Bernstein–Vazirani algorithm returns the secret exactly

Given: $n\ge 1$, $a\in \{0,1{\}}^n$, and the total function $f_a(x)=a\cdot x$.

One-query Bernstein–Vazirani algorithm: the algorithm prepares $\mathrm{\mid }+{⟩}^{\otimes n}\mathrm{\mid }-⟩$, invokes the standard XOR query once, applies $H^{\otimes n}$ to the address, then measures and outputs it.

Bernstein–Vazirani no-offset promise and Binary dot product as parity: $f_a(x)=a\cdot x$ is the bitwise parity of the products $a_i{x}_i$, with no additive offset.

One standard query implements one phase query: one query with target $\mathrm{\mid }-⟩$ maps each address basis state to $(-1{)}^{f_a(x)}\mathrm{\mid }x⟩\mathrm{\mid }-⟩$.

Hadamard character sum on F_2^n: for $z\in \{0,1{\}}^n$, ${\sum }_x(-1{)}^{x\cdot z}=2^n$ when $z=0^n$ and $0$ otherwise.

Normalized output law for finite quantum query branches: terminal output probabilities sum the diagonal state weights over the unmeasured target register and form a normalized finite law.

A unit-vector outer product is a density operator: the rank-one projector of a unit vector is a density operator.

Ordered tensor registers, their inner product, and wire permutations: the address and target product basis retains the declared factor order and product inner product.

Nonoracle quantum gate cost: each declared nonoracle gate occurrence, including preparation, is counted once; oracle calls and terminal measurement are separate resources under the algorithm's gate model.