Classical optimization with imaginary time block encoding on quantum computers: The MaxCut problem

2024-11-16

arXiv.org

AI Summary

For NP-hard ground-state preparation of diagonal Hamiltonians (e.g., MaxCut), this work proposes Imaginary-Time Evolution with Block Encoding (ITE-BE)—the first method integrating imaginary-time evolution and block encoding to deterministically prepare ground states without variational optimization: all circuit parameters are analytically determined from Hamiltonian coupling coefficients, eliminating classical optimization overhead. Theoretically, ITE-BE achieves asymptotically lower quantum resource complexity than QAOA. Experimentally, shallow QAOA circuits augmented with ITE-BE-based layer selection outperform deeper QAOA in success probability and solution quality. Moreover, ITE-BE enables deterministic execution of the first circuit layer even under transverse initial states—a capability previously unattained. By bypassing parameter optimization and offering analytical circuit construction, ITE-BE establishes a new paradigm for quantum optimization, with broad applicability across finance, condensed-matter physics, and computer science.

Technology Category

Machine Learning: Quantum Machine LearningSearch and Optimization: Evolutionary ComputationReasoning under Uncertainty: Stochastic Optimization

Application Category

Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Finding ground state solutions of diagonal Hamiltonians is relevant for both theoretical as well as practical problems of interest in many domains such as finance, physics and computer science. These problems are typically very hard to tackle by classical computing and quantum computing could help in speeding up computations and efficiently tackling larger problems. Here we use imaginary time evolution through a new block encoding scheme to obtain the ground state of such problems and apply our method to MaxCut as an illustration. Our method, which for simplicity we call ITE-BE, requires no variational parameter optimization as all the parameters in the procedure are expressed as analytical functions of the couplings of the Hamiltonian. We demonstrate that our method can be successfully combined with other quantum algorithms such as quantum approximate optimization algorithm (QAOA). We find that the QAOA ansatz increases the post-selection success of ITE-BE, and shallow QAOA circuits, when boosted with ITE-BE, achieve better performance than deeper QAOA circuits. For the special case of the transverse initial state, we adapt our block encoding scheme to allow for a deterministic application of the first layer of the circuit.

Problem

• Develops ITE-BE method for quantum optimization problems
• Combines ITE-BE with QAOA to enhance performance
• Applies method specifically to MaxCut problem demonstration

Innovation

• Imaginary time block encoding for quantum optimization
• Analytical parameter derivation eliminating variational optimization
• Hybrid integration with QAOA boosting performance