Bernstein-Vazirani Algorithm with A CCNOT-Based Oracle

TL;DR

The paper tackles the Bernstein-Vazirani problem of recovering the secret string $|\underline{\bf \Gamma}\rangle$ from a function $f(\underline{\bf x})=|\underline{\bf x}\rangle \odot |\underline{\bf \Gamma}\rangle$ by introducing CCNOT-based quantum oracles. It defines a CCNOT-based oracle $T_f$ and a change-of-phase oracle $P_f$ that are unitary (and Hermitian) and uses them, together with an ancilla, to produce entanglement so that a single measurement yields $|\underline{\bf \Gamma}\rangle$; this preserves the polynomial speed-up over classical approaches. It also introduces a related oracle $S_f$ to solve a new problem PI, showing a two-step algorithm that recovers $|\underline{\bf \Gamma}\rangle$ from measurements of the top register and notes a link to Deutsch-Jozsa through a similar construction. The work highlights how entanglement and reversible gate-based oracles can extend BV-type algorithms to new problems and to generalizations with other reversible gates.

Abstract

We introduce a quantum algorithm to solve Bernstein-Vazirani problem to recover secret strings, using quantum oracles that are based on the Toffoli (CCNOT) logic gate. As in the known algorithm, the proposed algorithm is a polynomial speed-up algorithm. Moreover, the proposed approach allows us to solve new problems.

Figures (2)

• Figure 1: The circuit of the proposed Algorithm to solve BVP.
• Figure 2: The circuit of the proposed Algorithm (\ref{['eq07']}) to solve the PI problem.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2