## Description

Galois-qudit CSS code constructed from a CSS code and a classical code using a distance-balancing procedure based on a generalized homological product. The initial code is said to be unbalanced, i.e., tailored to noise biased toward either bit- or phase-flip errors, and the procedure can result in a code that is treats both types of errors on a more equal footing. The original distance-balancing procedure [1], later generalized [3; Thm. 4.2], can yield QLDPC codes [1; Thm. 1].

## Decoding

Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [5]. The distance balancing technique of Ref. [3] preserves effective distance [5].

## Fault Tolerance

Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [5]. The distance balancing technique of Ref. [3] preserves effective distance [5].

## Parents

## Cousins

- Homological product code — Distance balancing relies on taking a homological product of chain complexes corresponding to a classical and a quantum code.
- Subsystem qubit stabilizer code
- GKP CV-cluster-state code — Weight reduction has been studied in the context of GKP CV-cluster-state codes [4].
- Quantum LDPC (QLDPC) code — Lattice surgery techniques for QLDPC codes [6,7] utilize weight reduction. Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [5].
- Asymmetric quantum code — Distance balancing is a procedure that can convert an asymmetric CSS code into a less asymmetric one.
- Quantum locally testable code (QLTC) — Distance balancing and weight reduction are useful for constructing QLTCs [1,8,9].
- Fiber-bundle code — Fiber-bundle code constructions use distance balancing and weight reduction to increase distance.
- Quantum check-product code — Quantum check-product code constructions use distance balancing to increase distance [10].
- High-dimensional expander (HDX) code — Ramanujan tensor-product constructions use distance balancing to increase distance.
- Hemicubic code — Application of generalized distance balancing [3] to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n) t^2))\) soundness and order \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as order \(\Theta(t^2)\) [8].
- Hypersphere product code — The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance order \(\Theta(\sqrt{n})\). Application of generalized distance balancing [3] to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n)^2 t^2))\) soundness and order \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as order \(\Theta(t^2)\) [8].
- Balanced product (BP) code — Distance balancing is used to form balanced-product subsystem codes [11].

## References

- [1]
- M. B. Hastings, “Weight Reduction for Quantum Codes”, (2016) arXiv:1611.03790
- [2]
- M. B. Hastings, “On Quantum Weight Reduction”, (2023) arXiv:2102.10030
- [3]
- S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
- [4]
- E. Sabo et al., “Weight Reduced Stabilizer Codes with Lower Overhead”, (2024) arXiv:2402.05228
- [5]
- S. J. S. Tan and L. Stambler, “Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes”, (2024) arXiv:2409.02193
- [6]
- L. Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
- [7]
- Q. Xu et al., “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
- [8]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “General Distance Balancing for Quantum Locally Testable Codes”, (2023) arXiv:2305.00689
- [9]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “Tradeoff Constructions for Quantum Locally Testable Codes”, (2024) arXiv:2309.05541
- [10]
- A. Cross et al., “Quantum Locally Testable Code with Constant Soundness”, (2023) arXiv:2209.11405
- [11]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI

## Page edit log

- Victor V. Albert (2022-01-20) — most recent

## Cite as:

“Distance-balanced code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/distance_balanced