Revisiting Deutsch-Jozsa algorithm

• Published in: Information and computation Vol. 275; p. 104605
• Main Authors: Qiu, Daowen, Zheng, Shenggen
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2020
• Subjects: Deutsch-Jozsa problems; Exact quantum query algorithms; Promise problems; Query complexity; Symmetric Boolean functions; Exact quantum query algorithms; Promise problems; Deutsch-Jozsa problems; Query complexity; Symmetric Boolean functions
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1016/j.ic.2020.104605

•We present all symmetric partial Boolean functions with degree 1 and 2.•We prove the exact quantum query complexity of all symmetric partial Boolean functions with degree 1 and 2.•We prove Deutsch-Jozsa algorithm can compute any symmetric partial Boolean function f with exact quantum 1-query complexity. The Deutsch-Jozsa algorithm is essentially faster than any possible deterministic classical algorithm for solving a promise problem that is in fact a symmetric partial Boolean function, named as the Deutsch-Jozsa problem. The Deutsch-Jozsa problem can be equivalently described as a partial function DJn0:{0,1}n→{0,1} defined as: DJn0(x)=1 for |x|=n/2, DJn0(x)=0 for |x|=0,n, and it is undefined for the remaining cases, where n is even, and |x| is the Hamming weight of x. The Deutsch-Jozsa algorithm needs only one query to compute DJn0 but the classical deterministic algorithm requires n2+1 queries to compute it in the worse case. We present all symmetric partial Boolean functions with degree 1 and 2; We prove the exact quantum query complexity of all symmetric partial Boolean functions with degree 1 and 2. We prove Deutsch-Jozsa algorithm can compute any symmetric partial Boolean function f with exact quantum 1-query complexity.