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 to 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 wavefunction 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 sub-algebra (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.
Highlights
In digital quantum computing, one prepares a wave function of the simulated quantum system and any property of interest needs to be physically measured to obtain estimates that constitute the result of the computation
Based on these two approaches, we propose the following modifications of the full-rank optimization (FRO) that reduce the total number of measurements: (a) Greedy FRO (GFRO)
We provide a unifying framework for many recently suggested approaches to efficient partitioning of quantum operators into measurable fragments
Summary
One prepares a wave function of the simulated quantum system and any property of interest needs to be physically measured to obtain estimates that constitute the result of the computation. Two main requirements for these fragments are that not a large number of them is required to represent the whole O, and that Un is not difficult to obtain using a classical computer The introduced framework allows us to propose an extension of previous approaches that results in lower numbers of measurements required to accurately obtain an expectation value. This scheme requires efficient partitioning of the operator into unitary components and their implementation in controlled form.
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