Universal quantum computation in a semiconductor quantum wire network.
Universal quantum computation (UQC) using Majorana fermions on a 2D topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include any two-qubit gate and also the single-qubit phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, etc.), to which the Majorana fermions are bound in a TS medium, can be quantum coherent. We show that these obstacles can be overcome using a semiconductor quantum wire network in the vicinity of an -wave superconductor, by constructing topologically protected two-qubit gates and any arbitrary single-qubit phase gate in a topologically unprotected manner, which can be error corrected using magic state distillation. Thus our strategy, using a judicious combination of topologically protected and unprotected gate operations, realizes UQC on a quantum wire network with a remarkably high error threshold of as compared to to in ordinary unprotected quantum computation.
The decoherence of quantum states by the environment is the nemesis of any proposed quantum computation scheme. Topological quantum computation (TQC) proposes Kitaev (); nayak_RevModPhys'08 () an elegant way to solve this environmental decoherence problem by encoding quantum information in an intrinsically non-local way. Quantum information thus stored is expected to be essentially immune to any local perturbation due to the environment. A class of quantum many-body states, characterized by excitations with non-Abelian statistics (non-Abelian anyons), allow such non-local encoding of quantum information. In principle, the non-Abelian anyons can be moved (braided) around each other to exploit their statistics, which can be used to manipulate the stored quantum information and build quantum gates Parsa (); Zhang (); Freedman (); Clarke (). Therefore, TQC using non-Abelian excitations is intrinsically fault-tolerant, which holds considerable promise to be able to beat the environmental decoherence problem.
Statistics Wilczek2 () is defined as the unitary transformations on many-body wave functions by the pair-wise exchange of the particles’ quantum numbers. In -dimensions, if the many-body ground state wave function happens to be a linear combination of states from a degenerate subspace, a pair-wise exchange of the particles can unitarily rotate the wave function in the ground state subspace. In this case, the statistics is non-Abelian Kitaev (); nayak_RevModPhys'08 () and the system of such quantum particles is a non-Abelian system. Non-Abelian quantum systems in the so-called Ising topological class nayak_RevModPhys'08 () are characterized by topological excitations called Majorana fermions. In some topological superconducting (TS) systems schnyder (), Majorana fermions arise as non-degenerate zero-energy excitations bound to vortices of the superconducting order parameter. These topological excitations are protected from the higher-energy, non-topological, Bogoliubov excitations at the vortex cores Caroli () by the so-called mini-gap , where is the superconducting pair potential and is the Fermi energy. The second quantized operators, , corresponding to the Majorana excitations are self-hermitian, , which is in sharp contrast to ordinary fermionic (or bosonic) operators for which . Therefore, each Majorana particle is its own anti-particle Wilczek-3 () unlike Dirac fermions where electrons and positrons (or holes) are distinct. Majorana particles have been predicted to occur in some exotic many-body states such as the proposed Pfaffian states in the filling fraction fractional quantum Hall (FQH) system Moore (), spin-less chiral -wave superconductors/superfluids Read (); Ivanov (), the surface of 3D strong topological insulators (TI) fu_prl'08 (), and non-centrosymmetric superconductors Parag ().
Semiconductor as a non-Abelian system:
Recently, a semiconductor thin film with Rashba-type SO coupling was proposed to be a suitable platform for realizing a Majorana-carrying TS state by the proximity effect Sau et al. (2010); Ann (). It was shown that, in the presence of a -wave superconducting pair potential and a Zeeman splitting , both of which can be proximity-induced ( can also be induced by a parallel magnetic field when the SO coupling includes a Dresselhaus component alicea ()), the appropriate TS state is realized when the parameters satisfy where is the chemical potential in the semiconductor. Following this, it was quickly realized unpublished () that the 1D version of the same set up, a semiconducting quantum wire with zero-energy Majorana states trapped at the two ends, would be an easier system to explore the physics of Majorana fermions, since the relevant mini-gap at the wire-ends is of order (there are no other sub-gap states localized near the ends other than the Majorana states). It is important to note that -wave proximity effect on a InAs quantum wire (which has a sizable SO coupling) has possibly been already realized in experiments doh (). Moreover, the required Zeeman splitting in the wire can be introduced more easily than in the 2D case by a magnetic field parallel to the superconductor roman (); Gil (). For all these reasons, it seems that a Majorana-carrying TS state in a semiconductor quantum wire may be within experimental reach. A discussion of the SO coupled semiconductor as a non-Abelian platform in both 2D and 1D, along with STM signatures of Majorana modes from the wire-ends, can be found in Ref. [long-PRB, ].
Topological qubit using quantum wires:
Let us consider a semiconductor quantum wire in the TS state (). Each wire (shown as red segments in Fig. 1) has a pair of Majorana modes (shown as circles at the wire-ends) at the left (L) and right (R) ends. With wire we can associate a regular fermion state represented by the operator Thus, the wire naturally forms a two-state system consisting of states and , where . Since the wave function for is composed of a pair of non-overlapping Majorana states, it is unaffected by all local changes in the Hamiltonian. Thus, the wire in the TS state constitutes a decoherence-free two state system which can be used to build a topologically protected qubit. However, such a two-state system does not allow the superposition of the basis states, i.e., the states () do not exist, because they violate the conservation of fermion parity Bravyi (2006). To remedy this, a topological logical qubit can be defined Bravyi (2006) via a pair of quantum wires in the TS state, i.e., with the states (-states in both quantum wires unoccupied), and (-states in both quantum wires occupied). The superposition states, , are now allowed because the superconducting condenset only conserves fermion number modulo . Note also that these two states do not mix with the other two states of the two-wire system by any unitary operation that conserves fermion parity.
Quantum wire network and non-Abelian statistics:
Recently, a network of 1D semiconductor quantum wires has been proposed alicea1 () as a suitable platform to create, transport, and fuse Majorana fermions at the wire-ends. The wire network consists of wire segments in the TS state (shown in red in Fig. 1) connected by segments in the non-topological superconducting (NTS) state (shown in blue in Fig. 1). The Majorana fermion states are transported by shifting the end points of the TS segments by applying locally tunable external gate potentials (which control ). That pair-wise exchange alicea1 () of these Majorana fermions leads to the familiar non-Abelian statistics (i.e., but ) follows most simply from fermion parity conservation. Suppose is the unitary operator for exchange of Majorana fermions. Suppose also that do not pick up a (relative) sign under . then transforms the neutral fermion operator into . Applying to , where is the empty state, it is easy to see that , where is a proportionality constant. This contradicts fermion parity since is even and is odd under fermion parity. It then follows that there must be a relative sign whenever two Majorana fermions are exchanged. Note also that the fact that the Majorana fermions in the present case are situated at the ends of 1D wires (and not in a 2D system like a 2D chiral wave superconductor) does not make any difference, since they are essentially zero-dimensional objects. In the wire network, these zero-dimensional objects are being moved (braided) on the 2D substrate of the superconductor. A more microscopic derivation of this non-Abelian statistics using Kitaev’s 1D construction for a TS state Kitaev1 () has been given in Ref. [alicea1, ].
The exchange and braiding operations on the Majorana fermions lead to some of the quantum gates such as the single-qubit phase gate and the single-qubit Hadamard gate. However, it is well known that Bravyi (2006), for a system of Majorana fermions, the exchange or braiding operations alone fail to provide any two-qubit gate: the topological braiding operations allowed in a quantum wire network, as in its 2D FQH or chiral -wave Pfaffian counterpart, are not computationally sufficient. A system of Majorana fermions can be made computationally sufficient if the braiding-generated gate set is supplemented by a single-qubit phase gate and a two-qubit Controlled-Not, or CNOT, gate Boykin et al. (1999); Bravyi (2006).
Universal quantum computation with Majorana fermions:
A system of Majorana fermions can be made computationally sufficient in one of two ways Nayak (); Bravyi (2006): (1) by dynamically changing the topology of the platform, which allows the crucial extra gates to be obtained in a topologically protected manner, or (2) by implementing these gates in a topologically unprotected manner, which, provided the other gates are topologically protected, can also lead to universal quantum computation (UQC) with the aid of certain error correction protocols. At present, it is not clear how the topologically protected route can be implemented in any proposed TQC architecture including the quantum wire network. Therefore, in this paper, we take the second route to UQC as described above. In the proposal we will consider, only the phase gate will be implemented in a topologically unprotected way. The topologically protected single qubit gates implemented through the braiding operations can then be used to perform “magic state distillation” Bravyi (2005, 2006) to produce error-corrected -phase gates from noisy ones. This purification protocol (which has poly-log overhead) consumes several copies of a magic state, e.g. , and outputs a single qubit with higher polarization along a magic direction. Once a sufficiently pure magic state is produced, it may then be consumed to generate a -phase gate. This protocol permits a remarkably high error threshold of over 0.14 for the noisy gates, as compared to to in ordinary unprotected quantum computation. The simpler strategy of adopting the topologically unprotected route to UQC using a TS system still leaves us with a non-trivial problem. The principal reason why any system of Majorana fermions is not computationally sufficient is that two qubits cannot be entangled using the braiding operations alone. Any logical state of the two qubits, accessible by braiding one Majorana fermion around another, can always be written as a product of the logical states of the individual qubits. It has been shown Bravyi (2006) that a two-qubit CNOT gate can be created with Majorana fermions provided there is a supply of entangled pairs of two topological qubits. In the FQH context, it has been proposed Bravyi (2006); Nayak (); loss () that quantum interference of Majorana currents can be used to generate the two-qubit entanglement. However, in a TS system, in which the Majorana fermions are trapped in the order parameter defects (vortices or domain walls), it is not clear that the motion of these defects is a quantum process which will lead to the desired quantum interference. Further, creating a single-qubit phase gate in a TS system is also problematic. A simple method for this could be moving a pair of Majorana fermions in a given qubit near each other. Because of the overlap of the Majorana wave functions, the energy degeneracy of the and the states are then split. The dynamic phase in the resulting time evolution could then be used to produce arbitrary single-qubit phase gates, were it not for the fact that the Majorana wave functions in the TS medium are oscillatory in space, which results in corresponding oscillations in the energy splitting as a function of inter-Majorana distance Cheng et al. (2009). We show in this paper that both of these obstacles can be overcome in the quantum wire network, which therefore allows a concrete realization of a UQC architecture.
UQC in the quantum wire network:
We first show that entangled pairs of qubits can be generated by the set-up shown in Fig. 1. The two superconducting islands B and C in Fig. 1, together with the main superconductor A (which holds the wire network) constitute a three-island Josephson junction flux qubit, which when biased with half a flux quantum, has a degenerate pair of states composed of clock-wise supercurrent (CW) and counter-clock-wise supercurrent (CCW). The charging energy of the islands leads to tunneling between these two states leading to a splitting of the degeneracy with the new eigenstates (CWCCW). This splitting between the energies is also sensitive to a Berry phase contribution which can be controlled by gate electrodes in the vicinity of the islands Tiwari and Stroud (2007). In addition, as we show on the last page, the splitting also depends on the parity of the number of neutral fermions shared between the pairs of Majorana fermions at the ends of TS segments on island B (here we have assumed that the capacitance of the main island A is large enough so we can ignore its charging energy). Using this, the system can be tuned such that an even number of neutral fermions on island B leads to an exact degeneracy, while an odd number of neutral fermions leads to a splitting between the states (CWCCW). This splitting can be measured by coupling the system to an rf circuit Tiwari and Stroud (2007), which can be used beenakker1 () to perform a non-demolition measurement of the state () of a pair of Majorana fermions on island B. As has been emphasized in Ref. [beenakker1, ], this provides an explicitly quantum mechanical method for the charge measurement of a pair of Majorana fermions. Note that in the analogous method of charge measurement using quantum interference of currents in a TS state, it is not clear if the motion of order parameter defects (vortices, domain walls) is a quantum mechanical process.
Let us now show how to use the quantum superposition states of the flux qubit to also create quantum entanglement between two topological qubits. The entangled state between the two qubits can then be used Bravyi (2006) as the ancillary two-qubit states to construct a two-qubit quantum gate. To generate an entangled pair of qubits, we first create a pair of qubits composed of TS segments (, ) and (, ) both in the state on the main island A. By applying a Hadamard gate to both, we then transform the states of both qubits to . The combined state of the two-qubit system is now . We then transfer the TS segments and to island B by applying external gate potentials. If the parity of neutral fermions on segments and is even (odd), the degeneracy of the states (CWCCW) is split (not split). By an rf measurement, one can then collapse the quantum states of the two qubits as
which is the desired entangled pair. If in the rf measurement, no splitting is observed ( chance), the process has to be repeated until a splitting is observed producing the desired entangled pair. Therefore, this method provides entangled pairs of qubits with a success rate deterministically.
In addition to two-qubit entanglement and a CNOT gate, for UQC, one needs a single-qubit phase gate. As discussed earlier Bravyi (2006); Nayak (), a simple way to create such a gate could be to bring a pair of Majorana fermions from a topological qubit near each other and let the microscopic physics split the degeneracy between the and states. This has also been discussed in the context of general anyons.bonderson3 () Any arbitrary single-qubit phase gate can then be created in principle by accumulating the relative dynamic phase between the and states over a finite period of time. However, it now appears that such a scheme does not work in both TS and FQH systems, because the splitting between the and states oscillates with distance between the Majorana fermions in a given pair because of the oscillatory nature of the wave functions Cheng et al. (2009); Simon (). Recently an interferometric proposal has been suggested to avoid these oscillations in the FQH system. Clarke () At first glance, it appears that the same problem would arise in our quantum wire network, because the Majorana wave functions oscillate in the TS segments as a function of distance from the domain wall (oscillating solid black lines in Fig. 2a) just like in the case of a chiral -wave superconductor. However, it is important to note that these functions do not oscillate and in fact monotonically decay with a decay length inversely proportional to the gap (which is essentially proportional to the gate voltage for ) in the non-topological segments of the wire network (decaying solid black lines in Fig. 2a). Further, in the wire network, to induce splitting between the states and between a pair of Majorana end states, one does not need to physically move these states near each other (a process which is prone to errors). Instead, one could simply reduce in the non-topological segments to generate the required overlap of the Majorana fermion wave functions on the two ends.
The presence of oscillations in the wave-functions in the TS and the absence thereof in the NTS can be understood from simple considerations of the asymptotic wave-functions. The TS segment consists of a superconducting state on a Fermi-liquid like state with Fermi wave-vector . Therefore the wave-function of a zero-energy Majorana state has an asymptotic form where the coherence length .long-PRB () The overlaps of this wave-function clearly has oscillations at the wave-vector as is the case with -wave superconductors.Cheng et al. (2009) In contrast the NTS part of the system is depleted and therefore has a vanishing Fermi-wave-vector . Thus the overlaps across the NTS has a purely decaying form as is clear from Fig. 2. Below, with the help of Fig. 2, we explicitly show how an arbitrary single-qubit gate can be constructed.
From the plot of the wave function (solid black line) in Fig. 2(a), it is clear that the wave functions no longer oscillate in the NTS segment. To generate a phase-shift, we first distort the TS segment 1 in the qubit in Fig. 2(b) into a U shape. This step is necessary to bring the two Majorana modes at the two ends of segment 1 separated by a NTS region. The gate voltage in the NTS segment is still kept large such that the overlap of the wave functions is still negligible. We now lower the gate voltage in the NTS segment to increase the decay lengths of the wave functions. This causes overlap between the Majorana fermion wave functions at the ends of the NTS segment. As seen in Fig 2(b), this leads to a clear non-oscillatory dependence of the energy splitting between the and states as a function of the applied gate voltage.
A phase gate is obtained by applying a pulse over an appropriate length of time. The appropriate length of time can be estimated by experimentally determining the energy splitting between the and states. The rate of phase shift can be determined by creating a test state and applying the sequence (Hadamard phase gate inverse Hadamard) in order. After this process, the probability of making a transition to the state is given by
By experimentally determining this probability at time , one gets for a given and . Arbitrary single-qubit phase gates can then be constructed by calculating the required time and accumulating the correct phase shift for a given .
Majorana Berry phase in the flux qubit:
To understand the origin of the Berry phase term generated by Majorana fermions it is necessary to revert back to the fermion description of the system. This description is in terms of the fermion fields for the electrons in both the semiconducting wires and on the superconducting island . The pairing interactions leading to Cooper-pairing on the islands together with Coulomb interaction induced capacitance and gate potential can be described in terms of an action
where and are the effective Hubbard-Stratonovich fields which may be interpreted as the time-dependent mean-field electrostatic potential and pairing potential respectively. The relatively weak tunnelings between the islands and also the wires on the different islands will give rise to the Josephson couplings between the islands. For our calculation, we will consider a phase model such that . We will also assume that the capacitances of the islands is large enough so that we can assume the quantum fluctuations in and to be slowly varying. In this limit, the fermionic part of the fields that contribute to the partition function may be assumed to follow ground-state of the Hamiltonian
as and vary in time adiabatically. For systems containing Majorana fermions the ground state is degenerate. The Majorana fermion sector of the Hamiltonian may be described in terms of pairs of Majorana fermions combined into regular zero energy fermions that are localized on the island . The relevant states may then be charcterized by where denotes the ground state with all Majorana fermion states empty and is the operator that accounts for the Majorana state occupation. The transition matrix element between the various phase states and is given by
where is the unitary time evolution matrix for the fermionic state over a particular phase trajectory. The states and are groundstates of and respectively with appropriate Majorana state occupancy.
Since the zero-energy fermion operators are spatially separated, and localized on each island , they evolve with phase according to
Moreover, because of the absence of tunneling between the Majorana fermions, the occupation of each of these zero energy modes is conserved during the evolution of the Hamiltonian. Therefore under the time-evolution , evolves into where and is the winding number of the phase trajectory along a particular phase . Furthermore . Therefore the transition amplitude for a given fermion occupation on each island is
The factor is precisely the Berry phase term associated with Majorana fermions and can be accounted for by adding a term to the phase action of the system. Adding this to the phase action considered by Tiwari and Stroud Tiwari and Stroud (2007), which can be obtained by performing the integral, is
In the presence of an externally applied flux as shown in Fig. 1, we replace the phase differences in the above equation by the gauge-invariant phase differences , and . The Josephson couplings are taken to be and . The gauge potentials are chosen such that . Here the phase of the island , and .
The lowest Josephson energy configurations of this system are given by where .Tiwari and Stroud (2007) In the geometry considered, ignoring the large capacitances , the capacitance term can cause tunneling between the 2 minima.Friedman and Averin (2002) For the case , and starting at , there are 2 equivalent in energy low barrier tunneling paths to 2 equivalent points and which are related to each other by the symmetry .
The total tunneling matrix element using the instanton approachFriedman and Averin (2002), is now given by the imaginary time action for tunneling path as
where is the attempt frequency. The contributions to the tunneling amplitudes from the 2 paths are identical by symmetry except for the term which creates a difference in the 2 paths of
where is the total number of fermions on island . This leads to an interferometric dependence
and is the single path tunneling amplitude. The interference effect between the 2 paths may be interpreted as a flux tunneling around the charge and has been referred to as a Aharanov-Casher effect Friedman and Averin (2002). Tuning , this leads to a splitting for (mod 2)=1 and no splitting for (mod 2)=0.
The magnitude of the splitting , which contains information about the Aharanov-Casher phase, can be measured by applying a flux of . In this case the two minimum Josephson energy configurations are degenerate and any splitting that is measured is a result of the Aharanov Casher phase.
We show that a network of semiconductor quantum wires in the vicinity of an -wave superconductor allows universal quantum computation. To do this, we propose a scheme to generate entanglement between two topological qubits in the wire network with the assistance of a flux qubit. We also show that the wave functions of the Majorana fermion states at the end points of the topological segments do not oscillate in the adjoining non-topological segments, even though they have the familiar oscillatory behavior Cheng et al. (2009); Simon () in the topological segments. This fact can be used to create arbitrary single-qubit phase gates by controlled overlap of the Majorana wave functions via the non-topological segments of the wire network. Our schemes for deterministically generating two-qubit entanglement and arbitrary single-qubit phase gates establish the semiconductor wire network as a viable platform for universal quantum computation.
This work is supported by DARPA-QuEST, JQI-NSF-PFC, and LPS-NSA. We thank the Aspen Center for Physics where a part of this work was completed. ST acknowledges DOE/EPSCoR Grant # DE-FG02-04ER-46139 and Clemson University start up funds for support.
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