# Quantum Error Correction Via Codes Over Gf4

A puriﬁcation protocol that uses only a one-way classical channelbetween th e experimenters can be converted into a quantum-error-corr ecting code, and viceversa [5]. Danielsen On self-dual quantum codes, graphs, and Boolean functions Master's thesis, Department of Informatics, University of Bergen, Norway, March, 2005 quant-ph/0503236 [8] L.E. 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 qubits. In other words, we must determinethe coset eS. weblink

Although itwas not immediately apparent, these two discoveries tur ned out to be diﬀerent ways of lookingat the same phenomenon. Rev. Sloane40.5 · UnknownAbstractThe problem of finding **quantum error correcting** codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain Cannon, W.

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 A, 65 (1) (2002), p. 012308 quant-ph/0012111 [22] N.J.A. Some of the ideas in [16](although neither the connections with the Cliﬀord group nor with ﬁnite geometries or ﬁelds)were discovered indep en dently by Gottesman [36].The paper is arranged as follows. Theory},year = {1998},volume = {44},pages = {1369--1387}} Share OpenURL Abstract The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF (4)

- Theory, 44 (4) (1998), pp. 1369–1387 quant-ph/9608006 [5] J.
- Classical information can be dup...
- It therefore s uﬃces to determine the coset eS⊥.
- Bosma Handbook of Magma Functions Version 2.11 http://magma.maths.usyd.edu.au/ (May 2004) [6] L.E.
- Thismeasurement has no eﬀect on the state, since the state lies inside one of the eigenspaces.6 On the other hand, suppose e1and e2are two errors such that ¯e−11¯e2∈¯S⊥\¯S.
- Combin., 8 (3) (1987), pp. 231–244 [3] A.
- Schlingemann Stabilizer codes can be realized as graph codes Quantum Inf.
- After these discoveries, a number of improved quantum codes were soon found byvarious researchers.The setting in which quantum-error-correcting codes exist is the quantum state space of nqubits (qu antum bits, or
- Suppose an error e ∈ E has occurred.

IEEE Int. Inform. W. IntroductionThe relationship between **quantum information and** classical information is a subject cur-rently receiving much study.

Comput. 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 Generated Tue, 25 Oct 2016 02:48:27 GMT by s_wx1157 (squid/3.5.20) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.5670 ERis an extraspecial 2-group with order 22n+1and2We follow Bolt et al. ([6], [7]) in calling these Cliﬀord groups.

Since we also want S⊥topreserve the code, we take the code to b e one of the eigens paces for S, to be denoted by Q(say). The dimension of a totally isotropic su b space is at most n. Rains and P. Anycorrection procedure must take any state e1(v) ∈ e1(Q) to v.

Introduction The relationship between quantum information and classical information is a subject currently receiving much study. Huffman, J.-L. Shor N. of the 37th Allerton Conference on Communication, Control, and Computing, University of Illinois at Urbana-Champaign, 1999, pp. 535–544 [10] P.

Since χ is a homomorphism, it suﬃces to compute thecharacter on a basis for¯S. have a peek at these guys JavaScript is disabled on your browser. Although there are still a large **number of gaps in the table,** the upper andlower bounds are generally not too far apart and there are a considerable number of entrieswhere the Their basic properties are described in the remainderof Section 3, and Section 4 gives a number of general constructions.

B, 45 (1) (1988), pp. 58–76 [4] A.R. Suppose¯S is an (n −k)-dimensional linear subspace of¯E which is contained inits dual¯S⊥(with respect to the inner product (1)), and is such that there are no vectors ofweight < d in¯S⊥\¯S. We have now completed the proof ofTheorem1: Q maps k qubits into n qubits and can correct [(d −1)/2] errors.Deco ding. http://vealcine.com/quantum-error/quantum-error-correction-with-degenerate-codes-for-correlated-noise.php Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors.

Section 3 shows that these spaces in turn areequivalent to a certain class of additive codes over GF (4) (Theorem 2). The system returned: (22) Invalid argument The remote host or network may be down. However, since e−11e26∈ S, there is a state v ∈ Q such thate−11e2(v) is not proportional to v, and we have failed to correct e2.It follows from the Lemma that if

## Pless On additive GF(4) codes Codes and Association Schemes, DIMACS Ser.

**M. **For many ap plications it i Skip to Main Content IEEE.org IEEE Xplore Digital Library IEEE-SA IEEE Spectrum More Sites Cart(0) Create Account Personal Sign In Personal Sign In Username Password Soc., Providence, RI (2001), pp. 135–149 [11] D.G. Ann., 327 (2) (2003), pp. 227–255 math.CO/0005266 [17] J.-L.

The second stage converts the latter to a codingtheory problem.The ﬁnite geometry problem can be summarized as follows. US & Canada: +1 800 678 4333 Worldwide: +1 732 981 0060 Contact & Support About IEEE Xplore Contact Us Help Terms of Use Nondiscrimination Policy Sitemap Privacy & Opting Out Analogously,we have the following lemma.Lemma 1. this content Let¯E denote a 2n-dimens ionalbinary vector s pace, whose elements are wr itten (a|b) and which is equip ped with the inner3 product((a|b), (a′|b′)) = a · b′+ a′· b .

However, it is very hard to construct codes using the framewor k of [16]. Get Help About IEEE Xplore Feedback Technical Support Resources and Help Terms of Use What Can I Access? It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to The resulting pairs can then beused to teleport quantum information from one experimenter to the other [3].

W. Finally, we again corrected a number of minor errors Subjects: Quantum Physics (quant-ph) Citeas: arXiv:quant-ph/9608006 (or arXiv:quant-ph/9608006v5 for this version) Submission history From: Peter Shor [view email] [v1] Tue, 6 The finite fields are classified by size; there is exactly one finite field up to isomorphism of size p for each prime p and positive integer k. For the purposes ofquantum error correction, however, we need consider only the three types of errors σx, σyand σz, since any error-correctin g code which corrects t of these errors will