Universal quantum computation in a semiconductor quantum wire network

Universal quantum computation (UQC) using Majorana fermions on a two-dimensional 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 no single-qubit $\pi /8$ 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 $s$-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 $0.14$ as compared to ${10}^{-3}$ to ${10}^{-4}$ in ordinary unprotected quantum computation.

Figure 1

Schematic of entanglement generation in quantum wire topological qubits using superconductor Josephson junctions. The light blue background represents superconducting islands labeled A, B and C. The dark blue unnumbered segments are semiconductor quantum wires in the nontopological superconducting state, while the red numbered segments are semiconductor quantum wires in the topological superconducting state. The purple circles represent the Majorana fermions at the end of topological segments. The topological segments 1 and 2 form a representative topologically protected qubit. Entanglement is generated in the qubits formed by the topological segments $(3,4)$ and $(5,6)$ by transferring segments $4$ and $6$ to island B and then performing a quantum nondemolition measurement of the number of neutral fermions on island B using microwaves. $\Phi$ is the bias flux in the central hole. See text for details.

Figure 2

(a) Majorana wave functions (black lines) are fused across a nontopological wire segment (blue middle segment) with a Gaussian potential (red dashed line) with peak height 2.0 meV for an InAs nanowire. Note that there are no oscillations of the wave functions in the nontopological blue segment. The wave function, on the other hand, rapidly oscillates in the red (left and right) topological segments. The purely decaying wave functions in the nontopological segment lead to a well-controlled phase shift as a function of time. (b) Inset shows arbitrary single-qubit phase gate constructed by controlled overlap of Majorana fermion wave functions through the nontopological segment. The plot shows the dependence of the phase rotation period on the barrier height.