Home > Quantum Error > Quantum Error Correction Via Codes Over Gf4# Quantum Error Correction Via Codes Over Gf4

## Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively. Keywords Self-dual codes; Graphs; Local complementation Download full text

Classical information can be duplicated, while **quantum informationcannot [29], [77].It is well** known that classical information can be protected from degradation by the use ofclassical error-correcting codes [54]. M. Since the discussion in [16] was necessarily very condensed, we give the proof of thistheorem below.Theorem 1. Your cache administrator is webmaster. http://caribtechsxm.com/quantum-error/quantum-error-correction-codes.php

Classical inform ation cannot travel faster than light, while quan-tum in formation appears to in some circumstances (although proper deﬁnitions can resolvethis apparent paradox). Rains P. Institutional Sign In By Topic Aerospace Bioengineering Communication, Networking & Broadcasting Components, Circuits, Devices & Systems Computing & Processing Engineered Materials, Dielectrics & Plasmas Engineering Profession Fields, Waves & Electromagnetics General Use of this web site signifies your agreement to the terms and conditions. https://arxiv.org/abs/quant-ph/9608006

The resulting pairs can then beused to teleport quantum information from one experimenter to the other [3]. A. Inform. Copyright © 2016 ACM, Inc.

- M Rains (2), P.
- Van den Nest, Local equivalence of stabilizer states and codes, PhD thesis, K.U.
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- Since χ is a homomorphism, it suﬃces to compute thecharacter on a basis for¯S.
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Schlingemann, R.F. Section 3 shows that these spaces in turn areequivalent to a certain class of additive codes over GF (4) (Theorem 2). An additive quantum-error-correcting code Q with associated space¯S can correcta set of errors Σ ⊆ E prec i sely when ¯e−11¯e26∈¯S⊥\¯S for all e1, e2∈ Σ.Proof. J.

Shor N. The distance between two elements (a|b), (a′|b′) ∈¯E is deﬁned tobe the weight of their diﬀerence.Then we have the followin g theorem, which is an immediate consequence of Theorem 1of [16]. Rains , P. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.5670 The dimension of a totally isotropic su b space is at most n.

Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. Rains3rd P.M. Rains, P.M. Kim, V.

Quantum error correcting codesare subspaces oriented so that any error in a relatively small number of qubits moves the statein a direction perpendicular to the coded subspace, and thus can be Terms of Usage Privacy Policy Code of Ethics Contact Us Useful downloads: Adobe Reader QuickTime Windows Media Player Real Player Did you know the ACM DL App is Help Direct export Save to Mendeley Save to RefWorks Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only Citation and Abstract Export Advanced search Close This document The system returned: (22) Invalid argument The remote host or network may be down.

It is also the group generated by fault-tolerant bitwise operations performed on qubits that are encoded by certain quantum codes [17], [66], [72]. http://caribtechsxm.com/quantum-error/quantum-computation-quantum-error-correcting-codes-and-information-theory.php J. Huffman, J.-L. W.

Inthe present paper we develop the theory to the point where it is possible to apply standardtechniques f rom classical coding theory to construct quantum codes. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, coding theory and Quantum error correction. Use of this web site signifies your agreement to the terms and conditions. http://caribtechsxm.com/quantum-error/quantum-error-correction-with-degenerate-codes-for-correlated-noise.php This induces a decom-position of2ninto orthogonal eigenspaces.

We have now completed the proof ofTheorem1: Q maps k qubits into n qubits and can correct [(d −1)/2] errors.Deco ding. Gaborit On extremal additive F4 codes of length 10 to 18 J. Inform.

To determine which eigenspace contains a given state it is thereforeenough to determine this character. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubitsDiscover the world's research11+ million Sloane, S. Of course, an [[n, k, d]] code is also an ((n, 2k, d)) code.Readers who are most interested in the codes themselves could now proceed directly toSection 3.Proof.

Replaced Sept. 24, 1996, to correct a number of minor errors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. click site Generated Tue, 25 Oct 2016 02:48:27 GMT by s_wx1157 (squid/3.5.20)

morefromWikipedia Tools and Resources Save to Binder Export Formats: BibTeX EndNote ACMRef Share: | Contact Us | Switch to single page view (no tabs) **Javascript is not enabled and is required To m otivate the following discussion we begin by describing classical binary linearco des from a slightly unusual perspective. If we deﬁne ERto be the real sub group of E, then LRis the normalizer ofERin the orthogonal group O(2n). Although these central elements haveno eﬀect quantum-mechanically, we wish to work with a ﬁnite group.

Of course, as in classical coding theory,identifying the most likely error given the synd rome can be a diﬃcult problem. (There is notheoretical diﬃculty, since in p rinciple an exhaustive search