Composite

Propagators for operators that are Cartesian sums of circulant operators, where each component mixes over a subregister. Used for optimisation of discretised continuous multivariable functions and combinatorial problems with decision variables of arity three or higher.

Propagator Class

class quop_mpi.propagator.composite.Unitary(Ns, *args, **kwargs)

Bases: UnitaryBase

Unitary propagator for composite (multidimensional) circulant operators.

Handles tensor-product structure of circulant graphs across multiple coordinate dimensions.

assign_backend(backend): Assign the compute backend for composite propagation.

propagate(t): Apply the composite propagator with parameters t.

Operators

Pre-defined operator functions for composite propagators.

Predefined Operator Functions for quop_mpi.propagator.composite.unitary.

The composite propagator handles operators that are Cartesian sums of circulant operators acting on subregisters. This is used for multivariable optimization problems where the mixing operator decomposes as:

\[H = H_1 \otimes I \otimes \cdots + I \otimes H_2 \otimes \cdots + \cdots\]

where each \(H_i\) is a circulant matrix acting on a subregister.

Return Format

The Operator Function must return:

eigenvaluesndarray[float64, shape=(max(Ns), n_dims)]
A 2-D real array where eigenvalues[:N_i, i] contains the eigenvalues of the i-th circulant suboperator. The first dimension is max(Ns) (padded with zeros for smaller grids), and the second dimension corresponds to the number of grid dimensions.

Propagation Method

The composite propagator exploits the Cartesian sum structure using multi-dimensional FFT:

Forward multi-dimensional FFT transforms the state to the joint eigenbasis
Multiply by \(e^{-it(\lambda^{(1)}_{k_1} + \lambda^{(2)}_{k_2} + \cdots)}\) where \(\lambda^{(d)}_{k_d}\) are eigenvalues for each dimension
Inverse multi-dimensional FFT transforms back to the computational basis

This avoids constructing the full \(\prod_i N_i\) sized operator by exploiting the separable structure of Cartesian sums. The implementation uses FFTW with MPI for parallel multi-dimensional transforms.

quop_mpi.propagator.composite.operator.ith(Ns, Cs=None)

Generate the eigenvalues of a QMOA mixing unitary operator.

An Operator Function for quop_mpi.propagator.composite.unitary. The Cs keyword argument may be defined via a corresponding FunctionDict on initialisation of the receiving unitary instance.

Parameters:

Nslist[int]: the number of grid points in each dimension of the Cartesian grid, quop_mpi.propagator.composite.unitary attribute
Cslist[int], optional: specifies the i-th index of the circulant operators associated with each dimension of the Cartesian grid, complete graphs by default

Returns:

ndarray[complex128]: a 2-D complex array containing local_i eigenvalues of the QMOA mixing unitary with global index offset local_i_offset