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International Journal of Geometric Methods in Modern Physics

GEOMETRIC PHASE AND TOPOLOGY QUANTUM GATES USING PARALLEL TRANSPORTATION

Yu-Mei Gao, Xin-Ding Zhang, Lian Hu

Published Aug 1, 2008• 0 citations
Publisher

Abstract

We present a novel evolution method to obtain pure geometric phase with orthogonal superposed initial state method (OGSM). As examples we illustrate in detail the geometric evolution both in NMR and superconducting Josephson junction systems, which may be further designed to construct fault-tolerant geometric quantum gates. In the end we also propose a simple way to construct topological quantum gates based on OGSM.

DOI
10.1142/s0219887808003041

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We present a novel evolution method to obtain pure geometric phase with orthogonal superposed initial state method (OGSM). As examples we illustrate in detail the geometric evolution both in NMR and superconducting Josephson junction systems, which may be further designed to construct fault-tolerant geometric quantum gates. In the end we also propose a simple way to construct topological quantum gates based on OGSM.
This article summarizes the recent results about geometric and topologic quantum computation.Based on a rigorous theorem concerned on orthogonal superposition quantum state proved in this paper,an orthogonal state method is proposed to carry out the non-adiabatic geometric quantum computation.And these geometric quantum gates are demonstrated in detail by two two-level systems: one is the system of spin-1/2 particle under NMR uniform rotating magnetic field;the other is asymmetric nano-SQUID circuit.The paper presents a perfect explanation for the early controversial helical fiber experiment done by Chiao R.Y.,also proposes a scheme of constructing topologic gates and illustrates these gates still with above two two-level systems.Finally,a brief comment is given on the controllability of quantum gates with the asymmetric nano-SQUID circuit.
In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also introduce a number of examples that will help the reader understand the basic issues involved. In the second part we show how to perform a universal quantum computation using only geometric effects appearing in quantum phases. It is then finally discussed how this geometric way of performing quantum gates can lead to a stable, large scale, intrinsically fault-tolerant quantum computer.
We propose an experimentally feasible scheme to achieve quantum computation based on a pair of orthogonal cyclic states. In this scheme, quantum gates can be implemented based on the total phase accumulated in cyclic evolutions. In particular, geometric quantum computation may be achieved by eliminating the dynamic phase accumulated in the whole evolution. Therefore, both dynamic and geometric operations for quantum computation are workable in the present theory. Physical implementation of this set of gates is designed for NMR systems. Also interestingly, we show that a set of universal geometric quantum gates in NMR systems may be realized in one cycle by simply choosing specific parameters of the external rotating magnetic fields. In addition, we demonstrate explicitly a multiloop method to remove the dynamic phase in geometric quantum gates.
Abstract Geometric phases are robust to local noises and the nonadiabatic ones can reduce the evolution time, thus nonadiabatic geometric gates have strong robustness and can approach high fidelity. However, the advantage of geometric phase has not been fully explored in previous investigations. Here,a scheme is proposed for universal quantum gates with pure nonadiabatic and noncyclic geometric phases from smooth evolution paths. In the scheme, only geometric phase can be accumulated in a fast way, and thus it not only fully utilizes the local noise resistant property of geometric phase but also reduces the difficulty in experimental realization. Numerical results show that the implemented geometric gates have stronger robustness than dynamical gates and the geometric scheme with cyclic path. Furthermore, it proposes to construct universal quantum gate on superconducting circuits, with the fidelities of single‐qubit gate and nontrivial two‐qubit gate can achieve 99.97% and 99.87%, respectively. Therefore, these high‐fidelity quantum gates are promising for large‐scale fault‐tolerant quantum computation.
Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quantum gates, which is the necessity of fault-tolerant quantum computation. However, it is limited by cyclic and mutative evolution path, which usually requires longer gate-time and abrupt pulse control, and thus weaken the gate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadiabatic evolutions, which effectively shorten the gate duration, and universal quantum gates can be realized in one step without path mutation. Our numerical simulation shows that, comparing with the conventional dynamical gates, the constructed geometric gates have stronger resistant not only to noise but also to decoherence effect. In addition, our scheme can also be implemented on a superconducting circuit platform, with the fidelities of single-qubit and two-qubit gates higher than 99.97%\%% and 99.84%\%%, respectively. Therefore, our scheme provides a promising way to realize high-fidelity fault-tolerant quantum gates for scalable quantum computation.