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 topologica...
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ISBN:
(纸本)9798331541378
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 methods to the well-known hypergraphproduct and lifted-productcodes. 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 hypergraphproduct and qudit lifted-productcodes: the merge construction applied to the lifted productcode 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 give a construction of quantum LDPC codes of dimension Theta(logN) and distance Theta(N/logN) as the code length N -> infinity. Using a product of chain complexes this construction also provides a family of quan...
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We give a construction of quantum LDPC codes of dimension Theta(logN) and distance Theta(N/logN) as the code length N -> infinity. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance Omega(N1-alpha/2/logN) and dimension Omega(N-alpha logN), where 0 <= alpha < 1. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixedR < 1 there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least R with, in some sense, optimal circulant size Omega(N/logN) as the code length N -> infinity.
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