Discusses, for a product of several operators and in relation to commutativity, the requirements for the establishment of meaningful expectation values. A published example relating to the product of spin operators, where not all pairs commute, is critically examined on the basis of the strict properties of the Pauli operators.
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/4 and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.
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(1), O(n), and O(n) scaling.
The expectation value of the product of two noncommuting observables (POTNO) can be used to test the commutation relation and dig out the correlation between two observables. However, since a POTNO is generally a non-Hermitian operator, its average cannot be measured directly according to standard quantum mechanics. Recently, a theoretical scheme has been proposed to obtain POTNOs, but the observables are limited to projection operators. In this paper, we present a scheme that employs weak measurement to directly measure the expectation value of the POTNO, where the observables are arbitrary operators. Our scheme transforms the measurement of this expectation value into the measurements of several single observable weak values. Experimentally we demonstrate our scheme by measuring the expectation value of the product of two Pauli operators and verify their commutation and anticommutation relations in a trapped-ion system. Moreover, our scheme is applicable to the cases with more than two observables.