METHOD FOR PLANAR IMPLEMENTATION OF /8 GATE IN CHIRAL
TOPOLOGICAL SUPERCONDUCTORS
TECHNICAL FIELD
[0001J 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 /8 gate in chiral topological
superconductors.
BACKGROUND
[0002] 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:0912.21 15), 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 . If the CTS is not
p]anar, but configured as a surface of genus> 0, a significant stiffness term 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.
[0003] 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 , the term CTS is used herein to emphasize the presence of the order
parameter.
SUMMARY
[0004] Disclosed herein is a topologically protected
which becomes universal when combined with the gates available through quasi-particle
braiding and previously described planar quasi-particle interferometry:
MS 330525.03
1 0 0 0
1 0
1
1 1 0 1 0 0
P .
[0005] Key fe rlying CTS quasiparticle
excitation include:
• Excitations I ,,(trivial, twist, fermion),
• Nontrivial fusions = 1+and = 1,
1 2 1
• S-
• Tw and
• sp ( )
[0006] 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.
1
[0007] A measurement operator O with possible outcome vectors v ,...,vn
can
b
O l
where v i s the detector state corresponding to outcome v l
.
l=1
Measurement can be applied not just to a vector w but to a tensor T (corresponding to a
segregation of quantum h case
w a w O w v v v
l=1
becomes
[0008] Once the outcome of the measurement is observed, the system is in a
tensor product state. Thus, measuring v (i.e., observing some ' ) 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.
[0009] The situation for a (2+l)-dimensional TQFT, as shown in FIG. 3, is only
slightly more complicated. The 3-manifold Mplays the role of the tensor T, but its
valence is unspecified until the boundary ofMis 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 ofMis varied. As an example, take Mto be a solid torus
S D2 and the theory to be Ising. Decomposing into A in the following three
ways yields three different matrices (2-tensors) in the \,,basis, along the loops A B.
[0010] These calculations may now be illustrated. Case (2) is axiomatic:
products correspond to identity morphisms. The identity operator "glues up" to become
the vector in the longitudinal basis. Now
transforming by S to the meridial basis we obtain vm = S(v,) = (1,0,0) Vm and converting
to an operator we should divide entries by to obtain case (1). Finally, to
compute case (3) note that if A=S T S is the modular transformation sending (1) to (3),
then in this twisted basis (/), v becom
which converts to case (3).
[0011] 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 -charge Wilson loop
running along its core. In case ')
and the corresponding operator
[0012] It is known that the operation of an interferometer within an Ising system
using probe particles projects the topological charge state of the interferometry loop to a
charge sector 1, , or , along the interferometry loop up to an error exponentially small
in the number of probes.
[0013] The TQFT analog to partial trace is to glue a 2-handle of
space-time topological fluid along the measured loop . As shown in FIG. 4, the charge
line at the core of the 2-handle is precisely the measurement outcome =\,,(so if the
trivial particle is measured, the 2-handle has no Wilson line). Up to an overall scalar
(= S a ) which has no physical significance, if is a loop on a torus boundary component
T, we may glue not just a 2-handle but also a 3-handle (B3,6B3 as well (containing a
matching charge line), so that measuring along is equivalent to Dehn filling Twith a
solid torus S D2,*xdD2 gluing to and S' x * being a Wilson loop of charge . The
analogy with the Penrose TQFT: may now be completed. As shown in
FIG. 5, a measurement Dehn fills a solid torus along with a Wilson loop of charge at
its core - and in a disjoint system, we have a state recording the fact that was the
measurement outcome.
BRIEF DESCRIPTION OF THE DRAWINGS
[0014] FIGs. 1A and IB illustrate tensor contraction in Penrose notation.
[0015] FIG. 2 depicts a measurement using tensor contraction.
[0016] FIG. 3 depicts a (2+l)-dimensional TQFT.
[0017] FIG. 4 depicts gluing a 2-handle of space-time topological fluid along a
measured loop.
[0018] FIG. 5 depicts a measurement Dehn filling a solid torus.
J0019] FIG. 6A provides a sketch of a Beenakker-style interferometer in the
CTS-ISH context.
[0020] FIG. 6B depicts a twisted interferometer.
[0021] FIGs. 7A and 7B illustrate computation of operators in a twisted annular
basis.
[0022] FIG. 8 summarizes a protocol producing a topologically protected /8-
gate using twisted interferometry.
[0023] FIGs, 9A and 9B illustrate how a /8-gate may be obtained.
[0024] FIGs. 10A and 10B show Wilson loops in relation to charge lines
corresponding to an original qubit in various states.
[0025] FIG. 11 shows an example computing environment in which example
embodiments and aspects may be implemented.
DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS
[0026] 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 applications no.
12/979,778 and 12/979,856, the disclosures of which are incorporated herein by reference.
As disclosed herein, three tunneling amplitudes, t 1, t 2, and t 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
in this region and increasing the amplitude to tunnel (note that where
M is an effective mass, the bulk gap, and L is the length of the summary tunneling
junction,) or 3) any other electronic, optical, or magnetic intervention.
[0027] 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 t1 and t2 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 / .
[0028] 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 figure 6B, the distance from a to b along the
island (through tunneling junctions 1 and 3) should be approximately 1/5 the distance L
from a to b detouring the island and passing through tunneling junction 2. Tunneling is
allowed at , where t is the time for a wave packet
to propagate once around the island at group velocity.
[0029] 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, t1 is made substantial for a brief time (order 10-10 - 10 -9 seconds),
allowing a wave packet of -particles with support small with respect to L\ to transit to the
island with amplitude t1 while, with amplitude the wave packet continues along
the left edge. Knowing the geometry and the velocity of the edge mode, t2 is briefly -
again for a time = t0/10 - increased from zero precisely as this latter wave packet arrives at
the second tunneling junction so that the packet is transmitted with amplitude t2. 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.
[0030] At this moment, t3is 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 allows the two branches to arrive synchronously. In
order to maintain superposition of trivial topological charge and topological charge
aJong 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.
[0031] Up to errors exponentially small in the number of Ising - particles
admitted to the island, the twisted interferometer acts on the topological charge enclosed
within the interferometry loop by either
in the longitudinal basis according to whether |l) or \ is observed, where
[0032] In the untwisted (Beenakker) context, the measurement is in the basis of
topological charge enclosed in the untwisted interferometry loop (FIG. 6A).
The operator O is the sum of the two basis projectors tensored with the measurement
outcome , or less formally, is observed and
observed. One might expect in the twisted context to affect
conjugates of P P2 - 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, = twisted must be diagonal in the basis of topological charge.
[0033] As described above, making a charge measurement with trivial outcome
is equivalent to topologically Dehn filling the twisted interferometry loop with a solid
torus S D2 of pure topological ground state medium; whereas outcome \ is
equivalent to Dehn filling along a solid torus with a ^-charge loop (Wilson line loop)
along the core circle S 1x 0 S 1x D .
[0034] In terms of operators in twisted annular basis, we have:
\+ 0 =(\ + a>)\\)(l\ + (l - )\ )( 0 \ -
if |l) is measured, and
if is measured.
[0035] 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 / in FIGs. 6A and 6B, i.e., the
usual interference loop, by a A := S T 2S change of coordinates. Referring to FIGs. 7A
and 7B, it can be seen that the computation provided above computes O, .
[0036] Twisted interferometry employs a single burst of n co-propagating probe
particles ( < 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 1 10 seconds is 10 < n < 100 . The probes follow trajectories
mutually twisting - 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 |l) or \) along the (-2,1) interferometric loop.
[0037] Ball park estimates for ISH Majorana edge mode velocities are 104 m/s.
If L = 5and the wave packet is to have most of its amplitude supported along a
length with exponentially decaying tails, then the tunneling junction should be open for
10 ° . For reasonable tunneling currents, this would permit between 10 and 100 < ' s to
tunnel, adequate to effectively converge the interferometer.
[0038] FIG. 8 summarizes a protocol producing a topologically protected /8-
gate using twisted interferometry. FIG. 8 depicts a 1 or ^-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 . Third,
twisted interferometry is performed along . Note that ^should not be read as a Wilson
loop in FIG. 8, but rather as our notation for twisted interferometry.
[0039] The fourth event is a fusion of the -charged dots of the original qubit.
The fifth and final event is charge measurement along a . The charge orat the top can be
measured to be 1 or by ordinary (previously described) Beenakker quasi-particle
interferometry. Twisted interferometry is used to measure the charge around
I ' in earlier notation) with the twists recorded by the two kinks in y corresponding to
the two loops around the island of vacuum in FIG. 6B. This yields either states A (l) or
) via projective measurement. Using techniques of quantum topology, we will verify
that the initial r|l) + | ) is transformed by the matrix following "outcomes" in the four
measurement cases:
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 -gate (up to an
irrelevant overall phase).
[0040] Freedman, Nayak, and Walker (arxiv: 0512.072 and 0512.066) and our
U.S. patent applications no. 12/979,778 and 12/979,856, show that the may be
obtained by cutting along in FIG. 9A if = 1(and its inverse if = ) . Thickening
the surface in FIG. 9A results in FIG. 9B. Now the framed curve in FIG. 9A is
precisely the surgery required to send to the meridian labeled in FIG. 9B.
Measuring 1 along affects ordinary framed surgery, while measuring 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.
[0041] Since the original qubit has charges on its internal punctures, there will
be a -charge on , but compared to the original qubit at time t = 0 , the relative phase
between the two fusion channels 1 and is now changed by . The loop ' in FIG.
9B is simply a copy of transported across the product structure.
[0042] A (-1) Dehn twist on the loop ' throws ' to the meridian . Thus
Dehn filling on a bulk parallel to ' , with a (-1) additional twist in its framing compared
to the normal framing ' inherits from the boundary of the bulk, effectively endows the
bulk with a new product structure in which is connected by a cylinder to . , as
drawn in FIG. 8, is this (-1) framed bulk loop isotopic to '. Thus twisted interferometry
with |l) as outcome "teleports" the twist and non-time-slice qubit determined by cutting
the surface of FIG. 9A along to an untwisted time-slice qubit at the top of FIG. 9B
(within the dotted circle).
[0043] It remains to compute the effect of twisted interferometry if the outcome
is I. is not a possible outcome as the charge along =I ' = (1, 2) is obtained from
the charge along / by applying the matrix A. A does not mix the | ) and the
sectors and the charge along The effect of outcome \) is a Wilson
loop of charge \ parallel to ' (in the bulk) with no additional twist in its framing.
[0044] Using the calculational tools of modular tensor categories (MTC), FIG.
10A shows the Wilson loop " " in relation to the charge lines corresponding to the
original qubit in state |l) , and FIG. 10B shows the configuration with the original qubit in
state . It is evident that measuring \) is equivalent to the action of which, up
to an overall phase, is the square of a braid generator.
Example Computing Environment
[0045] 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.
[0046] 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.
[0047] 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.
[0048] With reference to FIG. 11, an exemplary system includes a general
purpose computing device in the form of a computer 10. Components of computer 110
may include, but are not limited to, a processing unit 120, a system memory 130, and a
system bus 12 1 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 12 1 may also be
implemented as a point-to-point connection, switching fabric, or the like, among the
communicating devices.
[0049] 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 directwired
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.
[0050] The system memory 130 includes computer storage media in the form of
volatile and/or nonvolatile memory such as read only memory (ROM) 1 1 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. illustrates operating
system 134, application programs 135, other program modules 136, and program data 137.
[0051] The computer 110 may also include other removable/non-removable,
volatile/nonvolatile computer storage media. By way of example only, FIG. 1 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 52, and an optical disk drive 5 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 4 1 is typically connected to the system
bus 1 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.
[0052] 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 1 7 and printer 196, which may be connected through an output peripheral
interface 195.
[0053] 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. 1 include a local
area network (LAN) 7 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.
[0054] 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 12 1
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.
Appendix A
As an independent check of the calculation of our protocol, we compute the two diagonal
entries of the protocol operator using the techniques of braided tensor categories. We
exhibit the case where both "twisted" and "ordinary" inerferometry yield the trivial
particle,
. Furthermore,
Now using three times and twice yields
Passing a through a gives an additional -1, so
(for unitarity,
Similarly,
and following the precedi calculation, the middle term becomes
Since there is a sphere (dotted) crossed by a single charge line.
Using , we see the first and second terms are equal. Then
The factor of 2 comes from computing in R3 and not S3, and also relates quantum
dimensions to components of . In general, 2 is replaced by the total quantum
dimension Thus our protocol effects the -gate" as claimed, in
the case where both interferometry measurements yield the trivial topological charge.
1 Twisted Interferometry Diagrammatic Analysis
In this section, we restrict our attention to Ising anyons, which has the possible
topological charges 1, , and . The topological twist factor of these charges
are 1 = 1, = ei /8 , and = —1, respectively. The monodromy matrix is
We begin by considering the effect of a single probe with anyonic charge 6 = ,
i.e. ( = |, ; 1) (,1|. For a particular component of the target anyons'
density matrix, the relevant diagram that must be evaluated for a single probe
measurement is
where U now includes the twisting in the leg of the interferometer separating
anyons A and anyons C . For the outcome s = - , this is
where we have defined
and have used the diagrammatic rules to remove the 6 loops. The topological
twist factor for anyons is = e" / 8 . A similar calculation for the s = †
outcome gives
From this, inserting the appropriate coefficients and normalization factors,
we find the reduced density matrix of the target anyons after a single probe
measurement with outcome s :
where the probability of measurement outcome s is found by additionally
taking the quantum trace of the target system, which projects onto the e = 1
components, giving
We note that
give a well-defined probability distribution
The quantity
determines the visibility of quantum interference in this experiment, where
varying allows one to observe the interference term modulation. The ampli
interference. For the rest of the paper, we will ignore this issue and assume
Q = 1, but it should always be kept in mind that success of any interferometry
experiment is crucially dependent on Q being made as large as possible.
If we were to send probes through the twisted interferometer one at a time, the
result for N (initially unentangled identical) probes would simply be obtained
by iterating the single probe calculation. The string of measurement outcomes
s , . . . , ) occurs with probability
and results in the measured target anyon reduced density matrix
It is apparent that the specific order of the measurement outcomes is not
important in the result, but that only the total number of outcomes of each
type matters, hence leading to a binomial distribution. We denote the total
number of Sj = —» in the string of measurement outcomes as n , and cluster
together all results with the same n . Denning (for arbitrary p and q )
the probability of measuring n of the N probes at the horizontal detector is
and these measurements produce the target anyon reduced density matrix
This differs from the non-twisted case only by the phase of the interference
term, and so clearly produces the same asymptotic measurement result, i.e.
projection onto the charge basis and decoherence of any non-trivial ent angle
ment between the interior and exterior of the interferometer.
However, if we send all the probes through the interferometer at once, in
such a way that all of the probes passing through the twisting leg of the
interferometer are twisting together (i.e. are all on the twisting island before
the the first one completes its first full twist) , then the probe anyons experience
a very special entangling operation that allows a much different outcome of
the twisted interferometry. In this case, when m probes pass together through
the twisting leg of the interferometer, they all braid around each other twice,
as depicted in Fig. XX.
For Ising anyons, this double twisting of m probes, each carrying topological
charge b , can be evaluated in a straightforward manner. By applying a
partition of unity to the probes we see that the double twisting of m lines
is equal to f(m) times m untwisted lines, where
We now recall the definitions
of a non-twisted interferometer, and consider the expansion of
in powers of 1, , M*, , and Meb, corresponding to the four possible ways a
probe line can link the density matrix basis elements:
where are the polynomial coefficients.
For the twisted interferometer with Ising anyons, we replace
in the superoperator's action on the density matrix with
The difference between the twisted and non-twisted case is the insertion of
the factor / ( ) when m probes that pass through the twisted leg of the
interferometer in the ket portion of the diagram and / * ( ') when m! probes
that pass through the twisted leg of the interferometer in the bra portion of
the diagram. In the expansion above, = m a + m e and ' m a>+ m . This
expression can be rewritten in terms of a sum over binomials by making use
of the Fourier decomposition
which gives
With this, we find that
where p are defined as before, and we have defined the additional quantities
Thus, the density matrix that results from sending N probes through the
twisted interferometer with n of them being measured with outcome s =-» is
where the probability of measuring n of the N probes at the horizontal detector
is
We can now consider the N - oo limit. Defining r = n/N and
we have
( )
where respectively. We also notice that
Thus, in the N —oo limit, we have
If we restrict to the case of measuring the state of a topological qubit, i.e.
measuring two of the four anyons that comprise a qubit, which has a = 1, ,
we find that r p with probability Pr (pi) sin2
and the resulting superoperator acting on the density matrix
Similarly, we have r with probability
and the resulting superoperator acting on the density matrix
If we start from the initial qubit state (which can be
obtained from |1) by applying Clifford gates, i.e. | ) = P _ 1H |1)), we have
Pr (Pi) = Pf = 1/2, with the two outcomes giving the "magic states"
We also note that with probability a n
resulting density matrix is the same as in the non-twisted case, i.e.
One might worry that the description of the probes passing through the
"twisted" leg of the interferometer may not include the individual probe
anyons' twist factors, but rather is simply described by the probes all braid
ing around each other twice. If this alternate description is correct, then it is
straightforward to see that the result is actually quite similar, indicating that
it is the collective braiding of the probes, not the twisting that is the crucial
component to the twisted interferometer. To see this, we can simply notice
that is we remove all the twistin factors we would instead
In this case, we would have precisely the same analysis,
which have additional phase factors on the interference terms coming from
If we restrict to the case of measuring the state of a topological qubit, i.e.
measuring two of the four anyons that comprise a qubit, which has a = , ,
we find that r with probability
and the resulting superoperator acting on the density matrix
Similarly, we have r with probability
and the resulting superoperator acting on the density matrix
f we start from the initial qubit state { (which can be
obtained from |1) by applying Clifford gates, i.e. , we have
Pr p = Pr ( ) = 1/2, with the two outcomes giving the "magic states"
We also note that r with probability and the
resulting density matrix is the same as in the non-twisted case, i.e.
One might worry that the description of the probes passing through the
"twisted" leg of the interferometer may not include the individual probe
anyons' twist factors, but rather is simply described by the probes all braid
ing around each other twice. If this alternate description is correct, then it is
straightforward to see that the result is actually quite similar, indicating that
it is the collective braiding of the probes, not the twisting that is the crucial
component to the twisted interferometer. To see this, we can simply notice
that is we remove all the twistin factors, we would instead
In this case, we would have precisely the same analysis,
which have additional phase factors on the interference terms coming from
What is Claimed:
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 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.
5. The interferometer of claim 1, wherein, while the first, second, and third tunneling
junctions are open, all current flows into the first drain.
6. The interferometer of claim 1, wherein a wave packet of -particles with support
small with respect to the first distance is allowed to transit to the second vacuum with a
first amplitude.
7. The interferometer of claim 6, wherein the wave packet is allowed to transit the
first vacuum with a second amplitude that is smaller than the first amplitude.
8. The interferometer of claim 7, wherein a second wave packet arrives at the second
tunneling junction such that the packet is transmitted with a third amplitude.
9. The interferometer of claim 8, wherein the first wave packet arrives at the third
tunneling junction precisely when the second wave packet also arrives at the third
tunneling junction.
10. A planar /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 ferri-magnetic insulator.
11. The gate of claim 10, wherein the electronic structure is capable of supporting
universal topological, fault tolerant quantum computation.
12. A topologically protected method for implementing a /8-gate in a chiraltopological-
superconductor (CTS), Ising-sandwich-heterostructure (ISH) device, the
method comprising:
implementing universal quantum computation in a topologically protected, faulttolerant
manner within the CTS-ISH device.
13. The method of claim 12, wherein the CTS-ISH device comprises a low mobility
electronic structure that is capable of supporting universal topological, fault tolerant
quantum computation.
14. The method of claim 13, wherein the low mobility electronic structure comprises a
spin-orbit coupled semiconductor.
15. The method of claim 14, wherein the low mobility electronic structure comprises
an ordinary superconductor.