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arXiv:quant-ph/0204117 · Physical Review A

Topological features in ion-trap holonomic computation

Jiannis K. Pachos

Published Oct 23, 2002• 29 citations
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Abstract

Topological features in quantum computing provide controllability and noise error avoidance in the performance of logical gates. While such resilience is favored in the manipulation of quantum systems, it is very hard to identify topological features in nature. This paper proposes a scheme where holonomic quantum gates have intrinsic topological features. An ion trap is employed where the vibrational modes of the ions are coherently manipulated with lasers in an adiabatic cyclic way producing geometrical holonomic gates. A crucial ingredient of the manipulation procedures is squeezing of the vibrational modes, which effectively suppresses exponentially any undesired fluctuations of the squeezing parameter, thus making the gates resilient to control errors.

DOI
10.1103/physreva.66.042318

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Topological features in quantum computing provide controllability and noise error avoidance in the performance of logical gates. While such resilience is favored in the manipulation of quantum systems, it is very hard to identify topological features in nature. This paper proposes a scheme where holonomic quantum gates have intrinsic topological features. An ion trap is employed where the vibrational modes of the ions are coherently manipulated with lasers in an adiabatic cyclic way producing geometrical holonomic gates. A crucial ingredient of the manipulation procedures is squeezing of the vibrational modes, which effectively suppresses exponentially any undesired fluctuations of the squeezing parameter, thus making the gates resilient to control errors.
The implementation of holonomic quantum computation is meaningful. We can effectively resist local and collective noise in the process of physical implementation by using the advantage of non-Abelian geometric phase. In this paper, we set out from the simplest and most fundamental Jaynes-Cummings model of ion trap system to implement single-qubit logical operations, and taking advantage of the interaction of two ions with a pair of laser pulses to implement two-qubit logical operations, respectively. Compared with the previous proposal, the whole process of our proposal are in tunable way and the universal holonomic quantum qubit gates can be made robust to systematic error and decay which pushes the gate fidelities in the presence of decoherence and systematic error to well high level. Thus our scheme affords an experimentally feasible and simple way to make realizing the robust nonadiabatic holonomic quantum computation.
Holonomic quantum computation (HQC) is materialized here with quantum optics components. Holonomies are the generalization of the Berry phases to unitary matrices. In a nonlinear Kerr medium the degenerate states of a laser beams are interpreted as qubits. Control manipulations with displacers, squeezers and two-mode interfering devices in a cyclic, adiabatic fashion produces holonomies. Hence, they are employed as logical gates for our HQC proposal. The effects of errors from imperfect control of classical parameters, the looping variation of which builds up holonomic gates, are also investigated. Additionally, the influence of quantum decoherence on holonomic tele-portation used as a computational primitive is studied, and particular features of control error resilience are discussed.
Holonomic quantum computation uses non-Abelian geometric phases to realize error resilient quantum gates. Nonadiabatic holonomic gates are particularly suitable to avoid unwanted decoherence effects, as they can be performed at high speed. By letting the computational system interact with a structured environment, we show that the scope of error resilience of nonadiabatic holonomic gates can be widened to include systematic parameter errors. Our scheme maintains the geometric properties of the evolution and results in an environment-assisted holonomic quantum map that can mimic the effect of a holonomic gate. We demonstrate that the sensitivity to systematic errors can be reduced in a proof-of-concept spin-bath model.
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.
Quantum holonomic gates hold built-in resilience to local noises and provide a promising approach for implementing fault-tolerant quantum computation. We propose to realize high-fidelity holonomic (N+1)(N+1)(N+1)-qubit controlled gates using Rydberg atoms confined in optical arrays or superconducting circuits. We identify the scheme, deduce the effective multibody Hamiltonian, and determine the working condition of the multiqubit gate. Uniquely, the multiqubit gate is immune to systematic errors, i.e., laser parameter fluctuations and motional dephasing, as the NNN control atoms largely remain in the very stable qubit space during the operation. We show that CN{C}_{N}CN​-not gates can reach the same level of fidelity at a given gate time for $N\ensuremath{\le}5$ under a suitable choice of parameters, and the gate tolerance against errors in systematic parameters can be further enhanced through optimal pulse engineering. In the case of Rydberg atoms, the proposed protocol is intrinsically different from typical schemes based on Rydberg blockade or antiblockade. Our study paves an alternative way to build robust multiqubit gates with Rydberg atoms trapped in optical arrays or with superconducting circuits. It contributes to current efforts to develop scalable quantum computation with trapped atoms and fabricable superconducting devices.