Symmetry Dilemmas in Quantum Computing for Chemistry: A Comprehensive Analysis

TL;DR

This work investigates the balance between universality, symmetry adaptation, and resource efficiency in quantum simulations for chemistry. It proves that enforcing spatial symmetry in the gate-efficient saGSpD pool destroys universality, and conducts ADAPT-VQE studies across three scenarios to derive practical guidelines: (1) universal pools that break symmetry can reach the global ground state but symmetry-preserving pools converge more predictably; (2) for crossings with multiple symmetry differences, enforce at least one non-shared symmetry; (3) for crossings differing in a single symmetry, enforce that symmetry to avoid variational collapse. The results inform the design of symmetry-aware operator pools and benchmark strategies for robust state targeting in molecular simulations, with implications for near-term quantum hardware. The study also highlights that non-universal but fully symmetry-adapted pools can succeed in favorable orbital-symmetry contexts, suggesting hybrid strategies combining symmetry-aware unitaries with projection or penalty techniques in future work.

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.

Figures (15)

• Figure 1: Quantum circuit performing a Givens rotation by an angle $\theta$ in the $\ket{1_p 1_q 0_r 0_s}$--$\ket{0_p 0_q 1_r 1_s}$ subspace of the four qubits labeled $p$, $q$, $r$, and $s$.
• Figure 2: The two lowest-energy potential energy curves for the systems of interest, computed at the FCI/STO-6G level of theory: (a) $D_{\infty\text{h}}$-symmetric dissociation of H6; (b) $C_\text{2v}$-symmetric bending of CH2; (c) $C_\text{s}$-symmetric bending of asymmetrically stretched BeH2; and (d) $C_{\infty \text{v}}$-symmetric dissociation of BO.
• Figure 3: Convergence of ADAPT-VQE/STO-6G simulations using the GSD, saGSpD-full, saGSpD, and pDint0 operator pools for the H6 linear chain at the $R_{\text{H--H}}=2.0\angs$ geometry, shown as functions of the number of parameters: (a) Energy errors relative to the FCI/STO-6G energy of the lowest-energy $\tensor*[^{1}]{\Sigma}{^{+}_{\text{g}}}$ state; (b) expectation values of the total spin squared operator and their standard deviations using the GSD pool; and (c) weight of the totally symmetric component using the saGSpD-full pool.
• Figure 4: Convergence of ADAPT-VQE/STO-6G simulations using the GSD, saGSpD-full, saGSpD, and pDint0 operator pools for the CH2 molecule at the $\theta_{\text{HCH}}= 60^\circ$ and $180^\circ$ geometries, shown as functions of the number of parameters: (a,b) Energy errors relative to the FCI/STO-6G energy of the lowest-energy $\tensor*[^{1}]{A}{_{1}}$ state ($\tensor*[^{1}]{A}{_{\text{g}}}$ for the linear geometry); (c,d) expectation values of the total spin squared operator and their standard deviations using the GSD pool; and (e,f) weight of the totally symmetric component using the saGSpD-full pool.
• Figure 5: Convergence of ADAPT-VQE/STO-6G simulations using the GSD, saGSpD-full, saGSpD, and pDint0 operator pools for the BeH2 molecule at the $\theta_{\text{HBeH}}= 0^\circ$, $50^\circ$, and $180^\circ$ geometries, shown as functions of the number of parameters: (a--c) Energy errors relative to the FCI/STO-6G energy of the lowest-energy $\tensor*[^{1}]{A}{^{\prime}}$ state ($\tensor*[^{1}]{A}{_{1}}$ for the linear geometries) and (d--f) expectation values of the total spin squared operator and their standard deviations using the GSD pool. Note that, in this case, the ADAPT-VQE-saGSpD-full simulations do not break spatial symmetry (see main text).