Interleaving schemes for multidimensional cluster errors

We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice

Folding, Tiling, and Multidimensional Coding

Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. We show how folding can be used to construct multidimensional error-correcting codes and to generate multidimensional pseudo-random arrays

Topological code Autotune

Many quantum systems are being investigated in the hope of building a large-scale quantum computer. All of these systems suffer from decoherence, resulting in errors during the execution of quantum gates. Quantum error correction enables reliable quantum computation given unreliable hardware. Unoptimized topological quantum error correction (TQEC), while still effective, performs very suboptimally, especially at low error rates. Hand optimizing the classical processing associated with a TQEC scheme for a specific system to achieve better error tolerance can be extremely laborious. We describe a tool Autotune capable of performing this optimization automatically, and give two highly distinct examples of its use and extreme outperformance of unoptimized TQEC. Autotune is designed to facilitate the precise study of real hardware running TQEC with every quantum gate having a realistic, physics-based error model.Comment: 13 pages, 17 figures, version accepted for publicatio

Creation of quantum error correcting codes in the ultrastrong coupling regime

We propose to construct large quantum graph codes by means of superconducting circuits working at the ultrastrong coupling regime. In this physical scenario, we are able to create a cluster state between any pair of qubits within a fraction of a nanosecond. To exemplify our proposal, creation of the five-qubit and Steane codes is numerically simulated. We also provide optimal operating conditions with which the graph codes can be realized with state-of-the-art superconducting technologies.Comment: Added a new appendix sectio

Cluster-based architecture for fault-tolerant quantum computation

We present a detailed description of an architecture for fault-tolerant quantum computation, which is based on the cluster model of encoded qubits. In this cluster-based architecture, concatenated computation is implemented in a quite different way from the usual circuit-based architecture where physical gates are recursively replaced by logical gates with error-correction gadgets. Instead, some relevant cluster states, say fundamental clusters, are recursively constructed through verification and postselection in advance for the higher-level one-way computation, which namely provides error-precorrection of gate operations. A suitable code such as the Steane seven-qubit code is adopted for transversal operations. This concatenated construction of verified fundamental clusters has a simple transversal structure of logical errors, and achieves a high noise threshold ~ 3 % for computation by using appropriate verification procedures. Since the postselection is localized within each fundamental cluster with the help of deterministic bare controlled-Z gates without verification, divergence of resources is restrained, which reconciles postselection with scalability.Comment: 16 pages, 34 figure