Measuring energy by measuring any other observable

TL;DR

This paper introduces a versatile method to estimate quantum observable outcomes and moments by measuring any other observable, enhancing measurement flexibility in quantum systems and aiding in energy estimation in quantum simulators.

Contribution

The paper presents a general approach to estimate quantum observable statistics using arbitrary measurements, including bounds and practical applications in quantum energy estimation.

Findings

Effective estimation of energy in quantum systems using arbitrary measurements.

Demonstrated high exclusion of energy ranges in Heisenberg and Ising models.

Applicable to quantum simulators with measurement constraints.

Abstract

We present a method to estimate the probabilities of outcomes of a quantum observable, its mean value, and higher moments by measuring any other observable. This method is general and can be applied to any quantum system. In the case of estimating the mean energy of an isolated system, the estimate can be further improved by measuring the other observable at different times. Intuitively, this method uses interplay and correlations between the measured observable, the estimated observable, and the state of the system. We provide two bounds: one that is looser but analytically computable and one that is tighter but requires solving a non-convex optimization problem. The method can be used to estimate expectation values and related quantities such as temperature and work in setups where performing measurements in a highly entangled basis is difficult, finding use in state-of-the-art quantum simulators. As a demonstration, we show that in Heisenberg and Ising models of ten sites in the localized phase, performing two-qubit measurements excludes 97.5% and 96.7% of the possible range of energies, respectively, when estimating the ground state energy.