We construct a family of quantum low-density parity-check codes locally equivalent to higher-dimensional quantum hypergraph-product (QHP) codes. Similarly to QHP codes, the proposed codes have highly redundant sets of low-weight stabilizer generators, which improves decoding accuracy in a fault-tolerant regime and gives them single-shot properties. The advantage of the construction is that it gives shorter codes. We derive simple expressions for the dimension of the proposed codes in two important special cases, give bounds on the distances and explicitly construct some relatively short codes. Circuit simulations for codes locally equivalent to four-dimensional toric codes show a (pseudo)threshold close to 1.1 % , better than for toric or surface codes with a similar noise model.
We construct a family of quantum low-density parity-check codes locally equivalent to higher-dimensional quantum hypergraph-product (QHP) codes. Similarly to QHP codes, the proposed codes have highly redundant sets of low-weight stabilizer generators, which improves decoding accuracy in a fault-tolerant regime and gives them single-shot properties. The advantage of the construction is that it gives shorter codes. We derive simple expressions for the dimension of the proposed codes in two important special cases, give bounds on the distances and explicitly construct some relatively short codes. Circuit simulations for codes locally equivalent to four-dimensional toric codes show a (pseudo)threshold close to 1.1 % , better than for toric or surface codes with a similar noise model.
Abelian multicycle (AMC) codes are compact quantum low-density parity-check codes whose multiblock chain-complex structure provides redundant low-weight stabilizers and supports single-shot error correction. We introduce Floquet AMC codes by deriving a quotient-lattice representation of a general level-j, D-dimensional AMC complex over a finite Abelian group algebra, lifting this lattice to spacetime, and rotating the circuit-time direction in the associated ZX network. When the check and data spiders have even valence and admit a time-oriented local port matching, the network decomposes into a periodic schedule of native two-qubit XX and ZZ measurements. We construct generalized-bicycle and level-2 AMC4 examples, determine their instantaneous stabilizer groups, and compute their embedded distances by minimizing over all inequivalent circuit cuts. For AMC4 instances locally equivalent to four-dimensional toric codes, we obtain Floquet memories with parameters [[108,6,5]], [[144,6,8]], and [[324,6,10]]. Local Pauli-web detector templates and beam-search decoding under the measurement-native EM3 noise model yield an estimated pseudothreshold of approximately 1.2%. These results provide compact measurement-only realizations of higher-dimensional homological redundancy without directly measuring the original weight-six stabilizers.
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
We present a family of quantum low-density parity-check codes, which we call radial codes, obtained from the lifted product of a specific subset of classical quasicyclic codes. The codes are defined using a pair of integers ( r , s ) and have parameters ⟦ 2 r 2 s , 2 ( r − 1 ) 2 , ≤ 2 s ⟧ , with numerical studies suggesting average-case distance linear in s . In simulations of circuit-level noise, we observe comparable error suppression to surface codes of similar distance while using approximately five times fewer physical qubits. This is true even when radial codes are decoded using a single-shot approach, which can allow for faster logical clock speeds and reduced decoding complexity. We describe an intuitive visual representation, canonical basis of logical operators and optimal-length stabilizer measurement circuits for these codes, and argue that their error correction capabilities, tunable parameters and small size make them promising candidates for implementation on near-term quantum devices.
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.
Quantum error correction is a fundamental primitive of fault-tolerant quantum computing. But in order for error correction to proceed, one must first prepare the codespace of the underlying error-correcting code. A popular method for encoding quantum low-density parity-check codes is transversal initialization, where one begins in a product state and measures a set of stabilizer generators. In the presence of measurement errors however, this procedure is generically not fault-tolerant, and so one typically needs to repeat the measurements many times, resulting in a deep initialization circuit. We present a protocol that prepares the codespace of constant-rate hypergraph product codes in constant depth with O(n) spatial overhead, and we show that the protocol is robust even in the presence of measurement errors. Our construction is inspired by dimension-jumping in topological codes and leverages two properties that arise from the homological product of codes. We provide some improvements to lower the spatial overhead and discuss applications to fault-tolerant architectures.
Quantum low-density parity-check (qLDPC) codes with high-performance and nonlocal stabilizers are an attractive alternative to the surface code because of their relatively high code rate and distance. However, unlike the surface code that has simple, geometrically local, stabilizer checks, qLDPC codes with nonlocal stabilizers are challenging to measure. Recent advances have shown how to deterministically perform high-fidelity, cavity-mediated many-body gates, enabling the encoding and decoding of nonlocal GHZ states. We show how to use this resource for constant-depth fault-tolerant stabilizer measurements in hypergraph product and lifted product codes. Using circuit-level noise simulations, including leakage and collective error from the noise-optimized cavity-mediated gate, we find promising thresholds of 0.84 % – 0.60 % for the hypergraph product code and pseudothreshold of 0.3 % – 0.4 % for the lifted product codes, with cavity cooperativities in the range C ∼ 10 4 – 10 6 . We propose a trilayer architecture, compatible with neutral atom platforms, which enhances circuit parallelizability.