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Solvay Conference on Physics

EFFECTS OF STATIC IMPERFECTIONS FOR QUANTUM COMPUTING

Giuliano Benenti, Giulio Casati

Published Aug 1, 2003• 0 citations
Publisher

Abstract

We model the quantum computer hardware as a two-dimensional lattice of qubits with static imperfections, i.e. fluctuations in individual qubit energies and residual short-range inter-qubit couplings. We show that these imperfections can lead to the emergence of quantum chaos and dynamical thermalization also in a quantum computer ideally decoupled from the environment. We discuss their effect on the stability of (i) the quantum computer hardware and (ii) an efficient quantum algorithm simulating a physical model with rich and complex dynamics described by the quantum sawtooth map.

DOI
10.1142/9789812704634_0010

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We model the quantum computer hardware as a two-dimensional lattice of qubits with static imperfections, i.e. fluctuations in individual qubit energies and residual short-range inter-qubit couplings. We show that these imperfections can lead to the emergence of quantum chaos and dynamical thermalization also in a quantum computer ideally decoupled from the environment. We discuss their effect on the stability of (i) the quantum computer hardware and (ii) an efficient quantum algorithm simulating a physical model with rich and complex dynamics described by the quantum sawtooth map.
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.
We show that in an isolated model of quantum computer the residual short-range inter-qubit couplings and fluctuations in individual qubit energies can lead to quantum chaos and quantum ergodicity of eigenstates. The onset of chaos results in the interaction induced dynamical thermalization and the 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 also show that a quantum computer operating with a small number of qubits can simulate the dynamical localization of classical chaos in a system described by the quantum sawtooth map model. The dynamics of the system is computed efficiently up to a time t ≥ l , and then the localization length l can be obtained with accuracy ν by means of order 1/ν 2 computer runs, followed by coarse grained projective measurements on the computational basis.
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 propose a quantum algorithm which uses the number of qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics described by the quantum sawtooth map. 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.
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.