On the Quantum Kolmogorov Complexity of Classical Strings

TL;DR

This paper demonstrates that classical and quantum Kolmogorov complexities of binary strings are essentially equivalent up to a constant, establishing quantum complexity as a natural extension of classical complexity even with probabilistic errors.

Contribution

It proves the equivalence of classical and quantum Kolmogorov complexities for binary strings, extending classical complexity to quantum states with a rigorous mathematical foundation.

Findings

Classical and quantum complexities differ by at most a constant.

Quantum complexity extends classical complexity to quantum states.

Equivalence holds even with small probabilistic errors.

Abstract

We show that classical and quantum Kolmogorov complexity of binary strings agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the corresponding string. It follows that quantum complexity is an extension of classical complexity to the domain of quantum states. This is true even if we allow a small probabilistic error in the quantum computer's output. We outline a mathematical proof of this statement, based on an inequality for outputs of quantum operations and a classical program for the simulation of a universal quantum computer.