431 research outputs found
LDPC Codes for 2D Arrays
Binary codes over 2D arrays are very useful in data storage, where each array column represents a storage device or unit that may suffer failure. In this paper, we propose a new framework for probabilistic construction of codes on 2D arrays. Instead of a pure combinatorial erasure model used in traditional array codes, we propose a mixed combinatorial-probabilistic model of limiting the number of column failures, and assuming a binary erasure channel in each failing column. For this model, we give code constructions and detailed analysis that allow sustaining a large number of column failures with graceful degradation in the fraction of erasures correctable in failing columns. Another advantage of the new framework is that it uses low-complexity iterative decoding. The key component in the analysis of the new codes is to analyze the decoding graphs induced by the failed columns, and infer the decoding performance as a function of the code design parameters, as well as the array size and failure parameters. A particularly interesting class of codes, called probabilistically maximum distance separable (MDS) array codes, gives fault-tolerance that is equivalent to traditional MDS array codes. The results also include a proof that the 2D codes outperform standard 1D low-density parity-check codes
Downlink Steered Space-Time Spreading Assisted Generalised Multicarrier DS-CDMA Using Sphere-Packing-Aided Multilevel Coding
This paper presents a novel generalised Multi-Carrier Direct Sequence Code Division Multiple Access (MC DS-CDMA) system invoking smart antennas for improving the achievable performance in the downlink, as well as employing multi-dimensional Sphere Packing (SP) modulation for increasing the achievable diversity product. In this contribution, the MC DS-CDMA transmitter considered employs multiple Antenna Arrays (AA) and each of the AAs consists of several antenna elements. Furthermore, the proposed system employs both time- and frequency- (TF) domain spreading for extending the achievable capacity, when combined with a novel user-grouping technique for reducing the effects of Multiuser Interference (MUI). Moreover, in order to further enhance the system’s performance, we invoke a MultiLevel Coding (MLC) scheme, whose component codes are determined using the so-called equivalent capacity based constituent-code rate-calculation procedure invoking a 4-dimensional bit-to-SP-symbol mapping scheme. Our results demonstrate an approximately 3.8 dB Eb/N0 gain over an identical throughput scheme dispensing with SP modulation at a BER of 10?5
Long-range-enhanced surface codes
The surface code is a quantum error-correcting code for one logical qubit,
protected by spatially localized parity checks in two dimensions. Due to
fundamental constraints from spatial locality, storing more logical qubits
requires either sacrificing the robustness of the surface code against errors
or increasing the number of physical qubits. We bound the minimal number of
spatially non-local parity checks necessary to add logical qubits to a surface
code while maintaining, or improving, robustness to errors. We asymptotically
saturate this bound using a family of hypergraph product codes, interpolating
between the surface code and constant-rate low-density parity-check codes.
Fault-tolerant protocols for logical operations generalize naturally to these
longer-range codes, based on those from ordinary surface codes. We provide
near-term practical implementations of this code for hardware based on trapped
ions or neutral atoms in mobile optical tweezers. Long-range-enhanced surface
codes outperform conventional surface codes using hundreds of physical qubits,
and represent a practical strategy to enhance the robustness of logical qubits
to errors in near-term devices.Comment: 16 pages, 12 figures; v2 changes: fixed typos and added citation
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
Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays
Quantum low-density parity-check (qLDPC) codes can achieve high encoding
rates and good code distance scaling, providing a promising route to
low-overhead fault-tolerant quantum computing. However, the long-range
connectivity required to implement such codes makes their physical realization
challenging. Here, we propose a hardware-efficient scheme to perform
fault-tolerant quantum computation with high-rate qLDPC codes on reconfigurable
atom arrays, directly compatible with recently demonstrated experimental
capabilities. Our approach utilizes the product structure inherent in many
qLDPC codes to implement the non-local syndrome extraction circuit via atom
rearrangement, resulting in effectively constant overhead in practically
relevant regimes. We prove the fault tolerance of these protocols, perform
circuit-level simulations of memory and logical operations with these codes,
and find that our qLDPC-based architecture starts to outperform the surface
code with as few as several hundred physical qubits at a realistic physical
error rate of . We further find that less than 3000 physical qubits
are sufficient to obtain over an order of magnitude qubit savings compared to
the surface code, and quantum algorithms involving thousands of logical qubits
can be performed using less than physical qubits. Our work paves the way
for explorations of low-overhead quantum computing with qLDPC codes at a
practical scale, based on current experimental technologies
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