Approximate locality for quantum systems on graphs.
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 Delta 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/Delta 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.
Osborne
Tobias J
TJ
Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom. Tobias.Osborne@rhul.ac.uk