On the Quantum Complexity of Classical Words

We show that classical and quantum Kolmogorov complexity of binary words 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 word. 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 some analytical estimates and a classical program for the simulation of a universal quantum computer.