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

TL;DR

This paper confirms that the measurement cost for VQE on molecular Hamiltonians can be reduced to O(N^3) by partitioning terms into commuting groups, and provides an efficient method to create these partitions.

Contribution

It validates empirical reductions in measurement complexity and introduces a fast algorithm for generating commuting partitions using flow networks.

Findings

Confirmed O(N^3) measurement cost for VQE with partitioning.

Developed a pre-computable algorithm for creating commuting partitions.

Linked the problem to a round-robin scheduling problem solved via flow networks.

Abstract

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 $O(N^4)$ terms. Several recent papers observed that this scaling may be reducible to $O(N^3)$ by partitioning 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, pre-computable procedure for creating linearly-sized commuting partitions by solving a round-robin scheduling problem via flow networks.