Symmetry Dilemmas in Quantum Computing for Chemistry: A Comprehensive Analysis

TL;DR

This paper analyzes the trade-offs between symmetry adaptation, universality, and efficiency in quantum algorithms for chemistry, providing theoretical proofs and numerical simulations to guide the design of operator pools.

Contribution

It offers a comprehensive theoretical and numerical analysis of symmetry dilemmas in quantum computing for chemistry, including proofs of non-universality and practical guidelines for operator pool design.

Findings

Gate-efficient singlet spin-adapted pools are not universal with spatial symmetry.

Symmetry-breaking pools can be used safely under certain conditions.

Enforcing specific symmetries prevents variational collapse.

Abstract

Symmetry adaptation, universality, and gate efficiency are central but often competing requirements in quantum algorithms for electronic structure and many-body physics. For example, fully symmetry-adapted universal operator pools typically generate long and deep quantum circuits, gate-efficient universal operator pools generally break symmetries, and gate-efficient fully symmetry-adapted operator pools may not be universal. In this work, we analyze such symmetry dilemmas both theoretically and numerically. On the theory side, we prove that the popular, gate-efficient operator pool consisting of singlet spin-adapted singles and perfect-pairing doubles is not universal when spatial symmetry is enforced. To demonstrate the strengths and weaknesses of the three types of pools, we perform numerical simulations using an adaptive algorithm paired with operator pools that are (i) fully symmetry-adapted and universal, (ii) fully symmetry-adapted and non-universal, and (iii) breaking a single symmetry and are universal. Our numerical simulations encompass three physically relevant scenarios in which the target state is (i) the global ground state, (ii) the ground state crossed by a state differing in multiple symmetry properties, and (iii) the ground state crossed by a state differing in a single symmetry property. Our results show when symmetry-breaking but universal pools can be used safely, when enforcing at least one distinguishing symmetry suffices, and when a particular symmetry must be rigorously preserved to avoid variational collapse. Together, the formal and numerical analysis provides a practical guide for designing and benchmarking symmetry-adapted operator pools that balance universality, resource requirements, and robust state targeting in quantum simulations for chemistry.