We give a construction of quantum LDPC codes of dimension $\Theta (\log N)$ and distance $\Theta (N/\log N)$ as the code length$N\to \infty $. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance$ \Omega (N^{1-\alpha /2}/\log N) $and dimension$ \Omega (N^\alpha \log N)$, where $0 \le \alpha < 1$ . 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 fixed $R < 1$ there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least $R$ with, in some sense, optimal circulant size $\Omega (N/\log N)$ as the code length$N\to \infty $.We give a construction of quantum LDPC codes of dimension$\Theta (\log N)$and distance$\Theta (N/\log N)$as the code length$N\to \infty $. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance$\Omega (N^{1-\alpha /2}/\log N)$and dimension$\Omega (N^\alpha \log N)$, where$0 \le \alpha < 1$. 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 fixed$R < 1$there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least$R$with, in some sense, optimal circulant size$\Omega (N/\log N)$as the code length$N\to \infty $.

arXiv:2012.04068IEEE Transactions on Information TheoryOct 11, 2021170 citations

Coset Lifted Product Codes: Constructing Quantum LDPC Codes from Non-Normal Subgroup Actions

Rajendra Prasad Rajpurohit, V. Lalitha, Shayan Srinivasa Garani

We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$ , a subgroup $H \le G$ that need not be normal, a protograph $A$ over $\mathbb F_2[G]$ , and a protograph $B$ over $\mathbb F_2[N_G(H)]$ , the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $ρ$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120 \le n \le 400$ yields certified codes that exceed the best published value of $Q=kd^2/n$ at check weights $6,7$ and $9$ , i.e., we obtain $[[360,20,20]]$ at $w=6$ , a $[[360,24,20]]$ code at $w=7$ , and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$ . Our search also produce high-distance codes with parameters $[[168,10,14]], [[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$ .We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$, a subgroup $H \le G$ that need not be normal, a protograph $A$ over $\mathbb F_2[G]$, and a protograph $B$ over $\mathbb F_2[N_G(H)]$, the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $ρ$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120 \le n \le 400$ yields certified codes that exceed the best published value of $Q=kd^2/n$ at check weights $6,7$ and $9$, i.e., we obtain $[[360,20,20]]$ at $w=6$, a $[[360,24,20]]$ code at $w=7$, and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$. Our search also produce high-distance codes with parameters $[[168,10,14]], [[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$.

quant-pharXiv:2610.12782Oct 12, 20260 citations

A Note on the Minimum Distance of Quantum LDPC Codes

Nicolas Delfosse, Zhentao Li, Stéphan Thomassé

arXiv:1404.6441Lecture notes in computer scienceJan 1, 20140 citations

Random Construction of Quantum LDPC Codes

Koki Okada, Kouki OKADA, Kenta Kasai

Jan 1, 20250 citations

Towards practical quantum LDPC codes

Joschka Roffe

Nov 30, 20213 citations

Study of Quantum LDPC Codes and their Decoding

Wouter Rozendaal

Quantum Low-Density Parity-Check (LDPC) codes provide a promising solution to protect quantum information from errors and are therefore considered an important step in the realisation of a fullscale fault-tolerant quantum computer. In this thesis, we study quantum LDPC codes under three different approaches, providing insights into renormalisation decoding for Kitaev’s toric code, establishing bounds on the minimum distance of geometrically-local codes, and constructing small instances of quantum Tanner codes. Kitaev’s toric code is a prominent quantum LDPC code that is currently the most widely pursued for experimental implementation. We focus on the probabilistic renormalisation decoders introduced by Duclos- Cianci and Poulin, a family of decoders that exhibit one of the best trade-offs between accuracy and efficiency. We study how they handle adversarial errors by introducing a deterministic renormalisation decoder that does not use a priori probabilities of the error model. We find that this renormalisation decoder allows for fractal-like uncorrectable error patterns, and we obtain a lower bound on the weight of such errors. Due to physical constraints, a lot of attention has gone into quantum LDPC codes whose stabilisers act only on a limited number of nearby qubits. We extend the well-known Bravyi-Terhal bound to quantum codes defined by local constraints on a lattice quotient, providing a crucial insight into the limitations of geometrically-local stabiliser codes. As an application, we provide upper bounds on the minimum distance of Abelian Two-Block Group Algebra codes, a family of quantum LDPC codes for which the minimum distance is currently unknown. In recent years, asymptotically good quantum LDPC codes were constructed, but provable results on their minimum distances only apply to impractically long codes. Following the effort in the design of near-term implementable codes, we construct short instances of quantum Tanner codes using square complexes obtained from lifted products of regular graphs. Several explicit instances of these codes exhibit minimum distances surpassing the square root of the code length.Quantum Low-Density Parity-Check (LDPC) codes provide a promising solution to protect quantum information from errors and are therefore considered an important step in the realisation of a fullscale fault-tolerant quantum computer. In this thesis, we study quantum LDPC codes under three different approaches, providing insights into renormalisation decoding for Kitaev’s toric code, establishing bounds on the minimum distance of geometrically-local codes, and constructing small instances of quantum Tanner codes. Kitaev’s toric code is a prominent quantum LDPC code that is currently the most widely pursued for experimental implementation. We focus on the probabilistic renormalisation decoders introduced by Duclos- Cianci and Poulin, a family of decoders that exhibit one of the best trade-offs between accuracy and efficiency. We study how they handle adversarial errors by introducing a deterministic renormalisation decoder that does not use a priori probabilities of the error model. We find that this renormalisation decoder allows for fractal-like uncorrectable error patterns, and we obtain a lower bound on the weight of such errors. Due to physical constraints, a lot of attention has gone into quantum LDPC codes whose stabilisers act only on a limited number of nearby qubits. We extend the well-known Bravyi-Terhal bound to quantum codes defined by local constraints on a lattice quotient, providing a crucial insight into the limitations of geometrically-local stabiliser codes. As an application, we provide upper bounds on the minimum distance of Abelian Two-Block Group Algebra codes, a family of quantum LDPC codes for which the minimum distance is currently unknown. In recent years, asymptotically good quantum LDPC codes were constructed, but provable results on their minimum distances only apply to impractically long codes. Following the effort in the design of near-term implementable codes, we construct short instances of quantum Tanner codes using square complexes obtained from lifted products of regular graphs. Several explicit instances of these codes exhibit minimum distances surpassing the square root of the code length.

Dec 2, 20250 citations

We give a construction of quantum LDPC codes of dimension

\Theta (\log N)

and distance

\Theta (N/\log N)

as the code length$N\to \infty

. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance

\Omega (N^{1-\alpha /2}/\log N)

and dimension

\Omega (N^\alpha \log N)$, where

0 \le \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 fixed

R < 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/\log N)

as the code length$N\to \infty $.

G

, a subgroup

H \le G

that need not be normal, a protograph

A

over

\mathbb F_2[G]

, and a protograph

B

over

\mathbb F_2[N_G(H)]

, the two protographs act on the cosets

G/H

from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations

ρ

G

of hypergraph product dimensions. A search over

14,843

pairs

(G,H)

with

H

restricted to be non-normal and

120 \le n \le 400

yields certified codes that exceed the best published value of

Q=kd^2/n

at check weights

6,7

and

9

, i.e., we obtain

[[360,20,20]]

w=6

, a

[[360,24,20]]

code at

w=7

, and codes

[[252,16,18]]

and

[[288,16,20]]

w=9

. Our search also produce high-distance codes with parameters

[[168,10,14]], [[240,8,20]]

w=6

and

[[216,8,18]]

w=7