We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then F(f) is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then F(f) is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.

arXiv:2007.10924EntropyFeb 3, 20216 citations

Characterization of exact one-query quantum algorithms

Weijiang Chen, Zekun Ye, Lvzhou Li

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 $f(x)={x}_{{i}_{1}}$ 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 $f$ .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 $f(x)={x}_{{i}_{1}}$ 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 $f$.

arXiv:1912.03493Physical Review AFeb 20, 202014 citations

Parameterized query complexity in quantum computation

Parijat Prashun Purohit

Jan 1, 20170 citations

Characterization of exact one-query quantum algorithms (ii): for partial functions

Zekun Ye, Lvzhou Li

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.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.

arXiv:2008.11998Aug 27, 20201 citation

On Exact Quantum Query Complexity

Ashley Montanaro, Richard Jozsa, Graeme Mitchison

arXiv:1111.0475AlgorithmicaAug 29, 201334 citations

f(x)={x}_{{i}_{1}}

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

f