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arXiv:quant-ph/0012106 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Avoiding quantum chaos in quantum computation

G. P. Berman, F. Borgonovi, Felix M. Izrailev, V. I. Tsifrinovich

Published Dec 20, 2001• 22 citations
arXivHTMLPDF

Abstract

We study a one-dimensional chain of nuclear 1/2 spins in an external time-dependent magnetic field, considered as a possible candidate for experimental realization of quantum computation. According to the general theory of interacting particles, one of the most dangerous effects is quantum chaos that can destroy the stability of quantum operations. The standard viewpoint is that the threshold for the onset of quantum chaos due to an interaction between spins (qubits) strongly decreases with an increase of the number of qubits. Contrary to this opinion, we show that the presence of a nonhomogeneous magnetic field can strongly reduce quantum chaos effects. We give analytical estimates that explain this effect, together with numerical data supporting our analysis.

DOI
10.1103/physreve.65.015204

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We study a one-dimensional chain of nuclear 1/2 spins in an external time-dependent magnetic field, considered as a possible candidate for experimental realization of quantum computation. According to the general theory of interacting particles, one of the most dangerous effects is quantum chaos that can destroy the stability of quantum operations. The standard viewpoint is that the threshold for the onset of quantum chaos due to an interaction between spins (qubits) strongly decreases with an increase of the number of qubits. Contrary to this opinion, we show that the presence of a nonhomogeneous magnetic field can strongly reduce quantum chaos effects. We give analytical estimates that explain this effect, together with numerical data supporting our analysis.
We study the properties of spectra and eigenfunctions for a chain of 1/2 spins (qubits) in an external time-dependent magnetic field and under the conditions of nonselective excitation (when the amplitude of the magnetic field is large). This model is known as a possible candidate for experimental realization of quantum computation. We present the theory for finding delocalization transitions and show that for the interaction between nearest qubits, the transition is very different from that in quantum chaos. We explain this phenomena by showing that in the considered region of parameters our model is close to an integrable one. According to a general opinion, the threshold for the onset of quantum chaos due to the interqubit interaction decreases with an increase of the number of qubits. Contrary to this expectation, for a magnetic field with constant gradient we have found that chaos border does not depend on the number of qubits. We give analytical estimates that explain this effect, together with numerical data supporting our analysis. Random models with long-range interactions have been studied as well. In particular, we show that in this case the delocalization and quantum chaos borders coincide.
The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n . As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.
The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n. As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.
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.
Recently it was found that the dynamics in a Heisenberg spin-chain subjected to a sequence of periodic pulses from an external, parabolic, magnetic field can have a close correspondence with the quantum kicked rotor (QKR). The QKR is a key paradigm of quantum chaos; it has as its classical limit the well-known Standard Map. It was found that a single spin excitation could be converted into a pair of non-dispersive, counter-propagating spin coherent states equivalent to the accelerator modes of the Standard Map. Here we consider how other types of quantum chaotic systems such as a double-kicked quantum rotor or a quantum rotor with a double-well potential might be realized with spin chains; we discuss the possibilities regarding manipulation of the one-magnon spin waves.