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

Pranav Gokhale, Frederic T. Chong|arXiv (Cornell University)|Aug 30, 2019

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proves that the measurement cost for Variational Quantum Eigensolver (VQE) on molecular Hamiltonians encoded via Jordan-Wigner can be reduced from $O(N^4)$ to $O(N^3)$ by partitioning Pauli terms into commuting families of size $O(N)$. The method uses a constructive proof based on Baranyai's theorem and flow networks to generate these partitions in $O(N^5 \log N)$ time, enabling pre-computation independent of molecular coefficients.

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.

Motivation & Objective

• To resolve the high measurement cost of VQE on molecular Hamiltonians, which scales as $O(N^4)$ due to the need to measure each Pauli string separately.
• To confirm empirical observations that $O(N^3)$ measurement cost is achievable by grouping commuting Pauli strings for simultaneous measurement.
• To develop an efficient, pre-computable method for generating such commuting partitions that depends only on $N$, not on molecular-specific coefficients.
• To provide a constructive, scalable solution with low classical runtime to avoid undermining quantum advantage.

Proposed method

• Uses Baranyai's theorem to construct a partition of all 4-qubit Pauli terms into $O(N^3)$ families, each of size $O(N)$, where terms in each family commute.
• Models the partitioning problem as a round-robin scheduling task, encoded as a flow network with $O(N^3)$ vertices and $O(N^4)$ edges.
• Applies flow-rounding techniques to compute an integral maximum flow in $O(N^4 \log N)$ time per iteration, reducing the total runtime to $O(N^5 \log N)$.
• Constructs the partition via $N$ iterations of flow network resolution, each yielding a disjoint set of $N/4$ size-4 subsets of indices.
• The resulting commuting families allow simultaneous measurement using a shared eigenbasis, requiring only $O(N)$ quantum gates.
• The method is pre-computable: the partitioning depends only on $N$, not on the specific $h_{pq}$ or $h_{pqrs}$ coefficients of the molecular Hamiltonian.

Experimental results

Research questions

• RQ1 Can the $O(N^4)$ measurement cost of VQE on molecular Hamiltonians be rigorously reduced to $O(N^3)$ using commuting partitions?
• RQ2 Is there a constructive, efficient method to generate such partitions that is independent of molecular-specific parameters?
• RQ3 Can the classical cost of partitioning be kept low enough to preserve quantum advantage in near-term VQE implementations?
• RQ4 Does the use of Baranyai’s theorem and flow networks enable scalable, pre-computable partitioning for arbitrary $N$-qubit molecular systems?

Key findings

• The measurement cost of VQE on Jordan-Wigner encoded molecular Hamiltonians is provably reducible from $O(N^4)$ to $O(N^3)$ by grouping Pauli terms into commuting families.
• Each commuting family contains $O(N)$ terms, enabling simultaneous measurement and reducing the number of required measurement rounds to $O(N^3)$.
• The partitioning is constructed in $O(N^5 \log N)$ time using a flow network-based implementation of Baranyai’s theorem, making it pre-computable for any $N$.
• The method depends only on $N$, not on the specific $h_{pq}$ or $h_{pqrs}$ coefficients, so a single partitioning can be reused across all $N$-qubit molecular Hamiltonians.
• The simultaneous measurement circuits require only $O(N)$ quantum gates, as the shared eigenbasis is a tensor product over 4-qubit blocks.
• The approach enables asymptotic reduction in measurement cost and may improve other quantum chemistry tasks like Hamiltonian simulation via optimized Trotterization.