Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation

Gumaro Rendon1, Jacob Watkins2, and Nathan Wiebe3,4

1Zapata Computing Inc., Boston, MA 02110, USA
2Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA
3Department of Computer Science, University of Toronto, Toronto, ON M5S 2E4, Canada
4Pacific Northwest National Laboratory, Richland, WA 99352, USA

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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.

Quantum computers have the potential to enhance our understanding of chemistry, materials, nuclear physics, and other scientific disciplines through improved quantum simulation. There are several available quantum algorithms for this task, and among these, Trotter formulas are often preferred due to their simplicity and low up-front costs. Unfortunately, Trotter formulas are, in theory, relatively inaccurate compared to their newer and more sophisticated competitors. Though more computational time may help, this strategy become quickly unmanageable on the noisy quantum devices of today, with limited ability to perform long, uninterrupted calculations.

To mitigate errors in Trotter simulations without increasing the quantum processing time, we use polynomials to learn the relationship between error and step size. By collecting data for different choices of step size, we can interpolate, i.e. thread, the data with a polynomial, then estimate the expected behavior for very small step sizes. We prove mathematically that our approach yields asymptotic accuracy improvements over standard Trotter for two fundamental tasks: estimating eigenvalues and estimating expectation values.

Our method is simple and practical, requiring only standard techniques in quantum and classical computation. We believe our work provides a strong theoretical foothold for further investigations of algorithmic error mitigation. Extensions of this work could occur in several directions, from eliminating artificial assumptions in our analysis to demonstrating improved quantum simulations.

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[1] S. Lloyd, Universal quantum simulators, Science 273 (1996) 1073.

[2] M. Reiher, N. Wiebe, K.M. Svore, D. Wecker and M. Troyer, Elucidating reaction mechanisms on quantum computers, Proceedings of the National Academy of Sciences 114 (2017) 7555.

[3] J.D. Whitfield, J. Biamonte and A. Aspuru-Guzik, Simulation of electronic structure hamiltonians using quantum computers, Molecular Physics 109 (2011) 735.

[4] J. Lee, D.W. Berry, C. Gidney, W.J. Huggins, J.R. McClean, N. Wiebe et al., Even more efficient quantum computations of chemistry through tensor hypercontraction, PRX Quantum 2 (2021) 030305.

[5] V. von Burg, G.H. Low, T. Häner, D.S. Steiger, M. Reiher, M. Roetteler et al., Quantum computing enhanced computational catalysis, Physical Review Research 3 (2021) 033055.

[6] S.P. Jordan, K.S. Lee and J. Preskill, Quantum algorithms for quantum field theories, Science 336 (2012) 1130.

[7] A.F. Shaw, P. Lougovski, J.R. Stryker and N. Wiebe, Quantum algorithms for simulating the lattice schwinger model, Quantum 4 (2020) 306.

[8] N. Klco, M.J. Savage and J.R. Stryker, Su (2) non-abelian gauge field theory in one dimension on digital quantum computers, Physical Review D 101 (2020) 074512.

[9] A.M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Info. Comput. 12 (2012) 901–924.

[10] G.H. Low, V. Kliuchnikov and N. Wiebe, Well-conditioned multiproduct hamiltonian simulation, arXiv:1907.11679 (2019).

[11] D.W. Berry, A.M. Childs, R. Cleve, R. Kothari and R.D. Somma, Simulating hamiltonian dynamics with a truncated taylor series, Physical review letters 114 (2015) 090502.

[12] G.H. Low and N. Wiebe, Hamiltonian simulation in the interaction picture, arXiv:1805.00675 (2018).

[13] M. Kieferová, A. Scherer and D.W. Berry, Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series, Physical Review A 99 (2019) 042314.

[14] G.H. Low and I.L. Chuang, Hamiltonian Simulation by Qubitization, Quantum 3 (2019) 163.

[15] R. Babbush, C. Gidney, D.W. Berry, N. Wiebe, J. McClean, A. Paler et al., Encoding electronic spectra in quantum circuits with linear t complexity, Physical Review X 8 (2018) 041015.

[16] D.W. Berry, G. Ahokas, R. Cleve and B.C. Sanders, Efficient quantum algorithms for simulating sparse hamiltonians, Communications in Mathematical Physics 270 (2006) 359–371.

[17] N. Wiebe, D.W. Berry, P. Høyer and B.C. Sanders, Simulating quantum dynamics on a quantum computer, Journal of Physics A: Mathematical and Theoretical 44 (2011) 445308.

[18] A.M. Childs, Y. Su, M.C. Tran, N. Wiebe and S. Zhu, Theory of trotter error with commutator scaling, Physical Review X 11 (2021) 011020.

[19] J. Haah, M.B. Hastings, R. Kothari and G.H. Low, Quantum algorithm for simulating real time evolution of lattice hamiltonians, SIAM Journal on Computing (2021) FOCS18.

[20] M. Hagan and N. Wiebe, Composite quantum simulations, arXiv:2206.06409 (2022).

[21] G.H. Low, Y. Su, Y. Tong and M.C. Tran, On the complexity of implementing trotter steps, arXiv:2211.09133 (2022).

[22] G.H. Low and I.L. Chuang, Optimal hamiltonian simulation by quantum signal processing, Physical Review Letters 118 (2017).

[23] S. Endo, Q. Zhao, Y. Li, S. Benjamin and X. Yuan, Mitigating algorithmic errors in a hamiltonian simulation, Phys. Rev. A 99 (2019) 012334.

[24] A.C. Vazquez, R. Hiptmair and S. Woerner, Enhancing the quantum linear systems algorithm using richardson extrapolation, ACM Transactions on Quantum Computing 3 (2022).

[25] A.C. Vazquez, D.J. Egger, D. Ochsner and S. Woerner, Well-conditioned multi-product formulas for hardware-friendly hamiltonian simulation, Quantum 7 (2023) 1067.

[26] M. Suzuki, General theory of fractal path integrals with applications to many‐body theories and statistical physics, Journal of Mathematical Physics 32 (1991) 400.

[27] A. Gilyén, Y. Su, G.H. Low and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pp. 193–204, 2019, DOI.

[28] C. Yi and E. Crosson, Spectral analysis of product formulas for quantum simulation, npj Quantum Information 8 (2022) 37.

[29] A. Quarteroni, R. Sacco and F. Saleri, Numerical mathematics, vol. 37, Springer Science & Business Media (2010), 10.1007/​b98885.

[30] F. Piazzon and M. Vianello, Stability inequalities for lebesgue constants via markov-like inequalities, Dolomites Research Notes on Approximation 11 (2018).

[31] A.P. de Camargo, On the numerical stability of newton’s formula for lagrange interpolation, Journal of Computational and Applied Mathematics 365 (2020) 112369.

[32] L. Trefethen, Six myths of polynomial interpolation and quadrature, (2011).

[33] W. Gautschi, How (un)stable are vandermonde systems? asymptotic and computational analysis, in Lecture Notes in Pure and Applied Mathematics, pp. 193–210, Marcel Dekker, Inc, 1990.

[34] N.J. Higham, The numerical stability of barycentric lagrange interpolation, IMA Journal of Numerical Analysis 24 (2004) 547.

[35] J.C. Mason and D.C. Handscomb, Chebyshev polynomials, CRC press (2002), 10.1201/​9781420036114.

[36] G. Rendon, T. Izubuchi and Y. Kikuchi, Effects of cosine tapering window on quantum phase estimation, Physical Review D 106 (2022) 034503.

[37] L.N. Trefethen, Approximation Theory and Approximation Practice, Extended Edition, SIAM (2019), 10.1137/​1.9781611975949.

[38] F.L. Bauer and C.T. Fike, Norms and exclusion theorems, Numer. Math. 2 (1960) 137–141.

[39] S. Blanes, F. Casas, J.-A. Oteo and J. Ros, The magnus expansion and some of its applications, Physics reports 470 (2009) 151.

[40] N. Klco and M.J. Savage, Minimally entangled state preparation of localized wave functions on quantum computers, Physical Review A 102 (2020).

[41] J.J. García-Ripoll, Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations, Quantum 5 (2021) 431.

[42] W. Górecki, R. Demkowicz-Dobrzański, H.M. Wiseman and D.W. Berry, $\pi$-corrected heisenberg limit, Physical review letters 124 (2020) 030501.

[43] D. Grinko, J. Gacon, C. Zoufal and S. Woerner, Iterative quantum amplitude estimation, npj Quantum Information 7 (2021) 52 [1912.05559].

[44] N. Wiebe, D. Berry, P. Høyer and B.C. Sanders, Higher order decompositions of ordered operator exponentials, Journal of Physics A: Mathematical and Theoretical 43 (2010) 065203.

[45] R.A. Horn and C.R. Johnson, Matrix analysis, Cambridge university press (2012), 10.1017/​CBO9780511810817.

[46] M. Chiani, D. Dardari and M.K. Simon, New exponential bounds and approximations for the computation of error probability in fading channels, IEEE Transactions on Wireless Communications 2 (2003) 840.

[47] J.M. Borwein and P.B. Borwein, Pi and the AGM: a study in the analytic number theory and computational complexity, Wiley-Interscience (1987).

[48] B.L. Higgins, D.W. Berry, S.D. Bartlett, H.M. Wiseman and G.J. Pryde, Entanglement-free Heisenberg-limited phase estimation, Nature 450 (2007) 393.

[49] R.B. Griffiths and C.-S. Niu, Semiclassical Fourier Transform for Quantum Computation, Physical Review Letters 76 (1996) 3228.

[50] A.Y. Kitaev, Quantum measurements and the abelian stabilizer problem, quant-ph/​9511026 (1995).

[51] D.S. Abrams and S. Lloyd, Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors, Physical Review Letters 83 (1999) 5162.

[52] J. Watkins, N. Wiebe, A. Roggero and D. Lee, Time-dependent hamiltonian simulation using discrete clock constructions, arXiv:2203.11353 (2022).

[53] T.D. Ahle, Sharp and simple bounds for the raw moments of the binomial and poisson distributions, Statistics & Probability Letters 182 (2022) 109306.

[54] T. Rivlin, Chebyshev Polynomials, Dover Books on Mathematics, Dover Publications (2020).

Cited by

[1] Dean Lee, "Quantum techniques for eigenvalue problems", European Physical Journal A 59 11, 275 (2023).

[2] Tatsuhiko N. Ikeda, Hideki Kono, and Keisuke Fujii, "Trotter24: A precision-guaranteed adaptive stepsize Trotterization for Hamiltonian simulations", arXiv:2307.05406, (2023).

[3] Hans Hon Sang Chan, Richard Meister, Matthew L. Goh, and Bálint Koczor, "Algorithmic Shadow Spectroscopy", arXiv:2212.11036, (2022).

[4] Sergiy Zhuk, Niall Robertson, and Sergey Bravyi, "Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation", arXiv:2306.12569, (2023).

[5] Zhicheng Zhang, Qisheng Wang, and Mingsheng Ying, "Parallel Quantum Algorithm for Hamiltonian Simulation", Quantum 8, 1228 (2024).

[6] 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).

[7] Gumaro Rendon and Peter D. Johnson, "Low-depth Gaussian State Energy Estimation", arXiv:2309.16790, (2023).

[8] Gregory Boyd, "Low-Overhead Parallelisation of LCU via Commuting Operators", arXiv:2312.00696, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-04-12 03:18:30). The list may be incomplete as not all publishers provide suitable and complete citation data.

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