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-This is a comprehensive catalog of quantum algorithms. If you notice any errors or -omissions, please email me at stephen.jordan@microsoft.com. (Alternatively, you may -submit a pull request to the repository on github.) -Your help is appreciated and will be acknowledged.
-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 -\( \widetilde{O} (n^3) \) time -[82,125]. -The fastest known classical algorithm for integer factorization is the -general number field sieve, which is believed to run in time \( -2^{\widetilde{O}(n^{1/3})} \). The best rigorously proven upper bound -on the classical complexity of factoring is \( O(2^{n/4+o(1)}) \) via the Pollard-Strassen algorithm -[252, 362]. Shor's -factoring algorithm breaks RSA public-key encryption and the -closely related quantum algorithms for discrete logarithms break the DSA -and ECDSA digital signature schemes and the Diffie-Hellman -key-exchange protocol. A quantum algorithm even faster than Shor's for -the special case of factoring “semiprimes”, which are -widely used in cryptography, is given in [271]. -If small factors exist, Shor's algorithm can be beaten by a quantum -algorithm using Grover search to speed up the elliptic curve factorization -method [366]. Additional optimized versions of Shor's algorithm are given in [384, 386]. There are proposed classical public-key -cryptosystems not believed to be broken by quantum algorithms, cf. -[248]. At the core of Shor's factoring algorithm is -order finding, which can be reduced to the Abelian hidden subgroup problem, -which is solved using the quantum Fourier transform. A number of other -problems are known to reduce to integer factorization including the -membership problem for matrix groups over fields of odd order -[253], and certain diophantine problems relevant to -the synthesis of quantum circuits [254]. -
- -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, 14]. A further optimization to Shor's algorithm is given in [385]. The superpolynomial quantum speedup has also been extended to the discrete -logarithm problem on semigroups [203, -204]. 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]. A quantum - algorithm solving the principal ideal problem in number fields of arbitrary degree - (i.e. scaling polynomially in the degree) was subsequently given in - [329]. 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. Hallgren - and collaborators subsequently discovered how to achieve polynomial scaling - in the degree [213]. See also [329]. - The algorithms rely on solving Abelian hidden subgroup problems over the additive - group of real numbers. -
- -Algorithm: Class 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 - \( \mathbb{Q}(\theta) \), which is a Dedekind domain. For a Dedekind domain, the nonzero - fractional ideals modulo the nonzero principal ideals form a group called the class group. - As shown by Hallgren [50], a quantum computer can find - a set of generators for the class group of the ring of - integers of any constant degree number field, given a description of \( \theta \), in time - poly(log(\( | \mathcal{O} | \))). An improved quantum algorithm, whose runtime is also - polynomial in d was subsequently given in [329]. - No polynomial time classical algorithm for these problems are 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:Primality Proving
-Speedup:Polynomial
-Description: Given an n-bit number, return a proof of its primality. The fastest classical -algorithms are AKS, the best versions of which [393, 394] have -essentially-quartic complexity, and ECPP, where the heuristic complexity of the fastest version -[395] is also essentially quartic. The fastest known quantum algorithm for this -problem is the method of Donis-Vela and Garcia-Escartin [396], with complexity -\( O(n^2 (\log \ n)^3 \log \ \log \ n) \). This improves upon a prior factoring-based quantum algorithm -for primality proving [397] that has complexity \( O(n^3 \log \ n \ \log \ \log \ n) \). -A recent result of Harvey and Van Der Hoeven [398] can be used to improve the -complexity of the factoring-based quantum algorithm for primality proving to \( O(n^3 \log n) \) -and it may be possible to similarly reduce the complexity of the Donis-Vela-Garcia-Escartin algorithm -to \( O(n^2 (\log \ n)^3) \) [399]. -
- -Algorithm:Solving Exponential Congruences
-Speedup:Polynomial
-Description: - We are given \( a,b,c,f,g \in \mathbb{F}_q \). We must find integers \(x,y\) - such that \( a f^x + b g^y = c \). As shown in [111], - quantum computers can solve this problem in \( \widetilde{O}(q^{3/8}) \) time whereas the best - classical algorithm requires \( \widetilde{O}(q^{9/8}) \) 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. Conjugating the regular representation of a group by the -quantum Fourier transform circuit over that group yields a direct sum -of the group's irreducible representations. Thus, the efficient -quantum Fourier transform over the symmetric -group [196], together with the Hadamard -test, yields a fast quantum algorithm for additively approximating -individual matrix elements of the arbitrary irreducible -representations of \( S_n \). Similarly, using the quantum Schur -transform [197], one can efficiently approximate matrix elements of -the irreducible representations of SU(n) that have polynomial weight. -Direct implementations of individual irreducible representations for -the groups U(n), SU(n), SO(n), and \( A_n \) by efficient quantum -circuits are given in [106]. Instances that appear -to be exponentially hard for known classical algorithms are also -identified in [106].
- -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]. -
- -Algorithm: Subset-sum
-Speedup: Polynomial
-Description: Given a list of integers \( x_1,\ldots,x_n \), and -a target integer s, the subset-sum problem is to determine -whether the sum of any subset of the given integers adds up -to s. This problem is NP-complete, and therefore is unlikely to -be solvable by classical or quantum algorithms with polynomial -worst-case complexity. In the hard instances the given integers -are of order \( 2^n \) and much research on subset sum focuses on -average case instances in this regime. In [178], a quantum -algorithm is given that solves such instances in time \( 2^{0.241n} \), -up to polynomial factors. This quantum algorithm works by applying a -variant of Ambainis's quantum walk algorithm for -element-distinctness [7] to speed up a sophisticated -classical algorithm for this problem due to Howgrave-Graham and -Joux. The fastest known classical algorithm for such instances of subset-sum runs in time -\( 2^{0.291n} \), up to polynomial factors [404].
- -Algorithm: Decoding
-Speedup: Varies
-Description: Classical error correcting codes allow the -detection and correction of bit-flips by storing data -reduntantly. Maximum-likelihood decoding for arbitrary linear codes is -NP-complete in the worst case, but for structured codes or bounded -error efficient decoding algorithms are known. Quantum algorithms have -been formulated to speed up the decoding of convolutional codes -[238] and simplex codes -[239].
- -Algorithm: Quantum Cryptanalysis
-Speedup: Various
-Description: It is well-known that Shor's algorithms for -factoring and discrete logarithms -[82,125] -completely break the RSA and Diffie-Hellman cryptosystems, as well as -their elliptic-curve-based variants [109, 14]. -(A number of "post-quantum" public-key cryptosystems have been proposed to replace these -primitives, which are not known to be broken by quantum attacks.) -Beyond Shor's algorithm, there is a growing body of work on quantum -algorithms specifically designed to attack cryptosystems. These -generally fall into three categories. The first is quantum algorithms -providing polynomial or sub-exponential time attacks on cryptosystems -under standard assumptions. In particular, the algorithm of Childs, -Jao, and Soukharev for finding isogenies of elliptic curves breaks -certain elliptic curve based cryptosystems in subexponential time -that were not already broken by Shor's algorithm -[283]. The second category is quantum algorithms -achieving polynomial improvement over known classical cryptanalytic -attacks by speeding up parts of these classical algorithms using -Grover search, quantum collision finding, etc. Such attacks -on private-key [284, 285, 288, 315, 316] -and public-key [262, 287] primitives, do not -preclude the use of the associated cryptosystems but may influence -choice of key size. The third category is attacks that make use of quantum -superposition queries to block ciphers. These attacks in many cases completely -break the cryptographic primitives [286, 289, -290, 291, 292]. -However, in most practical situations such superposition queries are -unlikely to be feasible. - - -
- - -
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], which is - optimal [216]. 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, 255], take advantage of alternative - initial states [100] or nonuniform probabilistic priors - [123], work with oracles whose runtime varies - between inputs [138], approximate - definite integrals [77], and converge to a - fixed-point - [208, 209]. Considerations on optimizing - the depth of quantum search circuits are given in [405]. 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]. For further examples of application of - Grover search and amplitude amplification see - [261, 262]. A problem closely - related to, but harder than, Grover search, is spatial search, in - which database queries are limited by some graph structure. On - sufficiently well-connected graphs, \(O(\sqrt{n})\) quantum query - complexity is still achievable [274,275,303, -304, 305, 306, 330]. -
- - -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]. It is normally assumed in period finding that the - function \(f(x) \neq f(y) \) unless \( x-y = s \), where \( s \) is the period. - A quantum algorithm which applies even when this restiction is relaxed is given in - [388]. Period finding has been generalized to apply to - oracles which provide only the few most significant bits about the underlying - function in [389]. -
- - -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 -[81,55,72,53,9,22,56,71,57,43,44,28,126,207,273]. - 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, except in - special cases [312]. However, - Kuperberg [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]. A - further improvement in the asymptotic scaling of the required number - of qubits is obtained in [218]. Quantum query speedups (though - not necessarily efficient quantum algorithms in terms of gate count) for - somewhat more general problems of testing for isomorphisms of functions under sets of - permutations are given in [311] -
- -Algorithm: Bernstein-Vazirani
-Speedup: Polynomial Directly, Superpolynomial Recursively
-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 recursive versions of - this problem, called recursive Fourier sampling, such that quantum computers require - exponentially fewer queries than classical computers - [11]. See - [256, 257] for related work -on the ubiquity of quantum speedups from generic quantum circuits and -[258, 270] for related work on a quantum query speedup -for detecting correlations between the an oracle function and the -Fourier transform of another. -
- -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 related, more recent, pedagogical - example is given in [259].) 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], - [165], and [269] 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: 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 would break a certain - cryptographic pseudorandom generator given the ability to make - quantum queries to the generator [89]. - A quantum speedup for hidden shift problems of difference sets is given in [312], - and this also subsumes the Legendre symbol problem as a special case. - 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]. A recent result, building upon [408, 43], - achieves exponential quantum speedups for some generalizations of the Hidden shift problem including - the hidden multiple shift problem, in which one has query access to \(f_s(x) = f(x-hs) \) - over some allowed range of s and one wishes to infer h [407]. -
- -Algorithm: Polynomial interpolation
-Speedup: Varies
-Description: Let \( p(x) = a_d x^d + \ldots + a_1 x + a_0 \) be a polynomial over the finite field \( \mathrm{GF}(q) \). One is given access to an oracle that, given \( x \in \mathrm{GF}(q) \), returns \( p(x) \). The polynomial reconstruction problem is, by making queries to the oracle, to determine the coefficients \( a_d,\ldots,a_0 \). Classically, \( d + 1 \) queries are necessary and sufficient. (In some sources use the term reconstruction instead of interpolation for this problem.) Quantumly, \( d/2 + 1/2 \) queries are necessary and \( d/2 + 1 \) queries are sufficient [360,361]. For multivariate polynomials of degree d in n variables the interpolation problem has classical query complexity \( \binom{n+d}{d} \). As shown in [387], the quantum query complexity is \( O \left( \frac{1}{n+1} \binom{n+d}{d} \right) \) over \( \mathbb{R} \) and \( \mathbb{C} \) and it is \( O \left( \frac{d}{n+d} \binom{n+d}{d} \right) \) over \( \mathbb{F}_q \) for sufficiently large q. Quantum algorithms have also been discovered for the case that the oracle returns \( \chi(f(x)) \) where \( \chi \) is a quadratic character of \( \mathrm{GF}(q) \) [390], and the case where the oracle returns \( f(x)^e \) [392]. These generalize the hidden shift algorithm of [89] and achieve an exponential speedup over classical computation. A quantum algorithm for reconstructing rational functions over finite fields given noisy and incomplete oracle access to the function values is given in [391]. -
- -Algorithm: Pattern matching
-Speedup: Superpolynomial
-Description: Given strings T of length n -and P of length m < n, both from some finite -alphabet, the pattern matching problem is to find an occurrence -of P as a substring of T or to report that P is -not a substring of T. More generally, T and P -could be d-dimensional arrays rather than one-dimensional -arrays (strings). Then, the pattern matching problem is to return the -location of P as a \(m \times m \times \ldots \times m\) -block within the \(n \times n \times \ldots \times n\) array T -or report that no such location exists. The \( \Omega(\sqrt{N}) \) -query lower bound for unstructured search [216] -implies that the worst-case quantum query complexity of this problem -is \( \Omega ( \sqrt{n} + \sqrt{m} ) \). A quantum algorithm achieving -this, up to logarithmic factors, was obtained in -[217]. This quantum algorithm works through the -use of Grover's algorithm together with a classical method called -deterministic sampling. More recently, Montanaro showed that -superpolynomial quantum speedup can be achieved on average case -instances of pattern matching, provided that m is greater than -logarithmic in n. Specifically, the quantum algorithm given in -[215] solves average case pattern matching in -\( \widetilde{O}((n/m)^{d/2} 2^{O(d^{3/2} \sqrt{\log m})})\) time. This -quantum algorithm is constructed by generalizing Kuperberg's quantum -sieve algorithm [66] for dihedral hidden subgroup and -hidden shift problems so that it can operate in d dimensions -and accomodate small amounts of noise, and then classically reducing the -pattern matching problem to this noisy d-dimensional version of -hidden shift. -
- -Algorithm: Ordered Search
-Speedup: Constant factor
-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 \( \frac{\ln 2}{\pi} \log_2 N \) - quantum queries has been proven for this problem - [219, 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. 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 \( \widetilde{O}(n^{3/2}) \), and only logarithmically many qubits, as shown in [317], building upon [13, 272, 318]. A span-program-based quantum algorithm for detecting trees of a given - size as minors in \( \widetilde{O}(n) \) time is given in - [240]. 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. According to the widely-believed Aanderaa-Karp-Rosenberg conjecture, all of the above problems have \( \Omega(n^2) \) classical query complexity. 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 \( O(n^{5/4}) \) - quantum queries [319], improving upon - [276, 175, 171, - 70, 152, - 21]. Stronger quantum query complexity upper bounds are known when the graphs are sufficiently sparse [319, 320]. 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 \( \widetilde{O}\left( n^{\frac{3}{2}-\frac{1}{\mathrm{vc}(H)+1}} \right) \) - queries, provided G is sparse, where vc(H) is the size of the minimal - vertex cover of H. A quantum algorithm for finding - constant-sized sub-hypergraphs over 3-uniform hypergraphs in \( - O(n^{1.883}) \) queries is given in [241]. -
- -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 \( \widetilde{O}(N^{1/3}) \). Also in [144], - it is shown that distinguishing expander graphs from graphs that are far from - being expanders has quantum complexity \( \widetilde{O}(N^{1/3}) \) and - \( \widetilde{\Omega}(N^{1/4}) \), whereas the classical complexity is - \( \widetilde{\Theta}(\sqrt{N}) \). The key quantum algorithmic tool is Ambainis' algorithm for - element distinctness. 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 - \( \widetilde{\Theta}(\sqrt{NM}) \) 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 \( \widetilde{\Theta}(\sqrt{NM}) \). In [317] quantum algorithms are given for st-connectivity, deciding bipartiteness, and deciding whether a graph is a forest, which run in \( \widetilde{O}(N \sqrt{d}) \) time and use only logarithmically many qubits. -
- -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 and Element Distinctness
-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]. (See also [315].) Removing the - promise that f is two-to-one yields a problem called element distinctness, which - has \( \Theta(N) \) 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 [7, 374]. The problem of - deciding whether any k-fold collisions exist is - called k-distinctness. Improving upon - [7,154], - the best quantum query complexity for k-distinctness is - \( O(n^{3/4 - 1/(4(2^k-1))}) \) - [172, 173]. For k=2,3 this is also the time-complexity, up to logarithmic - factors, by [7]. For \( k > 3\) the fastest known quantum algorithm has time complexity \( O(n^{(k-1)/k}) \) [363]. - Given two functions f and g, on domains of - size N and M, respectively a claw is a pair x,y such that f(x) = - g(y). In the case that N=M, the algorithm of [7] - solves claw-finding in \( O(N^{2/3}) \) queries, improving on the previous \( O(N^{3/4} \log N) \) quantum algorithm - of [21]. Further work gives improved query complexity for various parameter regimes -in which \(N \neq M\) [364, 365]. More generally, a related problem to - element distinctness, is, given oracle access to a sequence, to - estimate the \(k^{\mathrm{th}}\) frequency moment \(F_k = \sum_j n_j^k - \), where \(n_j\) is the number of times that j occurs in the - sequence. An approximately quadratic speedup for this problem is - obtained in [277]. See also graph collision. -
- - -Algorithm: Graph Collision
-Speedup: Polynomial
-Description: -We are given an undirected graph of n vertices and oracle -access to a labelling of the vertices by 1 and 0. The graph collision -problem is, by querying this oracle, to decide whether there exist a -pair of vertices, connected by an edge, both of which are labelled -1. One can embed Grover's unstructured search problem as an instance -of graph collision by choosing the star graph, labelling the center 1, -and labelling the remaining vertices by the database entries. Hence, -this problem has quantum query complexity \( \Omega(\sqrt{n}) \) and -classical query complexity \( \Theta (n) \). In -[70], Magniez, Nayak, and Szegedy gave -a \( O(N^{2/3}) \)-query quantum algorithm for graph collision on -general graphs. This remains the best upper bound on quantum query -complexity for this problem on general graphs. However, stronger upper -bounds have been obtained for several special classes of -graphs. Specifically, the quantum query complexity on a -graph G is \( \widetilde{O}(\sqrt{n} + \sqrt{l}) \) where l -is the number of non-edges in G [161], -\(O(\sqrt{n} \alpha^{1/6}) \) where \(\alpha\) is the size of the -largest independent set of G [172], -\(O(\sqrt{n} + \sqrt{\alpha^*})\) where \( \alpha^* \) is the maximum -total degree of any independent set of G -[200], and \(O(\sqrt{n} t^{1/6}) \) where t is -the treewidth of G [201]. Furthermore, the -quantum query complexity is \( \widetilde{O}(\sqrt{n}) \) with high -probability for random graphs in which the presence or absence of an -edge between each pair of vertices is chosen independently with fixed -probability, (i.e. Erdős-Rényi graphs) -[200]. See [201] for a -summary of these results as well as new upper bounds for two -additional classes of graph that are too complicated to describe here. -
- -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 \( \widetilde{\Theta}(k^{2/3}) \) - [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, 212]. -
- -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. Zatloukal has recently given an exponential -quantum speedup for some instances of a problem closely related to -group isomorphism, namely testing equivalence of group extensions -[202]. -
- -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]. A further improved quantum algorithm for - this task is obtained in [265]. -
- -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: Matrix Multiplication over Semirings
-Speedup: Polynomial
-Description: A semiring is a set endowed with addition and -multiplication operations that obey all the axioms of a ring except -the existence additive inverses. Matrix multiplication over various -semirings has many applications to graph problems. Matrix -multiplication over semirings can be sped up by straightforward Grover -improvements upon schoolbook multiplication, yielding a quantum -algorithm that multiplies a pair of \(n \times n\) matrices in \( -\widetilde{O}(n^{5/2}) \) time. For some semirings this algorithm -outperforms the fastest known classical algorithms and for some -semirings it does not [206]. -A case of particular interest is the Boolean semiring, in which OR -serves as addition and AND serves as multiplication. No quantum -algorithm is known for Boolean semiring matrix multiplication in the -general case that beats the best classical algorithm, which has -complexity \( n^{2.373} \). However, for sparse input our output, -quantum speedups are known. Specifically, 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], -the best known upper bound on quantum query complexity is -\(\widetilde{O}(n \sqrt{l}) \), as shown in -[161]. If instead the input matrices are sparse, a -quantum speedup over the fastest known classical algorithm also has -been found in a certain regime [206]. -For detailed comparison to classical algorithms, see -[155, 206]. Quantum -algorithms have been found to perform matrix multiplication over the -(max,min) semiring in \(\widetilde{O}(n^{2.473})\) time and over the -distance dominance semiring in \(\widetilde{O}(n^{2.458})\) time -[206]. The fastest known classical -algorithm for both of these problems has \(\widetilde{O}(n^{2.687})\) -complexity. -
- -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, i.e. \( ((x_1,f(x_1)),\ldots,(x_k,f(x_k))) - \in P \), or reject if none exists. As usual, we wish to do this - with the minimum number of queries to f. 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].
- -Algorithm: Search with Wildcards
-Speedup: Polynomial
-Description: -The search with wildcards problem is to identify a hidden n-bit -string x by making queries to an oracle f. Given \( S -\subseteq \{1,2,\ldots,n\} \) and \( y \in \{0,1\}^{|S|} \), f -returns one if the substring of x specified by S is equal -to y, and returns zero otherwise. Classically, this problem has -query complexity \(\Theta(n)\). As shown in -[167], the quantum query -complexity of this problem is \( \Theta(\sqrt{n}) \). Interestingly, -this quadratic speedup is achieved not through amplitude -amplification or quantum walks, but rather through use of the -so-called Pretty Good Measurement. The paper -[167] also gives a quantum speedup -for the related problem of combinatorial group testing. This result -and subsequent faster quantum algorithms for group testing are -discussed in the entry on Junta Testing and Group Testing. -
- -Algorithm: Network flows
-Speedup: Polynomial
-Description: A network is a directed graph whose edges are -labeled with numbers indicating their carrying capacities, and two of -whose vertices are designated as the source and the sink. A flow on a -network is an assignment of flows to the edges such that no flow -exceeds that edge's capacity, and for each vertex other than the -source and sink, the total inflow is equal to the total outflow. The -network flow problem is, given a network, to find the flow that maximizes -the total flow going from source to sink. For a network with n -vertices, m edges, and integer capacities of maximum -magnitude U, [168] -gives a quantum algorithm to find the maximal flow in time \( O(\min -\{n^{7/6} \sqrt{m} \ U^{1/3}, \sqrt{nU}m\} \times \log n) \). The -network flow problem is closely related to the problem of finding a -maximal matching of a graph, that is, a maximal-size subset of edges -that connects each vertex to at most one other vertex. The paper -[168] gives algorithms for finding -maximal matchings that run in time \( O(n \sqrt{m+n} \log n) \) if -the graph is bipartite, and \( O(n^2 ( \sqrt{m/n} + \log n) \log n) \) -in the general case. The core of these algorithms is Grover search. -The known upper bounds on classical complexity of the network flow and -matching problems are complicated to state because different classical -algorithms are preferable in different parameter regimes. However, in -certain regimes, the above quantum algorithms beat all known classical -algorithms. (See [168] for -details.)
- -Algorithm: Electrical Resistance
-Speedup: Exponential
-Description: -We are given oracle access to a weighted graph of n vertices and maximum -degree d whose edge weights are to be interpreted as electrical resistances. -Our task is to compute the resistance between a chosen pair of vertices. Wang gave -two quantum algorithms in [210] for this task that run in time -\(\mathrm{poly}( \log n, d, 1/\phi, 1/\epsilon) \), where \( \phi \) is the expansion -of the graph, and the answer is to be given to within a factor of \( 1+\epsilon \). -Known classical algorithms for this problem are polynomial in n rather than -\( \log n \). One of Wang's algorithms is based on a novel use of -quantum walks. The other is based on the quantum algorithm of -[104] for solving linear systems of equations. The first quantum -query complexity upper bounds for the electrical resistance problem in the adjacency -query model are given in [280] using approximate span programs. -
- -Algorithm: Junta Testing and Group Testing
-Speedup: Polynomial
-Description: -A function \(f:\{0,1\}^n \to \{0,1\}\) is a k-junta if it -depends on at most k of its input bits. The k-junta -testing problem is to decide whether a given function is -a k-junta or is \( \epsilon \)-far from -any k-junta. Althoug it is not obvious, this problem is closely -related to group testing. A group testing problem is defined by a -function \(f:\{1,2,\ldots,n\} \to \{0,1\}\). One is given oracle -access to F, which takes as input subsets of -\( \{1,2,\ldots,n\} \). F(S) = 1 if there exists \(x \in - S \) such that f(x) = 1 and -F(S) = 0 otherwise. In [266] a -quantum algorithm is given solving the k-junta problem using -\( \widetilde{O}(\sqrt{k/\epsilon}) \) queries and -\( \widetilde{O}(n \sqrt{k/\epsilon}) \) time. This is a quadratic -speedup over the classical complexity, and improves upon a previous -quantum algorithm for k-junta testing given in -[267]. A polynomial speedup for a gapped -(i.e. approximation) version of group testing is also given in -[266], improving upon the earlier results of -[167,268]. -
- - -
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 - general classes of Hamiltonians - [25,95,92,5,12,170,205,211,244,245,278,293,294,295,372,382], - chemical dynamics - [63,68,227,310,375], condensed - matter physics - [1,99, 145], relativistic quantum mechanics (the Dirac and Klein-Gordon equations) [367,369,370,371], - open quantum systems [376, 377,378,379], - and quantum field theory - [107,166,228,229,230,368]. - 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 -[102,231,232,233,234,235,308,321,322,380,381] -as well as thermal states [132,121,281,282,307] -for some classes of Hamiltonians and equilibrium states for some classes of master equations [430]. Efficient quantum algorithms have been also obtained for preparing certain classes of tensor network states -[323,324,325,326,327,328]. -
- -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.) Polynomial-time quantum algorithms have also been - discovered for additively approximating link invariants arising from - quantum doubles of finite groups - [174]. (This problem is not known to - be BQP-hard.) 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]. A method based on quantum walks, achieving - polynomial speedup for evaluating partition functions is given in - [265]. -
- -Algorithm: Quantum Approximate Optimization
-Speedup: Superpolynomial
-Description: For many combinatorial optimization problems, -finding the exact optimal solution is NP-complete. There are also hardness-of-approximation -results proving that finding an approximation with sufficiently -small error bound is NP-complete. For certain problems there is a gap -between the best error bound achieved by a polynomial-time classical -approximation algorithm and the error bound proven to be NP-hard. In -this regime there is potential for exponential speedup by quantum -computation. In [242] a new quantum algorithmic -technique called the Quantum Approximate Optimization Algorithm (QAOA) -was proposed for finding approximate solutions to -combinatorial optimization problems. In [243] it -was subsequently shown that QAOA solves a combinatorial -optimization problem called Max E3LIN2 with a better approximation -ratio than any polynomial-time classical algorithm known at the -time. However, an efficient classical algorithm achieving an even -better approximation ratio (in fact, the approximation ratio -saturating the limit set by hardness-of-approximation) was -subsequently discovered [260]. Presently, the power of -QAOA relative to classical computing is an active area of research -[300, 301, 302, -314]. -
- -Algorithm: Semidefinite Programming
-Speedup: Polynomial (with some exceptions)
-Description: Given a list of m + 1 Hermitian \(n \times n \) -matrices \(C, A_1, A_2, \ldots, A_m\) and m numbers -\(b_1, \ldots, b_m \), the problem of semidefinite programming is to -find the positive semidefinite \( n \times n \) -matrix X that maximizes tr(CX) subject to the -constraints \( \mathrm{tr} (A_j X) \leq b_j \) for \( j = 1,2,\ldots, -m \). Semidefinite programming has many applications in operations -research, combinatorial optimization, and quantum information, and it -includes linear programming as a special case. Introduced in -[313], and subsequently improved in [383, 425], -quantum algorithms are now known that can approximately solve semidefinite programs to within \( \pm \epsilon \) in time -\( O (\sqrt{m} \log m \cdot \mathrm{poly}(\log n, r, \epsilon^{-1})) \), where r is -the rank of the semidefinite program. This constitutes a quadratic speedup over the fastest -classical algorithms when r is small compared to n. The quantum algorithm is -based on amplitude amplification and quantum Gibbs sampling [121, 307]. -In a model in which input is provided in the form of quantum states the quantum algorithm -for semidefinite programming can achieve superpolynomial speedup, as discussed in [383], -although recent dequantization results [421] delineate limitations on -the context in which superpolynomial quantum speedup for semidefinite programs is possible. -
- -Algorithm: Zeta Functions
-Speedup: Superpolynomial
-Description: Let f(x,y) be a degree-d -polynomial over a finite field \( \mathbb{F}_p \). Let \( N_r \) be -the number of projective solutions to f(x,y = 0 over the -extension field \( \mathbb{F}_{p^r} \). The zeta function for f -is defined to be \( Z_f(T) = \exp \left( \sum_{r=1}^\infty -\frac{N_r}{r} T^r \right) \). Remarkably, \( Z_f(T) \) always has the -form \( Z_f(T) = \frac{Q_f(T)}{(1-pT)(1-T)} \) where \( Q_f(T) \) is a -polynomial of degree 2g and \(g = \frac{1}{2} (d-1)(d-2) \) is -called the genus of f. Given \( Z_f(T) \) one can easily -compute the number of zeros of f over any extension field \( -\mathbb{F}_{p^r} \). One can similarly define the zeta function -when the original field over which f is defined does not have -prime order. As shown by Kedlaya [64], -quantum computers can determine the zeta function of a genus g -curve over a finite field \( \mathbb{F}_{p^r} \) in -\( \mathrm{poly}(\log p, r, g) \) time. The fastest known classical algorithms -are all exponential in either log(p) or g. In a -diffent, but somewhat related contex, van Dam has conjectured that due -to a connection between the zeros of Riemann 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]. -
- -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], Somma et al. have - shown [84, 177] that - quantum computers can sample from \( \pi_n \) with a runtime - proportional to \( 1/\sqrt{\delta} \). Additional methods by which - classical Markov chain Monte Carlo algorithms can be sped up using - quantum walks are given in [265]. -
- -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. -
-
- - -
Optimization, Numerics, and Machine Learning
- -Algorithm: Constraint Satisfaction-Speedup: Polynomial
-Description: Constraint satisfaction problems, many of which -are NP-hard, are ubiquitous in computer science, a canonical example -being 3-SAT. If one wishes to satisfy as many constraints as possible -rather than all of them, these become combinatorial optimization -problems. (See also adiabatic algorithms.) The brute force -solution to constraint satisfaction problems can be quadratically sped -up using Grover's algorithm. However, most constraint satisfaction -problems are solvable by classical algorithms that (although still -exponential-time) run more than quadratically faster than brute force -checking of all possible solutions. Nevertheless, a polynomial quantum -speedup over the fastest known classical algorithm for 3-SAT is given -in [133], and polynomial quantum -speedups for some other constraint satisfaction problems are given in -[134, 298]. In [423] a quadratic -quantum speedup for approximate solutions to homogeneous QUBO/Ising problems -is obtained by building upon the quantum algorithm for semidefinite programming. -A commonly used classical algorithm for constraint satisfaction is -backtracking, and for some problems this algorithm is the fastest known. -A general quantum speedup for backtracking algorithms is given in -[264] and further improved in [422].
- - -Algorithm: Adiabatic Algorithms
-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 at worst as \(1/ \gamma^3 \) where \( \gamma \) -is the minimum eigenvalue gap between the ground state -and the first excited state [185]. If the -Hamiltonian is varied sufficiently smoothly, one can improve this to -\( \widetilde{O}(1/\gamma^2) \) [247]. Adiabatic -quantum computation was first proposed by Farhi et al. as a -method for solving NP-complete combinatorial optimization problems -[96, 186]. Adiabatic quantum algorithms for -optimization problems typically use "stoquastic" Hamiltonians, which -do not suffer from the sign problem. Such algorithms are sometimes -referred to as quantum annealing. Adiabatic quantum computation with -non-stoquastic Hamiltonians is as powerful as the quantum circuit -model [97]. Adiabatic algorithms -using stoquastic Hamiltonians are probably less powerful -[183], but are likely more powerful -than classical computation [429]. The asymptotic runtime of adiabatic -optimization algorithms is notoriously difficult to analyze, but some -progress has been achieved -[179,180,181,182,187,188,189,190,191,226]. (Also -relevant is an earlier literature on quantum annealing, which -originally referred to a classical optimization algorithm that works by -simulating a quantum process, much as simulated annealing is a -classical optimization algorithm that works by simulating a thermal -process. See e.g. -[199, 198].) Adiabatic quantum -computers can perform a process somewhat analogous -to Grover search in \( O(\sqrt{N}) \) time -[98]. Adiabatic quantum algorithms -achieving quadratic speedup for a more general class of problems are -constructed in [184] by adapting techniques from -[85]. Adiabatic quantum algorithms have been -proposed for several specific problems, including PageRank -[176], machine learning -[192, 195], finding Hadamard -matrices [406], and graph problems [193, 194]. -Some quantum simulation algorithms also use adiabatic state -preparation. -
- -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 given in -[147,148,223]. -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. Remarkably general results in [418,419,420] -give quantum speedups for convex optimization and volume estimation of convex bodies, as well -as query complexity lower bounds. Roughly speaking these results show that for convex -optimization and volume estimation in d dimensions one gets a quadratic speedup in d -just as was found earlier for the special case of minimizing quadratic forms. -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: 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]. Exponentially improved -scaling of runtime with precision was obtained in -[263]. Some methods to extend this algorithm to apply to non-sparse matrices -have been proposed [309,402], although these require certain partial sums of the matrix elements to -be pre-computed. Extensions of this quantum algorithm have been applied to problems of estimating electromagnetic -scattering crossections [249] (see also [369] for a different approach), solving linear -differential equations [156, 296], estimating -electrical resistance of networks [210], least-squares -curve-fitting [169], solving Toeplitz systems -[297], and machine learning -[214,222,250,251,309]. -However, the linear-systems-based quantum algorithms for recommendation systems [309] -and principal component analysis [250] were subsequently "dequantized" by Tang -[400, 401]. -That is, Tang obtained polynomial time classical randomized algorithms for these problems, thus proving -that the proposed quantum algorithms for these tasks do not achieve exponential speedup. -Some limitations of the quantum machine learning algorithms based on -linear systems are nicely summarized in [246]. -In [220] it was shown that quantum -computers can invert well-conditioned \( n \times n \) matrices -using only \( O( \log n ) \) qubits, whereas the best classical -algorithm uses order \( \log^2 n \) bits. Subsequent improvements to -this quantum algorithm are given in [279]. -
- -Algorithm: Machine Learning
-Speedup: Varies
-Description: -Maching learning encompasses a wide variety of computational problems -and can be attacked by a wide variety of algorithmic techniques. This -entry summarizes quantum algorithmic techniques for improved machine -learning. Many of the quantum algorithms here are cross-listed under -other headings. In -[214,222,250,251,309,338,339,359,403], -quantum algorithms for solving linear systems [104] are -applied to speed up cluster-finding, principal component analysis, -binary classification, training of neural networks, and various forms of regression, provided the data satisfies certain conditions. However, a number of quantum machine learning algorithms based on -linear systems have subsequently been "dequantized". Specifically, Tang showed in [400, 401] that the -problems of recommendation systems and principal component analysis solved by the quantum algorithms of [251,309] can in fact also be solved in polynomial time randomized classical algorithms. -A cluster-finding method not based on the linear systems algorithm of [104] is given in [336]. The papers -[192,195,344,345,346,348] -explore the use of adiabatic optimization techniques to speed up the training of -classifiers. In [221], a method is proposed for -training Boltzmann machines by manipulating coherent quantum states -with amplitudes proportional to the Boltzmann weights. Polynomial speedups -can be obtained by applying Grover search and related techniques such as amplitude amplification to amenable -subroutines of state of the art classical machine learning algorithms. See, -for example -[358,340,341,342,343]. -Other quantum machine learning algorithms not falling into one of the above categories include -[337,349]. -Some limitations of quantum machine learning algorithms are nicely -summarized in [246]. Many other quantum query algorithms that extract hidden structure of the -black-box function could be cast as machine learning algorithms. See for example -[146,23,11,31,212]. -Query algorithms for learning the majority and "battleship" functions are -given in [224]. Large quantum advantages for -learning from noisy oracles are given in [236,237]. In [428] quantum kernel estimation is used to implement -a support-vector classifier solving a learning problem that is provably as hard as discrete logarithm. Several recent review articles -[299,332,333] -and a book [331] are available which summarize the state of the field. -There is a related body of work, not strictly within the standard setting of quantum algorithms, -regarding quantum learning in the case that the data itself is quantum coherent. -See e.g. [350,334,335,351,352,353,354,355,356,357]. -
- -Algorithm: Tensor Principal Component Analysis
-Speedup: Polynomial (quartic)
-Description: -In [424] a quantum algorithm is given for an idealized problem motivated by machine learning -applications on high-dimensional data sets. Consider \(T = \lambda v_{\mathrm{sig}}^{\otimes p} + G \) where -G is a p-index tensor of Gaussian random variables, symmetrized over all permutations of indices, and -\(v_{\mathrm{sig}}\) is an N-dimensional vector of magnitude \(\sqrt{N}\). The task is to recover \(v_{\mathrm{sig}}\). -Consider \( \lambda = \alpha N^{-p/4}\). The best classical algorithms succeed when \( \alpha \gg 1\) and have time and -space complexity that scale exponentially in \( \alpha^{-1}\). The quantum algorithm of [424] solves -this problem in polynomial space and with runtime scaling quartically better in \( \alpha^{-1} \) than the classical spectral -algorithm. The quantum algorithm works by encoding the problem into the eigenspectrum of a many-body Hamiltonian and applying -phase estimation together with amplitude amplification. -
- -Algorithm: Solving Differential Equations
-Speedup: Superpolynomial
-Description: Consider linear first order differential equation \( \frac{d}{dt} \mathbf{x} = A(t) \mathbf{x} + \mathbf{b}(t) \), where -\( \mathbf{x} \) and \( \mathbf{b} \) are N-dimensional vectors and A is an \(N \times N\) matrix. Given an initial condition -\( \mathbf{x}(0) \) one wishes to compute the solution \( \mathbf{x}(t) \) at some later time t to some precision \( \epsilon \) in -the sense that the normalized vector \( x(t)/\|x(t)\| \) produced has distance at most \( \epsilon \) from the exact solution. In [156], -Berry gives a quantum algorithm for this problem that runs in time \( O(t^2 \mathrm{poly}(1/\epsilon) \mathrm{poly log} N) \), whereas the fastest classical algorithms -run in time \( O ( t \mathrm{poly} N ) \). The final result is produced in the form of a quantum superposition state on \( O(log N) \) qubits whose amplitudes contain -the components of \( \mathbf{x}(t) \). The algorithm works by reducing the problem to linear algebra via a high-order finite difference method and applying -the quantum linear algebra primitive of [104]. In [410] an improved quantum algorithm for this problem was given -which brings the epsilon dependence down to \( \mathrm{poly log}(1/\epsilon) \). A quantum algorithm for solving nonlinear differential equations (again in the -sense of obtaining a solution encoded in the amplitudes) is described in [411], which has exponential scaling in t. -In [426,427] quantum algorithms are given for solving nonlinear differential equations that scale as \( O(t^2) \). -These are applicable to a restricted class of nonlinear differential equations. In particular, their solutions must not grow or shrink in magnitude too rapidly. -Partial differential equations can be reduced to ordinary differential equations through discretization, and higher order differential equations can be -reduced to first order through additiona of auxiliary variables. Consequently, these more general problems can be solved through the methods of -[156, 104]. However, quantum algorithms designed to solve these problems directly may be more -efficient (and for specific problems one may analyze the complexity of tasks that are unspecified in a more general formulation such as -preparation of relevant initial states). In [249] a quantum algorithm is given which solves the wave equation by -applying finite-element methods to reduce it to linear algebra and then applying the quantum linear algebra algorithm of [104] -with preconditioning. In [369] a quantum algorithm is given for solving the wave equation by discretizing it -with finite differences and massaging it into the form of a Schrodinger equation which is then simulated using the method of [245]. The -problem solved by [369] is not equivalent to that solved by [249] because in [249] the problem -is reduced to a time-indepent one through assuming sinusoidal time dependence and applying separation of variables, whereas [369] solves -the time-dependent problem. The quantum speedup achieved over classical methods for solving the wave equation in d-dimensiona is polynomial for fixed -d but expontial in d. Concrete resource estimates for quantum algorithms to solve differential equations are given in -[412, 413, 414]. A quantum algorithm for solving linear partial differential equations using -continuous-variable quantum computing is given in [415]. In [296] quantum finite element methods are analyzed in -a general setting. A quantum spectral method for solving differential equations is given in [416]. A quantum algorithm for solving the Vlasov equation is -given in [417]. -
- -Algorithm: Quantum Dynamic Programming
-Speedup: Polynomial
-Description: In [409] the authors introduce a problem called path-in-the-hypercube. In this problem, one given -a subgraph of the hypercube and asked whether there is a path along this subgraph that starts from the all zeros vertex, -ends at the all ones vertex, and makes only Hamming weight increasing moves. (The vertices of the hypercube graph correspond -to bit strings of length n and the hypercube graph joins vertices of Hamming distance one.) Many NP-complete problems -for which the best classical algorithm is dynamic programming can be modeled as instances of path-in-the-hypercube. By -combining Grover search with dynamic programming methods, a quantum algorithm can solve path-in-the-hypercube in time -\( O^*(1.817^n) \), where the notation \( O^* \) indicates that polynomial factors are being omitted. The fastest known -classical algorithm for this problem runs in time \( O^*(2^n) \). Using this primitive quantum algorithms can be constructed -that solve vertex ordering problems in \( O^*(1.817^n) \) vs. \( O^* (2^n) \) classically, graph bandwidth in \( O^*(2.946^n) \) -vs. \( O^*(4.383^n) \) classically, travelling salesman and feedback arc set in \( O^*(1.729^n) \) vs. \( O^*(2^n) \) classically, -and minimum set cover in \( O( \mathrm{poly}(m,n) 1.728^n ) \) vs. \( O(nm2^n) \) classically. -
- -
- - -
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 -
- Lov Grover -
- Eduin H. Serna -
- Charles Greathouse -
- Juani Bermejo-Vega -
- Luis Kowada -
- Keith Britt -
- Aram Harrow -
- Zafer Gedik -
- David Cornwell -
- Cedric Lin -
- Shelby Kimmel -
- Jeremy Singer -
- Dan Boneh -
- Rich Schroeppel -
- Yuan Su -
- Tim Stevens -
- Martin Ekerå -
- Igor Shparlinski -
- Timothy Chase -
- Alejandro Pozas-Kerstjens -
- Nikhil Vyas -
- Kevin Lui -
- Vladimir Korepin -
- Andriyan Suksmono -
- Jack Hidari -
- Donny Greenberg -
- Nicola Vitucci -
- Kunal Marwaha -
- José Ignacio Espinoza Camacho -
- Vincenzo Savona -
- Barry Sanders -
- Jeremy Wright -
- Sarah Keiser -
- Benjamin Tokgöz -
- -
- - -
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