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arXiv:2502.01864

Quantum Codes with Addressable and Transversal Non-Clifford Gates

Zhiyang He, Vinod Vaikuntanathan, Adam Wills, Rachel Zhang

Published Feb 3, 2025• 0 citations
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Abstract

The development of quantum codes with good error correction parameters and useful sets of transversal gates is a problem of major interest in quantum error-correction. Abundant prior works have studied transversal gates which are restricted to acting on all logical qubits simultaneously. In this work, we study codes that support transversal gates which induce $\textit{addressable}$ logical gates, i.e., the logical gates act on logical qubits of our choice. As we consider scaling to high-rate codes, the study and design of low-overhead, addressable logical operations presents an important problem for both theoretical and practical purposes. Our primary result is the construction of an explicit qubit code for which $\textit{any}$ triple of logical qubits across one, two, or three codeblocks can be addressed with a logical $\mathsf{CCZ}$ gate via a depth-one circuit of physical $\mathsf{CCZ}$ gates, and whose parameters are asymptotically good, up to polylogarithmic factors. The result naturally generalizes to other gates including the $\mathsf{C}^{\ell} Z$ gates for $\ell \neq 2$. Going beyond this, we develop a formalism for constructing quantum codes with $\textit{addressable and transversal}$ gates. Our framework, called $\textit{addressable orthogonality}$, encompasses the original triorthogonality framework of Bravyi and Haah (Phys. Rev. A 2012), and extends this and other frameworks to study addressable gates. We demonstrate the power of this framework with the construction of an asymptotically good qubit code for which $\textit{pre-designed}$, pairwise disjoint triples of logical qubits within a single codeblock may be addressed with a logical $\mathsf{CCZ}$ gate via a physical depth-one circuit of $\mathsf{Z}$, $\mathsf{CZ}$ and $\mathsf{CCZ}$ gates. In an appendix, we show that our framework extends to addressable and transversal $T$ gates, up to Clifford corrections.

DOI
10.48550/arxiv.2502.01864

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The development of quantum codes with good error correction parameters and useful sets of transversal gates is a problem of major interest in quantum error-correction. Abundant prior works have studied transversal gates which are restricted to acting on all logical qubits simultaneously. In this work, we study codes that support transversal gates which induce addressable\textit{addressable}addressable logical gates, i.e., the logical gates act on logical qubits of our choice. As we consider scaling to high-rate codes, the study and design of low-overhead, addressable logical operations presents an important problem for both theoretical and practical purposes. Our primary result is the construction of an explicit qubit code for which any\textit{any}any triple of logical qubits across one, two, or three codeblocks can be addressed with a logical CCZ\mathsf{CCZ}CCZ gate via a depth-one circuit of physical CCZ\mathsf{CCZ}CCZ gates, and whose parameters are asymptotically good, up to polylogarithmic factors. The result naturally generalizes to other gates including the CℓZ\mathsf{C}^{\ell} ZCℓZ gates for ℓ≠2\ell \neq 2ℓ=2. Going beyond this, we develop a formalism for constructing quantum codes with addressable and transversal\textit{addressable and transversal}addressable and transversal gates. Our framework, called addressable orthogonality\textit{addressable orthogonality}addressable orthogonality, encompasses the original triorthogonality framework of Bravyi and Haah (Phys. Rev. A 2012), and extends this and other frameworks to study addressable gates. We demonstrate the power of this framework with the construction of an asymptotically good qubit code for which pre-designed\textit{pre-designed}pre-designed, pairwise disjoint triples of logical qubits within a single codeblock may be addressed with a logical CCZ\mathsf{CCZ}CCZ gate via a physical depth-one circuit of Z\mathsf{Z}Z, CZ\mathsf{CZ}CZ and CCZ\mathsf{CCZ}CCZ gates. In an appendix, we show that our framework extends to addressable and transversal TTT gates, up to Clifford corrections.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S‾\mathrm{\overline{S}}S gate in the rotated surface code and a depth-one intra-block CZ‾\mathrm{\overline{CZ}}CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3L\ge3L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]][[n,\sqrt{n},Θ({n^β})]][[n,n​,Θ(nβ)]] where β≈0.2823β\approx 0.2823β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S‾\mathrm{\overline{S}}S, SHS‾=X\mathrm{\overline{SHS}} = \sqrt{X}SHS=X​, and CZ‾\mathrm{\overline{CZ}}CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]][[n_0, 2, d_0]][[n0​,2,d0​]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
Constructing quantum codes with good parameters and useful transversal gates is a central problem in quantum error correction. In this paper, we continue our work in arXiv:2502.01864 and construct the first family of asymptotically good quantum codes (over qubits) supporting transversally addressable non-Clifford gates. More precisely, given any three logical qubits across one, two, or three codeblocks, the logical CCZ\mathsf{CCZ}CCZ gate can be executed on those three logical qubits via a depth-one physical circuit of CCZ\mathsf{CCZ}CCZ gates. This construction is based on the transitive, iso-orthogonal algebraic geometry codes constructed by Stichtenoth (IEEE Trans. Inf. Theory, 2006). This improves upon our construction from arXiv:2502.01864, which also supports transversally addressable CCZ\mathsf{CCZ}CCZ gates and has inverse-polylogarithmic rate and relative distance.
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.
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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.
Transversal gates are the simplest form of fault-tolerant gates and are relatively easy to implement in practice. Yet designing codes that support useful transversal operations -- especially non-Clifford or addressable gates -- remains difficult within the stabilizer formalism or CSS constructions alone. We show that these limitations can be overcome using tensor-network frameworks such as the quantum lego formalism, where transversal gates naturally appear as global or localized symmetries. Within the quantum lego formalism, small codes carrying desirable symmetries can be "glued" into larger ones, with operator-flow rules guiding how logical symmetries are preserved. This approach enables the systematic construction of codes with addressable transversal single- and multi-qubit gates targeting specific logical qubits regardless of whether the gate is Clifford or not. As a proof of principle, we build new finite-rate code families that support strongly transversal TTT, CCZCCZCCZ, SHSHSH, and Gottesman's K3K_3K3​ gates, structures that are challenging to realize with conventional methods. We further construct holographic and fractal-like codes that admit addressable transversal inter-, meso-, and intra-block TTT, CSCSCS, and CℓZC^\ell ZCℓZ gates. As a corollary, we demonstrate that the heterogeneous holographic Steane-Reed-Muller black hole code also supports fully addressable transversal inter- and intra-block CZCZCZ gates, significantly lowering the overhead for universal fault-tolerant computation.