Categories: Algorithms, Quantum information.

In quantum information, the **Bernstein-Vazirani algorithm** proves the supremacy of quantum computers over classical deterministic or probabilistic computers. It is extremely similar to the Deutsch-Jozsa algorithm, and even uses the same circuit.

It solves a very artificial problem: we are given a “black box” function \(f(x)\) that takes an \(N\)-bit \(x\) and returns a single bit, which we are promised is the lowest bit of the bitwise dot product of \(x\) with an unknown \(N\)-bit string \(s\):

\[\begin{aligned} f(x) = s \cdot x \:\:(\bmod 2) = (s_1 x_1 + s_2 x_2 + \:...\: + s_N x_N) \:\:(\bmod 2) \end{aligned}\]

The goal is to find \(s\). To solve this problem, a classical computer would need to call \(f(x)\) exactly \(N\) times with \(x = 2^n\) for \(n \in \{ 0, ..., N \!-\! 1\}\). However, the Bernstein-Vazirani algorithm allows a quantum computer to do it with only a single query. It uses the following circuit:

Where \(U_f\) is a phase oracle, whose action is defined as follows, where \(\ket{x} = \ket{x_1} \cdots \ket{x_N}\):

\[\begin{aligned} \ket{x} \quad \to \boxed{U_f} \to \quad (-1)^{f(x)} \ket{x} = (-1)^{s \cdot x} \ket{x} \end{aligned}\]

That is, it introduces a phase flip based on the value of \(f(x)\). For an example implementation of such an oracle, see the Deutsch-Jozsa algorithm: its circuit is identical to this one, but describes \(U_f\) in a different (but equivalent) way.

Starting from the state \(\ket{0}^{\otimes N}\), applying the Hadamard gate \(H\) to all qubits yields:

\[\begin{aligned} \ket{0}^{\otimes N} \quad \to \boxed{H^{\otimes N}} \to \quad \ket{+}^{\otimes N} = \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} \ket{x} \end{aligned}\]

This is an equal superposition of all candidates \(\ket{x}\), which we feed to the oracle:

\[\begin{aligned} \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} \ket{x} \quad \to \boxed{U_f} \to \quad \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} (-1)^{s \cdot x} \ket{x} \end{aligned}\]

Then, thanks to the definition of the Hadamard transform, a final set of \(H\)-gates leads us to:

\[\begin{aligned} \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} (-1)^{s \cdot x} \ket{x} \quad \to \boxed{H^{\otimes N}} \to \quad \ket{s} = \ket{s_1} \cdots \ket{s_N} \end{aligned}\]

Which, upon measurement, gives us the desired binary representation of \(s\). For comparison, the Deutsch-Jozsa algorithm only cares whether \(s = 0\) or \(s \neq 0\), whereas this algorithm is interested in the exact value of \(s\).

- J.S. Neergaard-Nielsen,
*Quantum information: lectures notes*, 2021, unpublished. - S. Aaronson,
*Introduction to quantum information science: lecture notes*, 2018, unpublished.

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