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 $F = HS^\dagger$, 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.
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.
Code-switching offers a route to universal, fault-tolerant quantum computation by circumventing the limitation implied by the Eastin-Knill theorem against a universal transversal gate set within a single quantum code. Here, we present a fault-tolerant code-switching protocol between two versions of the [[8,3,2]] code. One version supports weakly fault-tolerant single-qubit Clifford gates, while the other supports a logical CCZ gate via transversal T/T† together with logical CZ, CNOT, and SWAP gates. Because both codes have distance 2, the protocol operates in a postselected, error-detecting regime: single faults lead to detectable outcomes, and accepted runs exhibit quadratic suppression of logical error rates. This yields a universal scheme for postselected fault-tolerant computation. We validate the protocol numerically through simulations of state preparation, code switching, and a three-logical-qubit implementation of Grover's search.
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.
Transversal gates on quantum error correction codes have been a promising approach for fault-tolerant quantum computing, but are limited by the Eastin-Knill no-go theorem. Existing solutions such as gate teleportation and magic state distillation are resource-intensive. We present a measurement-free code-switching protocol for universal quantum computation, switching between a stabilizer code for transversal Clifford gates and a permutation-invariant (PI) code for transversal non-Clifford gates that are logical Z rotations for any rational multiple of π . The novel non-Clifford gates enabled by this code-switching protocol provide for a lower gate count implementation of a universal gate set relative to the Clifford + T gate set. To achieve this, we present a protocol for performing controlled- gates between the codes using near-term quantum control operations that employ a catalytic bosonic mode. We also present a new class of PI codes with tunable code distance, supporting transversal non-Clifford gates, and demonstrate their reduced gate count overhead relative to a comparable stabilizer code to stabilizer code-switching scheme.
A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical T gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer Z-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical Z-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of Z-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of r-orthogonal quantum codes, r≥2, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical Z-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of Z-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.
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.