# Quantum Error Correcting Codes Using Qudit Graph States

The concept of a stabilizer **is extended to general D, andshown** to provide a dual representation of an additive graph code.PACS numbers: 03.67.PpI. Exhaustively checking all possibilities israther time consuming, somewhat like solving an optimalpacking problem.In practice what we do is to ﬁrst construct a lookuptable containing the Dn−1 Pauli distances from |Gi to We fully characterized how much information is left on the remaining carrier qudits using concepts like types of information and information groups. Some authors preferthe term MDS, but as it is not clear to us how the conceptof “maximum distance separable,” as explained in [20],carries over to quantum codes, we prefer to use weblink

IV C and IV D. III C, whilevarious results in terms o f speciﬁc codes are the subjectof Sec.IV.In Sec.IV B we show how to construct graph codeswith δ = 2 that saturate the quantum Singleton Physical Review A™ is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. Applied Phys.

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- Griffiths40.66 · Carnegie Mellon UniversityAbstractGraph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$, and simple graphical methods are used to construct both additive and nonadditive
- Phys.
- 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
- We also show the three-part problem above can be mapped to studying correlations of entanglement of three-part stabilizer states.
- Math.
- Bibliographic Code:2011PhDT.......110L Abstract Graph states were first introduced to construct quantum error-correcting codes.

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. Lidar,Todd **A. **We shall refer to codeswhich saturate this bound (the inequality is an equality)as quantum Singleton (QS) codes. C Phys.

We extended Bravyi et al.'s proof [J. Phys. 47, 062106 (2006)] on entanglement of three part stabilizer states from qubits ( D = 2) to general squarefree D, i.e. As the states Zν|+i, 0 ≤ ν ≤ D − 1, are anorthonormal basis for a single qudit, their products forman orthonormal basis for n qudits. https://arxiv.org/abs/0712.1979v1 This leadsto a considerable simpliﬁcation of the problem along w iththe possibility of treating nonadditive graph codes onexactly the same basis as additive or stabilizer codes.

We also consider the distinction be-tween degenerate and nondegenerate codes. For this reason, we shall omitthe G in G-additive except in cases where it is essentialto make the distinction. For both D = 2 and D = 3 we have studied nonde-generate codes on sequences of cycle and wheel graphs,in Secs. While P isnot Abelian, it has **the property thatP Q =** ωµQP, (4)where µ is an integer that depends on P and Q. (WhenD = 2 and ω = −1 it

INTRODUCTIONQuantum erro r correction is an important part ofvarious schemes for quantum computation and quan-tum communication, and hence quantum error correctingcodes, ﬁrst introduced about a decade ago [1, 2, 3] havereceived Scalable quantum computers require a far-reaching theory of fault-tolerant quantum computation. Rev. (Series I) Physics Volume: Article: × Cornell University Library We gratefully acknowledge support fromthe Simons Foundation and member institutions arXiv.org > quant-ph > arXiv:0712.1979v1 Search or Article-id (Help | Advanced In this per-sp e c tive the stabilizer is a dual repre sentation of a codewhich is eq ually well represented by its codewords.

The book is not limited to a single approach, but reviews many different methods to control quantum errors, including topological codes, dynamical decoupling and decoherence-free subspaces. http://vealcine.com/quantum-error/quantum-error-correcting-codes-from-the-compression-formalism.php The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an additive graph code. E Phys. It was subsequently shown to be a powerful formalism as it includes additive (or stabilizer) as well nonadditive codes.

Distance δ = 2; bar and star graphsIt was shown in [19] that for D = 2 one can constructδ = 2 QS codes for any even n, and similar codes D is not divisible by a perfect square.

AdviserRobert B. 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 check over here Wecan only say the code is maximal in the sense thatno codeword can be added without violating (13).Absence of this s uperscript implies no code witha larger K exists for thisEduc. 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 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

## 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

The APS Physics logo and Physics logo are trademarks of the American Physical Society. Computer searches have produced a number of cod es with distances 3 and4, some previously kn own and some new. The original graphstate |Gi is |0, 0, . . . , 0i in this notation. A Phys.

D Phys. It follows from its deﬁnition that Clmcom-mutes with Zland Zm, and thus with Zpfor any q uditp.B. Phys. this content Sincethe Clmfor diﬀerent l and m commute with ea ch other,and also with Zpfor any p, the or der of the operators onthe right side of (7) is unimportant.X≡Z3Z2ZX2≡Z6= Z2Z4= IZ2FIG.

Thisprocess also yields the diagonal distance ∆′. Applying the unitaryU to this basis yields the orthonormal graph basis. 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 $D$,