Measurement reduction for expectation values via fine-grained commutativity

TL;DR

The paper introduces a new concept called $k$-commutativity for operators on tensor product spaces, enabling more efficient measurement of expectation values in quantum circuits by reducing the number of measurements needed.

Contribution

It proposes $k$-commutativity, a novel operator commutativity notion, and demonstrates its application in reducing measurement complexity in quantum expectation value estimation.

Findings

Reduction in measurement count at increased circuit depth

Examples of $k$-commutativity scaling as O(1), O(√n), and O(n)

Discussion of asymptotic measurement complexity for various Hamiltonians

Abstract

We introduce a notion of commutativity between operators on a tensor product space, nominally Pauli strings on qubits, that interpolates between qubit-wise commutativity and (full) commutativity. We apply this notion, which we call $k$-commutativity, to measuring expectation values of observables in quantum circuits and show a reduction in the number measurements at the cost of increased circuit depth. Last, we discuss the asymptotic measurement complexity of $k$-commutativity for several families of $n$-qubit Hamiltonians, showing examples with $O(1)$, $O(\sqrt{n})$, and $O(n)$ scaling.