Description
Galois-qudit CSS code obtained from a CSS code by increasing the smaller of the \(X\)- and \(Z\)-distances using a homological-product-based balancing step or one of its generalizations. 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 treats both types of errors on a more equal footing.
In the original construction [1; Sec. 4], if \(C\) is a QLDPC CSS code then applying the balancing step with parameter \(l\) yields \(\tilde K=K\), \(\tilde d_X=l d_X\), \(\tilde d_Z=d_Z\), and \(\tilde N=O(Nl)\), so choosing \(l\approx d_Z/d_X\) balances the two distances. In the generalized construction [3; Thm. 4.2], combining a component quantum code \(\mathcal{Q}\) with a classical code \(C\) yields a new code with \(K=k(\mathcal{Q})k(C)\), \(D_X=d_X(\mathcal{Q})d(C)\), and \(D_Z=d_Z(\mathcal{Q})\), so choosing \(d(C)\approx d_Z/d_X\) balances the two distances. The original distance-balancing procedure [1], later generalized in this way, can yield QLDPC codes [1; Thm. 1].
Weight reduction: Various procedures performing weight reduction [1,2,4] take in a stabilizer code and output a longer code with bounded stabilizer-generator weight. Hastings’ original construction [1] makes a qubit CSS code QLDPC while preserving the number of logical qubits and keeping the block length polynomial in the original one. The weight reduction procedure of Ref. [4] has been extended to subsystem qubit stabilizer codes [5]. The coning step of Hastings’ procedure [2] is an explicit use of a height-1 mapping cone [6; Prop. V.2]. Because this cone is regular, the coning step preserves the logical operators of the input code [6; Sec. V.A]. Weight reduction has been extended to non-CSS qubit stabilizer codes via the symplectic cone framework [7].
Decoding
If the auxiliary classical LDPC code corrects all error patterns of weight \(<\alpha |A|\), then the resulting product code has a polynomial-time decoder for \(X\)-errors of weight \(< \alpha |A| d_X/2\), where \(d_X\) is the \(X\)-distance of the component quantum code [3].If the component 2D complex has a polynomial-time decoder for \(Z\)-errors of weight \(< w\), then the resulting distance-balanced code also has a polynomial-time decoder for \(Z\)-errors of weight \(< w\) [3].The effective distance of single-ancilla syndrome extraction QLDPC code circuits can be preserved under weight reduction [8]. The distance balancing technique of Ref. [3] preserves the effective distance of single-ancilla syndrome extraction circuits [8].Fault Tolerance
Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [8]. The distance balancing technique of Ref. [3] preserves effective distance [8].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— The weight reduction procedure of Ref. [4] has been extended to subsystem qubit stabilizer codes [5].
- GKP CV-cluster-state code— Weight reduction has been studied in the context of GKP CV-cluster-state codes [4].
- QLDPC code— Lattice surgery techniques for QLDPC codes [9,10] utilize weight reduction. Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [8].
- Asymmetric quantum code (AQC)— 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 [1,11,12].
- Mapping cone code— The coning step of the weight reduction procedure of Ref. [2] is a height-1 cone [6; Sec. V.A]. The triangulation and thickening steps used in this procedure are regular height-2 cones [6; Sec. V.B].
- Fiber-bundle code— Fiber-bundle code constructions use distance balancing and weight reduction to increase distance.
- High-dimensional expander (HDX) code— Ramanujan tensor-product constructions use distance balancing to increase distance.
- Quantum check-product code— Quantum check-product code constructions use distance balancing to increase distance [13].
- Symplectic cone code— Symplectic cones extend weight reduction to non-CSS input codes [7; Thm. 53]. For an \(n\)-qubit input of check weight \(w\) and total qubit degree \(q\), the output has block length \(O(w^4q^4)n\). Its maximum stabilizer-generator weight is nine and its total qubit degree is at most eight [7; Thm. 53].
- Hemicubic code— Application of generalized distance balancing [3] to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(\Omega( 1/(\log(n) t^2) )\) soundness and order \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and increasing the dimension to order \(\Theta(t^2)\) [11].
- Hypersphere product code— Application of generalized distance balancing [3] to hypersphere product 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 \(\Theta(t^2)\) [11].
- Balanced product (BP) code— Applying distance balancing to the explicit subsystem balanced-product family of Ref. [14] yields an LDPC code family with \(k \in \Theta(n^{4/5})\) and \(d \in \Omega(n^{3/5})\).
Primary Hierarchy
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, L. G. Gunderman, B. Ide, M. Vasmer, and G. Dauphinais, “Weight-Reduced Stabilizer Codes with Lower Overhead”, PRX Quantum 5, (2024) arXiv:2402.05228 DOI
- [5]
- N. Baspin and D. Williamson, “Wire Codes”, Quantum 10, 2083 (2026) arXiv:2410.10194 DOI
- [6]
- A. C. Yuan, “Unified framework for quantum code embedding”, Physical Review A 113, (2026) arXiv:2507.05361 DOI
- [7]
- A. C. Yuan and N. Baspin, “Non-CSS Quantum Code Embedding”, (2026) arXiv:2608.16995
- [8]
- S. J. S. Tan and L. Stambler, “Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes”, Quantum 9, 1897 (2025) arXiv:2409.02193 DOI
- [9]
- L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
- [10]
- Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
- [11]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “General Distance Balancing for Quantum Locally Testable Codes”, (2023) arXiv:2305.00689
- [12]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “Tradeoff Constructions for Quantum Locally Testable Codes”, (2024) arXiv:2309.05541
- [13]
- A. Cross, Z. He, A. Natarajan, M. Szegedy, and G. Zhu, “Quantum Locally Testable Code with Constant Soundness”, Quantum 8, 1501 (2024) arXiv:2209.11405 DOI
- [14]
- 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 (2026-09-24) — most recent
- Victor V. Albert (2026-08-22)
- Victor V. Albert (2026-07-29)
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2022-01-20)
Cite as:
“Distance-balanced code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/distance_balanced, arXiv:2606.11484