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https://quantum-journal.org/papers/q-2024-02-20-1259/pdf/
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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.[7]
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.[8]
Algorithms based on the quantum Fourier transform
[edit]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.[citation needed]
Deutsch–Jozsa algorithm
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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
[edit]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
[edit]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
[edit]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
[edit]Shor's algorithm solves the?discrete logarithm?problem and the?integer factorization?problem in polynomial time,[9]?whereas the best known classical algorithms take super-polynomial time. These problems are not known to be 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
[edit]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.[10]?The more general hidden subgroup problem, where the group isn't 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[11]?and the?dihedral group, which would solve certain lattice problems.[12]
Estimating Gauss sums
[edit]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.[13]
Fourier fishing and Fourier checking
[edit]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).[14]
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