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Register a new userEntanglement-Assisted Quantum Quasi-Cyclic Low-Density Parity-Check Codes
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We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.
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Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial $g(x)$ of degree $r$ over $mathbb{F}_{q}$ satisfying $gamma g(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=left(begin{array}{cc} a&b1&dend{array}right)in PGL_2(Bbb F_{q})$, $mathrm{ord}(A)$ is a prime, $g(a) e 0$, and $0 e gammain Bbb F_q$. And we give a complete characterization of irreducible polynomials $g(x)$ of degree $2s$ or $3s$ as above, where $s$ is a positive integer. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.
Coherent parity check (CPC) codes are a new framework for the construction of quantum error correction codes that encode multiple qubits per logical block. CPC codes have a canonical structure involving successive rounds of bit and phase parity checks, supplemented by cross-checks to fix the code distance. In this paper, we provide a detailed introduction to CPC codes using conventional quantum circuit notation. We demonstrate the implementation of a CPC code on real hardware, by designing a [[4,2,2]] detection code for the IBM 5Q superconducting qubit device. Whilst the individual gate-error rates on the IBM device are too high to realise a fault tolerant quantum detection code, our results show that the syndrome information from a full encode-decode cycle of the [[4,2,2]] CPC code can be used to increase the output state fidelity by post-selection. Following this, we generalise CPC codes to other quantum technologies by showing that their structure allows them to be efficiently compiled using any experimentally realistic native two-qubit gate. We introduce a three-stage CPC design process for the construction of hardware-optimised quantum memories. As a proof-of-concept example, we apply our design process to an idealised linear seven-qubit ion trap. In the first stage of the process, we use exhaustive search methods to find a large set of [[7,3,3]] codes that saturate the quantum Hamming bound for seven qubits. We then optimise over the discovered set of codes to meet the hardware and layout demands of the ion trap device. We also discuss how the CPC design process will generalise to larger-scale codes and other qubit technologies.
Photonic circuits in which stateful components are coupled via guided electromagnetic fields are natural candidates for native implementation of iterative stochastic algorithms based on propagation of information around a graph. Conversely, such message passing algorithms suggest novel circuit architectures for signal processing and computation that are well matched to nanophotonic device physics. Here we construct and analyze a quantum optical model of a photonic circuit for iterative decoding of a class of low-density parity-check (LDPC) codes called expander codes. Our circuit can be understood as an open quantum system whose autonomous dynamics map straightforwardly onto the subroutines of an LDPC decoding scheme, with several attractive features: it can operate in the ultra-low power regime of photonics in which quantum fluctuations become significant, is robust to noise and component imperfections, achieves comparable performance to known iterative algorithms for this class of codes, and provides an instructive example of how nanophotonic cavity quantum electrodynamic components can enable useful new information technology even if the solid-state qubits on which they are based are heavily dephased and cannot support large-scale entanglement.
Consider transmission over a binary additive white gaussian noise channel using a fixed low-density parity check code. We consider the posterior measure over the code bits and the corresponding correlation between two codebits, averaged over the noise realizations. We show that for low enough noise variance this average correlation decays exponentially fast with the graph distance between the code bits. One consequence of this result is that for low enough noise variance the GEXIT functions (further averaged over a standard code ensemble) of the belief propagation and optimal decoders are the same.
We consider the effect of log-likelihood ratio saturation on belief propagation decoder low-density parity-check codes. Saturation is commonly done in practice and is known to have a significant effect on error floor performance. Our focus is on threshold analysis and stability of density evolution. We analyze the decoder for standard low-density parity-check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation. Stability of density evolution is, on the other hand, rather strongly effected by saturation and the asymptotic qualitative effect of saturation is similar to reduction by one of variable node degree. We also show under what conditions the block threshold for the saturated belief propagation corresponds with the bit threshold.
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