[Paper Review] Minimizing State Preparations in Variational Quantum Eigensolver by Partitioning into Commuting Families

Pranav Gokhale, Olivia Angiuli+6 more|arXiv (Cornell University)|Jul 31, 2019

Quantum Computing Algorithms and ArchitectureComputer Science74 references79 citations

TL;DR

The paper reduces the number of state preparations in VQE by partitioning Pauli-string terms into commuting families, via GC-based clique-cover approximations and a linear-time structure-aware partitioning, with experimental validation on IBM Q.

ABSTRACT

Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum machines. VQE aims to approximate the lowest eigenvalue of an exponentially sized matrix in polynomial time. It minimizes quantum resource requirements both by co-processing with a classical processor and by structuring computation into many subproblems. Each quantum subproblem involves a separate state preparation terminated by the measurement of one Pauli string. However, the number of such Pauli strings scales as $N^4$ for typical problems of interest--a daunting growth rate that poses a serious limitation for emerging applications such as quantum computational chemistry. We introduce a systematic technique for minimizing requisite state preparations by exploiting the simultaneous measurability of partitions of commuting Pauli strings. Our work encompasses algorithms for efficiently approximating a MIN-COMMUTING-PARTITION, as well as a synthesis tool for compiling simultaneous measurement circuits. For representative problems, we achieve 8-30x reductions in state preparations, with minimal overhead in measurement circuit cost. We demonstrate experimental validation of our techniques by estimating the ground state energy of deuteron on an IBM Q 20-qubit machine. We also investigate the underlying statistics of simultaneous measurement and devise an adaptive strategy for mitigating harmful covariance terms.

Motivation & Objective

• Reduce quantum state preparation overhead in VQE by grouping commuting Pauli strings that can be measured together.
• Develop efficient, scalable algorithms to approximate MIN-COMMUTING-PARTITION for molecular Hamiltonians.
• Provide a circuit synthesis tool for simultaneous measurement across commuting Pauli strings.
• Analyze the statistics of simultaneous measurement and mitigate harmful covariance terms.
• Validate the approach through benchmarks, simulations, and experiments on quantum hardware.

Proposed method

• Formulate MIN-COMMUTING-PARTITION as partitioning Pauli strings into commuting families to minimize the number of state preparations.
• Extend commutativity to General Commutativity (GC) and contrast with Qubit-Wise Commutativity (QWC).
• Map the problem to MIN-CLIQUE-COVER and apply approximation algorithms (Bron–Kerbosch, Boppana–Halldórsson) and a linear-time, structure-aware partitioning exploiting molecular Hamiltonians.
• Provide a circuit synthesis tool for constructing simultaneous measurement circuits for commuting Pauli strings.
• Analyze measurement statistics and devise strategies to guard against covariance terms that degrade accuracy.

Experimental results

Research questions

• RQ1 What is the minimal number of state preparations needed to estimate the molecular Hamiltonian energy in VQE when exploiting commuting groups of Pauli strings?
• RQ2 How can Pauli-string commutativity (GC vs QWC) be leveraged to reduce measurement overhead without incurring prohibitive classical or quantum costs?
• RQ3 Can we design a linear-time, structure-aware partitioning algorithm tailored to molecular Hamiltonians that outperforms general clique-cover approaches?
• RQ4 What is the impact of simultaneous measurement on measurement circuits, and how can covariance terms be mitigated adaptively?

Key findings

• 8x reduction in partitions for asymptotically dominant O(N4) Hamiltonian terms using JW encoding (and up to 8x for certain term families).
• GC-based partitioning yields denser commutation graphs and smaller clique covers than QWC, enabling more substantial simultaneous-measurement gains.
• Linear-time, structure-aware partitioning exploiting molecular Hamiltonian structure achieves practical overheads and remains below quantum invocation costs.
• A circuit synthesis tool enables efficient simultaneous measurement circuits for commuting Pauli strings.
• Experimental validation on an IBM Q 20-qubit machine demonstrates ground-state energy estimation for the deuteron, supporting practical viability.
• Adaptive strategies are proposed to mitigate harmful covariance terms arising in simultaneous measurements.