Bounds on quantum evolution complexity via lattice cryptography

TL;DR

This paper introduces a new method to estimate the complexity of quantum evolution operators using lattice cryptography, providing practical bounds that distinguish between integrable and chaotic quantum systems.

Contribution

It proposes an upper bound on quantum complexity based on lattice problems, enabling efficient computation and differentiation of quantum system types.

Findings

Upper bounds effectively distinguish integrable from chaotic systems.

Algorithms provide practical estimates for systems up to 10^4 dimensions.

Lattice cryptography techniques are applicable to quantum complexity analysis.

Abstract

We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries. (An appropriate `complexity metric' must be used that takes into account the relative difficulty of performing `nonlocal' operations that act on many degrees of freedom at once.) While simply formulated and geometrically attractive, this notion of complexity is numerically intractable save for toy models with Hilbert spaces of very low dimensions. To bypass this difficulty, we trade the exact definition in terms of geodesics for an upper bound on complexity, obtained by minimizing the distance over an explicitly prescribed infinite set of curves, rather than over all possible curves. Identifying this upper bound turns out equivalent to the closest vector problem (CVP) previously studied in integer optimization theory, in particular, in relation to lattice-based cryptography. Effective approximate algorithms are hence provided by the existing mathematical considerations, and they can be utilized in our analysis of the upper bounds on quantum evolution complexity. The resulting algorithmically implemented complexity bound systematically assigns lower values to integrable than to chaotic systems, as we demonstrate by explicit numerical work for Hilbert spaces of dimensions up to ~10^4.