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Gottesman–Knill Theorem
In quantum computing, the Gottesman–Knill theorem is a theoretical result by Daniel Gottesman and Emanuel Knill that states that stabilizer circuits–circuits that only consist of gates from the normalizer of the qubit Pauli group, also called Clifford group–can be perfectly simulated in polynomial time on a probabilistic classical computer. The Clifford group can be generated solely by using the controlled NOT, Hadamard, and phase gates (CNOT, ''H'' and ''S''); and therefore stabilizer circuits can be constructed using only these gates. The reason for the speed up of quantum computers compared to classical ones is not yet fully understood. The Gottesman-Knill theorem proves that all quantum algorithms whose speed up relies on entanglement that can be achieved with CNOT and Hadamard gates do not achieve any computational advantage relative classical computers, due to the classical simulability of such algorithms (and the particular types of entangled states they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computing
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations Exponential growth, exponentially faster than any modern "classical" computer. Theoretically a large-scale quantum computer could post-quantum cryptography, break some widely used encryption schemes and aid physicists in performing quantum simulator, physical simulations; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications. The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in classical computing. However, unlike a classical bit, which can be in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat, and Picard. He obtained his doctorate in 1892 and in the same year was awarded the for his essay on the Riemann zeta function. In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stabilizer Code
Stabilizer, stabiliser, stabilisation or stabilization may refer to: Chemistry and food processing * Stabilizer (chemistry), a substance added to prevent unwanted change in state of another substance ** Polymer stabilizers are stabilizers used specifically in plastic or other polymers * Stabilizer (food), a type of food additive * Wood stabilization, a wood preservation process to prevent distortion caused by moisture * Clarification and stabilization of wine Mathematics * Stabilization (category theory) * Stabilizer subgroup Technology * Buoyancy compensator (diving) adjusts buoyancy. * Stabilizer (aircraft), surfaces to help keep aircraft under control. Includes: ** Vertical stabilizer of airplanes ** Tailplane or horizontal stabilizer * Stabilizer (ship), fins on ships to counteract roll * Stabilizer, another name for bicycle training wheels * Stabilizers, the extendable legs mounted on a land vehicle which are folded out when stabilization is required; see Outrigger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic State Distillation
Magic state distillation is a method for creating more accurate quantum states from multiple noisy ones, which is important for building fault tolerant quantum computers. It has also been linked to quantum contextuality, a concept thought to contribute to quantum computers' power. The technique was first proposed by Emanuel Knill in 2004, and further analyzed by Sergey Bravyi and Alexei Kitaev the same year. Thanks to the Gottesman–Knill theorem, it is known that some quantum operations (operations in the Clifford group) can be perfectly simulated in polynomial time on a classical computer. In order to achieve universal quantum computation, a quantum computer must be able to perform operations outside this set. Magic state distillation achieves this, in principle, by concentrating the usefulness of imperfect resources, represented by mixed states, into states that are conducive for performing operations that are difficult to simulate classically. A variety of qubit magic sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph State
In quantum computing, a graph state is a special type of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. In particular, they are a convenient way of representing certain types of entangled states. Graph states are useful in quantum error-correcting codes, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models. A graph state is a particular case of a stabilizer state as well as a 2-uniform hypergraph state, a generalization where the edges have cardinality between 1 and N. Formal definition Quantum graph states can be defined in two equivalent ways: through the notion of quantum circuits and stabilizer formalism. Quantum circuit definition Given a graph G = (V, E), with the set of vertices V and the set of edges E, the corresponding graph state is defined as : =\prod _ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Error Correction
Quantum error correction (QEC) is a set of techniques used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum state preparation, and faulty measurements. Effective quantum error correction would allow quantum computers with low qubit fidelity to execute algorithms of higher complexity or greater circuit depth. Classical error correction often employs redundancy. The simplest albeit inefficient approach is the repetition code. A repetition code stores the desired (logical) information as multiple copies, and—if these copies are later found to disagree due to errors introduced to the system—determines the most likely value for the original data by majority vote. For instance, suppose we copy a bit in the one (on) state thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Distillation
Entanglement distillation (also called entanglement purification) is the transformation of ''N'' copies of an arbitrary Quantum entanglement, entangled state \rho into some number of approximately pure Bell pairs, using only LOCC, local operations and classical communication. Entanglement distillation can overcome the degenerative influence of noisy quantum channels by transforming previously shared, less-entangled pairs into a smaller number of Maximally entangled state, maximally-entangled pairs. History The limits for entanglement dilution and distillation are due to Charles H. Bennett (computer scientist), C. H. Bennett, H. Bernstein, Sandu Popescu, S. Popescu, and Benjamin Schumacher, B. Schumacher, who presented the first distillation protocols for pure states in 1996; entanglement distillation protocols for Mixed state (physics), mixed states were introduced by Bennett, Gilles Brassard, Popescu, Schumacher, John A. Smolin and William Wootters the same year. Bennett, David D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Entanglement
Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurement#Quantum mechanics, Measurements of physical properties such as position (vector), position, momentum, Spin (physics), spin, and polarization (waves), polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior ... [...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 spin states (left-handed and the right-handed circular polarization) can also be measured as horizontal and vertical linear 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 multiple 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 acknow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controlled NOT Gate
In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-''X'' gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer. It can be used to entangle and disentangle Bell states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations. The gate is sometimes named after Richard Feynman who developed an early notation for quantum gate diagrams in 1986. The CNOT can be expressed in the Pauli basis as: : \mbox = e^= e^. Being both unitary and Hermitian, CNOT has the property e^=(\cos \theta)I+(i\sin \theta) U and U =e^=e^, and is involutory. The CNOT gate can be further decomposed as products of rotation operator gates and exactly one two qubit interaction gate, for example : \mbox =e^R_(-\pi/2)R_(-\pi/2)R_(-\pi/2)R_(\pi/2)R_(\pi/2). In general, any sing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Gottesman
Daniel Gottesman is a physicist, known for his work regarding quantum error correction, in particular the invention of the stabilizer formalism for quantum error-correcting codes, Gottesman–Kitaev–Preskill code and the Gottesman–Knill theorem. He is a faculty member at the University of Maryland. Gottesman completed a B.A. in physics at Harvard University (1992) and a Ph.D. in physics at Caltech (1997). He is a Fellow of the American Physical Society (2013). In 2003, he was named to the MIT ''Technology Review'' TR100 as one of the top 100 innovators in the world under the age of 35. See also *Clifford gates *Continuous-variable quantum information Continuous-variable (CV) quantum information is the area of quantum information science that makes use of Observable, physical observables, like the strength of an electromagnetic field, whose numerical values belong to List of continuity-related m ... References External linksGottesman's homepage at the Perimeter Instit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action of a group G on a set (mathematics), set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformation (function), transformations form a group (mathematics), group under function composition; for example, the rotation (mathematics), rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a mathematical structure, structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
