Quantum Low-density Parity-check Codes Research Articles

Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes

We introduce and analyse an efficient decoder for quantum Tanner codes that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight \(O(\sqrt{n\log n})\) . We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.

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.

Decoding Quantum Tanner Codes

We introduce sequential and parallel decoders for quantum Tanner codes. When the Tanner code construction is applied to a sufficiently expanding square complex with robust local codes, we obtain a family of asymptotically good quantum low-density parity-check codes. In this case, our decoders provably correct arbitrary errors of weight linear in the code length, respectively in linear or logarithmic time. The same decoders are easily adapted to the expander lifted product codes of Panteleev and Kalachev. Along the way, we exploit recently established bounds on the robustness of random tensor codes to give a tighter bound on the minimum distance of quantum Tanner codes.

Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive.

Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi etal. [Phys. Rev. Lett. 104, 050503 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.050503] have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension k and distance d. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters k and d. In particular, in 2D we show that a quantum LDPC code with distance d∝n^{1/2+ϵ} requires Ω(n^{1/2+ϵ}) interactions of length Ω[over ˜](n^{ϵ}). Further, a code satisfying k∝n with distance d∝n^{α} requires Ω[over ˜](n) interactions of length Ω[over ˜](n^{α/2}). As an application of these results, we consider a model called a stacked architecture, which has previously been considered as a potential way to implement quantum LDPC codes. In this model, although most interactions are local, a few of them are allowed to be very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.

Constant-Overhead Quantum Error Correction with Thin Planar Connectivity.

Quantum low density parity check (LDPC) codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any Calderbank-Shor-Steane code with a degree-δ Tanner graph, we design stabilizer measurement circuits with depth at most (2δ+2) using at most ⌈δ/2⌉ layers. We observe a circuit-noise threshold of 0.28% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of 10^{-4}, this family reaches a logical error rate of 10^{-15} using fourteen times fewer physical qubits than the surface code.

Decodable Quantum LDPC Codes beyond the $\sqrt{n}$ Distance Barrier Using High-Dimensional Expanders

Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich--Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.

Connectivity constrains quantum codes

Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.

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.

Low-loss belief propagation decoder with Tanner graph in quantum error-correction codes

Quantum error-correction codes are immeasurable resources for quantum computing and quantum communication. However, the existing decoders are generally incapable of checking node duplication of belief propagation (BP) on quantum low-density parity check (QLDPC) codes. Based on the probability theory in the machine learning, mathematical statistics and topological structure, a GF(4) (the Galois field is abbreviated as GF) augmented model BP decoder with Tanner graph is designed. The problem of repeated check nodes can be solved by this decoder. In simulation, when the random perturbation strength p=0.0115–0.0116 and number of attempts N = 60–70, the highest decoding efficiency of the augmented model BP decoder is obtained, and the low-loss frame error rate (FER) decreases to 7.1975 ×10−5. Hence, we design a novel augmented model decoder to compare the relationship between GF(2) and GF(4) for quantum code [[450,200]] on the depolarization channel. It can be verified that the proposed decoder provides the widely application range, and the decoding performance is better in QLDPC codes.

Design of low-density-generator-matrix–based quantum codes for asymmetric quantum channels

Quantum low-density-generator-matrix (QLDGM) codes are known to exhibit great error correction capabilities, surpassing existing quantum low-density-parity-check (QLDPC) codes and other sparse-graph schemes over the depolarizing channel. Most of the research on QLDPC codes and quantum error correction (QEC) is conducted for the symmetric instance of the generic Pauli channel, which incurs bit flips, phase flips, or a combination of both with the same probability. However, due to the behavior of the materials they are built from, some quantum devices must be modelled using a different channel model capable of accurately representing asymmetric scenarios in which the likelihood of a phase flip is higher than that of a bit flip. In this work, we study the design of QLDGM CSS codes for such Pauli channels. We show how codes tailored to the depolarizing channel are not well suited to these asymmetric environments and we derive methods to aptly design QLDGM CSS codes for this paradigm.

A Rate Scalable Construction of Non-Homogeneous Quantum LDPC Codes of CSS Type Based on Balanced Incomplete Block Design

Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be 1-2/m approximately, where m, the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are 1-2/m', where m'=m/e and e ∈ {1, 2}. Note that here m' can be odd. Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.

Cellular automaton decoders for topological quantum codes with noisy measurements and beyond

We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is also essentially insensitive to the details of the lattice and noise model. Our work constitutes a step towards finding simple and high-performance decoding strategies for a wide range of quantum low-density parity-check codes.

A Performance Analysis of Quantum Low-Density Parity-Check Codes for Correcting Correlated Errors

In this paper we analyse the performance of quantum low-density parity-check (LDPC) codes over quantum memory channels for correcting correlated errors, where the quantum LDPC codes we use are the hypergraph product quantum LDPC codes. We generalize the classical Gilbert-Elliot markovian memory channel to the quantum Gilbert-Elliot channel and use it to model the memory effect in quantum memory channels. The simulation results show the memory effects in quantum GE channels has a bad effect to the performance of hypergraph product quantum LDPC codes. Then we propose a concatenation scheme by serially concatenating two hypergraph product quantum LDPC codes and it is shown that the serial concatenation scheme can improve the performance of hypergraph product quantum LDPC codes and lower down the error floor over quantum GE channels.

Modified belief propagation decoders for quantum low-density parity-check codes

Quantum low-density parity-check codes can be decoded using a syndrome based $\mathrm{GF}(4)$ belief propagation decoder. However, the performance of this decoder is limited both by unavoidable $4$-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of $4$-cycles can be reduced by breaking an error into an $X$ and $Z$ component and decoding each with an individual $\mathrm{GF}(2)$ based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of modified belief propagation decoders that address these issues. We propose a $\mathrm{GF}(2)$ based decoder for CSS codes that reintroduces error correlations by reattempting decoding with adjusted error probabilities. We also propose the use of an augmented decoder, which has previously been suggested for classical binary low-density parity-check codes. This decoder iteratively reattempts decoding on factor graphs that have a subset of their check nodes duplicated. The augmented decoder can be based on a $\mathrm{GF}(4)$ decoder for any code, a $\mathrm{GF}(2)$ decoder for CSS code, or even a supernode decoder for a dual-containing CSS code. For CSS codes, we further propose a $\mathrm{GF}(2)$ based decoder that combines the augmented decoder with error probability adjustment. We demonstrate the performance of these new decoders on a range of different codes, showing that they perform favorably compared to other decoders presented in literature.

Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes.

Belief-propagation (BP) decoders play a vital role in modern coding theory, but they are not suitable to decode quantum error-correcting codes because of a unique quantum feature called error degeneracy. Inspired by an exact mapping between BP and deep neural networks, we train neural BP decoders for quantum low-density parity-check codes with a loss function tailored to error degeneracy. Training substantially improves the performance of BP decoders for all families of codes we tested and may solve the degeneracy problem which plagues the decoding of quantum low-density parity-check codes.

Design of Quantum LDPC Codes From Quadratic Residue Sets

We design classes of quantum low-density parity-check (LDPC) codes, called quasi-cyclic stabilizer (QCS) codes, from conventional QC-LDPC codes. The proposed QCS codes belong to the family of non-Calderbank–Shor–Steane stabilizer codes. The QC-LDPC codes are self-orthogonal with respect to the symplectic inner product (SIP) and are constructed from submatrices of nonorthogonal Latin squares via array dispersion. The Latin squares are constructed using quadratic (non)-residue sets of prime modulus $p$ , where $p = 4n\pm 1$ . For $p = 4n-1$ , two constructions, namely, Type-I-A and Type-I-B QCS codes, are proposed based on matrix superposition and matrix concatenation, respectively. For $p = 4n+1$ , Type-II QCS codes are proposed based on permutations of a base matrix. We show that the parity-check matrix for Type-I-B and Type-II QCS codes is self-orthogonal with respect to the SIP for all orders of circulant permutation matrix. This resulting in ensembles of QCS codes characterized by a single base matrix. We show that the minimum distance of Type-II QCS codes can be lower bounded by the minimum distance of the QC-LDPC codes. Simulation results show that the proposed QCS codes outperform some codes in the literature with a noteworthy low-error floor, below 10−7, over quantum depolarizing channels.

Distance Verification for Classical and Quantum LDPC Codes

The techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes. Then, these techniques are considered for classical and quantum (stabilizer) low-density-parity-check (LDPC) codes. New complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of LDPC codes. These bounds are expressed in terms of the erasure-correcting capacity of the corresponding ensemble. We also present a new irreducible-cluster technique that can be applied to any LDPC code and takes advantage of parity-checks’ sparsity for both the classical and quantum LDPC codes. This technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances, which also include all known families of the quantum stabilizer LDPC codes.

Construction of Quantum LDPC Codes From Classical Row-Circulant QC-LDPCs

Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank–Shor–Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.