Def_2: Gauss's Law Operators #
Statement #
Given a connected graph $G = (V_G, E_G)$ whose vertices are identified with the qubits in the support of a logical operator $L = \prod_{v \in V_G} X_v$, the Gauss's law operators are the set $\mathcal{A} = \{A_v\}_{v \in V_G}$ where: $$A_v = X_v \prod_{e \ni v} X_e$$ Here $X_v$ is the Pauli-$X$ operator on the vertex qubit $v$, and $X_e$ is the Pauli-$X$ operator on the edge qubit $e$. The product $\prod_{e \ni v}$ is over all edges incident to vertex $v$.
The Gauss's law operators satisfy:
- Each $A_v$ is Hermitian with eigenvalues $\pm 1$.
- All $A_v$ mutually commute: $[A_v, A_{v'}] = 0$ for all $v, v' \in V_G$.
- $\prod_{v \in V_G} A_v = L \cdot \prod_{e \in E_G} X_e^{2} = L$ (since $X_e^2 = I$).
This last property is the key to the gauging measurement: measuring all $A_v$ and multiplying the outcomes yields the eigenvalue of $L$.
Main Definitions #
GaussLawOperator: The support of a Gauss law operator A_v as binary vectorsgaussLawOperator_vertexSupport: The vertex support (1 at v, 0 elsewhere)gaussLawOperator_edgeSupport: The edge support (1 at incident edges, 0 elsewhere)
Key Properties #
gaussLaw_hermitian: A_v² = I (implies Hermitian with eigenvalues ±1)gaussLaw_commute: [A_v, A_w] = 0 for all v, w (X-type operators always commute)gaussLaw_product_vertex_support: Product of vertex supports is the all-ones vector (= L)gaussLaw_product_edge_support: Product of edge supports is zero (edges cancel pairwise)
Corollaries #
gaussLaw_product_is_L: ∏_v A_v has vertex support = all 1s (represents L)gaussLaw_independent_count: |V| - 1 independent generators
Gauss Law Operator Support #
A Gauss law operator A_v is an X-type Pauli operator. We represent it by its support:
- On vertex qubits: 1 at position v, 0 elsewhere
- On edge qubits: 1 at edges incident to v, 0 elsewhere
In the binary vector representation (Rem_3), the support encodes where X operators act.
The vertex support of Gauss law operator A_v: a binary vector with 1 at v, 0 elsewhere. This represents X_v in the product A_v = X_v ∏_{e∋v} X_e.
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The edge support of Gauss law operator A_v: 1 at edges incident to v, 0 elsewhere. This represents ∏{e∋v} X_e in the product A_v = X_v ∏{e∋v} X_e.
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Basic Properties of Gauss Law Operator Support #
The edge support at an edge e
Property 1: A_v² = I (Hermitian with eigenvalues ±1) #
In the binary vector representation over ZMod 2:
- X² = I means the support XOR'd with itself gives 0
- Since x + x = 0 in ZMod 2 for any x, we have A_v² = I
This implies A_v is Hermitian (since X† = X, products of X are Hermitian) and has eigenvalues ±1 (from A² = I, any eigenvalue λ satisfies λ² = 1).
In ZMod 2, any element added to itself is 0
A_v² = I on vertex support: support + support = 0 (in ZMod 2)
A_v² = I on edge support: support + support = 0 (in ZMod 2)
Property 1: A_v is Hermitian with eigenvalues ±1. Represented by A_v² = I, which in ZMod 2 is: 2 • support = 0. If A|ψ⟩ = λ|ψ⟩ and A² = I, then λ² = 1, so λ ∈ {-1, +1}.
Property 2: All A_v mutually commute #
For Pauli operators, [A, B] = 0 iff the symplectic form ω(A, B) ≡ 0 (mod 2), where: ω(A, B) = |supp_X(A) ∩ supp_Z(B)| + |supp_Z(A) ∩ supp_X(B)|
Since Gauss law operators are X-type (only X, no Z), they have:
- supp_Z(A_v) = ∅ for all v
Therefore ω(A_v, A_w) = |supp_X(A_v) ∩ ∅| + |∅ ∩ supp_X(A_w)| = 0.
The Z-support of a Gauss law operator is empty (X-type operators have no Z component)
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- _G.gaussLaw_ZSupport_vertex _v = ∅
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The Z-support on edges is also empty
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- _G.gaussLaw_ZSupport_edge _v = ∅
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The symplectic form between two Gauss law operators. For X-type operators: ω(A_v, A_w) = |X_v ∩ Z_w| + |Z_v ∩ X_w| = 0 + 0 = 0
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- G.gaussLaw_symplectic v w = (G.gaussLaw_ZSupport_vertex w).card + (G.gaussLaw_ZSupport_vertex v).card
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The symplectic form is zero for X-type operators
Property 2: Two Gauss law operators commute. [A_v, A_w] = 0 since ω(A_v, A_w) = 0 (X-type operators always commute).
Property 3: ∏_{v ∈ V} A_v = L #
The product of all Gauss law operators:
- On vertices: Each vertex v contributes 1 at position v, so sum = all 1s = support of L
- On edges: Each edge e is incident to exactly 2 vertices, so appears twice in the sum. In ZMod 2, 1 + 1 = 0, so edge contribution cancels.
Therefore ∏_v A_v = L (the logical operator with support on all vertices).
Sum of all vertex supports: ∑_v (A_v vertex support)
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- G.gaussLaw_product_vertexSupport = ∑ v : V, G.gaussLawOperator_vertexSupport v
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Sum of all edge supports: ∑_v (A_v edge support)
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- G.gaussLaw_product_edgeSupport = ∑ v : V, G.gaussLawOperator_edgeSupport v
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Each vertex w appears in exactly one A_v (namely when v = w), so the sum at w is 1
The product of all A_v has vertex support equal to the all-ones vector. This is the support of L = ∏_{v ∈ V} X_v.
Each edge e is incident to exactly 2 vertices (its endpoints). In ZMod 2, 1 + 1 = 0, so edge contributions cancel.
The product of all A_v has edge support equal to zero (edges cancel pairwise). This represents X_e² = I for each edge.
Property 3: The product of all Gauss law operators equals L.
- Vertex support: all 1s (= L = ∏_v X_v)
- Edge support: all 0s (since X_e² = I, edge terms cancel)
This is the key property: measuring all A_v and multiplying outcomes gives the eigenvalue of L.
Relationship to Coboundary Map #
The edge support of A_v equals δ(v) where δ is the coboundary map from Def_1. This connects Gauss law operators to the chain complex structure.
The edge support of A_v equals the coboundary of the basis vector at v
The sum of all edge supports equals the coboundary of the all-ones vector
Group Structure: |V| - 1 Independent Generators #
The Gauss law operators satisfy one constraint: ∏_v A_v = L. This means of the |V| generators, only |V| - 1 are independent. The abelian group generated by {A_v} has order 2^{|V|-1}.
Number of Gauss law operators equals number of vertices
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Number of constraints among Gauss law operators (the single product constraint)
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Number of independent Gauss law generators = |V| - 1
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The independent count is |V| - 1
The abelian group generated by {A_v} has order 2^{|V|-1}
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The constraint: sum of all generators (in ZMod 2) equals the all-ones vector
There exists a generator that is a linear combination of the others (linear dependency)
Support Size and Degree #
The support size of A_v is 1 + degree(v):
- 1 for the vertex qubit v
- degree(v) for the edge qubits incident to v
The degree of vertex v in the graph
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- G.vertexDegree v = (G.incidentEdges v).card
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The support size of A_v on edge qubits equals the degree of v
Total support size of A_v is 1 (vertex) + degree(v) (edges)
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- G.gaussLawOperator_totalSupport v = 1 + G.vertexDegree v
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Equivalence with Coboundary in Chain Complex #
The Gauss law operators are closely related to the coboundary map δ. Specifically, A_v has edge support δ(e_v) where e_v is the basis vector at v.
The Gauss law operator A_v corresponds to:
- Vertex support: e_v (basis vector at v)
- Edge support: δ(e_v) (coboundary of basis vector at v) This shows A_v = X_v · ∏_{e∋v} X_e in the binary vector representation.
Measurement Property #
The key property for the gauging measurement: Measuring all A_v and multiplying the outcomes (±1) gives the eigenvalue of L.
In our ZMod 2 representation:
- Each measurement outcome is represented as 0 (for +1) or 1 (for -1)
- The product of outcomes corresponds to XOR (addition in ZMod 2)
- The total measurement result equals the eigenvalue of L
Convert a measurement outcome to ZMod 2: +1 → 0, -1 → 1
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Product of measurement outcomes as ZMod 2 sum
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- GraphWithCycles.measurementProduct outcomes = ∑ v : V, GraphWithCycles.measurementToZMod2 (outcomes v)
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The measurement of all A_v determines the eigenvalue of L. The XOR of all measurement outcomes (in ZMod 2) equals the L eigenvalue: 0 means L eigenvalue +1, 1 means L eigenvalue -1. Sum = 0 iff even number of -1 outcomes.
If all outcomes are +1, the product is 0 (L eigenvalue +1)
Summary #
The Gauss law operators formalize the key concept from the gauging measurement protocol:
Definition: A_v = X_v ∏_{e∋v} X_e represented as binary vectors over ZMod 2
- Vertex support: basis vector e_v (1 at v, 0 elsewhere)
- Edge support: coboundary δ(e_v) (1 at incident edges, 0 elsewhere)
Property 1 (Hermitian): A_v² = I
- In ZMod 2: support + support = 0
- Implies eigenvalues are ±1
Property 2 (Commutativity): [A_v, A_w] = 0 for all v, w
- X-type operators have zero symplectic form
- All Gauss law operators mutually commute
Property 3 (Product Constraint): ∏_v A_v = L
- Vertex support: all 1s (= support of L = ∏_v X_v)
- Edge support: 0 (edges cancel since each appears twice, and X² = I)
Group Structure: |V| generators with 1 constraint give |V|-1 independent generators
- The generated abelian group has order 2^{|V|-1}
Measurement: Multiplying all A_v outcomes gives the L eigenvalue
- Key to the gauging measurement protocol