To be able to perform useful computations employing quantum computers we need to make quantum information more robust against noise. One promising approach is to protect the information using quantum LDPC codes, however they are useless without a powerful decoder. In this thesis we will study the decoding of LDPC quantum codes, and in particular of hypergraph product codes with several decoders. We first study a decoder solely focused on performance obtained from the combination of two already existing decoders, BP and SSF. We show through simulations that it has improved performance on random hypergraph product codes compared to each of the decoders alone. Later we concentrate on a decoder easier to analyse which exhibits interesting theoretical guarantees by continuing the work started by Li and Vontobel of importing the LP decoder to quantum error correction. We manage to prove negative results affecting the asymptotic performance on several LDPC quantum codes, while showing that it still performs well on hypergraph product codes of realistic size. Finally we focus on local decoders for the decoding of quantum LDPC code. In particular we numerically study a parallel variant of SSF in its fastest and most local implementation.
To be able to perform useful computations employing quantum computers we need to make quantum information more robust against noise. One promising approach is to protect the information using quantum LDPC codes, however they are useless without a powerful decoder. In this thesis we will study the decoding of LDPC quantum codes, and in particular of hypergraph product codes with several decoders. We first study a decoder solely focused on performance obtained from the combination of two already existing decoders, BP and SSF. We show through simulations that it has improved performance on random hypergraph product codes compared to each of the decoders alone. Later we concentrate on a decoder easier to analyse which exhibits interesting theoretical guarantees by continuing the work started by Li and Vontobel of importing the LP decoder to quantum error correction. We manage to prove negative results affecting the asymptotic performance on several LDPC quantum codes, while showing that it still performs well on hypergraph product codes of realistic size. Finally we focus on local decoders for the decoding of quantum LDPC code. In particular we numerically study a parallel variant of SSF in its fastest and most local implementation.
We introduce a decoder for quantum CSS codes that is based on linear programming. Our definition is a priori slightly different from the one proposed by Li and Vontobel as we have a syndrome oriented approach instead of an error oriented one, but we show that the success condition is equivalent. Although we prove that this decoder fails for quantum codes that do not have good soundness property (i.e., having large errors with syndrome of small weight) such as the toric code, we obtain good results from simulations. We run our decoder for hypergraph products of two random LDPC codes, showing that it performs better than belief propagation, even combined with the small-set-flip decoder that can provably correct a constant fraction of random errors.
Décodage de codes quantiques LDPC Afin de pouvoir réaliser des calculs utiles grâce à des ordinateurs quantiques nous devons rendre l’information quantique plus robuste contre le bruit. Une approche prometteuse repose sur l’utilisation de code quantiques LDPC, qui nécessitent d’être associés à un décodeur. Dans cette thèse nous étudions le décodage de codes quantiques LDPC, et en particulier de codes produits d’hypergraphes, par diverse décodeurs. Nous commençons par étudier un décodeur focalisé sur les performances et obtenus en combinant deux décodeurs existants, BP et SSF. Nous montrons par des simulations que ses performances sont meilleurs que celles de chacun des deux décodeurs seuls sur des codes produits d’hypergraphes aléatoires. Nous nous concentrons ensuite sur un décodeur plus simple à analyser et ayant des garanties théoriques intéressantes, en continuant l’importation du décodeur LP qu cas quantiques, travail entamé par Li et Vontobel. Nous réussissons à prouver des résultats négatifs limitant les performances asymptotiques du décodeur sur plusieurs codes quantiques LDPC, tout en obtenant de bons résultats sur des codes produits d’hypergraphes de tailles réalistes. Enfin nous étudions le décodage local de codes quantiques LDPC, et plus particulièrement nous analysons par des simulations les performances de l’implémentation la plus rapide et la plus locale de SSF.
Hypergraph product codes are a class of constant-rate quantum low-density parity-check (LDPC) codes equipped with a linear-time decoder called small-set-flip (SSF). This decoder displays sub-optimal performance in practice and requires very large error correcting codes to be effective. In this work, we present new hybrid decoders that combine the belief propagation (BP) algorithm with the SSF decoder. We present the results of numerical simulations when codes are subject to independent bit-flip and phase-flip errors. We provide evidence that the threshold of these codes is roughly 7.5% assuming an ideal syndrome extraction, and remains close to 3% in the presence of syndrome noise. This result subsumes and significantly improves upon an earlier work by Grospellier and Krishna (arXiv:1810.03681). The low-complexity high-performance of these heuristic decoders suggests that decoding should not be a substantial difficulty when moving from zero-rate surface codes to constant-rate LDPC codes and gives a further hint that such codes are well-worth investigating in the context of building large universal quantum computers.
Finding good quantum low density parity check (LDPC) codes is an essential step for the design of efficient quantum fault-tolerant protocols. The hypergraph product construction [1] is a combinatorial way to build an LDPC CSS code with parameters [[n, Θ(n), Θ(√n)]] from a well chosen classical code. Moreover, the hypergraph product obtained from a classical expander code [2] can be used to perform fault-tolerant quantum computation with constant overhead in space and has the single-shot error correction property [3, 4]. Although these codes exhibit good features asymptotically, it is unknown whether they are suitable for small block lengths [5, 6, 7]. In this work, we take steps towards addressing this problem. Our motivations stem from classical coding theory. Sipser and Spielman discovered a linear time decoder for classical expander codes [2]. This inspired the quantum decoding algorithm small-set-flip introduced in [8]. The bit-flip algorithm, however, is never used in practice because it is outperformed by message passing algorithms such as belief propagation. Naively generalizing belief propagation to the quantum LDPC codes is ineffective due to degeneracy [9, 10, 11]. In particular, the belief propagation decoder does not perform well on hypergraph product codes [7]. In this work, we show that running the BP algorithm followed by the small-set-flip decoder significantly improves the decoding threshold and the finite-length behavior. In [5] it has been shown that an exponential time decoder achieves a threshold near 7% and [6] shows that the small-set-flip algorithm has a threshold near 4.5%. This work shows that combining the belief propagation and the small-set-flip decoder, leads to a threshold above 7% for hypergraph product codes with rate 4% constructed from (3, 4)-regular codes. In addition, running the small-set-flip decoder alone requires quantum codes with generators of weight at least 11 [6] whereas combining the two decoders allows us to use codes with generators of weight 7. Reducing the generators weight improves the running time of the decoder and is required to perform the corresponding quantum measurements on real devices.
Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction of qLDPC codes based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev and Kalachev, 2020) achieved a near-linear distance, and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects a near-linear number of adversarial errors. En route, we give a similar algorithm for the hypergraph product version these codes, which are simpler but have distance growing only as the square root of the block length. Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.
Qudits offer significant advantages over qubit-based architectures, including more efficient gate compilation, reduced resource requirements, improved error-correction primitives, and enhanced capabilities for quantum communication and cryptography. Yet, one of the most promising families of quantum error correction codes, namely quantum low-density parity-check (LDPC) codes, have so far been mostly restricted to qubits. Here, we generalize recent advancements in LDPC codes from qubits to qudits. We introduce a general framework for finding qudit LDPC codes and apply our formalism to several promising types of LDPC codes. We generalize bivariate bicycle codes, including their coprime variant; hypergraph product codes, including the recently proposed La-cross codes; subsystem hypergraph product (SHYPS) codes; high-dimensional expander codes, which make use of Ramanujan complexes; and fiber bundle codes. Using the qudit generalization formalism, we then numerically search for and decode several novel qudit codes compatible with near-term hardware. Our results highlight the potential of qudit LDPC codes as a versatile and hardware-compatible pathway toward scalable quantum error correction.