Abstract We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance Ω(N2/3/loglogN) which can improve Hastings’s recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman- Zémor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quan- tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√N)]] and they also belong to the Bacon-Shor codes. We show that Q can be encoded very efficiently by circuits of size O(N) and depth O(√N), and can correct any adversarial error of weight up to half the minimum distance bound in O(√N) time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.
Abstract We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance Ω(N2/3/loglogN) which can improve Hastings’s recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman- Zémor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quan- tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√N)]] and they also belong to the Bacon-Shor codes. We show that Q can be encoded very efficiently by circuits of size O(N) and depth O(√N), and can correct any adversarial error of weight up to half the minimum distance bound in O(√N) time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.
In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes Lk and Lk+, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters [[6k2,k2,d]]. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies d=3 for k=3 and d=4 for k≥4, establishing a new framework for structured quantum LDPC code design and optimization.
In classical coding theory code concatenation is successfully used to construct good errorcorrecting codes and most of the asymptotically good codes known so far are based on concatenation. In this paper we present some classes of asymptotically good concatenated quantum codes, which are a quantum analogue of classical concatenated codes, and derive lower bounds on the minimum distance and the rate of the codes. Our bounds improve on the best lower bound of Ashikhmin–Litsyn–Tsfasman and Matsumoto for rates smaller than about one half. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.
In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes L_{k} and L_{k}^{+} , defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters [[6k^{2} , k^{2} , d]]. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies d = 3 for k = 3 and d = 4 for k ≥ 4, establishing a new framework for structured quantum LDPC code design and optimization.
In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parametersQ1=[[N,Ω(N),Ω(N)]], andQ2=[[N,Ω(N/logN),Ω(N/logN)/Ω(logN)]]. We show thatQ1can be encoded very efficiently by circuits of sizeO(N)and depthO(N). For an input error syndrome,Q1can correct any adversarial error of weight up to half the minimum distance bound inO(N)time.Q1can also be decoded in parallel inO(N)time by usingO(N)classical processors. For an input error syndrome, we proved thatQ2can correct a linear number ofX-errors with high probability and an almost linear number ofZ-errors inO(N)time. Moreover,Q2can be decoded in parallel inO(log(N))time by usingO(N)classical processors.
Quantum computation has shown advantages in several problems over the corresponding classical algorithms. The noisy intermediate-size\n quantum devices with dozens of qubits make it more promising in the\n past decade, as the quantum advantage was demonstrated\n experimentally. Before entering the era of scalable quantum\n computation, one has to resolve the errors in a quantum many-body\n system, which is inevitable due to the environment interaction during\n quantum control processes. The goal of quantum error correction is to\n reduce such errors and increase decoherence time. The most successful candidates of quantum error correction codes are topological codes, especially surface codes, which were discovered by\n Alexei Kitaev. The ordered phase of the 2D\n Ising model on a torus ensures that toric codes are fault-tolerant\n below a critical error probability. Other than the FT threshold,\n the topological codes feature efficient encoding and\n decoding, local measurements, but suffer from asymptotically zero\n code rate. To encode more logical qubits with finite resources, one can loosen the condition on locality and extend to a broader class of quantum\n low-density parity-check (LDPC)codes. There are many known algebraic\n constructions for such codes, but only a precious few of them have\n finite code rates and meet the fault-tolerant condition:\n the stabilizer weight is bounded and the distance scales at least\n logarithmically with the code size. In this thesis, I construct the\n higher-dimensional quantum hypergraph product (HQHP) codes, which\n generalize quantum hypergraph product (QHP) codes and toric codes in\n all dimensions. The HQHP codes projected into a single space gives\n subsystem product codes, which can then be gauge fixed to\n concatenated codes or homological product codes. Those include some\n common CSS codes, like Shor’s codes, Bacon Shor’s codes, and\n subsystem QHP codes. The HQHP codes can be mapped to the tensor\n product of chain complexes, which provides an algebraic framework to\n construct quantum LDPC codes with finite code rates, square root\n distances, FT thresholds and single-shot properties with redundant\n checks. Meanwhile, their rich connection to other codes are very\n instructive and may lead to further optimizations. Regarding the\n remained procedures towards fault-tolerant universal quantum\n computation through quantum LDPC codes, I will discuss the\n fault-tolerant condition for each code, including distance,\n stabilizer weight, decoding, fault-tolerant gates and measurement\n protocols.\n
Constructing quantum 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 generalises a construction of Hastings and also the Tillich-Zemor 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 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance n1/2logn, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n1/2log1/2n. 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.