Unbounded-error quantum query complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve—Wigderson [STOC’98] is optimal even in the unbounded-error setting.We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q+log(1/2(p−1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative Θ(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.     

Nonadaptive quantum query complexity

We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes a total boolean function depending on n variables must make Ω(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using queries to solve the same problem. Thus, in the nonadaptive setting, quantum algorithms for these tasks can achieve at most a very limited speed-up over classical query algorithms.     

The quantum query complexity of the hidden subgroup problem is polynomial

We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log|G|) number of calls to the oracle. This is exponentially better than the best classical algorithm. However our quantum algorithm requires exponential time, as in the classical case. Our algorithm utilizes a new technique for constructing error-free algorithms for non-decision problems on quantum computers.     

A note on the search for k elements via quantum walk

In this paper we use the quantum walk search scheme by Magniez et al. (2007) [13] to find k solutions of a search problem. We show that the quantum query complexity is at most of order times the number of queries to find one solution.     

Self-protected quantum algorithms based on quantum state tomography

Only a few classes of quantum algorithms are known which provide a speed-up over classical algorithms. However, these and any new quantum algorithms provide important motivation for the development of quantum computers. In this article new quantum algorithms are given which are based on quantum state tomography. These include an algorithm for the calculation of several quantum mechanical expectation values and an algorithm for the determination of polynomial factors. These quantum algorithms are important in their own right. However, it is remarkable that these quantum algorithms are immune to a large class of errors. We describe these algorithms and provide conditions for immunity.      

Quantum and Classical Query Complexities of Local Search Are Polynomially Related

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).     

基于连续时间量子游走的链路预测方法研究*

链路预测是复杂网络研究的基础问题之一。目前研究者们已经提出了许多链路预测的方法,其中大量的链路预测方法是基于经典随机游走。量子游走是经典随机游走的量子模拟。大量研究表明,在诸如图匹配、搜索等很多领域,基于量子游走的量子算法的性能远优于其对应的经典随机游走算法。但目前关于基于量子游走的链路预测算法几乎没有研究报道。本文提出了一种基于连续时间量子游走的链路预测方法。实验结果表明,连续时间量子游走链路预测结果的AUC值和经典随机游走的结果非常接近。而在Precision和Recall指标上,远优于经典随机游走的链路预测结果。     

On the uselessness of quantum queries

Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged after the oracle responds to the queries. A familiar example is the parity of a uniformly random Boolean-valued function over {1,2,…,N}, for which N−1 classical queries are useless. We prove that if 2k classical queries are useless for some oracle problem, then k quantum queries are also useless. For such problems, which include classical threshold secret sharing schemes, our result also gives a new way to obtain a lower bound on the quantum query complexity, even in cases where neither the function nor the property to be determined is Boolean.     

Quantum zero-error algorithms cannot be composed

We exhibit two black-box problems, both of which have an efficient quantum algorithm with zero-error, yet whose composition does not have an efficient quantum algorithm with zero-error. This shows that quantum zero-error algorithms cannot be composed. In oracle terms, we give a relativized world where ZQP^ZQP≠ZQP, while classically we always have ZPP^ZPP=ZPP.     

A review of procedures to evolve quantum algorithms

There exist quantum algorithms that are more efficient than their classical counterparts; such algorithms were invented by Shor in 1994 and then Grover in 1996. A lack of invention since Grover’s algorithm has been commonly attributed to the non-intuitive nature of quantum algorithms to the classically trained person. Thus, the idea of using computers to automatically generate quantum algorithms based on an evolutionary model emerged. A limitation of this approach is that quantum computers do not yet exist and quantum simulation on a classical machine has an exponential order overhead. Nevertheless, early research into evolving quantum algorithms has shown promise. This paper provides an introduction into quantum and evolutionary algorithms for the computer scientist not familiar with these fields. The exciting field of using evolutionary algorithms to evolve quantum algorithms is then reviewed.

The Quantum Black-Box Complexity of Majority

   Abstract. We describe a quantum black-box network computing the majority of N bits with zero-sided error ɛ using only

Optimal computation with non-unitary quantum walks

Quantum versions of random walks on the line and the cycle show a quadratic improvement over classical random walks in their spreading rates and mixing times, respectively. Non-unitary quantum walks can provide a useful optimisation of these properties, producing a more uniform distribution on the line, and faster mixing times on the cycle. We investigate the interplay between quantum and random dynamics by comparing the resources required, and examining numerically how the level of quantum correlations varies during the walk. We show numerically that the optimal non-unitary quantum walk proceeds such that the quantum correlations are nearly all removed at the point of the final measurement. This requires only O(logT)$O(logT)$ random bits for a quantum walk of T$T$ steps.     

随机行走的电路分析应用及并行化改进

论述了随机行走算法的基本原理,理论分析了给定允许误差和置信概率下,随机行走算法的结束条件;讨论了随机行走算法在电路分析中的应用,并结合应用实例分析了算法的性能;讨论了算法的时间复杂性和影响算法执行时间的主要因素,重点分析了算法的并行特征,提出了采用并行计算技术提高算法性能的新方法,通过与串行算法的实验比较,表明了并行计算技术是提高随机行走算法执行速度的有效方法,比现有的方法适应性更广。     

基于多权重相似度的随机漫步推荐算法

在传统个性化推荐算法的基础上,提出了一种基于多权重相似度的随机漫步推荐算法。为了解决传统协同过滤算法中忽略了社交网络、热门项目以及共同评分项目之间影响等问题,通过引入万有引力公式计算社交网络中的用户相似度,并对传统协同过滤算法中的相似度进行改进,采用权重因子结合这两者相似度,最后开拓性地结合随机漫步算法进行商品推荐。实验结果表明,提出的算法具有比其他推荐算法更好的推荐性能。     

基于一维三态量子游走的量子聚类算法

量子游走具有与经典随机游走不同的特性,因此它已经被用来解决包括元素区分、组合优化、图同构等问题。考虑量子游走和聚类两个领域,提出了一个基于一维三态离散量子游走的聚类算法。在该算法中,将数据点看作游走粒子;然后,这些粒子执行三态量子游走,接着根据粒子的测量结果更新数据点的属性值;最后,属于同一簇的数据点将会聚集,而属于不同簇的数据点将会分离。仿真实验结果表明了所提算法的有效性。     

Exponential Separation of Quantum and Classical Online Space Complexity

Although quantum algorithms realizing an exponential time speed-up over the best known classical algorithms exist, no quantum algorithm is known performing computation using less space resources than classical algorithms. In this paper, we study, for the first time explicitly, space-bounded quantum algorithms for computational problems where the input is given not as a whole, but bit by bit. We show that there exist such problems that a quantum computer can solve using exponentially less work space than a classical computer. More precisely, we introduce a very natural and simple model of a space-bounded quantum online machine and prove an exponential separation of classical and quantum online space complexity, in the bounded-error setting and for a total language. The language we consider is inspired by a communication problem (the disjointness function) that Buhrman, Cleve and Wigderson used to show an almost quadratic separation of quantum and classical bounded-error communication complexity. We prove that, in the framework of online space complexity, the separation becomes exponential.