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arXiv:quant-ph/0301086 · Physical Review A

Entanglement versus relaxation and decoherence in a quantum algorithm for quantum chaos

Stefano Bettelli, Dima L. Shepelyansky

Published May 27, 2003• 37 citations
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Abstract

We study analytically and numerically the behavior of the concurrence (a measure of the entanglement of formation) of a pair of qubits in a quantum computer operating an efficient algorithm for quantum chaos. Our results show that in an ideal algorithm the entanglement decays exponentially with the diffusive relaxation rate induced by classical chaos. This decay reaches a residual level which drops exponentially with increasing number of qubits ${n}_{q}.$ Decoherence destroys the residual entanglement with a rate exponential in ${n}_{q}.$

DOI
10.1103/physreva.67.054303

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We study analytically and numerically the behavior of the concurrence (a measure of the entanglement of formation) of a pair of qubits in a quantum computer operating an efficient algorithm for quantum chaos. Our results show that in an ideal algorithm the entanglement decays exponentially with the diffusive relaxation rate induced by classical chaos. This decay reaches a residual level which drops exponentially with increasing number of qubits nq.{n}_{q}.nq​. Decoherence destroys the residual entanglement with a rate exponential in nq.{n}_{q}.nq​.
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, the quantum ergodic regime, and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates, and per gate dissipation rate induced by external decoherence. In the limit of strong dissipation the quantum algorithm generates a quantum attractor which may have a complex or simple structure. We also compare the effects of dissipative decoherence with the effects of static imperfections.
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.
Abstract. Using the methods of quantum trajectories we study effects of dissipative decoherence on the accuracy of the Grover quantum search algorithm. The dependence on the number of qubits and dissipation rate are determined and tested numerically with up to 16 qubits. As a result, our numerical and analytical studies give the universal law for decay of fidelity and probability of searched state which are induced by dissipative decoherence effects. This law is in agreement with the results obtained previously for quantum chaos algorithms. PACS. 03.67.Lx Quantum Computation – 03.65.Yz Decoherence; open systems; quantum statistical meth-ods – 24.10.Cn Many-body theory QICS. 11.40.+s Search – 13.10.+n Effects of noise and imperfections – 11.50.+m Quantum maps, quantum chaos Nowadays the quantum computing attracts a great inter-est of the scientific community [1]. The main reason of that is due to the fact that certain quantum algorithms allow to perform computations much faster than the usual classical algorithms. The famous example is the Shor algo-
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This paper reviews recent work related to the interplay between quantum information and computation on the one hand and classical and quantum chaos on the other. First, we present several models of quantum chaos that can be simulated efficiently on a quantum computer. Then a discussion of information extraction shows that such models can give rise to complete algorithms including measurements that can achieve an increase in speed compared with classical computation. It is also shown that models of classical chaos can be simulated efficiently on a quantum computer, and again information can be extracted efficiently from the final wave function. The total gain can be exponential or polynomial, depending on the model chosen and the observable measured. The simulation of such systems is also economical in the number of qubits, allowing implementation on present-day quantum computers, some of these algorithms having been already experimentally implemented. The second topic considered concerns the analysis of errors on quantum computers. It is shown that quantum chaos algorithms can be used to explore the effect of errors on quantum algorithms, such as random unitary errors or dissipative errors. Furthermore, the tools of quantum chaos allows a direct analysis of the effects of static errors on quantum computers. Finally, we consider the different resources used by quantum information, and show that quantum chaos has some precise consequences on entanglement generation, which becomes close to maximal. For another resource, interference, a proposal is presented for quantifying it, enabling a discussion on entanglement and interference generation in quantum algorithms.