Maximum Likelihood Decoding Research Articles

Hardness results for decoding the surface code with Pauli noise

Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit.In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent X and Z noise.

Decoding error probability of random parity-check matrix ensemble over the erasure channel

AbstractIn this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

Fast erasure decoder for hypergraph product codes

We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in O(N2) bit operations where N is the length of the quantum code. A probabilistic version of this decoder can be implemented in O(N1.5) bit operations.

Classical product code constructions for quantum Calderbank-Shor-Steane codes

Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3-fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

Application of STBC in MIMO-OFDM Broadband Wireless Communications

The Space Time Block Coding (STBC) is applied in multiple transmit antennas using data transfer for fading channels. After being encoded, the data is broken into n streams of continuously broadcast strings over n transmit antennas using STBC. The received signal is the combination of n broadcast signals and it becomes erroneous due to the entry of noise at receiver end. The probabilistic decoding approach is applied instead of joint detection approach for recovering data via decoupling of signals delivered from various antennas. The maximum likelihood decoding scheme employs the orthogonal structure of STBC called as OSTBC (Orthogonal Space Time Block Code) for providing the maximum likelihood decoding algorithm depending upon linear computation at receiver. In the present work, a MATLAB/SIMULINK environment of OSTBC has been used in order to acquire the highest dimension of variability for specific number of broadcasts as well as receive antennas utilizing a simple decoding techniques. With or without Grey Coding, OSTBC / STBC is used in MATLAB / SIMULINK blocs. The OSTBC algorithm calculates the highest achievable transmission rate for every amount of transmit antennas in any constellations including M-PSK array. M-PSK STBCs are used for different complex constellations to achieve ½ as well as ¾ of highest allowable transmission rate for Multiple Input Multiple Output (MIMO) transmit antennas utilizing various constellations

Rate 5/6 TCM Code Having 64 States with 64 QAM for Fading Channel

Trellis Coded Modulation (TCM) is powerful technique employed in digital communication systems to improve the reliability and efficiency of data transmission. TCM combines error control coding and modulation schemes to achieve superior performance in challenging channel conditions. In TCM, information bits are encoded using a trellis encoder, which generates a sequence of encoded symbols. These symbols are then mapped onto a modulation scheme, such as Quadrature Amplitude Modulation (QAM) or Phase Shift Keying (PSK), to create the modulated signal. At the receiver, the received signal is demodulated and decoded using a trellis decoder, which employs maximum likelihood decoding to recover the original information bits. The trellis structure allows for efficient error correction and makes TCM particularly suitable for channels with fading, noise, and other impairments. TCM has been widely used in various communication standards and applications to enhance data transmission reliability and spectral efficiency. TCM is a coding technique that merges coding and modulation to attain substantial coding gain without compromising bandwidth efficiency. The emergence of TCM occurred in the late 1970s, and ever since, it has found extensive utilization in various contemporary information transmission systems. In this paper, a new methodology is introduced for the design of a 64-state rate 5/6 TCM code optimized for fading channels. The outcomes obtained from the study are remarkably encouraging, demonstrating coding gain of around 10 dB in comparison to uncoded 32 PSK. This substantial gain plays a pivotal role in boosting data rates, especially in situations where bandwidth-limited channels are involved.

Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder

Recent work [M. J. Gullans , ] has shown that quantum error correcting codes defined by random Clifford encoding circuits can achieve a nonzero encoding rate in correcting errors even if the random circuits on n qubits, embedded in one spatial dimension (1D), have a logarithmic depth d=O(logn). However, this was demonstrated only for a simple erasure noise model. In this work, we discover that, for the same class of codes, this desired property indeed holds for the conventional Pauli noise model. Specifically, we numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with d=O(n)-depth random encoding circuits, can be attained for the above codes even when the circuit depth is restricted to d=O(logn) in 1D for depolarizing noise of various strengths. This analysis is made possible with our development of a tensor-network maximum-likelihood decoding algorithm that works efficiently for log-depth encoding circuits in 1D. Published by the American Physical Society 2024

Probabilistic Searching for MIMO Detection Based on Lattice Gaussian Distribution

In this paper, a deterministic sampling decoding strategy for multiple-input multiple output (MIMO) systems is studied, which performs probabilistic searching according to a probability threshold in the lattice Gaussian distribution. Motivated by model probabilistic twin (MPT), the randomness in obtaining the target decoding solution is overcome by the proposed probabilistic searching decoding (PSD) algorithm, which brings considerable decoding gains in both performance and complexity. Specifically, the decoding radius of PSD is derived while the decoding complexity in terms of the number of visited nodes during the searching is also upper bounded, leading to an explicit decoding trade-off. Meanwhile, we generalize PSD by the mechanism of candidate protection so that it enjoys a flexible performance between the suboptimal successive interference cancelation (SIC) decoding and the optimal maximum likelihood (ML) decoding by adjusting the initial search size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> . Methods for further optimization and complexity reduction of the proposed PSD algorithm are also given. Finally, simulation results based on MIMO detection are presented to confirm the tractable and flexible decoding trade-off of the proposed PSD algorithm.

Joint Partial Response Equalization with Mapping-Rule-Based Simplified Maximum Likelihood Decoder for intra-DCIs

Joint Partial Response Equalization with Mapping-Rule-Based Simplified Maximum Likelihood Decoder for intra-DCIs

Neural layered min-sum decoders for cyclic codes

This paper proposes a low-complexity neural network decoder based on the layered min-sum algorithm to decode cyclic codes. By generalizing the layered min-sum algorithm to its neural network counterpart, the number of network weights decreases while retaining a good error correction performance. The Bose–Chaudhuri–Hocquenghem (BCH) codes, quadratic residue (QR) codes, and punctured Reed–Muller (RM) codes are selected as three exemplary binary cyclic codes. Simulation results show that the proposed neural decoder achieves superior performance with less computational complexity compared with the state-of-the-art neural network decoder. Further, a neural decoder incorporating the modified random redundant decoding (mRRD) algorithm is investigated to approach the performance of maximum-likelihood decoding for some short codes.

LoRa Backscatter Communications: Temporal, Spectral, and Error Performance Analysis

LB communication systems can be considered as a potential candidate for ultra LPWAN because of their low cost and low power consumption. In this paper, we comprehensively analyze LB modulation from various aspects, i.e., temporal, spectral, and error performance characteristics. First, we propose a signal model for LB signals that accounts for the limited number of loads in the tag. Then, we investigate the spectral properties of LB signals, obtaining a closed-form expression for the power spectrum. Finally, we derived the SER of LB with two decoders, i.e., the ML and FFT decoders, in both AWGN and double Nakagami-m fading channels. The spectral analysis shows that out-of-band emissions for LB satisfy the ETSI regulation only when considering a relatively large number of loads. For the error performance, unlike conventional LoRa, the FFT decoder is not optimal. Nevertheless, the ML decoder can achieve a performance similar to conventional LoRa with a moderate number of loads.

Analysis of Rateless Multiple Access Scheme With Maximum Likelihood Decoding in an AWGN Channel

The rateless multiple access (RMA) scheme is a promising distributed multiple access scheme to achieve simultaneous high reliability, low latency and massive connectivity. In this paper, we investigate the maximum likelihood (ML) decoding performance of the RMA scheme in an Additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) modulation. For the first time, this paper derives the ensemble weight distribution of the RMA scheme. We derive an upper bound on the decoding error performance of the RMA scheme under ML decoding in an AWGN channel with BPSK modulation. Using the derived bound as the fitness function, we adopt the continuous genetic algorithm to optimize the parameters of the RMA scheme. Simulation results show the tightness of the derived bound and the superiority of the optimized degree distribution over the conventional degree distributions.

Nonbinary LDPC Coded QAM Signals With Optimized Mapping: Bounds and Simulation Results

This paper studies specific properties of nonbinary low-density parity-check (NB LDPC) codes when used in coded modulation systems. The paper is focused on the practically important NB LDPC codes over the Galois extension fields GF(2m) with m ≤ 6 used with QAM signaling. Performance of NB LDPC coded transmission strongly depends on the mapping of nonbinary symbols to signal constellation points. We obtain a random coding bound on the maximum-likelihood decoding error probability for an ensemble of random irregular NB LDPC codes used with QAM signaling for specific symbol-to-signal point mappings. This bound is based on the ensemble average squared Euclidean distance spectra derived for these mappings. The simulation results for the belief-propagation decoding in the coded modulation schemes with the NB quasi-cyclic (QC)-LDPC codes under different mappings are given. Comparisons with the optimized binary QC-LDPC codes in the WiFi and 5G standards, as well as with the new bound, are performed.

Low-Rate and Short-Block-Length Random Permutation-Coded Modulations Achieve Finite-Length Bounds

Ultra-reliable low-latency communication (URLLC) is an important feature brought by 5G New Radio (NR). By using a block code, the low-latency requirement in general requires a short block length in order to reduce the latency of transmitting the entire code block. The ultra-reliable communication requires a low-rate code to achieve high decoding reliability even under low signal-to-noise ratios (SNRs). This paper proposes a new class of coding and modulation schemes, termed permutation-coded modulations, for low-rate and short-block-length applications, such as URLLC. We show that permutation-coded modulations under maximum-likelihood (ML) decoding have remarkable performance levels that achieve the dispersion bounds with normal approximation (NA) for short block lengths. However, their encoding and ML-decoding complexities are prohibitive if the number of message bits transmitted per code block increases. To reduce the encoding and decoding complexities, we propose a trapezoidal permutation-coded modulation scheme which can be decoded efficiently by successive cancellation list (SCL) decoders. We also show that the trapezoidal permutation-coded modulations under SCL decoding can achieve the dispersion bounds with NA for short block lengths.

Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison

Concatenating bosonic error-correcting codes with qubit codes can substantially boost the error-correcting power of the original qubit codes. It is not clear how to concatenate optimally, given there are several bosonic codes and concatenation schemes to choose from, including the recently discovered GKP-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020)}] that allow protection of a logical bosonic mode from fluctuations of the mode's conjugate variables. We develop efficient maximum-likelihood decoders for and analyze the performance of three different concatenations of codes taken from the following set: qubit stabilizer codes, analog/Gaussian stabilizer codes, GKP codes, and GKP-stabilizer codes. We benchmark decoder performance against additive Gaussian white noise, corroborating our numerics with analytical calculations. We observe that the concatenation involving GKP-stabilizer codes outperforms the more conventional concatenation of a qubit stabilizer code with a GKP code in some cases. We also propose a GKP-stabilizer code that suppresses fluctuations in both conjugate variables without extra quadrature squeezing, and formulate qudit versions of GKP-stabilizer codes.

Decoding the projective transverse field Ising model

The competition between noncommuting projective measurements in discrete quantum circuits can give rise to entanglement transitions. It separates a regime where initially stored quantum information survives the time evolution from a regime where the measurements destroy the quantum information. Here we study one such system---the projective transverse field Ising model---with a focus on its capabilities as a quantum error correction code. The idea is to interpret one type of measurement as an error and the other type as a syndrome measurement. We demonstrate that there is a finite threshold below which quantum information encoded in an initially entangled state can be retrieved reliably. In particular, we implement the maximum likelihood decoder to demonstrate that the error correction threshold is distinct from the entanglement transition. This implies that there is a finite regime where quantum information is protected by the projective dynamics, but cannot be retrieved by using syndrome measurements.

Decoding of the extended Golay code by the simplified successive-cancellation list decoder adapted to multi-kernel polar codes

This paper describes an adaptation of a polar code decoding technique in favor of the extended Golay code. Based on the bridge provided by a permutation matrix between the code words of these two classes of codes, the Golay code can be decoded by any polar code technique. Contrary to the successive-cancellation list technique which is characterized by a serial estimation of the bits, we propose in this work an adaptation of the simplified successive-cancellation list technique to polar codes equivalent to the Golay code. The simulations have achieved the performance of a maximum likelihood decoding, with the low decoding complexity of polar codes, compared to one of the universal decoders of linear codes most known in the literature.

Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders

Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.

Design of Polar Codes and PAC Codes for SCL Decoding

The performance of a code under the maximum-likelihood (ML) decoder highly depends on the weight enumerating function (WEF). However, how to compute efficiently the WEF of a polar code or a polarization-adjusted convolutional (PAC) code is still an open problem. For the design of stand-alone polar codes, we consider enumerating the number of minimum-weight non-zero codewords of the polar code. The block error rate (BLER) under the ML decoder can be approximated as a function of the minimum weight of non-zero codewords and its multiplicity. On the other hand, for the design of PAC codes, we consider enumerating the WEF averaged over the ensemble of random PAC codes. The ML-BLER upper bound can be represented as a function of the WEF. The bit-channel selection algorithms for polar codes and PAC codes are proposed, which take both utilization of the polarization effect and the ML decoding performance as selection criteria. Simulation results show that the proposed stand-alone polar codes are competitive when the block length gets larger. Also, simulation results show that the proposed PAC codes yield excellent performance for a wide range of code rates and block lengths and outperform the 5G polar codes.

Neural decoders with permutation invariant structure

Neural decoders were introduced as a generalization of the classic Belief Propagation (BP) decoding algorithms. In this work, we propose several neural decoders with different permutation invariant structures for BCH codes and punctured RM codes. Firstly, we propose the cyclically equivariant neural decoder which makes use of the cyclically invariant structure of these two codes. Next, we propose an affine equivariant neural decoder utilizing the affine invariant structure of those two codes. Both these two decoders outperform previous neural decoders when decoding cyclic codes. The affine decoder achieves a smaller decoding error probability than the cyclic decoder, but it usually requires a longer running time. Similar to using the property of the affine invariant property of extended BCH codes and RM codes, we propose the list decoding version of the cyclic decoder that can significantly reduce the frame error rate(FER) for these two codes. For certain high-rate codes, the gap between the list decoder and the Maximum Likelihood decoder is less than 0.1 dB when measured by FER.