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:2610.09341

Building codes with transversal CCZ using projective geometry and SAT solvers

Bohan Lu, Kenneth R. Brown

quant-phcs.ITSubmitted Oct 7, 2026• Announced Oct 8, 2026• 0 citations• Revised Oct 16, 2026 (v2)
v1 · Oct 7, 2026v2 · Oct 16, 2026
arXivHTMLPDF

Abstract

We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical $T/T^\dagger$ gates and whose distance is at least 3. These codes start with 8-divisible $X$-stabilizers defined by projective geometry, which guarantees $d_Z \geq 3$. A SAT solver then finds three $X$-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from $n=48$ to $n=496$, with $d_Z=3$ or 4, and illustrate the method with the [[48,3,3]] code $Q_{48}$. Additionally, we prove that no CSS code with three logical qubits, $Z$-distance at least 3, and a quasi-transversal CCZ gate (transversal $T$ with a diagonal Clifford correction) exists below $n=39$. The [[47,3,3]] code of Jacinto et al. attains the minimum length in this class, as established by the classification of Wills, Jain, and Singh.

Comments
40 pages, 3 figures. Code and data: https://github.com/jerrylvx/gate2code-public

Related papers

Good Binary Quantum Codes with Transversal CCZ Gate

Quynh T. Nguyen

We give an asymptotically good family of quantum CSS codes on qubits with a transversal CCZ gate, meaning that the parallel logical CCZ on all logical qubits is performed by parallel physical CCZs on (a subset of) physical qubits. The construction is based on the observation that any classical code satisfying a multiplication property can be used to construct a quantum CSS code with transversal (qudit) CCZ. To obtain a constant-rate and linear-distance family, we then instantiate this construction with a classical good family of algebraic-geometry codes on a non-binary, but constant-sized, alphabet. Finally, we use a technique from the arithmetic secret sharing literature to reduce the alphabet to binary. As a corollary, the constructed code family provides a magic state distillation scheme with constant space overhead.

arXiv:2408.10140•ACM Symposium on Theory of Computing (STOC)•Jun 15, 2025•8 citations

Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS Codes

K. Sai Mineesh Reddy, Navin Kashyap

Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled $Z$-rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled $Z$-rotations.

quant-pharXiv:2608.19094•Aug 20, 2026•0 citations

Classical Coding Problem from Transversal T Gates

Narayanan Rengaswamy, Robert Calderbank, Michael Newman +1 more

Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T†gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle Z-rotations in https://arxiv.org/abs/1910.09333.

arXiv:2001.04887•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2020•17 citations

Transversal AND in Quantum Codes

Christine Li, Lia Yeh

The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to $n$ qubits with T-count $3n-3$. Our main result is the construction of a novel qutrit $\mathopen{[\![} 6,2,2 \mathclose{]\!]}$ quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a $\mathopen{[\![} 48,2,4 \mathclose{]\!]}$ code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.

quant-pharXiv:2603.04548•Lecture notes in computer science•Mar 6, 2026•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

Asymptotically Good Quantum Codes with Addressable Transversal T Gates

Tongyin Lin, Bujiao Wu, Bin Cheng

Designing quantum codes with both transversal non-Clifford gates and good error-correcting parameters is an important goal in fault-tolerant quantum computation. Here, we construct an explicit family of binary CSS codes with constant rate and linear distance that admit fully addressable transversal $T$ gates. Specifically, any prescribed tensor product of logical powers of $T$ is implemented by a tensor product of physical powers of $T$, without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal $T$ gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.

quant-pharXiv:2609.35129•Sep 29, 2026•0 citations
We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical T/T†T/T^\daggerT/T† gates and whose distance is at least 3. These codes start with 8-divisible XXX-stabilizers defined by projective geometry, which guarantees dZ≥3d_Z \geq 3dZ​≥3. A SAT solver then finds three XXX-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from n=48n=48n=48 to n=496n=496n=496, with dZ=3d_Z=3dZ​=3 or 4, and illustrate the method with the [[48,3,3]] code Q48Q_{48}Q48​. Additionally, we prove that no CSS code with three logical qubits, ZZZ-distance at least 3, and a quasi-transversal CCZ gate (transversal TTT with a diagonal Clifford correction) exists below n=39n=39n=39. The [[47,3,3]] code of Jacinto et al. attains the minimum length in this class, as established by the classification of Wills, Jain, and Singh.
We give an asymptotically good family of quantum CSS codes on qubits with a transversal CCZ gate, meaning that the parallel logical CCZ on all logical qubits is performed by parallel physical CCZs on (a subset of) physical qubits. The construction is based on the observation that any classical code satisfying a multiplication property can be used to construct a quantum CSS code with transversal (qudit) CCZ. To obtain a constant-rate and linear-distance family, we then instantiate this construction with a classical good family of algebraic-geometry codes on a non-binary, but constant-sized, alphabet. Finally, we use a technique from the arithmetic secret sharing literature to reduce the alphabet to binary. As a corollary, the constructed code family provides a magic state distillation scheme with constant space overhead.
Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆C1​, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled ZZZ-rotations.
Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T†gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle Z-rotations in https://arxiv.org/abs/1910.09333.
The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to nnn qubits with T-count 3n−33n-33n−3. Our main result is the construction of a novel qutrit [ ⁣[6,2,2] ⁣]\mathopen{[\![} 6,2,2 \mathclose{]\!]}[[6,2,2]] quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a [ ⁣[48,2,4] ⁣]\mathopen{[\![} 48,2,4 \mathclose{]\!]}[[48,2,4]] code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.
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.
Designing quantum codes with both transversal non-Clifford gates and good error-correcting parameters is an important goal in fault-tolerant quantum computation. Here, we construct an explicit family of binary CSS codes with constant rate and linear distance that admit fully addressable transversal TTT gates. Specifically, any prescribed tensor product of logical powers of TTT is implemented by a tensor product of physical powers of TTT, without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal TTT gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.