Quantum Circuits for Sparse Isometries

Emanuel Malvetti1, Raban Iten2, and Roger Colbeck3

1Department of Chemistry, Technische Universität München, Lichtenbergstraße 4, 85747 Garching, Germany
2ETH Zürich, 8093 Zürich, Switzerland
3Department of Mathematics, University of York, YO10 5DD, UK

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Abstract

We consider the task of breaking down a quantum computation given as an isometry into C-NOTs and single-qubit gates, while keeping the number of C-NOT gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.

In order to realise a quantum computation on a quantum computer it is useful to break it down into a sequence of elementary gates. The number of elementary gates required for an arbitrary quantum computation scales exponentially in the number of qubits the computation acts on. However, useful quantum computations often come from a much smaller class and hence we would like decomposition methods that can exploit additional structure. In this work we introduce decomposition schemes that work on the set of sparse computations, showing that the number of elementary gates required dramatically decreases with the sparseness. Minimizing the number of elementary gates is particularly important for near term quantum devices, since each additional gate adds noise to the computation. We optimize our decomposition schemes for sparse computations on such devices, i.e., for a small number of qubits. The decomposition schemes have been implemented in UniversalQCompiler, an open source software package for compiling quantum computations.

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Cited by

[1] Carsten Blank and Francesco Petruccione, "1. Quantum Applications - Fachbeitrag: Vielversprechend: Monte-Carlo-ähnliche Methoden auf dem Quantencomputer ", Digitale Welt 5 4, 40 (2021).

[2] Davide Orsucci and Vedran Dunjko, "On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number", arXiv:2101.11868.

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