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

TL;DR

This paper proves that quantum imaginary time evolution converges to the global minimum efficiently for bounded order systems, offering a promising method for quantum circuit optimization and combinatorial problems with strong convergence guarantees.

Contribution

It provides the first convergence and efficiency proof for quantum imaginary time evolution applied to bounded order systems, ensuring global convergence without critical slowing down.

Findings

Guarantees convergence to the global minimum

Achieves efficient compilation up to arbitrary error thresholds

Provides success probability estimates for combinatorial problems

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.