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arXiv:quant-ph/0201064 · Physical Review Letters

Experimental Implementation of the Quantum Baker’s Map

Yaakov S. Weinstein, Seth Lloyd, Joseph Emerson, David G. Cory

Published Sep 24, 2002• 81 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.89.157902

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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.
The most significant achievements of this project were: (1) Development and validation of the hardware and software needed to implement "strongly modulating pulses", by which high precision quantum gates can be obtain in realistic systems of up to 12 qubits; (2) A detailed analysis and evaluation of a 3-qubit quantum Fourier transform via full quantum process tomography; (3) An extensive set of mathematical and computational techniques to accomplish these goals, including Hadamard products, the real density matrix, and methods of fitting superoperators to experimental data; (4) Creation of a 3-qubit "noiseless subsystem", of a Bell state on two 2-qubit decoherence-free subspaces (DFS), and the invention of "partial" pseudopure states which will enable us to demonstrate robust methods for controlling multi-DFS-qubit systems by NMR; (5) Implementation of several quantum chaotic maps, and the discovery that these provide a scalable approach to determining the magnitude and kind of errors present in complex quantum computations; (6) The invention of a "spin amplifier", by which entanglement can be used to enable single spin measurement, and a small-scale demonstration by NMR; (7) Experiments demonstrating that the foregoing advances enable implementation of complex entangling unitary and decoherent operations, culminating in creation of a 12-qubit CAT state.
Being able to quantify the level of coherent control in a proposed device implementing a quantum information processor (QIP) is an important task for both comparing different devices and assessing a device's prospects with regards to achieving fault-tolerant quantum control. We implement in a liquid-state nuclear magnetic resonance QIP the randomized benchmarking protocol presented by Knill et al (2008 Phys. Rev. A 77 012307). We report an error per randomized π/2 pulse of 1.3±0.1 × 10 −4 with a single-qubit QIP and show an experimentally relevant error model where the randomized benchmarking gives a signature fidelity decay which is not possible to interpret as a single error per gate. We explore and experimentally investigate multi-qubit extensions of this protocol and report an average error rate for one- and two-qubit gates of 4.7±0.3 × 10 −3 for a three-qubit QIP. We estimate that these error rates are still not decoherence limited and thus can be improved with modifications to the control hardware and software.
The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive-although complete positivity is required for any quantum operation. By comparison with the results of computer simulations, the lack of complete positivity was shown to be largely a consequence of the incoherent errors which occurred over the full quantum process tomography procedure. These simulations further showed that coherent, incoherent, and decoherent errors can often be identified by their distinctive effects on the spectrum of the overall superoperator. The gate fidelity of the experimentally determined superoperator was 0.64, while the correlation coefficient between experimentally determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cumulative effect of small errors in the single qubit gates.
In this work we study several models of decoherence and how different quantum maps and algorithms react when perturbed by them. Following closely Ref. [1], generalizations of the three paradigmatic one single qubit quantum channels (these are the depolarizing channel, the phase damping channel and the amplitude damping channel) for the case of an arbitrarily-sized finite-dimensional Hilbert space are presented, as well as other types of noise in phase space. More specifically, Grover's search algorithm's response to decoherence is analyzed; together with those of a family of quantum versions of chaotic and regular classical maps (the baker's map and the cat maps). A relationship between how sensitive to decoherence a quantum map is and the degree of complexity in the dynamics of its associated classical counterpart is observed; resulting in a clear tendency to react the more decoherently the more complex the associated classical dynamics is.