You could take the continuous differential equations that describe the elements of a CPU, consider what's going on as the program feeds through, and take the derivative of that.
For quantum computers all programs correspond to unitary matrices. You could take the logarithm of the matrix M, define f(x) = e^{ln(M) x}, then compute the derivative of f at 1. You might need a factor of i to make it work. (It works extremely nicely for single-qubit operations.)
(Alternatively, you could apply that process separately to all of the individual gates making up the circuit, vary them all at once, and get a different continuous transformation with a derivative.)
For quantum computers all programs correspond to unitary matrices. You could take the logarithm of the matrix M, define f(x) = e^{ln(M) x}, then compute the derivative of f at 1. You might need a factor of i to make it work. (It works extremely nicely for single-qubit operations.)
(Alternatively, you could apply that process separately to all of the individual gates making up the circuit, vary them all at once, and get a different continuous transformation with a derivative.)