neuralqx.operators.computational.Euclidean4d.numba.area_difference module

class AreaDifferenceSquaredOperator(H, edges)

Bases: ComputationalOperator

Diagonal operator implementing the squared area difference between two edges in the U(1)^3 model:

\[(\hat A_{e_1} - \hat A_{e_2})^2 \;=\; \hat A_{e_1}^2 + \hat A_{e_2}^2 - 2\,\hat A_{e_1}\hat A_{e_2},\]

where for a single edge (e), (hat A_e = |vec m_e|_2 = sqrt{(m_e^{(1)})^2 + (m_e^{(2)})^2 + (m_e^{(3)})^2}).

Since this operator is diagonal in the (|vec mrangle) basis, its matrix element on a configuration (sigma) is simply

\[\big(\|\vec m_{e_1}\|_2 - \|\vec m_{e_2}\|_2\big)^2 \;=\; \|\vec m_{e_1}\|_2^2 + \|\vec m_{e_2}\|_2^2 \;-\; 2\,\|\vec m_{e_1}\|_2\,\|\vec m_{e_2}\|_2.\]

The operator precomputes the three component indices for each of the two edges using the graph’s edge_to_index and static gauge offsets. This avoids any Python-side control-flow inside jitted code and prevents recompilation.

Notes

• Assumes U(1)^3 (gauge_dim == 3) and unit overall normalization (8πℓₚ² absorbed)

• If both edges are the same, the operator correctly yields zero everywhere

property is_hermitian: bool

This function must return either True or False based on whether your operator is Hermitian or not. Note that unlike for the case of LocalOperator types, you must specify by-hand whether this operator is Hermitian or not, there is no implementation to deduce that information for you as there is no matrices stored in this operator type.

This property plays a role in determining the computational path to be taken when computing gradients. If you specify that the operator is Hermitian while in reality it is not, the computed gradients will be incorrect.

Returns:

True if this operator is Hermitian, False otherwise

property dtype

Specify a JAX NumPy or NumPy dtype for this operator (that is, what is the dtype of the matrix elements returned by this operator)

class AreaDifferenceSquaredSurfacesOperator(H, surfaces)

Bases: ComputationalOperator

Diagonal operator implementing the squared area difference between two surfaces (each a list of edges) in the U(1)^3 model:

\[(\hat A(S_1) - \hat A(S_2))^2 \;=\; \hat A(S_1)^2 + \hat A(S_2)^2 - 2\,\hat A(S_1)\hat A(S_2),\]

where for a surface (S = {e_i}),

\[\hat A(S) = \sum_{e_i \in S} \sqrt{ (m_{e_i}^{(1)})^2 + (m_{e_i}^{(2)})^2 + (m_{e_i}^{(3)})^2 }.\]

This operator is diagonal in the (|vec mrangle) basis, and so its matrix element on a configuration (sigma) is simply

\[\big(\hat A(S_1) - \hat A(S_2)\big)^2.\]

Notes

• Assumes U(1)^3 (gauge_dim == 3) and unit normalization (8πℓₚ² absorbed).

• If both surfaces are identical, the operator yields zero everywhere.

property is_hermitian: bool

This function must return either True or False based on whether your operator is Hermitian or not. Note that unlike for the case of LocalOperator types, you must specify by-hand whether this operator is Hermitian or not, there is no implementation to deduce that information for you as there is no matrices stored in this operator type.

This property plays a role in determining the computational path to be taken when computing gradients. If you specify that the operator is Hermitian while in reality it is not, the computed gradients will be incorrect.

Returns:

True if this operator is Hermitian, False otherwise

property dtype

Specify a JAX NumPy or NumPy dtype for this operator (that is, what is the dtype of the matrix elements returned by this operator)