Strong empiricalAlgorithmsAlgorithmAmplitude amplification2 toy q
Grover unstructured search
The canonical search primitive, paired with the linear classical baseline it is meant to improve.
Every quantum algorithm worth knowing about, written down the same way: what it takes, what it returns, what it costs, and who proved it.
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27 public entries
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The canonical search primitive, paired with the linear classical baseline it is meant to improve.
Let G be a finitely generated Abelian group and let H be a subgroup of G such that G/H is finite, and let f be a function on G with the promise that f(g1) = f(g2) if and only if g1 and g2 lie in the same coset of H. The task is to find H, that is, a set of generators for H, by making queries to f. Mosca and Ekert work order finding and period finding as G = Z with hidden subgroup rZ, Simon's problem as G = Z_2^l, and the discrete logarithm as G = Z_r × Z_r.
A query circuit that recovers the hidden bit string in f(x) = s · x.
Given N coins of which exactly k are counterfeit, where the real coins all share one weight and the counterfeit coins all share another, and given a pan balance that can compare the weight of any pair of subsets of the coins but reports only whether they balance or tilt, identify all of the counterfeit coins in as few weighings as possible.
A one-query promise-problem circuit that separates constant from balanced Boolean oracles.
Given oracle access to two unknown probability distributions P and Q on an N-element set — in the Zoo's setup, black boxes A and B whose domain is the integers 1 through T and whose range is the integers 1 through N, with the distribution over outputs induced by choosing uniformly at random among allowed inputs — approximate the L1 distance between the two distributions to constant precision.
Given oracle access to a weighted graph on N vertices of maximum degree d whose edge weights are conductances, so that an edge of weight w carries resistance 1/w, estimate the effective resistance between a chosen pair of vertices s and t to within a factor of 1 + ε. Wang poses it as ENA-ER: for a network with |V| = N, deg(G) ≤ d, edge conductances normalized to 1 ≤ w_e ≤ c for every edge — equivalently edge resistances in [1/c, 1] — and spectral gap λ2(L_G) ≥ λ > 0, estimate R_eff(s, t) up to multiplicative error ε, succeeding with probability at least 2/3, given a procedure Pv that on input a vertex index and a number k returns the k-th edge incident to that vertex and a procedure Pe that on input an edge index returns that edge's two endpoints and its weight, both of which Wang assumes can be implemented in time poly(log(N)).
We are given an undirected graph G on n vertices, known explicitly in advance, together with oracle access to a labeling of the vertices by 1 and 0; the graph collision problem is to decide, by querying that labeling, whether there exist a pair of vertices, connected by an edge, both of which are labeled 1. Only the labeling is queried and never the graph, which is why Magniez, Santha and Szegedy can restate the task as deciding whether the set of vertices of value 1 forms an independent set in G. It is not either of the two problems it is easily confused with: collision finding asks for two arguments of a two-to-one function that share a value, and element distinctness drops that promise and asks whether any two of N items are equal. Those two share a single other Zoo entry, Collision Finding and Element Distinctness, and a single record here, element-distinctness; neither of them has a graph or an edge relation in it.
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.
Given only an oracle that, for a vertex of an N-vertex graph of degree d together with an index j in {1, 2, ..., d}, returns that vertex's j-th neighbor or null when the vertex has degree less than d, decide in as few queries as possible whether the graph is bipartite or far from bipartite — far meaning that a constant fraction of the edges would have to be removed to achieve bipartiteness — and, in the same model, whether the graph is an expander or far from being one.
Given a list of k generators for a group G together with black-box access to group multiplication, decide whether G is commutative using as few queries to that black box as possible.
Given a finite group G presented only through an oracle that takes an ordered pair of element labels and returns the label of their product, compute the order of G from the labels of a set of generators, and decide whether a given bitstring is the label of a group element — in the constructive form of the membership question, exhibiting that element as a product of the generators.
Given oracle access to a hidden subset over a finite field that is not a lattice, that is, a hidden nonlinear structure, identify that subset.
Given oracle access to a function f on the integers mod N together with the promise that f(x) = g(x+s) for a known function g and an unknown shift s, recover s.
Given a subgraph of the Boolean hypercube on bit strings of length n, whose edges all run from smaller to larger Hamming weight, decide whether it contains a path from the all-zeros vertex 0^n to the all-ones vertex 1^n.
Given oracle access to k matrices of size n × n, where a query names a matrix index x together with a pair of indices i, j and returns the ij entry of the x-th matrix, decide whether all k of the matrices commute with one another.
Compute the product of two n × n matrices over a semiring rather than a ring — the (max, min) product, the distance product, and the Boolean product — where the absence of subtraction rules out the fast algebraic algorithms that make ring matrix multiplication cost n^ω. The question the paper poses is whether anything beats the Õ(n^(5/2)) that follows from applying Grover search or quantum minimum-finding entry by entry.
Given three n×n matrices A, B and C over a field, decide whether AB = C, rather than computing the product and comparing it entry by entry.
Given oracle access to the integer entries of an n×m matrix A, determine the rank of A.
Determine the value at the root of a read-once Boolean formula, canonically a NAND tree on N variables, given only oracle access to the variables and using as few queries as possible.
Given oracle access to a list of N numbers held in order from least to greatest, together with a number x, determine where in the list x would fit.
Given an oracle that returns the value p(x) of an unknown degree-d polynomial p over the finite field GF(q) at any queried point x, determine the coefficients of p using as few queries as possible.
Given oracle access to a spherically symmetric function f from R^d to an arbitrary set S, locate its center of symmetry to a fixed precision using as few queries as possible.
Identify a hidden n-bit string x using an oracle that, for a chosen subset S of the n positions and a string y of length |S|, returns one when the substring of x specified by S equals y and zero otherwise.
Given a text T of length n and a pattern P of length m < n, both over a finite alphabet, find an occurrence of P as a substring of T or report that P is not a substring of T. The Zoo also states the problem for d-dimensional arrays rather than strings, where the task is to return the location of P as an m × m × ... × m block within the n × n × ... × n array T or report that no such location exists.
Given oracle access to N items carrying no promise about the underlying function, find two items that are equal, or establish that all N are distinct.
Given oracle access to a graph built by joining two depth-n binary trees with a random weld, so that every node except the two roots has degree three, start from the label of one root and find the label of the other root.