Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes

The paper in detail

Transversal gates act qubit by qubit and so do not spread faults, but by the Eastin–Knill theorem they cannot be universal on a code that detects single-qubit errors. Stabilizer codes have finite Pauli descriptions that make classification tractable. Nonadditive codes come in continuous families, and a transversal symmetry adds continuous phases of its own. The SSLP method fixes a finite-order diagonal gate \(D(a;q)=\bigotimes_j\operatorname{diag}(1,\omega_q^{a_j})\), \(\omega_q=e^{2\pi i/q}\). Basis strings then fall into sectors \(S_b=\{x:a\cdot x\equiv b\pmod q\}\), and the logical eigenvectors must live in distinct sectors. The diagonal Knill–Laflamme (KL) conditions become a linear program on squared amplitudes, and the remaining conditions an exact algebraic completion.

Two gaps stood between that method and completeness. A code-preserving diagonal gate can have irrational phases, and so infinite physical order. And the least physical order realizing a logical action can exceed its logical order. Theorem III.2 closes the first gap using only the single-qubit conditions \(PZ_jP=\lambda_jP\). Preservation is an integer-linear condition on the phases. Directions in its real kernel act as logical scalars, and a rational section lemma replaces any solution by one with a fixed denominator. The diagonal logical group is therefore finite abelian, with finite-order representatives satisfying \(m\mid q\). A seven-qubit example shows the second gap is real: a logical action of order 6 whose physical representatives all have order divisible by 12.

Theorem III.4 shows that every such code has an SSLP normal form, and that every exact completion of SSLP data is a code. Theorem III.9 bounds the needed order. All logical eigenspaces share the same first moment \(\mu\), so the phases solve a binary linear system, and the Smith normal form bounds its denominators: \(q\le\beta_{n+\ell-1}\), with \(\beta_5,\ldots,\beta_9=5,9,18,42,120\). At distance three with one logical qubit, the second moments force the two supports to share an affine span, sharpening the bound to \(q\le\beta_n\), which is 9 for six qubits.

For exact ((6,2,3)) codes the search gives 66 candidate classes. Exact no-go proofs exclude 51, and explicit codes realize 15: 11 of logical order 2, 3 of order 3, and 1 of order 5. In the order-five code, \(D(a^\star;5)\) with \(a^\star=(1,1,1,1,2,3)\) acts as \(\operatorname{diag}(1,\zeta^4)\), \(\zeta=e^{2\pi i/5}\). Every code with this symmetry is equivalent to it, and since it has \(A_1=0\), no shorter code can carry an order-five transversal action. Exact weight enumerators are determined for 13 of the 15 classes. The case analysis was produced by agents using TeXRA with GPT-5.6 and checked in Lean.

Transferable methods

1. Strip continuous phases that act only as logical scalars

   Single-qubit error detection makes every kernel direction of the phase constraints act as a scalar on the code, so continuous phase families collapse to a finite set of logical actions.

   \(D(t)\) multiplies \(\ket x\) by \(e^{2\pi i\,x\cdot t}\), so \((D(t)PD(t)^\dagger)_{xy}=e^{2\pi i(x-y)\cdot t}P_{xy}\), and the code is preserved if and only if \((x-y)\cdot t\in\mathbb Z\) whenever \(P_{xy}\ne0\). If \((x-y)\cdot h=0\) for all such pairs, then \(H_h=\sum_jh_jN_j\), with \(N_j=(I-Z_j)/2\), commutes with \(P\). Detection gives \(PN_jP=\mu_jP\), so \(H_hP=PH_hP=(h\cdot\mu)P\) and \(D(h)P=e^{2\pi i\,h\cdot\mu}P\). Subtracting such an \(h\) changes the logical action only by a scalar. With \(B=A^{\mathsf T}A\), a rational \(G\) with \(BGB=B\) and \(Q_AG\) integral, \(a=Q_AGA^{\mathsf T}(At)\) satisfies \(A(t-a/Q_A)=0\), because \(\ker A^{\mathsf T}A=\ker A\).

   Example III.3: \(\theta(t)=\tfrac1{12}(3,3,3,3,4,4,4)+t(1,-1,0,0,1,-1,0)\) preserves a seven-qubit distance-2 code for every real \(t\), with logical action \(\operatorname{diag}(1,e^{2\pi i/6})\). For irrational \(t\) the physical gate never repeats. Any finite-order representative has phase \(a/4\) with \(a\) odd on qubit 3 and \(b/3\) with \(b\equiv1\) on qubit 7, so \(12\mid q\) although \(m=6\). Editorial check: at \(t=0\) and \(t=\sqrt2/7\) the gate preserves both codewords with relative phase exactly \(1/6\) turn, and \(\langle Z_1Z_2\rangle\) is \(1\) and \(-\tfrac13\) on the two codewords, confirming distance two.

   How to reuse it

   When a symmetry search has continuous parameters, find the directions that act trivially on the object of interest and quotient them out. What remains is a rational lattice problem whose denominators a Smith normal form can bound. Do not assume the smallest physical order equals the logical order; search over its multiples.

2. Subset-sum sectors and moment matching as a staged filter

   A diagonal gate sorts basis strings by phase; codewords must live in single sectors, and the diagonal error conditions become cheap linear constraints before the hard amplitude equations are solved.

   With \(\ket{0_L}\) supported in \(S_0\) and \(\ket{1_L}\) in \(S_{q/m}\), off-diagonal elements of every diagonal Pauli \(Z_J\) vanish automatically. The diagonal conditions \(\sum_xp_0(x)\prod_{j\in J}(1-2x_j)=\sum_yp_1(y)\prod_{j\in J}(1-2y_j)\) for \(|J|<d\) are equivalent, via \(z_j=1-2x_j\), to equal binary moments up to degree \(d-1\). At distance three this means \(\phi_2(x)=(1,x_i,x_ix_j)\) must have a common average, a linear-programming test: the convex hulls of \(\phi_2(S_0)\) and \(\phi_2(S_{q/m})\) must intersect. Passing it is necessary, not sufficient, since the \(X\)- and \(Y\)-type conditions remain.

   Editorial check on the order-five code with \(a^\star=(1,1,1,1,2,3)\): the strings of \(\ket{\psi_0}\) have subset sums \(0\), \(3+2\) and \(2+3\), all \(\equiv0\pmod5\). Those of \(\ket{\psi_1}\) have sums \(4\), \(4+5\), \(1+3\) and \(2+2\), all \(\equiv4\pmod5\), so the gate acts as \(\operatorname{diag}(1,\zeta^4)\). Numerically, all Paulis of weight at most two satisfy the KL conditions to within \(2\times10^{-16}\).

   How to reuse it

   To search for codes with a prescribed diagonal symmetry, stage the problem: discrete supports from the gate’s phase sectors, then convex moment conditions, then the exact complex equations. The cheap stages prune most candidates, and the staging does not depend on stabilizer structure.

Source: Section II.C (Eqs. 16–25); Section III.A (Lemma III.1, Theorem III.2, Example III.3); Sections III.B–III.C; Section IV.B. Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Turns the subset-sum–linear-programming (SSLP) construction into a complete classification procedure, with no stabilizer or additivity assumption. Continuous families of nonadditive codes can therefore be searched exhaustively for transversal symmetries.

Limitations. Diagonality refers to one fixed product basis, so noncommuting transversal generators that cannot be diagonalized together are not covered. A code-level classification is complete only in the order-five sector, and two of the fifteen weight-enumerator sets remain open. The case-by-case proofs and Lean verification were produced by AI agents and audited by the authors; the structural theorems are the authors’ own.

Description based on the full text of the linked arXiv version, read 2026-10-22; equations follow the paper’s notation.