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arXiv:0906.4444 · Journal of the Physical Society of Japan

Quantum Computing with p-Wave Superfluid Vortices

Tetsuo Ohmi, Mikio Nakahara

Published Oct 12, 2010• 5 citations
arXivHTMLPDF

Abstract

It is shown that Majorana fermions trapped in three vortices in a p-wave superfluid form a qubit in a topological quantum computing (TQC). Several similar ideas have already been proposed: Ivanov [Phys. Rev. Lett. {\bf 86}, 268 (2001)] and Zhang {\it et al.} [Phys. Rev. Lett. {\bf 99}, 220502 (2007)] have proposed schemes in which a qubit is implemented with two and four Majorana fermions, respectively, where a qubit operation is performed by exchanging the positions of Majorana fermions. The set of gates thus obtained is a discrete subset of the relevant unitary group. We propose, in this paper, a new scheme, where three Majorana fermions form a qubit. We show that continuous 1-qubit gate operations are possible by exchanging the positions of Majorana fermions complemented with dynamical phase change. 2-qubit gates are realized through the use of the coupling between Majorana fermions of different qubits.

DOI
10.1143/jpsj.79.104602

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It is shown that Majorana fermions trapped in three vortices in a p-wave superfluid form a qubit in a topological quantum computing (TQC). Several similar ideas have already been proposed: Ivanov [Phys. Rev. Lett. {\bf 86}, 268 (2001)] and Zhang {\it et al.} [Phys. Rev. Lett. {\bf 99}, 220502 (2007)] have proposed schemes in which a qubit is implemented with two and four Majorana fermions, respectively, where a qubit operation is performed by exchanging the positions of Majorana fermions. The set of gates thus obtained is a discrete subset of the relevant unitary group. We propose, in this paper, a new scheme, where three Majorana fermions form a qubit. We show that continuous 1-qubit gate operations are possible by exchanging the positions of Majorana fermions complemented with dynamical phase change. 2-qubit gates are realized through the use of the coupling between Majorana fermions of different qubits.
Quantum computers are in hot-spot with the potential to handle more complex problems than classical computers can. Realizing the quantum computation requires the universal quantum gate set {T, H, CNOT} so as to perform any unitary transformation with arbitrary accuracy. Here we first briefly review the Majorana fermions and then propose the realization of arbitrary two-qubit quantum gates based on chiral Majorana fermions. Elementary cells consist of a quantum anomalous Hall insulator surrounded by a topological superconductor with electric gates and quantum-dot structures, which enable the braiding operation and the partial exchange operation. After defining a qubit by four chiral Majorana fermions, the single-qubit T and H quantum gates are realized via one partial exchange operation and three braiding operations, respectively. The entangled CNOT quantum gate is performed by braiding six chiral Majorana fermions. Besides, we design a powerful device with which arbitrary two-qubit quantum gates can be realized and take the quantum Fourier transform as an example to show that several quantum operations can be performed with this space-limited device. Thus, our proposal could inspire further utilization of mobile chiral Majorana edge states for faster quantum computation.
We propose to use the recently predicted two-dimensional "weak-pairing" px + ipy superfluid state of fermionic cold atoms as a platform for topological quantum computation. In the core of a vortex, this state supports a zero-energy Majorana mode, which moves to finite energy in the corresponding topologically trivial "strong-pairing" state. By braiding vortices in the "weak-pairing" state, unitary quantum gates can be applied to the Hilbert space of Majorana zero modes. For readout of the topological qubits, we propose realistic schemes suitable for atomic superfluids.
We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum-dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot that is tunnel coupled to two topological superconductors. The effective spin-Majorana exchange facilitates a hybrid cnot gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a cnot gate is implemented that acts on neighboring Majorana qubits and eliminates the necessity of interqubit braiding. In both setups, the spin-Majorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a swap or hybrid swap gate which is sufficient for universal quantum computation without projective measurements.
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 $\ensuremath{\pi}/8$ 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 sss-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 0.140.140.14 as compared to ${10}^{\ensuremath{-}3}$ to ${10}^{\ensuremath{-}4}$ in ordinary unprotected quantum computation.
Abstract Majorana-based quantum gates are not complete for performing universal topological quantum computation while Fibonacci-based gates are difficult to be realized electronically and hardly coincide with the conventional quantum circuit models. In reference Hu and Kane (2018 Phys. Rev. Lett. 120 066801), it has been shown that a strongly correlated Majorana edge mode in a chiral topological superconductor can be decomposed into a Fibonacci anyon τ and a thermal operator anyon ɛ in the tricritical Ising model. The deconfinement of τ and ɛ via the interaction between the fermion modes yields the anyon collisions and gives the braiding of either τ or ɛ. With these braidings, the complete members of a set of universal gates, the Pauli gates, the Hadamard gate and extra phase gates for one-qubit as well as controlled-NOT (CNOT) gate for two-qubits, are topologically assembled. Encoding quantum information and reading out the computation results can be carried out through electric signals. With the sparse-dense mixed encodings, we set up the quantum circuit where the CNOT gate turns out to be a probabilistic gate and design the corresponding devices with thin films of the chiral topological superconductor. As an example of the universal topological quantum computing, we show the application to Shor’s integer factorization algorithm.