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Solovay–Kitaev Theorem
In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubit quantum gates generates a dense subset of SU(2), then that set can be used to approximate any desired quantum gate with a relatively short sequence of gates. This theorem is considered one of the most significant results in the field of quantum computation and was first announced by Robert M. Solovay in 1995 and independently proven by Alexei Kitaev in 1997. Michael Nielsen and Christopher M. Dawson have noted its importance in the field. A consequence of this theorem is that a quantum circuit of m constant-qubit gates can be approximated to \varepsilon error (in operator norm) by a quantum circuit of O(m\log^c(m/\varepsilon)) gates from a desired finite universal gate set. By comparison, just knowing that a gate set is universal only implies that constant-qubit gates can be approximated by a finite circuit from the gate set, with no bound on its length. So, the Solova ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing. Etymology The coining of the term ''qubit'' is attributed to Benjamin Schumacher. In the acknowledgments of his 1995 paper, Schumacher states that the term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits. Unlike many classical logic gates, quantum logic gates are reversible. It is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits. Quantum gates are unitary operators, and are described as unitary matrices relative to some basis. Usually we use the ''computational basis'', which unless we compare it with something, just means that for a ''d''-level quantum system (such as a qubit, a quantum register ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra (structure), Boolean algebra under the subset relation, in which the join and meet are given by Intersection (set theory), intersection and Union (set theory), union, and the subset relation itself is the Inclusion (Boolean algebra), Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n qubits and thus 2^n basis states. (Alternatively, the more genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though current quantum computers may be too small to outperform usual (classical) computers for practical applications, larger realizations are believed to be capable of solving certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. There are several models of quantum computation with the most widely used being quantum circuits. Other models include the quantum Turing machine, quantum annealing, and adiabatic quantum computation. Most models are based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. A qubit can be in a 1 or 0 quantum s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert M
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexei Kitaev
Alexei Yurievich Kitaev (russian: Алексей Юрьевич Китаев; born August 26, 1963) is a Russian–American professor of physics at the California Institute of Technology and permanent member of the Kavli Institute for Theoretical Physics. He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at the Landau Institute for Theoretical Physics. For this work, he was awarded a MacArthur Fellowship in 2008. He is also known for introducing the complexity class QMA and showing the 2-local Hamiltonian problem is QMA-complete, the most complete result for k-local Hamiltonians. Kitaev is also known for contributions to research on a model relevant to researchers of the AdS/CFT correspondence started by Subir Sachdev and Jinwu Ye; this model is known as the Sachdev–Ye–Kitaev (SYK) model. Kitaev was educated in Russia, receiving an M.Sc. from the Moscow Institute of Physics and Technolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Nielsen
Michael Aaron Nielsen (born January 4, 1974) is a quantum physicist, science writer, and computer programming researcher living in San Francisco. Work In 1998, Nielsen received his PhD in physics from the University of New Mexico. In 2004, he was recognized as Australia's "youngest academic" and was awarded a Federation Fellowship at the University of Queensland. During this fellowship, he worked at the Los Alamos National Laboratory, Caltech, at the Perimeter Institute for Theoretical Physics. Alongside Isaac Chuang, Nielsen co-authored a Quantum Computation and Quantum Information (book), popular textbook on quantum computing, which has been cited more than 36,000 times as of December 2019. In 2007, Nielsen shifted his focus from quantum information and computation to “the development of new tools for scientific collaboration and publication”, including the Polymath project with Timothy Gowers, which aims to facilitate "massively collaborative mathematics." Besides wri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \mbox v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, one can take the infimum of the numbers c such that the above i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Logic Gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits. Unlike many classical logic gates, quantum logic gates are reversible. It is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits. Quantum gates are unitary operators, and are described as unitary matrices relative to some basis. Usually we use the ''computational basis'', which unless we compare it with something, just means that for a ''d''-level quantum system (such as a qubit, a quantum register ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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