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Hamiltonian-Informed Point Group Symmetry-Respecting Ansatz for Variational Quantum Eigensolver

Runhong He, Arapat Ablimit, Xin Hong, Qiaozhen Chai, Junyuan Zhou, Ji Guan, Guolong Cui, Shenggang Ying

TL;DR

The paper tackles the challenge of designing compact, symmetry-respecting ansatzes for variational quantum eigensolver (VQE) applications in quantum chemistry. It introduces HiUCCSD, a Hamiltonian-informed pruning method that removes symmetry-violating excitations, with a theoretical guarantee for Abelian point groups and empirical evidence extending to non-Abelian cases. Compared to SymUCCSD, HiUCCSD achieves equivalent performance for Abelian groups and greater robustness for non-Abelian systems, while significantly reducing both parameter counts (18%-83%) and ADAPT-VQE operator pools (27%-84%). These reductions translate into lower quantum resource requirements and broader applicability of VQE to larger, more complex molecules on NISQ devices. Overall, HiUCCSD offers a more robust and scalable symmetry-respecting ansatz for practical quantum chemical simulations.

Abstract

Solving molecular energy levels via the Variational Quantum Eigensolver (VQE) algorithm represents one of the most promising applications for demonstrating practically meaningful quantum advantage in the noisy intermediate-scale quantum (NISQ) era. To strike a balance between ansatz complexity and computational stability in VQE calculations, we propose the HiUCCSD, a novel symmetry-respecting ansatz engineered from the intrinsic information of the Hamiltonian. We theoretically prove the effectiveness of HiUCCSD within the scope of Abelian point groups. Furthermore, we compare the performance of HiUCCSD and the established SymUCCSD via VQE and Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT)-VQE numerical experiments on ten molecules with distinct point groups. The results show that HiUCCSD achieves equivalent performance to SymUCCSD for Abelian point group molecules, while avoiding the potential performance failure of SymUCCSD in the case of non-Abelian point group molecules. Across the studied molecular systems, HiUCCSD cuts the parameter count by 18%-83% for VQE and reduces the excitation operator pool size by 27%-84% for ADAPT-VQE, as compared with the UCCSD ansatz. With enhanced robustness and broader applicability, HiUCCSD offers a new ansatz option for advancing large-scale molecular VQE implementation.

Hamiltonian-Informed Point Group Symmetry-Respecting Ansatz for Variational Quantum Eigensolver

TL;DR

The paper tackles the challenge of designing compact, symmetry-respecting ansatzes for variational quantum eigensolver (VQE) applications in quantum chemistry. It introduces HiUCCSD, a Hamiltonian-informed pruning method that removes symmetry-violating excitations, with a theoretical guarantee for Abelian point groups and empirical evidence extending to non-Abelian cases. Compared to SymUCCSD, HiUCCSD achieves equivalent performance for Abelian groups and greater robustness for non-Abelian systems, while significantly reducing both parameter counts (18%-83%) and ADAPT-VQE operator pools (27%-84%). These reductions translate into lower quantum resource requirements and broader applicability of VQE to larger, more complex molecules on NISQ devices. Overall, HiUCCSD offers a more robust and scalable symmetry-respecting ansatz for practical quantum chemical simulations.

Abstract

Solving molecular energy levels via the Variational Quantum Eigensolver (VQE) algorithm represents one of the most promising applications for demonstrating practically meaningful quantum advantage in the noisy intermediate-scale quantum (NISQ) era. To strike a balance between ansatz complexity and computational stability in VQE calculations, we propose the HiUCCSD, a novel symmetry-respecting ansatz engineered from the intrinsic information of the Hamiltonian. We theoretically prove the effectiveness of HiUCCSD within the scope of Abelian point groups. Furthermore, we compare the performance of HiUCCSD and the established SymUCCSD via VQE and Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT)-VQE numerical experiments on ten molecules with distinct point groups. The results show that HiUCCSD achieves equivalent performance to SymUCCSD for Abelian point group molecules, while avoiding the potential performance failure of SymUCCSD in the case of non-Abelian point group molecules. Across the studied molecular systems, HiUCCSD cuts the parameter count by 18%-83% for VQE and reduces the excitation operator pool size by 27%-84% for ADAPT-VQE, as compared with the UCCSD ansatz. With enhanced robustness and broader applicability, HiUCCSD offers a new ansatz option for advancing large-scale molecular VQE implementation.
Paper Structure (9 sections, 1 theorem, 18 equations, 7 figures, 2 tables)

This paper contains 9 sections, 1 theorem, 18 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. VQE Algorithm
  4. Point Group Symmetry and SymUCCSD
  5. HiUCCSD
  6. Numerical Results
  7. VQE Implementation with Reduced Ansatz
  8. ADAPT-VQE Implementation with Reduced Operator Pools
  9. Conclusion

Key Result

Theorem 1

For any Abelian point group, the electronic integrals in the Hamiltonian will vanish if the associated excitation operators fail to satisfy the molecular point group symmetry requirements.

Figures (7)

  • Figure 1: The symmetry operations of the H$_2$O molecule (which belongs to the $C_{2\text{v}}$ point group) include the $C_{2}$ rotation and the $\sigma_{\text{v}}/\sigma'_{\text{v}}$ reflections.
  • Figure 2: The character table (a) and the irrep direct product table (b) of the $C_{2\text{v}}$ point group for the H$_{2}$O molecule.
  • Figure 3: Isosurfaces of the molecular orbital wavefunctions (isovalues $=$ 0.08 arbitrary units).
  • Figure 4: Electron configurations of the $\mathrm{H_2O}$ molecule: (a) Reference state $|\Psi_{0}\rangle$ (Hartree-Fock state with $\mathrm{A}_1$ irrep); (b) Excited state $\hat{t}_{4,6}^{10,12}|\Psi_{0}\rangle$, which shares the same irrep with the reference state; (c) Excited state $\hat{t}_{6,8}^{10,12}|\Psi_{0}\rangle$ with an irrep distinct from that of the reference state. Here, the irrep of a molecular orbital is labeled in lowercase (e.g., $\mathrm{a_1}$), while that of a quantum state is denoted in uppercase (e.g., $\mathrm{A_1}$).
  • Figure 5: Pseudocode of the HiUCCSD method for constructing symmetry-respecting ansatz.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof