Better product formulas for quantum phase estimation

TL;DR

This paper improves the understanding of error bounds in quantum energy estimation using product formulas, introducing a new 9-term second-order formula with significantly reduced error and potential for quadratic speedup.

Contribution

It generalizes error analysis for product formulas in quantum energy estimation and introduces a novel 9-term second-order formula with better accuracy.

Findings

A 9-term second-order product formula with quadratically improved error.

Tighter bounds for energy estimation error in low-energy subspace.

Potential quadratic speedup for Hamiltonians with locality and positivity.

Abstract

Quantum phase estimation requires simulating the evolution of the Hamiltonian, for which product formulas are attractive due to their smaller qubit cost and ease of implementation. However, the estimation of the error incurred by product formulas is usually pessimistic and task-agnostic, which poses problems for assessing their performance in practice for problems of interest. In this work, we study the error of product formulas for the specific task of quantum energy estimation. To this end, we employ the theory of Trotter error with a Magnus-based expansion of the effectively simulated Hamiltonian. The result is a generalization of previous energy estimation error analysis of gapped eigenstates to arbitrary order product formulas. As an application, we discover a 9-term second-order product formula with an energy estimation error that is quadratically better than Trotter-Suzuki. Furthermore, by leveraging recent work on low-energy dynamics of product formulas, we provide tighter bounds for energy estimation error in the low-energy subspace. We show that for Hamiltonians with some locality and positivity properties, the cost can achieve up to a quadratic asymptotic speedup in terms of the target error.