Volumes of parent Hamiltonians for benchmarking quantum simulators

TL;DR

This paper analyzes the measure of parent Hamiltonians with a given ground state, providing bounds and numerical estimates to benchmark quantum simulators' performance.

Contribution

It introduces a framework to quantify the volume of parent Hamiltonians with a specific ground state, including local and translationally invariant cases, with numerical estimates for the Ising model.

Findings

Relative volume is zero for exact ground states.

Volume increases with Hilbert space dimension for approximate states.

Numerical estimates for the Ising chain ground state volume.

Abstract

We investigate the relative volume of parent Hamiltonians having a target ground state up to some fixed error $\epsilon$, a quantity which sets a benchmark on the performance of quantum simulators. For vanishing error, this relative volume is of measure zero, whereas for a generic $\epsilon$ we show that it increases with the dimension of the Hilbert space. We also address the volume of parent Hamiltonians when they are restricted to be local. For translationally invariant Hamiltonians, we provide an upper bound to their relative volume. Finally, we estimate numerically the relative volume of parent Hamiltonians when the target state is the ground state of the Ising chain in a transverse field.