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arXiv:quant-ph/0402157 · Physical Review A

Pseudorandom operators of the circular ensembles

Yaakov S. Weinstein, C. Stephen Hellberg

Published Jan 28, 2005• 7 citations
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Abstract

We demonstrate quantum algorithms to implement pseudorandom operators that closely reproduce statistical properties of random matrices from the three universal classes: unitary, symmetric, and symplectic. Modified versions of the algorithms are introduced for the less experimentally challenging quantum cellular automata. For implementing pseudorandom symplectic operators we provide gate sequences for the unitary part of the time-reversal operator.

DOI
10.1103/physreva.71.014303

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In close analogy to the fundamental role of random numbers in classical information theory, random operators are a basic component of quantum information theory. Unfortunately, the implementation of random unitary operators on a quantum processor is exponentially hard. Here we introduce a method for generating pseudo-random unitary operators that can reproduce those statistical properties of random unitary operators most relevant to quantum information tasks. This method requires exponentially fewer resources, and hence enables the practical application of random unitary operators in quantum communication and information processing protocols. Using a nuclear magnetic resonance quantum processor, we were able to realize pseudorandom unitary operators that reproduce the expected random distribution of matrix elements.
We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble. Our study includes the full distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates which scales at most as ${n}_{q}{[\mathrm{ln}({n}_{q}∕ϵ)]}^{\ensuremath{\nu}}$ with the number of qubits nq{n}_{q}nq​ for a given fixed precision ϵϵϵ and $\ensuremath{\nu}>0$.
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