qml.math.binary_solve_linear_system

binary_solve_linear_system(A, b)

Solve the linear system of equations \(Ax=b\) over the Booleans/\(\mathbb{Z}_2\), where \(A\) is assumed to be regular, i.e., non-singular. The implementation is based on Gaussian elimination to bring the augmented matrix \((A|b)\) into (reduced) row echelon form via binary_finite_reduced_row_echelon().

Parameters:

• A (np.ndarray) – Square matrix of the linear system of equations with binary entries.

• b (np.ndarray) – Binary coefficient vector with same length as A.

Returns:

Binary solution vector with same length as A and b.

Return type:

np.ndarray

Warning

This function is currently not compatible with JAX.

Example

Consider a simple regular Boolean matrix A and a coefficient vector b:

>>> A = np.array([
...     [1, 0, 0],
...     [0, 1, 1],
...     [1, 0, 1]
... ])
>>> b = np.array([1, 1, 1])

Then we can solve the system A@x=b for x over \(\mathbb{Z}_2\):

>>> x = qml.math.binary_solve_linear_system(A, b)
>>> print(x)
[1 1 0]

Internally, this is done with the Gauss-Jordan elimination of the extended matrix (A | b). Indeed, we can verify that A @ x = b (over \(\mathbb{Z}_2\)):

>>> print(np.allclose((A @ x) % 2, b))
True

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