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Register a new userGeneralized Haah Codes and Fracton Models
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Entanglement renormalization group flow of the Haah cubic code produces another fracton model with 4 qubits per lattice site, dubbed as the Haah B-code. We provide a schema that generalizes both models to stabilizer codes on any finite group with 2q qubits per site and labeled by multi-subsets of the finite group and a symmetric binary matrix.
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Haah codes represent a singularly interesting gapped Hamiltonian schema that has resisted a natural generalization, although recent work shows that the closely related type I fracton models are more commonplace. These type I siblings of Haah codes are better understood, and a generalized topological quantum field theory framework has been proposed. Following the same conceptual framework, we outline a program to generalize Haah codes to all 3-manifolds using Hastings LR stabilizer codes for finite groups.
Fracton topological phases have a large number of materialized symmetries that enforce a rigid structure on their excitations. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fracton phase enable us to design decoding algorithms. Here we propose and implement decoding algorithms for the three-dimensional X-cube model. In our example, decoding is parallelized into a series of two-dimensional matching problems, thus significantly simplifying the most time consuming component of the decoder. We also find that the rigid structure of its point excitations enable us to obtain high threshold error rates. Our decoding algorithms bring to light some key ideas that we expect to be useful in the design of decoders for general topological stabilizer codes. Moreover, the notion of parallelization unifies several concepts in quantum error correction. We conclude by discussing the broad applicability of our methods, and we explain the connection between parallelizable codes and other methods of quantum error correction. In particular we propose that the new concept represents a generalization of single-shot error correction.
We introduce the concept of generalized concatenated quantum codes. This generalized concatenation method provides a systematical way for constructing good quantum codes, both stabilizer codes and nonadditive codes. Using this method, we construct families of new single-error-correcting nonadditive quantum codes, in both binary and nonbinary cases, which not only outperform any stabilizer codes for finite block length, but also asymptotically achieve the quantum Hamming bound for large block length.
We consider a notion of relative homology (and cohomology) for surfaces with two types of boundaries. Using this tool, we study a generalization of Kitaevs code based on surfaces with mixed boundaries. This construction includes both Bravyi and Kitaevs and Freedman and Meyers extension of Kitaevs toric code. We argue that our generalization offers a denser storage of quantum information. In a planar architecture, we obtain a three-fold overhead reduction over the standard architecture consisting of a punctured square lattice.
Streaming codes represent a packet-level FEC scheme for achieving reliable, low-latency communication. In the literature on streaming codes, the commonly-assumed Gilbert-Elliott channel model, is replaced by a more tractable, delay-constrained, sliding-window (DCSW) channel model that can introduce either random or burst erasures. The known streaming codes that are rate optimal over the DCSW channel model are constructed by diagonally embedding a scalar block code across successive packets. These code constructions have field size that is quadratic in the delay parameter $tau$ and have a somewhat complex structure with an involved decoding procedure. This led to the introduction of simple streaming (SS) codes in which diagonal embedding is replaced by staggered-diagonal embedding (SDE). The SDE approach reduces the impact of a burst of erasures and makes it possible to construct near-rate-optimal streaming codes using Maximum Distance Separable (MDS) code having linear field size. The present paper takes this development one step further, by retaining the staggered-diagonal feature, but permitting the placement of more than one code symbol from a given scalar codeword within each packet. These generalized, simple streaming codes allow us to improve upon the rate of SS codes, while retaining the simplicity of working with MDS codes. We characterize the maximum code rate of streaming codes under a constraint on the number of contiguous packets over which symbols of the underlying scalar code are dispersed. Such a constraint leads to simplified code construction and reduced-complexity decoding.
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