An Algorithmic Construction of Quantum Circuits of High Descriptive Complexity

We discuss an algorithmic construction which, for any finite but universal set of computable quantum gates and a given measurement basis, will produce a rational quantum circuit whose shortest ϵ-approximations from products of instances of the gates have sizes which grow at least exponentially in the input sizes of the circuits and logarithmically in the reciprocal of ϵ. We also discuss the constructive content of the Solovay-Kitaev theorem by considering the algorithmic enumeration of all quantum circuits of a given input size.