We establish that transversal diagonal gates on exact qubit codes of distance at least two can be exhaustively classified within a finite search space. Although code-preserving physical gates can have continuous phases, any projective logical action induced by a transversal diagonal gate can always be realized by a physical gate of finite order. Crucially, we prove that this physical order has an upper bound determined solely by the code parameters. This order bound enables the subset-sum-linear-programming (SSLP) framework introduced in arXiv:2504.20847 to exhaustively enumerate all candidate discrete symmetries, after which the Knill-Laflamme equations determine the remaining continuous code parameters. Applying this method to the nontrivial diagonal sector of exact $((6,2,3))$ codes, SSLP yields $66$ candidate symmetry classes, of which exactly $15$ are realizable, supporting full projective diagonal logical cyclic groups of orders $2$, $3$, and $5$. In particular, for order $5$, the code is unique up to local unitaries and qubit permutations, equivalent to the code constructed in arXiv:2504.20847. This code also attains the minimum length $n=6$ among exact $((n, 2, \ge 3))$ codes whose full projective transversal group cannot be conjugated into the projective Clifford group.
We establish that transversal diagonal gates on exact qubit codes of distance at least two can be exhaustively classified within a finite search space. Although code-preserving physical gates can have continuous phases, any projective logical action induced by a transversal diagonal gate can always be realized by a physical gate of finite order. Crucially, we prove that this physical order has an upper bound determined solely by the code parameters. This order bound enables the subset-sum-linear-programming (SSLP) framework introduced in arXiv:2504.20847 to exhaustively enumerate all candidate discrete symmetries, after which the Knill-Laflamme equations determine the remaining continuous code parameters. Applying this method to the nontrivial diagonal sector of exact ((6,2,3)) codes, SSLP yields 66 candidate symmetry classes, of which exactly 15 are realizable, supporting full projective diagonal logical cyclic groups of orders 2, 3, and 5. In particular, for order 5, the code is unique up to local unitaries and qubit permutations, equivalent to the code constructed in arXiv:2504.20847. This code also attains the minimum length n=6 among exact ((n,2,≥3)) codes whose full projective transversal group cannot be conjugated into the projective Clifford group.
Universal fault-tolerant quantum computing combines versatile but expensive operations with specialized but cheap ones. Its efficiency depends on how much computation can be pushed onto the cheap operations and on the size of the code needed to do so. Automorphism gates provide such cheap operations using only physical single-qubit Clifford gates and qubit permutations. Yet no general theory characterizes their maximum logical power or the minimum code size needed to attain it. We develop such a theory. For stabilizer codes encoding k≥3 logical qubits, we show that the largest logical group attainable by automorphisms is generated by all addressable S and CX gates, and we construct codes attaining it. While this group contains exponentially fewer gates than the full Clifford group, adding one suitable non-Clifford gate yields universality. We further classify the largest logical groups attainable using qubit permutations, physical single-qubit Cliffords, or both across general stabilizer and CSS codes, and derive refined bounds for self-dual CSS subclasses. Achieving the maximum-size logical group through automorphisms requires n=Θ(2k) physical qubits. By contrast, all addressable diagonal Clifford gates, generated by S and CZ, require only n=Θ(k2) physical qubits when implemented using physical single-qubit Cliffords alone. Both bounds are tight. This polynomial qubit cost extends beyond Cliffords to all addressable diagonal gates at any fixed level of the Clifford hierarchy, using physical single-qubit diagonal gates. Thus, for full addressability, the sharpest physical-qubit cost divide lies between diagonal and CX-type gates, not between Clifford and non-Clifford gates.
Transversal gates play a crucial role in suppressing error propagation in fault-tolerant quantum computation, yet they are intrinsically constrained: any nontrivial code encoding a single logical qubit admits only a finite subgroup of SU(2) as its transversal operations. We introduce a systematic framework for searching codes with specified transversal groups by parametrizing their logical subspaces on the Stiefel manifold and minimizing a composite loss that enforces both the Knill-Laflamme conditions and a target transversal-group structure. Applying this method, we uncover a new ((6,2,3)) code admitting a transversal Z(52π) gate (transversal group C10), the smallest known distance 3 code supporting non-Clifford transversal gates, as well as several new ((7,2,3)) codes realizing the binary icosahedral group 2I. We further propose the \emph{Subset-Sum-Linear-Programming} (SS-LP) construction for codes with transversal \emph{diagonal} gates, which dramatically shrinks the search space by reducing to integer partitions subject to linear constraints. In a more constrained form, the method also applies directly to the binary-dihedral groups BD2m. Specializing to n=7, the SS-LP method yields codes for all BD2m with 2m≤36, including the first ((7,2,3)) examples supporting transversal T gate (BD16) and T gate (BD32), improving on the previous smallest examples ((11,2,3)) and ((19,2,3)). Extending the SS-LP approach to ((8,2,3)), we construct new codes for 2m>36, including one supporting a transversal T1/4 gate (BD64). These results reveal a far richer landscape of nonadditive codes than previously recognized and underscore a deeper connection between quantum error correction and the algebraic constraints on transversal gate groups.
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice.
We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group.
For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.
This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical diagonal gates on the underlying qubits that compose the code are restricted to have entries of the form e iπc/2 k along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. As such, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Moreover, we show that for a given stabilizer code, the two-qubit diagonal transversal gates must belong to the same level of Clifford hierarchy as the single-qubit diagonal transversal gates available for the given code. We use this result to prove a conjecture about arbitrary transversal gates made by Zeng et al. in 2007.
We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical T/T† gates and whose distance is at least 3. These codes start with 8-divisible X-stabilizers defined by projective geometry, which guarantees dZ≥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 dZ=3 or 4, and illustrate the method with the [[48,3,3]] code Q48. 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.
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.