Abstract:
Computing bus devices that enable quantum information to be coherently transferred between topological and conventional qubits are disclosed. A concrete realization of such a topological quantum bus acting between a topological qubit in a Majorana wire network and a conventional semiconductor double quantum dot qubit is described. The disclosed device measures the joint (fermion) parity of the two different qubits by using the Aharonov-Casher effect in conjunction with an ancillary superconducting flux qubit that facilitates the measurement. Such a parity measurement, together with the ability to apply Hadamard gates to the two qubits, allows for the production of states in which the topological and conventional qubits are maximally entangled, and for teleporting quantum states between the topological and conventional quantum systems.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This patent application is a divisional application of co-pending, commonly owned U.S. patent application Ser. No. 13/292,217, filed Nov. 9, 2011, which claims the benefit of provisional U.S. patent application Ser. No. 61/412,225, filed Nov. 10, 2010, which applications are hereby incorporated herein by reference. 
     This application is related to U.S. patent application Ser. No. 13/292,220, filed on Nov. 9, 2011, entitled “Coherent Quantum Information Transfer Between Conventional Qubits,” the disclosure of which is hereby incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     Generally, this application relates to quantum computational systems. More specifically, the application relates to devices that can entangle and coherently transfer quantum information between topological and conventional quantum media. 
     BACKGROUND 
     The topological approach to quantum information processing obtains exceptional fault-tolerance by encoding and manipulating information in non-local (i.e., topological) degrees of freedom of topologically ordered systems. Such non-local degrees of freedom typically do not couple to local operations. Consequently, the error rates for topological qubits and computational gates may be exponentially suppressed with distance between anyons and inverse temperature. 
     This suppression of error rates may provide an advantage over conventional quantum computing platforms. However, it also makes it challenging to coherently transfer quantum information into and out of topological systems. Not only is coupling the non-local degrees of freedom in the topological system to an external system required, but it must be done in a controlled and coherent manner. Consequently, it is desirable to create quantum entanglement between the topological and conventional states. 
     SUMMARY 
     Described herein is a device that can entangle and coherently transfer quantum information between topological and conventional quantum media. Such a device may be referred to herein as a “topological quantum bus.” The device allows for harnessing the relative strengths of the different quantum media, and may be implemented in connection with quantum computation. 
     An example of how a topological quantum bus may be useful stems from the understanding that a computationally universal gate set cannot be produced for Ising anyons using topologically-protected braiding operations alone. Unless a truly topologically ordered Ising system is provided (which is not the case for superconductor-based systems, including Majorana wires), and can perform certain well-known topology changing operations, braiding operations need to be supplemented with topologically unprotected operations. Fortunately, these can be error-corrected for an error-rate threshold of approximately 0.14 by using the topologically-protected Ising braiding gates to perform “magic-state distillation.” 
     Within a topological system, unprotected gates can be generated by, for example, bringing non-Abelian anyons close to each other, which generically splits the energy degeneracy of the topological state space and, hence, dynamically gives rise to relative phases, or by using interfering currents of anyons, which can have an equivalent effect. However, besting even such a high error threshold may still prove difficult using unprotected operations within a topological system, as a result of significant non-universal effects. A topological quantum bus allows for the desired topologically unprotected gates to be imported from conventional quantum systems, for which error rates below 0.14% of logical operations have already been achieved. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  depicts a semiconductor nanowire coupled by proximity with an s-wave superconductor in the presence of an in-plane magnetic field. 
         FIG. 2  is a plot of energy dispersion for the semiconductor depicted in  FIG. 1 , with spin-orbit coupling in the magnetic field. 
         FIG. 3  is a schematic of a device for entangling topological and conventional qubits. 
         FIG. 4  is a schematic of a device for coherent quantum information transfer between topological and conventional semiconductor qubits using joint parity measurements. 
         FIG. 5  is a flowchart of a method for coherent quantum information transfer between topological and conventional semiconductor qubits using joint parity measurements. 
     
    
    
     DETAILED DESCRIPTION 
     A method for implementing a topological quantum bus may include the use of measurements in an entangled basis, e.g., Bell state measurements. For a topological quantum bus, this can be achieved by a measurement of the joint parity of a topological-conventional qubit pair, given the ability to perform Hadamard gates on any qubit. 
     Joint parity measurement corresponds to the two orthogonal projectors:
 
Π 0 =|00                     00|+|11                   11|,  (1)
 
Π 1 =|01                   01|+|10                   10|,  (2)
 
where |0          and |1          and are the logical basis states of the qubits. Topological systems, however, tend to be rather obstructive to such hybridization with external systems. For example, quantum Hall states (the archetypal topological systems) require a large background magnetic field, which “destroys” superconductivity and eliminates the possibility of coupling to Josephson-junction qubits.

     Fortunately, recently proposed implementations of Majorana nanowires appear promising for overcoming such obstacles. These wires localize zero energy Majorana fermions at their endpoints and, as such, provide a one-dimensional topologically protected two-level system. It may appear that such a system might provide a topological qubit, but lack quantum information processing. However, a network of Majorana wires may be formed, and manipulated using gate electrodes in a manner that performs braiding exchanges of their endpoints (and, hence, their respective Majorana fermions). This generates the topologically-protected braiding operations of Ising anyons (up to an overall phase) on the topological state space. 
     It follows from this that Majorana wire networks can be utilized as Ising anyons for topologically-protected quantum information processing. A concrete realization of a topological quantum bus that uses the Aharonov-Casher effect to coherently transfer quantum information between a topological qubit in a Majorana wire system and a conventional semiconductor double-dot qubit will now be described in detail. 
     The Aharonov-Casher effect involves interference of particles with magnetic moment (vortices) moving around a line charge. It may enable performance of non-local measurements of charge in a region by utilizing the interference of vortices traveling through two different paths around the region. For superconducting systems it is natural to try to use Abrikosov vortices in this context. However, Abrikosov vortices in s-wave superconductors may have rather large mass due to the large number of subgap states localized in their cores. As a result, these vortices may behave classically. 
     An alternative is to use Josephson vortices (fluxons), which arise due to phase-slip events in Josephson junctions. Their effective mass is determined by the charging and Josephson energies of the junction, and can be much smaller than that of Abrikosov vortices, allowing them to behave quantum-mechanically. Indeed, the Aharonov-Casher effect with Josephson vortices has been experimentally observed, and several proposals have been made to utilize it in the context of topological quantum information processing. 
     The basic element in the implementation of Majorana wires is a semiconductor nanowire with strong spin-orbit inter-actions, coupled with an s-wave superconductor.  FIG. 1  depicts a semiconductor nanowire coupled by proximity with an s-wave superconductor, in the presence of an in-plane magnetic field. The Hamiltonian (with            =1) for such a nanowire is:
                     H   0     =       ∫       -   L     /   2       L   /   2       ⁢           ⁢       ⅆ   x     ⁢           ⁢       ψ   σ   †     ⁡     (   x   )       ⁢       (       -       ∂   x   2       2   ⁢   m   *         -   μ   +     i   ⁢           ⁢     ασ   y     ⁢       ∂   x     ⁢     +     V   x         ⁢     σ   x         )     σσ′     ⁢     ψ       σ′   ⁡     (   x   )       ,                   (   3   )               
where m*, μ, and α are the effective mass, chemical potential, and strength of spin-orbit Rashba interaction, respectively, and L is the length of the wire, which is taken to be much longer than the effective superconducting coherence length ξ in the semiconductor.
 
     An in-plane magnetic field B x  leads to spin splitting V x =g SMμB B x /2, where g SM  is the g-factor in the semiconductor, and μ B  is the Bohr magneton.  FIG. 2  is a plot of energy dispersion for the semiconductor, with spin-orbit coupling in the magnetic field B x , which opens a gap in the spectrum. When the chemical potential μ is in this gap, the nanowire coupled with the s-wave superconductor is driven into topological phase. 
     In other words, when coupled with an s-wave superconductor, the nanowire can be driven into a non-trivial topological phase with Majorana zero-energy states localized at the ends when the chemical potential is properly adjusted and lies in the gap. In the simplest case of a single-channel nanowire, the topological phase corresponds to |V x |&gt;√{square root over (μ 2 +Δ 2 )}, where Δ is the proximity-induced pairing potential. Multi-channel use is described in R. M. Lutchyn, T. Stanescu, and S. Das Sarma, Phys. Rev. Lett. 106, 127001 (2011), arXiv:1008.0629, incorporated herein by reference. 
     As seen in  FIGS. 3 and 4 , two Majorana fermions γ 1  and γ 2  residing at the ends of a wire  10  constitute a topological qubit  20 , since they give rise to a two-level system that is degenerate up to O(e −L/ξ ) corrections that are exponentially suppressed with the length of the wire  10 . Indeed, a non-local Dirac fermion operator can be formally defined as c=γ 1 +iγ 2 , and then the two logical states of the qubit correspond to the state in which this Dirac fermion is unoccupied |0           ≡|n p =0          and occupied |1         ≡|n p =1         , where c|n p =1         =n p =0         ,c|n p =0         =0, and c†c|n p =n p |n p           . Thus, the topological qubit states are characterized by fermion parity n p =0,1. As previously mentioned, in a network of such wires, these Majorana fermions γ 1  and γ 2  behave as Ising non-Abelian anyons when they are translocated, e.g., using gate electrodes.
     Topological and conventional qubits can be entangled by measuring the fermion parity on the superconducting island using the Aharonov-Casher effect. FIG.  3  is a schematic of a device for entangling a topological (e.g., Majorana wire) qubit  20  and a conventional (e.g., semiconductor double-dot) qubit  30 . A flux qubit Φ having three Josephson junctions (the strips labeled J 1 , J 2 , and J 3 ) supports clockwise or counter-clockwise supercurrent. When E H1 =E J3 , there is interference between quantum phase slips across junctions J 1  and J 3 . These phase slips correspond to Josephson vortex tunneling encircling the superconducting islands as shown by the dashed line. Via the Aharonov-Casher effect, quantum interference of vortices around the islands produces an energy splitting for the flux qubit (at its degeneracy point) that strongly depends on the state of the topological and conventional semiconductor qubits  20 ,  30 . The nanowire  10  may have topological  10 T and non-topological  10 N segments. The latter can be achieved by driving the wire  10  into the insulating or trivial superconducting phases. 
     More specifically, consider the superconducting flux qubit Φ with Josephson junctions designed to have left-right symmetry such that Josephson coupling energies E J1 =E J3 ≡E J . The two current-carrying states, clockwise |                      and counter-clockwise |                   , form the basis states of the flux qubit Φ. When the applied external flux piercing the flux qubit Φ is equal to a half flux quantum, i.e., Φ=h/4e, there is a degeneracy between the two current-carrying states. This degeneracy is lifted by the macroscopic quantum tunneling between the state |                    and |                    due to the presence of a finite charging energy of the islands, which tends to delocalize the phase. Thus, the new eigenstates of the qubit are |±         =(|                   ±|                   )/√{square root over (2)}. For the device shown in  FIG. 3 , the energy splitting between states |±          depends on the quantum interference of the fluxon trajectories. Indeed, the total Josephson energy of the qubit is
                         U   J       E   J       =     -     [       cos   ⁢           ⁢     φ   1       +     cos   ⁢           ⁢     φ   2       +         E     J   2         E   J       ⁢     cos   ⁡     (       2   ⁢           ⁢   π   ⁢     Φ     Φ   0         -     φ   1     -     φ   2       )           ]         ,           (   4   )               
where it is assumed that E J1 &gt;E J , in contrast with values typically used for flux qubits.
 
     The potential U J  reaches its minima at two inequivalent points (φ 1 ,φ 2 )=(±φ*+2πm,∓φ*∓2πn) for a given n and m which correspond to clockwise and counter-clockwise circulating currents, and φ*=cos −1 (E J /E J2 ). Starting, for example, from the configuration with (φ*−φ*), there are two paths to tunnel to a different flux state: (φ* ,−φ*)→(φ*−2π,φ*), and (φ* ,−φ*)→(−φ*,φ*+2π), which correspond to a phase slip through junction J 1  or J 3 , respectively. As a result, there is an interference between the two paths that encircle the middle islands in the system shown in  FIG. 3 . Note that the amplitude for the phase slips across the middle junction is suppressed in this setup since E J2 &gt;E J . This interference is sensitive to the total charge enclosed by the paths, i.e., the charge residing on the two islands, and is determined by the Berry phase contribution. 
     For the device shown in  FIG. 3 , the splitting energy is given by Δ=Δ 0  cos(∅ AC /2), where ∅ AC =πq/e is the Aharonov-Casher phase for total charge on the islands given by q=en p +q ext , where n p  is the fermion occupation of the Majorana wire  10  and q ext  is the induced gate charge on the islands. Given that the qubit splitting energy now depends on the fermion occupation number, the state of a topological qubit can be efficiently read out using, for example, the well-known radio-frequency (RF) reflectometry technique, which can be carried out with sub-microsecond resolution times. It is assumed that superconducting islands have the same charging energy yielding the same tunneling amplitude Δ 0 . Assuming E J /E c ≈10 and E J2 /E J ≈1.25, WKB approximation gives Δ 0 ≈0.02hv a , where v a  is the attempt frequency, which is estimated to be v a ˜0.1−1 GHz. 
     A situation where q ext  has a quantum component corresponding to coherent electron tunneling inside the area enclosed by the vortex circulation can be realized, for example, by coupling the flux qubit Φ to a semiconductor double quantum dot (DQD) qubit  30 , as shown in  FIG. 3 . Galvanic isolation may exist between the superconductor and semiconductor so that there is no charge transfer between them. DQD qubits may be realized using indium arsenide (InAs) nanowires, which may thus serve as a dual-purpose component (i.e., also being used for the Majorana nanowires). 
     If there is a single electron in the DQD, the logical qubit basis states can be defined to be |0           ≡|0           U           |1           L , where the electron occupies the lower quantum dot, and |1         ≡|01           U           |0           L , where the upper quantum dot is occupied. This situation corresponds to a semiconductor charge qubit. If there are two electrons in the DQD, then the logical qubit basis states can be defined to be |0         ≡|0           U           |2           L  and |1         ≡|1           U           |1           L , where the electron spins are in the singlet and triplet states, respectively. This situation corresponds to the semiconductor spin qubit.
     Both these qubits share a common feature that can be exploited: the qubit basis states correspond to the electron parity on the upper dot enclosed by the vortex circulation. If the evolution of the semi-conductor qubit is much slower than the measurement time and fluxon tunneling rate, then the flux qubit Φ can be used to entangle topological and conventional qubits  20 ,  30  via the Aharonov-Casher effect. Indeed, the flux qubit splitting energy Δ is the same for combined topological-DQD qubit states with equal joint-parity, i.e., the combined states |00            and |11          have the same splitting, and |01          and |10          have the same splitting. Thus, measurement of the flux qubit splitting energy Δ is equivalent to a joint parity measurement corresponding to the projectors Π 0  and Π 1  from Eqs. (1) and (2) acting on the topological-DQD qubit pair.
     If the topological and conventional qubits  20 ,  30  are initially prepared in the superposition states |ψ T             =α T |0         +β T |1          and |ψ C           =α C |0         +β B |1         , respectively, then application of the even or odd parity projectors gives the (unnormalized) states
 
Π 0 ([ψ T                     |ψ C           )=α T α C |00         +β T β C |11           (5)
 
Π 1 ([ψ T                     |ψC         )=α T β C |01         +β T α C |10         ,  (6)
 
It should be understood that the flux qubit Φ acts as an interferometer that enables this measurement.

     Qubits can be entangled and coherent quantum information transfer performed using parity measurements with the help of two flux qubits. The maximally entangled Bell states (which can be used as entanglement resources) may be denoted as
 
|Φ μ             ≡(                   σ μ )(|01         −|10         )√{square root over (2)},  (7)
 
for μ=0,1,2,3(σ μ =         ). The ability to perform measurements in the Bell basis allows for the teleportation of quantum states, and hence, for the transfer quantum information. It should be understood from Eqs. (5) and (6) that joint parity measurements can produce entangled states, such as Bell states. More generally, however, it should be understood that
 
Π 0 =|Φ 1                     Φ 1 |+|Φ 2                     Φ 2 |  (8)
 
Π 1 =|Φ 0                     Φ 0 |+|Φ 3                     Φ 3 |  (9)
 
Π ˜   0 =( H             H )Π 0 ( H             H )=|Φ 2                     Φ 2 |+|Φ 3                     Φ 3 |  (10)
 
Π ˜   1 =( H             H )Π 1 ( H             H )=|Φ 0                     Φ 0 |+|Φ 1                     Φ 1 |  (11)
 
where the (single-qubit) Hadamard gate is given by

                   H   =         1     2       ⁡     [         1       1           1         -   1           ]       .             (   12   )               
Hence, joint parity measurements combined with Hadamard gates fully resolves the Bell basis.
 
     Hadamard gates can be generated (with topological protection) by braiding Ising anyons and through standard methods for conventional qubits. As described above, the device depicted in  FIG. 3  can be used to implement a joint parity measurement of a topological-conventional qubit pair  20 ,  30 . It can also be used to implement joint parity measurements of topological-topological and conventional-conventional qubit pairs. 
     Specifically, consider the device  100  shown in  FIG. 4  where there are three flux qubits Φ 1 , Φ 2 , and Φ 3 . One of these, Φ 3 , is coupled to two semi-conductor DQD qubits  30 A,  30 B. Again, it is assumed that there is right-left symmetry (E J7 =E J9 ) so that fluxon tunneling in the superconducting qubit that allows for measuring the combined charge parity for the conventional-conventional qubit pair, as explained above. The other flux qubit Φ 2 , with (E J4 =E J6 ) allows for joint parity measurements on topological-topological qubit pairs. The combined device  100  allows quantum information to be transferred between topological and conventional qubits  20 ,  30 . Finally, by tuning the external fluxes Φ away from the degeneracy point, flux and conventional or topological qubits can be decoupled. 
     It should be understood that the joint parity measurement device  100  depicted in  FIG. 4  not only allows for coherent entanglement and transfer of information between topological and conventional systems, but also provides a new method of entangling conventional qubits, e.g., semiconductor charge or spin qubits, with each other, and hence could also be useful for purely conventional systems. 
     It should be understood from the foregoing description that a topological quantum bus may process is accordance with the method  200  of  FIG. 5 . That is, a method  200  for coherent coupling of topological and conventional qubits may include, at step  202 , providing a topological qubit and a conventional qubit that are coupled to one another via a plurality of Josephson junctions. 
     At  204 , a joint parity measurement may be performed. The joint parity measurement may enable quantum information to be coherently transferred between the topological and conventional qubits. The joint parity of the topological and conventional qubits may be measured via the Aharonov-Casher effect. The joint parity measurement may be performed in the Bell basis. Hadamard gates may be applied to the topological and conventional quabits. The Hadamard gates may be generated by braiding Ising anyons. 
     At  206 , quantum information may be coherently transferred between the topological and conventional qubits.