Quantum LDPC Research Articles

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.

Quantum LDPC Codes From Intersecting Subsets

Quantum LDPC Codes From Intersecting Subsets

New quantum LDPC codes based on projective geometry

With the progress in techniques for correcting errors in quantum computing, quantum low density parity check (QLDPC) codes have gained increasing significance within the field of quantum error correction. This paper focuses on different strategies for the construction of QLDPC codes, which are based on all points and lines as well as partial points and lines from projective geometry. Finally, a series of simulation analyses are presented.

Check-Agnosia based Post-Processor for Message-Passing Decoding of Quantum LDPC Codes

The inherent degeneracy of quantum low-density parity-check codes poses a challenge to their decoding, as it significantly degrades the error-correction performance of classical message-passing decoders. To improve their performance, a post-processing algorithm is usually employed. To narrow the gap between algorithmic solutions and hardware limitations, we introduce a new post-processing algorithm with a hardware-friendly orientation, providing error correction performance competitive to the state-of-the-art techniques. The proposed post-processing, referred to as check-agnosia, is inspired by stabilizer-inactivation, while considerably reducing the required hardware resources, and providing enough flexibility to allow different message-passing schedules and hardware architectures. We carry out a detailed analysis for a set of Pareto architectures with different tradeoffs between latency and power consumption, derived from the results of implemented designs on an FPGA board. We show that latency values close to one microsecond can be obtained on the FPGA board, and provide evidence that much lower latency values can be obtained for ASIC implementations. In the process, we also demonstrate the practical implications of the recently introduced t-covering layers and random-order layered scheduling.

New quantum LDPC codes based on Euclidean Geometry

With the advancements in quantum error correction technique, quantum low density parity check (QLDPC) codes have emerged as a promising area in the field of quantum error correction. This paper focuses on the requirement of QLDPC codes based on points excluding the origin, and lines that do not pass through the origin of Euclidean Geometry. It also explores QLDPC codes based on all the points and lines. Finally, a series of simulation analyses are presented.

Entanglement Purification with Quantum LDPC Codes and Iterative Decoding

Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond the application of DQC since entanglement distribution and purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder with a sequential schedule for distilling 3-qubit GHZ states using a rate 0.118 family of lifted product QLDPC codes and obtain an input fidelity threshold of ≈0.7974 under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of 0.118 for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of 3-qubit GHZ states to construct a scalable GHZ purification protocol.

Soft syndrome iterative decoding of quantum LDPC codes and hardware architectures

In practical quantum error correction implementations, the measurement of syndrome information is an unreliable step—typically modeled as a binary measurement outcome flipped with some probability. However, the measured syndrome is in fact a discretized value of the continuous voltage or current values obtained in the physical implementation of the syndrome extraction. In this paper, we use this “soft” or analog information to benefit iterative decoders for decoding quantum low-density parity-check (QLDPC) codes. Syndrome-based iterative belief propagation decoders are modified to utilize the soft syndrome to correct both data and syndrome errors simultaneously. We demonstrate the advantages of the proposed scheme not only in terms of comparison of thresholds and logical error rates for quasi-cyclic lifted-product QLDPC code families but also with faster convergence of iterative decoders. Additionally, we derive hardware (FPGA) architectures of these soft syndrome decoders and obtain similar performance in terms of error correction to the ideal models even with reduced precision in the soft information. The total latency of the hardware architectures is about 600 ns (for the QLDPC codes considered) in a 20 nm CMOS process FPGA device, and the area overhead is almost constant—less than 50% compared to min-sum decoders with noisy syndromes.

Quantum LDPC Codes Based on Cocyclic Block Matrices.

Motivated by a family of binary cocyclic block matrices over GF(2), we proposed a construction method to gain the stabilizer of long-length quantum error-correction codes (QECCs). Stabilizer quantum codes (SQCs) can be obtained by the different rows of the yielded circulant permutation matrices; hence, the quantum codes have the virtue of a fast construction algorithm. The recursive relation of a block matrix is employed in the proposed approach, so that the generator matrix of quantum cocyclic codes with long length can be constructed easily. Furthermore, the obtained quantum codes have the low-density advantage of there being no 4-cycles in the Tanner graph.

Quantum Low-Density Parity-Check Codes for Modular Architectures

In efforts to scale the size of quantum computers, modularity plays a central role across most quantum computing technologies. In the light of fault tolerance, this necessitates designing quantum error-correcting codes that are compatible with the connectivity arising from the architectural layouts. In this paper, we aim to bridge this gap by giving a novel way to view and construct quantum LDPC codes tailored for modular architectures. We demonstrate that if the intra- and inter-modular qubit connectivity can be viewed as corresponding to some classical or quantum LDPC codes, then their hypergraph product code fully respects the architectural connectivity constraints. Finally, we show that relaxed connectivity constraints that allow twists of connections between modules pave a way to construct codes with better parameters.

Семейство асимптотически хороших квантовых и локально тестируемых классических LDPC-кодов

In this paper we study classical and quantum LDPC codes with constant rate obtained using tensor product operation over noncommutative group algebra for chain complexes. It is shown that these families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. The existence of asymptotically good families of locally testable codes with constant parameters of locality and soundness is also shown, which gives the solution of the known conjecture in the field of classical locally testable codes.

Quantum XYZ Product Codes

We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N2/3). While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.

Quantum LDPC Codes With Almost Linear Minimum Distance

We give a construction of quantum LDPC codes of dimension $\Theta(\log N)$ and distance $\Theta(N/\log N)$ as the code length $N\to\infty$. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance $\Omega(N^{1-\alpha/2}/\log N)$ and dimension $\Omega(N^\alpha \log N)$, where $0 \le \alpha < 1$. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed $R < 1$ there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least $R$ with, in some sense, optimal circulant size $\Omega(N/\log N)$ as the code length $N\to\infty$.

Hybrid CV-DV Quantum Communications and Quantum Networks

Quantum information processing (QIP) opens new opportunities for high-performance computing, high-precision sensing, and secure communications. Among various QIP features, the entanglement is a unique one. To take full advantage of quantum resources, it will be necessary to interface quantum systems based on different encodings of information both discrete and continuous. The goal of this paper is to lay the groundwork for the development of a robust and efficient hybrid continuous variable-discrete variable (CV-DV) quantum network, enabling the distribution of a large number of entangled states over hybrid DV-CV multi-hop nodes in an arbitrary topology. The proposed hybrid quantum communication network (QCN) can serve as the backbone for a future quantum Internet, thus providing extensive long-term impacts on the economy and national security through QIP, distributed quantum computing, quantum networking, and distributed quantum sensing. By employing the photon addition and photon subtraction modules we describe how to generate the hybrid DV-CV entangled states and how to implement their teleportation and entanglement swapping through entangling measurements. We then describe how to extend the transmission distance between nodes in hybrid QCN by employing macroscopic light states, noiseless amplification, and reconfigurable quantum LDPC coding. We further describe how to enable quantum networking and distributed quantum computing by employing the deterministic cluster state concept introduced here. Finally, we describe how the proposed hybrid CV-DV states can be used in an entanglement-based hybrid QKD.

Degenerate Quantum LDPC Codes With Good Finite Length Performance

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

Trapping Sets of Quantum LDPC Codes

Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.

Syndrome-Based Min-Sum vs OSD-0 Decoders: FPGA Implementation and Analysis for Quantum LDPC Codes

Quantum processors need to improve their reliability to scale up the number of qubits and increase the number of algorithms that can execute. To reduce the logical error rate of the quantum systems, the use of error correction codes and decoders has been established as a low-cost and feasible approach, with good results from a theoretical perspective, for mid and long-term architectures. While most of the authors are focused on the algorithms to improve the correction capability of quantum computers, without taking into account a fundamental implementation aspect for their deployment in a real system, i.e. , their latency must be bounded to avoid the qubit decoherence, only a few propose hardware architectures and they just include time estimations of their decoding latency. However, a real implementation has not been shown yet. In this work, we analyze from the point of view of hardware implementation two algorithmic options based on quantum low-density parity-check (QLDPC) codes: a) belief propagation min-sum decoders combined with codes with good error-floor behavior and b) belief propagation min-sum decoders concatenated with ordered statistics decoders (OSDs) for codes with early error-floor. The bounds for the maximum clock frequency required by the decoders to decode within the qubit coherence time are established as a parameter to show if a practical implementation is possible with the present or near future FPGA technology. Furthermore, real implementation results for a Xilinx FPGA device are provided, showing that some solutions can meet the timing constraints set up by the state-of-the-art quantum processors.

The Need for Structure in Quantum LDPC Codes

The existence of quantum LDPC codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information. Despite years of research good quantum LDPC codes are not known to exist, but at the very least it is known they cannot be defined on very regular topologies, like low-dimensional grids. In this work we establish a complementary result, showing that good quantum CSS codes which are sparsely generated require “structure” in the local terms that constrain the code space so as not to be “too-random” in a well-defined sense. To show this, we prove a weak converse to a theorem of Krasikov and Litsyn on weight distributions of classical codes due to which may be of independent interest: subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight, tantamount to having a non-negligible “approximate” minimal distance. While they may not have a large minimal non-zero weight, they still have very few words of low Hamming weight.

Golden codes: quantum LDPC codes from regular tessellations of hyperbolic 4-manifolds

We adapt a construction of Guth and Lubotzky \cite{GL14} to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like n^{0.1} where n is the number of physical qubits. Similarly as in Ref.~\cite{GL14}, our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.

A theory of single-shot error correction for adversarial noise

Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code’s measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.

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%.