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arXiv:2410.07595 · IEEE Transactions on Information Theory

Geometric Structure and Transversal Logic of Quantum Reed–Muller Codes

Alexander Barg, Nolan J. Coble, Dominik Hangleiter, Christopher Kang

Published Jul 25, 2025• 6 citations
arXivHTMLPDF

Abstract

Designing efficient and noise-tolerant quantum computation protocols generally begins with an understanding of quantum error-correcting codes and their native logical operations. The simplest class of native operations are transversal gates, which are naturally fault-tolerant. In this paper, we aim to characterize the transversal gates of quantum Reed–Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. We start our work by establishing a new geometric characterization of quantum RM codes via the Boolean hypercube and its associated subcube complex. More specifically, a set of stabilizer generators for a quantum RM code can be described via transversalXandZoperators acting on subcubes of particular dimensions. This characterization leads us to definesubcube operatorscomposed of single-qubit$\pi /2^{k}~Z$-rotations that act on subcubes of given dimensions. We first characterize the action of subcube operators on the code space: depending on the dimension of the subcube, these operators either (1) act as a logical identity on the code space, (2) implement non-trivial logic, or (3) rotate a state away from the code space. Second, and more remarkably, we uncover that the logic implemented by these operators corresponds to circuits of multi-controlled-Zgates that have an explicit and simple combinatorial description. Overall, this suite of results yields a comprehensive understanding of a class of natural transversal operators for quantum RM codes.

DOI
10.1109/tit.2025.3592631

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Designing efficient and noise-tolerant quantum computation protocols generally begins with an understanding of quantum error-correcting codes and their native logical operations. The simplest class of native operations are transversal gates, which are naturally fault-tolerant. In this paper, we aim to characterize the transversal gates of quantum Reed–Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. We start our work by establishing a new geometric characterization of quantum RM codes via the Boolean hypercube and its associated subcube complex. More specifically, a set of stabilizer generators for a quantum RM code can be described via transversalXandZoperators acting on subcubes of particular dimensions. This characterization leads us to definesubcube operatorscomposed of single-qubitπ/2k Z\pi /2^{k}~Zπ/2k Z-rotations that act on subcubes of given dimensions. We first characterize the action of subcube operators on the code space: depending on the dimension of the subcube, these operators either (1) act as a logical identity on the code space, (2) implement non-trivial logic, or (3) rotate a state away from the code space. Second, and more remarkably, we uncover that the logic implemented by these operators corresponds to circuits of multi-controlled-Zgates that have an explicit and simple combinatorial description. Overall, this suite of results yields a comprehensive understanding of a class of natural transversal operators for quantum RM codes.
Reed-Muller (RM) codes form a classic family studied for its interesting algebraic and combinatorial properties as well as from the perspective of information transmission. They achieve Shannon capacity of the basic binary channel models such as channels with independent erasures or flip errors. They have numerous applications to the theory of computational complexity and have well-understood local testability. They also give rise to a family of quantum codes admitting transversal logical operators in increasing levels of the Clifford hierarchy, a property of codes considered necessary for future fault-tolerant quantum computations. In this dissertation, we show that, despite decades of research surrounding RM codes, new directions sprouting from their simple definition continue to be possible. We will begin by considering an alternate description of RM codes in terms of faces of the Boolean hypercube, and will see that by replacing the hypercube with a more general algebraic object we obtain a large class of previously undiscovered classical error-correcting codes sharing many structural properties with RM codes. These classical codes directly lead to an even larger class of new quantum error-correcting codes with explicitly determined parameters; we will demonstrate that these quantum codes admit many natural transversal logical gates. We finish by specializing to the class of quantum Reed-Muller codes and provide a complete characterization of the logic implemented by these transversal operators, yielding a broad generalization of prior work on the subject.
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.
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.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉ for distance-ddd codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed TTT gate on the [[16,4,4]][[16,4,4]][[16,4,4]] BB code, TTT-type rotations on the [[64,8,8]][[64,8,8]][[64,8,8]] BB code, and CCZ-, CS-, and TTT-type operations on the [[36,6,6]][[36,6,6]][[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields Cr{}^rrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
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.
In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal T and T†gates preserve the codespace. Our Heisenberg perspective reduces this question to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries of this result: (a) For any non-degenerate [n, k, d] stabilizer code supporting a physical transversal T (which might not be logical T), there exists an [n, k, d]] CSS code with the same property; (b) Triorthogonal codes form the most general family of CSS codes that realize logical transversal T via physical transversal T; (c) Triorthogonality is necessary for physical transversal T on a CSS code to realize the logical identity. The main tool we use is a recent characterization of a particular family of diagonal gates in the Clifford hierarchy that are efficiently described by symmetric matrices over rings of integers [N. Rengaswamy et al., Phys. Rev. A 100, 022304]. We refer to these operations as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing stabilizer code constructions that realize logical gates via transversal T. We provide several examples of codes and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal π/2ℓZ-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal π/2ℓZ-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal 1- or 2-local diagonal gate.