Approximate locality for quantum systems on graphs
| dc.creator | Osborne, Tobias J. | |
| dc.date | 2006-11-22 | |
| dc.date | 2006-11-23 | |
| dc.date.accessioned | 2026-07-07T10:07:34Z | |
| dc.date.available | 2026-07-07T10:07:34Z | |
| dc.description | In this Letter we make progress on a longstanding open problem of Aaronson and Ambainis [Theory of Computing 1, 47 (2005)]: we show that if A is the adjacency matrix of a sufficiently sparse low-dimensional graph then the unitary operator e^{itA} can be approximated by a unitary operator U(t) whose sparsity pattern is exactly that of a low-dimensional graph which gets more dense as |t| increases. Secondly, we show that if U is a sparse unitary operator with a gap Δin 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/Δincreases. These two results can be interpreted as a way to convert between local continuous-time and local discrete-time processes. As an example we show that the discrete-time coined quantum walk can be realised as an approximately local continuous-time quantum walk. Finally, we use our construction to provide a definition for a fractional quantum fourier transform. | |
| dc.description | 5 pages, 2 figures, corrected typo | |
| dc.identifier | https://arxiv.org/abs/quant-ph/0611231 | |
| dc.identifier | http://arxiv.org/abs/quant-ph/0611231 | |
| dc.identifier | Phys. Rev. Lett. 101, 140503 (2008) | |
| dc.identifier | doi:10.1103/PhysRevLett.101.140503 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/170681 | |
| dc.subject | Quantum Physics | |
| dc.title | Approximate locality for quantum systems on graphs | |
| dc.type | text |