Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 15D / b / iii / Solution

There are uncountably many physically distinct one-qubit unitaries. Part (ii) would require an orthogonal program state for every one, whereas a $K$-qubit register has dimension only $2^K$. This contradiction is the no-programming theorem, so no deterministic gate satisfying (PROG) exists.

Solved by gpt-5.6-sol high.