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Steane Code
The Steane code is a tool in quantum error correction introduced by Andrew Steane in 1996. It is a CSS code (Calderbank-Shor-Steane), using the classical binary ,4,3Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors. Its check matrix in standard form is : \begin H & 0 \\ 0 & H \end where H is the parity-check matrix of the Hamming code and is given by : H = \begin 1 & 0 & 0 & 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1 & 0 & 1\\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end. The 7,1,3 Steane code is the first in the family of quantum Hamming codes, codes with parameters 2^r-1, 2^r-1-2r, 3 for integers r \geq 3. It is also a quantum color code. Expression in the stabilizer formalism In a quantum error-correcting code, the codespace is the subspace of the overall Hilbert space where all logical states li ... [...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] |
Andrew Steane
Andrew Martin Steane is Professor of physics at the University of Oxford. He is also a fellow of Exeter College, Oxford. He was a student at St Edmund Hall, Oxford where he obtained his MA and DPhil. His major works to date are on error correction in quantum information processing, including Steane codes. He was awarded the Maxwell Medal and Prize of the Institute of Physics The Institute of Physics (IOP) is a UK-based not-for-profit learned society and professional body that works to advance physics education, physics research, research and applied physics, application. It was founded in 1874 and has a worldwide ... in 2000. Papers * "Quantum Computing" Reports on Progress in Physics 61: 117–173. Steane, A.M. (1998) * "A Quantum Computer Needs Only One Universe" Studies in History and Philosophy of Modern Physics 34B: 469–478, Steane, A.M. (2003) Books * * 'Relativity Made Relatively Easy' is a text that follows closely to the 'Symmetry and Relativity' course that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CSS Code
In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of stabilizer code constructed from classical codes with some special properties. An example of a CSS code is the Steane code. Construction Let C_1 and C_2 be two (classical) ,k_1/math>, ,k_2/math> codes such, that C_2 \subset C_1 and C_1 , C_2^\perp both have minimal distance \geq 2t+1, where C_2^\perp is the code dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual number, a nu ... to C_2. Then define \text(C_1,C_2), the CSS code of C_1 over C_2 as an ,k_1 - k_2, d/math> code, with d \geq 2t+1 as follows: Define for x \in C_1 : x + C_2 \rangle := 1 / \sqrt \sum_ x + y \rangle, where + is bitwise addition modulo 2. Then \text( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Code
In computer science and telecommunications, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming(7,4) code which adds three parity bits to four bits of data. In mathematical terms, Hamming codes are a class of binary linear code. For each integer there is a code-word with block length and message length . Hence the rate of Hamming codes is , which is the highest p ... [...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] |
Check Matrix
Check or cheque, may refer to: Places * Check, Virginia Arts, entertainment, and media * ''Check'' (film), a 2021 Indian Telugu-language film * "The Check" (''The Amazing World of Gumball''), a 2015 episode of ''The Amazing World of Gumball'' * '' The Checks'' (episode), a 1996 TV episode of ''Seinfeld'' Games and sports * Check (chess), a threat to capture the king or general * Check (poker), declining to bet * Checking (ice hockey), several techniques * Casino chip, less commonly referred to as a check Music * ''The Checks'' (band), a New Zealand band * "Check" (Meek Mill song), 2015 * "Check" (Young Thug song), 2015 * "Check", a song by bbno$, 2025 * "Check", a song by Chris Janson from '' Real Friends'', 2019 * "Check", a song by E-40 from '' The D-Boy Diary: Book 1'', 2016 * "Check", a song by Flo from '' Access All Areas'', 2024 * "Check", a song by Kojo Funds featuring Raye from ''Golden Boy'', 2018 * "Check", a song by Lil Durk from '' Lil Durk 2X'', 2016 * "C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers as equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity-check Matrix
In coding theory, a parity-check matrix of a linear block code ''C'' is a matrix which describes the linear relations that the components of a codeword must satisfy. It can be used to decide whether a particular vector is a codeword and is also used in decoding algorithms. Definition Formally, a parity check matrix ''H'' of a linear code ''C'' is a generator matrix of the dual code, ''C''⊥. This means that a codeword c is in ''C ''if and only if the matrix-vector product (some authors would write this in an equivalent form, c''H''⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. That is, they show how linear combinations of certain digits (components) of each codeword equal zero. For example, the parity check matrix :H = \left \begin 0&0&1&1\\ 1&1&0&0 \end \right, compactly represents the parity check equations, :\begin c_3 + c_4 &= 0 \\ c_1 + c_2 &= 0 \end, that must be satisfied for the vector (c_1, c_2, c_3, c_4) to be a ... [...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] |
Generator (mathematics)
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. List of generators A list of examples of generating sets follow. * Generating set or spanning set of a vector space: a set that spans the vector space * Generating set of a group: a subset of a group that is not contained in any sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
