setup_approximate_model

setup_approximate_model(lse, *, max_l1_norm=10.0)[source]

Construct a cvxpy.Problem for finding approximate static MPF coefficients.

The optimization problem constructed by this class is defined as follows:

  • the cost function minimizes the sum of squares (sum_squares()) of the distances to an exact solution for all equations of the LSE:

    \[\sum_i \left( \sum_j A_{ij} x_j - b_i \right)^2\]

  • two constraints are set:

    1. the variables must sum to 1: \(\sum_i x_i == 1\)

    2. the L1-norm (norm1) of the variables is bounded by max_l1_norm

Here is an example:

>>> from qiskit_addon_mpf.static import setup_lse, setup_approximate_model
>>> lse = setup_lse([1,2,3], order=2, symmetric=True)
>>> problem, coeffs = setup_approximate_model(lse, max_l1_norm=3.0)
>>> print(problem)
minimize quad_over_lin(Vstack([1. 1.     1.]         @ x + -1.0,
                              [1. 0.25   0.11111111] @ x + -0.0,
                              [1. 0.0625 0.01234568] @ x + -0.0), 1.0)
subject to Sum(x, None, False) == 1.0
           norm1(x) <= 3.0

You can then solve the problem and extract the expansion coefficients like so:

>>> final_cost = problem.solve()
>>> print(coeffs.value)  
[ 0.03513467 -1.          1.96486533]
Parameters:

  • lse (LSE) – the linear system of equations from which to build the model.

  • max_l1_norm (float) – the upper limit to use for the constrain of the L1-norm of the variables.

Returns:

The optimization problem and coefficients variable.

Return type:

tuple[Problem, Variable]

References

[1]: S. Zhuk et al., arXiv:2306.12569 (2023).

https://arxiv.org/abs/2306.12569