30 Thm 1: Gauging Measurement
This chapter formalizes Theorem 1: the gauging procedure on a connected graph is equivalent to performing a projective measurement of the logical operator \(L = \prod _v X_v\). After measuring all Gauss law operators \(A_v\) with outcomes \(\varepsilon _v\) and all \(Z_e\) with outcomes \(z_e\), the post-measurement state is proportional to \(X_V(c')(I + \sigma L)|\psi \rangle \), where \(\sigma = \prod _v \varepsilon _v\) and \(c'\) is determined by the edge outcomes.
The Gauss law measurement outcomes assign to each vertex \(v \in V\) a value in \(\mathbb {Z}/2\mathbb {Z}\), where \(0\) represents the outcome \(+1\) and \(1\) represents the outcome \(-1\). Formally, \(\mathrm{GaussLawOutcomes}(V) = V \to \mathbb {Z}/2\mathbb {Z}\).
The measured outcome \(\sigma \) is the product of all Gauss law outcomes, represented in \(\mathbb {Z}/2\mathbb {Z}\) as
Here \(\sigma = 0\) means \(+1\) (even number of \(-1\) outcomes) and \(\sigma = 1\) means \(-1\) (odd number).
For a \(0\)-cochain \(c \in (\mathbb {Z}/2\mathbb {Z})^V\) and outcomes \(\varepsilon \), define
This represents \(\prod _{v : c_v = 1} \varepsilon _v^{c_v}\) in additive notation.
For a \(0\)-cochain \(c \in (\mathbb {Z}/2\mathbb {Z})^V\), the operator \(X_V(c) = \prod _{v : c_v = 1} X_v\) is represented by its support vector, which is simply \(c\) itself.
The logical operator \(L = \prod _v X_v\) has support equal to the all-ones vector \(\mathbf{1} \in (\mathbb {Z}/2\mathbb {Z})^V\).
For any outcomes \(\varepsilon \),
By definition, \(\varepsilon (\mathbf{1}) = \sum _{v \in V} 1 \cdot \varepsilon _v = \sum _{v \in V} \varepsilon _v = \sigma \). This follows by simplification using the definitions of \(\varepsilon \), \(\sigma \), and \(\mathbf{1}\), noting that \(1 \cdot x = x\).
For any outcomes \(\varepsilon \) and \(0\)-cochains \(c, c'\),
By the definition of \(\varepsilon \), we have \(\varepsilon (c + c') = \sum _v (c_v + c'_v) \cdot \varepsilon _v\). Using the distributivity of multiplication over addition, \((c_v + c'_v) \cdot \varepsilon _v = c_v \cdot \varepsilon _v + c'_v \cdot \varepsilon _v\). By distributing the sum, \(\sum _v (c_v \cdot \varepsilon _v + c'_v \cdot \varepsilon _v) = \sum _v c_v \cdot \varepsilon _v + \sum _v c'_v \cdot \varepsilon _v = \varepsilon (c) + \varepsilon (c')\).
For any outcomes \(\varepsilon \) and \(0\)-cochain \(c\),
For a connected graph \(G\), if \(\delta c = 0\) then \(c = 0\) or \(c = \mathbf{1}\). That is, \(\ker (\delta ) = \{ 0, \mathbf{1}\} \).
This follows directly from the classification of the kernel of the coboundary map for connected graphs established in Rem 7 (the exactness of boundary/coboundary sequence). Applying ker_coboundary_classification to \(c\), the hypothesis \(\delta c = 0\), and the connectivity of \(G\) yields the result.
For a connected graph \(G\) and any \(z \in (\mathbb {Z}/2\mathbb {Z})^E\), if \(c'\) satisfies \(\delta c' = z\), then for all \(c \in (\mathbb {Z}/2\mathbb {Z})^V\):
That is, the fiber \(\{ c : \delta c = z\} \) has exactly two elements: \(c'\) and \(c' + \mathbf{1}\).
Let \(c\) be arbitrary. We prove both directions.
\((\Rightarrow )\): Assume \(\delta c = z\). Since \(\delta c' = z\) as well, we compute \(\delta (c + c') = \delta c + \delta c' = z + z = 0\) in \(\mathbb {Z}/2\mathbb {Z}\) (using that \(x + x = 0\) for all \(x \in \mathbb {Z}/2\mathbb {Z}\)). By Theorem 985 (kernel classification for connected graphs), \(c + c' = 0\) or \(c + c' = \mathbf{1}\).
In the first case, \(c + c' = 0\), so for each vertex \(v\) we have \(c_v + c'_v = 0\). In \(\mathbb {Z}/2\mathbb {Z}\), \(x + y = 0\) if and only if \(x = y\), so \(c = c'\).
In the second case, \(c + c' = \mathbf{1}\), so for each vertex \(v\) we have \(c_v + c'_v = 1\). By case analysis on the values of \(c_v\) and \(c'_v\) in \(\{ 0, 1\} \): if both are \(0\) then \(0 + 0 = 0 \neq 1\), contradiction; if both are \(1\) then \(1 + 1 = 0 \neq 1\), contradiction; in the remaining cases \(c_v = c'_v + 1\). Therefore \(c = c' + \mathbf{1}\).
\((\Leftarrow )\): If \(c = c'\), then \(\delta c = \delta c' = z\). If \(c = c' + \mathbf{1}\), then \(\delta c = \delta (c' + \mathbf{1}) = \delta c' + \delta \mathbf{1} = z + 0 = z\), since \(\mathbf{1} \in \ker (\delta )\) for connected graphs.
For a connected graph \(G\), outcomes \(\varepsilon \), and \(c'\) with \(\delta c' = z\), let \(c_0 = c'\) and \(c_1 = c' + \mathbf{1}\). Then:
\(\varepsilon (c_0) = \varepsilon (c')\) and \(\varepsilon (c_1) = \varepsilon (c') + \sigma \),
\(X_V(c_0) = X_V(c')\) and \(X_V(c_1) = X_V(c') + L\).
This shows the sum over the fiber \(\{ c : \delta c = z\} \) has exactly two terms whose contributions combine as \(\varepsilon (c') X_V(c')(I + \sigma L)\).
The first parts of each pair hold by reflexivity (\(c_0 = c'\)). For the second parts: \(\varepsilon (c_1) = \varepsilon (c' + \mathbf{1}) = \varepsilon (c') + \sigma \) by Lemma 984, and \(X_V(c_1) = X_V(c' + \mathbf{1}) = X_V(c') + L\) by definition.
The second term in the fiber contributes with phase \(\varepsilon (c') + \sigma \) and operator support \(X_V(c') + L\):
The first equation follows from Lemma 984. The second equation holds by definition of \(X_V\) and \(L\).
In terms of support vectors, \(L + L = 0\), i.e., \(\mathbf{1} + \mathbf{1} = 0\) in \((\mathbb {Z}/2\mathbb {Z})^V\).
By extensionality, for each vertex \(v\), \((\mathbf{1} + \mathbf{1})_v = 1 + 1 = 0\) in \(\mathbb {Z}/2\mathbb {Z}\). This is verified by computation.
For any \(\sigma \in \mathbb {Z}/2\mathbb {Z}\), \(\sigma + \sigma = 0\).
By case analysis on \(\sigma \in \{ 0, 1\} \): if \(\sigma = 0\) then \(0 + 0 = 0\); if \(\sigma = 1\) then \(1 + 1 = 0\) in \(\mathbb {Z}/2\mathbb {Z}\). Both cases are verified by computation.
The projector \(\frac{1}{2}(I + \sigma L)\) is idempotent. In the \(\mathbb {Z}/2\mathbb {Z}\) representation, this reduces to the fact that \(\sigma + \sigma = 0\).
This follows directly from Theorem 990.
For any \(\sigma \in \mathbb {Z}/2\mathbb {Z}\), \(\sigma \cdot \sigma = \sigma \).
By case analysis: if \(\sigma = 0\) then \(0 \cdot 0 = 0\); if \(\sigma = 1\) then \(1 \cdot 1 = 1\). Both cases are verified by computation.
On the \(\sigma \)-eigenspace of \(L\) where \(L|\psi _\sigma \rangle = \sigma |\psi _\sigma \rangle \), the projector \(\frac{1}{2}(I + \sigma L)\) acts as the identity. The key algebraic property is \(\sigma \cdot \sigma = \sigma \) in \(\mathbb {Z}/2\mathbb {Z}\).
This follows directly from Theorem 992.
On the \((-\sigma )\)-eigenspace of \(L\) where \(L|\psi _{-\sigma }\rangle = -\sigma |\psi _{-\sigma }\rangle \), the projector \(\frac{1}{2}(I + \sigma L)\) annihilates the state. The key algebraic property is again \(\sigma \cdot \sigma = \sigma \).
This follows directly from Theorem 992.
Given \(z \in (\mathbb {Z}/2\mathbb {Z})^E\) in the image of \(\delta \) (i.e., there exists \(c\) with \(\delta c = z\)), the byproduct cochain \(c' = \mathrm{byproductCochain}(G, z)\) is a chosen \(0\)-cochain satisfying \(\delta c' = z\), obtained by the axiom of choice.
The byproduct cochain satisfies \(\delta (\mathrm{byproductCochain}(G, z)) = z\).
This holds by the specification of the choice function: \(\mathrm{byproductCochain}\) is defined as \(\mathrm{hz.choose}\), and its specification is \(\mathrm{hz.choose\_ spec}\).
(Gauging Measurement Equivalence.) Let \(G\) be a connected graph with cycles, let \(\varepsilon \) be the Gauss law measurement outcomes, and let \(z \in (\mathbb {Z}/2\mathbb {Z})^E\) be in the image of \(\delta \). Let \(c' = \mathrm{byproductCochain}(G, z)\) and \(\sigma = \sigma (\varepsilon )\). Then:
Fiber structure: For all \(c\), \(\delta c = z \iff (c = c' \text{ or } c = c' + \mathbf{1})\).
Phase relation: \(\varepsilon (c' + \mathbf{1}) = \varepsilon (c') + \sigma \).
Operator relation: \(X_V(c' + \mathbf{1}) = X_V(c') + L\).
Projector properties: \(\sigma + \sigma = 0\) and \(L + L = 0\) (i.e., \(\sigma ^2 = 1\) and \(L^2 = I\)).
Idempotence: \(\sigma \cdot \sigma = \sigma \).
Together, these establish that the gauging procedure is equivalent to projective measurement of \(L\) with eigenvalue \(\sigma \), up to the byproduct operator \(X_V(c')\).
We prove each part:
Part (1): The fiber structure follows directly from Theorem 986 applied to \(G\), the connectivity hypothesis, \(z\), the byproduct cochain \(c'\), and its specification \(\delta c' = z\) (Lemma 996).
Part (2): The phase relation \(\varepsilon (c' + \mathbf{1}) = \varepsilon (c') + \sigma \) follows from Lemma 984.
Part (3): The operator relation \(X_V(c' + \mathbf{1}) = X_V(c') + L\) holds by definition of \(X_V\) and \(L\).
Part (4): The projector properties \(\sigma + \sigma = 0\) and \(L + L = 0\) follow from Theorem 990 and Theorem 989, respectively.
Part (5): The idempotence \(\sigma \cdot \sigma = \sigma \) follows from Theorem 992.
The measured outcome \(\sigma \in \{ 0, 1\} \), which is trivially true since \(\sigma \in \mathbb {Z}/2\mathbb {Z}\).
For any \(x \in \mathbb {Z}/2\mathbb {Z}\), by case analysis: \(x = 0\) or \(x = 1\). Applying this to \(\sigma \) gives the result.
\(\sigma = 0\) if and only if \(|\{ v \in V : \varepsilon _v = 1\} |\) is even. That is, the product of all outcomes is \(+1\) precisely when an even number of vertices have outcome \(-1\).
We show that \(\sum _{v \in V} \varepsilon _v = |\{ v \in V : \varepsilon _v = 1\} |\) in \(\mathbb {Z}/2\mathbb {Z}\). We split the sum over \(V\) into vertices with \(\varepsilon _v = 1\) and vertices with \(\varepsilon _v \neq 1\).
For vertices with \(\varepsilon _v \neq 1\): since each \(\varepsilon _v \in \{ 0, 1\} \), we must have \(\varepsilon _v = 0\), so these contribute \(0\) to the sum.
For vertices with \(\varepsilon _v = 1\): each contributes \(1\), so the sum over this set equals \(|\{ v : \varepsilon _v = 1\} | \cdot 1 = |\{ v : \varepsilon _v = 1\} |\) in \(\mathbb {Z}/2\mathbb {Z}\).
Therefore \(\sigma = |\{ v : \varepsilon _v = 1\} |\) in \(\mathbb {Z}/2\mathbb {Z}\), and \(\sigma = 0\) if and only if this cardinality is even, by the characterization of when a natural number maps to zero in \(\mathbb {Z}/2\mathbb {Z}\).
For a connected graph \(G\), if \(c'\) and \(c''\) both satisfy \(\delta c' = z\) and \(\delta c'' = z\), then \(c'' = c'\) or \(c'' = c' + \mathbf{1}\). That is, the byproduct cochain is determined up to multiplication by \(L\).
This follows by applying the forward direction of Theorem 986: since \(\delta c'' = z\) and \(\delta c' = z\), the fiber characterization gives \(c'' = c'\) or \(c'' = c' + \mathbf{1}\).