcome "teleports" the twist and
non-time-slice qubit determined by cutting the surface of FIG. 9A
along .beta. to an untwisted time-slice qubit at the top of FIG. 9B
(within the dotted circle).
[0049] It remains to compute the effect of twisted interferometry
if the outcome is |.psi.. |.sigma. is not a possible outcome as the
charge along .gamma.=l'=(1, 2) is obtained from the charge along l
by applying the matrix A. A does not mix the |.sigma. and the
{|1,|.psi.} sectors and the charge along l=.alpha.|1+.beta.|.psi..)
The effect of outcome |.psi. is a Wilson loop of charge |.psi.
parallel to .gamma.' (in the bulk) with no additional twist in its
framing.
[0050] Using the calculational tools of modular tensor categories
(MTC), FIG. 10A shows the Wilson loop ".gamma." in relation to the
charge lines corresponding to the original qubit in state |1, and
FIG. 10B shows the configuration with the original qubit in state
|.psi.. It is evident that measuring |.psi. is equivalent to the
action of
- 1 0 0 1 ##EQU00026##
which, up to an overall phase, is the square of a braid
generator.
Example Computing Environment
[0051] FIG. 11 shows an example computing environment in which
example embodiments and aspects may be implemented. The computing
system environment 100 is only one example of a suitable computing
environment and is not intended to suggest any limitation as to the
scope of use or functionality. Neither should the computing
environment 100 be interpreted as having any dependency or
requirement relating to any one or combination of components
illustrated in the exemplary operating environment 100.
[0052] Numerous other general purpose or special purpose computing
system environments or configurations may be used. Examples of well
known computing systems, environments, and/or configurations that
may be suitable for use include, but are not limited to, personal
computers, server computers, hand-held or laptop devices,
multiprocessor systems, microprocessor-based systems, set top
boxes, programmable consumer electronics, network PCs,
minicomputers, mainframe computers, embedded systems, distributed
computing environments that include any of the above systems or
devices, and the like.
[0053] Computer-executable instructions, such as program modules,
being executed by a computer may be used. Generally, program
modules include routines, programs, objects, components, data
structures, etc. that perform particular tasks or implement
particular abstract data types. Distributed computing environments
may be used where tasks are performed by remote processing devices
that are linked through a communications network or other data
transmission medium. In a distributed computing environment,
program modules and other data may be located in both local and
remote computer storage media including memory storage devices.
[0054] With reference to FIG. 11, an exemplary system includes a
general purpose computing device in the form of a computer 110.
Components of computer 110 may include, but are not limited to, a
processing unit 120, a system memory 130, and a system bus 121 that
couples various system components including the system memory to
the processing unit 120. The processing unit 120 may represent
multiple logical processing units such as those supported on a
multi-threaded processor. The system bus 121 may be any of several
types of bus structures including a memory bus or memory
controller, a peripheral bus, and a local bus using any of a
variety of bus architectures. By way of example, and not
limitation, such architectures include Industry Standard
Architecture (ISA) bus, Micro Channel Architecture (MCA) bus,
Enhanced ISA (EISA) bus, Video Electronics Standards Association
(VESA) local bus, and Peripheral Component Interconnect (PCI) bus
(also known as Mezzanine bus). The system bus 121 may also be
implemented as a point-to-point connection, switching fabric, or
the like, among the communicating devices.
[0055] Computer 110 typically includes a variety of computer
readable media. Computer readable media can be any available media
that can be accessed by computer 110 and includes both volatile and
nonvolatile media, removable and non-removable media. By way of
example, and not limitation, computer readable media may comprise
computer storage media and communication media. Computer storage
media includes both volatile and nonvolatile, removable and
non-removable media implemented in any method or technology for
storage of information such as computer readable instructions, data
structures, program modules or other data. Computer storage media
includes, but is not limited to, RAM, ROM, EEPROM, flash memory or
other memory technology, CDROM, digital versatile disks (DVD) or
other optical disk storage, magnetic cassettes, magnetic tape,
magnetic disk storage or other magnetic storage devices, or any
other medium which can be used to store the desired information and
which can accessed by computer 110. Communication media typically
embodies computer readable instructions, data structures, program
modules or other data in a modulated data signal such as a carrier
wave or other transport mechanism and includes any information
delivery media. The term "modulated data signal" means a signal
that has one or more of its characteristics set or changed in such
a manner as to encode information in the signal. By way of example,
and not limitation, communication media includes wired media such
as a wired network or direct-wired connection, and wireless media
such as acoustic, RF, infrared and other wireless media.
Combinations of any of the above should also be included within the
scope of computer readable media.
[0056] The system memory 130 includes computer storage media in the
form of volatile and/or nonvolatile memory such as read only memory
(ROM) 131 and random access memory (RAM) 132. A basic input/output
system 133 (BIOS), containing the basic routines that help to
transfer information between elements within computer 110, such as
during start-up, is typically stored in ROM 131. RAM 132 typically
contains data and/or program modules that are immediately
accessible to and/or presently being operated on by processing unit
120. By way of example, and not limitation, FIG. 11 illustrates
operating system 134, application programs 135, other program
modules 136, and program data 137.
[0057] The computer 110 may also include other
removable/non-removable, volatile/nonvolatile computer storage
media. By way of example only, FIG. 11 illustrates a hard disk
drive 140 that reads from or writes to non-removable, nonvolatile
magnetic media, a magnetic disk drive 151 that reads from or writes
to a removable, nonvolatile magnetic disk 152, and an optical disk
drive 155 that reads from or writes to a removable, nonvolatile
optical disk 156, such as a CD ROM or other optical media. Other
removable/non-removable, volatile/nonvolatile computer storage
media that can be used in the exemplary operating environment
include, but are not limited to, magnetic tape cassettes, flash
memory cards, digital versatile disks, digital video tape, solid
state RAM, solid state ROM, and the like. The hard disk drive 141
is typically connected to the system bus 121 through a
non-removable memory interface such as interface 140, and magnetic
disk drive 151 and optical disk drive 155 are typically connected
to the system bus 121 by a removable memory interface, such as
interface 150.
[0058] The drives and their associated computer storage media
discussed above and illustrated in FIG. 11, provide storage of
computer readable instructions, data structures, program modules
and other data for the computer 110. In FIG. 11, for example, hard
disk drive 141 is illustrated as storing operating system 144,
application programs 145, other program modules 146, and program
data 147. Note that these components can either be the same as or
different from operating system 134, application programs 135,
other program modules 136, and program data 137. Operating system
144, application programs 145, other program modules 146, and
program data 147 are given different numbers here to illustrate
that, at a minimum, they are different copies. A user may enter
commands and information into the computer 20 through input devices
such as a keyboard 162 and pointing device 161, commonly referred
to as a mouse, trackball or touch pad. Other input devices (not
shown) may include a microphone, joystick, game pad, satellite
dish, scanner, or the like. These and other input devices are often
connected to the processing unit 120 through a user input interface
160 that is coupled to the system bus, but may be connected by
other interface and bus structures, such as a parallel port, game
port or a universal serial bus (USB). A monitor 191 or other type
of display device is also connected to the system bus 121 via an
interface, such as a video interface 190. In addition to the
monitor, computers may also include other peripheral output devices
such as speakers 197 and printer 196, which may be connected
through an output peripheral interface 195.
[0059] The computer 110 may operate in a networked environment
using logical connections to one or more remote computers, such as
a remote computer 180. The remote computer 180 may be a personal
computer, a server, a router, a network PC, a peer device or other
common network node, and typically includes many or all of the
elements described above relative to the computer 110, although
only a memory storage device 181 has been illustrated in FIG. 11.
The logical connections depicted in FIG. 11 include a local area
network (LAN) 171 and a wide area network (WAN) 173, but may also
include other networks. Such networking environments are
commonplace in offices, enterprise-wide computer networks,
intranets and the Internet.
[0060] When used in a LAN networking environment, the computer 110
is connected to the LAN 171 through a network interface or adapter
170. When used in a WAN networking environment, the computer 110
typically includes a modem 172 or other means for establishing
communications over the WAN 173, such as the Internet. The modem
172, which may be internal or external, may be connected to the
system bus 121 via the user input interface 160, or other
appropriate mechanism. In a networked environment, program modules
depicted relative to the computer 110, or portions thereof, may be
stored in the remote memory storage device. By way of example, and
not limitation, FIG. 11 illustrates remote application programs 185
as residing on memory device 181. It will be appreciated that the
network connections shown are exemplary and other means of
establishing a communications link between the computers may be
used.
* * * * *