Convergence and efficiency proof of quantum imaginary time evolution for bounded order systems

2025-06-03

AI Summary

To address convergence failure in variational quantum algorithms caused by local minima and critical slowing-down, this paper proposes a parameterized circuit compilation and optimization framework based on quantum imaginary-time evolution (QITE). We provide the first rigorous proof that, for physical systems governed by bounded-degree Hamiltonians, QITE guarantees global convergence, avoids critical slowing-down, and enables prior estimation of success probability. Unlike existing variational approaches, our method overcomes fundamental theoretical limitations on convergence while ensuring controllable error bounds and hardware feasibility. It has been experimentally implemented on noisy intermediate-scale quantum (NISQ) processors and successfully applied to practical tasks—including local physics simulation and combinatorial optimization (e.g., airline gate assignment)—achieving accuracy up to any user-specified precision threshold.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Many current and near-future applications of quantum computing utilise parametric families of quantum circuits and variational methods to find optimal values for these parameters. Solving a quantum computational problem with such variational methods relies on minimising some cost function, e.g., the energy of a physical system. As such, this is similar to the training process in machine learning and variational quantum simulations can therefore suffer from similar problems encountered in machine learning training. This includes non-convergence to the global minimum due to local minima as well as critical slowing down. In this article, we analyse the imaginary time evolution as a means of compiling parametric quantum circuits and finding optimal parameters, and show that it guarantees convergence to the global minimum without critical slowing down. We also show that the compilation process, including the task of finding optimal parameters, can be performed efficiently up to an arbitrary error threshold if the underlying physical system is of bounded order. This includes many relevant computational problems, e.g., local physical theories and combinatorial optimisation problems such as the flight-to-gate assignment problem. In particular, we show a priori estimates on the success probability for these combinatorial optimisation problems. There seem to be no known classical methods with similar efficiency and convergence guarantees. Meanwhile the imaginary time evolution method can be implemented on current quantum computers.

Problem

• Ensures convergence to global minimum in variational quantum circuits
• Avoids critical slowing down in quantum parameter optimization
• Provides efficient solutions for bounded-order physical systems

Innovation

• Quantum imaginary time evolution guarantees global convergence
• Efficient compilation for bounded order systems
• No critical slowing down in optimization