$O(N^3)$ Measurement Cost for Variational Quantum Eigensolver on Molecular Hamiltonians

Variational quantum eigensolver (VQE) is a promising algorithm for near-term quantum machines. It can be used to estimate the ground state energy of a molecule by performing separate measurements of <inline-formula><tex-math notation="LaTeX">$O(N^4)$</tex-math></inline-formula> terms. This quartic scaling appears to be a significant obstacle to practical applications. However, we note that it empirically reduces to <inline-formula><tex-math notation="LaTeX">$O(N^3)$</tex-math></inline-formula> when we partition the terms into linear-sized commuting families that can be measured simultaneously. We confirm these empirical observations by studying the MIN-COMMUTING-PARTITION problem at the level of the fermionic Hamiltonian and its encoding into qubits. Moreover, we provide a fast, precomputable procedure for creating linearly sized commuting partitions by solving a round-robin scheduling problem via flow networks. In addition, we demonstrate how to construct the quantum circuits necessary for simultaneous measurement, and we discuss the statistical implication of simultaneous measurement. Our results are experimentally validated by a ground state estimation of deuteron with low shot budget on a 20-qubit IBM machine.

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