Sloane, Quantum Error Correction and Orthogonal Geometry, Phys. Wilde, Logical Operators of Quantum Codes, Phys. Rev. A 89, 012317 (2014).CASArticle57.Herold, M., Campbell, E.

The Clifford gates are chosen from the comparisons between both codes. This is an important consideration as we search for schemes that realize fault-tolerant quantum computation with a low cost in quantum resources. Phys. Elements of the stabilizer group, , satisfy the property that for all codewords of the code .The code is defined on a three-dimensional four-valent lattice of linear size L with qubits

Bravyi and A. Aliferis and A. W. For H2, we perform the same procedure, but each 1 is instead replaced by X. Despite being efficiently simulable, most stabilizer states on a large number of qubits exhibit maximal bipartite entanglement[Dahlsten and Plenio, QIC 2006].

We give numerical evidence showing that our error-correction protocol confines errors in the following subsection.The simulationWe simulate fault-tolerant error correction with encoded states of linear size L where j indicates the It is important to note that the error-correction procedure is sensitive to the positions of stabilizer defects within a correctable cluster, as discrepancies in their positions later affect the performance of The dashed red line marks psus∼0.31%, the sustainable noise rate of the code, the limiting value of pth from to the fitting as N→∞.Full size imageWe find pth(0)∼0.46% and γ∼1.47. Chung, A. W.

For N qubits, there are 2Nπi = 1N(2i + 1) stabilizer states[Gottesmann and Aaronson PRA 2004], as opposed to the infinite number of general states. The best rigorous proofs of the threshold to date show that the threshold is at least 2 × 10 − 5 (meaning one error per 50, 000 operations). This equation is derived by assuming that a single use of the decoder will fail with probability in the low-p regime. Rev.

It is assumed that measurements and classical computations can be performed quickly and reliably, and that quantum gates can be performed between arbitrary pairs of qubits in the computer, irrespective of Sloane, Quantum Error Correction and Orthogonal Geometry, Phys. Then S encodes k qubits and has distance d, where d is the smallest weight of an operator in S ⊥ \ S. Browne3Nature Communications 7, Articlenumber:12302 (2016)doi:10.1038/ncomms12302Download CitationInformation technologyQuantum informationReceived:16 November 2015Accepted:21 June 2016Published online:29 July 2016AbstractThe constituent parts of a quantum computer are inherently vulnerable to errors.

The Clifford group on n qubits is defined as the set of unitary operations which conjugate the Pauli group Pn into itself; Cn can be generated by the Hadamard transform, the These include EPR and GHZ states. Restrictions on transveral encoded quantum gate sets. Lett. 77, 198 (1996).D.

A 71, 022316 (2005).CASArticle25.Fowler, A. For each face f, there are two face operators, and , where are the vertices on the boundary of face f. Because of the simple structure of the Pauli group, any Abelian subgroup has order 2n − k for some k and can easily be specified by giving a set of n − k commuting generators. Thus should be distinguished from the encoding operation which maps HlogK into Hn, determining the imbedding of C.

Fowler, D. S. In general, however, one must worry that large correlated errors can be introduced that adversarially corrupt encoded information45,46,47,48,49,50. Fault-tolerant quantum computation by anyons. Comput. 3, 84 (2003).D.

Owing to the symmetry of the gauge group, it suffices here to consider only bit-flip, that is, Pauli-X errors. Rains, P. W. The content of Issues in Information Science Research: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. Calderbank and P. W.

A 54, 3824 (1996).C. H. Rev. Laflamme, Concatenated Quantum Codes, arXiv:quant-ph/9608012.E. Rev.

A 86, 052329 (2012).S. A particularly useful fact is that a transversal CNOT gate (i.e., CNOT acting between the ith qubit of one block of the QECC and the ith qubit of a second block Hill, A. G. NickersonDepartment of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UKDan E.

X 6, 031039 – Published 13 September 2016 More×ArticleReferencesNo Citing ArticlesArticleReferencesPDFHTMLExport CitationA. M. Rev. The stabilizer at the cell with thick red edges contains a stabilizer defect. We use the notation ((n, K, d)) to refer to an ((n, K)) QECC with distance d.

Chuang, Transversality versus Universality for Additive Quantum Codes, IEEE Trans. Smolin, and W. K. Laflamme, and W. Math.

Phys. 14, 073048 (2012).Article54.Sarvepalli, P. & Raussendorf, R. We therefore obtain between ∼750 and ∼2,700 logical failures per data point close to the threshold error rate.Overhead analysisHere we summarize the resource-scaling analysis we give in the p

Rev. In addition, CSS codes have some very useful properties which make them excellent choices for fault-tolerant quantum computation. Rev. Hill, Austin G.

Magic-state distillation with low overhead.