[Paper Review] Building spatial symmetries into parameterized quantum circuits for faster training

Frédéric Sauvage, Martín Larocca+2 more|arXiv (Cornell University)|Jul 28, 2022

Quantum Computing Algorithms and Architecture62 references4 citations

TL;DR

This paper introduces ORB, a parameterized quantum circuit construction that embeds spatial symmetries—specifically, the automorphism group of the problem Hamiltonian—into the parameter structure of variational quantum algorithms. By correlating parameters within the same symmetry orbit, ORB achieves higher expressibility per layer, enabling shorter circuits with fewer parameters and improved trainability compared to both HVA-like and free-parameter ansatzes, particularly in ground state preparation and Max-Cut optimization tasks.

ABSTRACT

Practical success of quantum learning models hinges on having a suitable structure for the parameterized quantum circuit. Such structure is defined both by the types of gates employed and by the correlations of their parameters. While much research has been devoted to devising adequate gate-sets, typically respecting some symmetries of the problem, very little is known about how their parameters should be structured. In this work, we show that an ideal parameter structure naturally emerges when carefully considering spatial symmetries (i.e., the symmetries that are permutations of parts of the system under study). Namely, we consider the automorphism group of the problem Hamiltonian, leading us to develop a circuit construction that is equivariant under this symmetry group. The benefits of our novel circuit structure, called ORB, are numerically probed in several ground-state problems. We find a consistent improvement (in terms of circuit depth, number of parameters required, and gradient magnitudes) compared to literature circuit constructions.

Motivation & Objective

• To address the challenge of balancing expressibility and trainability in parameterized quantum circuits (PQCs) for near-term quantum devices.
• To identify a principled method for structuring gate parameters based on spatial symmetries of the problem Hamiltonian, rather than relying on ad hoc parameter correlations.
• To develop a circuit architecture that combines the flexibility of hardware-efficient ansatzes with the problem-specific structure of HVA-like ansatzes.
• To demonstrate that symmetry-aware parameter design leads to significant improvements in circuit depth, parameter count, and gradient magnitudes during optimization.
• To challenge the conventional use of correlated parameters in QAOA by showing that independent parameters can be more effective when symmetries are properly encoded.

Proposed method

• The ORB ansatz is constructed by identifying the automorphism group of the problem Hamiltonian, which defines spatial symmetries of the system.
• Gate parameters are grouped into orbits under the action of this symmetry group, and parameters within each orbit are made identical, enforcing equivariance of the circuit under the symmetry group.
• This construction ensures that the quantum circuit respects the spatial symmetries of the target state, reducing the search space while preserving relevant physical structure.
• The method leverages group-theoretic tools to systematically identify and enforce parameter correlations based on symmetry orbits, avoiding arbitrary or heuristic parameter sharing.
• The approach is general and can be combined with any gate set, including those used in QAOA or HVA, and can be decoupled from internal symmetries such as SU(2) invariance.
• Numerical benchmarks are performed using ground state preparation tasks on spin chains and Max-Cut problems on random 3-regular graphs, comparing ORB to HVA-like and free-parameter ansatzes.

Experimental results

Research questions

• RQ1 How can spatial symmetries of a problem Hamiltonian be systematically encoded into the parameter structure of a parameterized quantum circuit to improve training efficiency?
• RQ2 Can a symmetry-aware parameter structure lead to a significant reduction in circuit depth and number of parameters while maintaining or improving expressibility?
• RQ3 Does enforcing symmetry via parameter orbit correlation outperform both independent-parameter ansatzes and HVA-like correlated-parameter constructions in terms of trainability and convergence?
• RQ4 In what regimes does the ORB construction provide a clear advantage over standard QAOA parameterization, especially when symmetries are present or absent?
• RQ5 Can the ORB framework be generalized to other quantum machine learning and variational quantum simulation tasks involving spatial or graph symmetries?

Key findings

• ORB circuits achieve up to a quadratic reduction in circuit depth compared to HVA-like ansatzes for the same expressibility, due to a higher number of free parameters per layer.
• The ORB construction packs significantly more independent parameters per layer than HVA-like circuits, while maintaining similar expressibility measured via the Lie algebra dimension.
• For Max-Cut problems on random 3-regular graphs, ORB circuits with independent parameters outperform standard QAOA, suggesting that independent parameterization is preferable when symmetries are properly encoded.
• Gradient magnitudes in ORB circuits are consistently larger than in HVA-like ansatzes, indicating improved trainability at scale.
• The ORB framework naturally recovers the ma-QAOA circuit in the absence of symmetries, validating its consistency with existing approaches.
• The method is scalable and effective even in the regime of large circuit depths (L > 3), where prior works on symmetry-preserving circuits were limited to shallow circuits.