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تسجيل مستخدم جديدHomological Product Codes
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Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most $sqrt{n}$ qubits, where $n$ is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight $n^a$ for any $a>0$. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.
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Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product
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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 wh
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We provide a numerical investigation of two families of subsystem quantum codes that are related to hypergraph product codes by gauge-fixing. The first family consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal code parameters for subsys
tem quantum codes local in 2-dimensions. The second family consists of the constant rate generalized Shor codes of Bacon and Cassicino cite{bacon2006quantum}, which we re-brand as subsystem hypergraph product (SHP) codes. We show that any hypergraph product code can be obtained by entangling the gauge qubits of two SHP codes. To evaluate the performance of these codes, we simulate both small and large examples. For circuit noise, a $[[21,4,3]]$ BBS code and a $[[49,16,3]]$ SHP code have pseudthresholds of $2times10^{-3}$ and $8times10^{-4}$, respectively. Simulations for phenomenological noise show that large BBS and SHP codes start to outperform surface codes with similar encoding rate at physical error rates $1times 10^{-6}$ and $4times10^{-4}$, respectively.
Product perfect codes have been proven to enhance the performance of the F5 steganographic method, whereas perfect Z2Z4-linear codes have been recently introduced as an efficient way to embed data, conforming to the +/-1-steganography. In this paper,
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