Exponential Separation Criteria for Quantum Iterative Power Algorithms

TL;DR

This paper critically examines the claimed exponential speedup of Quantum Iterative Power Algorithms (QIPA) over varQITE, proving that such speedups are theoretically unachievable under certain conditions, but polynomial improvements remain feasible in practice.

Contribution

The study introduces criteria for exponential separation between QIPA and varQITE, proves these separations are unachievable in theory, and demonstrates practical polynomial improvements through preprocessing.

Findings

Exponential separation criteria require specific problem properties.

Preprocessing can enforce these properties, leading to exponential error growth.

Polynomial enhancements are still practically achievable, as shown by experiments.

Abstract

In the vast field of Quantum Optimization, Quantum Iterative Power Algorithms (QIPA) has been introduced recently with a promise of exponential speedup over an already established and well-known method, the variational Quantum Imaginary Time Evolution (varQITE) algorithm. Since the convergence and error of varQITE are known, the promise of QIPA also implied certain collapses in the complexity hierarchy - such as NP $\subseteq$ BQP, as we show in our study. However the original article of QIPA explicitly states the algorithm does not cause any collapses. In this study we prove that these collapses indeed do not occur, and with that, prove that the promised exponential separation is practically unachievable. We do so by introducing criteria for the exponential separation between QIPA that uses a double exponential function and varQITE, and then showing how these criteria require certain properties in problem instances. After that we introduce a preprocessing step that enforces problems to satisfy these criteria, and then we show that the algorithmic error blows up exponentially for these instances, as there is an inverse polynomial term between speedup and the error. Despite the theoretical results, we also show that practically relevant polynomial enhancement is still possible, and show experimental results on a small problem instance, where we used our preprocessing step to achieve the improvement.