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 is a sparse unitary operator with a gap in its spectrum, then there exists an approximate logarithm of which is also sparse. The sparsity pattern of gets more dense as 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.
- Received 3 May 2007
DOI:https://doi.org/10.1103/PhysRevLett.101.140503
©2008 American Physical Society