Approximate Locality for Quantum Systems on Graphs

Tobias J. Osborne
Phys. Rev. Lett. 101, 140503 – Published 3 October 2008

Abstract

In this Letter we make progress on a long-standing open problem of Aaronson and Ambainis [Theory Comput. 1, 47 (2005)]: we show that if U is a sparse unitary operator with a gap Δ in its spectrum, then there exists an approximate logarithm H of U which is also sparse. The sparsity pattern of H gets more dense as 1/Δ increases. This result can be interpreted as a way to convert between local continuous-time and local discrete-time quantum processes. As an example we show that the discrete-time coined quantum walk can be realized stroboscopically from an approximately local continuous-time quantum walk.

  • Figure
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  • Received 3 May 2007

DOI:https://doi.org/10.1103/PhysRevLett.101.140503

©2008 American Physical Society

Authors & Affiliations

Tobias J. Osborne*

  • Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom

  • *Tobias.Osborne@rhul.ac.uk

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Vol. 101, Iss. 14 — 3 October 2008

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