Product codes are a class of quantum error correcting codes built from two or more constituent codes. They have recently gained prominence for a breakthrough yielding quantum low-density parity-check (qLDPC) codes with favorable scaling of both code distance and encoding rate. However, despite its powerful algebraic formulation, the physical mechanism for assembling a general product code from its constituents remains unclear. In this letter, we show that the tensor and balanced product codes admit an intuitive coupled-layer construction by taking a stack of one code and condensing a set of excitations in the pattern given by the checks of the other code. We also make a connection to concatenated codes by showing that the tensor product code can be obtained by gauging large-weight logicals in concatenated codes, making them qLDPC. Our framework accommodates both classical or quantum CSS input codes, unifies known physical mechanisms for constructing higher dimensional topological phases via anyon condensation, and naturally extends to non-topological codes.
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5+26 pages, 5+20 figures. v2: References added. Expanded supplemental section on relation to concatenated and subsystem codes, and introduced the balanced version
Product codes are a class of quantum error correcting codes built from two or more constituent codes. They have recently gained prominence for a breakthrough yielding quantum low-density parity-check (qLDPC) codes with favorable scaling of both code distance and encoding rate. However, despite its powerful algebraic formulation, the physical mechanism for assembling a general product code from its constituents remains unclear. In this letter, we show that the tensor and balanced product codes admit an intuitive coupled-layer construction by taking a stack of one code and condensing a set of excitations in the pattern given by the checks of the other code. We also make a connection to concatenated codes by showing that the tensor product code can be obtained by gauging large-weight logicals in concatenated codes, making them qLDPC. Our framework accommodates both classical or quantum CSS input codes, unifies known physical mechanisms for constructing higher dimensional topological phases via anyon condensation, and naturally extends to non-topological codes.
Product codes are an important class of quantum error-correcting codes constructed from multiple input codes, which can give rise to asymptotically good quantum low-density parity check codes. In previous work, we showed how the product between two codes can be physically implemented by coupling layers of the first code using checks of the second code. In this work, we further unify product code constructions with coupled-layer constructions of phases of matter by introducing coupled-layer codes. The essential strategy is to condense general excitations created by Pauli operators among multiple decoupled layers of the first code. The condensation is specified by an excitation algebra, which encodes the excitations, along with an algebra-preserving map. This coupling between layers generalizes the notion of gauging in physics as well as the mapping cone in homological algebra, and can be used to produce non-CSS codes. As examples, we show how coupled-layer codes reproduce the X-cube and Chamon models. We further generalize the balanced product code by allowing a unitary transformation in addition to a group action by free permutation and describe its corresponding coupled-layer construction. In particular, we show how balancing by a ZX-duality can reproduce non-CSS codes such as the fermionic toric code and the 3-fermion Walker-Wang model in 3D.
Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to establish code families with constant encoding rate and linear distance, simultaneously achieving optimal asymptotic scaling in both parameters. In particular, we characterize explicit logical representatives of the full gauged code space, and provide a fundamental treatment of non-Abelian code distance in the sense of Knill--Laflamme based on combining expansion with cleaning to establish protection against arbitrary errors. We further construct an almost-good family with entire code spaces exhibiting long-range magic. Moreover, we use linearly addressable cup product to directly generate linearly many encoded magic states from gauging measurements of Clifford operators. These results extend good qLDPC codes beyond the Pauli stabilizer setting, lay the foundation for pursuing the no low-energy trivial magic conjecture and shed light on non-Abelian quantum phases and orders beyond geometric locality.
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.
We realize a broad class of code transformations, including Kramers-Wannier duality, tensor product, and check product, as quantum processes consisting of ancilla initialization, local unitaries, and projective measurements. Using ZX-calculus, we formulate these code transformations diagrammatically and develop a systematic algorithm for extracting their quantum-process realizations. Central to our framework is the observation that the physical content of a classical LDPC code is captured by the operator algebra associated with its Tanner graph, and that code transformations correspond to maps between such algebras. Kramers-Wannier duality then admits a natural interpretation as gauging, while tensor and check products correspond to coupled-layer constructions in which interlayer coupling and projection implement a quotient on stacked operator algebras. Together, these results establish a unified framework connecting code transformations, quantum circuits, and mappings between distinct quantum phases of matter.
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
Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.