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Composite Quantum Simulations

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Matthew Hagan1 and Nathan Wiebe2,3,4

1Department of Physics, University of Toronto, Toronto ON, Canada
2Department of Computer Science, University of Toronto, Toronto ON, Canada
3Pacific Northwest National Laboratory, Richland Wa, USA
4Canadian Institute for Advanced Study, Toronto ON, Canada

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Abstract

In this paper we provide a framework for combining multiple quantum simulation methods, such as Trotter-Suzuki formulas and QDrift into a single Composite channel that builds upon older coalescing ideas for reducing gate counts. The central idea behind our approach is to use a partitioning scheme that allocates a Hamiltonian term to the Trotter or QDrift part of a channel within the simulation. This allows us to simulate small but numerous terms using QDrift while simulating the larger terms using a high-order Trotter-Suzuki formula. We prove rigorous bounds on the diamond distance between the Composite channel and the ideal simulation channel and show under what conditions the cost of implementing the Composite channel is asymptotically upper bounded by the methods that comprise it for both probabilistic partitioning of terms and deterministic partitioning. Finally, we discuss strategies for determining partitioning schemes as well as methods for incorporating different simulation methods within the same framework.

► BibTeX data

► References

[1] James D Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik. “Simulation of electronic structure hamiltonians using quantum computers”. Molecular Physics 109, 735–750 (2011). url: https:/​/​doi.org/​10.1080/​00268976.2011.552441.
https:/​/​doi.org/​10.1080/​00268976.2011.552441

[2] Stephen P Jordan, Keith SM Lee, and John Preskill. “Quantum algorithms for quantum field theories”. Science 336, 1130–1133 (2012). url: https:/​/​doi.org/​10.1126/​science.1217069.
https:/​/​doi.org/​10.1126/​science.1217069

[3] Markus Reiher, Nathan Wiebe, Krysta M Svore, Dave Wecker, and Matthias Troyer. “Elucidating reaction mechanisms on quantum computers”. Proceedings of the national academy of sciences 114, 7555–7560 (2017). url: https:/​/​doi.org/​10.1073/​pnas.1619152114.
https:/​/​doi.org/​10.1073/​pnas.1619152114

[4] Ryan Babbush, Dominic W. Berry, and Hartmut Neven. “Quantum simulation of the sachdev-ye-kitaev model by asymmetric qubitization”. Phys. Rev. A 99, 040301 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.040301

[5] Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush. “Fault-tolerant quantum simulations of chemistry in first quantization”. PRX Quantum 2, 040332 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040332

[6] Thomas E. O’Brien, Michael Streif, Nicholas C. Rubin, Raffaele Santagati, Yuan Su, William J. Huggins, Joshua J. Goings, Nikolaj Moll, Elica Kyoseva, Matthias Degroote, Christofer S. Tautermann, Joonho Lee, Dominic W. Berry, Nathan Wiebe, and Ryan Babbush. “Efficient quantum computation of molecular forces and other energy gradients”. Phys. Rev. Res. 4, 043210 (2022).
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.043210

[7] Dorit Aharonov and Amnon Ta-Shma. “Adiabatic quantum state generation and statistical zero knowledge”. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. Pages 20–29. (2003). url: https:/​/​doi.org/​10.1145/​780542.780546.
https:/​/​doi.org/​10.1145/​780542.780546

[8] Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. “Efficient quantum algorithms for simulating sparse hamiltonians”. Communications in Mathematical Physics 270, 359–371 (2007). url: https:/​/​doi.org/​10.1007/​s00220-006-0150-x.
https:/​/​doi.org/​10.1007/​s00220-006-0150-x

[9] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. “Simulating hamiltonian dynamics with a truncated taylor series”. Phys. Rev. Lett. 114, 090502 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.090502

[10] Andrew M. Childs, Aaron Ostrander, and Yuan Su. “Faster quantum simulation by randomization”. Quantum 3, 182 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-02-182

[11] Guang Hao Low and Isaac L. Chuang. “Hamiltonian Simulation by Qubitization”. Quantum 3, 163 (2019).
https:/​/​doi.org/​10.22331/​q-2019-07-12-163

[12] Guang Hao Low, Vadym Kliuchnikov, and Nathan Wiebe. “Well-conditioned multiproduct hamiltonian simulation” (2019). url: https:/​/​doi.org/​10.48550/​arXiv.1907.11679.
https:/​/​doi.org/​10.48550/​arXiv.1907.11679

[13] Guang Hao Low and Nathan Wiebe. “Hamiltonian simulation in the interaction picture” (2019). arXiv:1805.00675.
arXiv:1805.00675

[14] Earl Campbell. “Random compiler for fast hamiltonian simulation”. Phys. Rev. Lett. 123, 070503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.070503

[15] Nathan Wiebe, Dominic Berry, Peter Høyer, and Barry C Sanders. “Higher order decompositions of ordered operator exponentials”. Journal of Physics A: Mathematical and Theoretical 43, 065203 (2010).
https:/​/​doi.org/​10.1088/​1751-8113/​43/​6/​065203

[16] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. “Theory of trotter error with commutator scaling”. Phys. Rev. X 11, 011020 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.011020

[17] Dominic W. Berry, Andrew M. Childs, Yuan Su, Xin Wang, and Nathan Wiebe. “Time-dependent Hamiltonian simulation with $L^1$-norm scaling”. Quantum 4, 254 (2020).
https:/​/​doi.org/​10.22331/​q-2020-04-20-254

[18] Dave Wecker, Bela Bauer, Bryan K. Clark, Matthew B. Hastings, and Matthias Troyer. “Gate-count estimates for performing quantum chemistry on small quantum computers”. Physical Review A 90 (2014).
https:/​/​doi.org/​10.1103/​physreva.90.022305

[19] David Poulin, Matthew B Hastings, Dave Wecker, Nathan Wiebe, Andrew C Doherty, and Matthias Troyer. “The trotter step size required for accurate quantum simulation of quantum chemistry” (2014). url: https:/​/​doi.org/​10.48550/​arXiv.1406.4920.
https:/​/​doi.org/​10.48550/​arXiv.1406.4920

[20] Ian D Kivlichan, Christopher E Granade, and Nathan Wiebe. “Phase estimation with randomized hamiltonians” (2019). arXiv:1907.10070.
arXiv:1907.10070

[21] Abhishek Rajput, Alessandro Roggero, and Nathan Wiebe. “Hybridized Methods for Quantum Simulation in the Interaction Picture”. Quantum 6, 780 (2022).
https:/​/​doi.org/​10.22331/​q-2022-08-17-780

[22] Yingkai Ouyang, David R. White, and Earl T. Campbell. “Compilation by stochastic hamiltonian sparsification”. Quantum 4, 235 (2020).
https:/​/​doi.org/​10.22331/​q-2020-02-27-235

[23] Shi Jin and Xiantao Li. “A partially random trotter algorithm for quantum hamiltonian simulations” (2021). url: https:/​/​doi.org/​10.48550/​arXiv.2109.07987.
https:/​/​doi.org/​10.48550/​arXiv.2109.07987

[24] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. “Low-depth quantum simulation of materials”. Phys. Rev. X 8, 011044 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.011044

[25] Masuo Suzuki. “Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations”. Physics Letters A 146, 319–323 (1990).
https:/​/​doi.org/​10.1016/​0375-9601(90)90962-N

[26] Andrew M Childs and Nathan Wiebe. “Hamiltonian simulation using linear combinations of unitary operations” (2012). url: https:/​/​doi.org/​10.26421/​QIC12.11-12.
https:/​/​doi.org/​10.26421/​QIC12.11-12

[27] Paul K Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert. “Randomizing multi-product formulas for improved hamiltonian simulation” (2021). url: https:/​/​ui.adsabs.harvard.edu/​link_gateway/​2022Quant…6..806F/​doi:10.48550/​arXiv.2101.07808.
https:/​/​ui.adsabs.harvard.edu/​link_gateway/​2022Quant…6..806F/​doi:10.48550/​arXiv.2101.07808

[28] Dominic W. Berry, Andrew M. Childs, and Robin Kothari. “Hamiltonian simulation with nearly optimal dependence on all parameters”. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science. Pages 792–809. (2015).
https:/​/​doi.org/​10.1109/​FOCS.2015.54

[29] Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, and Joel A. Tropp. “Concentration for random product formulas”. PRX Quantum 2 (2021).
https:/​/​doi.org/​10.1103/​prxquantum.2.040305

Cited by

[1] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, “Quantum algorithms: A survey of applications and end-to-end complexities”, arXiv:2310.03011, (2023).

[2] Etienne Granet and Henrik Dreyer, “Continuous Hamiltonian dynamics on noisy digital quantum computers without Trotter error”, arXiv:2308.03694, (2023).

[3] Almudena Carrera Vazquez, Daniel J. Egger, David Ochsner, and Stefan Woerner, “Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation”, Quantum 7, 1067 (2023).

[4] Matthew Pocrnic, Matthew Hagan, Juan Carrasquilla, Dvira Segal, and Nathan Wiebe, “Composite QDrift-Product Formulas for Quantum and Classical Simulations in Real and Imaginary Time”, arXiv:2306.16572, (2023).

[5] Nicholas H. Stair, Cristian L. Cortes, Robert M. Parrish, Jeffrey Cohn, and Mario Motta, “Stochastic quantum Krylov protocol with double-factorized Hamiltonians”, Physical Review A 107 3, 032414 (2023).

[6] Gumaro Rendon, Jacob Watkins, and Nathan Wiebe, “Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation”, arXiv:2212.14144, (2022).

[7] Zhicheng Zhang, Qisheng Wang, and Mingsheng Ying, “Parallel Quantum Algorithm for Hamiltonian Simulation”, arXiv:2105.11889, (2021).

[8] Maximilian Amsler, Peter Deglmann, Matthias Degroote, Michael P. Kaicher, Matthew Kiser, Michael Kühn, Chandan Kumar, Andreas Maier, Georgy Samsonidze, Anna Schroeder, Michael Streif, Davide Vodola, and Christopher Wever, “Quantum-enhanced quantum Monte Carlo: an industrial view”, arXiv:2301.11838, (2023).

[9] Alireza Tavanfar, S. Alipour, and A. T. Rezakhani, “Does Quantum Mechanics Breed Larger, More Intricate Quantum Theories? The Case for Experience-Centric Quantum Theory and the Interactome of Quantum Theories”, arXiv:2308.02630, (2023).

[10] Pei Zeng, Jinzhao Sun, Liang Jiang, and Qi Zhao, “Simple and high-precision Hamiltonian simulation by compensating Trotter error with linear combination of unitary operations”, arXiv:2212.04566, (2022).

[11] Oriel Kiss, Michele Grossi, and Alessandro Roggero, “Importance sampling for stochastic quantum simulations”, Quantum 7, 977 (2023).

[12] Lea M. Trenkwalder, Eleanor Scerri, Thomas E. O’Brien, and Vedran Dunjko, “Compilation of product-formula Hamiltonian simulation via reinforcement learning”, arXiv:2311.04285, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-11-14 23:25:35). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-11-14 23:25:33).

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