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On the robustness of holonomic quantum computation

Cosmo Lupo

Published Jan 1, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.6092/unina/fedoa/1789

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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.
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.
Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum information processing. In the HQC strategy information is encoded in degenerate eigen-spaces of a parametric family of Hamiltonians. The computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops the nontrivial curvature of the underlying bundle geometry gives rise to unitary transformations i.e., holonomies that implement the desired unitary transformations. Conditions necessary for universal QC are stated in terms of the curvature associated to the non-abelian gauge potential (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gates are performed makes in principle HQC an appealing way towards universal fault-tolerant QC.
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)].
Geometric and holonomic quantum computation utilizes intrinsic geometric properties of quantum-mechanical state spaces to realize quantum logic gates. Since both geometric phases and quantum holonomies are global quantities depending only on the evolution paths of quantum systems, quantum gates based on them possess built-in resilience to certain kinds of errors. This review provides an introduction to the topic as well as gives an overview of the theoretical and experimental progress for constructing geometric and holonomic quantum gates and how to combine them with other error-resistant techniques.
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.