# Unitary partitioning approach to the measurement problem in the Variational Quantum Eigensolver method

## TL;DR

This paper introduces a unitary partitioning method for the VQE algorithm that reduces the number of measurement groups from N^4 to N, enabling more efficient electronic energy estimation on current quantum hardware.

## Contribution

The authors propose a novel circuit formulation of VQE using unitary partitioning, allowing measurement of fully anti-commuting Hamiltonian terms in fewer steps.

## Key findings

- Achieves N-fold reduction in measurement groups compared to previous methods
- Enables measurement of fully anti-commuting Hamiltonian terms in a single measurement series
- Improves scalability of VQE for larger quantum systems

## Abstract

To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and thus, only parts of the Hamiltonian which form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as $N^4$, where $N$ is the number of qubits, and thus puts serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anti-commuting terms of the Hamiltonian in a single series of single-qubit measurements. Numerical comparison of the unitary partitioning to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity shows an $N$-fold reduction in the number of measurable groups.