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arXiv:quant-ph/0106150 · Journal of Physics A Mathematical and General

Can quantum chaos enhance the stability of quantum computation?

Tomaž Prosen, Marko Žnidarič

Published Nov 21, 2001• 60 citations
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Abstract

We consider the stability of a general quantum algorithm (QA) with respect to a fixed but unknown residual interaction between qubits, and show a surprising fact, namely that the average fidelity of quantum computation increases on decreasing the average time correlation function of the perturbing operator in sequences of consecutive quantum gates. Our thinking is applied to the quantum Fourier transformation, where an alternative `less regular' QA is devised, which is qualitatively more robust against static random residual n -qubit interaction.

DOI
10.1088/0305-4470/34/47/103

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We consider the stability of a general quantum algorithm (QA) with respect to a fixed but unknown residual interaction between qubits, and show a surprising fact, namely that the average fidelity of quantum computation increases on decreasing the average time correlation function of the perturbing operator in sequences of consecutive quantum gates. Our thinking is applied to the quantum Fourier transformation, where an alternative `less regular' QA is devised, which is qualitatively more robust against static random residual n -qubit interaction.
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We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength εεε. The critical border εcε_cεc​ between two phases drops polynomially with the number of qubits nqn_qnq​ as εc∼nq−3/2ε_c \sim n_q^{-3/2}εc​∼nq−3/2​. In the regular phase (ε<εc)(ε< ε_c)(ε<εc​) the algorithm remains robust against imperfections showing the efficiency gain εc/εε_c / εεc​/ε for ε>2−nq/2ε> 2^{-n_q/2}ε>2−nq​/2. In the chaotic phase (ε>εc)(ε> ε_c)(ε>εc​) the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.
The stability of quantum computating of the kicked Harper model with various perturbations is investigated. Above a certain threshold of the imperfections, quantum chaos sets in. The effects of the noise errors and the static imperfections on the quantum computation are analyzed by comparing the statistical ergodic properties and the Husimi functions of the perturbed eigenstates with the ideal eigenstates of the Floquet operator. It is shown that the fidelity decay with static imperfections is exponential while it is Gaussian with noise errors. The time scales of reliable computation with these perturbations are obtained through numerical simulations. Due to the errors in classical computation the distance of two initial states increases exponentially, while the stability of quantum computation is independent of the integrable or chaotic nature of the underlying dynamics.
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We study the effects of static interqubit interactions on the accuracy of various quantum algorithms. Extensive numerical simulations show that their effect is stronger compared to external decoherence. Analytical approach based on Random Matrix Theory is developed. It gives universal law for fidelity decay induced by interqubit static interactions. This determines the time scale for reliable quantum computation in presence of realistic static imperfections and external decoherence. New polynomial algorithms are developed for simulation of complex dynamics in the regime of classical and quantum chaos, and Anderson metal- insulator transition. A generic quantum error correction method is developed; it allows to eliminate coherent effect of static errors. The theoretical results are confirmed by numerical computations with up to 28 qubits.
We model an isolated quantum computer as a two-dimensional lattice of qubits (spin halves) with fluctuations in individual qubit energies and residual short-range inter-qubit couplings. We show that above a critical inter-qubit coupling strength, quantum chaos sets in and this results in the interaction induced dynamical thermalization and occupation numbers well described by the Fermi–Dirac distribution. This thermalization destroys the noninteracting qubit structure and sets serious requirements for the quantum computer operability. We then construct a quantum algorithm which uses qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics. The numerical study of the effect of static imperfections in the quantum computer hardware shows that the main elements of the phase space structures are accurately reproduced up to a time scale which is polynomial in the number of qubits. The errors generated by these imperfections are more significant than the errors of random noise in gate operations.