# Quantum low-weight check (QLWC) code[1]

## Description

Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes for which the number of sites participating in each stabilizer generator is bounded by a constant as \(n\to\infty\).

## Parents

## Child

- Quantum LDPC (QLDPC) code — QLDPC codes are QLWC codes for which the number of stabilizer generators that each site participates in is bounded by a constant as \(n\to\infty\).

## Cousins

- Unary code — A family of approximate non-stabilizer QLWC codes with linear distance and rate has been constructed [1] using unary codes that arise from the Feynman-Kitaev clock construction [2].
- Approximate quantum error-correcting code (AQECC) — A family of approximate non-stabilizer QLWC codes with linear distance and rate has been constructed [1] using unary codes that arise from the Feynman-Kitaev clock construction [2].
- Circuit-to-Hamiltonian approximate code — The circuit-to-Hamiltonian code construction yields approximate codes whose distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) [3; Thm. 3.1]. These codes are approximate non-stabilizer QLWC codes since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by order \(O(\text{polylog}(n)\) projectors.

## References

- [1]
- C. Nirkhe, U. Vazirani, and H. Yuen, “Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States”, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany (2018) arXiv:1802.07419 DOI
- [2]
- A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (American Mathematical Society, 2002) DOI
- [3]
- T. C. Bohdanowicz et al., “Good approximate quantum LDPC codes from spacetime circuit Hamiltonians”, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) arXiv:1811.00277 DOI

## Page edit log

- Victor V. Albert (2023-11-16) — most recent

## Cite as:

“Quantum low-weight check (QLWC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qlwc