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Multi-product Hamiltonian simulation with explicit commutator scaling

Junaid Aftab, Dong An, Konstantina Trivisa

TL;DR

A rigorous complexity analysis of the well-conditioned MPF is conducted, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time, and presenting several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision.

Abstract

The well-conditioned multi-product formula (MPF), proposed by [Low, Kliuchnikov, and Wiebe, 2019], is a simple high-order time-independent Hamiltonian simulation algorithm that implements a linear combination of standard product formulas of low order. While the MPF aims to simultaneously exploit commutator scaling among Hamiltonians and achieve near-optimal time and precision dependence, its lack of a rigorous error bound on the nested commutators renders its practical advantage ambiguous. In this work, we conduct a rigorous complexity analysis of the well-conditioned MPF, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time. Using our improved complexity analysis, we present several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision, compared to second-order and even higher-order product formulas. Compared to post-Trotter methods, the MPF based on a second-order product formula can achieve polynomially better scaling in system size, with only poly-logarithmic overhead in evolution time and precision.

Multi-product Hamiltonian simulation with explicit commutator scaling

TL;DR

A rigorous complexity analysis of the well-conditioned MPF is conducted, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time, and presenting several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision.

Abstract

The well-conditioned multi-product formula (MPF), proposed by [Low, Kliuchnikov, and Wiebe, 2019], is a simple high-order time-independent Hamiltonian simulation algorithm that implements a linear combination of standard product formulas of low order. While the MPF aims to simultaneously exploit commutator scaling among Hamiltonians and achieve near-optimal time and precision dependence, its lack of a rigorous error bound on the nested commutators renders its practical advantage ambiguous. In this work, we conduct a rigorous complexity analysis of the well-conditioned MPF, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time. Using our improved complexity analysis, we present several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision, compared to second-order and even higher-order product formulas. Compared to post-Trotter methods, the MPF based on a second-order product formula can achieve polynomially better scaling in system size, with only poly-logarithmic overhead in evolution time and precision.
Paper Structure (27 sections, 10 theorems, 120 equations, 1 table)

This paper contains 27 sections, 10 theorems, 120 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications and comparison
  4. Sketch of the proof
  5. MPF based on high-order product formula
  6. Related works
  7. Discussions and open questions
  8. Organization
  9. Preliminaries
  10. Notation
  11. Product formulas
  12. Multi-product formulas (MPF)
  13. Quantum implementation of MPF
  14. MPF based on second-order product formula
  15. BCH formula and its convergence
  16. ...and 12 more sections

Key Result

Lemma 3

LowKliuchnikovWiebe2019 Time evolution operator of a time-independent Hamiltonian simulation problem can be approximated by the $2$nd-based MPF with error at most $\epsilon$ and probability at least $1 - \Omega(\epsilon)$, using $O(T\lambda \log^2(T\lambda/\epsilon))$ controlled-$U_2$ queries, where

Theorems & Definitions (16)

  • Lemma 3
  • Lemma 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • ...and 6 more