Given access only to an oracle that, for a pair of integers in {1, 2, ..., n}, says whether the corresponding vertices of an n-vertex graph are joined by an edge, decide a property of that graph or find a structure in it, such as connectivity, a minimum spanning tree, a lowest weight path, or a triangle, using as few queries as possible.
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Given access only to an oracle that, for a pair of integers in {1, 2, ..., n}, says whether the corresponding vertices of an n-vertex graph are joined by an edge, decide a property of that graph or find a structure in it, such as connectivity, a minimum spanning tree, a lowest weight path, or a triangle, using as few queries as possible. Durr, Heiligman, Hoyer and Mhalla treat these tasks in two input models, the adjacency matrix model in which the oracle answers about a pair of vertices and an adjacency list-like array model, and they give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree and Single Source Shortest Paths. Their abstract states that the upper bounds utilize search procedures for finding minima of functions under various conditions, without naming them; an earlier algorithm of Durr and Hoyer, catalogued here alongside that paper, finds the index of the minimum entry of a table of size N in time O(c√N) with probability at least 1 - 1/2^c. The Zoo places this work inside a wider body of results in the same model: bipartiteness, cycle detection and st-connectivity in Õ(n^(3/2)) queries and quantum gates using only logarithmically many qubits, and Childs and Kothari's Θ(n^(2/3)) for sparse graph properties, sparse meaning that some constant bounds the ratio of edges to vertices for every graph with the property, provided the property cannot be characterized by a list of forbidden subgraphs. Finding a subgraph is treated separately there, the simplest case being the triangle, for which the Zoo reports O(n^(5/4)) quantum queries as the fastest known.
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Durr, Heiligman, Hoyer and Mhalla treat these tasks in two input models, the adjacency matrix model in which the oracle answers about a pair of vertices and an adjacency list-like array model, and they give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree and Single Source Shortest Paths. Their abstract states that the upper bounds utilize search procedures for finding minima of functions under various conditions, without naming them; an earlier algorithm of Durr and Hoyer, catalogued here alongside that paper, finds the index of the minimum entry of a table of size N in time O(c√N) with probability at least 1 - 1/2^c. The Zoo places this work inside a wider body of results in the same model: bipartiteness, cycle detection and st-connectivity in Õ(n^(3/2)) queries and quantum gates using only logarithmically many qubits, and Childs and Kothari's Θ(n^(2/3)) for sparse graph properties, sparse meaning that some constant bounds the ratio of edges to vertices for every graph with the property, provided the property cannot be characterized by a list of forbidden subgraphs. Finding a subgraph is treated separately there, the simplest case being the triangle, for which the Zoo reports O(n^(5/4)) quantum queries as the fastest known. This record's speedup class, "Polynomial", is a secondary source's classification of the oracular algorithms it files this under — not a claim its primary paper makes. Not checked against the primary source yet. Reported cost: Θ(n^(3/2)) quantum queries in the adjacency matrix model for Minimum Spanning Tree and for Connectivity, against Θ(√(nm)) and Θ(n) respectively in the array model; the Zoo adds O(n^(3/2) log² n) for finding lowest weight paths, Õ(n^(3/2)) queries and quantum gates for bipartiteness, cycle detection and st-connectivity, Θ(n^(2/3)) for sparse graph properties that cannot be characterized by a list of forbidden subgraphs, and O(n^(5/4)) queries for finding a triangle. On the classical side the Zoo puts every problem it lists ahead of triangle finding at Ω(n²) queries only under the widely-believed Aanderaa-Karp-Rosenberg conjecture, whereas for triangle finding it states Ω(n²) classical queries outright. Durr and Hoyer's minimum-finding algorithm runs in time O(c√N) on a table of size N and returns the index of the minimum with probability at least 1 - 1/2^c..
ALGORITHM: Quantum query complexity of graph properties in the adjacency matrix model
PROBLEM: Given access only to an oracle that, for a pair of integers in {1, 2, ..., n}, says whether the corresponding vertices of an n-vertex graph are joined by an edge, decide a property of that graph or find a structure in it, such as connectivity, a minimum spanning tree, a lowest weight path, or a triangle, using as few queries as possible.
IDEA: Durr, Heiligman, Hoyer and Mhalla treat these tasks in two input models, the adjacency matrix model in which the oracle answers about a pair of vertices and an adjacency list-like array model, and they give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree and Single Source Shortest Paths. Their abstract states that the upper bounds utilize search procedures for finding minima of functions under various conditions, without naming them; an earlier algorithm of Durr and Hoyer, catalogued here alongside that paper, finds the index of the minimum entry of a table of size N in time O(c√N) with probability at least 1 - 1/2^c. The Zoo places this work inside a wider body of results in the same model: bipartiteness, cycle detection and st-connectivity in Õ(n^(3/2)) queries and quantum gates using only logarithmically many qubits, and Childs and Kothari's Θ(n^(2/3)) for sparse graph properties, sparse meaning that some constant bounds the ratio of edges to vertices for every graph with the property, provided the property cannot be characterized by a list of forbidden subgraphs. Finding a subgraph is treated separately there, the simplest case being the triangle, for which the Zoo reports O(n^(5/4)) quantum queries as the fastest known.
REPORTED COST: Θ(n^(3/2)) quantum queries in the adjacency matrix model for Minimum Spanning Tree and for Connectivity, against Θ(√(nm)) and Θ(n) respectively in the array model; the Zoo adds O(n^(3/2) log² n) for finding lowest weight paths, Õ(n^(3/2)) queries and quantum gates for bipartiteness, cycle detection and st-connectivity, Θ(n^(2/3)) for sparse graph properties that cannot be characterized by a list of forbidden subgraphs, and O(n^(5/4)) queries for finding a triangle. On the classical side the Zoo puts every problem it lists ahead of triangle finding at Ω(n²) queries only under the widely-believed Aanderaa-Karp-Rosenberg conjecture, whereas for triangle finding it states Ω(n²) classical queries outright. Durr and Hoyer's minimum-finding algorithm runs in time O(c√N) on a table of size N and returns the index of the minimum with probability at least 1 - 1/2^c.
BASIS: abstract of arXiv:quant-ph/0401091 (plain-text TeX rendered as Unicode, and widetilde O written as Õ where quoted from the Zoo): "we show that the query complexity of Minimum Spanning Tree is in Θ(n^(3/2)) in the matrix model and in Θ(√(nm)) in the array model, while the complexity of Connectivity is also in Θ(n^(3/2)) in the matrix model, but in Θ(n) in the array model", and "The upper bounds utilize search procedures for finding minima of functions under various conditions."; Quantum Algorithm Zoo entry "Graph Properties in the Adjacency Matrix Model": "finding lowest weight paths has O(n^(3/2) log² n) quantum query complexity", "Deciding whether a graph is bipartite, detecting cycles, and deciding whether a given vertex can be reached from another (st-connectivity) can all be achieved using a number of queries and quantum gates that both scale as Õ(n^(3/2)), and only logarithmically many qubits", "all sparse graph properties have query complexity Θ(n^(2/3)) if they cannot be characterized by a list of forbidden subgraphs and o(n^(2/3)) (little-o) if they can", "The fastest known quantum algorithm for this finds a triangle in O(n^(5/4)) quantum queries", "According to the widely-believed Aanderaa-Karp-Rosenberg conjecture, all of the above problems have Ω(n²) classical query complexity", and "Classically, triangle finding requires Ω(n²) queries"; abstract of arXiv:quant-ph/9607014: "We give a quantum algorithm to find the index y in a table T of size N such that in time O(c sqrt N), T[y] is minimum with probability at least 1-1/2^c."
PRIMARY SOURCE: Christoph Durr, Mark Heiligman, Peter Hoyer, Mehdi Mhalla (2004), Quantum query complexity of some graph problems — https://arxiv.org/abs/quant-ph/0401091
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Primary reference for quantum query complexity of graph properties in the adjacency matrix model; the source of this record's cost claim.
arxiv.org/abs/quant-ph/0401091 ↗A minimum-finding algorithm, catalogued alongside the graph paper rather than as a component of it. Durr and Hoyer give a quantum algorithm that finds the index y in a table T of size N such that T[y] is minimum, in time O(c√N) and with probability at least 1 - 1/2^c. The graph paper's abstract states that its upper bounds utilize search procedures for finding minima of functions under various conditions; it does not name them, so consult both papers before treating this one as the subroutine used there.
arxiv.org/abs/quant-ph/9607014 ↗