Unverified 2026

Charge-conserving Z3 neural block

Implementation & benchmark of arXiv:2607.08425 — Quantum and Classical Potts Criticality in Driven-Dissipative Bosonic Lattices

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Architecture Accuracy Generalization Sample efficiency Graph nn Mlp Attention Regularization Representation theory Algebra Dynamical systems

Source paper: Quantum and Classical Potts Criticality in Driven-Dissipative Bosonic Lattices arXiv:2607.08425 ⓘ · analyzed Aug 30, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Represent each feature as belonging to one of three $\mathbb{Z}_3$ charge sectors and constrain every linear and multiplicative operation to obey charge addition modulo 3. Add invariant cubic gates such as $x_1x_2x_3$ or $x_q^3$, which can express the same phase-insensitive interaction selected by the paper's three-photon drive. This should improve data efficiency and exact cyclic-augmentation consistency when the task has a genuine ternary symmetry.

Formulas

$$\hat{H}=\sum_j\left(-\Delta\hat{a}_j^\dagger\hat{a}_j+\frac{U}{2}\hat{a}_j^{\dagger 2}\hat{a}_j^2+G_3\hat{a}_j^{\dagger 3}+G_3^*\hat{a}_j^3\right)-\sum_{\langle j,j'\rangle}\frac{J}{z}\left(\hat{a}_j^\dagger\hat{a}_{j'}+\mathrm{h.c.}\right)$$

$$\omega=e^{2\pi i/3},\qquad \hat a_j\mapsto\omega\hat a_j,\qquad \hat a_j^\dagger\mapsto\omega^{-1}\hat a_j^\dagger,\qquad \omega^3=1$$

$$x^{(q)}\mapsto \omega^q x^{(q)},\qquad q\in\{0,1,2\};\qquad x^{(q)}y^{(r)}\mapsto\omega^{q+r}x^{(q)}y^{(r)}$$

$$\begin{aligned}h^{(0)}&=W_0x^{(0)}+A_0\big(x^{(1)}x^{(2)}\big)+B_0\operatorname{Re}\big((x^{(1)})^3\big),\\ h^{(1)}&=W_1x^{(1)}+A_1\big(x^{(0)}x^{(1)}\big)+B_1\big((x^{(2)})^2\big),\\ h^{(2)}&=W_2x^{(2)}+A_2\big(x^{(0)}x^{(2)}\big)+B_2\big((x^{(1)})^2\big),\\ y^{(q)}&=h^{(q)}\,\phi\!\left(\frac{|h^{(q)}|}{\sqrt{\epsilon+|s|}}\right),\qquad s=h^{(0)}+\operatorname{Re}(h^{(1)}h^{(2)})+\operatorname{Re}((h^{(1)})^3),\end{aligned}$$

Mathematical statement

The model Hamiltonian contains the three-photon terms $G_3\hat a_j^{\dagger 3}+G_3^*\hat a_j^3$, where $\hat a_j$ is the bosonic annihilation operator at site $j$ and $G_3$ is the drive amplitude. Let $\omega=e^{2\pi i/3}$, so $\omega^3=1$. The transformation $\hat a_j\mapsto\omega\hat a_j$, $\hat a_j^\dagger\mapsto\omega^{-1}\hat a_j^\dagger$ leaves every term of the Hamiltonian invariant: number terms have charge zero, hopping pairs $\hat a_i^\dagger\hat a_j$ have net charge zero, and $\hat a_j^3$ has charge $3\equiv0\pmod 3$. Transfer this representation-theoretic structure to neural features. Assign a feature $x^{(q)}$ charge $q\in\{0,1,2\}$, transforming as $x^{(q)}\mapsto\omega^q x^{(q)}$. A linear map is equivariant only when it preserves charge, while a product obeys charge addition: $x^{(q)}y^{(r)}$ has charge $q+r\pmod 3$. A scalar invariant can therefore be formed from $x^{(0)}$, $x^{(1)}x^{(2)}$, or cubic terms $(x^{(1)})^3$ and $(x^{(2)})^3$. The exact equivariance condition is $F(\rho(g)x)=\rho(g)F(x)$ for $g\in\mathbb{Z}_3$, with $\rho(g)x^{(q)}=\omega^{qg}x^{(q)}$.

Implementation notes

Integrate this as a replacement for an MLP or graph-message-passing block. Store each node or token feature as three complex tensors $x0,x1,x2$, with charges 0, 1, and 2; real scalar channels can be included inside $x0$. For each layer, compute charge-preserving linear maps $W_qx_q$, message-passing or attention aggregation independently within each charge sector, and the charge-valid products $x1*x2\to q=0$, $x0*x1\to q=1$, $x0*x2\to q=2$, $x2*x2\to q=1$, and $x1*x1\to q=2$. Form an invariant gate $s=x0+\operatorname{Re}(x1\overline{x1})+\operatorname{Re}(x1^3)$, pass it through a real SiLU or sigmoid, and multiply every charged output by that gate. Pseudocode is: h0=W0(x0)+A0*(x1*x2)+B0*real(x1**3); h1=W1(x1)+A1*(x0*x1)+B1*(x2**2); h2=W2(x2)+A2*(x0*x2)+B2*(x1**2); gate=silu(real(h0)+real(h1*conj(h2))+real(h1**3)); yq=hq*gate. Estimate no mathematical constants; all $W_q,A_q,B_q$ are learned. Validate equivariance numerically by applying $x_q\leftarrow\omega^q x_q$ and checking relative error $\|F(\rho x)-\rho F(x)\|/\|F(x)\|$. First run a small 3-class cyclic image or graph benchmark with random $\mathbb{Z}_3$ phase rotations, comparing this block with a parameter-matched MLP/GNN. Measure clean accuracy, accuracy after unseen cyclic rotations, sample efficiency at 10%, 25%, and 100% data, and parameter/FLOP overhead. Success means near-machine-precision equivariance and materially better rotated-test accuracy or lower data requirement without increasing inference cost by more than roughly 2x.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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