Quantum XYZ Product Codes
- URL: http://arxiv.org/abs/2011.09746v3
- Date: Tue, 12 Jul 2022 15:48:19 GMT
- Title: Quantum XYZ Product Codes
- Authors: Anthony Leverrier, Simon Apers, Christophe Vuillot
- Abstract summary: 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 which might display a large minimum distance.
- Score: 0.3222802562733786
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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 which might display a large minimum distance. The simplest
instance of this construction, corresponding to the product of 3 repetition
codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon
code. The general construction was introduced in Denise Maurice's PhD thesis,
but has remained poorly understood so far. The reason is that while hypergraph
product codes can be analyzed with combinatorial tools, the XYZ product codes
also depend crucially on the algebraic properties of the parity-check matrices
of the three classical codes, making their analysis much more involved.
Our main motivation for studying XYZ product codes is that the natural
representatives of logical operators are two-dimensional objects. This
contrasts with standard hypergraph product codes in 3 dimensions which always
admit one-dimensional logical operators. In particular, specific instances of
XYZ product codes with constant rate might display a minimum distance as large
as $\Theta(N^{2/3})$. While we do not prove this result here, we obtain the
dimension of a large class of XYZ product codes, and when restricting to codes
with dimension 1, we reduce the problem of computing the minimum distance to a
more elementary combinatorial problem involving binary 3-tensors. We also
discuss in detail some families of XYZ product codes that can be embedded in
three dimensions with local interaction. Some of these codes seem to share
properties with Haah's cubic codes and might be interesting candidates for
self-correcting quantum memories with a logarithmic energy barrier.
Related papers
- SSIP: automated surgery with quantum LDPC codes [55.2480439325792]
We present Safe Surgery by Identifying Pushouts (SSIP), an open-source lightweight Python package for automating surgery between qubit CSS codes.
Under the hood, it performs linear algebra over $mathbbF$ governed by universal constructions in the category of chain complexes.
We show that various logical measurements can be performed cheaply by surgery without sacrificing the high code distance.
arXiv Detail & Related papers (2024-07-12T16:50:01Z) - Equivalence Classes of Quantum Error-Correcting Codes [49.436750507696225]
Quantum error-correcting codes (QECC's) are needed to combat the inherent noise affecting quantum processes.
We represent QECC's in a form called a ZX diagram, consisting of a tensor network.
arXiv Detail & Related papers (2024-06-17T20:48:43Z) - Maximally Extendable Sheaf Codes [5.439020425819001]
We study sheaf codes, a type of linear codes with a fixed hierarchical collection of local codes.
We introduce a new property of a sheaf code, called maximalibility, which ensures that within class of codes on the same coded space, we encounter as few obstructions as possible.
arXiv Detail & Related papers (2024-03-06T12:20:49Z) - Small Quantum Codes from Algebraic Extensions of Generalized Bicycle
Codes [4.299840769087443]
Quantum LDPC codes range from the surface code, which has a vanishing encoding rate, to very promising codes with constant encoding rate and linear distance.
We devise small quantum codes that are inspired by a subset of quantum LDPC codes, known as generalized bicycle (GB) codes.
arXiv Detail & Related papers (2024-01-15T10:38:13Z) - Lift-Connected Surface Codes [0.0]
We use the recently introduced lifted product to construct a family of Quantum Low Density Parity Check Codes (QLDPC codes)
The codes we obtain can be viewed as stacks of surface codes that are interconnected, leading to the name lift-connected surface (LCS) codes.
arXiv Detail & Related papers (2024-01-05T17:22:49Z) - Homological Quantum Rotor Codes: Logical Qubits from Torsion [51.9157257936691]
homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code.
We show that the $0$-$pi$-qubit as well as Kitaev's current-mirror qubit are indeed small examples of such codes.
arXiv Detail & Related papers (2023-03-24T00:29:15Z) - Classical product code constructions for quantum Calderbank-Shor-Steane codes [1.4699455652461726]
We introduce a new product code construction which is a natural generalisation of classical product codes to quantum codes.
We show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors.
arXiv Detail & Related papers (2022-09-27T15:48:37Z) - Morphing quantum codes [77.34726150561087]
We morph the 15-qubit Reed-Muller code to obtain the smallest known stabilizer code with a fault-tolerant logical $T$ gate.
We construct a family of hybrid color-toric codes by morphing the color code.
arXiv Detail & Related papers (2021-12-02T17:43:00Z) - KO codes: Inventing Nonlinear Encoding and Decoding for Reliable
Wireless Communication via Deep-learning [76.5589486928387]
Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes.
In this paper, we construct KO codes, a computationaly efficient family of deep-learning driven (encoder, decoder) pairs.
KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding.
arXiv Detail & Related papers (2021-08-29T21:08:30Z) - Finding the disjointness of stabilizer codes is NP-complete [77.34726150561087]
We show that the problem of calculating the $c-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete.
We provide bounds on the disjointness for various code families, including the CSS codes,$d codes and hypergraph codes.
Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.
arXiv Detail & Related papers (2021-08-10T15:00:20Z) - Decoding Across the Quantum LDPC Code Landscape [4.358626952482686]
We show that belief propagation combined with ordered statistics post-processing is a general decoder for quantum low density parity check codes.
We run numerical simulations of the decoder applied to three families of hypergraph product code: topological codes, fixed-rate random codes and a new class of codes that we call semi-topological codes.
arXiv Detail & Related papers (2020-05-14T14:33:08Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.