7 The homological framework
This chapter is the abstract heart of the development. A length-three chain complex over \(\mathbb {F}_2\) induces a CSS code: \(1\)-chains are the physical qubits, \(\partial _1 \partial _2 = 0\) is the commutation of the \(X\)- and \(Z\)-checks, and — the slogan — logical operators are homology classes. Both the toric family and the gross code are instances, and both inherit their distance argument from theorem 58.
The abstract input to the CSS-from-homology machine: a length-three chain complex of \(\mathbb {F}_2\) vector spaces \(C_2 \xrightarrow {\partial _2} C_1 \xrightarrow {\partial _1} C_0\) with \(\partial _1 \partial _2 = 0\), where \(C_1\) is indexed by the physical qubits, \(C_0\) by the \(Z\)-checks and \(C_2\) by the \(X\)-checks. Every topological code in this development is an instance.
\(\partial _1(\partial _2 c) = 0\) for every \(c \in C_2\).
A field of definition 46, restated pointwise. It is exactly the statement that the \(X\)- and \(Z\)-checks of the induced CSS code commute.
\(Z_1 = \ker \partial _1 \le C_1\). Under definition 53 these are exactly the chains whose \(X\) operator commutes with every \(Z\)-check.
\(B_1 = \operatorname {im} \partial _2 \le C_1\): the chains obtained from an \(X\)-check, i.e. the products of stabilizer generators.
\(B_1 \le Z_1\).
Immediate from theorem 47.
\(H_1 = Z_1 / B_1\) (definition 48, definition 49). The slogan of the whole framework is *logical operators are homology classes*: nontrivial logical operators correspond to nonzero classes in \(H_1\), and the code distance is the minimum weight of a chain representing a nonzero class.
\(\dim H_1 = \dim Z_1 - \dim B_1\), so the number of encoded qubits is computed by two rank calculations.
Rank-nullity for the quotient of finite-dimensional spaces, using theorem 50 to know the quotient is well-formed.
7.1 From chains to Paulis
The \(X\)-type Pauli operator supported on the qubits in a \(1\)-chain: place \(X\) where the chain is \(1\) and \(I\) elsewhere. This is the dictionary between \(\mathbb {F}_2\) linear algebra and \(\mathcal{P}_n\).
The \(Z\)-type counterpart of definition 53, used for the dual side of the CSS code.
\(X(c + c') = X(c) \cdot X(c')\): addition of chains over \(\mathbb {F}_2\) corresponds to multiplication of \(X\)-type Paulis.
Qubitwise. \(X \cdot X = I\) matches \(1 + 1 = 0\) in \(\mathbb {F}_2\), and the phase stays trivial because \(X\)-type operators commute.
The \(X\) operator of a chain commutes with every \(Z\)-check if and only if the chain is a cycle (definition 48).
Commutation of \(X(c)\) with the \(Z\)-check at vertex \(v\) is, by theorem 11, the parity of the overlap between \(c\) and the edges meeting \(v\) — which is exactly the \(v\)-component of \(\partial _1 c\). Requiring this for all \(v\) says \(\partial _1 c = 0\).
7.2 Distance from homology
If a Pauli operator is a nontrivial logical operator of the induced CSS code, then its \(X\)-chain and its \(Z\)-chain cannot both be boundaries.
Suppose both were. Then the corresponding \(X\)- and \(Z\)-type operators are each products of stabilizer generators, so their product \(g_X g_Z\) lies in \(\mathcal{S}\) and has the same operator part as \(g\). That contradicts the third clause of definition 31 — the clause whose presence is precisely what makes this argument go through.
A lower bound on the weight of every chain representing a nonzero class in \(H_1\) (and dually) transfers to a lower bound on the weight of every nontrivial logical operator.
Let \(g\) be a nontrivial logical. By theorem 57 at least one of its two chains is a non-boundary cycle, and by theorem 56 it is a cycle. The hypothesised chain bound applies to it, and the weight of \(g\) dominates the weight of each of its chains since the two supports are contained in the support of \(g\). This theorem is the reason a purely homological argument settles a question about Pauli operators, and it is used unchanged by both the toric family and the gross code.
Theorem 58 is the load-bearing step: it is what allows a statement about minimum weights of homology representatives — a question in \(\mathbb {F}_2\) linear algebra — to settle a statement about the weights of Pauli operators. Every concrete distance theorem below is an application of it.