# Asymptotically Good Quantum and Locally Testable Classical LDPC Codes

@article{Panteleev2021AsymptoticallyGQ, title={Asymptotically Good Quantum and Locally Testable Classical LDPC Codes}, author={Pavel Panteleev and Gleb Kalachev}, journal={ArXiv}, year={2021}, volume={abs/2111.03654} }

We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes.

## 18 Citations

Quantum Tanner codes

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2022

This work proves a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et al. code.

Fold-Transversal Clifford Gates for Quantum Codes

- Computer Science
- 2022

We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in…

Explicit Abelian Lifts and Quantum LDPC Codes

- Computer ScienceITCS
- 2022

The following explicit constructions of expanders obtained via abelian lifts are shown, including explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes.

Distance bounds for generalized bicycle codes

- Computer Science
- 2022

An exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8 is done; the observed distance scaling is consistent with A ( w ) n 1 / 2 + B ( w ), where n is the code length and B is increasing with w .

c3-Locally Testable Codes from Lossless Expanders

- Computer ScienceArXiv
- 2022

This work constructs a new LTC family using 1-sided lossless expanders and balanced products and proves that there are c-LTCs which are LTCs with constant rate, constant relative distance and constant locality.

QDistRnd: A GAP package for computing the distance of quantum error-correcting codes

- Computer ScienceJ. Open Source Softw.
- 2022

A format for storing matrices associated with q-ary quantum codes is introduced and implemented via the provided import/export functions, based on the well established MaTrix market eXchange (MTX) Coordinate format, and is designed for full backward compatibility with this format.

Locally Testable Codes with constant rate, distance, and locality

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2021

This work constructs LTCs with constant rate, constant distance, and constant locality based on a new two-dimensional complex which they call a left-right Cayley complex, which is essentially a graph which has vertices and edges and also has squares.

Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

- Computer ScienceArXiv
- 2021

It is observed that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves.

Partitioning qubits in hypergraph product codes to implement logical gates

- Computer Science
- 2022

It is demonstrated that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection.

Relaxed Locally Decodable and Correctable Codes: Beyond Tensoring

- Computer Science
- 2022

This work constructs asymptotically-good RLDC and RLCC with an improved query complexity of p log n q O p log log log nq and devise a mechanism– an alternative to the tensor product–that squares the length of a given code.

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