Mario Szegedy
We show that in the quantum query model the complexity of detecting a triangle in an undirected graph on $n$ nodes can be done using $O(n^{1+{3\over 7}}\log^{2}n)$ quantum queries. The same complexity bound applies for outputting the triangle if there is any. This improves upon the earlier bound of $O(n^{1+{1\over 2}})$.
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Quantum Query Lower Bounds for Triangle-Listing and Spanners
本文通过扩展量子查询记录框架，证明了三角形列举和乘法生成子图构造的量子查询下界，首次给出三角形列举的非平凡量子下界，并匹配围长猜想。
We show that the quantum query complexity of detecting if an $n$-vertex graph contains a triangle is $O(n^{9/7})$. This improves the previous best algorithm of Belovs making $O(n^{35/27})$ queries. For the problem of determining if an operation $\circ : S \times S \rightarrow S$ is associative, we give an algorithm making $O(|S|^{10/7})$ queries, the first improvement to the trivial $O(|S|^{3/2})$ application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be di…
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work by M.O.Renou, et al., in [1], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In [1], the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios is yet to be…
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the $k$-distinctness problems. * We prove tight lower bounds for the $k$-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and s…
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and spare instances.For dense graphs on $n$ vertices, we get a query complexity of $O(n^{5/4})$ without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with $m\geq n^{5/4}$ edges, we get a query complexity of $O(n^{11/12}m^{1/6}\sqrt{\log n})$, which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when $m \geq n^{3/2}$. We also obtain an algorithm with query complexity ${O}(n^{5/6}(…
We present two new quantum algorithms that either find a triangle (a copy of $K_{3}$) in an undirected graph $G$ on $n$ nodes, or reject if $G$ is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes $\tilde{O}(n^{10/7})$ queries. The second algorithm uses $\tilde{O}(n^{13/10})$ queries, and it is based on a design concept of Ambainis~\cite{amb04} that incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The first algorithm uses only $O(\log n)$ qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Pr…
We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $ n $-vertex graph contains a triangle, clique or star of some size. For a general subgraph $ H $ with $ k $ vertices, we show that $ H $ containment can be solved with quantum query complexity $ O(n^{2-\frac{2}{k}-g(H)}) $, with $ g(H) $ a strictly positive function of $ H $. This is better than $ \tilde{O}\s{n^{2-2/k}} $ by Magniez et al. These results are obtained in the learning graph model of Belovs.
Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the t…
In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where $n$ denotes the number of vertices in the graph. This improves the previous upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires $Ω(n^{9/7}/\sqrt{\log n})…
We show that an improvement to the best known quantum lower bound for GRAPH-COLLISION problem implies an improvement to the best known lower bound for TRIANGLE problem in the quantum query complexity model. In GRAPH-COLLISION we are given free access to a graph $(V,E)$ and access to a function $f:V\rightarrow \{0,1\}$ as a black box. We are asked to determine if there exist $(u,v) \in E$, such that $f(u)=f(v)=1$. In TRIANGLE we have a black box access to an adjacency matrix of a graph and we have to determine if the graph contains a triangle. For both of these problems the known lower bounds a…
This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the recent $\tilde O(n^{5/4})$-query algorithm given by Le Gall [FOCS 2014] for triangle finding over dense graphs (here $n$ denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with $O(n^{5/4-ε})$ queries for some constant $ε>0$ whenever the graph has at most $O(n^{2-c})$ edges for some constant $c>0$.
We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an op…
Detecting the purely imaginary Fisher zeros of an Ising spin system on a quantum computer
本文提出一种利用量子计算机探测伊辛模型纯虚费舍尔零点的协议，通过将纯虚温度配分函数与演化算符对应，在 ibm-lagos 上验证了 3 自旋链、三角形团簇及 7 自旋团簇情形。
Recently, Ambainis gave an O(N^(2/3))-query quantum walk algorithm for element distinctness, and more generally, an O(N^(L/(L+1)))-query algorithm for finding L equal numbers. We point out that this algorithm actually solves a much more general problem, the problem of finding a subset of size L that satisfies any given property. We review the algorithm and give a considerably simplified analysis of its query complexity. We present several applications, including two algorithms for the problem of finding an L-clique in an N-vertex graph. One of these algorithms uses O(N^(2L/(L+1))) edge queries…
Quantum Algorithms on Edge Lists: Hiding, Shuffling, and Cycle Finding
本文研究边列表模型下三角形寻找问题的三种变体的量子查询复杂度，并给出紧或近乎紧的结果，同时开发新技术证明一般图的环寻找下界。
Quantum algorithms for hypergraph simplex finding
本文研究超图单纯形查找的量子查询算法，利用秩约化性质、嵌套 Johnson 图量子游走与自适应学习图的转换，提出 α-对称学习图框架，并将秩-4 超图中 4-单纯形查找的量子查询复杂度从 O(n^2.5) 改进到 O(n^2.46)。
A Perfectly Distributable Quantum-Classical Algorithm for Estimating Triangular Balance in a Signed Edge Stream
本文提出一种量子-经典算法，用于高效估计有向边流中不同符号配置的三角数量，通过量子草图寄存器和经典估计器结合，实现空间优势。
Superlinear Quantum Query Lower Bounds for Subgraph Detection
本文证明了子图检测问题（通过邻接矩阵查询判断图是否包含固定子图）的首个无条件超线性量子查询下界，针对团和完全二分图给出了具体指数下界，并引入了基于压缩预言机和随机植入证书的新技术。
Pseudo-deterministic Quantum Algorithms
研究提出伪确定性量子算法，展示其在查询复杂度上的分离结果，并识别可实现伪确定性的量子搜索问题类。
The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case. In this paper we present a framework for fast quantum distribut…
Quantum Algorithm for Path-Edge Sampling
本研究提出一种路径边采样量子算法，其查询复杂度与路径检测复杂度渐近一致，可作为子例程用于st路径与割集查找，核心贡献是生成与张成程序正见证向量成正比量子态的算法。
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