Home > Quantum Error > Quantum Error Correcting Codes Using Qudit Graph States# Quantum Error Correcting Codes Using Qudit Graph States

## Graph states can also be used to construct families of equally-entangled bases on two or more parts.

**Rev. **We will see an example ofthis in Sec.IV for D = 2.The quantum Singleton (QS) bound [2]n ≥ logDK + 2(δ − 1 ) or K ≤ Dn−2(δ−1)(18)is a simple but Publication Date:00/2011 Category:Physics, Quantum;Physics, Theory Origin:UMI Comment:Publication Number: AAT 3476126; ISBN: 9781124926445; Advisor: Griffiths, Robert B. IntroductionResults obtained using methods described above arereported here for vario us sequences of graphs, each se-quence containing graphs of increasing n while preservingcertain basic properties. http://caribtechsxm.com/quantum-error/quantum-computation-quantum-error-correcting-codes-and-information-theory.php

V we show that what we call G-additive codesare stabilizer codes (hence “additive” in the sens e usuallyemployed in the literature), using a suitable generaliza-tion of the stabilizer formalism to general These include a number of caseswhich saturate the QS bo und for δ = 2 a nd 3, and o therswith δ = 3 and 4 which are the largest possible Rev. Rev. check this link right here now

He has worked on quantum control theory for the past 15 years and is well known for his contributions to quantum error correction, in particular the development of decoherence-free subspaces. Codes of distance 2 saturating the quantumSingleton bound for arbitrarily large n and D are constructed using simple graphs, except when nis odd and D is even. An operator of the formP = ωλXµ11Zν11Xµ22Zν22· · · XµnnZνnn, (3)where λ, and µland νlfor 1 ≤ l ≤ n, are integers in therange 0 to D − 1, will be Applying these two rules andkeeping track of powers of ω resulting from interchang-ing Xland Zl, see (2), allows one to easily evaluate theaction of any Pauli product on any |ai in

- Graph states can also be used to construct families of equally-entangled bases on two or more parts.
- B Phys.
- The concept of a stabilizer is extended to general D, and shown to provide a dual representation of an additive graph code.Received 12 February 2008DOI:https://doi.org/10.1103/PhysRevA.78.042303©2008 American Physical SocietyAuthors & Affiliations Shiang
- BrunCambridge University Press, 12 Σεπ 2013 - 592 σελίδες 0 Κριτικέςhttps://books.google.gr/books/about/Quantum_Error_Correction.html?hl=el&id=fafqAAAAQBAJQuantum computation and information is one of the most exciting developments in science and technology of the last twenty years.
- SeeSec.
- Next pair the vertices by connectingeach vertex in V1by a single edge to a vertex in V2, withone additional edge when n is odd, as shown in the ﬁgure.(Multiple edges are
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Skip to Main ContentJournalsPhysical Review LettersPhysical Review XReviews of Modern PhysicsPhysical Review APhysical Review BPhysical Review CPhysical Review DPhysical Review EPhysical Review AppliedPhysical Review FluidsPhysical Review Accelerators and BeamsPhysical Review Physics Of those, over 1.7 million are available in PDF format. As the states Zν|+i, 0 ≤ ν ≤ D − 1, are anorthonormal basis for a single qudit, their products forman orthonormal basis for n qudits. Griffiths (Submitted on 12 Dec 2007 (v1), last revised 11 Nov 2008 (this version, v4)) Abstract: Graph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$,

We fully characterized how much information is left on the remaining carrier qudits using concepts like types of information and information groups. We shall refer to codeswhich saturate this bound (the inequality is an equality)as quantum Singleton (QS) codes. Rev. see it here Phys.

Beams Phys. They s atisfyZD= I = XD, XZ = ωZX, ω := e2π i/D. (2)We shall refer to the collection of D2operators {XµZν},µ, ν = 0, 1, . . . , D Thisprocess also yields the diagonal distance ∆′. To achieve large scale quantum computers and communication networks it is essential not only to overcome noise in stored quantum information, but also in general faulty quantum...https://books.google.gr/books/about/Quantum_Error_Correction.html?hl=el&id=fafqAAAAQBAJ&utm_source=gb-gplus-shareQuantum Error CorrectionΗ βιβλιοθήκη μουΒοήθειαΣύνθετη

It follows from its deﬁnition that Clmcom-mutes with Zland Zm, and thus with Zpfor any q uditp.B. D−1 be an orthonormal basis forthe D-dimensio nal Hilbert space of a qudit, and deﬁne 2the unitary operators [17]Z =D−1Xj=0ωj|jihj| , X =D−1Xj=0|jihj ⊕ 1| , (1)with ⊕ denoting addition mo The size ofan operator R is deﬁned as the number of qudits in itsbase, i.e., the number on which it acts in a nontrivialfashion. Inthe present paper we follow [9] in focusing on qudits withgeneral D, thought of as elements of the additive groupZDof integers mod D.

Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. http://caribtechsxm.com/quantum-error/quantum-error-correcting-codes-from-the-compression-formalism.php Bibliographic Code:2011PhDT.......110L Abstract Graph states were first introduced to construct quantum error-correcting codes. Skip to Main ContentJournalsPhysical Review LettersPhysical Review XReviews of Modern PhysicsPhysical Review APhysical Review BPhysical Review CPhysical Review DPhysical Review EPhysical Review AppliedPhysical Review FluidsPhysical Review Accelerators and BeamsPhysical Review Physics Theﬁnal Sec.VI has a summary of our results and indicatesdirections in which they might be extended.II.

Physical Review A™ is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. A and illustratedin Fig.1, the eﬀect of applying Xlto |Gi is the s ame asapplying (Zm)Γlmto each of the qudits corresponding toneighbors of l in the gra ph. Rev. More about the author Therefore, if (12) is satisﬁed for some Q anda collection {|cqi} of codewords, the same will be truefor the same Q and the collection {|cq⊕ ai} (with anappropriate change in f(Q)).

Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. D Phys. We know of no ge neral principles for making thischoice, tho ugh it is helpful to note, see App.A, that thediagonal distance ∆′cannot exceed 1 plus the minimumover all vertices of

Of the nearly 4 million graduate works included in the database, ProQuest offers more than 2.5 million in full text formats. That this col-lection forms a n orthonormal basis follows from the factthat the Zpoperators commute with the Clmoperators ,so can be moved thro ugh the unitary U on the right sideof Graph codesWhen each basis vector |cqi is a member of the graphbasis, of the form (10) for some graph G, we shall saythat the corresponding code is a graph code associatedwith Mod.

Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. D is not divisible by a perfect square. The additiveproperty allows one to express all codewords as “linearcombinations” of k suitably chosen codeword generators.This implies an additive code must have K = Dr, r aninteger, whenever D is prime. click site Table I shows themaximum number K of codewords for codes o f distanceδ = 2 and δ = 3 for both D = 2 qubits and D = 3qutrits.

For example, the base of P = ω2X21X4Z4(as-suming D ≥ 3) is {1, 4} and its size is 2; whereas the sizeof R = X1+ 0.5X2Z22Z3+ iX4is 4.For two distinct qudits Res. A 78, 042303 (2008) DOI: 10.1103/PhysRevA.78.042303 Citeas: arXiv:0712.1979 [quant-ph] (or arXiv:0712.1979v4 [quant-ph] for this version) Submission history From: Shiang Yong Looi [view email] [v1] Wed, 12 Dec 2007 17:02:31 GMT If you have questions, please feel free to visit the ProQuest Web site - http://www.proquest.com - or contact ProQuest Support.

For a detailed dis-cussion see Ch. 10 of [4]. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement. The approach used here was developed indepen-dently in [13] and [14] for D = 2, and in [15] for D > 2;thus several of our results are similar to those reportedin

Physical Review A™ is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. When the Knill-Laﬂamme [2] conditionhcq| Q |cri = f(Q)δqr(12)is satisﬁed for all q and r between 0 and K − 1, andevery operato r Q on H such that 1 ≤ Phys. The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an additive graph code.Discover the world's research11+ million members100+ million publications100k+ research projectsJoin

Sometimes thisrevealed a pattern which could be further ana ly z e d usinganalytic arguments or known bounds on the number ofcodewords.In the c ase of distance δ = 2 we Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement. PAULI OPERATORS AND GRAPH STATESA. Both must hold for all op e ratorsof s iz e up to and including δ −1, but need not be satisﬁedfor larger operators.In the coding literature it is customa ry

Phys. 47, 062106 (2006)] on entanglement of three part stabilizer states from qubits ( D = 2) to general squarefree D, i.e.