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Operator Commutativity Screening and Progressive Operator Block Reordering toward Many-body Inspired Quantum State Preparation

Dibyendu Mondal, Debaarjun Mukherjee, Rahul Maitra

TL;DR

The paper addresses the challenge of performing accurate molecular energies with VQE on NISQ devices by designing a compact yet expressive dynamic ansatz. It introduces COMPASS-PRO, which screens dominant operator blocks via commutativity and energy criteria, augments blocks with non-commuting scatterers to capture higher-order correlations, and progressively reorders blocks through local micro-cycles and global macro-cycles, yielding a disentangled ansatz $\ket{\psi(\theta)}=\prod_s e^{\tau_s(\theta_s)} \prod_\alpha U_\alpha\ket{\phi_{HF}}$ with $U_\alpha=[\prod_{\mu}\overline{e^{\sigma_{\mu}(\theta_{\mu})}e^{\tau_I(\theta_I)}}]_{\alpha}$. Across BeH$_2$, H$_2$O, and BH, COMPASS-PRO achieves chemical accuracy with far fewer parameters than UCCSD/UCCSDT and ADAPT-VQE, and remains robust in bond-dissociation regimes where other methods stall due to gradient troughs or local minima. The approach thereby reduces quantum-resource requirements while maintaining high accuracy, offering a practical path for reliable molecular simulations on NISQ devices and a foundation for extensions to excited states and qubit-based depth reductions with error mitigation.

Abstract

In the field of quantum chemistry, the variational quantum eigensolver (VQE) has emerged as a highly promising approach to determine molecular energies and properties within the noisy intermediate-scale quantum (NISQ) era. The central challenges of this approach lie in the design of an expressive ansatz capable of representing the exact ground state wavefunction while concurrently being efficient to avoid numerical instabilities during the classical optimization. Owing to the constraints of current quantum hardware, the ansatz must remain sufficiently compact while retaining the flexibility to capture essential correlation effects. To address these challenges, we propose a systematic dynamic ansatz construction strategy in which the dominant operator blocks are initially identified through commutativity screening, combined with an energy sorting criteria. Subsequently, the ansatz is progressively expanded in a stepwise manner via iterative operator block reordering. To minimize the overhead, the higher order correlation terms are incorporated via reduced lower-body tensor factorization in each operator block, while the adaptive construction strategy ensures that the optimization is guided along the optimal trajectory to mitigate potential numerical instabilities due to the presence of local traps. Benchmark applications to various molecular systems demonstrate that this strategy of progressive operator-block addition achieves accurate energetics with significantly fewer parameters while efficiently bypassing local traps. Moreover, in strongly correlated regions, such as bond dissociation, the method successfully reproduces the ground state, where other contemporary approaches often fail.

Operator Commutativity Screening and Progressive Operator Block Reordering toward Many-body Inspired Quantum State Preparation

TL;DR

The paper addresses the challenge of performing accurate molecular energies with VQE on NISQ devices by designing a compact yet expressive dynamic ansatz. It introduces COMPASS-PRO, which screens dominant operator blocks via commutativity and energy criteria, augments blocks with non-commuting scatterers to capture higher-order correlations, and progressively reorders blocks through local micro-cycles and global macro-cycles, yielding a disentangled ansatz with . Across BeH, HO, and BH, COMPASS-PRO achieves chemical accuracy with far fewer parameters than UCCSD/UCCSDT and ADAPT-VQE, and remains robust in bond-dissociation regimes where other methods stall due to gradient troughs or local minima. The approach thereby reduces quantum-resource requirements while maintaining high accuracy, offering a practical path for reliable molecular simulations on NISQ devices and a foundation for extensions to excited states and qubit-based depth reductions with error mitigation.

Abstract

In the field of quantum chemistry, the variational quantum eigensolver (VQE) has emerged as a highly promising approach to determine molecular energies and properties within the noisy intermediate-scale quantum (NISQ) era. The central challenges of this approach lie in the design of an expressive ansatz capable of representing the exact ground state wavefunction while concurrently being efficient to avoid numerical instabilities during the classical optimization. Owing to the constraints of current quantum hardware, the ansatz must remain sufficiently compact while retaining the flexibility to capture essential correlation effects. To address these challenges, we propose a systematic dynamic ansatz construction strategy in which the dominant operator blocks are initially identified through commutativity screening, combined with an energy sorting criteria. Subsequently, the ansatz is progressively expanded in a stepwise manner via iterative operator block reordering. To minimize the overhead, the higher order correlation terms are incorporated via reduced lower-body tensor factorization in each operator block, while the adaptive construction strategy ensures that the optimization is guided along the optimal trajectory to mitigate potential numerical instabilities due to the presence of local traps. Benchmark applications to various molecular systems demonstrate that this strategy of progressive operator-block addition achieves accurate energetics with significantly fewer parameters while efficiently bypassing local traps. Moreover, in strongly correlated regions, such as bond dissociation, the method successfully reproduces the ground state, where other contemporary approaches often fail.
Paper Structure (15 sections, 7 equations, 4 figures)

This paper contains 15 sections, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Choice of Operators in UCC Ansatz
  4. Construction of Operator Block Snippets and a Road-map toward a Dynamic Ansatz Design
  5. COMPASS with Progressive block Re-Ordering (COMPASS-PRO)
  6. Compression of Two-body Operator Block through Removal of Redundant Pathways:
  7. Progressive Ansatz Construction via Operator Block Reordering and Step-wise Optimization
  8. Results
  9. General Numerical Considerations
  10. Molecular Potential Energy Profile and Parameter Count:
  11. Optimization Landscape and Distribution of Local Minima:
  12. Progressive Ansatz Construction and Optimization with Random Initialization: The Burrowing Mechanism
  13. Adaptive Operator Block Augmentation vs Gradient Based Ansatz Construction: a Comparative Study of Energy Accuracy and Wavefunction Overlap
  14. Conclusions and Future Outlooks
  15. Acknowledgment

Figures (4)

  • Figure 1: Accuracy as a function of the bond length parameter for UCCSD, UCCSDT, COMPASS, ADAPT-VQE (SD) and COMPASS-PRO is plotted with respect to FCI: (a) linear $BeH_2$, (b) $H_2O$, and (c) $BH$. The shaded region indicates chemical accuracy. Sub-figures (d), (e) and (f) estimate parameter counts for $BeH_2$, $H_2O$ and $BH$, respectively.
  • Figure 2: Energy error across successive macro-cycles is shown for COMPASS-PRO (warm), COMPASS-PRO (HF), and COMPASS (stepwise) ansatze. The dashed lines represent the optimized energy error achieved at each intermediate ansatz length for COMPASS-PRO (warm) and COMPASS (stepwise), where the parameters are initialized from random starting values at the global VQE macro-cycle.
  • Figure 3: Energy accuracy with respect to FCI for 10 independent runs is plotted at each macro-cycle of COMPASS-PRO for two distinct geometries of linear $BeH_2$. In each macro-cycle, the global VQE optimization is performed, initialized from a randomly chosen set of parameter values. While the random initialization may occasionally trap the intermediate ansatz in local minima, the incorporation of the most stabilizing operator block in the subsequent cycle deepens these minima and guides the optimization along a quasi-optimal pathway toward the global minimum.
  • Figure 4: In (a), (b) and (c) accuracy with respect to FCI is depicted for three geometries of linear $H_4$ at each macro-cycle of COMPASS-PRO, ADAPT-VQE (SD) and ADAPT-VQE (gSD). In (a), (b) and (c) the horizontal brown and yellow dotted-dashed lines indicate the energy difference with FCI GS for several low-lying FCI ES. The yellow color is for first $^1A_{1g}$ ES. Plot (d), (e) and (f) show the corresponding overlap convergence of COMPASS-PRO, ADAPT-VQE (SD) and ADAPT-VQE (gSD) with $^1A_{1g}$ ground state (GS) and first $^1A_{1g}$ excited state (ES)..