Def_4: Deformed Operator #
Statement #
Let $P$ be a Pauli operator that commutes with the logical operator $L = \prod_{v \in V_G} X_v$. Write $P = i^{\sigma} \prod_{v \in \mathcal{S}_X} X_v \prod_{v \in \mathcal{S}_Z} Z_v$ where $\mathcal{S}_X$ is the $X$-type support and $\mathcal{S}_Z$ is the $Z$-type support of $P$. Since $P$ commutes with $L$, we have $|\mathcal{S}_Z| \equiv 0 \pmod{2}$.
The deformed operator $\tilde{P}$ is defined as: $$\tilde{P} = P \prod_{e \in \gamma} Z_e$$ where $\gamma$ is an edge-path in $G$ satisfying $\partial \gamma = \mathcal{S}_Z \cap V_G$ (i.e., the boundary of the edge-set $\gamma$ equals the $Z$-type support of $P$ restricted to the vertices of $G$). Such a path $\gamma$ exists because $|\mathcal{S}_Z \cap V_G|$ is even.
Convention: We typically choose $\gamma$ to be a minimum-weight path (shortest collection of edges) satisfying the boundary condition.
Note: If $P$ does not commute with $L$, then $|\mathcal{S}_Z \cap V_G|$ is odd, no valid path $\gamma$ exists, and $P$ has no well-defined deformed version.
Main Definitions #
DeformablePauliOperator: A Pauli operator P that commutes with L (has deformable property)EdgePath: An edge-set γ (subset of E) in the graphIsValidDeformingPath: An edge-path γ satisfying ∂γ = S_Z ∩ V_GDeformedOperator: The deformed operator P̃ = P · ∏_{e ∈ γ} Z_e
Key Properties #
zSupport_even_of_valid_path_exists: If a valid deforming path exists, then |S_Z ∩ V_G| is evenno_valid_path_if_odd: If |S_Z| is odd, no valid deforming path existsdeformed_commutes_with_gaussLaw: P̃ commutes with all Gauss law operators A_v
Corollaries #
no_deformed_if_odd_zSupport: If P doesn't commute with L, no valid deforming path existsdeformed_xSupportOnV_eq: P̃ has the same X-type support as P on verticesdeformed_zSupportOnV_eq: P̃ has the same Z-type support as P on vertices
Z-Type Support Restricted to Graph Vertices #
In the gauging context, we have:
- Original code qubits (some identified with graph vertices V_G)
- Edge qubits E_G
- A Pauli operator P on the original code
The Z-type support of P restricted to vertices V_G determines the boundary condition.
A Pauli operator's Z-type support restricted to the graph vertices. This is S_Z ∩ V_G in the paper notation. We represent it as a binary vector over V.
Equations
- _G.zSupportOnVertices zSupport = zSupport
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The binary vector representation of Z-support restricted to V
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Edge Paths as Binary Vectors #
An edge-path γ in the graph is represented as a subset of edges (or equivalently, a binary vector over E). The boundary map ∂ gives the vertices that are endpoints of an odd number of edges in γ.
An edge-path in the graph, represented as a subset of edges
Equations
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The edge-path as a binary vector over edges
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The boundary of an edge-path: ∂γ = sum of boundaries of individual edges
Equations
- G.edgePathBoundary γ = G.boundaryMap (G.edgePathVector γ)
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Alternative characterization: the boundary at vertex v counts (mod 2) edges in γ incident to v
The boundary is 1 at v iff an odd number of edges in γ are incident to v
The boundary is 0 at v iff an even number of edges in γ are incident to v
Valid Deforming Paths #
A valid deforming path γ for Z-support S_Z is one satisfying the boundary condition: ∂γ = S_Z (as binary vectors over V).
For such a path to exist, |S_Z| must be even (since ∂ is the boundary map and boundaries in Z₂-homology have even cardinality).
A valid deforming path γ for Z-support S is one where ∂γ = S (as binary vectors)
Equations
- G.IsValidDeformingPath zSupport γ = (G.edgePathBoundary γ = G.zSupportVector zSupport)
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Characterization: γ is valid iff ∂γ = S at each vertex
At vertices in S, the boundary is 1
At vertices not in S, the boundary is 0
Even Cardinality Condition #
The boundary map ∂ : Z₂^E → Z₂^V has the property that the sum of ∂γ over all vertices is 0 (since each edge contributes to exactly 2 vertices, and 1+1=0 in Z₂).
This means: if ∂γ = S (as vectors), then |S| must be even for a solution to exist.
The sum of boundary values over all vertices is always 0. This is because each edge is incident to exactly 2 vertices.
If a valid deforming path exists for S, then |S| is even
The contrapositive: if |S| is odd, no valid deforming path exists
Deformable Pauli Operators #
A Pauli operator P is deformable if it commutes with the logical operator L = ∏_v X_v. This ensures that |S_Z ∩ V_G| is even, so a valid deforming path exists.
A Pauli operator (represented by its supports) that is deformable. The key condition is that it commutes with L = ∏_{v ∈ V} X_v, which means the Z-support restricted to V has even cardinality.
- xSupportOnV : Finset V
The X-type support on vertices: {v | P acts as X or Y on v}
- zSupportOnV : Finset V
The Z-type support on vertices: {v | P acts as Z or Y on v}
- xSupportOnE : Finset E
The X-type support on edges (typically empty for original code checks)
- zSupportOnE : Finset E
The Z-type support on edges (typically empty for original code checks)
- phase : ZMod 4
The global phase (represented as ZMod 4: 0=+1, 1=+i, 2=-1, 3=-i)
The key deformability condition: |S_Z ∩ V| is even
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The deformability condition as stated: P commutes with L = ∏_v X_v. This is equivalent to having even Z-support on V.
Construction of the Deformed Operator #
Given a deformable operator P and a valid deforming path γ, the deformed operator P̃ is:
- Same X-support on V as P
- Same phase as P
- Z-support on V: S_Z (same as P)
- Z-support on E: S_Z^E ∪ γ (XOR in Z₂, which is symmetric difference)
In the binary vector representation:
- P̃_X^V = P_X^V (X-support on vertices unchanged)
- P̃_Z^V = P_Z^V (Z-support on vertices unchanged; deformation only affects edges)
- P̃_X^E = P_X^E (X-support on edges unchanged)
- P̃_Z^E = P_Z^E + γ (Z-support on edges is symmetric difference with γ)
The deformed operator P̃ = P · ∏_{e ∈ γ} Z_e. Given P and a valid deforming path γ, construct P̃.
Equations
- One or more equations did not get rendered due to their size.
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The deformed operator's Z-support on edges (alternative formulation using XOR)
The deformed operator's edge Z-support as binary vector is P's + γ's
The Deformed Operator Commutes with Gauss Law Operators #
The key property: if γ is a valid deforming path (∂γ = S_Z^V), then P̃ commutes with all Gauss law operators A_v.
Recall: A_v = X_v ∏_{e∋v} X_e The symplectic form ω(P̃, A_v) counts:
- |S_Z^V(P̃) ∩ {v}| = 1 if v ∈ S_Z^V, else 0 (from Z-support on V)
- |S_Z^E(P̃) ∩ edges(v)| = |edges in γ incident to v| (from Z-support on E, assuming P had no Z on E)
For commutation: ω(P̃, A_v) ≡ 0 (mod 2)
If ∂γ = S_Z^V, then:
- v ∈ S_Z^V ⟹ (∂γ)_v = 1 ⟹ odd number of edges in γ incident to v
- v ∉ S_Z^V ⟹ (∂γ)_v = 0 ⟹ even number of edges in γ incident to v
So ω(P̃, A_v) = |S_Z^V ∩ {v}| + |γ ∩ edges(v)| ≡ 0 (mod 2) in both cases.
The symplectic form between the deformed operator and a Gauss law operator. This counts anticommuting pairs: positions where P̃ has Z-type and A_v has X-type.
Equations
- G.deformed_gaussLaw_symplectic P γ v = (if v ∈ P.zSupportOnV then 1 else 0) + ((G.DeformedOperator P γ).zSupportOnE ∩ G.incidentEdges v).card
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Assuming P has no Z-support on edges originally, simplification
Equations
- G.deformed_gaussLaw_symplectic_simple zSupportOnV γ v = (if v ∈ zSupportOnV then 1 else 0) + Finset.card (γ ∩ G.incidentEdges v)
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The key theorem: if γ is a valid deforming path, P̃ commutes with all A_v. Proof outline:
- If v ∈ S_Z^V: contributes 1 from vertex, and (∂γ)_v = 1 means odd edges, so total even
- If v ∉ S_Z^V: contributes 0 from vertex, and (∂γ)_v = 0 means even edges, so total even
Corollary: the full deformed operator (assuming P originally had no Z-support on edges) commutes with all Gauss law operators
Non-Commuting Operators Cannot Be Deformed #
If P does not commute with L (i.e., |S_Z ∩ V| is odd), then no valid deforming path exists, and P has no well-defined deformed version.
If |S_Z^V| is odd, P doesn't commute with L and cannot be deformed
Alternative statement: if ∃ valid path, then P commutes with L (even Z-support)
Properties of the Deformed Operator #
The X-support on vertices is unchanged after deformation
The Z-support on vertices is unchanged after deformation
The X-support on edges is unchanged after deformation
The phase is unchanged after deformation
The deformability condition is preserved
Minimum Weight Path Convention #
The paper mentions choosing γ to be a minimum-weight path. This is a convention for making the choice canonical and minimizing the overhead of the deformation.
The weight of an edge-path is its cardinality
Equations
- _G.edgePathWeight γ = Finset.card γ
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A minimum-weight valid deforming path
Equations
- One or more equations did not get rendered due to their size.
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If any valid path exists, a minimum-weight one exists (by finiteness of E)
Summary of Deformed Operator #
The deformed operator construction captures:
- Input: A Pauli operator P with Z-support S_Z on vertices
- Condition: P must commute with L = ∏_v X_v, equivalently |S_Z| even
- Construction: Choose edge-path γ with ∂γ = S_Z, form P̃ = P · ∏_{e∈γ} Z_e
- Result: P̃ has the same vertex supports as P, with additional Z on edges in γ
- Key Property: P̃ commutes with all Gauss law operators A_v
- Non-existence: If P doesn't commute with L (|S_Z| odd), no deformed version exists
- Convention: Choose γ to be minimum-weight among valid paths
Summary #
The deformed operator formalization captures:
Deformability Condition: A Pauli operator P is deformable iff it commutes with L = ∏_v X_v, equivalently iff |S_Z ∩ V_G| is even (Z-support restricted to graph vertices).
Valid Deforming Path: An edge-path γ satisfying ∂γ = S_Z ∩ V_G (the boundary equals the Z-support on vertices). Such paths exist iff |S_Z ∩ V_G| is even.
Deformed Operator P̃: Given P and valid path γ, form P̃ = P · ∏_{e∈γ} Z_e.
- Same X-support on V and E as P
- Same Z-support on V as P
- Z-support on E = P's Z-support on E ⊕ γ (symmetric difference)
- Same phase as P
Key Property: P̃ commutes with all Gauss law operators A_v. This is because at each vertex v, the contribution from Z-support on V and edges in γ incident to v always sums to 0 (mod 2) by the boundary condition.
Non-Commuting Case: If P doesn't commute with L (|S_Z| odd), no valid deforming path exists, and P has no well-defined deformed version.
Minimum Weight Convention: Typically choose γ to minimize |γ| among valid paths.
This construction is fundamental to the gauging measurement: it shows how original code operators transform during the gauging procedure while maintaining compatibility with the new Gauss law constraints.