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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 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 qubit flip errors (X errors) and the dual (mathema .... 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 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] |
Quantum Error Correction
Quantum error correction (QEC) is 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 preparation, and faulty measurements. Classical error correction employs redundancy. The simplest albeit inefficient approach is the repetition code. The idea is to store the information multiple times, and—if these copies are later found to disagree—take a majority vote; e.g. suppose we copy a bit in the one state three times. Suppose further that a noisy error corrupts the three-bit state so that one of the copied bits is equal to zero but the other two are equal to one. Assuming that noisy errors are independent and occur with some sufficiently low probability ''p'', it is most likely that the error is a single-bit error and the tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Calderbank
Robert Calderbank (born 28 December 1954) is a professor of Computer Science, Electrical Engineering, and Mathematics and director of the Information Initiative at Duke University. He received a BSc from Warwick University in 1975, an MSc from Oxford in 1976, and a PhD from Caltech in 1980, all in mathematics. He joined Bell Labs in 1980, and retired from AT&T Labs in 2003 as Vice President for Research and Internet and network systems. He then went to Princeton as a professor of Electrical Engineering, Mathematics and Applied and Computational Mathematics, before moving to Duke in 2010 to become Dean of Natural Sciences. His contributions to coding and information theory won the IEEE Information Theory Society Paper Award in 1995 and 1999. He was elected as a member into the US National Academy of Engineering in 2005 for leadership in communications research, from advances in algebraic coding theory to signal processing for wire-line and wireless modems. He also becam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Shor
Peter Williston Shor (born August 14, 1959) is an American professor of applied mathematics at MIT. He is known for his work on quantum computation, in particular for devising Shor's algorithm, a quantum algorithm for factoring exponentially faster than the best currently-known algorithm running on a classical computer. Early life and education Shor was born in New York City to Joan Bopp Shor and S. W. Williston Shor, of Jewish descent. He grew up in Washington, D.C. and Mill Valley, California. While attending Tamalpais High School, he placed third in the 1977 USA Mathematical Olympiad. After graduation that year, he won a silver medal at the International Math Olympiad in Yugoslavia (the U.S. team achieved the most points per country that year). He received his B.S. in Mathematics in 1981 for undergraduate work at Caltech, and was a Putnam Fellow in 1978. He earned his PhD in Applied Mathematics from MIT in 1985. His doctoral advisor was F. Thomson Leighton, and his thesi ... [...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 learned society and professional body that works to advance physics education, research and application. It was founded in 1874 and has a worldwide membership of over 20,000. The IOP is the Physica ... 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] |
Stabilizer Code
The theory of quantum error correction plays a prominent role in the practical realization and engineering of quantum computing and quantum communication devices. The first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance. Quantum error-correcting codes restore a noisy, decohered quantum state to a pure quantum state. A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors. A quantum error-correcting code makes quantum computation and quantum communication practical by providing a way for a sender and receiver to simulate a noiseless qubit channel given a noisy qubit channel whose noise conforms to a particular error model. The stabilizer theory of quantum error correction allows one to import some class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 qubit flip errors (X errors) and the dual of the Hamming code, the ,3,4code, to correct for 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Code
In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of ''all'' block codes in a unified way. Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, and Reed–Muller codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using boolean polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Code
In coding theory, the dual code of a linear code :C\subset\mathbb_q^n is the linear code defined by :C^\perp = \ where :\langle x, c \rangle = \sum_^n x_i c_i is a scalar product. In linear algebra terms, the dual code is the annihilator of ''C'' with respect to the bilinear form \langle\cdot\rangle. The dimension of ''C'' and its dual always add up to the length ''n'': :\dim C + \dim C^\perp = n. A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code. Self-dual codes A self-dual code is one which is its own dual. This implies that ''n'' is even and dim ''C'' = ''n''/2. If a self-dual code is such that each codeword's weight is a multiple of some constant c > 1, then it is of one of the following four types: *Type I codes are binary self-dual codes which are not doubly even. Type I codes are always even (every codeword has even Hamming weight). *Type II codes are binar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Footnotes
A note is a string of text placed at the bottom of a page in a book or document or at the end of a chapter, volume, or the whole text. The note can provide an author's comments on the main text or citations of a reference work in support of the text. Footnotes are notes at the foot of the page while endnotes are collected under a separate heading at the end of a chapter, volume, or entire work. Unlike footnotes, endnotes have the advantage of not affecting the layout of the main text, but may cause inconvenience to readers who have to move back and forth between the main text and the endnotes. In some editions of the Bible, notes are placed in a narrow column in the middle of each page between two columns of biblical text. Numbering and symbols In English, a footnote or endnote is normally flagged by a superscripted number immediately following that portion of the text the note references, each such footnote being numbered sequentially. Occasionally, a number between brack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computation And Quantum Information (book)
''Quantum Computation and Quantum Information'' is a textbook about quantum information science written by Michael Nielsen and Isaac Chuang, regarded as a standard text on the subject. It is informally known as "Mike and Ike", after the candies of that name. The book assumes minimal prior experience with quantum mechanics and with computer science, aiming instead to be a self-contained introduction to the relevant features of both. (Lov Grover recalls a postdoc disparaging it with the remark, "The book is too elementary – it starts off with the assumption that the reader does not even know quantum mechanics.") The focus of the text is on theory, rather than the experimental implementations of quantum computers, which are discussed more briefly. , the book has been cited over 39,000 times on Google Scholar. In 2019, Nielsen adapted parts of the book for his ''Quantum Country'' project. Table of Contents (Tenth Anniversary Edition) * Chapter 1: Introduction and Overview ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
