Solution
= Solution
There are <uncountable set>[uncountably] many physically distinct one-qubit <unitary operators>[unitaries]. Part (ii) would require an <orthogonal vectors>[orthogonal] program state for every one, whereas a $K$-qubit register has <dimension of a vector space>[dimension] only $2^K$. This contradiction is the <no-programming theorem>, so no deterministic gate satisfying (PROG) exists.