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arXiv:1209.1693 · New Journal of Physics

On the stability of quantum holonomic gates

Paolo Solinas, Maura Sassetti, Piero Truini, Nino Zanghı̀

Published Sep 5, 2012• 55 citations
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Abstract

We provide a unified geometrical description for analyzing the stability of holonomic quantum gates in the presence of imprecise driving controls (parametric noise). We consider the situation in which these fluctuations do not affect the adiabatic evolution but can reduce the logical gate performance. Using the intrinsic geometric properties of the holonomic gates, we show under which conditions of the noise's correlation time and strength the fluctuations in the driving field cancel out. In this way, we provide theoretical support for previous numerical simulations. We also briefly comment on the error due to the mismatch between the real and the nominal time of the period of the driving fields and show that it can be reduced by suitably increasing the adiabatic time.

DOI
10.1088/1367-2630/14/9/093006

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We provide a unified geometrical description for analyzing the stability of holonomic quantum gates in the presence of imprecise driving controls (parametric noise). We consider the situation in which these fluctuations do not affect the adiabatic evolution but can reduce the logical gate performance. Using the intrinsic geometric properties of the holonomic gates, we show under which conditions of the noise's correlation time and strength the fluctuations in the driving field cancel out. In this way, we provide theoretical support for previous numerical simulations. We also briefly comment on the error due to the mismatch between the real and the nominal time of the period of the driving fields and show that it can be reduced by suitably increasing the adiabatic time.
We present a numerical study of the robustness of a specific class of non-Abelian holonomic quantum gates. We take into account the parametric noise due to stochastic fluctuations of the control fields which drive the time-dependent Hamiltonian along an adiabatic loop. The performance estimator used is the state fidelity between noiseless and noisy holonomic gates. We carry over our analysis with different correlation times and we find out that noisy holonomic gates seem to be close to the noiseless ones for ``short'' and ``long'' noise correlation times. This result can be interpreted as a consequence of the geometric nature of the holonomic operator. Our simulations have been performed by using parameters relevant to the excitonic proposal for the implementation of holonomic quantum computation [P. Solinas et al., Phys. Rev. B 67, 121307 (2003)].
Holonomic gates for quantum computation are commonly considered to be robust against certain kinds of parametric noise, the cause of this robustness being the geometric character of the transformation achieved in the adiabatic limit. On the other hand, the effects of decoherence are expected to become more and more relevant when the adiabatic limit is approached. Starting from the system described by Florio et al. [Phys. Rev. A 73, 022327 (2006)], here we discuss the behavior of nonideal holonomic gates at finite operational time, i.e., long before the adiabatic limit is reached. We have considered several models of parametric noise and studied the robustness of finite-time gates. The results obtained suggest that the finite-time gates present some effects of cancellation of the perturbations introduced by the noise which mimic the geometrical cancellation effect of standard holonomic gates. Nevertheless, a careful analysis of the results leads to the conclusion that these effects are related to a dynamical instead of a geometrical feature.
We study entanglement and fidelity of a two-qubit system when a noisy holonomic, non-Abelian, transformation is applied to one of them. The source of noise we investigate is of two types: one due to a stochastic error representing an imprecise control of the fields driving the evolution; the other due to an interaction between the two near qubits. The peculiar level structure underlying the holonomic operator leads us to introduce the reduced logical entanglement which is the fraction of entanglement in the logical space. The comparison between entanglement and fidelity shows how they are differently affected by the noise and that, in general, the first is more robust than the latter. We find a range of physical parameters for which both fidelity and reduced logical entanglement are well preserved.
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.
A practical quantum computer must be capable of performing high fidelity quantum gates on a set of quantum bits (qubits). In the presence of noise, the realization of such gates poses daunting challenges. Geometric phases, which possess intrinsic noise-tolerant features, hold the promise for performing robust quantum computation. In particular, quantum holonomies, i.e., non-Abelian geometric phases, naturally lead to universal quantum computation due to their non-commutativity. Although quantum gates based on adiabatic holonomies have already been proposed, the slow evolution eventually compromises qubit coherence and computational power. Here, we propose a general approach to speed up an implementation of adiabatic holonomic gates by using transitionless driving techniques and show how such a universal set of fast geometric quantum gates in a superconducting circuit architecture can be obtained in an all-geometric approach. Compared with standard non-adiabatic holonomic quantum computation, the holonomies obtained in our approach tends asymptotically to those of the adiabatic approach in the long run-time limit and thus might open up a new horizon for realizing a practical quantum computer.
Holonomic quantum computation exploits the geometric evolution of eigenspaces of a degenerate Hamiltonian to implement unitary evolution of computational states. In this work we introduce a framework for performing scalable quantum computation in atom experiments through a universal set of fully holonomic adiabatic gates. Through a detailed differential geometric analysis, we elucidate the geometric nature of these gates and their inherent robustness against classical control errors and other noise sources. The concepts that we introduce here are expected to be widely applicable to the understanding and design of error robustness in generic holonomic protocols. To underscore the practical feasibility of our approach, we contextualize our gate design within recent advancements in Rydberg-based quantum computing and simulation.