# Measurement Optimization in the Variational Quantum Eigensolver Using a Minimum Clique Cover

## TL;DR

This paper introduces a method to optimize measurement in the Variational Quantum Eigensolver by using a minimum clique cover approach to group Hamiltonian terms, significantly reducing measurement complexity.

## Contribution

It formulates the measurement grouping problem as a minimum clique cover problem and tests heuristic algorithms to improve measurement efficiency in VQE.

## Key findings

- Grouping reduces measurement operators by a factor of three on average.
- Qubit-wise commutativity can be represented as a graph for optimization.
- Heuristic algorithms effectively approximate the minimum clique cover.

## Abstract

Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves measurement of the Hamiltonian expectation value. Current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as $O(N^4)$ with the size of the system, $N$. To optimize VQE measurement one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms that commute within corresponding single-qubit subspaces or qubit-wise commuting. We found that qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent of finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times compared to the total number of terms in the considered Hamiltonians.