Tailoring Term Truncations for Electronic Structure Calculations Using a Linear Combination of Unitaries

Richard Meister1, Simon C. Benjamin1, and Earl T. Campbell2,3

1Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
2Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
3AWS Center for Quantum Computing, Pasadena, CA 91125, USA

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Abstract

A highly anticipated use of quantum computers is the simulation of complex quantum systems including molecules and other many-body systems. One promising method involves directly applying a linear combination of unitaries (LCU) to approximate a Taylor series by truncating after some order. Here we present an adaptation of that method, optimized for Hamiltonians with terms of widely varying magnitude, as is commonly the case in electronic structure calculations. We show that it is more efficient to apply LCU using a truncation that retains larger magnitude terms as determined by an iterative procedure. We obtain bounds on the simulation error for this generalized truncated Taylor method, and for a range of molecular simulations, we report these bounds as well as exact numerical results. We find that our adaptive method can typically improve the simulation accuracy by an order of magnitude, for a given circuit depth.

Simulating the time evolution of complex quantum systems is of interest for many applications in physics. Quantum computers promise to offer the ability to calculate such dynamics much more efficiently than classical computers. One approach to achieve this is to approximate the time evolution operator with a truncated Taylor series, where all terms above a certain order are discarded. As sums of operators are not native instructions on quantum hardware, this requires an elaborate construction known as linear combination of unitaries. In this paper, we modify the well-known method of the truncated Taylor series to use a different truncation scheme. Instead of dropping all terms above some order in the expansion, our approach retains them depending on their magnitude. We show that this method of truncating is advantageous for many Hamiltonians describing the electronic structures of molecules, typically reducing the simulation error by roughly one order of magnitude compared to the canonical method for a given circuit depth.

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[4] Earl T. Campbell, "Early fault-tolerant simulations of the Hubbard model", arXiv:2012.09238.

[5] Sam McArdle, Earl Campbell, and Yuan Su, "Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian", Physical Review A 105 1, 012403 (2022).

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