Quantum circuits computing unitary transformations

Let $H_1, H_2$ be Hilbert spaces of the same finite dimension $ge2$, and $C$ an arbitrary quantum circuit with (principal) input state in $H_1$ and (principal) output state in $H_2$. $C$ may use ancillas and produce garbage which is traced out. $C$ may employ classical channels and measurement gates. If $C$ computes, for each computation path $mu$ through the circuit, a unitary transformation $U_mu: H_1 to H_2$ then, for each $mu$, the probability that a computation takes path $mu$ is independent of the input.