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 -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 . 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.