Query complexity of unitary operation discrimination

Discrimination of unitary operations is fundamental in quantum computation and information. A lot of quantum algorithms including the well-known Deutsch-Jozsa algorithm, Simon algorithm, and Grover algorithm can essentially be regarded as discriminating among individual, or sets of unitary operations (oracle operators). The problem of discriminating between two unitary operations $U$ and $V$ can be described as: Given $X\in\{U, V\}$, determine which one $X$ is. If $X$ is given with multiple copies, then one can design an adaptive procedure that takes multiple queries to $X$ to output the identification result of $X$. In this paper, we consider the problem: How many queries are required for achieving a desired failure probability $\epsilon$ of discrimination between $U$ and $V$. We prove in a uniform framework: (i) if $U$ and $V$ are discriminated with bound error $\epsilon$ , then the number of queries $T$ must satisfy $T\geq \left\lceil\frac{2\sqrt{1-4\epsilon(1-\epsilon)}}{\Theta (U^\dagger V)}\right\rceil$, and (ii) if they are discriminated with one-sided error $\epsilon$, then there is $T\geq \left\lceil\frac{2\sqrt{1-\epsilon}}{\Theta (U^\dagger V)}\right\rceil$, where $\Theta(W)$ denotes the length of the smallest arc containing all the eigenvalues of $W$ on the unit circle

Deterministic quantum search with adjustable parameters: implementations and applications

Grover's algorithm provides a quadratic speedup over classical algorithms to search for marked elements in an unstructured database. The original algorithm is probabilistic, returning a marked element with bounded error. There are several schemes to achieve the deterministic version, by using the generalized Grover's iteration $G(\alpha,\beta):=S_r(\beta)\, S_o(\alpha)$ composed of phase oracle $S_o(\alpha)$ and phase rotation $S_r(\beta)$. However, in all the existing schemes the value range of $\alpha$ and $\beta$ is limited; for instance, in the three early schemes $\alpha$ and $\beta$ are determined by the proportion of marked states $M/N$. In this paper, we break through this limitation by presenting a search framework with adjustable parameters, which allows $\alpha$ or $\beta$ to be arbitrarily given. The significance of the framework lies not only in the expansion of mathematical form, but also in its application value, as we present two disparate problems which we are able to solve deterministically using the proposed framework, whereas previous schemes are ineffective.Comment: The title and the abstract have been slightly revise

Super-exponential quantum advantage for finding the center of a sphere

This article considers the geometric problem of finding the center of a sphere in vector space over finite fields, given samples of random points on the sphere. We propose a quantum algorithm based on continuous-time quantum walks that needs only a constant number of samples to find the center. We also prove that any classical algorithm for the same task requires approximately as many samples as the dimension of the vector space, by a reduction to an old and basic algebraic result -- Warning's second theorem. Thus, a super-exponential quantum advantage is revealed for the first time for a natural and intuitive geometric problem