Quantum complexity and the virial theorem

TL;DR

This paper explores a novel approach to understanding quantum complexity by applying the virial theorem to the group manifold of unitary operators, linking classical entropy to quantum circuit complexity.

Contribution

It introduces a method to relate Kolmogorov and computational complexity using the virial theorem in the geometric formulation of quantum computing.

Findings

Derived a relation between Kolmogorov and computational complexity at thermal equilibrium.

Proposed a geometric framework connecting classical entropy with quantum circuit complexity.

Abstract

It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computational complexity in the thermal equilibrium.