Spin glass reflection of the decoding transition for quantum error correcting codes

We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette $\Z2$ gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite.Comment: 15 pages, 2 figure

Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes

We discuss error-correction properties for families of quantum low-density parity check (LDPC) codes with relative distance that tends to zero in the limit of large blocklength. In particular, we show that any family of LDPC codes, quantum or classical, where distance scales as a positive power of the block length, $d \propto n^\alpha$, $\alpha>0$, can correct all errors with certainty if the error rate per (qu)bit is sufficiently small. We specifically analyze the case of LDPC version of the quantum hypergraph-product codes recently suggested by Tillich and Z\'emor. These codes are a finite-rate generalization of the toric codes, and, for sufficiently large quantum computers, offer an advantage over the toric codes.Comment: 4.5 pages, 1 figur

Numerical Techniques for Finding the Distances of Quantum Codes

We survey the existing techniques for calculating code distances of classical codes and apply these techniques to generic quantum codes. For classical and quantum LDPC codes, we also present a new linked-cluster technique. It reduces complexity exponent of all existing deterministic techniques designed for codes with small relative distances (which include all known families of quantum LDPC codes), and also surpasses the probabilistic technique for sufficiently high code rates.Comment: 5 pages, 1 figure, to appear in Proceedings of ISIT 2014 - IEEE International Symposium on Information Theory, Honolul

Numerical and analytical bounds on threshold error rates for hypergraph-product codes

We study analytically and numerically decoding properties of finite rate hypergraph-product quantum LDPC codes obtained from random (3,4)-regular Gallager codes, with a simple model of independent X and Z errors. Several non-trival lower and upper bounds for the decodable region are constructed analytically by analyzing the properties of the homological difference, equal minus the logarithm of the maximum-likelihood decoding probability for a given syndrome. Numerical results include an upper bound for the decodable region from specific heat calculations in associated Ising models, and a minimum weight decoding threshold of approximately 7%.Comment: 14 pages, 5 figure

A VLBA survey of the core shift effect in AGN jets I. Evidence for dominating synchrotron opacity

The effect of a frequency dependent shift of the VLBI core position (known as the "core shift") was predicted more than three decades ago and has since been observed in a few sources, but often within a narrow frequency range. This effect has important astrophysical and astrometric applications. To achieve a broader understanding of the core shift effect and the physics behind it, we conducted a dedicated survey with NRAO's Very Long Baseline Array (VLBA). We used the VLBA to image 20 pre-selected sources simultaneously at nine frequencies in the 1.4-15.4 GHz range. The core position at each frequency was measured by referencing it to a bright, optically thin feature in the jet. A significant core shift has been successfully measured in each of the twenty sources observed. The median value of the core shift is found to be 1.21 mas if measured between 1.4 and 15.4 GHz, and 0.24 mas between 5.0 and 15.4 GHz. The core position, r, as a function of frequency, n, is found to be consistent with an r n^-1 law. This behavior is predicted by the Blandford & Koenigl model of a purely synchrotron self-absorbed conical jet in equipartition. No systematic deviation from unity of the power law index in the r(n) relation has been convincingly detected. We conclude that neither free-free absorption nor gradients in pressure and/or density in the jet itself and in the ambient medium surrounding the jet play a significant role in the sources observed within the 1.4-15.4 GHz frequency range. These results support the interpretation of the parsec-scale core as a continuous Blandford-Koenigl type jet with smooth gradients of physical properties along it.Comment: 31 pages, 6 figures, 5 tables; accepted to Astronomy & Astrophysic