Algebraic and Number Theoretic Algorithms

Algorithm: Factoring
Speedup: Superpolynomial
Description: Given an n-bit integer, find the prime factorization. The quantum algorithm of Peter Shor solves this in poly(n) time [82,125]. The fastest known classical algorithm requires time superpolynomial in n. Shor's algorithm breaks the RSA cryptosystem. At the core of this algorithm is order finding, which can be reduced to the Abelian hidden subgroup problem.

Algorithm: Discrete-log
Speedup: Superpolynomial
Description: We are given three n-bit numbers a, b, and N, with the promise that \( b = a^s \mod N \) for some s. The task is to find s. As shown by Shor [82], this can be achieved on a quantum computer in poly(n) time. The fastest known classical algorithm requires time superpolynomial in n. By similar techniques to those in [82], quantum computers can solve the discrete logarithm problem on elliptic curves, thereby breaking elliptic curve cryptography [109]. See also Abelian hidden subgroup.

Algorithm: Pell's Equation
Speedup: Superpolynomial
Description: Given a positive nonsquare integer d, Pell's equation is \( x^2 - d y^2 = 1 \). For any such d there are infinitely many pairs of integers (x,y) solving this equation. Let \( (x_1,y_1) \) be the pair that minimizes \( x+y\sqrt{d} \). If d is an n-bit integer (i.e. \( 0 \leq d \lt 2^n \) ), \( (x_1,y_1) \) may in general require exponentially many bits to write down. Thus it is in general impossible to find \( (x_1,y_1) \) in polynomial time. Let \( R = \log(x_1+y_1 \sqrt{d}) \). \( \lfloor R \rceil \) uniquely identifies \( (x_1,y_1) \). As shown by Hallgren [49], given a n-bit number d, a quantum computer can find \( \lfloor R \rceil \) in poly(n) time. No polynomial time classical algorithm for this problem is known. Factoring reduces to this problem. This algorithm breaks the Buchman-Williams cryptosystem. See also Abelian hidden subgroup.

Algorithm: Principal Ideal
Speedup: Superpolynomial
Description: We are given an n-bit integer d and an invertible ideal I of the ring \( \mathbb{Z}[\sqrt{d}] \). I is a principal ideal if there exists \( \alpha \in \mathbb{Q}(\sqrt{d}) \) such that \( I = \alpha \mathbb{Z}[\sqrt{d}] \). \( \alpha \) may be exponentially large in d. Therefore \( \alpha \) cannot in general even be written down in polynomial time. However, \( \lfloor \log \alpha \rceil \) uniquely identifies \( \alpha \). The task is to determine whether I is principal and if so find \( \lfloor \log \alpha \rceil \) As shown by Hallgren, this can be done in polynomial time on a quantum computer [49]. A modified quantum algorithm for this problem using fewer qubits was given in [131]. Factoring reduces to solving Pell's equation, which reduces to the principal ideal problem. Thus the principal ideal problem is at least as hard as factoring and therefore is probably not in P. See also Abelian hidden subgroup.

Algorithm: Unit Group
Speedup: Superpolynomial
Description: The number field \( \mathbb{Q}(\theta) \) is said to be of degree d if the lowest degree polynomial of which \( \theta \) is a root has degree d. The set \( \mathcal{O} \) of elements of \( \mathbb{Q}(\theta) \) which are roots of monic polynomials in \( \mathbb{Z}[x] \) forms a ring, called the ring of integers of \( \mathbb{Q}(\theta) \). The set of units (invertible elements) of the ring \( \mathcal{O} \) form a group denoted \( \mathcal{O}^* \). As shown by Hallgren [50], and independently by Schmidt and Vollmer [116], for any \( \mathbb{Q}(\theta) \) of fixed degree, a quantum computer can find in polynomial time a set of generators for \( \mathcal{O}^* \) given a description of \( \theta \). No polynomial time classical algorithm for this problem is known. See also Abelian hidden subgroup.

Algorithm: Class Group
Speedup: Superpolynomial
Description: The number field The number field \( \mathbb{Q}(\theta) \) is said to be of degree d if the lowest degree polynomial of which \( \theta \) is a root has degree d. The set \( \mathcal{O} \) of elements of \( \mathbb{Q}(\theta) \) which are roots of monic polynomials in \( \mathbb{Z}[x] \) forms a ring, called the ring of integers of \( \mathbb{Q}(\theta) \). For a ring, the ideals modulo the prime ideals form a group called the class group. As shown by Hallgren [50], a quantum computer can find in polynomial time a set of generators for the class group of the ring of integers of any constant degree number field, given a description of \( \theta \). No polynomial time classical algorithm for this problem is known. See also Abelian hidden subgroup.

Algorithm: Gauss Sums
Speedup: Superpolynomial
Description: Let \( \mathbb{F}_q \) be a finite field. The elements other than zero of \( \mathbb{F}_q \) form a group \( \mathbb{F}_q^\times \) under multiplication, and the elements of \( \mathbb{F}_q \) form an (Abelian but not necessarily cyclic) group \( \mathbb{F}_q^+ \) under addition. We can choose some character \( \chi^\times \) of \( \mathbb{F}_q^\times \) and some character \( \chi^+ \) of \( \mathbb{F}_q^+ \). The corresponding Gauss sum is the inner product of these characters: \( \sum_{x \neq 0 \in \mathbb{F}_q} \chi^+(x) \chi^\times(x) \) As shown by van Dam and Seroussi [90], Gauss sums can be estimated to polynomial precision on a quantum computer in polynomial time. Although a finite ring does not form a group under multiplication, its set of units does. Choosing a representation for the additive group of the ring, and choosing a representation for the multiplicative group of its units, one can obtain a Gauss sum over the units of a finite ring. These can also be estimated to polynomial precision on a quantum computer in polynomial time [90]. No polynomial time classical algorithm for estimating Gauss sums is known. Discrete log reduces to Gauss sum estimation [90]. Certain partition functions of the Potts model can be computed by a polynomial-time quantum algorithm related to Gauss sum estimation [47].

Algorithm:Solving Exponential Congruences
Speedup:Polynomial
Description: We are given \( a,b,c,f,g \in \mathbb{F}_q \). We must find \( x,y \in \mathbb{F}_q \) such that \( a f^x + b g^y = c \). As shown in [111], quantum computers can solve this problem in \( \tilde{O}(q^{3/8}) \) time whereas the best classical algorithm requires \( \tilde{O}(q^{9/8}) \) time. The quantum algorithm of [111] is based on the quantum algorithms for discrete logarithms and searching.

Algorithm:Matrix Elements of Group Representations
Speedup:Superpolynomial
Description: All representations of finite groups and compact linear groups can be expressed as unitary matrices given an appropriate choice of basis. Quantum circuits can efficiently implement unitary versions of all the irreducible representations of the symmetric and alternating groups, and all the irreducible representations of unitary, special unitary, and orthogonal groups with polynomial highest weight. As discussed in [106], using the Hadamard test one can thus approximate individual matrix elements of the irreducible representations of these groups to polynomial precision in polynomial time, whereas for certain hard instances, classical computers probably cannot achieve this.

Algorithm: Verifying Matrix Products
Speedup: Polynomial
Description: Given three \( n \times n \) matrices, A,B, and C, the matrix product verification problem is to decide whether AB=C. Classically, the best known algorithm achieves this in time \( O(n^2) \), whereas the best known classical algorithm for matrix multiplication runs in time \( O(n^{2.373}) \). Ambainis et al. discovered a quantum algorithm for this problem with runtime \( O(n^{7/4}) \) [6]. Subsequently, Buhrman and Špalek improved upon this, obtaining a quantum algorithm for this problem with runtime \( O(n^{5/3}) \) [19]. This latter algorithm is based on results regarding quantum walks that were proven in [85].

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Oracular Algorithms

Algorithm: Searching
Speedup: Polynomial
Description: We are given an oracle with N allowed inputs. For one input w ("the winner") the corresponding output is 1, and for all other inputs the corresponding output is 0. The task is to find w. On a classical computer this requires \( \Omega(N) \) queries. The quantum algorithm of Lov Grover achieves this using \( O(\sqrt{N}) \) queries [48].This has algorithm has subsequently been generalized to search in the presence of multiple "winners" [15], evaluate the sum of an arbitrary function [15,16,73], find the global minimum of an arbitrary function [35,75], take advantage of alternative initial states [100] or nonuniform probabilistic priors [123], work with oracles whose runtime varies between inputs [138], and approximate definite integrals [77]. The generalization of Grover's algorithm known as amplitude estimation [17] is now an important primitive in quantum algorithms. Amplitude estimation forms the core of most known quantum algorithms related to collision finding and graph properties. One of the natural applications for Grover search is speeding up the solution to NP-complete problems such as 3-SAT. Doing so is nontrivial, because the best classical algorithm for 3-SAT is not quite a brute force search. Nevertheless, amplitude amplification enables a quadratic quantum speedup over the best classical 3-SAT algorithm, as shown in [133]. Quadratic speedups for other constraint satisfaction problems are obtained in [134].

Algorithm: Abelian Hidden Subgroup
Speedup: Superpolynomial
Description: Let G be a finitely generated Abelian group, and let H be some subgroup of G such that G/H is finite. Let f be a function on G such that for any \( g_1,g_2 \in G \), \( f(g_1) = f(g_2) \) if and only if \( g_1 \) and \( g_2 \) are in the same coset of H. The task is to find H (i.e. find a set of generators for H) by making queries to f. This is solvable on a quantum computer using \( O(\log \vert G\vert) \) queries, whereas classically \( \Omega(|G|) \) are required. This algorithm was first formulated in full generality by Boneh and Lipton in [14]. However, proper attribution of this algorithm is difficult because, as described in chapter 5 of [76], it subsumes many historically important quantum algorithms as special cases, including Simon's algorithm [108], which was the inspiration for Shor's period finding algorithm, which forms the core of his factoring and discrete-log algorithms. The Abelian hidden subgroup algorithm is also at the core of the Pell's equation, principal ideal, unit group, and class group algorithms. In certain instances, the Abelian hidden subgroup problem can be solved using a single query rather than order \( \log(\vert G\vert) \), as shown in [30].

Algorithm: Non-Abelian Hidden Subgroup
Speedup: Superpolynomial
Description: Let G be a finitely generated group, and let H be some subgroup of G that has finitely many left cosets. Let f be a function on G such that for any \( g_1, g_2 \), \( f(g_1) = f(g_2) \) if and only if \( g_1 \) and \( g_2 \) are in the same left coset of H. The task is to find H (i.e. find a set of generators for H) by making queries to f. This is solvable on a quantum computer using \( O(\log(|G|) \) queries, whereas classically \( \Omega(|G|) \) are required [37,51]. However, this does not qualify as an efficient quantum algorithm because in general, it may take exponential time to process the quantum states obtained from these queries. Efficient quantum algorithms for the hidden subgroup problem are known for certain specific non-Abelian groups [81,55,72,53,9,22,56,71,57,43,44,28,126]. A slightly outdated survey is given in [69]. Of particular interest are the symmetric group and the dihedral group. A solution for the symmetric group would solve graph isomorphism. A solution for the dihedral group would solve certain lattice problems [78]. Despite much effort, no polynomial-time solution for these groups is known. However, Kuperburg [66] found a time \( 2^{O( \sqrt{\log N})}) \) algorithm for finding a hidden subgroup of the dihedral group \( D_N \). Regev subsequently improved this algorithm so that it uses not only subexponential time but also polynomial space [79].

Algorithm: Bernstein-Vazirani
Speedup: Polynomial
Description: We are given an oracle whose input is n bits and whose output is one bit. Given input \( x \in \{0,1\}^n \), the output is \( x \odot h \), where h is the "hidden" string of n bits, and \( \odot \) denotes the bitwise inner product modulo 2. The task is to find h. On a classical computer this requires n queries. As shown by Bernstein and Vazirani [11], this can be achieved on a quantum computer using a single query. Furthermore, one can construct a recursive version of this problem, called recursive Fourier sampling, such that quantum computers require exponentially fewer queries than classical computers [11].

Algorithm: Deutsch-Jozsa
Speedup: Exponential over P, none over BPP
Description: We are given an oracle whose input is n bits and whose output is one bit. We are promised that out of the \( 2^n \) possible inputs, either all of them, none of them, or half of them yield output 1. The task is to distinguish the balanced case (half of all inputs yield output 1) from the constant case (all or none of the inputs yield output 1). It was shown by Deutsch [32] that for n=1, this can be solved on a quantum computer using one query, whereas any deterministic classical algorithm requires two. This was historically the first well-defined quantum algorithm achieving a speedup over classical computation. A single-query quantum algorithm for arbitrary n was developed by Deutsch and Jozsa in [33]. Although probabilistically easy to solve with O(1) queries, the Deutsch-Jozsa problem has exponential worst case deterministic query complexity classically.

Algorithm: Formula Evaluation
Speedup: Polynomial
Description: A Boolean expression is called a formula if each variable is used only once. A formula corresponds to a circuit with no fanout, which consequently has the topology of a tree. By Reichardt's span-program formalism, it is now known [158] that the quantum query complexity of any formula of O(1) fanin on N variables is \( \Theta(\sqrt{N}) \). This result culminates from a long line of work [27,8,80,159,160], which started with the discovery by Farhi et al. [38] that NAND trees on \( 2^n \) variables can be evaluated on quantum computers in time \( O(2^{0.5n}) \) using a continuous-time quantum walk, whereas classical computers require \( \Omega(2^{0.753n}) \) queries. In many cases, the quantum formula-evaluation algorithms are efficient not only in query complexity but also in time-complexity. The span-program formalism also yields quantum query complexity lower bounds [149]. Although originally discovered from a different point of view, Grover's algorithm can be regarded as a special case of formula evaluation in which every gate is OR. The quantum complexity of evaluating non-boolean formulas has also been studied [29], but is not as fully understood. Childs et al. have generalized to the case in which input variables may be repeated (i.e. the first layer of the circuit may include fanout) [101]. They obtained a quantum algorithm using \( O(\min \{N,\sqrt{S},N^{1/2} G^{1/4} \}) \) queries, where N is the number of input variables not including multiplicities, S is the number of inputs counting multiplicities, and G is the number of gates in the formula. References [164] and [165] consider special cases of the NAND tree problem in which the number of NAND gates taking unequal inputs is limited. Some of these cases yield superpolynomial separation between quantum and classical query complexity.

Algorithm: Gradients, Structured Search, and Learning Polynomials
Speedup: Polynomial
Description: Suppose we are given a oracle for computing some smooth function \( f:\mathbb{R}^d \to \mathbb{R} \). The inputs and outputs to f are given to the oracle with finitely many bits of precision. The task is to estimate \( \nabla f \) at some specified point \( \mathbf{x}_0 \in \mathbb{R}^d \). As shown in [61], a quantum computer can achieve this using one query, whereas a classical computer needs at least d+1 queries. In [20], Bulger suggested potential applications for optimization problems. As shown in appendix D of [62], a quantum computer can use the gradient algorithm to find the minimum of a quadratic form in d dimensions using O(d) queries, whereas, as shown in [94], a classical computer needs at least \( \Omega(d^2) \) queries. Single query quantum algorithms for finding the minima of basins based on Hamming distance were earlier given in [147,148]. The quantum algorithm of [62] can also extract all \( d^2 \) matrix elements of the quadratic form using O(d) queries, and more generally, all \( d^n \) nth derivatives of a smooth function of d variables in \( O(d^{n-1}) \) queries. As shown in [130,146], quadratic forms and multilinear polynomials in d variables over a finite field may be extracted with a factor of d fewer quantum queries than are required classically.

Algorithm: Hidden Shift
Speedup: Superpolynomial
Description: We are given oracle access to some function f on \( \mathbb{Z}_N \). We know that f(x) = g(x+s) where g is a known function and s is an unknown shift. The hidden shift problem is to find s. By reduction from Grover's problem it is clear that at least \( \sqrt{N} \) queries are necessary to solve hidden shift in general. However, certain special cases of the hidden shift problem are solvable on quantum computers using O(1) queries. In particular, van Dam et al. showed that this can be done if f is a multiplicative character of a finite ring or field [89]. The previously discovered shifted Legendre symbol algorithm [88,86] is subsumed as a special case of this, because the Legendre symbol \( \left(\frac{x}{p} \right) \) is a multiplicative character of \( \mathbb{F}_p \). No classical algorithm running in time O(polylog(N)) is known for these problems. Furthermore, the quantum algorithm for the shifted Legendre symbol problem breaks certain classical cryptosystems [89]. Roetteler has found exponential quantum speedups for finding hidden shifts of certain nonlinear Boolean functions [105,130]. Building on this work, Gavinsky, Roetteler, and Roland have shown [142] that the hidden shift problem on random boolean functions \( f:\mathbb{Z}_2^n \to \mathbb{Z}_2 \) has O(n) average case quantum complexity, whereas the classical query complexity is \( \Omega(2^{n/2}) \). The results in [143], though they are phrased in terms of the hidden subgroup problem for the dihedral group, imply that the quantum query complexity of the hidden shift problem for an injective function on \( \mathbb{Z}_N \) is O(log n), whereas the classical query complexity is \( \Theta(\sqrt{N}) \). However, the best known quantum circuit complexity for injective hidden shift on \( \mathbb{Z}_N \) is \( O(2^{C \sqrt{\log N}}) \), achieved by Kuperberg's sieve algorithm [66].

Algorithm: Linear Systems
Speedup: Superpolynomial
Description: We are given oracle access to an \( n \times n \) matrix A and some description of a vector b. We wish to find some property of f(A)b for some efficiently computable function f. Suppose A is a Hermitian matrix with O(polylog n) nonzero entries in each row and condition number k. As shown in [104], a quantum computer can in \( O(k^2 \log n) \) time compute to polynomial precision various expectation values of operators with respect to the vector f(A)b (provided that a quantum state proportional to b is efficiently constructable). For certain functions, such as f(x)=1/x, this procedure can be extended to non-Hermitian and even non-square A. The runtime of this algorithm was subsequently improved to \( O(k \log^3 k \log n) \) in [138].

Algorithm: Ordered Search
Speedup: Constant
Description: We are given oracle access to a list of N numbers in order from least to greatest. Given a number x, the task is to find out where in the list it would fit. Classically, the best possible algorithm is binary search which takes \( \log_2 N \) queries. Farhi et al. showed that a quantum computer can achieve this using 0.53 \( \log_2 N \) queries [39]. Currently, the best known deterministic quantum algorithm for this problem uses 0.433 \( \log_2 N \) queries [103]. A lower bound of \( \frac{1}{\pi} \log_2 N \) quantum queries has been proven for this problem [24]. In [10], a randomized quantum algorithm is given whose expected query complexity is less than \( \frac{1}{3} \log_2 N \).

Algorithm: Graph Properties in the Adjacency Matrix Model
Speedup: Polynomial
Description: Let G be a graph of n vertices. We are given access to an oracle, which given a pair of integers in {1,2,...,n} tells us whether the corresponding vertices are connected by an edge. Building on previous work [35,52,36], Dürr et al. [34] show that the quantum query complexity of finding a minimum spanning tree of weighted graphs, and deciding connectivity for directed and undirected graphs have \( \Theta(n^{3/2}) \) quantum query complexity, and that finding lowest weight paths has \( O(n^{3/2}\log^2 n) \) quantum query complexity. Berzina et al. [13] show that deciding whether a graph is bipartite can be achieved using \( O(n^{3/2}) \) quantum queries. All of these problems are thought to have \( \Omega(n^2) \) classical query complexity. A graph property is sparse if there exists a constant c such that every graph with the property has a ratio of edges to vertices at most c. Childs and Kothari have shown that all sparse graph properties have query complexity \( \Theta(n^{2/3}) \) if they cannot be characterized by a list of forbidden subgraphs and \( o(n^{2/3}) \) (little-o) if they can [140]. The former algorithm is based on Grover search, the latter on the quantum walk formalism of [141]. By Mader's theorem, sparse graph properties include all nontrivial minor-closed properties. These include planarity, being a forest, and not containing a path of given length. Another interesting computational problem is finding a subgraph H in a given graph G. The simplest case of this finding the triangle, that is, the clique of size three. The fastest known quantum algorithm for this finds a triangle in \( O(n^{35/27}) \) quantum queries [152], improving on the previous algorithms of [70] and [21]. Classically, triangle finding requires \( \Omega(n^2) \) queries [21]. More generally, a quantum computer can find an arbitrary subgraph of k vertices using \( O(n^{2-2/k-t}) \) queries where \( t=(2k-d-3)/(k(d+1)(m+2)) \) and d and m are such that H has a vertex of degree d and m+d edges [153]. This improves on the previous algorithm of [70]. In some cases, this query complexity is beaten by the quantum algorithm of [140], which finds H using \( \tilde{O}\left( n^{\frac{3}{2}-\frac{1}{\mathrm{vc}(H)+1}} \right) \) queries, provided G is sparse, where vc(H) is the size of the minimal vertex cover of H.

Algorithm: Graph Properties in the Adjacency List Model
Speedup: Polynomial
Description: Let G be a graph of N vertices, M edges, and degree d. We are given access to an oracle which, when queried with the label of a vertex and \( j \in \{1,2,\ldots,d\} \) outputs the jth neighbor of the vertex or null if the vertex has degree less than d. Suppose we are given the promise that G is either bipartite or is far from bipartite in the sense that a constant fraction of the edges would need to be removed to achieve bipartiteness. Then, as shown in [144], the quantum complexity of deciding bipartiteness is \( \tilde{O}(N^{1/3}) \). Also in [144], it is shown that distinguishing expander graphs from graphs that are far from being expanders has quantum complexity \( \tilde{O}(N^{1/3}) \) and \( \tilde{\Omega}(N^{1/4}) \), whereas the classical complexity is \( \tilde{\Theta}(\sqrt{N}) \). The key quantum algorithmic tool is Ambainis' algorithm for collision finding. In [34], it is shown that finding a minimal spanning tree has quantum query complexity \( \Theta(\sqrt{NM}) \), deciding graph connectivity has quantum query complexity \( \Theta(N) \) in the undirected case, and \( \tilde{\Theta}(\sqrt{NM}) \) in the directed case, and computing the lowest weight path from a given source to all other vertices on a weighted graph has quantum query complexity \( \tilde{\Theta}(\sqrt{NM}) \).

Algorithm: Welded Tree
Speedup: Superpolynomial
Description: Some computational problems can be phrased in terms of the query complexity of finding one's way through a maze. That is, there is some graph G to which one is given oracle access. When queried with the label of a given node, the oracle returns a list of the labels of all adjacent nodes. The task is, starting from some source node (i.e. its label), to find the label of a certain marked destination node. As shown by Childs et al. [26], quantum computers can exponentially outperform classical computers at this task for at least some graphs. Specifically, consider the graph obtained by joining together two depth-n binary trees by a random "weld" such that all nodes but the two roots have degree three. Starting from one root, a quantum computer can find the other root using poly(n) queries, whereas this is provably impossible using classical queries.

Algorithm: Collision Finding
Speedup: Polynomial
Description: Suppose we are given oracle access to a two to one function f on a domain of size N. The collision problem is to find a pair \( x,y \in \{1,2,\ldots,N\} \) such that f(x) = f(y). The classical randomized query complexity of this problem is \( \Theta(\sqrt{N}) \), whereas, as shown by Brassard et al., a quantum computer can achieve this using \(O(N^{1/3}) \) queries [18]. Removing the promise that f is two-to-one yields a problem called element distinctness, which has \( \Theta(N\log N) \) classical query complexity. Improving upon [21], Ambainis gives a quantum algorithm with query complexity of \( O(N^{2/3}) \) for element distinctness, which is optimal. Ambainis' algorithm can also detect k-fold collisions using \( O(N^{k/(k+1)}) \) queries [7]. A subsequent learning-graph based algorithm of Belovs and Lee can beat this given some prior knowledge on the input. Specifically, given the number of t-fold collisions for each t < k, one can decide the existence of a k-fold collision using \( O(N^{1-2^{k-2}/(2^k-1)}) \) queries [154]. Given two functions f and g, each on a domain of size N, a claw is a pair x,y such that f(x) = g(y). The algorithm of [7] solves claw-finding in \( O(N^{2/3}) \) queries as a special case, improving on the previous \( O(N^{3/4} \log N) \) quantum algorithm of [21]. One can also define collision finding on a graph: a value is associated with each vertex, and a collision only counts if the vertices share an edge. As shown by Magniez, Nayak, and Szegedy, this can be solved using \( O(N^{2/3}) \) quantum queries [70].

Algorithm: Matrix Commutativity
Speedup: Polynomial
Description: We are given oracle access to k matrices, each of which are \( n \times n \). Given integers \( i,j \in \{1,2,\ldots,n\} \), and \( x \in \{1,2,\ldots,k\} \) the oracle returns the ij matrix element of the \( x^{\mathrm{th}} \) matrix. The task is to decide whether all of these k matrices commute. As shown by Itakura [54], this can be achieved on a quantum computer using \( O(k^{4/5}n^{9/5}) \) queries, whereas classically this requires \( \Omega( k n^2 ) \) queries.

Algorithm: Group Commutativity
Speedup: Polynomial
Description: We are given a list of k generators for a group G and access to a blackbox implementing group multiplication. By querying this blackbox we wish to determine whether the group is commutative. The best known classical algorithm is due to Pak and requires O(k) queries. Magniez and Nayak have shown that the quantum query complexity of this task is \( \tilde{\Theta}(k^{2/3}) \) [139].

Algorithm: Hidden Nonlinear Structures
Speedup: Superpolynomial
Description: Any Abelian group G can be visualized as a lattice. A subgroup H of G is a sublattice, and the cosets of H are all the shifts of that sublattice. The Abelian hidden subgroup problem is normally solved by obtaining superposition over a random coset of the Hidden subgroup, and then taking the Fourier transform so as to sample from the dual lattice. Rather than generalizing to non-Abelian groups (see non-Abelian hidden subgroup), one can instead generalize to the problem of identifying hidden subsets other than lattices. As shown by Childs et al. [23] this problem is efficiently solvable on quantum computers for certain subsets defined by polynomials, such as spheres. Decker et al. showed how to efficiently solve some related problems in [31].

Algorithm: Center of Radial Function
Speedup: Polynomial
Description: We are given an oracle that evaluates a function f from \( \mathbb{R}^d \) to some arbitrary set S, where f is spherically symmetric. We wish to locate the center of symmetry, up to some precision. (For simplicity, let the precision be fixed.) In [110], Liu gives a quantum algorithm, based on a curvelet transform, that solves this problem using a constant number of quantum queries independent of d. This constitutes a polynomial speedup over the classical lower bound, which is \( \Omega(d) \) queries. The algorithm works when the function f fluctuates on sufficiently small scales, e.g., when the level sets of f are sufficiently thin spherical shells. The quantum algorithm is shown to work in an idealized continuous model, and nonrigorous arguments suggest that discretization effects should be small.

Algorithm: Group Order and Membership
Speedup: Superpolynomial
Description: Suppose a finite group G is given oracularly in the following way. To every element in G, one assigns a corresponding label. Given an ordered pair of labels of group elements, the oracle returns the label of their product. There are several classically hard problems regarding such groups. One is to find the group's order, given the labels of a set of generators. Another task is, given a bitstring, to decide whether it corresponds to a group element. The constructive version of this membership problem requires, in the yes case, a decomposition of the given element as a product of group generators. Classically, these problems cannot be solved using polylog(|G|) queries even if G is Abelian. For Abelian groups, quantum computers can solve these problems using polylog(|G|) queries by reduction to the Abelian hidden subgroup problem, as shown by Mosca [74]. Furthermore, as shown by Watrous [91], quantum computers can solve these problems using polylog(|G|) queries for any solvable group. For groups given as matrices over a finite field rather than oracularly, the order finding and constructive membership problems can be solved in polynomial time by using the quantum algorithms for discrete log and factoring [124]. See also group isomorphism.

Algorithm: Group Isomorphism
Speedup: Superpolynomial
Description: Let G be a finite group. To every element of G is assigned an arbitrary label (bit string). Given an ordered pair of labels of group elements, the group oracle returns the label of their product. Given access to the group oracles for two groups G and G', and a list of generators for each group, we must decide whether G and G' are isomorphic. For Abelian groups, we can solve this problem using poly(log |G|, log |G'|) queries to the oracle by applying the quantum algorithm of [127], which decomposes any Abelian group into a canonical direct product of cyclic groups. The quantum algorithm of [128] solves the group isomorphism problem using poly(log |G|, log |G'|) queries for a certain class of non-Abelian groups. Specifically, a group G is in this class if G has a normal Abelian subgroup A and an element y of order coprime to |A| such that G = A, y>.

Algorithm: Statistical Difference
Speedup: Polynomial
Description: Suppose we are given two black boxes A and B whose domain is the integers 1 through T and whose range is the integers 1 through N. By choosing uniformly at random among allowed inputs we obtain a probability distribution over the possible outputs. We wish to approximate to constant precision the L1 distance between the probability distributions determined by A and B. Classically the number of necessary queries scales essentially linearly with N. As shown in [117], a quantum computer can achieve this using \( O(\sqrt{N}) \) queries. Approximate uniformity and orthogonality of probability distributions can also be decided on a quantum computer using \( O(N^{1/3}) \) queries. The main tool is the quantum counting algorithm of [16].

Algorithm: Finite Rings and Ideals
Speedup: Superpolynomial
Description: Suppose we are given black boxes implementing the addition and multiplication operations on a finite ring R, not necessarily commutative, along with a set of generators for R. With respect to addition, R forms a finite Abelian group (R,+). As shown in [119], on a quantum computer one can find in poly(log |R|) time a set of additive generators \( \{h_1,\ldots,h_m\} \subset R \) such that \( (R,+) \simeq \langle h_1 \rangle \times \ldots \times \langle h_M \rangle\) and m is polylogarithmic in |R|. This allows efficient computation of a multiplication tensor for R. As shown in [118], one can similarly find an additive generating set for any ideal in R. This allows one to find the intersection of two ideals, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. As shown in [120], one can also use a quantum computer to efficiently decide whether a given polynomial is identically zero on a given finite black box ring. Known classical algorithms for these problems scale as poly(|R|).

Algorithm: Counterfeit Coins
Speedup: Polynomial
Description: Suppose we are given N coins, k of which are counterfeit. The real coins are all of equal weight, and the counterfeit coins are all of some other equal weight. We have a pan balance and can compare the weight of any pair of subsets of the coins. Classically, we need \( \Omega(k \log(N/k)) \) weighings to identify all of the counterfeit coins. We can introduce an oracle such that given a pair of subsets of the coins of equal cardinality, it outputs one bit indicating balanced or unbalanced. Building on previous work by Terhal and Smolin [137], Iwama et al. have shown [136] that on a quantum computer, one can identify all of the counterfeit coins using \( O(k^{1/4}) \) queries. The core techniques behind the quantum speedup are amplitude amplification and the Bernstein-Vazirani algorithm.

Algorithm: Matrix Rank
Speedup: Polynomial
Description: Suppose we are given oracle access to the (integer) entries of an \( n \times m \) matrix A. We wish to determine the rank of the matrix. Classically this requires order nm queries. Building on [149], Belovs [150] gives a quantum algorithm that can use fewer queries given a promise that the rank of the matrix is at least r. Specifically, Belovs' algorithm uses \( O(\sqrt{r(n-r+1)}LT) \) queries, where L is the root-mean-square of the reciprocals of the r largest singular values of A and T is a factor that depends on the sparsity of the matrix. For general A, \( T = O(\sqrt{nm}) \). If A has at most k nonzero entries in any row or column then \( T = O(k \log(n+m)) \). (To achieve the corresponding query complexity in the k-sparse case, the oracle must take a column index as input, and provide a list of the nonzero matrix elements from that column as output.) As an important special case, one can use these quantum algorithms for the problem of determining whether a square matrix is singular, which is sometimes referred to as the determinant problem. For general A the quantum query complexity of the determinant problem is no lower than the classical query complexity [151]. However, [151] does not rule out a quantum speedup given a promise on A, such as sparseness or lack of small singular values.

Algorithm: Boolean Matrix Multiplication
Speedup: Polynomial
Description: Boolean variables form a semiring with logical OR serving as addition and logical AND serving as multiplication. Multiplication of matrices with Boolean entries can be defined correspondingly. Let A,B be n by n Boolean matrices. Let C=AB, and let l be the number of entries of C that are equal to 1 (i.e. TRUE). Improving upon [19,155,157,156], the best known upper bound on quantum query complexity is \( \tilde{O}(n \sqrt{l}) \), as shown in [161]. For discussion of quantum time-complexity, and a comparison to classical algorithms, see [155].

Algorithm: Subset finding
Speedup: Polynomial
Description: We are oracle access to a function \( f:D \to R \) where D and R are finite sets. Some property \( P \subset (D \times R)^k \) is specified, for example as an explicit list. Our task is to find a size-k subset of D satisfying P by querying f or reject if none exists. Generalizing the result of [7], it was shown in [162] that this can be achieved using \(O(|D|^{k/(k+1)}) \) quantum queries. As an noteworthy special case, this algorithm solves the k-subset-sum problem of finding k numbers from a list with some desired sum. A matching lower bound for the quantum query complexity is proven in [163].

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Approximation and Simulation Algorithms

Algorithm: Quantum Simulation
Speedup: Superpolynomial
Description: It is believed that for any physically realistic Hamiltonian H on n degrees of freedom, the corresponding time evolution operator \( e^{-i H t} \) can be implemented using poly(n,t) gates. Unless BPP=BQP, this problem is not solvable in general on a classical computer in polynomial time. Many techniques for quantum simulation have been developed for different applications [25,95,92,5,1,12,63,68,99,107,145,166]. The exponential complexity of classically simulating quantum systems led Feynman to first propose that quantum computers might outperform classical computers on certain tasks [40]. Although the problem of finding ground energies of local Hamiltonians is QMA-complete and therefore probably requires exponential time on a quantum computer in the worst case, quantum algorithms have been developed to approximate ground and thermal states for some classes of Hamiltonians [102,132,121].

Algorithm: Knot Invariants
Speedup: Superpolynomial
Description: As shown by Freedman [42, 41], et al., finding a certain additive approximation to the Jones polynomial of the plat closure of a braid at \( e^{i 2 \pi/5} \) is a BQP-complete problem. This result was reformulated and extended to \( e^{i 2 \pi/k} \) for arbitrary k by Aharonov et al. [4,2]. Wocjan and Yard further generalized this, obtaining a quantum algorithm to estimate the HOMFLY polynomial [93], of which the Jones polynomial is a special case. Aharonov et al. subsequently showed that quantum computers can in polynomial time estimate a certain additive approximation to the even more general Tutte polynomial for planar graphs [3]. It is not fully understood for what range of parameters the approximation obtained in [3] is BQP-hard. (See also partition functions.) As shown in [83], the problem of finding a certain additive approximation to the Jones polynomial of the trace closure of a braid at \( e^{i 2 \pi/5} \) is DQC1-complete.

Algorithm: Three-manifold Invariants
Speedup: Superpolynomial
Description: The Turaev-Viro invariant is a function that takes three-dimensional manifolds as input and produces a real number as output. Homeomorphic manifolds yield the same number. Given a three-manifold specified by a Heegaard splitting, a quantum computer can efficiently find a certain additive approximation to its Turaev-Viro invariant, and this approximation is BQP-complete [129]. Earlier, in [114], a polynomial-time quantum algorithm was given to additively approximate the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation. Squaring the WRT invariant yields the Turaev-Viro invariant. However, it is unknown whether the approximation achieved in [114] is BQP-complete. A suggestion of a possible link between quantum computation and three-manifold invariants was also given in [115].

Algorithm: Partition Functions
Speedup: Superpolynomial
Description: For a classical system with a finite set of states S the partition function is \( Z = \sum_{s \in S} e^{-E(s)/kT} \), where T is the temperature and k is Boltzmann's constant. Essentially every thermodynamic quantity can be calculated by taking an appropriate partial derivative of the partition function. The partition function of the Potts model is a special case of the Tutte polynomial. A quantum algorithm for approximating the Tutte polynomial is given in [3]. Some connections between these approaches are discussed in [67]. Additional algorithms for estimating partition functions on quantum computers are given in [112,113,45,47]. A BQP-completeness result (where the "energies" are allowed to be complex) is also given in [113]. A method for approximating partition functions by simulating thermalization processes is given in [121]. A quadratic speedup for the approximation of general partition functions is given in [122].

Algorithm: Adiabatic Optimization
Speedup: Unknown
Description: In adiabatic quantum computation one starts with an initial Hamiltonian whose ground state is easy to prepare, and slowly varies the Hamiltonian to one whose ground state encodes the solution to some computational problem. By the adiabatic theorem, the system will track the instantaneous ground state provided the variation of the Hamiltonian is sufficiently slow. The runtime of an adiabatic algorithm scales as a polynomial in 1/g, where g is the minimum eigenvalue gap between the ground state and the first excited state. Adiabatic quantum computation was first proposed by Farhi et al. as a method for solving NP-complete combinatorial optimization problems [96]. Despite much effort, the scaling of the eigenvalue gap, and hence the asymptotic runtime of this class of adiabatic quantum algorithms remains unknown. It is known however, that adiabatic quantum computation is equally powerful as the standard quantum circuit model. That is, each can simulate the other with polynomial overhead [97]. Furthermore, adiabatic quantum computers can perform a process somewhat analogous to Grover search in \( O(\sqrt{N}) \) time [98]. See also quantum simulation, as some quantum simulation algorithms use adiabatic state preparation.

Algorithm: Zeta Functions
Speedup: Superpolynomial
Description: As shown by Kedlaya [64], quantum computers can determine the zeta function of a genus g curve over a finite field \( \mathbb{F}_q \) in time polynomial in g and log(q). No polynomial time classical algorithm for this problem is known. More speculatively, van Dam has conjectured that due to a connection between the zeros of zeta functions and the eigenvalues of certain quantum operators, quantum computers might be able to efficiently approximate the number of solutions to equations over finite fields [87]. Some evidence supporting this conjecture is given in [87].

Algorithm: Weight Enumerators
Speedup: Superpolynomial
Description: Let C be a code on n bits, i.e. a subset of \( \mathbb{Z}_2^n \). The weight enumerator of C is \( S_C(x,y) = \sum_{c \in C} x^{|c|} y^{n-|c|} \) where |c| denotes the Hamming weight of c. Weight enumerators have many uses in the study of classical codes. If C is a linear code, it can be defined by \( C = \{c: Ac = 0\} \) where A is a matrix over \( \mathbb{Z}_2 \) In this case \( S_C(x,y) = \sum_{c:Ac=0} x^{|c|} y^{n-|c|} \). Quadratically signed weight enumerators (QWGTs) are a generalization of this: \( S(A,B,x,y) = \sum_{c:Ac=0} (-1)^{c^T B c} x^{|c|} y^{n-|c|} \). Now consider the following special case. Let A be an \( n \times n \) matrix over \( \mathbb{Z}_2 \) such that diag(A)=I. Let lwtr(A) be the lower triangular matrix resulting from setting all entries above the diagonal in A to zero. Let l,k be positive integers. Given the promise that \( |S(A,\mathrm{lwtr}(A),k,l)| \geq \frac{1}{2} (k^2+l^2)^{n/2} \) the problem of determining the sign of \( S(A,\mathrm{lwtr}(A),k,l) \) is BQP-complete, as shown by Knill and Laflamme in [65]. The evaluation of QWGTs is also closely related to the evaluation of Ising and Potts model partition functions [67,45,46].

Algorithm: Simulated Annealing
Speedup: Polynomial
Description: In simulated annealing, one has a series of Markov chains defined by stochastic matrices \( M_1, M_2,\ldots,M_n \). These are slowly varying in the sense that their limiting distributions \( pi_1, \pi_2, \ldots, \pi_n \) satisfy \( |\pi_{t+1} -\pi_t| \lt \epsilon \) for some small \( \epsilon \). These distributions can often be thought of as thermal distributions at successively lower temperatures. If \( \pi_1 \) can be easily prepared, then by applying this series of Markov chains one can sample from \( \pi_n \). Typically, one wishes for \( \pi_n \) to be a distribution over good solutions to some optimization problem. Let \( \delta_i \) be the gap between the largest and second largest eigenvalues of \( M_i \). Let \( \delta = \min_i \delta_i \). The run time of this classical algorithm is proportional to \( 1/\delta \). Building upon results of Szegedy [135,85], Sommaet al. have shown [84] that quantum computers can sample from \( \pi_n \) with a runtime proportional to \( 1/\sqrt{\delta} \).

Algorithm: String Rewriting
Speedup: Superpolynomial
Description: String rewriting is a fairly general model of computation. String rewriting systems (sometimes called grammars) are specified by a list of rules by which certain substrings are allowed to be replaced by certain other substrings. For example, context free grammars, are equivalent to the pushdown automata. In [59], Janzing and Wocjan showed that a certain string rewriting problem is PromiseBQP-complete. Thus quantum computers can solve it in polynomial time, but classical computers probably cannot. Given three strings s,t,t', and a set of string rewriting rules satisfying certain promises, the problem is to find a certain approximation to the difference between the number of ways of obtaining t from s and the number of ways of obtaining t' from s. Similarly, certain problems of approximating the difference in number of paths between pairs of vertices in a graph, and difference in transition probabilities between pairs of states in a random walk are also BQP-complete [58].

Algorithm: Matrix Powers
Speedup: Superpolynomial
Description: Quantum computers have an exponential advantage in approximating matrix elements of powers of exponentially large sparse matrices. Suppose we are have an \( N \times N \) symmetric matrix A such that there are at most polylog(N) nonzero entries in each row, and given a row index, the set of nonzero entries can be efficiently computed. The task is, for any 1 < i < N, and any m polylogarithmic in N, to approximate \( (A^m)_{ii} \) the \( i^{\mathrm{th}} \) diagonal matrix element of \( A^m \). The approximation is additive to within \( b^m \epsilon \) where b is a given upper bound on |A| and \( \epsilon \) is of order 1/polylog(N). As shown by Janzing and Wocjan, this problem is PromiseBQP-complete, as is the corresponding problem for off-diagonal matrix elements [60]. Thus, quantum computers can solve it in polynomial time, but classical computers probably cannot.

---

Acknowledgments

I thank the following people for contributing their expertise (in chronological order).

• Daniel Lidar
• Wim van Dam
• Geordie Rose
• Yi-Kai Liu
• Robin Kothari
• Martin Schwarz
• Dorit Aharonov
• Alessandro Cosentino
• Andrew Childs
• Stacey Jeffery

---

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78

Oded Regev.
Quantum computation and lattice problems.
In Proceedings of the 43rd Symposium on Foundations of Computer Science, 2002.
arXiv:cs/0304005.

79

Oded Regev.
A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space.
arXiv:quant-ph/0406151, 2004.

80

Ben Reichardt and Robert Špalek.
Span-program-based quantum algorithm for evaluating formulas.
Proceedings of STOC 2008
arXiv:0710.2630.

81

Martin Roetteler and Thomas Beth.
Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups.
arXiv:quant-ph/9812070, 1998.

82

Peter W. Shor.
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.
SIAM Journal on Computing, 26(5):1484-1509, 2005.
arXiv:quant-ph/9508027.

83

Peter W. Shor and Stephen P. Jordan.
Estimating Jones polynomials is a complete problem for one clean qubit.
Quantum Information and Computation, 8(8/9):681-714, 2008.
arXiv:0707.2831.

84

R. D. Somma, S. Boixo, and H. Barnum.
Quantum simulated annealing.
arXiv:0712.1008, 2007.

85

M. Szegedy.
Quantum speed-up of Markov chain based algorithms.
In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pg. 32, 2004.

86

Wim van Dam.
Quantum algorithms for weighing matrices and quadratic residues.
Algorithmica, 34(4):413-428, 2002.
arXiv:quant-ph/0008059.

87

Wim van Dam.
Quantum computing and zeros of zeta functions.
arXiv:quant-ph/0405081, 2004.

88

Wim van Dam and Sean Hallgren.
Efficient quantum algorithms for shifted quadratic character problems.
arXiv:quant-ph/0011067, 2000.

89

Wim van Dam, Sean Hallgren, and Lawrence Ip.
Quantum algorithms for some hidden shift problems.
SIAM Journal on Computing, 36(3):763-778, 2006.
arXiv:quant-h/0211140.

90

Wim van Dam and Gadiel Seroussi.
Efficient quantum algorithms for estimating Gauss sums.
arXiv:quant-ph/0207131, 2002.

91

John Watrous.
Quantum algorithms for solvable groups.
In Proceedings of the 33rd ACM Symposium on Theory of Computing, pages 60-67, 2001.
arXiv:quant-ph/0011023.

92

Stephen Wiesner.
Simulations of many-body quantum systems by a quantum computer.
arXiv:quant-ph/9603028, 1996.

93

Pawel Wocjan and Jon Yard.
The Jones polynomial: quantum algorithms and applications in quantum complexity theory.
Quantum Information and Computation 8(1/2):147-180, 2008.
arXiv:quant-ph/0603069.

94

Andrew Yao.
On computing the minima of quadratic forms.
In Proceedings of the 7th ACM Symposium on Theory of Computing, pages 23-26, 1975.

95

Christof Zalka.
Efficient simulation of quantum systems by quantum computers.
Proceedings of the Royal Society of London Series A, 454:313, 1996.
arXiv:quant-ph/9603026.

96

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser.
Quantum computation by adiabatic evolution.
arXiv:quant-ph/0001106, 2000.

97

Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev.
Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation.
SIAM Journal on Computing, 37(1):166-194, 2007.
arXiv:quant-ph/0405098

98

Jeremie Roland and Nicolas J. Cerf
Quantum search by local adiabatic evolution.
Physical Review A, 65(4):042308, 2002.
arXiv:quant-ph/0107015

99

L.-A. Wu, M.S. Byrd, and D. A. Lidar
Polynomial-Time Simulation of Pairing Models on a Quantum Computer.
Physical Review Letters, 89(6):057904, 2002.
arXiv:quant-ph/0108110

100

Eli Biham, Ofer Biham, David Biron, Markus Grassl, and Daniel Lidar
Grover's quantum search algorithm for an arbitrary initial amplitude distribution.
Physical Review A, 60(4):2742, 1999.
arXiv:quant-ph/9807027 and arXiv:quant-ph/0010077

101

Andrew Childs, Shelby Kimmel, and Robin Kothari
The quantum query complexity of read-many formulas
arXiv:1112.0548, 2011.

102

Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon
Simulated quantum computation of molecular energies.
Science, 309(5741):1704-1707, 2005.
arXiv:quant-ph/0604193

103

A. M. Childs, A. J. Landahl, and P. A. Parrilo
Quantum algorithms for the ordered search problem via semidefinite programming.
Physical Review A, 75 032335, 2007.
arXiv:quant-ph/0608161

104

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd
Quantum algorithm for solving linear systems of equations.
Phys. Rev. Lett. 15(103):150502, 2009.
arXiv:0811.3171.

105

Martin Roetteler
Quantum algorithms for highly non-linear Boolean functions.
Proceedings of SODA 2010
arXiv:0811.3208.

106

Stephen P. Jordan
Fast quantum algorithms for approximating the irreducible representations of groups.
arXiv:0811.0562, 2008.

107

Tim Byrnes and Yoshihisa Yamamoto
Simulating lattice gauge theories on a quantum computer.
Physical Review A, 73, 022328, 2006.
arXiv:quant-ph/0510027.

108

D. Simon
On the Power of Quantum Computation.
In Proceedings of the 35th Symposium on Foundations of Computer Science, pg. 116-123, 1994.

109

John Proos and Christof Zalka
Shor's discrete logarithm quantum algorithm for elliptic curves.
Quantum Information and Computation Vol. 3, No. 4, pg.317-344, 2003.
arXiv:quant-ph/0301141.

110

Yi-Kai Liu
Quantum algorithms using the curvelet transform.
Proceedings of STOC 2009 pg. 391-400.
arXiv:0810.4968.

111

Wim van Dam and Igor Shparlinski
Classical and quantum algorithms for exponential congruences.
Proceedings of TQC 2008 pg. 1-10.
arXiv:0804.1109.

112

Itai Arad and Zeph Landau
Quantum computation and the evaluation of tensor networks.
arXiv:0805.0040, 2008.

113

M. Van den Nest, W. Dür, R. Raussendorf, and H. J. Briegel
Quantum algorithms for spin models and simulable gate sets for quantum computation.
arXiv:0805.1214, 2008.

114

Silvano Garnerone, Annalisa Marzuoli, and Mario Rasetti
Efficient quantum processing of 3-manifold topological invariants.
arXiv:quant-ph/0703037, 2007.

115

Louis H. Kauffman and Samuel J. Lomonaco Jr.
q-deformed spin networks, knot polynomials and anyonic topological quantum computation.
Journal of Knot Theory, Vol. 16, No. 3, pg. 267-332, 2007.
arXiv:quant-ph/0606114.

116

Arthur Schmidt and Ulrich Vollmer
Polynomial time quantum algorithm for the computation of the unit group of a number field.
In Proceedings of the 37th Symposium on the Theory of Computing, pg. 475-480, 2005.

117

Sergey Bravyi, Aram Harrow, and Avinatan Hassidim
Quantum algorithms for testing properties of distributions.
arXiv:0907.3920, 2009.

118

Pawel M. Wocjan, Stephen P. Jordan, Hamed Ahmadi, and Joseph P. Brennan
Efficient quantum processing of ideals in finite rings.
arXiv:0908.0022, 2009.

119

V. Arvind, Bireswar Das, and Partha Mukhopadhyay
The complexity of black-box ring problems.
In Proceedings of COCCOON 2006, pg 126-145.

120

V. Arvind and Partha Mukhopadhyay
Quantum query complexity of multilinear identity testing.
In Proceedings of STACS 2009, pg. 87-98.

121

David Poulin and Pawel Wocjan
Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer.
Phys. Rev. Lett. 103:220502, 2009.
arXiv:0905.2199

122

Pawel Wocjan, Chen-Fu Chiang, Anura Abeyesinghe, and Daniel Nagaj
Quantum speed-up for approximating partition functions.
Phys. Rev. A 80:022340, 2009.
arXiv:0811.0596

123

Ashley Montanaro
Quantum search with advice.
In Proceedings of the 5th conference on Theory of quantum computation, communication, and cryptography (TQC 2010)
arXiv:0908.3066

124

Laszlo Babai, Robert Beals, and Akos Seress
Polynomial-time theory of matrix groups.
In Proceedings of STOC 2009, pg. 55-64.

125

Peter Shor
Algorithms for Quantum Computation: Discrete Logarithms and Factoring.
In Proceedings of FOCS 1994, pg. 124-134.

126

Aaron Denney, Cristopher Moore, and Alex Russell
Finding conjugate stabilizer subgroups in PSL(2;q) and related groups.
arXiv:0809.2445, 2008.

127

Kevin K. H. Cheung and Michele Mosca
Decomposing finite Abelian groups.
Quantum Information and Computation 1(2):26-32, 2001.
arXiv:cs/0101004.

128

François Le Gall
An efficient quantum algorithm for some instances of the group isomorphism problem.
In Proceedings of STACS 2010.
arXiv:1001.0608.

129

Gorjan Alagic, Stephen Jordan, Robert Koenig, and Ben Reichardt
Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation.
Phys. Rev. A 82, 040302(R) (2010)
arXiv:1003.0923

130

Martin Rötteler
Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm.
In Proceedings of MFCS 2009, pg 663-674.
arXiv:0911.4724.

131

Arthur Schmidt
Quantum Algorithms for many-to-one Functions to Solve the Regulator and the Principal Ideal Problem.
arXiv:0912.4807, 2009.

132

K. Temme, T.J. Osborne, K.G. Vollbrecht, D. Poulin, and F. Verstraete
Quantum Metropolis Sampling.
Nature, Vol. 471, pg. 87-90, 2011.
arXiv:0911.3635.

133

Andris Ambainis
Quantum Search Algorithms.
SIGACT News, 35 (2):22-35, 2004.
arXiv:quant-ph/0504012

134

Nicolas J. Cerf, Lov K. Grover, and Colin P. Williams
Nested quantum search and NP-hard problems.
Applicable Algebra in Engineering, Communication and Computing, 10 (4-5):311-338, 2000.

135

Mario Szegedy
Spectra of Quantized Walks and a \( \sqrt{\delta \epsilon} \) rule.
arXiv:quant-ph/0401053, 2004.

136

Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Junichi Teruyama
Quantum Counterfeit Coin Problems.
In Proceedings of 21st International Symposium on Algorithms and Computation (ISAAC2010), LNCS 6506, pp.73-84 (2010)
arXiv:1009.0416

137

Barbara Terhal and John Smolin
Single quantum querying of a database.
Phys. Rev. A 58:1822, 1998.
arXiv:quant-ph/9705041

138

Andris Ambainis
Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations.
arXiv:1010.4458, 2010.

139

Frédéric Magniez and Ashwin Nayak
Quantum complexity of testing group commutativity.
In Proceedings of 32nd International Colloquium on Automata, Languages and Programming. LNCS 3580, pg. 1312-1324, 2005.
arXiv:quant-ph/0506265

140

Andrew Childs and Robin Kothari
Quantum query complexity of minor-closed graph properties.
In Proceedings of the 28th Symposium on Theoretical Aspects of Computer Science (STACS 2011), pg. 661-672
arXiv:1011.1443

141

Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha
Search via quantum walk.
In Proceedings of the 39th ACM Symposium on the Theory of Computing (STOC 2007), pg. 575-584.
arXiv:quant-ph/0608026

142

Dmitry Gavinsky, Martin Roetteler, and Jérémy Roland
Quantum algorithm for the Boolean hidden shift problem.
In Proceedings of the 17th annual international conference on Computing and combinatorics (COCOON '11), 2011
arXiv:1103.3017

143

Mark Ettinger and Peter Høyer
On quantum algorithms for noncommutative hidden subgroups.
Advances in Applied Mathematics, Vol. 25, No. 3, pg. 239-251, 2000.
arXiv:quant-ph/9807029

144

Andris Ambainis, Andrew Childs, and Yi-Kai Liu
Quantum property testing for bounded-degree graphs.
In Proceedings of RANDOM '11: Lecture Notes in Computer Science 6845, pp. 365-376, 2011.
arXiv:1012.3174

145

G. Ortiz, J.E. Gubernatis, E. Knill, and R. Laflamme
Quantum algorithms for Fermionic simulations.
Phys. Rev. A 64: 022319, 2001.
arXiv:cond-mat/0012334

146

Ashley Montanaro
The quantum query complexity of learning multilinear polynomials.
arXiv:1105.3310, 2011.

147

Tad Hogg
Highly structured searches with quantum computers.
Phys. Rev. Lett. 80: 2473, 1998.

148

Markus Hunziker and David A. Meyer
Quantum algorithms for highly structured search problems.
Quantum Information Processing, Vol. 1, No. 3, pg. 321-341, 2002.

149

Ben Reichardt
Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function.
arXiv:0904.2759
In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS '09), pg. 544-551, 2009.

150

Aleksandrs Belovs
Span-program-based quantum algorithm for the rank problem.
arXiv:1103.0842, 2011.

151

Sebastian Dörn and Thomas Thierauf
The quantum query complexity of the determinant.
Information Processing Letters Vol. 109, No. 6, pg. 305-328, 2009.

152

Aleksandrs Belovs
Span programs for functions with constant-sized 1-certificates.
arXiv:1105.4024, 2011.

153

Troy Lee, Frédéric Magniez, and Mikos Santha
A learning graph based quantum query algorithm for finding constant-size subgraphs.
arXiv:1109.5135, 2011.

154

Aleksandrs Belovs and Troy Lee
Quantum algorithm for k-distinctness with prior knowledge on the input.
arXiv:1108.3022, 2011.

155

François Le Gall
Improved output-sensitive quantum algorithms for Boolean matrix multiplication.
In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12), 2012.

156

Stacey Jeffery and Frédéric Magniez
Improving quantum query complexity of Boolean matrix multiplication using graph collision.
arXiv:1112.5855, 2011.

157

Virginia Vassilevska Williams and Ryan Williams
Subcubic equivalences between path, matrix, and triangle problems.
In 51st IEEE Symposium on Foundations of Computer Science (FOCS '10) pg. 645 - 654, 2010.

158

Ben W. Reichardt
Reflections for quantum query algorithms.
In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pg. 560-569, 2011.
arXiv:1005.1601

159

Ben W. Reichardt
Span-program-based quantum algorithm for evaluating unbalanced formulas.
arXiv:0907.1622, 2009.

160

Ben W. Reichardt
Faster quantum algorithm for evaluating game trees.
In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pg. 546-559, 2011.
arXiv:0907.1623

161

Stacey Jeffery, Robin Kothari, and Frédéric Magniez
Improving quantum query complexity of Boolean matrix multiplication using graph collision.
arXiv:1112.5855, 2012.

162

Andrew M. Childs and Jason M. Eisenberg
Quantum algorithms for subset finding.
Quantum Information and Computation 5(7):593-604, 2005.
arXiv:quant-ph/0311038.

163

Aleksandrs Belovs and Robert Špalek
Adversary lower bound for the k-sum problem.
arXiv:1206.6528, 2012.

164

Bohua Zhan, Shelby Kimmel, and Avinatan Hassidim
Super-polynomial quantum speed-ups for Boolean evaluation trees with hidden structure.
ITCS 2012: Proceedings of the 3rd Innovations in Theoretical Computer Science, ACM, Pg. 249-265.
arXiv:1101.0796

165

Shelby Kimmel
Quantum adversary (upper) bound.
39th International Colloquium on Automata, Languages and Programming - ICALP 2012 Volume 7391, p. 557-568.
arXiv:1101.0797

166

Stephen Jordan, Keith Lee, and John Preskill
Quantum algorithms for quantum field theories
Science, Vol. 336, pg. 1130-1133. (2012)
arXiv:1111.3633

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Terminology

If there exists a positive constant \( \alpha \) such that the runtime \( C(n) \) of the best known classical algorithm and the runtime \( Q(n) \) of the quantum algorithm satisfy \( C = 2^{\Omega(Q^\alpha)} \) then I call the speedup superpolynomial. Otherwise I call it polynomial. For a review of the \( O, \Omega, \Theta, \tilde{O}, ... \) notations see the Wikipedia article.

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Author:

Stephen Jordan

Applied and Computational Mathematics Division, NIST

Last updated: August 27, 2012
Date created: April 22, 2011

See also: ITL Quantum Information

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