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:2410.23263

Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays

Anqi Gong, Joseph M. Renes

Published Oct 30, 2024• 0 citations
arXivHTMLPDF

Abstract

We give a fault tolerant construction for error correction and computation using two punctured quantum Reed-Muller (PQRM) codes. In particular, we consider the $[[127,1,15]]$ self-dual doubly-even code that has transversal Clifford gates (CNOT, H, S) and the triply-even $[[127,1,7]]$ code that has transversal T and CNOT gates. We show that code switching between these codes can be accomplished using Steane error correction. For fault-tolerant ancilla preparation we utilize the low-depth hypercube encoding circuit along with different code automorphism permutations in different ancilla blocks, while decoding is handled by the high-performance classical successive cancellation list decoder. In this way, every logical operation in this universal gate set is amenable to extended rectangle analysis. The CNOT exRec has a failure rate approaching $10^{-9}$ at $10^{-3}$ circuit-level depolarizing noise. Furthermore, we map the PQRM codes to a 2D layout suitable for implementation in arrays of trapped atoms and try to reduce the circuit depth of parallel atom movements in state preparation. The resulting protocol is strictly fault-tolerant for the $[[127,1,7]]$ code and practically fault-tolerant for the $[[127,1,15]]$ code. Moreover, each patch requires a permutation consisting of $7$ sub-hypercube swaps only. These are swaps of rectangular grids in our 2D hypercube layout and can be naturally created with acousto-optic deflectors (AODs). Lastly, we show for the family of $[[2^{2r},{2r\choose r},2^r]]$ QRM codes that the entire logical Clifford group can be achieved using only permutations, transversal gates, and fold-transversal gates.

DOI
10.48550/arxiv.2410.23263

Related papers

Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates

Theerapat Tansuwannont, Tim Chan, Ryuji Takagi

To build large-scale quantum computers while minimizing resource requirements, one may want to use high-rate quantum error-correcting codes that can efficiently encode information. However, realizing an addressable gate$\unicode{x2014}$a logical gate on a subset of logical qubits within a high-rate code$\unicode{x2014}$in a fault-tolerant manner can be challenging and may require ancilla qubits. Transversal and fold-transversal gates could provide a means to fault-tolerantly implement logical gates using a constant-depth circuit without ancilla qubits, but available gates of these types could be limited depending on the code and might not be addressable. In this work, we study a family of $[\![n=2^m,k={m \choose m/2}\approx n/\sqrt{π\log_2(n)/2},d=2^{m/2}=\sqrt{n}]\!]$ self-dual quantum Reed$\unicode{x2013}$Muller codes, where $m$ is a positive even number. For any code in this family, we construct a generating set of the full logical Clifford group comprising only transversal and fold-transversal gates, thus enabling the implementation of any addressable Clifford gate. To our knowledge, this is the first known construction of the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a family of codes in which $k$ grows near-linearly in $n$.

quant-pharXiv:2602.09788•Feb 11, 2026•0 citations

Efficient recursive encoders for quantum Reed-Muller codes towards Fault tolerance

Praveen Jayakumar, Priya J. Nadkarni, Shayan Srinivasa Garani

Transversal gates are logical gate operations on encoded quantum information that are efficient in gate count and depth, and are designed to minimize error propagation. Efficient encoding circuits for quantum codes that admit transversal gates are thus crucial to reduce noise and realize useful quantum computers. The class of punctured Quantum Reed-Muller codes admit transversal gates. We construct resource efficient recursive encoders for the class of quantum codes constructed from Reed-Muller and punctured Reed-Muller codes. These encoders on $n$ qubits have circuit depth of $O(\log n)$ and lower gate counts compared to previous works. The number of CNOT gates in the encoder across bi-partitions of the qubits is found to be equal to the entanglement entropy across these partitions, demonstrating that the encoder is optimal in terms of CNOT gates across these partitions. Finally, connecting these ideas, we explicitly show that entanglement can be extracted from QRM codewords.

arXiv:2405.14549•May 23, 2024•0 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

A New Family of Fault Tolerant Quantum Reed-Muller Codes

Harrison Eggers

Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.

•Jan 1, 2020•0 citations

Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays

Willers Yang, Casey Duckering, Arpit Dua

Achieving useful fault-tolerant applications requires co-designing the quantum error-correcting code for footprint, logical instruction set, and hardware compatibility simultaneously. Recent breakthroughs in ultra-high-rate codes have the potential to reduce footprint of large scale quantum applications but their practical utility is limited without an addressable logical instruction set and efficient implementation on near term hardware. Reconfigurable neutral atom arrays, for example, support only hundreds of qubits with constrained reconfigurability from a few crossed acousto-optic deflectors (AOD). We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that allows such co-design by construction. GALA lifts over a product group $G = H_k \times C_m$ (or $H_k \ltimes C_m^k$); the small non-abelian factor $H_k$ supplies active orthogonality that improves qubit overhead, reaching 1/2 encoding rate with above-weight distance, while the large abelian factor $C_m$ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Logical addressibility and hardware compatibility follow directly from construction, rather than verified post-hoc. The GALA family contains several known rate-1/2 Kasai codes of Ref.[arXiv:2601.08824, arXiv:2604.16209] while exposing simpler parameter bounds, logical operations, and ZX-dual variants with AOD-compatible fold-transversal Clifford gates. In particular, we find a compact self-dual $[[132,30,12]]$ with $1.0^{+0.9}_{-0.6}\times 10^{-9}$ per-round per-qubit logical error rate at $10^{-3}$ physical error rate, equipped with 3.1 ms syndrome-extraction cycles and an addressable logical instruction set; a ZX-dual, girth-6, rate-1/2 $[[672,336,12]]$ with a 6.76 ms cycles; and barrier-breaking $[[1752,880,14]]$ and $[[2232,1120,16]]$ smaller than previously known hardware-compatible rate-1/2 codes.

quant-pharXiv:2608.07431•Aug 10, 2026•0 citations

Code conversion with the quantum Golay code for a universal transversal gate set

Matthew Sullivan

The [[7,1,3]] Steane code and [[23,1,7]] quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates but require some other scheme for fault-tolerant $T$ gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal $T$ gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal $T$ gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.

arXiv:2307.14425•Physical Review A•Apr 18, 2024•7 citations
We give a fault tolerant construction for error correction and computation using two punctured quantum Reed-Muller (PQRM) codes. In particular, we consider the [[127,1,15]][[127,1,15]][[127,1,15]] self-dual doubly-even code that has transversal Clifford gates (CNOT, H, S) and the triply-even [[127,1,7]][[127,1,7]][[127,1,7]] code that has transversal T and CNOT gates. We show that code switching between these codes can be accomplished using Steane error correction. For fault-tolerant ancilla preparation we utilize the low-depth hypercube encoding circuit along with different code automorphism permutations in different ancilla blocks, while decoding is handled by the high-performance classical successive cancellation list decoder. In this way, every logical operation in this universal gate set is amenable to extended rectangle analysis. The CNOT exRec has a failure rate approaching 10−910^{-9}10−9 at 10−310^{-3}10−3 circuit-level depolarizing noise. Furthermore, we map the PQRM codes to a 2D layout suitable for implementation in arrays of trapped atoms and try to reduce the circuit depth of parallel atom movements in state preparation. The resulting protocol is strictly fault-tolerant for the [[127,1,7]][[127,1,7]][[127,1,7]] code and practically fault-tolerant for the [[127,1,15]][[127,1,15]][[127,1,15]] code. Moreover, each patch requires a permutation consisting of 777 sub-hypercube swaps only. These are swaps of rectangular grids in our 2D hypercube layout and can be naturally created with acousto-optic deflectors (AODs). Lastly, we show for the family of [[22r,(2rr),2r]][[2^{2r},{2r\choose r},2^r]][[22r,(r2r​),2r]] QRM codes that the entire logical Clifford group can be achieved using only permutations, transversal gates, and fold-transversal gates.
To build large-scale quantum computers while minimizing resource requirements, one may want to use high-rate quantum error-correcting codes that can efficiently encode information. However, realizing an addressable gate$\unicode{x2014}$a logical gate on a subset of logical qubits within a high-rate code$\unicode{x2014}$in a fault-tolerant manner can be challenging and may require ancilla qubits. Transversal and fold-transversal gates could provide a means to fault-tolerantly implement logical gates using a constant-depth circuit without ancilla qubits, but available gates of these types could be limited depending on the code and might not be addressable. In this work, we study a family of [ ⁣[n=2m,k=(mm/2)≈n/πlog⁡2(n)/2,d=2m/2=n] ⁣][\![n=2^m,k={m \choose m/2}\approx n/\sqrt{π\log_2(n)/2},d=2^{m/2}=\sqrt{n}]\!][[n=2m,k=(m/2m​)≈n/πlog2​(n)/2​,d=2m/2=n​]] self-dual quantum Reed$\unicode{x2013}$Muller codes, where mmm is a positive even number. For any code in this family, we construct a generating set of the full logical Clifford group comprising only transversal and fold-transversal gates, thus enabling the implementation of any addressable Clifford gate. To our knowledge, this is the first known construction of the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a family of codes in which kkk grows near-linearly in nnn.
Transversal gates are logical gate operations on encoded quantum information that are efficient in gate count and depth, and are designed to minimize error propagation. Efficient encoding circuits for quantum codes that admit transversal gates are thus crucial to reduce noise and realize useful quantum computers. The class of punctured Quantum Reed-Muller codes admit transversal gates. We construct resource efficient recursive encoders for the class of quantum codes constructed from Reed-Muller and punctured Reed-Muller codes. These encoders on nnn qubits have circuit depth of O(log⁡n)O(\log n)O(logn) and lower gate counts compared to previous works. The number of CNOT gates in the encoder across bi-partitions of the qubits is found to be equal to the entanglement entropy across these partitions, demonstrating that the encoder is optimal in terms of CNOT gates across these partitions. Finally, connecting these ideas, we explicitly show that entanglement can be extracted from QRM codewords.
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.
Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.
Achieving useful fault-tolerant applications requires co-designing the quantum error-correcting code for footprint, logical instruction set, and hardware compatibility simultaneously. Recent breakthroughs in ultra-high-rate codes have the potential to reduce footprint of large scale quantum applications but their practical utility is limited without an addressable logical instruction set and efficient implementation on near term hardware. Reconfigurable neutral atom arrays, for example, support only hundreds of qubits with constrained reconfigurability from a few crossed acousto-optic deflectors (AOD). We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that allows such co-design by construction. GALA lifts over a product group G=Hk×CmG = H_k \times C_mG=Hk​×Cm​ (or Hk⋉CmkH_k \ltimes C_m^kHk​⋉Cmk​); the small non-abelian factor HkH_kHk​ supplies active orthogonality that improves qubit overhead, reaching 1/2 encoding rate with above-weight distance, while the large abelian factor CmC_mCm​ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Logical addressibility and hardware compatibility follow directly from construction, rather than verified post-hoc. The GALA family contains several known rate-1/2 Kasai codes of Ref.[arXiv:2601.08824, arXiv:2604.16209] while exposing simpler parameter bounds, logical operations, and ZX-dual variants with AOD-compatible fold-transversal Clifford gates. In particular, we find a compact self-dual [[132,30,12]][[132,30,12]][[132,30,12]] with 1.0−0.6+0.9×10−91.0^{+0.9}_{-0.6}\times 10^{-9}1.0−0.6+0.9​×10−9 per-round per-qubit logical error rate at 10−310^{-3}10−3 physical error rate, equipped with 3.1 ms syndrome-extraction cycles and an addressable logical instruction set; a ZX-dual, girth-6, rate-1/2 [[672,336,12]][[672,336,12]][[672,336,12]] with a 6.76 ms cycles; and barrier-breaking [[1752,880,14]][[1752,880,14]][[1752,880,14]] and [[2232,1120,16]][[2232,1120,16]][[2232,1120,16]] smaller than previously known hardware-compatible rate-1/2 codes.
The [[7,1,3]] Steane code and [[23,1,7]] quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates but require some other scheme for fault-tolerant TTT gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal TTT gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal TTT gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.