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Kluwer Academic Publishers eBooks

Hypersensitivity to Perturbation on an NMR Quantum Computer

Rüdiger Schack, Todd A. Brunb

Published Dec 19, 2005• 1 citation
Publisher

Abstract

By numerically simulating an implementation of the quantum baker’s map on a 3-qubit NMR quantum computer based on the molecule trichloroethylene, we demonstrate the feasibility of quantum chaos experiments on present-day quantum computers. In particular, we describe an experiment investigating the phenomenon of hypersensitivity to perturbation.

DOI
10.1007/0-306-47097-7_34

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By numerically simulating an implementation of the quantum baker’s map on a 3-qubit NMR quantum computer based on the molecule trichloroethylene, we demonstrate the feasibility of quantum chaos experiments on present-day quantum computers. In particular, we describe an experiment investigating 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.
Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.