Hamiltonian simulation using linear combinations of unitary operations

Quantum Information and Computation

Volumn 12, Issue 11-12, 2012, Pages 901-924

  Childs, Andrew M ^a   Wiebe, Nathan ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Quantum computing; Quantum simulation

Indexed keywords

QUANTUM CHEMISTRY; QUANTUM COMPUTERS;

HAMILTONIAN DYNAMICS; LINEAR COMBINATIONS; NEW APPROACHES; PERFORMANCE; QUANTUM COMPUTING; QUANTUM SIMULATIONS; SIMULATION ALGORITHMS; SIMULATION ERROR; SIMULATION TECHNIQUE; UNITARY OPERATION;

HAMILTONIANS;

EID: 84866747423     PISSN: 15337146     EISSN: None     Source Type: Journal    
DOI: 10.26421/qic12.11-12-1     Document Type: Article

References (27)

1. D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev (2007), Adiabatic quantum computation is equivalent to standard quantum computation, SIAM J. Comput., 37, pp. 166-194. Preliminary version in FOCS 2004.
2. A. M. Childs (2009), Universal computation by quantum walk, Phys. Rev. Lett., 102, 180501.
3. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser (2000), Quantum computation by adiabatic evolution, arXiv:quant-ph/0001106.
4. A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman (2003), Exponential algorithmic speedup by quantum walk, In Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 59-68.
5. E. Farhi, J. Goldstone, and S. Gutmann (2008), A quantum algorithm for the Hamiltonian NAND tree, Theory of Computing, 4, pp. 169-190.
6. A. W. Harrow, Avinatan Hassidim, and S. Lloyd (2009), Quantum algorithm for linear systems of equations, Phys. Rev. Lett., 103, 150502.
7. D. W. Berry (2010), Quantum algorithms for solving linear differential equations, arXiv:1010.2745.
8. S. Lloyd (1996), Universal quantum simulators, Science, 273, pp. 1073-1078.
9. D. Aharonov and A. Ta-Shma (2003), Adiabatic quantum state generation and statistical zero knowledge, In Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 20-29.
10. A. M. Childs (2004), Quantum information processing in continuous time. Ph.D. thesis, Mas-sachusetts Institute of Technology.
11. D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders (2007), Efficient quantum algorithms for simulating sparse Hamiltonians, Commun. Math. Phys., 270, pp. 359-371.
12. A. Papageorgiou and C. Zhang (2012), On the efficiency of quantum algorithms for Hamiltonian simulation to appear in Quantum Information Processing.
13. A. M. Childs and R. Kothari (2011), Simulating sparse Hamiltonians with star decompositions, In Theory of Quantum Computation, Communication, and Cryptography, volume 6519 of Lecture Notes in Computer Science, pp. 94-103, Springer.
14. N. Wiebe, D. W. Berry, P. Høyer, and B. C. Sanders (2011), Simulating quantum dynamics on a quantum computer, J. Phys. A, 44, 445308.
15. D. Poulin, A. Qarry, R. Somma, and F. Verstraete (2011), Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space, Phys. Rev. Lett., 106,170501.
16. A. M. Childs (2009), On the relationship between continuous-and discrete-time quantum walk, Commun. Math. Phys., 294, pp. 581-603.
17. D. W. Berry and A. M. Childs (2009), Black-box Hamiltonian simulation and unitary implementation, Quantum Information and Computation, 12, pp. 29-62.
18. M. Suzuki (1991), General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys., 32, pp. 400-407.
19. S. Blanes, F. Casas, and J. Ros (1999), Extrapolation of symplectic integrators, Celestial Mechanics and Dynamical Astronomy, 75, pp. 149-161.
20. N. Wiebe, D. W. Berry, P. Høyer, and B. C. Sanders (2010), Higher order decompositions of ordered operator exponentials, J. Phys. A, 43, 065203.
21. R. Cleve, D. Gottesman, M. Mosca, R. D. Somma, and D. Yonge-Mallo (2009), Efficient discretetime simulations of continuous-time quantum query algorithm, In Proceedings of the 41st ACM Symposium on Theory of Computing, pp. 409-416.
22. S. Chin (2010), Multi-product splitting and Runge-Kutta-Nyström integrators, Celestial Mechanics and Dynamical Astronomy, 106, pp. 391-406.
23. L. F. Richardson (1911), The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London Series A, 210, pp. 307-357.
24. Q. Sheng (1989), Solving linear partial differential equations by exponential splitting, IMA Journal of Numerical Analysis, 9, pp. 199-212.
25. M. Abramowitz and I. A. Stegun (1964), Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of Applied Mathematics Series, U.S. Government Printing Office.
26. S. A. Chin and J. Geiser (2011), Multi-product operator splitting as a general method of solving autonomous and nonautonomous equations, IMA Journal of Numerical Analysis, 31, pp. 1552-1577.
27. M. E. A. El-Mikkawy (2003), Explicit inverse of a generalized Vandermonde matrix, Applied Mathematics and Computation, 146, pp. 643-651.