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arXiv:quant-ph/9807050 · Physical Review A

Realizing the quantum baker’s map on a NMR quantum computer

Todd A. Brun, Rüdiger Schack

Published Apr 1, 1999• 39 citations
arXivHTMLPDF

Abstract

By numerically simulating an implementation of the quantum baker's map on a NMR quantum computer based on the molecule trichloroethylene, we demonstrate the feasibility of quantum chaos experiments on present day quantum computers. We give detailed descriptions of proposed experiments that investigate (a) the rate of entropy increase due to decoherence and (b) the phenomenon of hypersensitivity to perturbation.

DOI
10.1103/physreva.59.2649

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•Jan 1, 2001•0 citations
By numerically simulating an implementation of the quantum baker's map on a NMR quantum computer based on the molecule trichloroethylene, we demonstrate the feasibility of quantum chaos experiments on present day quantum computers. We give detailed descriptions of proposed experiments that investigate (a) the rate of entropy increase due to decoherence and (b) the phenomenon of hypersensitivity to perturbation.
This Letter reports on the experimental implementation of the quantum baker's map via a three bit nuclear magnetic resonance quantum information processor. The experiments tested the sensitivity of the quantum chaotic map to perturbations. In the first experiment, the map was iterated forward and then backward to provide benchmarks for intrinsic errors and decoherence. In the second set of experiments, the least significant qubit was perturbed in between the iterations to test the sensitivity of the quantum chaotic map to controlled perturbations. These experiments can be used to investigate existing theoretical predictions for quantum chaotic dynamics.
Quantum error correction is required to compensate for the fragility of the state of a quantum computer. We report the first experimental implementations of quantum error correction and confirm the expected state stabilization. A precise analysis of the decay behavior is performed in alanine and a full implementation of the error correction procedure is realized in trichloroethylene. In NMR computing, however, a net improvement in the signal to noise would require very high polarization. The experiment implemented the three-bit code for phase errors using liquid state NMR.
We analyze a randomly perturbed quantum version of the baker's transformation, a prototype of an area-conserving chaotic map. By simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this grows very rapidly and becomes larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker's map displays a hypersensitivity to perturbations analogous to behavior found in the classical case. This hypersensitivity characterizes ``quantum chaos'' in a way that is relevant to statistical physics.
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.