Momentum

Propagators where the state evolves in momentum space. Used for gradient-descent search algorithms inspired by continuous-variable photonic quantum computing. See Theoretical Background for details.

Propagator Class

class quop_mpi.propagator.momentum.unitary(Ns: list[int], minsq: list[float], minsk: list[float], deltasq: list[float], deltask: list[float], *args, **kwargs)

Bases: Unitary

Implements the QOWE mixing unitary.

This propagator uses Fourier transforms to evolve the quantum state in momentum space, applying kinetic energy evolution.

Warning

unitary instances of type 'momentum require that the size of the MPI communicator associated with quop_mpi.Ansatz class be a factor of the first grid dimension (Ns[0] % size == 0).

Inheritance Diagram:

See quop_mpi.Unitary.

Parameters:

Nslist[int]

the number of grid points in each dimension of the Cartesian grid

minsqlist[float]

the minimum of each Cartesian coordinate in position space

minsklist[float]

the minimum of each Cartesian coordinate in momentum space

deltasqlist[float]

the step-size in each Cartesian coordinate in position space

deltasklist[float]

the step-size in each Cartesian coordinate in momentum space

*args and **kwargs:

passed to the initialisation method of quop_mpi.Unitary

Attributes:

unitary_type

'momentum'

planner

True

unitary_n_params

len(Ns)

assign_backend(backend)

Assign the Fortran backend for momentum propagation.

propagate(t)

Apply momentum-space evolution.

Parameters:

tndarray

Evolution times for each dimension

Operators

Pre-defined operator functions for momentum propagators.

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

The momentum propagator implements kinetic energy evolution for continuous-variable quantum optimization (QOWE - Quantum Optimization with Wavepacket Evolution).

Return Format

The Operator Function must return:

momentumsndarray[complex128, shape=(max(Ns), n_dims)]

A 2-D complex array where momentums[:N_i, i] contains the squared momentum values for the i-th dimension. The first dimension is max(Ns) (padded with zeros for smaller grids), and the second dimension corresponds to the number of grid dimensions.

The momentum values are computed from the grid parameters (minsk, deltask) specified when constructing the unitary.

Propagation Method

The momentum propagator computes \(e^{-itT}|\psi\rangle\) where \(T\) is the kinetic energy operator, using multi-dimensional FFT:

1. Forward multi-dimensional FFT transforms the state from position to momentum space

2. Multiply by \(e^{-it(k_1^2 + k_2^2 + \cdots)}\) (kinetic phase)

3. Inverse multi-dimensional FFT transforms back to position space

The implementation uses FFTW with MPI for parallel multi-dimensional transforms. Grid parameters (Ns, minsq, deltasq, etc.) are specified when constructing the unitary.

quop_mpi.propagator.momentum.operator.magnitude_squared(Ns: list[int], minsk: list[float], deltask: list[float]) → np.ndarray[np.complex128]

Generate the QMOA mixing unitary operator.

An Operator Function for quop_mpi.propagator.momentum.unitary.

Parameters:

Nslist[int]

the number of grid points in each dimension of the Cartesian grid in position and momentum space, quop_mpi.propagator.momentum.unitary attribute

minsklist[float]

the minimum of each Cartesian coordinate in momentum space, quop_mpi.propagator.momentum.unitary attribute

deltasklist[float]

the step-size in each Cartesian coordinate in momentum space, quop_mpi.propagator.momentum.unitary attribute

Returns:

np.ndarray[np.complex128]

a 1-D complex array of local_i elements of the QOWE diagonal momentum-space operator with global index offset local_i_offset