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What can exact one-query quantum algorithms do?

Weijiang Chen, Zekun Ye, Lvzhou li

Published Dec 7, 2019• 0 citations
OpenAlex

Abstract

Quantum query models are one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including Deutsch-Jozsa algorithm, Simon algorithm, Grover algorithm and so on. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ can be exactly computed by a one-query quantum algorithm if and only if $f(x)=x_{i_1}$ or $f(x)=x_{i_1} \oplus x_{i_2} $ (up to isomorphism). Note that unlike most work in the literature based on the polynomial method, our proof does not resort to any knowledge about the polynomial degree of $f$.

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Physical Review A•Jun 20, 2018•31 citations
Quantum query models are one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including Deutsch-Jozsa algorithm, Simon algorithm, Grover algorithm and so on. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function f:{0,1}n→{0,1}f:\{0,1\}^n \rightarrow \{0,1\}f:{0,1}n→{0,1} can be exactly computed by a one-query quantum algorithm if and only if f(x)=xi1f(x)=x_{i_1}f(x)=xi1​​ or f(x)=xi1⊕xi2f(x)=x_{i_1} \oplus x_{i_2} f(x)=xi1​​⊕xi2​​ (up to isomorphism). Note that unlike most work in the literature based on the polynomial method, our proof does not resort to any knowledge about the polynomial degree of fff.
The quantum query model is one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including the Deutsch-Jozsa algorithm, the Simon algorithm, the Grover algorithm, and others. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function $f:{{0,1}}^{n}\ensuremath{\rightarrow}{0,1}$ can be exactly computed by a one-query quantum algorithm if and only if f(x)=xi1f(x)={x}_{{i}_{1}}f(x)=xi1​​ or ${x}_{{i}_{1}}\ensuremath{\bigoplus}{x}_{{i}_{2}}$ (up to isomorphism). Note that, unlike most work in the literature based on the polynomial method, our proof does not resort to any knowledge about the polynomial degree of fff.
The query model (or black-box model) has attracted much attention from the communities of both classical and quantum computing. Usually, quantum advantages are revealed by presenting a quantum algorithm that has a better query complexity than its classical counterpart. For example, the well-known quantum algorithms including Deutsch-Jozsa algorithm, Simon algorithm and Grover algorithm all show a considerable advantage of quantum computing from the viewpoint of query complexity. Recently we have considered in (Phys. Rev. A. {\bf 101}, 02232 (2020)) the problem: what functions can be computed by an exact one-query quantum algorithm? This problem has been addressed for total Boolean functions but still open for partial Boolean functions. Thus, in this paper we continue to characterize the computational power of exact one-query quantum algorithms for partial Boolean functions by giving several necessary and sufficient conditions. By these conditions, we construct some new functions that can be computed exactly by one-query quantum algorithms but have essential difference from the already known ones. Note that before our work, the known functions that can be computed by exact one-query quantum algorithms are all symmetric functions, whereas the ones constructed in this papers are generally asymmetric.
Early in 1992, Deutsch-Jozsa algorithm computed a symmetric partial Boolean function with a single quantum query, and thus achieved the best separation between classical deterministic and exact quantum query complexity. Recently, it was clarified that all symmetric partial Boolean functions with a single quantum query can be computed exactly by Deutsch-Jozsa algorithm. For the general partial Boolean functions with a single quantum query, the latest characterizations is complex and not very satisfactory. Based on this, this paper proves and discovers three new results: (1) Under a new equivalence, each partial Boolean function with a single quantum query can be transformed to a simple partial Boolean function whose polynomial degree is just one; (2) For partial Boolean functions up to four bits, there are only 10 non-trivial partial Boolean functions with a single quantum query; (3) For each quantum 1-query algorithm with undefined measurement, there exists a constructive method for finding out all partial Boolean functions that can be computed exactly by the algorithm.
We give and prove an optimal exact quantum query algorithm with complexity k+1k+1k+1 for computing the promise problem (i.e., symmetric and partial Boolean function) DJnkDJ_n^kDJnk​ defined as: DJnk(x)=1DJ_n^k(x)=1DJnk​(x)=1 for ∣x∣=n/2|x|=n/2∣x∣=n/2, DJnk(x)=0DJ_n^k(x)=0DJnk​(x)=0 for ∣x∣|x|∣x∣ in the set {0,1,…,k,n−k,n−k+1,…,n}\{0, 1,\ldots, k, n-k, n-k+1,\ldots,n\}{0,1,…,k,n−k,n−k+1,…,n}, and it is undefined for the rest cases, where nnn is even, ∣x∣|x|∣x∣ is the Hamming weight of xxx. The case of k=0k=0k=0 is the well-known Deutsch-Jozsa problem. We outline all symmetric (and partial) Boolean functions with degrees 1 and 2, and prove their exact quantum query complexity. Then we prove that any symmetrical (and partial) Boolean function fff has exact quantum 1-query complexity if and only if fff can be computed by the Deutsch-Jozsa algorithm. We also discover the optimal exact quantum 2-query complexity for distinguishing between inputs of Hamming weight {⌊n/2⌋,⌈n/2⌉}\{ \lfloor n/2\rfloor, \lceil n/2\rceil \}{⌊n/2⌋,⌈n/2⌉} and Hamming weight in the set {0,n}\{ 0, n\}{0,n} for all odd nnn. In addition, a method is provided to determine the degree of any symmetrical (and partial) Boolean function.
Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.
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 DJn0(x)=1{\text{DJ}}_{n}^{0}(x)=1DJn0​(x)=1 for ∣x∣=n/2,DJn0(x)=0|x|=n/2, {\text{DJ}}_{n}^{0}(x)=0∣x∣=n/2,DJn0​(x)=0 for ∣x∣=0,n|x|=0,n∣x∣=0,n, and it is undefined for the rest of the cases, where nnn is even, and ∣x∣|x|∣x∣ is the Hamming weight of xxx. The optimal quantum algorithm needs only one query to compute DJn0{\text{DJ}}_{n}^{0}DJn0​ 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 DJnk(x)=1{\text{DJ}}_{n}^{k}(x)=1DJnk​(x)=1 for ∣x∣=n/2,DJnk(x)=0|x|=n/2, {\text{DJ}}_{n}^{k}(x)=0∣x∣=n/2,DJnk​(x)=0 for ∣x∣|x|∣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+1k+1k+1 for computing the generalized Deutsch-Jozsa problem DJnk{\text{DJ}}_{n}^{k}DJnk​. It is clear that the case of k=0k=0k=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.