Let \(H\) be a finite group acting on a graph with group action \(X\), and let \(\Lambda \) be a cell labeling of \(X\) with \(s\) labels. For a vertex \(v \in C_0(X)\) (a 0-cell of \(X\)), the cell local view at \(v\) is the linear map

defined by \(\operatorname {cellLocalView}(v)(c)(i) = c\! \left(\Lambda (v)^{-1}(i)\right)\), where \(\Lambda (v)^{-1}(i)\) denotes the preimage of \(i\) under the labeling bijection at \(v\).

The Tanner differential is the linear map

defined by \(\partial ^{\mathrm{Tan}}(c)(v) = \operatorname {cellLocalView}(v)(c)\), i.e.,

This map assembles the local views at all vertices.

The Tanner code chain complex \(C(X, \Lambda )\) is the chain complex over \(\mathbb {F}_2\) defined by:

where

• \(C_1(X,\Lambda ) = (C_1(X) \to \mathbb {F}_2)\), the \(\mathbb {F}_2\)-module of functions on 1-cells of \(X\),

• \(C_0(X,\Lambda ) = (C_0(X) \to \operatorname {Fin}(s) \to \mathbb {F}_2)\), the \(\mathbb {F}_2\)-module of labeled functions on 0-cells,

• the differential at degrees \((1,0)\) is \(\partial ^{\mathrm{Tan}}\), and all other differentials are zero,

• all modules outside degrees \(0\) and \(1\) are the trivial module \(\{ *\} \).

The chain complex condition \(\partial ^{\mathrm{Tan}} \circ \partial ^{\mathrm{Tan}} = 0\) holds trivially since the complex is concentrated in two consecutive degrees.

The \(H\)-representation on \(C_1(X,\Lambda )\) is the \(\mathbb {F}_2\)-linear group action

defined by \((\rho _1(h) \cdot c)(e) = c(h^{-1} \cdot e)\) for \(h \in H\) and \(e \in C_1(X)\). This is the standard permutation representation by precomposition with the inverse group action.

The \(H\)-representation on \(C_0(X,\Lambda )\) is the \(\mathbb {F}_2\)-linear group action

defined by \((\rho _0(h) \cdot f)(v) = f(h^{-1} \cdot v)\) for \(h \in H\) and \(v \in C_0(X)\).

Given an \(H\)-invariant labeling \(\Lambda \) (i.e., \(\Lambda \) satisfies the invariance condition \(\operatorname {IsInvariantLabeling}(\Lambda )\)), the Tanner code chain complex \(C(X, \Lambda )\) equipped with the \(H\)-representations \(\rho _1\) on \(C_1\) and \(\rho _0\) on \(C_0\) forms an \(H\)-equivariant chain complex \(\widetilde{C}(X,\Lambda ) \in \operatorname {HEquivariantChainComplex}(H)\).

The equivariance condition requires that \(\partial ^{\mathrm{Tan}} \circ \rho _1(h) = \rho _0(h) \circ \partial ^{\mathrm{Tan}}\) for all \(h \in H\), i.e., the Tanner differential commutes with the \(H\)-action.

For \(\ell \geq 1\), the cycle graph chain complex \(C(C_\ell )\) is the chain complex over \(\mathbb {F}_2\) defined by:

where both degree-\(1\) and degree-\(0\) components equal \((\operatorname {Fin}(\ell ) \to \mathbb {F}_2)\), the differential at \((1,0)\) is the cycle graph differential \(\partial _{C_\ell }\), and all modules outside degrees \(0\) and \(1\) are the trivial module.

A cycle compatible \(H\)-action on \(\operatorname {Fin}(\ell )\) is an \(H\)-action satisfying the compatibility condition: for all \(h \in H\) and \(i \in \operatorname {Fin}(\ell )\),

where \(\langle k \rangle \) denotes the element of \(\operatorname {Fin}(\ell )\) with value \(k\). In other words, the \(H\)-action commutes with taking the predecessor modulo \(\ell \).

The cycle chain \(H\)-representation is the \(\mathbb {F}_2\)-linear group action

defined by \((\rho _C(h) \cdot f)(i) = f(h^{-1} \cdot i)\) for \(h \in H\) and \(i \in \operatorname {Fin}(\ell )\).

Given a cycle compatible \(H\)-action on \(\operatorname {Fin}(\ell )\), the cycle graph chain complex \(C(C_\ell )\) equipped with the \(H\)-representation \(\rho _C\) on both degrees \(0\) and \(1\) forms an \(H\)-equivariant chain complex \(\widetilde{C}(C_\ell ) \in \operatorname {HEquivariantChainComplex}(H)\).

The equivariance condition requires \(\partial _{C_\ell } \circ \rho _C(h) = \rho _C(h) \circ \partial _{C_\ell }\) for all \(h \in H\), which follows from the cycle compatible action hypothesis.

The balanced product Tanner cycle code is the chain complex

defined as the balanced product complex of the \(H\)-equivariant Tanner code complex and the \(H\)-equivariant cycle graph complex:

This requires \(\ell \geq 3\), \(\ell \) odd, an \(H\)-invariant labeling \(\Lambda \), and a cycle compatible \(H\)-action on \(\operatorname {Fin}(\ell )\).

The total complex has three non-trivial degrees:

where physical qubits correspond to \(C_1\), Z-checks are given by \(\partial _2 : C_2 \to C_1\), and X-checks are given by \(\partial _1 : C_1 \to C_0\).

The canonical inclusion map

is the map into the total complex at degree \(1\) corresponding to the \((1,0)\)-summand of the balanced product double complex.

The canonical inclusion map

is the map into the total complex at degree \(1\) corresponding to the \((0,1)\)-summand of the balanced product double complex.

The canonical inclusion map

is the map into the total complex at degree \(2\) corresponding to the \((1,1)\)-summand of the balanced product double complex.

The canonical inclusion map

is the map into the total complex at degree \(0\) corresponding to the \((0,0)\)-summand of the balanced product double complex.

The Z-check map of the balanced product Tanner cycle code is the differential

i.e., the chain complex differential from degree \(2\) to degree \(1\). Z-errors are detected by this map.

The X-check map of the balanced product Tanner cycle code is the differential

i.e., the chain complex differential from degree \(1\) to degree \(0\). X-errors are detected by this map.