From torus polynomials to a CSS chain complex

This page fixes the convention used by constructor.py and pays the commutation argument in full. The purpose is not just to remember the formula $$H_X=[A|B]$$ , $$H_Z=[B^T|A^T]$$ , but to see why that particular formula is the natural one.

1. The regular torus representation

Let $e_{i,j}$ be the standard basis vector at $(i,j)\in\mathbb Z_l\times\mathbb Z_m$ . The constructor's cyclic matrix $$S_l$$ has a one in row $$i$$ , column $$i+1$$ , with indices modulo $$l$$ . With the flattened index $$im+j$$ , define

x=S_l\otimes I_m,\qquad y=I_l\otimes S_m.

The first operator changes only the $$i$$ coordinate; the second changes only the $$j$$ coordinate. Therefore

x^l=I,\qquad y^m=I,\qquad xy=yx.

These matrices give the regular representation of

R=\mathbb F_2[x,y]/(x^l-1,y^m-1).

In characteristic two, $$x^l-1=x^l+1$$ , so the plus-sign convention used in some papers describes the same quotient.

Every monomial $$x^a y^b$$ is a permutation matrix. Its transpose is its inverse:

(x^a y^b)^T=x^{-a}y^{-b},

with negative exponents reduced modulo $$l,m$$ . This is why transposition has a geometric meaning: it reverses a torus translation.

2. Sparse polynomials give LDPC incidence

For a binary polynomial

A=\sum_{(a,b)\in T_A} x^a y^b,

each row contains one entry contributed by each distinct monomial. If the reduced exponent pairs in $$T_A$$ are distinct, every row and every column has weight $$|T_A|$$ . Repeated terms cancel over $\mathbb F_2$ ; the constructor therefore reduces exponent pairs modulo the two cyclic orders before applying XOR.

The gross inputs are

A=x^3+y+y^2,\qquad B=y^3+x+x^2

on $$l=12,m=6$$ . Both matrices have row and column weight three. No geometric plaquette was separately specified: the check graph is the incidence graph of these translations.

3. Why the swapped transpose is forced

Place two qubits at each torus site. The first $$lm$$ columns form the left data block and the second $$lm$$ columns the right block. Define

H_X=[A\mid B].

A CSS Z-check matrix must obey $$H_XH_Z^T=0$$ . Choose

H_Z=[B^T\mid A^T].

Then

\begin{aligned} H_XH_Z^T &=[A\mid B] \begin{bmatrix}B\\A\end{bmatrix}\\ &=AB+BA\\ &=AB+AB\\ &=0. \end{aligned}

The third line uses $$AB=BA$$ , which follows from the abelian torus algebra. The last line uses characteristic two.

Both the transpose and the swap have jobs. Transpose reverses the Z-side incidence so that it is the dual map. The swap routes the overlap through $$AB+BA$$ . If the candidate were instead $$[A^T|B^T]$$ , its product with $$H_X$$ would be $$A^2+B^2$$ , which is not constrained to vanish.

4. The three-term complex

The same calculation is the boundary-of-a-boundary identity in

R \xrightarrow{\;\partial_2=[B;A]\;} R^2 \xrightarrow{\;\partial_1=[A|B]\;} R.

Here

\partial_1\partial_2=AB+BA=0.

In the binary CSS convention, $\partial_1=H_X$ and $\partial_2^T=H_Z$ . An X-type Pauli that commutes with every Z check is a cycle; multiplying by an X stabilizer adds a boundary. The nontrivial X-logical classes are therefore a quotient of cycles by boundaries. The Z-logical classes are the dual quotient.

This is the exact place where the homological language earns its keep: “commutes with the checks” and “is a stabilizer” are different subspaces, and logical information is the gap between them. The general binary dictionary is in CSS chain complexes.

5. The toric anchor

Set $$l=m=L$$ and choose

A=1+x,\qquad B=1+y.

Each polynomial has two terms, so each check has weight four. The resulting complex is the periodic square-lattice toric complex in the BB column convention: two qubits per unit cell, one X check and one Z check per cell.

For the shipped $$L=3$$ instance, the constructor produces two $9\times18$ check matrices. Both have rank eight, leaving

\[ k=18-8-8=2. \]

The Atlas exact CSS enumerator finds distance three. This [[18,2,3]] anchor tests the shift orientation, swapped transpose, ranks, and exact-distance path on a small instance.

It does not prove that an arbitrary BB block is locally Clifford equivalent to independent toric codes. sahay2026matching uses a symmetry reduction to obtain toric-like cylinders for decoding, while chen2025anyon finds topological frustration in more general BB data. Those statements concern different equivalences.

6. What the executable artifact certifies

pedagogy/reproduce.py independently reconstructs $$x,y,A,B$$ , then checks:

the two periods and $$xy=yx$$ ;
the exact gross polynomials;
$$H_Z=[B^T|A^T]$$ ;
$$H_XH_Z^T=0$$ ;
the two small toric ranks and exact distance.

The card's construction-replay tree now binds the two exponent lists into versioned bivariate_polynomial@1 factors and assembles them with bivariate_bicycle@1. The verifier compares the complete output arrays, not only ranks or row spaces, so this page's block and translation conventions are part of the machine witness rather than an equivalence-only assertion.

Provenance

The polynomial and syndrome-circuit presentation follows bravyi2024bicycle; the group-algebra placement follows wang2023abelian and lin2023twoblock. Every displayed matrix identity for the shipped instances is rerun locally. The general topological caveat is literature-sourced to chen2025anyon and sahay2026matching.