Generalized Quantum Deutsch-Jozsa Algorithm

Hao Li, Daowen Qiu, Le Luo

Zhengjun Cao, Jeffrey K. Uhlmann, Lihua Liu

Daowen Qiu, Shenggen Zheng

The Deutsch-Jozsa algorithm is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. It was proposed by Deutsch and Jozsa in 1992 with improvements by Cleve, Ekert, Macchiavello, and Mosca in 1998. The Deutsch-Jozsa problem is a promise problem and we can equivalently describe it as a partial function ${\text{DJ}}_{n}^{0}:{{0,1}}^{n}\ensuremath{\rightarrow}{0,1}$ defined as ${\text{DJ}}_{n}^{0}(x)=1$ for $|x|=n/2, {\text{DJ}}_{n}^{0}(x)=0$ for $|x|=0,n$, and it is undefined for the rest of the cases, where $n$ is even, and $|x|$ is the Hamming weight of $x$. The optimal quantum algorithm needs only one query to compute ${\text{DJ}}_{n}^{0}$ but the classical deterministic algorithm requires ${2}^{n\ensuremath{-}1}+1$ queries to compute it in the worse case. In this article, we generalize the Deutsch-Jozsa problem as ${\text{DJ}}_{n}^{k}(x)=1$ for $|x|=n/2, {\text{DJ}}_{n}^{k}(x)=0$ for $|x|$ in the set ${0,1,...,k,n\ensuremath{-}k,n\ensuremath{-}k+1,...,n}$, and it is undefined for the rest of the cases, where $0\ensuremath{\le}k<n/2$. In particular, we give and prove an optimal exact quantum query algorithm with complexity $k+1$ for computing the generalized Deutsch-Jozsa problem ${\text{DJ}}_{n}^{k}$. It is clear that the case of $k=0$ is in accordance with the Deutsch-Jozsa problem. Also, we give a method for finding the approximate and exact degrees of symmetric partial Boolean functions.

E. S. Guerra

The Deutsch-Jozsa algorithm is a generalization of the Deutsch algorithm which was the first algorithm written. We present schemes to implement the Deutsch algorithm and the Deutsch-Jozsa algorithm via cavity QED.

Elton Ballhysa

Marcin Zwierz, Carlos A. Pérez-Delgado, Pieter Kok

We reveal a close relationship between quantum metrology and the Deutsch-Jozsa algorithm on continuous-variable quantum systems. We develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. Depending on which parameter we keep constant, the procedure implements either the parameter-estimation protocol or the Deutsch-Jozsa algorithm. The parameter-estimation part of the procedure attains the Heisenberg limit and is therefore optimal. Due to the use of approximate normalizable continuous-variable eigenstates, the Deutsch-Jozsa algorithm is probabilistic. The procedure estimates a value of an unknown parameter and solves the Deutsch-Jozsa problem without the use of any entanglement.