Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle finding problem (quant-ph/0310134) and matrix verification problem (quant-ph/0409035) to compare the powers of the two different walks. We also present Ambainis's method of lower bounding technique (quant-ph/0002066) to obtain a lower bound for this problem. It turns out Szegedy's algorithm can be generalized to solve similar problems. Therefore we use Szegedy's theorem to analyze the problem of matrix set commutativity. We give an O(k^{4/5}n^{9/5}) algorithm as well as a lower bound of Omega(k^{1/2}n). We generalize the technique used in coming up with the upper bound to solve a broader range of similar problems. This is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter, here, k and n.
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle finding problem (quant-ph/0310134) and matrix verification problem (quant-ph/0409035) to compare the powers of the two different walks. We also present Ambainis's method of lower bounding technique (quant-ph/0002066) to obtain a lower bound for this problem. It turns out Szegedy's algorithm can be generalized to solve similar problems. Therefore we use Szegedy's theorem to analyze the problem of matrix set commutativity. We give an O(k^{4/5}n^{9/5}) algorithm as well as a lower bound of Omega(k^{1/2}n). We generalize the technique used in coming up with the upper bound to solve a broader range of similar problems. This is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter, here, k and n.
We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^{2/3}). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^{2/3}), we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.
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, and the other uses \\tilde{O}(N^((5L-2)/(2L+4))), which is an improvement for L <= 5. The latter algorithm generalizes a recent result of Magniez, Santha, and Szegedy, who considered the case L=3 (finding a triangle). We also pose two open problems regarding continuous time quantum walk and lower bounds.
This thesis provides an introduction to the quantum walk based search method which is the quantum analogue of the classical Markov chain based search. We develop the basic theory of the Markov chain based search together with its quantisation due to Mario Szegedy[Sz2]. We deeply analyse their relationship, and sketch some of the possible algorithmic applications. The thesis also addresses the question of the implementation, which is largely hidden by the high level description. We show how to implement the quantum walk based search algorithm in the case of the element distinctness problem. We implement the necessary operations using only very basic data structures, mainly arrays. Also while we use a weak and basic computational model we could still significantly reduce the exponent of the polylog terms arising from counting non oracle calls. Another interesting achievement is that we could parallelise the Setup operation reducing its circuit depth form O(N ) to O(polylog(N)). Probably the most important tool we introduced for the implementation is the use of reversible sorting networks, but we introduce some other tricks as well. The optimal quantum query algorithm for the element distinctness problem was first described by Andris Ambainis [Amb3] and provided the basis of the generalised method. In this seminal paper Ambainis also addressed the question of implementation. However that implementation method does not fully cast into the general framework which was introduced later. The operations described by Ambainis use some involved data structures, such as skip lists and hash table. Also it uses a strong computational model for example, a qRAM query has cost only 1. In contrast with Ambainis’s single node implementation method ours uses bipartite style approach fitting the general framework of Szegedy. At first sight it may seem unnecessary since it results in a duplication of a large amount of data stored during the walk steps, but in fact it simplifies the implementation process. On the structure of the thesis: Section 1 covers some background on quantum walks and basic techniques related to the quantum walk based search method. Section 2-5 gives a comprehensive description of the general scheme introduced by Szegedy and derives the main theorems showing the quadratic speed-up compared to the classical case. These Sections roughly follow the structure of Szegedy’s original paper [Sz2], but the proofs are restructured, so that I could state the main theorem in a slightly stronger form with a much improved constant. Section 6 addresses the implementation issues already mentioned. Finally, Section 7 gives a brief overview of the further generalised scheme introduced by Frederic Magniez, Ashwin Nayak, Jeremie Roland and Miklos Santha. Quantum walk based search methods and algorithmic applications 2
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph H is present in an input graph G, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length k. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly k, or at most k. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes.
When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity O(n3/2−αk), where αk∈Θ(c−k) and c=3+17/2≈1.33, beating the previous best upper bound O(n3/2) on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an O(n) query algorithm.
We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether S is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group. We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha that improves the previous bounds by Ambainis and Szegedy. We also present several lower bounds for testing algebraic properties.
In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.