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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Abstract

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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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).

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[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).

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