Abstract
We study algorithms simulating a system evolving with Hamiltonian \({H = \sum_{j=1}^m H_j}\) , where each of the H j , j = 1, . . . ,m, can be simulated efficiently. We are interested in the cost for approximating \({e^{-iHt}, t \in \mathbb{R}}\) , with error \({\varepsilon}\) . We consider algorithms based on high order splitting formulas that play an important role in quantum Hamiltonian simulation. These formulas approximate e −iHt by a product of exponentials involving the H j , j = 1, . . . ,m. We obtain an upper bound for the number of required exponentials. Moreover, we derive the order of the optimal splitting method that minimizes our upper bound. We show significant speedups relative to previously known results.
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References
Feynman R.P.: Simulating physics with computers. Int. J. Theoret. Phys. 21, 467–488 (1982)
Lloyd S.: Universal quantum simulators. Science 273, 1073–1078 (1996)
Kassal I., Jordan S.P., Love P.J., Mohseni M., Aspuru-Guzik A.: Polynomial-time quantum algorithm for the simulation of chemical dynamics. Proc. Nat. Acad. Sci. 105, 18681 (2008)
Buluta I., Nori F.: Quantum simulators. Science 326, 108 (2009)
Zalka C.: Simulating quantum systems on a quantum computer. Proc. R. Soc. Lond. A 454, 313 (1998)
Zalka C.: Efficient simulation of quantum systems by quantum computers. Fortschritte der Physik 46, 877 (1998)
Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pp. 20–29 (2003)
Farhi, E., Goldstone, J., Gutmann, S., Sipser M.: Quantum computation by adiabatic evolution, quant-ph/0001106 (2000)
Childs A., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. J. Quant. Inf. Proc. 1, 35–43 (2002)
Farhi, E., Goldstone, J., Gutmann, S.: A Quantum algorithm for the Hamiltonian NAND Tree, quant-ph/0702144 (2007)
Childs A.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)
Childs A.: On the relationship between continuous- and discrete-time quantum walk. Commun. Math. Phys 294, 581–603 (2010)
Berry D.W., Ahokas G., Cleve R., Sanders B.C.: Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys. 270, 359 (2007)
Suzuki M.: Fractal decomposition of exponential operators with application to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323 (1990)
Suzuki M.: General theory of fractal path integrals with application to many-body theories and statistical physics. J. Math. Phys. 32, 400–407 (1991)
Wiebe N., Berry D., Hoyer P., Sanders B.C.: Higher order decompositions of ordered operator exponentials. J. Phys. A: Math. Theor. 43, 065203 (2010)
Klar B.: Bounds on tail probabilities of discrete distributions. Probab. Eng. Inf. Sci. 14, 161–171 (2000)
Abramowitz M., Stegun A.: Handbook of Mathematical Functions. Dover, New York (1972)
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Papageorgiou, A., Zhang, C. On the efficiency of quantum algorithms for Hamiltonian simulation. Quantum Inf Process 11, 541–561 (2012). https://doi.org/10.1007/s11128-011-0263-9
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DOI: https://doi.org/10.1007/s11128-011-0263-9