Sequential Hamiltonian Assembly: Enhancing the training of combinatorial optimization problems on quantum computers

TL;DR

This work addresses the training challenges of parameterized quantum circuits under global loss by introducing Sequential Hamiltonian Assembly (SHA), which builds the loss from local Hamiltonian components to mitigate vanishing gradients. SHA improves training by iteratively assembling Ĥ = ∑_i Ĥ_i through problem-informed partitions, enabling locality-aware learning and integration with methods like Layer-VQE, LL, and, to a limited extent, QAOA. Empirical results on graph coloring and Max-Cut show substantial gains in mean accuracy and solution quality, with up to 43.89% improvement over standard training on Max-Cut and notable synergy with L-VQE. The findings suggest SHA as a practical approach to address locality-based gradient issues in PQCs, though the benefits with QAOA are not uniform, motivating larger-scale studies and exploration of other combinatorial problems.

Abstract

A central challenge in quantum machine learning is the design and training of parameterized quantum circuits (PQCs). Much like in deep learning, vanishing gradients pose significant obstacles to the trainability of PQCs, arising from various sources. One such source is the presence of non-local loss functions, which require the measurement of a large subset of qubits involved. To address this issue and facilitate parameter training for quantum applications using global loss functions, we propose Sequential Hamiltonian Assembly (SHA). SHA iteratively approximates the loss by assembling it from local components. To further demonstrate the feasibility of our approach, we extend our previous case study by introducing a new partitioning strategy, a new merger between QAOA and SHA, and an evaluation of SHA onto the Max-Cut optimization problem. Simulation results show that SHA outperforms conventional parameter training by 43.89% and the empirical state-of-the-art, Layer-VQE by 29.08% in the mean accuracy for Max-Cut. This paves the way for locality-aware learning techniques, mitigating vanishing gradients for a large class of practically relevant problems.

Figures (6)

• Figure 1: Accuracy over all graphs and circuit architectures per assembly strategy.
• Figure 2: Solution quality averaged over all graphs and circuit architectures for all approaches. Stein_Roshani
• Figure 3: Solution quality averaged over all graphs and circuit architectures when combining different approaches. From Stein_Roshani
• Figure 4: Solution quality averaged over all graphs and circuit architectures when combining QAOA and SHA.
• Figure 5: Number of optimization iterations. From Stein_Roshani