We describe a family of quantum error-correcting codes which generalize both the quantum hypergraph-product codes by Tillich and Zémor and all families of toric codes on m-dimensional hypercubic lattices. Parameters of the constructed codes, including the minimum distances, are given explicitly in terms of those of binary codes associated with the matrices used in the construction.
We describe a family of quantum error-correcting codes which generalize both the quantum hypergraph-product codes by Tillich and Zémor and all families of toric codes on m-dimensional hypercubic lattices. Parameters of the constructed codes, including the minimum distances, are given explicitly in terms of those of binary codes associated with the matrices used in the construction.
We suggest several techniques to improve the toric codes and the finite-rate generalized toric codes (quantum hypergraph-product codes) recently introduced by Tillich and Zémor. For the usual toric codes, we introduce the rotated lattices specified by two integer-valued periodicity vectors. These codes include the checkerboard codes, and the family of minimal single-qubit-encoding toric codes with block length n = t2+ (t+1)2and distance d = 2t + 1, t = 1, 2, ... We also suggest several related algebraic constructions which increase the rate of the existing hypergraph-product codes by up to four times.
The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.
We introduce an ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Z\'emor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both the lower and the upper bounds on the minimum distance; they scale as a square root of the block length. Many thus defined codes have a finite rate and limited-weight stabilizer generators, an analog of classical low-density parity-check (LDPC) codes. Compared to the hypergraph-product codes, hyperbicycle codes generally have a wider range of parameters; in particular, they can have a higher rate while preserving the estimated error threshold.
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 low-density parity-check (QLDPC) codes are an important class of quantum error-correcting codes that have low-weight stabilizer generators and typically offer encoding rates much higher than popular topological quantum codes such as surface and toric codes. While recent constructions of QLDPC codes have been aimed at two-level quantum systems, several hardware platforms for quantum computing, including superconducting and photonic systems, support a much larger Hilbert space. This makes the design of qudit-based quantum LDPC codes an issue of paramount significance in harnessing the increased flexibility afforded by high -dimensional quantum systems. In this paper, we forge the first steps in addressing this gap and explore the generalization of QLDPC constructions to qudit systems. We review two methods of generalizing binary quantum codes to qudit codes, namely, the stack and merge constructions, and investigate the application of these two meth-ods to the well-known hypergraph product and lifted-product codes. Subsequently, we provide upper and lower bounds for the encoding rates of the resulting qudit codes. We also prove an interesting relationship between the qudit hypergraph product and qudit lifted-product codes: the merge construction applied to the lifted product code is equivalent to the stack construction applied to the hypergraph product code under certain conditions. We evaluate the performance of these qudit codes under non-binary belief-propagation decoding and observe that qudit codes offer us a greater degree of flexibility in optimizing their rate-performance trade-off. We also demonstrate that, under certain conditions, qudit codes can simultaneously achieve higher rates and higher frame error rate (FER) performance than their binary counterparts.
Hypergraph product codes introduced by Tillich and Zémor are a class of quantum LDPC codes with constant rate and distance scaling with the square-root of the block size. Quantum expander codes, a subclass of these codes, can be decoded using the linear time small-set-flip algorithm of Leverrier, Tillich and Zémor. In this paper, we numerically estimate the performance for the hypergraph product codes under independent bit and phase flip noise. We focus on two families of hypergraph product codes. The first family has rate 1/61∼1.6%, has qubits of weight 10 or 12 and stabilizers of weight 11. We report a threshold near 4.6% for the small-set-flip decoder. We also show that for similar rate, the performance of the hypergraph product is better than the performance of the toric code as soon as we deal with more than 500 logical qubits and that for 3600 logical qubits, the logical error rate for the hypergraph product code is several orders of magnitude smaller. The second family has rate 0.2, qubits of weight 10 and 20 and stabilizers of weight 15. We report a threshold near 2% for the small-set-flip decoder.