Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.09601

Simplified Quantum Weight Reduction with Optimal Bounds

Min-Hsiu Hsieh, Xingjian Li, Ting-Chun Lin

Published Oct 10, 2025• 0 citations
arXivHTMLPDF

Abstract

Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. This problem is important in practice, since low-weight measurements are necessary for reliable implementations of quantum error correction on physical hardware. It is also important theoretically, as it can be used to construct constant-locality versions of quantum locally testable codes, which may be relevant to the quantum PCP conjecture. We give a streamlined geometric procedure for quantum weight reduction based entirely on coning and treating X and Z checks symmetrically, which simplifies Hastings' previous approach. In particular, given an arbitrary $[[n,k,d]]$ quantum code with weight $w$, our method produces a code with parameters $[[O(n w^2 \log w), k, Ω(d w)]]$, check weight $5$, and qubit weight $6$; these bounds are optimal or close to optimal within the current geometric framework. Applied to random dense CSS codes, our procedure yields explicit quantum codes surpassing the square-root distance barrier, with parameters $[[n, \tilde Ω(n^{1/3}), \tilde Ω(n^{2/3})]]$. These codes also admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound, recovering the layer-code result. Our weight reduction technique also improves fault-tolerant logical-operator measurements by reducing the required number of ancilla qubits.

DOI
10.48550/arxiv.2510.09601

Related papers

Weight-Reduced Stabilizer Codes with Lower Overhead

Eric Sabo, Lane G. Gunderman, Benjamin Ide +2 more

Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024

arXiv:2402.05228•PRX Quantum•Oct 4, 2024•9 citations

Quantum Weight Reduction with Layer Codes

Andrew C. Yuan, Nouédyn Baspin, Dominic J. Williamson

Quantum weight reduction procedures ease the implementation of quantum codes by sparsifying them, resulting in low-weight checks and low-degree qubits. However, to date, only few quantum weight reduction methods have been explored. In this work we introduce a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead. Our quantum weight reduction procedure replaces each qubit and check in an arbitrary Calderbank-Shor-Steane code with an ample patch of surface code, these patches are then joined together to form a geometrically nonlocal Layer Code. This is a quantum analog of the simple classical weight reduction procedure where each bit and check is replaced by a repetition code. Due to the simplicity of our weight reduction procedure, bounds on the weight and degree of the resulting code follow directly from the Layer Code construction and hence are easily verified by inspection. Our procedure is well suited for implementation in modular architectures that consist of surface code patches networked via long-range interconnects.

quant-pharXiv:2603.04883•Mar 6, 2026•0 citations

Check-weight-constrained quantum codes: Bounds and examples

Lily Wang, Andy Zeyi Liu, Ray Li +2 more

Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.

quant-pharXiv:2601.15446•Jan 23, 2026•0 citations

Discovering highly efficient low-weight quantum error-correcting codes with reinforcement learning

He, Austin Yubo, Zi-Wen Liu

The realization of scalable fault-tolerant quantum computing is expected to hinge on quantum error-correcting codes. In the quest for more efficient quantum fault tolerance, a critical code parameter is the weight of measurements that extract information about errors to enable error correction: as higher measurement weights require higher implementation costs and introduce more errors, it is important in code design to optimize measurement weight. This underlies the surging interest in quantum low-density parity-check (qLDPC) codes, the study of which has primarily focused on the asymptotic (large-code-limit) properties. In this work, we introduce a versatile and computationally efficient approach to stabilizer code weight reduction based on reinforcement learning (RL), which produces new low-weight codes that substantially outperform the state of the art in practically relevant parameter regimes, extending significantly beyond previously accessible small distances. For example, our approach demonstrates savings in physical qubit overhead compared to existing results by 1 to 2 orders of magnitude for weight 6 codes and brings the overhead into a feasible range for near-future experiments. We also investigate the interplay between code parameters using our RL framework, offering new insights into the potential efficiency and power of practically viable coding strategies. Overall, our results demonstrate how RL can effectively advance the crucial yet challenging problem of quantum code discovery and thereby facilitate a faster path to the practical implementation of fault-tolerant quantum technologies.

arXiv:2502.14372•Feb 20, 2025•0 citations

On Quantum Weight Reduction

M. B. Hastings

We give a general procedure for weight reducing quantum codes. This corrects a previous work\cite{owr}, and introduces a new technique that we call "coning" to effectively induce high weight stabilizers in an LDPC code. As one application, any LDPC code (with arbitrary $O(1)$ stabilizer weights) may be turned into a code where all stabilizers have weight at most $5$ at the cost of at most a constant factor increase in number of physical qubits and constant factor reduction in distance. Also, by applying this technique to a quantum code whose $X$-stabilizers are derived from a classical log-weight random code and whose $Z$-stabilizers have linear weight, we construct an LDPC quantum code with distance $\tilde Ω(N^{2/3})$ and $\tildeΩ(N^{2/3})$ logical qubits.

arXiv:2102.10030•Feb 19, 2021•9 citations

Improved Quantum Codes with Transversal T Gates

Adam Wills

In this work, we study quantum CSS codes with transversal $T$ gates. Here, $T$ gate transversality is meant in the strongest sense; the application of physical $T$ to every physical qubit yields logical $T$ on every logical qubit, without Clifford corrections. Despite the importance of the $T$ gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal $T$ gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal $T$ gate. These are also the first codes achieving $γ\to 0$ for a code with a transversal $T$ gate, where $γ$ is the overhead exponent of magic state distillation. To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.

quant-pharXiv:2608.24000•Aug 26, 2026•0 citations
Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. This problem is important in practice, since low-weight measurements are necessary for reliable implementations of quantum error correction on physical hardware. It is also important theoretically, as it can be used to construct constant-locality versions of quantum locally testable codes, which may be relevant to the quantum PCP conjecture. We give a streamlined geometric procedure for quantum weight reduction based entirely on coning and treating X and Z checks symmetrically, which simplifies Hastings' previous approach. In particular, given an arbitrary [[n,k,d]][[n,k,d]][[n,k,d]] quantum code with weight www, our method produces a code with parameters [[O(nw2log⁡w),k,Ω(dw)]][[O(n w^2 \log w), k, Ω(d w)]][[O(nw2logw),k,Ω(dw)]], check weight 555, and qubit weight 666; these bounds are optimal or close to optimal within the current geometric framework. Applied to random dense CSS codes, our procedure yields explicit quantum codes surpassing the square-root distance barrier, with parameters [[n,Ω~(n1/3),Ω~(n2/3)]][[n, \tilde Ω(n^{1/3}), \tilde Ω(n^{2/3})]][[n,Ω~(n1/3),Ω~(n2/3)]]. These codes also admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound, recovering the layer-code result. Our weight reduction technique also improves fault-tolerant logical-operator measurements by reducing the required number of ancilla qubits.
Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024
Quantum weight reduction procedures ease the implementation of quantum codes by sparsifying them, resulting in low-weight checks and low-degree qubits. However, to date, only few quantum weight reduction methods have been explored. In this work we introduce a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead. Our quantum weight reduction procedure replaces each qubit and check in an arbitrary Calderbank-Shor-Steane code with an ample patch of surface code, these patches are then joined together to form a geometrically nonlocal Layer Code. This is a quantum analog of the simple classical weight reduction procedure where each bit and check is replaced by a repetition code. Due to the simplicity of our weight reduction procedure, bounds on the weight and degree of the resulting code follow directly from the Layer Code construction and hence are easily verified by inspection. Our procedure is well suited for implementation in modular architectures that consist of surface code patches networked via long-range interconnects.
Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.
The realization of scalable fault-tolerant quantum computing is expected to hinge on quantum error-correcting codes. In the quest for more efficient quantum fault tolerance, a critical code parameter is the weight of measurements that extract information about errors to enable error correction: as higher measurement weights require higher implementation costs and introduce more errors, it is important in code design to optimize measurement weight. This underlies the surging interest in quantum low-density parity-check (qLDPC) codes, the study of which has primarily focused on the asymptotic (large-code-limit) properties. In this work, we introduce a versatile and computationally efficient approach to stabilizer code weight reduction based on reinforcement learning (RL), which produces new low-weight codes that substantially outperform the state of the art in practically relevant parameter regimes, extending significantly beyond previously accessible small distances. For example, our approach demonstrates savings in physical qubit overhead compared to existing results by 1 to 2 orders of magnitude for weight 6 codes and brings the overhead into a feasible range for near-future experiments. We also investigate the interplay between code parameters using our RL framework, offering new insights into the potential efficiency and power of practically viable coding strategies. Overall, our results demonstrate how RL can effectively advance the crucial yet challenging problem of quantum code discovery and thereby facilitate a faster path to the practical implementation of fault-tolerant quantum technologies.
We give a general procedure for weight reducing quantum codes. This corrects a previous work\cite{owr}, and introduces a new technique that we call "coning" to effectively induce high weight stabilizers in an LDPC code. As one application, any LDPC code (with arbitrary O(1)O(1)O(1) stabilizer weights) may be turned into a code where all stabilizers have weight at most 555 at the cost of at most a constant factor increase in number of physical qubits and constant factor reduction in distance. Also, by applying this technique to a quantum code whose XXX-stabilizers are derived from a classical log-weight random code and whose ZZZ-stabilizers have linear weight, we construct an LDPC quantum code with distance Ω~(N2/3)\tilde Ω(N^{2/3})Ω~(N2/3) and Ω~(N2/3)\tildeΩ(N^{2/3})Ω~(N2/3) logical qubits.
In this work, we study quantum CSS codes with transversal TTT gates. Here, TTT gate transversality is meant in the strongest sense; the application of physical TTT to every physical qubit yields logical TTT on every logical qubit, without Clifford corrections. Despite the importance of the TTT gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal TTT gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal TTT gate. These are also the first codes achieving γ→0γ\to 0γ→0 for a code with a transversal TTT gate, where γγγ is the overhead exponent of magic state distillation. To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.