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arXiv:2609.27801

Improved Transversal Non-Clifford Gates from Cup Products

Louis Golowich, Itzhak Tamo, Guanyu Zhu

quant-phcs.ITSubmitted Aug 17, 2026• Announced Aug 19, 2026• 0 citations
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Abstract

It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford $C^{r-1}Z$ gate, for every constant $r\geq 3$. In particular, we obtain length-$n$ quantum LDPC codes (with constant-weight stabilizers) of polynomial distance $d\geq n^{(1-ε)/r}$ supporting transversal $C^{r-1}Z$ gates on a close-to-linear number $k\geq n^{1-ε}$ of disjoint tuples of logical qubits, for arbitrarily small $ε>0$. Our construction is the first with constant-weight stabilizers that obtains $dk\gg n$, and as a consequence achieves arbitrarily small magic state overhead exponent $γ=\log(n/k)/\log(d)>0$. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many $k=Ω(n)$ logical $C^{r-1}Z$ gates, though with stabilizer weight and physical circuit depth $n^ε$. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal $C^{r-1}Z$. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.

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It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1ZC^{r-1}ZCr−1Z gate, for every constant r≥3r\geq 3r≥3. In particular, we obtain length-nnn quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥n(1−ε)/rd\geq n^{(1-ε)/r}d≥n(1−ε)/r supporting transversal Cr−1ZC^{r-1}ZCr−1Z gates on a close-to-linear number k≥n1−εk\geq n^{1-ε}k≥n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0ε>0ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫ndk\gg ndk≫n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log⁡(n/k)/log⁡(d)>0γ=\log(n/k)/\log(d)>0γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n)k=Ω(n)k=Ω(n) logical Cr−1ZC^{r-1}ZCr−1Z gates, though with stabilizer weight and physical circuit depth nεn^εnε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits.
To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1ZC^{r-1}ZCr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
For every integer r≥ 2 and every є>0, we construct an explicit infinite family of quantum LDPC codes supporting a transversal Cr−1Z gate with length N, dimension K≥ N1−є, distance D≥ N1/r/poly(logN), and stabilizer weight w≤ poly(logN). The previous state of the art construction (in most parameter regimes) was the r-dimensional color code, which has only constant dimension K=O(1), and otherwise has the same parameters up to polylogarithmic factors. Our construction provides the first known codes with low-weight stabilizers that are capable of magic state distillation with arbitrarily small yield parameter γ=log(N/K)/log(D)>0. A classical analogue of transversal Cr−1Z gates is given by the multiplication property, which requires component-wise products of classical codewords to belong to another similar code. As a byproduct of our techniques, we also obtain a new construction of classical locally testable codes with such a multiplication property. We construct our codes as products of chain complexes associated to classical LDPC codes, which in turn we obtain by imposing local Reed-Solomon codes on a specific spectral expander that we construct. We prove that our codes support the desired transversal Cr−1Z gates by using the multiplication property to combine local circuits based on the topological structure.
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.
Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.
It is a major challenge to construct good quantum codes supporting fault-tolerant (e.g. transversal) non-Clifford gates with low-weight parity-check measurements. In this paper, we construct the first known quantum codes with linear dimension and distance supporting transversal non-Clifford gates that have sublinear locality (i.e. parity-check weight). Specifically, we construct codes with transversal CCZ gates that have dimension and distance growing linearly in the block length, and have locality growing as the square root of the block length. We furthermore design an efficient decoding algorithm for these codes. The alphabet size of these codes grows as the square root of the block length, but it can be reduced to a constant (e.g. binary) while incurring a polylogarithmic loss in other parameters. We also show how to decrease the locality to the cube root of the block length, albeit with a larger alphabet size and slightly lower distance.We construct these codes as products of classical codes with appropriate algebraic structure. While our quantum codes are subsystem codes with non-commuting gauge operators, we show they nevertheless permit error correction from noisy syndrome measurements.As byproducts, we prove multiple technical results of independent interest. In particular, our efficient decoder can be viewed as a new multivariate generalization of Prony’s method for reconstructing a function from partial access to its Fourier transform. Meanwhile, our distance analysis involves new connections to the classical study of maximally recoverable codes. Our results on product codes also resolve a conjecture of Bravyi & Hastings (2014) in the large-alphabet regime, by providing a new construction of quantum codes with linear dimension and distance and small polynomial locality.
A major goal in quantum computing is to build a fault-tolerant quantum computer. One approach involves quantum low-density parity-check (qLDPC) codes that support transversal non-Clifford gates. In this work, we provide a large family of such codes. The key insight is to interpret the logical operators of qLDPC codes as geometric surfaces and use the intersection number of these surfaces to define the non-Clifford operation. At a more abstract level, this construction is based on defining the cup product on the chain complex induced from a sheaf.
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.