Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation

Rendon, Gumaro; Watkins, Jacob; Wiebe, Nathan

doi:10.22331/q-2024-02-26-1266

Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation

Journal Article · Mon Feb 26 00:00:00 EST 2024 · Quantum

DOI:https://doi.org/10.22331/q-2024-02-26-1266· OSTI ID:2311917

Rendon, Gumaro ^[1]; Watkins, Jacob ^[2]; Wiebe, Nathan ^[3]

Zapata Computing Inc., Boston, MA (United States)
Michigan State Univ., East Lansing, MI (United States)
Univ. of Toronto, ON (Canada); Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Quantum metrology allows for measuring properties of a quantum system at the optimal Heisenberg limit. However, when the relevant quantum states are prepared using digital Hamiltonian simulation, the accrued algorithmic errors will cause deviations from this fundamental limit. In this work, we show how algorithmic errors due to Trotterized time evolution can be mitigated through the use of standard polynomial interpolation techniques. Our approach is to extrapolate to zero Trotter step size, akin to zero-noise extrapolation techniques for mitigating hardware errors. We perform a rigorous error analysis of the interpolation approach for estimating eigenvalues and time-evolved expectation values, and show that the Heisenberg limit is achieved up to polylogarithmic factors in the error. Our work suggests that accuracies approaching those of state-of-the-art simulation algorithms may be achieved using Trotter and classical resources alone for a number of relevant algorithmic tasks.

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Research Organization:: Brookhaven National Laboratory (BNL), Upton, NY (United States); Michigan State Univ., East Lansing, MI (United States)

Sponsoring Organization:: USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Nuclear Physics (NP)

Grant/Contract Number:: SC0012704; SC0023658

OSTI ID:: 2311917

Alternate ID(s):: OSTI ID: 3006233

Journal Information:: Quantum, Journal Name: Quantum Vol. 8; ISSN 2521-327X

Publisher:: Quantum Science Open CommunityCopyright Statement

Country of Publication:: United States

Language:: English

References (46)

Norms and exclusion theorems Bauer, F. L.; Fike, C. T. Numerische Mathematik, Vol. 2, Issue 1 https://doi.org/10.1007/BF01386217	journal	December 1960
Numerical Mathematics Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto Texts in Applied Mathematics https://doi.org/10.1007/b98885	book	January 2007
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard Communications in Mathematical Physics, Vol. 270, Issue 2 https://doi.org/10.1007/s00220-006-0150-x	journal	December 2006
On the numerical stability of Newton’s formula for Lagrange interpolation de Camargo, André Pierro Journal of Computational and Applied Mathematics, Vol. 365 https://doi.org/10.1016/j.cam.2019.112369	journal	February 2020
The Magnus expansion and some of its applications Blanes, S.; Casas, F.; Oteo, J. A. Physics Reports, Vol. 470, Issue 5-6 https://doi.org/10.1016/j.physrep.2008.11.001	journal	January 2009
Sharp and simple bounds for the raw moments of the binomial and Poisson distributions Ahle, Thomas D. Statistics & Probability Letters, Vol. 182 https://doi.org/10.1016/j.spl.2021.109306	journal	March 2022
Matrix analysis Horn, Roger A.; Johnson, Charles R. https://doi.org/10.1017/CBO9780511810817	book	January 1985
Entanglement-free Heisenberg-limited phase estimation Higgins, B. L.; Berry, D. W.; Bartlett, S. D. Nature, Vol. 450, Issue 7168 https://doi.org/10.1038/nature06257	journal	November 2007
Iterative quantum amplitude estimation Grinko, Dmitry; Gacon, Julien; Zoufal, Christa npj Quantum Information, Vol. 7, Issue 1 https://doi.org/10.1038/s41534-021-00379-1	journal	March 2021
Spectral analysis of product formulas for quantum simulation Yi, Changhao; Crosson, Elizabeth npj Quantum Information, Vol. 8, Issue 1 https://doi.org/10.1038/s41534-022-00548-w	journal	April 2022
General theory of fractal path integrals with applications to many‐body theories and statistical physics Suzuki, Masuo Journal of Mathematical Physics, Vol. 32, Issue 2 https://doi.org/10.1063/1.529425	journal	February 1991
Simulation of electronic structure Hamiltonians using quantum computers Whitfield, James D.; Biamonte, Jacob; Aspuru-Guzik, Alán Molecular Physics, Vol. 109, Issue 5 https://doi.org/10.1080/00268976.2011.552441	journal	March 2011
Higher order decompositions of ordered operator exponentials Wiebe, Nathan; Berry, Dominic; Høyer, Peter Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 6 https://doi.org/10.1088/1751-8113/43/6/065203	journal	January 2010
Simulating quantum dynamics on a quantum computer Wiebe, Nathan; Berry, Dominic W.; Høyer, Peter Journal of Physics A: Mathematical and Theoretical, Vol. 44, Issue 44 https://doi.org/10.1088/1751-8113/44/44/445308	journal	October 2011
The numerical stability of barycentric Lagrange interpolation Higham, N. J. IMA Journal of Numerical Analysis, Vol. 24, Issue 4 https://doi.org/10.1093/imanum/24.4.547	journal	October 2004
Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction Lee, Joonho; Berry, Dominic W.; Gidney, Craig PRX Quantum, Vol. 2, Issue 3 https://doi.org/10.1103/PRXQuantum.2.030305	journal	July 2021
Complexity of Implementing Trotter Steps Low, Guang Hao; Su, Yuan; Tong, Yu PRX Quantum, Vol. 4, Issue 2 https://doi.org/10.1103/PRXQuantum.4.020323	journal	May 2023
Mitigating algorithmic errors in a Hamiltonian simulation Endo, Suguru; Zhao, Qi; Li, Ying Physical Review A, Vol. 99, Issue 1 https://doi.org/10.1103/PhysRevA.99.012334	journal	January 2019
Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series Kieferová, Mária; Scherer, Artur; Berry, Dominic W. Physical Review A, Vol. 99, Issue 4 https://doi.org/10.1103/PhysRevA.99.042314	journal	April 2019
SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers Klco, Natalie; Savage, Martin J.; Stryker, Jesse R. Physical Review D, Vol. 101, Issue 7 https://doi.org/10.1103/PhysRevD.101.074512	journal	April 2020
Effects of cosine tapering window on quantum phase estimation Rendon, Gumaro; Izubuchi, Taku; Kikuchi, Yuta Physical Review D, Vol. 106, Issue 3 https://doi.org/10.1103/PhysRevD.106.034503	journal	August 2022
Simulating Hamiltonian Dynamics with a Truncated Taylor Series Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard Physical Review Letters, Vol. 114, Issue 9 https://doi.org/10.1103/PhysRevLett.114.090502	journal	March 2015
π -Corrected Heisenberg Limit Górecki, Wojciech; Demkowicz-Dobrzański, Rafał; Wiseman, Howard M. Physical Review Letters, Vol. 124, Issue 3 https://doi.org/10.1103/PhysRevLett.124.030501	journal	January 2020
Semiclassical Fourier Transform for Quantum Computation Griffiths, Robert B.; Niu, Chi-Sheng Physical Review Letters, Vol. 76, Issue 17 https://doi.org/10.1103/PhysRevLett.76.3228	journal	April 1996
Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors Abrams, Daniel S.; Lloyd, Seth Physical Review Letters, Vol. 83, Issue 24 https://doi.org/10.1103/PhysRevLett.83.5162	journal	December 1999
Quantum computing enhanced computational catalysis von Burg, Vera; Low, Guang Hao; Häner, Thomas Physical Review Research, Vol. 3, Issue 3 https://doi.org/10.1103/PhysRevResearch.3.033055	journal	July 2021
Theory of Trotter Error with Commutator Scaling Childs, Andrew M.; Su, Yuan; Tran, Minh C. Physical Review X, Vol. 11, Issue 1 https://doi.org/10.1103/PhysRevX.11.011020	journal	February 2021
Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity Babbush, Ryan; Gidney, Craig; Berry, Dominic W. Physical Review X, Vol. 8, Issue 4 https://doi.org/10.1103/PhysRevX.8.041015	journal	October 2018
Minimally entangled state preparation of localized wave functions on quantum computers Klco, Natalie; Savage, Martin J. Physical Review A, Vol. 102, Issue 1 https://doi.org/10.1103/physreva.102.012612	journal	July 2020
Optimal Hamiltonian Simulation by Quantum Signal Processing Low, Guang Hao; Chuang, Isaac L. Physical Review Letters, Vol. 118, Issue 1 https://doi.org/10.1103/physrevlett.118.010501	journal	January 2017
New exponential bounds and approximations for the computation of error probability in fading channels Chiani, M.; Dardari, D.; Simon, M. K. IEEE Transactions on Wireless Communications, Vol. 24, Issue 5 https://doi.org/10.1109/TWC.2003.814350	journal	May 2003
Quantum Algorithms for Quantum Field Theories Jordan, S. P.; Lee, K. S. M.; Preskill, J. Science, Vol. 336, Issue 6085 https://doi.org/10.1126/science.1217069	journal	May 2012
Universal Quantum Simulators Lloyd, S. Science, Vol. 273, Issue 5278 https://doi.org/10.1126/science.273.5278.1073	journal	August 1996
Approximation Theory and Approximation Practice, Extended Edition Trefethen, Lloyd N. https://doi.org/10.1137/1.9781611975949	book	January 2019
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics Gilyén, András; Su, Yuan; Low, Guang Hao Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019 https://doi.org/10.1145/3313276.3316366	conference	January 2019
Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation Vazquez, Almudena Carrera; Hiptmair, Ralf; Woerner, Stefan ACM Transactions on Quantum Computing, Vol. 3, Issue 1 https://doi.org/10.1145/3490631	journal	March 2022
Hamiltonian Simulation by Qubitization Low, Guang Hao; Chuang, Isaac L. Quantum, Vol. 3 https://doi.org/10.22331/q-2019-07-12-163	journal	July 2019
Quantum Algorithms for Simulating the Lattice Schwinger Model Shaw, Alexander F.; Lougovski, Pavel; Stryker, Jesse R. Quantum, Vol. 4 https://doi.org/10.22331/q-2020-08-10-306	journal	August 2020
Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations García-Ripoll, Juan José Quantum, Vol. 5 https://doi.org/10.22331/q-2021-04-15-431	journal	April 2021
Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation Carrera Vazquez, Almudena; Egger, Daniel J.; Ochsner, David Quantum, Vol. 7 https://doi.org/10.22331/q-2023-07-25-1067	journal	July 2023
Composite Quantum Simulations Hagan, Matthew; Wiebe, Nathan Quantum, Vol. 7 https://doi.org/10.22331/q-2023-11-14-1181	journal	November 2023
Hamiltonian simulation using linear combinations of unitary operations Childs, Andrew M.; Wiebe, Nathan Quantum Information and Computation, Vol. 12, Issue 11&12 https://doi.org/10.26421/QIC12.11-12-1	journal	November 2012
Hamiltonian Simulation in the Interaction Picture Low, Guang Hao; Wiebe, Nathan arXiv https://doi.org/10.48550/arXiv.1805.00675	preprint	January 2018
Well-conditioned multiproduct Hamiltonian simulation Low, Guang Hao; Kliuchnikov, Vadym; Wiebe, Nathan arXiv https://doi.org/10.48550/arXiv.1907.11679	preprint	January 2019
Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions Watkins, Jacob; Wiebe, Nathan; Roggero, Alessandro arXiv https://doi.org/10.48550/arXiv.2203.11353	preprint	January 2022
Quantum measurements and the Abelian Stabilizer Problem Kitaev, A. Yu. arXiv https://doi.org/10.48550/arXiv.quant-ph/9511026	preprint	January 1995

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