In quantum computing, a quantum algorithm is an algorithm that runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is generally reserved for algorithms that seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement.
Problems that are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms because the quantum superposition and quantum entanglement that quantum algorithms exploit generally cannot be efficiently simulated on classical computers (see quantum supremacy).
The best-known quantum algorithms are Shor's algorithm for factoring and Grover's algorithm for searching an unstructured database or an unordered list. Shor's algorithm would, if implemented, run much (almost exponentially) faster than the most efficient known classical algorithm for factoring, the general number field sieve. Likewise, Grover's algorithm would run quadratically faster than the best possible classical algorithm for the same task, a linear search.
Overview Quantum algorithms are usually described, in the commonly used circuit model of quantum computation, by a quantum circuit that acts on some input qubits and terminates with a measurement. A quantum circuit consists of simple quantum gates, each of which acts on some finite number of qubits. Quantum algorithms may also be stated in other models of quantum computation, such as the Hamiltonian oracle model.
Quantum algorithms can be categorized by the main techniques involved in the algorithm. Some commonly used techniques/ideas in quantum algorithms include phase kick-back, phase estimation, the quantum Fourier transform, quantum walks, amplitude amplification and topological quantum field theory. Quantum algorithms may also be grouped by the type of problem solved; see, e.g., the survey on quantum algorithms for algebraic problems.
Algorithms based on the quantum Fourier transform The quantum Fourier transform is the quantum analogue of the discrete Fourier transform, and is used in several quantum algorithms. The Hadamard transform is also an example of a quantum Fourier transform over an n-dimensional vector space over the field F2. The quantum Fourier transform can be efficiently implemented on a quantum computer using only a polynomial number of quantum gates.
Deutsch–Jozsa algorithm
The Deutsch–Jozsa algorithm solves a black-box problem that requires exponentially many queries to the black box for any deterministic classical computer, but can be done with a single query by a quantum computer. However, when comparing bounded-error classical and quantum algorithms, there is no speedup, since a classical probabilistic algorithm can solve the problem with a constant number of queries with small probability of error. The algorithm determines whether a function f is either constant (0 on all inputs or 1 on all inputs) or balanced (returns 1 for half of the input domain and 0 for the other half).
Bernstein–Vazirani algorithm
The Bernstein–Vazirani algorithm is the first quantum algorithm that solves a problem more efficiently than the best known classical algorithm. It was designed to create an oracle separation between BQP and BPP.
Simon's algorithm
Simon's algorithm solves a black-box problem exponentially faster than any classical algorithm, including bounded-error probabilistic algorithms. This algorithm, which achieves an exponential speedup over all classical algorithms that we consider efficient, was the motivation for Shor's algorithm for factoring.
Quantum phase estimation algorithm
The quantum phase estimation algorithm is used to determine the eigenphase of an eigenvector of a unitary gate, given a quantum state proportional to the eigenvector and access to the gate. The algorithm is frequently used as a subroutine in other algorithms.
Shor's algorithm
Shor's algorithm solves the discrete logarithm problem and the integer factorization problem in polynomial time, whereas the best known classical algorithms take super-polynomial time. It is unknown whether these problems are in P or NP-complete. It is also one of the few quantum algorithms that solves a non-black-box problem in polynomial time, where the best known classical algorithms run in super-polynomial time.
Hidden subgroup problem The abelian hidden subgroup problem is a generalization of many problems that can be solved by a quantum computer, such as Simon's problem, solving Pell's equation, testing the principal ideal of a ring R and factoring. There are efficient quantum algorithms known for the Abelian hidden subgroup problem. The more general hidden subgroup problem, where the group is not necessarily abelian, is a generalization of the previously mentioned problems, as well as graph isomorphism and certain lattice problems. Efficient quantum algorithms are known for certain non-abelian groups. However, no efficient algorithms are known for the symmetric group, which would give an efficient algorithm for graph isomorphism and the dihedral group, which would solve certain lattice problems.
Estimating Gauss sums A Gauss sum is a type of exponential sum. The best known classical algorithm for estimating these sums takes exponential time. Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely. However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.
Fourier fishing and Fourier checking Consider an oracle consisting of n random Boolean functions mapping n-bit strings to a Boolean value, with the goal of finding n n-bit strings z1,..., zn such that for the Hadamard-Fourier transform, at least 3/4 of the strings satisfy
and at least 1/4 satisfy
This can be done in bounded-error quantum polynomial time (BQP).
Algorithms based on amplitude amplification Amplitude amplification is a technique that allows the amplification of a chosen subspace of a quantum state. Applications of amplitude amplification usually lead to quadratic speedups over the corresponding classical algorithms. It can be considered as a generalization of Grover's algorithm.
Grover's algorithm
Grover's algorithm searches an unstructured database (or an unordered list) with N entries for a marked entry, using only queries instead of the queries required classically. Classically, queries are required even allowing bounded-error probabilistic algorithms.
Theorists have considered a hypothetical generalization of a standard quantum computer that could access the histories of the hidden variables in Bohmian mechanics. (Such a computer is completely hypothetical and would not be a standard quantum computer, or even possible under the standard theory of quantum mechanics.) Such a hypothetical computer could implement a search of an N-item database in at most steps. This is slightly faster than the steps taken by Grover's algorithm. However, neither search method would allow either model of quantum computer to solve NP-complete problems in polynomial time.
Quantum counting Quantum counting solves a generalization of the search problem. It solves the problem of counting the number of marked entries in an unordered list, instead of just detecting whether one exists. Specifically, it counts the number of marked entries in an -element list with an error of at most by making only queries, where is the number of marked elements in the list. More precisely, the algorithm outputs an estimate for , the number of marked entries, with accuracy .
Algorithms based on quantum walks
A quantum walk is the quantum analogue of a classical random walk. A classical random walk can be described by a probability distribution over some states, while a quantum walk can be described by a quantum superposition over states. Quantum walks are known to give exponential speedups for some black-box problems. They also provide polynomial speedups for many problems. A framework for the creation of quantum walk algorithms exists and is a versatile tool.
Boson sampling problem
The Boson Sampling Problem in an experimental configuration assumes an input of bosons (e.g., photons) of moderate number that are randomly scattered into a large number of output modes, constrained by a defined unitarity. When individual photons are used, the problem is isomorphic to a multi-photon quantum walk. The problem is then to produce a fair sample of the probability distribution of the output that depends on the input arrangement of bosons and the unitarity. Solving this problem with a classical computer algorithm requires computing the permanent of the unitary transform matrix, which may take a prohibitively long time or be outright impossible. In 2014, it was proposed that existing technology and standard probabilistic methods of generating single-photon states could be used as an input into a suitable quantum computable linear optical network and that sampling of the output probability distribution would be demonstrably superior using quantum algorithms. In 2015, investigation predicted the sampling problem had similar complexity for inputs other than Fock-state photons and identified a transition in computational complexity from classically simulable to just as hard as the Boson Sampling Problem, depending on the size of coherent amplitude inputs.
Element distinctness problem
The element distinctness problem is the problem of determining whether all the elements of a list are distinct. Classically, queries are required for a list of size ; however, it can be solved in queries on a quantum computer. The optimal algorithm was put forth by Andris Ambainis, and Yaoyun Shi first proved a tight lower bound when the size of the range is sufficiently large. Ambainis and Kutin independently (and via different proofs) extended that work to obtain the lower bound for all functions.
Triangle-finding problem
The triangle-finding problem is the problem of determining whether a given graph with N vertices contains a triangle (a clique of size 3). Given oracle access to the adjacency matrix of the graph, the classical query complexity is , since for a graph with only one triangle this is the query complexity needed to find any edge at all. Meanwhile, for quantum algorithms the lower bound is , the number of queries needed to find any edge with Grover's algorithm. However, finding any edge does not guarantee finding a triangle if there exists any. A Grover search over all potential triangles does solve the problem, but the query complexity, O(N3/2), can be improved upon.
While remains the best-known lower bound for quantum algorithms, the best algorithm known requires O(N5/4) queries, an improvement over the previous best O(N1.3) queries.
Formula evaluation A formula is a tree with a gate at each internal node and an input bit at each leaf node. The problem is to evaluate the formula, which is the output of the root node, given oracle access to the input.