The Second Law of Quantum Complexity

TL;DR

This paper proposes a thermodynamic framework for quantum complexity, establishing a Second Law of Complexity, and links quantum circuit complexity to classical entropy, suggesting uncomplexity as a computational resource and relating it to black hole physics.

Contribution

It introduces a thermodynamics of quantum complexity, connecting quantum circuit complexity with classical entropy and proposing uncomplexity as a resource linked to black hole horizons.

Findings

Quantum complexity growth parallels classical entropy increase.

Uncomplexity can be used as a resource for quantum computation.

A novel interpretation of uncomplexity relates to spacetime volume behind black holes.

Abstract

We give arguments for the existence of a thermodynamics of quantum complexity that includes a "Second Law of Complexity". To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of $K$ qubits, and the positional entropy of a related classical system with $2^K$ degrees of freedom. We also argue that the kinetic entropy of the classical system is equivalent to the Kolmogorov complexity of the quantum Hamiltonian. We observe that the expected pattern of growth of the complexity of the quantum system parallels the growth of entropy of the classical system. We argue that the property of having less-than-maximal complexity (uncomplexity) is a resource that can be expended to perform directed quantum computation. Although this paper is not primarily about black holes, we find a surprising interpretation of the uncomplexity-resource as the accessible volume of spacetime behind a black hole horizon.