Towards enhancing quantum expectation estimation of matrices through partial Pauli decomposition techniques and post-processing

We introduce an approach for estimating the expectation values of arbitrary n-qubit matrices M ∈ℂ^2^n× 2^n on a quantum computer. In contrast to conventional methods like the Pauli decomposition that utilize 4^n distinct quantum circuits for this task, our technique employs at most 2^n unique circuits, with even fewer required for matrices with limited bandwidth. Termed the partial Pauli decomposition, our method involves observables formed as the Kronecker product of a single-qubit Pauli operator and orthogonal projections onto the computational basis. By measuring each such observable, one can simultaneously glean information about 2^n distinct entries of M through post-processing of the measurement counts. This reduction in quantum resources is especially crucial in the current noisy intermediate-scale quantum era, offering the potential to accelerate quantum algorithms that rely heavily on expectation estimation, such as the variational quantum eigensolver and the quantum approximate optimization algorithm.