We introduce a notion of commutativity between operators on a tensor product space, nominally Pauli strings on qubits, that interpolates between qubitwise 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. We discuss the asymptotic measurement complexity of $k$-commutativity for several families of $n$-qubit Hamiltonians, showing examples with $O\left(1\right)$, $O\left(\sqrt{n}\right)$, and $O\left(n\right)$ scaling.

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• Quantum circuits
• Quantum information processing
• Quantum simulation