Strength-two signed trades define high-rate error-detecting qudit codes whose surviving third moment is a transversal logical cubic phase. The paper studies conic-matching examples through exact Hessian-rank and Pauli spectra. The ten-qudit example over the field of eleven elements is not Clifford-equivalent to a product across any bipartition. For the six-qudit example over the field of seven elements, exact enumerators prove a resource advantage over specified independent primitive-distillation menus for independent uniform Z noise, ideal Clifford operations, and input error between zero and one percent. A chordal restriction identifies the Clifford action of geometric sign choices. The scholarly supplement records finite computations and their trust boundaries; no Lean coverage is claimed.
Strength-two signed trades define high-rate error-detecting qudit codes whose surviving third moment is a transversal logical cubic phase. The paper studies conic-matching examples through exact Hessian-rank and Pauli spectra. The ten-qudit example over the field of eleven elements is not Clifford-equivalent to a product across any bipartition. For the six-qudit example over the field of seven elements, exact enumerators prove a resource advantage over specified independent primitive-distillation menus for independent uniform Z noise, ideal Clifford operations, and input error between zero and one percent. A chordal restriction identifies the Clifford action of geometric sign choices. The scholarly supplement records finite computations and their trust boundaries; no Lean coverage is claimed.
The isotropic 4D cubic toric code on T^4 with sides (3,3,3,3) — the [[486,6,9]] code — has a transversal CNOT and a transversal Hadamard. Its non-Clifford completion had not previously been costed. This note re-verifies the code from an independent construction (n=486, rank(Hx)=rank(Hz)=240, k=6, logical weight 9), classifies its transversal gates, and derives the distillation cost. Two no-gos, both computed rather than assumed No transversal phase gate exists at all. Over all 2^486 subsets R of the qubits, the only product of S gates preserving the stabiliser group is the empty product. This is a complete classification, not a sample: the admissible subsets form a GF(2) nullspace, computed to have dimension 0. No transversal T. The matrix [Hx ; L_X] is not triorthogonal (3132 of 54285 pairs and 2268 of 5935160 triples have odd overlap), verified against a positive control on [[15,1,3]] which passes the identical test. Distillation cost The k=6 structure turns an apparent 6x qubit penalty into six 15-to-1 factories running in lockstep for the price of one, because every operation in the protocol is a block-transversal CNOT between distinct blocks. With single-shot error correction (a 4D-specific property), the space-time cost per output magic state is 7.7x lower than a rotated-surface-code factory at the same distance, despite using 7.04x more physical qubits. Output error coefficient is exactly 35 (the A_3 coefficient of the [15,11,3] Hamming code); the break-even fixed point is 0.141480. Code switching, quantified and rejected The only natural same-lattice partner has d=L, so matching d=9 costs n=26244 qubits for k=4 — 54x the qubits for fewer logical qubits. Distillation wins decisively. Relation to prior work This supersedes the branch note describing S and T as "unresolved": S was already resolved by |Y⟩-state injection in the 2026-06-23 session records. The open question was the cost, which this supplies. Two results here independently reproduce earlier corpus numbers from a different derivation (exact Hamming weight enumerator rather than 2^15 enumeration plus fit). Erratum (2026-07-29): The rail-parallelism design result in §5.2 — six 15-to-1 factories running in lockstep on the six rails of this code, driven by block-transversal CNOT, for a 7.67x space-time advantage — is prior art and is withdrawn as a novel claim. Xu et al., arXiv:2407.18490 (2024) §V.C.2, state that k magic states can be distilled in parallel in [[n,k,d]] patches, that this “mostly requires inter-block Clifford operations”, and explicitly that “the same protocol can be applied to distilling |T⟩ states in parallel using, e.g. a 15-to-1 distillation protocol, with the same scaling of the space and the time overhead”; their Appendix A additionally constructs the parallel magic-state injection that §7 of this record identified as its own unconstructed assumption. Separately, this record omits a cost the lockstep layout necessarily incurs: six rails sharing one block cannot post-select independently without rail-level addressability, so a round yields six magic states with probability p^6 or none, a space-time multiplier of 1/p^5 (1.08 at eps_in = 1e-3, 2.12 at 1e-2). The exact distillation results — coefficient 35, eps* = 0.141480, the depth-14 Koenig schedule — and both transversal no-gos are unaffected and are re-verified in the erratum. See 10.5281/zenodo.21697731.
We study the algebraic compatibility between quantum homomorphic encryption and stabilizer quantum error correction in an honest-but-curious model, restricting the encryption to the transversal block-Pauli mask (XaZb)⊗n controlled by two secret key bits a and b. We prove a necessary and sufficient criterion for an [[n,1,d]] stabilizer code to remain compatible with this mask, namely that the transversal X and Z operators commute with every stabilizer generator, and we specialize it to a classical-code criterion for Calderbank--Shor--Steane codes. Combining code-space preservation with the logical action of the mask, every stabilizer code falls, with respect to each of the two transversal mask operators, into exactly one of three classes: incompatible, compatible but trivially acting, or compatible with a nontrivial logical Pauli action; the two operators need not fall into the same class. We verify this classification by exact symplectic computation for rotated and planar surface codes, toric codes, triangular and toric color codes, the [[15,1,3]] code and several small codes, and find that the class is fixed by the lattice size and the boundary presentation rather than by the topological phase: the surface-code family realizes all three classes, whereas color codes realize only the two compatible ones. For encrypted computation we show that the Shor code does not implement the logical T gate by a transversal T⊗9, whereas suitable triorthogonal codes do so up to a Clifford correction. We formulate logical-Pauli masking as a code-independent algebraic template. Syndrome extraction and recovery are key-independent whenever the mask preserves the code space, so that the client only needs the block size and the class of the code to decrypt.
We show that every intermediate state of a broad class of non-Clifford quantum error-correction circuits is a third-order phase-polynomial state, even under stochastic Pauli noise. The class includes magic state distillation and cultivation, code switching and gauge fixing, transversal non-Clifford gates with syndrome extraction, and injection of diagonal magic states. Third-order phase-polynomial states strictly generalize stabilizer states, and we prove that they are exactly the stabilizer states of the diagonal-Clifford-and-Pauli (DCP) stabilizer formalism that we introduce. We characterize these states and show that their representations can be updated in polynomial time. This yields an exact polynomial-time simulation algorithm for these circuits and, more broadly, a framework for reasoning about their internal states and about magic states in general. We provide an open-source implementation, \texttt{merlin}, and benchmark it against existing non-Clifford simulators on distillation, cultivation, and code switching circuits. The benchmarks demonstrate improved runtime and memory scaling as the number of logical outputs grows in Bravyi-Haah distillation, and simulation of a code switching circuit beyond the reach of all other tested simulators.
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all [[n,1]]3 qutrit stabilizer codes with a complete set of transversal Clifford gates for n≤23, and all [[n,1]]3 stabilizer codes with a transversal H2 gate with n≤9 qudits. For n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.
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.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4). At p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.