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arXiv:quant-ph/0604180 · Laser Physics

Robustness of optimal working points for nonadiabatic holonomic quantum computation

A. Trullo, Paolo Facchi A, Rosario Fazio, Giuseppe Florio, Vittorio Giovannetti, Saverio Pascazio

Published Sep 28, 2006• 6 citations
arXivHTMLPDF

Abstract

Geometric phases are an interesting resource for quantum computation in view of their robustness against decoherence effects. We study the effects of the environment on a class of one-qubit holonomic gates that have recently been shown to be characterized by “optimal” working times. We numerically analyze the behavior of these optimal points and focus on their robustness against noise.

DOI
10.1134/s1054660x06100094

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Geometric phases are an interesting resource for quantum computation in view of their robustness against decoherence effects. We study the effects of the environment on a class of one-qubit holonomic gates that have recently been shown to be characterized by “optimal” working times. We numerically analyze the behavior of these optimal points and focus on their robustness against noise.
Non-Abelian geometric phases are attracting increasing interest because of possible experimental application in quantum computation. We study the effects of the environment (modeled as an ensemble of harmonic oscillators) on a holonomic transformation and write the corresponding master equation. The solution is analytically and numerically investigated and the behavior of the fidelity analyzed: fidelity revivals are observed and an optimal finite operation time is determined at which the gate is most robust against noise.
Geometric phases are an interesting field of research in quantum mechanics. Recently both abelian and nonabelian geometric phases have been proposed as a useful resource for the experimental implementation of quantum computation. In this paper we focus on a particular physical model and study the effect of a bosonic bath on a class of holonomic transformations. We write a general master equation for time-dependent Hamiltonians and derive analytical and numerical solutions for the system considered. The fidelity is analyzed in the adiabatic and nonadiabatic regime. We also determine an optimal finite operation time for this class of gates.
Nonadiabatic holonomic quantum computation (NHQC) has been developed to shorten the construction times of geometric quantum gates. However, previous NHQC gates require the driving Hamiltonian to satisfy a set of rather restrictive conditions, reducing the robustness of the resulting geometric gates against control errors. Here we show that nonadiabatic geometric gates can be constructed in an extensible way, called NHQC+, for maintaining both flexibility and robustness against certain types of noises. Consequently, this approach makes it possible to incorporate most of the existing optimal control methods, such as dynamical decoupling, composite pulses, and a shortcut to adiabaticity, into the construction of single-looped geometric gates. Furthermore, this extensible approach of geometric quantum computation can be applied to various physical platforms such as superconducting qubits and nitrogen-vacancy centers. Specifically, we performed numerical simulation to show how the noise robustness in recent experimental implementations [Phys. Rev. Lett. 119, 140503 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.140503; Nat. Photonics 11, 309 (2017)NPAHBY1749-488510.1038/nphoton.2017.40] can be significantly improved by our NHQC+.approach. These results cover a large class of new techniques combing the noise robustness of both geometric phase and optimal control theory.
Besides standard dynamical schemes to realize a quantum computer, there are particular approaches which are based on intrinsic properties of quantum systems, leading to the definition of topological computation and holonomic, or geometric, computation. The holonomic approach can be viewed as the application of non-Abelian geometric phases to quantum information processing, it is believed to be fault-tolerant with respect of certain kind of parametric noise. Here we discuss the issue of robustness of holonomic quantum gates under parametric noise: we distinguish between geometric and dynamical effects of cancelation, which can appear in different contexts. A so-called standard argument in favor of the stability of noisy holonomic quantum gates is reviewed and extended to more general settings. New geometric effects that describe the behavior of noisy holonomic gates are presented. These effects lead to a refining of the optimal strategy to achieve a robust computation.
Nonadiabatic holonomic quantum computation (NHQC) is implemented by fast evolution processes in a geometric way to withstand local noises. However, recent works of implementing NHQC are sensitive to the systematic noise and error. Here, we present a path-optimized NHQC (PONHQC) scheme based on the non-Abelian geometric phase, and find that a geometric gate can be constructed by different evolution paths, which have different responses to systematic noises. Due to the flexibility of the PONHQC scheme, we can choose an optimized path that can lead to excellent gate performance. Numerical simulation shows that our optimized scheme can greatly outperform the conventional NHQC scheme, in terms of both fidelity and robustness of the gates. In addition, we propose to implement our strategy on superconducting quantum circuits with decoherence-free subspace encoding with the experiment-friendly two-body exchange interaction. Therefore, we present a flexible NHQC scheme that is promising for the future robust quantum computation.
Quantum computation has been shown to be superior to classical computation in solving some problems, and therefore can substantially change our life. However, the realization of quantum computation is still challenging, even if quantum technologies have been improved significantly. Two main obstacles to the realization of practical quantum computation are control errors and decoherence. Control errors are caused by inaccurate manipulations of quantum systems and decoherence is caused by the inevitable interaction between the system and its environment. Geometric phases are only dependent on evolution paths of quantum systems but independent of the evolution details and therefore quantum computation based on geometric phases, i.e., geometric quantum computation, is robust against control errors, benefiting the realization of practical quantum computation. The early proposals of geometric quantum computation are based on adiabatic geometric phases. These proposals require quantum systems to undergo adiabatic evolution, which makes quantum systems evolve for a long time. To circumvent this, nonadiabatic geometric quantum computation based on nonadiabatic Abelian geometric phases was proposed soon after. In 2012, nonadiabatic holonomic quantum computation based on nonadiabatic non-Abelian geometric phases was proposed, which also circumvents long-time evolutions. Moreover, compared with nonadiabatic geometric quantum computation that uses the geometric phase as one parameter of a quantum gate, nonadiabatic holonomic quantum computation uses the holonomic matrix itself as a quantum gate. This makes nonadiabatic holonomic quantum computation possess whole-geometric property. Due to the merits of both geometric robustness and high-speed implementation without the limit of adiabatic evolution, nonadiabatic holonomic quantum computation has been attracting much attention. Until now, much progress has been achieved in the field of nonadiabatic holonomic quantum computation. On one hand, various methods have been proposed to design more efficient nonadiabatic holonomic gates. Nonadiabatic holonomic gates were first realized by using resonant laser fields to drive a three-level system. After this, the single-shot proposal and the single-loop proposal were proposed, allowing us to realize one-qubit gates by a shorter path and thereby reducing the exposure time of nonadiabatic holonomic gates to the environment. To further shorten the exposure time, the path-shortening protocol was put forward, where nonadiabatic holonomic gates can be realized based on a class of extended evolution paths that are shorter than the former ones. Recently, a general approach of constructing Hamiltonians for nonadiabatic holonomic quantum computation was put forward, by using which one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. On the other hand, various methods have been proposed to combine nonadiabatic holonomic gates and various decoherence-resilient methods, making the resulting schemes robust against both control errors and decoherence. The first proposal in this aspect is combining nonadiabatic holonomic gates and decoherence-free subspaces. Gradually, proposals combining nonadiabatic holonomic gates with noiseless subsystems, dynamical decoupling and surface codes were put forward. Last but not least, various proposals suitable for specific physical systems have been proposed and particularly various experimental platforms have been used to demonstrate nonadiabatic holonomic gates. This also significantly improves the development of nonadiabatic holonomic quantum computation. In this paper, we review the above research advances on nonadiabatic holonomic quantum computation, aiming to help readers understand the main developments of nonadiabatic holonomic quantum computation.