Deutsch-Jozsa algorithm as a generalization of Bernstein-Vazirani

Question

It is known that both algorithms use the same gates: $H^{\oplus n}U_fH^{\oplus n}$.

After the circuit, the qubits are in the state $\sum_y \left( \sum_x (-1)^{f(x)+xy} \right) |y\rangle $.

In DJ's algorithm, one measures the amplitude of $|0\rangle^{\oplus n}$. It can only be $0$ or $\pm 1$, and the function is balanced iff it is 0 (for any balanced or constant function).

In Bernstein-Vazirani, one measures every qubit in its $Z$ basis and deduces the bits of the dot-product function.

It seems that one can apply Bernstein-Vazirani to any balanced or constant function (not just a dot-product function) to get a bit vector, and state that the function is constant iff the vector is zero.

What seems strange to me is that all the vector returned by BV for a DJ input is not necessarily all zero or all one, so I don't understand the criterion.

How are the measurements of each qubit and of the state $|0\rangle^{\oplus n}$ related?

Answer 1

Finally, I found the answer myself there.

The only interesting thing is the amplitude of $|0\rangle^{\oplus n}$. If the function is constant, it is $\pm 1$ and if the function is balanced, it is $0$.

Hence, in the first case we are sure to measure all qubits in the $|0\rangle$ state and in the second case, we cannot measure all of them in this state (else we would have the state $|0\rangle^{\oplus n}$ that has amplitude $0$).