Recent investigations of theoretical and experimental possibilities of quantum information processing have made the idea of quantum computation [1] very attractive and important (see e.g. [2] for a review). Having the apparatus which is capable of manipulation and measurement on pure states of individual quantum systems one can make use of massive intrinsic parallelism of coherent quantum time evolution. The main idea of quantum computation is the following: Consider a many-body system of n elementary twolevel quantum excitations — qubits, which is called the quantum register, store the data for quantum computation in the initial state of a register |r〉 which is a superposition of an exponential number N = 2 of basic qubit states, then perform certain unitary transformation U by decomposing U = U(T ) · · ·U(2)U(1) into a sequence of T elementary one-qubit and two-qubit quantum gates U(t), t = 1, 2, . . . , T , such decomposition being called a quantum algorithm (QA), and in the end obtain the results by performing measurements of qubits on a final register state U |r〉. QA is called efficient if the number of needed elementary gates T grows with at most polynomial rate in n = log2N , and only in this case it can generally be expected to outperform the best classical algorithms (in the limit n → ∞). At present only few efficient QAs are known, and perhaps the most generally useful is the Quantum Fourier transformation (QFT) [3]. There are two major obstacles for performing practical quantum computation: First, there is a problem of decoherence [4] resulting from an unavoidable time-dependent coupling between qubits and the environment. If the perturbation couples only a small number of qubits at a time then such errors can be eliminated at the expense of extra qubits by quantum error correcting codes [5] (see Ref. [6] for another approach). Second, even if one knows an efficient error correcting code or assumes that quantum computer is ideally decoupled from the environment, there will typically exist a small unknown or uncontrollable residual interaction among qubits which one may describe by a general static perturbation. Therefore, understanding the stability of QAs with respect to various types of perturbations is an important problem (see [7–9] for some results on this topic). Motivated by [10], we propose a new approach to the stability of quantum computation with respect to a static but incurable (perhaps unknown) perturbing interaction. We consider QA as a time-dependent dynamical system and relate its fidelity measuring the Hilbert space distance between computed states of exact and perturbed algorithm in terms of integrated time-autocorrelation of the perturbing operator (generalizing Ref. [11]). The derived relation looks very surprising: it tells that faster decay of time-correlations of the perturbation between sequences of successive quantum gates means larger fidelity, and vice versa. We propose to use our rule of thumb as a guide to devise or to improve QAs, either by introducing extra ’chaotic’ gates or by rewriting the gates in a different order in order to make time-evolution U(t) ’more chaotic’. As an important example, the well known QFT algorithm whose internal dynamics appears unpleasantly ’regular’ has been improved in a way that the modified algorithm becomes qualitatively more robust against static random perturbation of the gates. We think our effect should be considered in experimental realization of QFT which are underway [12]. Let us write the partial evolution operator for a sequence of consecutive gates from t′ to t, t′ < t, as U(t, t′) = U(t)U(t−1) · · ·U(t′+2)U(t′+1), with U(t, t) ≡ 1, and perturb the quantum gates by a (generally timedependent) perturbation of strength δ generated by hermitean operators V (t)

@inproceedings{ProsenCanQC,
title={Can quantum chaos enhance stability of quantum computation?},
author={Tomaz Prosen and Marko Znidaric}
}