Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 10E (Quantum Information and Computation)

Let $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. After the Hadamards and the oracle, Bernstein-Vazirani phase kickback gives

The Walsh-Hadamard identityand $H|-⟩=|1⟩$ give

A final oracle call evaluates $f$ on the definite input $a$:Since $a\cdot a$ modulo two is the parity of the Hamming weight of $a$, this form also determines the last-qubit measurement.

The first register of $|{\Phi }_3⟩$ is exactly $|a⟩$, so the probability of observing the string $a$ is

Measuring the first register does not disturb the product-state last register. Its state is

If $a$ has odd Hamming weight, then $a\cdot a=1$ and the final bit is $b$, so the probability is $1$. If $a$ has even Hamming weight, then $a\cdot a=0$ and the final bit is $1\oplus b$, so the probability is $0$. Thus