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H. Lamm, S. Lawrence, and Y. Yamauchi, “Quantum simulation of gauge theories,” (2019), arXiv:1903.08807
Original
1903
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2019

Nearly optimal lattice simulation by product formulas

Childs, Andrew M., Su, Yuan

Understand

We consider simulating an $n$-qubit Hamiltonian with nearest-neighbor interactions evolving for time $t$ on a quantum computer.

We show that this simulation has gate complexity $(nt)^{1+o(1)}$ using product formulas, a straightforward approach that has been demonstrated by several experimental groups.
While it is reasonable to expect this complexity---in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill---we are not aware of a straightforward proof.
Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and further simplified here.

Built on

H. Lamm, S. Lawrence, and Y. Yamauchi, “Quantum simulation of gauge theories,” (2019), arXiv:1903.08807
Original
1903
Earlier work this paper cites.
T. Peng, A. Harrow, M. Ozols, and X. Wu, “Simulating large quantum circuits on a small quantum computer,” (2019), arXiv:1904.00102
Original
1904
Earlier work this paper cites.
R. P. Feynman, International Journal of Theoretical Physics 21
1982
Earlier work this paper cites.
M. Suzuki, Journal of Mathematical Physics 32
1991
Earlier work this paper cites.
S. Lloyd, Science 273
1996
Earlier work this paper cites.
See Supplemental Material for details, which includes Refs. Auzinger and Herfort [2014] , Rossmann [2006] , Chakraborty et al. [2018] , Bellman [1997]
1997
Earlier work this paper cites.
R. Bellman, Introduction to matrix analysis (Society for Industrial and Applied Mathematics, 1997)
1997
Earlier work this paper cites.
F. Verstraete and J. I. Cirac, Journal of Statistical Mechanics 2005
2005
Earlier work this paper cites.
N. Hatano and M. Suzuki, in Quantum Annealing and Other Optimization Methods , edited by A. Das and B. K. Chakrabarti (Springer, 2005) pp. 37–68
2005
Earlier work this paper cites.
A. W. Knapp, Basic real analysis (Birkhëuser, 2005)
2005
Earlier work this paper cites.
K. R. Brown, R. J. Clark, and I. L. Chuang, Physical Review Letters 97
2006
Earlier work this paper cites.
R. Jozsa, “On the simulation of quantum circuits,” (2006), arXiv:quant-ph/0603163
2006
Earlier work this paper cites.
T. J. Osborne, Physical Review Letters 97
2006
Earlier work this paper cites.
W. Rossmann, Lie groups: An introduction through linear groups (Oxford University Press, 2006)
2006
Earlier work this paper cites.
D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Communications in Mathematical Physics 270
2007
Earlier work this paper cites.
S. Descombes and M. Thalhammer, BIT Numerical Mathematics 50
2010
Earlier work this paper cites.
N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders, Journal of Physics A 43
Original
2010
Earlier work this paper cites.
M. B. Hastings, “Locality in quantum systems,” (2010), arXiv:1008.5137
Original
2010
Earlier work this paper cites.
B. P. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F. Roos, Science 334
Original
2011
Earlier work this paper cites.
R. Machleidt and D. R. Entem, Physics Reports 503
Original
2011
Earlier work this paper cites.