Cartan Subalgebra Approach to Efficient Measurements of Quantum Observables

Abstract

An arbitrary operator corresponding to a physical observable cannot be measured in a single measurement on currently available quantum hardware. To obtain the expectation value of the observable, one needs to partition its operator into measurable fragments. However, the observable and its fragments generally do not share any eigenstates and thus the number of measurements needed to obtain the expectation value of the observable can grow rapidly even when the wave function prepared is close to an eigenstate of the observable. We provide a unified Lie-algebraic framework for developing efficient measurement schemes for quantum observables. It is based on two elements: (1) embedding the observable operator in a Lie algebra and (2) transforming Lie-algebra elements into those of a Cartan subalgebra (CSA) using unitary operators. The CSA plays the central role because all its elements are mutually commutative and thus can be measured simultaneously. We illustrate the framework on measuring expectation values of Hamiltonians appearing in the variational quantum eigensolver approach to quantum chemistry. The CSA approach puts many recently proposed methods for the measurement optimization within a single framework and allows one not only to reduce the number of measurable fragments but also the total number of measurements.