Abstract:
A twisted track interferometer (TTI) for producing magic states is disclosed. The spin of ½-vortices may be exploited to produce magic states. The disclosed “twisted track interferometer” is a “topological twist” on the conventional Pabre-Pero interferometer adapted to topological superconductors. In the disclosed TTI, the probe particles may be Josephson vortices (JVs). JVs are estimated to be light and will tunnel more easily than Abrikosov vortices. Also, the disclosed TTI does not require multiple tunneling events. Rather, the JVs are propelled down thin insulating tracks within a 2D topological p-wave superconductor by a Magnus force generated by a tunneling supercurrent across the tracks. The JVs encounter tunneling junctions as they pass into the arms of the TTI.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of provisional U.S. patent application No. 61/471,870, filed Apr. 5, 2011, the disclosure of which is incorporated herein by reference. 
     The subject matter disclosed herein is related to the subject matter disclosed and claimed in U.S. patent application Ser. No. 13/077,339, filed Mar. 31, 2011, and in U.S. patent application Ser. No. 13/111,828, filed May 19, 2011, each of which claims the benefit of provisional U.S. patent application No. 61/347,022, filed May 21, 2010 (collectively, “the 022 family”). The disclosures of the above-referenced patent applications are incorporated herein by reference. 
    
    
     BACKGROUND 
     It is known that all the so-called “Clifford operations” can be realized by braiding and/or interferometric measurement within Ising systems. The fractional quantum Hall state 
             v   =     5   2           
is thought to be an (Ising system)×(U(1) system). The U(1) sector does not affect braiding and interferometry. However, it is known to alter the statistics (e.g., “twist factors”) of the quasiparticles. There have been many proposals for synthesizing superconductor/semiconductor (SC/SM) systems to realize a physical two-dimensional, chiral, topological, (p x +ip y ) superconductor (referred to herein as a “topological SC”) whose topological characteristics are purely Ising.
 
     To extend beyond Clifford operations to universal quantum computation, it is sufficient to produce so-called “magic” states: 
                                     cos   ⁡     (     π   8     )       |   1     〉     +     sin   ⁡     (     π   8     )         |   Ψ     〉     ⁢           ⁢   or   ⁢           ⁢     sin   ⁡     (     π   8     )         |   1     〉     -     cos   ⁡     (     π   8     )         |   Ψ     〉     ,         
which differ from one another by the Pauli operator σ y . Given a magic state, Clifford operations, and measurement, one can build a π/8-gate, yielding, along with the Clifford gates, a universal gate set. An interferometer for producing magic states is, therefore, desirable.
 
     SUMMARY OF THE INVENTION 
     As disclosed herein, the spin, 
                 θ   σ     =     ⅇ       2   ⁢   π   ⁢           ⁢   ⅈ     16         ,     of   ⁢           ⁢     1   2     ⁢     -vortices             
(referred to herein as “σ&#39;s”) may be exploited to produce magic states. The disclosed “twisted track interferometer” is a “topological twist” on the conventional Fabre-Pero interferometer adapted to topological SC. It is well-known that there is significant non-topological physics in a topological SC (for example, the order parameter phase, Φ, magnetic B-fields, and screening currents). It may be desirable to account for such non-topological physics in the design of any device intended to extract topological information. The interaction energy of probe particles, for example, tends to wash out the interferometric signal if the interaction energy is not suppressed.
 
     The twisted track interferometer (TTI) disclosed herein bears a mathematical relationship to the twisted interferometer disclosed and claimed in the 022 family, though, physically, it may be very different. As disclosed in the 022 family, the probe particles may be Abrikosov vortices, each of which may undergo multiple tunneling events to accomplish the desired “twist.” 
     In the disclosed TTI, the probe particles may be Josephson vortices (JV) (also called fluxons). JVs are estimated to be light (e.g., having an effective mass less than one electron mass), and will tunnel more easily than Abrikosov vortices. Also, the disclosed TTI does not require multiple tunneling events. Rather, the JVs are propelled down thin (e.g., order one-nanometer) insulating tracks within a 2D topological p-wave superconductor by a Magnus force generated by a tunneling supercurrent J S  across the tracks. The JVs encounter tunneling junctions as they pass into the arms of the TTI. 
     The disclosed twisted track interferometer may enable construction of a universal gate set for quantum computation on a topological superconductor substrate, and, therefore, may enable topologically protected production of magic states in a topological quantum computer based on two-dimensional topological superconductors. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts an example twisted track interferometer. 
         FIG. 2  depicts an immersed track. 
         FIG. 3  depicts parallel tracks that effect JV-JV interaction. 
         FIG. 4  depicts pairing ±JVs into excitons. 
         FIG. 5  depicts a JV storage ring and gating. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  depicts an example twisted track interferometer  100 . The lines represent “tracks,” which may be formed as narrow insulating defects in the superconductivity, Δ, of a 2D topological superconductor. The mathematical similarity with the 022 family is net ±double twist in one arm of the interferometer. Diagrammatic calculations show term-by-term in agreement (after adjusting by an overall phase) with that the twisted interferometry diagrammatic analysis provided in the 022 family. 
     As shown in  FIG. 1 , probe anyons (not shown), which may be JVs (a.k.a., “fluxons”), for example, may be generated in a topological SC  102 . More generally, the probe anyons may be any physical realization of Ising anyons in the topological SC  102 . The topological SC  102  may be any physical device capable of generating such anyons. 
     The probe anyons may be entered onto a first, or “entry,” track  104  from the topological SC  102 , or from a JV storage ring  103 , or both. A more detailed description of a JV storage ring  103  is provided below in connection with  FIG. 4 . Thus, the fluxons may be generated in the topological SC  102 , or imported from the storage ring  103 . 
     The first track  104 , which may be a straight track, as shown, is provided to separate the particles being measured from high-energy events needed to enter the JVs onto the track. The length of the first track  104  may be chosen such that the particles being measured are separated sufficiently from the high-energy events. A time-varying tunnel barrier  106  may be provided in the first track. 
     The probe anyons may travel along the first track  104  and encounter a scattering junction  108 , which may be a balanced scattering junction. At this point, the tracks may split into two complementary tracks  110 A,  110 B. Along the complementary tracks  110 A,  110 B, a first half of a first qubit may be in an initial state, e.g., 
                     1     2       ⁢     (     ❘   1     〉       -     i   ⁢        Ψ   〉         )     .         
A second half of the first qubit may be sparsely encoded, absorbing the charge at infinity. The first qubit may be encoded in four anyons, such that each half of the qubit includes two anyons.
 
     The complementary tracks  110 A,  110 B may be contoured to bend away from each other beginning at the scattering junction. The complementary tracks  110 A,  110 B may be contoured to close in on each other opposite the scattering junction, with the distance between them being reduced. 
     A pair of straight racks  112 A,  112 B, or “race track”  112 , may extend from the complementary tracks  110 A,  110 B. A more detailed description of a race track  112  is provided below in connection with  FIG. 3 . 
     Beyond the race track  112 , the tracks may assume significantly different contours. The left track  114  (as shown in  FIG. 1 ) may define a twisted portion, which may include two “twisting loops”  114 A and  114 B, as shown. In each loop  114 A and  114 B, the probe anyons are physically spun 360° (i.e., “twisted”), for a total spin of 720°. From the mathematics provided in the 022 family, it should be understood that a spin of 720° is desirable to change the phase of the qubit between |1            and |Ψ         .
     The track  114  may loop back onto itself at a first angle α 1  to close the first loop  114 A. The track  114  may loop back onto itself at a second angle α 2  to close the second loop  114 B. The first angle α 1  and the second angle α 2  may each be much greater than zero. Note that the loops  114 A and  114 B may form any shape. Thus, the term “loop” as used herein should not be interpreted to imply that the loops must be circular or any other regular shape, though they could be. 
     The right track  116  is a delay track, having a total length sufficient to allow for the anyons to traverse the left track. The length of the right track  116  may be the same as the length of the left track  114 , though it need not be. The length of the right track  116  may be chosen to provide any desired delay. Note that the probe anyons do not spin as they traverse the right track  116 . 
     A straight track  118 A extends from the left track  114 . A straight track  118 B extends from the right track  118 . The pair of tracks  118 A,  118 B define a racetrack  118 , which, again, is described in detail below in connection with  FIG. 4 . 
     The interferometer  100  may include only one racetrack (e.g., racetrack  112 ) before the twisting loops  114 A,  114 B. The interferometer  100  may include only one racetrack (e.g., racetrack  118 ) after the twisting loops  114 A,  114 B. Or the interferometer  100  may include both a racetrack (e.g., racetrack  112 ) before the twisting loops  114 A,  114 B and a racetrack (e.g., racetrack  118 ) after the twisting loops  114 A,  114 B. 
     The tracks  118 A and  118   b  converge to form a single output track  120 , which feeds into a flux measuring device  122 . The flux measuring device  122  measures the current in the loop (i.e., the “flux”), which determines the final state of the qubit. 
     To preserve unitarity without the possibility of JV reflection from the “track fuse” point p, a JV sink may be provided near the “track fuse” point p for JVs traveling along each arm. Such a sink is not depicted in  FIG. 1 . It is represented in  FIG. 4  as tracks into the detector at points  1  and  3 . 
     Though the twisted track interferometer (“TTI”) disclosed herein is mathematically similar to the twisted interferometer disclosed and claimed in the 022 family, there are a number of physical differences between the two. For example, the TTI uses Josephson vortices (“JVs”) as probe anyons, rather than Abrikosov vortices. JVs, due to their smaller effective mass, provide increased output signal strength. Also, because the TTI employs insulating tracks to guide vortices, delicately tuned tunneling junctions can be avoided. And because trajectories along the twisted track are more deterministic than the previously disclosed multiple-tunneling design, the reduction of interferometric visibility due to undesired tunneling trajectories may be avoided 
     
       
         
           
             
               ( 
               
                 
                   by 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   a 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   factor 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   estimated 
                 
                 ≤ 
                 
                   4 
                   27 
                 
               
               ) 
             
             . 
           
         
       
     
     It should be understood that JV tracks need not be imbedded. They may cross, as seen in the “immersed track”  200  depicted in  FIG. 2  (that is, provided the supercurrent J S  is arranged to tunnel from left to right as seen from the frame of the propagating JV). According to the Josephson equation, I=I 0  sin(Φ). Accordingly, the desired phase differences may be achieved by producing phases π&gt;&gt;Φ 2 &gt;Φ 1 &gt;Φ 0 &gt;0 in the three complementary regions shown in  FIG. 2 . Effectively, the relative phases orient the track, so the JVs travel with the arrow of orientation. It may be desirable that the JVs do not “short-cut” the loop and turn right (with reference to  FIG. 2 ). The amplitude for this process is exponentially surpassed by the stiffness of the order parameter Φ provided the angle marked α satisfies α&lt;&lt;π, as the shortcut would then impose a demanding constraint on the Φ-field, and thus raise energy. 
     As described in the 022 family, twisted interferometry cannot operate in the low tunneling limit. In interferometry, the multiplicity of probe particles generally achieves fault tolerance with respect to certain design parameters. Because twisted interferometry typically requires mutual linking in space-time of probe world lines, it may be desirable for all probes to be sent in a short burst so that all probes passing through a twisting loop reside simultaneously on it at some point in time. Accordingly, it may be desirable to control the energy splitting between distinct left/right tunneling configurations (race tracks, for example), and the burst of probe anyons generated (via a JV storage ring, for example). 
     In electromagnetism, the base energy scales for magnetic interaction are much larger for magnetic flux than electric flux. The effective charge of a quantum ½-vortex is 
               Φ   0     =         2   ⁢   π   ⁢           ⁢   hc       2   ⁢           ⁢   e       .           
The ratio
 
                   Φ   ⁢           ⁢     0   /   4     ⁢   π     e     =     1     4   ⁢   α         ,       where   ⁢           ⁢   α   ⁢           ⁢   is   ⁢           ⁢   the   ⁢           ⁢   fine   ⁢           ⁢   structure   ⁢           ⁢   constant     ≈       1   137     .             
In the magnetic case, therefore, typical interaction energies E B  of the form
 
                     q   1     ⁢     q   2       γ     ⁢           ⁢   may   ⁢           ⁢   be   ⁢           ⁢       (     1     4   ⁢   α       )     2       ,         
which may be about 4×10 3  times larger than corresponding electric energies E E . For this reason, it may be desirable to minimize the JV-JV interaction energies, E JV-JV , which will typically differ according to the arrangement of the JVs between the left and right arms of the twisted track interferometer.
 
       FIG. 3  depicts an example race track  300 . Generally, the uniform division between right and left arms (R and L, respectively) will have lower interaction energy than highly skewed divisions. This can be compensated, in part, by an opposite effect referred to herein as “race track.” 
     The tracks R and L may be parallel to one another and close enough to one another for JV-JV interaction to exist. Through variations in the material structure of the tracks R, L, “slow” and “fast” regions may be alternatingly arranged in each track R, L. Fast racks are depicted in  FIG. 3  as solid lines; slow tracks are depicted in  FIG. 3  as solid lines. This is possible because the Josephson vertex group velocity v JV  depends sensitively on many parameters, including track width, height, and the local london penetration depth (3D λ L ) of the bulk superconductor. 
     The effect of velocity variation on parallel tracks is repeated passing of JV (in the manner of cars on a freeway traveling in traffic each within their respective lanes). Passing events cost energy, and these events are more common in cases where the JVs are most uniformly divided between the two tracks R and L. This provides a simple mechanism to balance the opposite effect of less JV-JV interaction (in the uniformly divided case) when the tracks R and L are well separated. 
       FIG. 4  depicts apparatus for pairing ±JVs into excitons. As shown in  FIG. 4 , the linear weaknesses in the superconductivity of a topological chiral SC (which are referred to herein as “tracks” and represented in the figures as lines) can be formed as “double tracks” that are spaced some large multiple of the correlation length apart (but still a small fraction of a micron). Instead of individual probe vortices, numerous ±pairs of vortices called “excitons” may be sent down these parallel tracks. The vortices in ±pairs still may be topologically uncorrelated. Features such as track splitting, joining, and crossing, for example, may be engineered in the double track case analogously to the single track case described above in connection with  FIG. 1 . 
     By using excitons (shown as Xs in  FIG. 4 ), X-X interaction may be dipolar and decays as 1/r 3 , which reduces the demands on screening that are present for JV-JV interactions, which decay only like 1/r. In order to reduce E JV-JV  to the 1/r 3  power law associated with dipolar interactions, oppositely oriented pairs of JVs may be paired into magnetic field loops that interact as dipoles, essentially by doubling the apparatus depicted in  FIG. 1 . 
     As shown in  FIG. 4 , vortex loops  402 , or “JV excitons,” including ±JV pairs (+JV, −JV) may be generated in a topological SC  404 . The loops  402  may be sent down a pair of parallel tracks  406 A,  406 B. At a point, S, two more parallel tracks,  406 C and  406 D, may be introduced, to form first and second pairs of tracks. 
     The first pair of tracks,  406 A and  406 C, may be contoured to form two twisting loops  408  A and  408 B, which case the 720° rotation described above. The second pair of tracks,  406 B and  406 D, may be contoured to provide the corresponding delay, as described above. 
     The four σ qubits, σ 1 , σ 2 , σ 3 , σ 4 , shown in  FIG. 4  may be “acted on,” though they are not actually measured in this approach. To achieve the topological properties of a single JV probe, one half of the pair (e.g., σ 3 , σ 4 ) may have its topological contribution neutralized. This may be accomplished by introducing a “5th” σ-particle, σ 5 , to achieve the “odd-part” of the well-known “odd-even” effect of Majorana interference. As shown in  FIG. 4 , the 5 th  particle, σ 5 . neutralizes the topological contribution from −JV. 
     Beyond the twisting loop  408 B, tracks  406 B and  406 C fuse at a first fusion point P 1  and emanate as a single track  4 . Tracks  406 A and  406 D fuse at a second fusion point P 2 , and emanate as three parallel tracks  1 ,  2 , and  3 . The tracks  1 ,  2 ,  3 , and  4  lead into a detector  412 . The detector  412  counts flux loops  410 B that bind tracks  2  and  4 . In an example embodiment, the flux loops  410 A binding tracks  1  and  4 , and the flux loops  410 C that bind tracks  3  and  4  are not measured. 
       FIG. 5  depicts a JV storage ring  500  and gating. An internal/external phase differential Φ 1 &gt;Φ 0  will keep in circulation n JVs, introduced by a flux solenoid into a circular track, i.e., a “ring”  500 . Then, a precisely timed pair of electrostatic top gates  510 ,  520  may simultaneously break the ring and lower the tunneling barrier to a second track  530  leading to the TTI. The gate pulses may be very sharp (e.g., having frequency components of at least one gigahertz) to release all n JVs in a group. Accordingly, it may be desirable to well-separate this event in distance from all stored quantum information, and in particular from the qubit residing in the TTI. 
     In an alternate mode, the TTI can be operated in a limit where only a single probe JV is used. This may be desirable as there will be no JV-JV interaction to possibly degrade the performance of the TTI. In the single probe case, neither the “race track” nor pairing into excitons is required. However, the magic states precision depends on numerous probe particles, though a single probe passing through a perfectly tuned TTI will produce a perfect magic state. As a practical matter, use of a single probe will produce a magic state of fidelity 1−ε. So-called “magic state distillation” takes as input many magic states of fidelity 1−ε and, using only protected Clifford operations and measurements in the charge basis {|1           , |Ψ         }, produces as output a single magic state of fidelity 1−ε′, where ε′≈35ε 3  when ε is small. The threshold for the initial ε is roughly ε&lt;0.3 and is therefore not extremely demanding, though the asymptotic regime is not reached until ε≈10 −3 .