We construct a two-parameter family of single-error-correcting seven-ququart codes with transversal $\tilde{A_7}$ symmetry, realizing the finite component of a two-qubit super-golden gate set. These $((7,4,3))_4$ codes encode two logical qubits and support non-Clifford operations by applying the same gate to each physical ququart.
Our method constructs invariant parent states rather than searching directly for a codespace. For $n$ systems of local dimension $q$, suppose the local symmetry is a unitary $t$-group and the permutation symmetry is $t$-transitive. Maximal mixing of all marginals on at most $t$ sites is then equivalent to linear equations in Schur--Weyl sector weights. Every feasible solution yields parents of pure distance at least $t+1$; puncturing one site produces a $q$-dimensional code of distance at least $t$ with the prescribed local symmetry acting transversally.
Steane and the nonstabilizer Fake Steane code reveal why the construction leaves continuous freedom. They share the same prescribed transversal Clifford and permutation symmetries, and their eight-qubit purifications lie on a circle: error correction fixes two invariant-sector norms but leaves the relative phase free. Keeping the same permutation group and replacing Clifford symmetry by $\tilde{A_7}$ $\leq SU(4)$ gives three fixed sector norms and two free relative phases, producing the new ququart family.
We construct a two-parameter family of single-error-correcting seven-ququart codes with transversal A7~ symmetry, realizing the finite component of a two-qubit super-golden gate set. These ((7,4,3))4 codes encode two logical qubits and support non-Clifford operations by applying the same gate to each physical ququart.
Our method constructs invariant parent states rather than searching directly for a codespace. For n systems of local dimension q, suppose the local symmetry is a unitary t-group and the permutation symmetry is t-transitive. Maximal mixing of all marginals on at most t sites is then equivalent to linear equations in Schur--Weyl sector weights. Every feasible solution yields parents of pure distance at least t+1; puncturing one site produces a q-dimensional code of distance at least t with the prescribed local symmetry acting transversally.
Steane and the nonstabilizer Fake Steane code reveal why the construction leaves continuous freedom. They share the same prescribed transversal Clifford and permutation symmetries, and their eight-qubit purifications lie on a circle: error correction fixes two invariant-sector norms but leaves the relative phase free. Keeping the same permutation group and replacing Clifford symmetry by A7~≤SU(4) gives three fixed sector norms and two free relative phases, producing the new ququart family.
Code switching moves an encoded qubit between two codes whose transversal gates together are universal, but the switching operation itself is usually not transversal and can spread a single fault into an uncorrectable error. We study switching between two permutation-invariant codes on the same qubits, one with transversal Clifford gates and one with transversal gates from the non-Clifford binary icosahedral group. Both are compatible with transversal X, Z, and F=HS†, and the encoders with this property form a small linear space. We show that its codes of distance at least three are cut out by at most two quadratic equations, and that all of them have the same error geometry in the sense that their single-qubit error states have the same inner products. It follows that a switching unitary commuting with qubit permutations and with the shared transversal gates at every instant, and tolerating one single-qubit fault at an unknown time, exists if and only if the two codes are joined by a continuous path of distance-three codes. Such a switch can be made error transparent, so that a fault at any moment reaches the target code as the same single-qubit error. For real encoders, the distance-three codes are the solutions of a single quadratic equation, and on thirteen qubits, they form an explicit circle through two Clifford codes and an icosahedral code. We construct the switching unitary along this circle as a collective operator, leaving its synthesis into physical control open, and verify that its recovery infidelity per fault is at machine precision, whereas two natural alternatives leave average infidelities of 0.13 and 0.85. A survey over odd qubit numbers up to 29 lists where such switches exist.
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→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.
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.
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 gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]] where β≈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, SHS=X, and 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]], 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 gate can be executed on those three logical qubits via a depth-one physical circuit of 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 gates and has inverse-polylogarithmic rate and relative distance.