43 Rem 21: Gross Code Gauging Measurement Construction
This chapter presents the explicit construction of the gauging graph \(G\) for measuring \(\bar{X}_\alpha = X(\alpha f, 0)\) in the Gross code, with degree bounds on the Tanner graph of the deformed code. The construction specifies the 12 vertices (support of \(f\)), 22 edges (18 matching + 4 expansion), and 7 independent cycles, yielding a total overhead of 41 additional checks and qubits.
The gauging vertices are the support of \(f\): the set of monomials \(\gamma \in \mathbb {Z}_{12} \times \mathbb {Z}_6\) such that \(f(\gamma ) \neq 0\):
The gauging graph has exactly 12 vertices: \(|V(G)| = 12\).
This is verified by computation (native_decide).
A monomial difference \(d\) is a matching difference if \(d = -B_i + B_j\) for some distinct terms \(B_i, B_j\) of \(B = y^3 + x^2 + x\), where the terms of \(B\) are \((0,3)\), \((2,0)\), and \((1,0)\) in \(\mathbb {Z}_{12} \times \mathbb {Z}_6\), and \(d \neq 0\).
Two monomials \(\gamma \) and \(\delta \) form a matching edge if both are in \(\operatorname {supp}(f)\) (i.e., \(f(\gamma ) \neq 0\) and \(f(\delta ) \neq 0\)) and their difference \(\gamma - \delta \) is a matching difference. This corresponds to \(\gamma \) and \(\delta \) sharing a \(Z\) check.
The 4 expansion edges ensuring the deformed code has distance 12, given as pairs of monomials:
A pair \((\gamma , \delta )\) is an expansion edge if it appears (in either order) in the list of 4 expansion edges.
The adjacency relation on the gauging graph: \(\gamma \) and \(\delta \) are adjacent if \(\gamma \neq \delta \) and either \((\gamma , \delta )\) is a matching edge or an expansion edge.
The gauging graph \(G\) for measuring \(\bar{X}_\alpha \) in the Gross code is the simple graph on \(\operatorname {GrossMonomial} = \mathbb {Z}_{12} \times \mathbb {Z}_6\) with adjacency given by \(\operatorname {gaugingAdj}\). The “active” vertices are \(\operatorname {supp}(f)\).
The gauging graph has exactly 22 edges: \(|E(G)| = 22\).
This is verified by computation (native_decide).
The 7 independent cycles for the flux checks, given as lists of vertices. Each list \([v_1, v_2, \ldots , v_k]\) represents the cycle \(v_1 \to v_2 \to \cdots \to v_k \to v_1\):
\([(9,0),\, (7,3),\, (8,0)]\)
\([(9,0),\, (11,3),\, (7,3)]\)
\([(6,0),\, (7,0),\, (5,3)]\)
\([(6,0),\, (2,0),\, (5,3)]\)
\([(3,0),\, (2,0),\, (5,3)]\)
\([(6,0),\, (7,0),\, (8,0),\, (7,3)]\)
\([(1,0),\, (2,0),\, (5,3),\, (11,3)]\)
There are exactly 7 independent cycles: \(|\text{cycles}| = 7\).
This holds by reflexivity (the list length is definitionally 7).
Every vertex appearing in any of the 7 cycles belongs to the gauging vertices \(V(G) = \operatorname {supp}(f)\).
This is verified by computation (native_decide).
Each cycle has at least 3 vertices.
This is verified by computation (native_decide).
For each cycle, consecutive vertices (including the wrap-around from the last vertex back to the first) are adjacent in the gauging graph.
This is verified by computation (native_decide).
All cycles have length 3 or 4.
This is verified by computation (native_decide).
Every cycle has length at most 4.
Let \(\mathrm{cyc}\) be any cycle in the list and let \(h_{\mathrm{mem}}\) be the proof that it belongs to the list. By the theorem that all cycles have length 3 or 4 (gaugingCycles_short), we obtain \(|\mathrm{cyc}| = 3\) or \(|\mathrm{cyc}| = 4\). In either case, \(|\mathrm{cyc}| \leq 4\) follows by integer arithmetic.
By Euler’s formula, the cycle space dimension satisfies \(|E| + 1 = |V| + 11\), i.e., \(22 + 1 = 12 + 11\).
Rewriting using \(|E(G)| = 22\) and \(|V(G)| = 12\), the equation \(22 + 1 = 12 + 11\) holds.
The number of additional Gauss’s law checks \(A_v\) (one per vertex of \(G\)) is 12.
This follows directly from the vertex count \(|V(G)| = 12\).
The number of additional flux checks \(B_p\) (one per independent cycle) is 7.
This follows directly from the cycle count of 7.
The number of additional edge qubits (one per edge of \(G\)) is 22.
This follows directly from the edge count \(|E(G)| = 22\).
The total overhead is \(|V(G)| + |\text{cycles}| + |E(G)| = 12 + 7 + 22 = 41\) additional checks and qubits.
Rewriting using \(|V(G)| = 12\), \(|\text{cycles}| = 7\), and \(|E(G)| = 22\), we obtain \(12 + 7 + 22 = 41\).
All 4 expansion edges are edges of the gauging graph: for every pair \((a,b) \in \text{expansionEdges}\), \(a\) and \(b\) are adjacent in \(G\).
This is verified by computation (native_decide).
All expansion edge endpoints belong to \(\operatorname {supp}(f)\): for every \((a,b) \in \text{expansionEdges}\), both \(a \in V(G)\) and \(b \in V(G)\).
This is verified by computation (native_decide).
The maximum degree in the gauging graph isat most 6: for every vertex \(v\), \(\deg (v) \leq 6\).
This is verified by computation (native_decide).
The number of matching edges (non-expansion edges) is 18.
This is verified by computation (native_decide).
The number of expansion edges is 4.
This is verified by computation (native_decide).
The number of \(Z\) checks adjacent to the support of \(\bar{X}_\alpha \) is 18: the number of monomials \(\beta \) such that there exists at least one \(\gamma \in V(G)\) with \(B(\beta - \gamma ) \neq 0\) is 18.
This is verified by computation (native_decide).
The gauging construction for \(\bar{X}_\alpha \) in the Gross code satisfies:
\(|V(G)| = 12\),
\(|E(G)| = 22\),
There are 7 independent cycles,
Total overhead: \(12 + 7 + 22 = 41\),
Maximum degree \(\leq 6\),
Maximum cycle length \(\leq 4\).
This follows directly by combining the previously established results: gaugingVertices_card, gaugingEdges_card, gaugingCycles_count, total_overhead, gaugingDegree_le_six, and gaugingCycles_max_length.