12 Rem 7: Codespace Dimension After Gauging
This remark analyzes how the codespace dimension changes when a stabilizer code is gauged. If the original code has parameters \([\! [n, k, d]\! ]\) with \(J\) checks and the gauging graph \(G = (V, E)\) has a cycle collection \(C\) generating the full cycle space (so \(|C| = |E| - |V| + 1\) for a connected graph), then the deformed code has \(n + |E|\) physical qubits and \(|V| + |C| + |J|\) checks, giving \(k_{\text{new}} = (n + |E|) - (|V| + |C| + |J|) = k - 1\). The gauging measurement consumes exactly one logical qubit.
Let \(G = (V, E)\) be a simple graph, let \(C\) be a collection of cycles satisfying the even-incidence condition, let \(J\) be a set of pairwise-commuting Pauli checks, and let data be the deformed code data. Then the number of physical qubits in the deformed stabilizer code is
This follows directly from deformedStabilizerCode_numQubits.
Under the same hypotheses as above, the number of checks in the deformed stabilizer code is
This follows directly from deformedStabilizerCode_numChecks.
The cycle rank property for a graph \(G = (V, E)\) with a cycle collection \(C\) is the assertion that
This holds when \(C\) is a complete cycle basis for a connected graph \(G\), and is equivalent to the statement that the cycle rank (first Betti number) satisfies \(|C| = |E| - |V| + 1\).
Let the original code act on \(n\) qubits with \(|J|\) checks, so that \(k = n - |J|\) is the nominal number of logical qubits with \(k \geq 1\). Suppose the gauging graph \(G = (V, E)\) with cycle collection \(C\) satisfies the cycle rank property \(|C| + |V| = |E| + 1\). Then the deformed stabilizer code satisfies
That is, gauging consumes exactly one logical qubit.
We rewrite the number of qubits and checks of the deformed code using the identities \(\operatorname {numQubits} = |V| + |E|\) and \(\operatorname {numChecks} = |V| + |C| + |J|\). Unfolding the cycle rank property gives \(|C| + |V| = |E| + 1\). We need to show \((|V| + |E|) - (|V| + |C| + |J|) = k - 1\), where \(k = n - |J|\) and \(n = |V|\). From the cycle rank property, \(|E| = |C| + |V| - 1\), so
This is verified by integer arithmetic (omega).
Let \(n = |V|\) and assume the cycle rank property holds. Then the total number of checks satisfies
That is, the net check increase minus the net qubit increase equals \(1\).
We unfold the cycle rank property \(|C| + |V| = |E| + 1\) and verify the equality by integer arithmetic (omega).
If the original code has \(k = 0\) logical qubits (i.e., \(|V| = |J|\)) and the cycle rank property holds, then the deformed code satisfies
That is, the natural number subtraction underflows to \(0\).
We rewrite using \(\operatorname {numQubits} = |V| + |E|\) and \(\operatorname {numChecks} = |V| + |C| + |J|\). Unfolding the cycle rank property \(|C| + |V| = |E| + 1\) and using the hypothesis \(|V| = |J|\), the inequality follows by integer arithmetic (omega).
Under the cycle rank property, the number of checks in the deformed code satisfies
We rewrite using \(\operatorname {numQubits} = |V| + |E|\) and \(\operatorname {numChecks} = |V| + |C| + |J|\). Unfolding the cycle rank property and applying integer arithmetic (omega) yields the result.
Let the original code have \(n\) qubits with \(|J|\) checks and \(k = n - |J| \geq 1\) logical qubits. Under the cycle rank property, the deformed code \(\mathrm{DC}\) satisfies
This follows directly by applying the theorem codespace_dimension_change_after_gauging with the given parameters.
Under the cycle rank property, the number of additional gauging checks (vertices plus cycles) exceeds the number of additional qubits (edges) by exactly one:
We unfold the cycle rank property \(|C| + |V| = |E| + 1\) and the result follows by integer arithmetic (omega).
Under the cycle rank property, the edge count satisfies
We unfold the cycle rank property and the equality follows by integer arithmetic (omega).
Now let me also check what the relevant definitions look like to understand the mathematical content better:Now I have a thorough understanding of the code. Let me produce the LaTeX translation: