9 Def 6: Subsystem CSS Code
Let \(Q\) be a CSS code with parity check matrices \(H_X\) and \(H_Z^T\). The submodule of boundaries inside cycles is
viewed as a submodule of \(\ker (H_X)\) via the comap of the subtype inclusion. Formally, it is the \(\mathbb {F}_2\)-submodule
The homology type of a CSS code \(Q\) is the quotient \(\mathbb {F}_2\)-vector space
i.e., the quotient of \(\ker (H_X)\) by the submodule of boundaries in cycles.
The submodule of coboundaries inside cocycles is
where \(H_Z = (H_Z^T)^T = \operatorname {dualMap}(H_Z^T)\) and \(H_X^T = \operatorname {dualMap}(H_X)\), viewed as a submodule of \(\ker (H_Z)\). Formally, it is
The cohomology type of a CSS code \(Q\) is the quotient \(\mathbb {F}_2\)-vector space
i.e., the quotient of \(\ker (\operatorname {dualMap}(H_Z^T))\) by the submodule of coboundaries in cocycles.
The homology quotient map is the canonical \(\mathbb {F}_2\)-linear surjection
defined as the module quotient map \(\operatorname {mkQ}\) for the submodule of boundaries in cycles.
The cohomology quotient map is the canonical \(\mathbb {F}_2\)-linear surjection
defined as the module quotient map \(\operatorname {mkQ}\) for the submodule of coboundaries in cocycles.
A subsystem CSS code with parameters \(n, r_X, r_Z \in \mathbb {N}\) is a CSS code \(Q\) (Definition 50) together with the following data:
A logical subspace \(H_0^L \subseteq H_0\): an \(\mathbb {F}_2\)-submodule of the homology \(H_0 = \ker (H_X)/\operatorname {im}(H_Z^T)\).
A gauge subspace \(H_0^G \subseteq H_0\): an \(\mathbb {F}_2\)-submodule of the homology.
A direct sum decomposition \(H_0 = H_0^L \oplus H_0^G\): the pair \((H_0^L, H_0^G)\) satisfies \(\operatorname {IsCompl}(H_0^L, H_0^G)\), meaning \(H_0^L \cap H_0^G = 0\) and \(H_0^L + H_0^G = H_0\).
A linear equivalence \(\phi : H_0 \xrightarrow {\; \sim \; } H^0\) between the homology and cohomology, encoding the nondegenerate pairing. The cohomology splitting is induced by \(\phi \): \(H^0_L = \phi (H_0^L)\) and \(H^0_G = \phi (H_0^G)\), ensuring that \(H_0^L\) pairs nondegenerately with \(H^0_L\), \(H_0^G\) pairs nondegenerately with \(H^0_G\), and the cross-pairings vanish.
Let \(S\) be a subsystem CSS code with equivalence \(\phi : H_0 \xrightarrow {\; \sim \; } H^0\). The logical cohomology subspace is
Let \(S\) be a subsystem CSS code with equivalence \(\phi : H_0 \xrightarrow {\; \sim \; } H^0\). The gauge cohomology subspace is
Let \(S\) be a subsystem CSS code. The cohomology splitting \(H^0 = H^0_L \oplus H^0_G\) holds, i.e., \(\operatorname {IsCompl}(H^0_L, H^0_G)\).
We prove both conditions of \(\operatorname {IsCompl}(H^0_L, H^0_G)\) separately.
Disjointness (\(H^0_L \cap H^0_G = 0\)): We use the definition of \(\operatorname {Submodule.disjoint_def}\). Let \(x \in H^0_L \cap H^0_G\). Since \(x \in H^0_L = \phi (H_0^L)\), there exists \(a \in H_0^L\) with \(\phi (a) = x\). Since \(x \in H^0_G = \phi (H_0^G)\), there exists \(b \in H_0^G\) with \(\phi (b) = x\). By injectivity of \(\phi \), we have \(a = b\). Therefore \(a \in H_0^L \cap H_0^G\). Since \(H_0 = H_0^L \oplus H_0^G\) (i.e., \(\operatorname {IsCompl}(H_0^L, H_0^G)\)), we have \(H_0^L \cap H_0^G = 0\), so \(a = 0\). Hence \(x = \phi (a) = \phi (0) = 0\) by the linear map property, and \(x = 0\) follows by simplification.
Codisjointness (\(H^0_L + H^0_G = H^0\)): Unfolding \(H^0_L = \phi (H_0^L)\) and \(H^0_G = \phi (H_0^G)\), we rewrite using codisjointness and compute
Since \(H_0 = H_0^L \oplus H_0^G\), we have \(H_0^L + H_0^G = H_0\) (i.e., \(\operatorname {sup_eq_top}\)). Therefore \(\phi (H_0) = \operatorname {Submodule.map_top}(\phi )\). By surjectivity of \(\phi \), \(\operatorname {LinearMap.range_eq_top}\) gives \(\operatorname {range}(\phi ) = H^0\), so \(H^0_L + H^0_G = H^0\).
The logical projection is the \(\mathbb {F}_2\)-linear map
defined as the linear projection onto \(H_0^L\) along \(H_0^G\), using the direct sum decomposition \(H_0 = H_0^L \oplus H_0^G\) (via \(\operatorname {Submodule.linearProjOfIsCompl}\)).
The logical cohomology projection is the \(\mathbb {F}_2\)-linear map
defined as the linear projection onto \(H^0_L\) along \(H^0_G\), using the induced cohomology direct sum decomposition \(H^0 = H^0_L \oplus H^0_G\) (via \(\operatorname {Submodule.linearProjOfIsCompl}\) and \(\operatorname {cohcompl}\)).
The number of logical qubits of a subsystem CSS code \(S\) is
i.e., the \(\mathbb {F}_2\)-dimension (finrank) of the logical subspace \(H_0^L\). In a subsystem code, only \(H_0^L\) encodes logical information, while \(H_0^G\) corresponds to gauge degrees of freedom.
The Z-distance \(d_Z\) of a subsystem CSS code \(S\) is the minimum Hamming weight of a representative \(z \in \ker (H_X)\) of a homology class \([z] \in H_0\) whose projection onto the logical subspace is nonzero:
where \(\operatorname {wt}(z) = \operatorname {hammingWeight}(z)\) denotes the Hamming weight of \(z \in \mathbb {F}_2^n\), \([z] = \pi _H(z)\) is the image under the homology quotient map, and \(\pi _L : H_0 \to H_0^L\) is the logical projection. By convention, \(d_Z = 0\) when no such class exists (i.e., \(H_0^L = 0\)).
The X-distance \(d_X\) of a subsystem CSS code \(S\) is the minimum Hamming weight (under the identification \(\operatorname {dotProductEquiv} : (\mathbb {F}_2^n)^* \cong \mathbb {F}_2^n\)) of a representative \(\zeta \in \ker (H_Z)\) of a cohomology class \([\zeta ] \in H^0\) whose projection onto \(H^0_L\) is nonzero:
where \(\phi : \mathbb {F}_2^n \xrightarrow {\; \sim \; } (\mathbb {F}_2^n)^*\) is the dot-product equivalence, \([\zeta ] = \pi _{H^0}(\zeta )\) is the image under the cohomology quotient map, and \(\pi _L^* : H^0 \to H^0_L\) is the logical cohomology projection. By convention, \(d_X = 0\) when \(H^0_L = 0\).
The distance of a subsystem CSS code \(S\) is
A subsystem CSS code \(S\) is an \([\! [n,k,d]\! ]\)-code (written \(\operatorname {IsNKDCode}(k, d)\)) if
i.e., it encodes \(k\) logical qubits with overall distance \(d\).
A subsystem CSS code \(S\) is an \([\! [n,k,d_X,d_Z]\! ]\)-code (written \(\operatorname {IsNKDXZCode}(k, d_X, d_Z)\)) if
i.e., it encodes \(k\) logical qubits with X-distance \(d_X\) and Z-distance \(d_Z\) separately specified.