Approximate Locality for Quantum Systems on Graphs
Abstract
In this Letter we make progress on a long-standing open problem of Aaronson and Ambainis [Theory Comput. 1, 47 (2005)1557-2862]: 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.
- Publication:
-
Physical Review Letters
- Pub Date:
- October 2008
- DOI:
- 10.1103/PhysRevLett.101.140503
- arXiv:
- arXiv:quant-ph/0611231
- Bibcode:
- 2008PhRvL.101n0503O
- Keywords:
-
- 03.67.-a;
- 05.40.Fb;
- 05.45.Mt;
- 05.60.Gg;
- Quantum information;
- Random walks and Levy flights;
- Quantum chaos;
- semiclassical methods;
- Quantum transport;
- Quantum Physics
- E-Print:
- 5 pages, 2 figures, corrected typo