Shot-noise reduction for lattice Hamiltonians

TL;DR

This paper introduces geometric partitioning to reduce measurement costs in estimating energies of lattice Hamiltonians on quantum computers, providing theoretical bounds and demonstrating advantages over traditional methods.

Contribution

It proposes a scalable geometric partitioning method that improves measurement efficiency for lattice Hamiltonians, with proven bounds and applicability to various models.

Findings

Lower bounds on sampling number improvement grow with subsystem size.

Advantage of geometric partitioning increases for more correlated states.

Method effectively reduces measurements in multiple lattice models.

Abstract

Efficiently estimating energy expectation values of lattice Hamiltonians on quantum computers is a serious challenge, where established techniques can require excessive sample numbers. Here we introduce geometric partitioning as a scalable alternative. It splits the Hamiltonian into subsystems that extend over multiple lattice sites, for which transformations between their local eigenbasis and the computational basis can be efficiently found. This allows us to reduce the number of measurements as we sample from a more concentrated distribution without diagonalizing the problem. For systems in an energy eigenstate, we prove a lower bound on the sampling number improvement over the "naive" mutually commuting local operator grouping, which grows with the considered subsystem size, consistently showing an advantage for our geometric partitioning strategy. Notably, our lower bounds do not decrease but increase for more correlated states (Theorem 1). For states that are weakly isotropically perturbed around an eigenstate, we show how the sampling number improvement translates to imperfect eigenstate improvements, namely measuring close to the true eigenbasis already for smaller perturbations (Theorem 2). We illustrate our findings on multiple two-dimensional lattice models incl. the transverse field XY- and Ising model as well as the Fermi Hubbard model.