Source code for strawberryfields.apps.points

# Copyright 2019 Xanadu Quantum Technologies Inc.

# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

# Unless required by applicable law or agreed to in writing, software
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# See the License for the specific language governing permissions and
r"""
Tools for building kernel matrices and generating point processes using GBS.

.. seealso::

:ref:apps-points-tutorial

Point processes
---------------

A point process is a mechanism that generates random point patterns among a set of possible
outcomes.
Point processes are statistical models that can replicate the stochastic
properties of natural phenomena, or be used as subroutines in statistical and machine learning
algorithms.

Several point processes rely on matrix functions to assign probabilities to different point
patterns. As shown in Ref. :cite:jahangiri2019point, GBS naturally gives rise to a *hafnian*
point process that employs the hafnian
<https://the-walrus.readthedocs.io/en/latest/hafnian.html>_ as the underlying matrix function.
This point process has the central property of generating clustered data points with
high probability. In this setting, a GBS device is programmed according to a *kernel* matrix that
encodes information about the similarity between points. When this kernel matrix is positive
semidefinite, it is possible to use GBS to implement a *permanental* point process and employ
fast classical algorithms to sample from the resulting distribution.

One choice of kernel matrix is the radial basis function (RBF) kernel whose elements are computed
as:

.. math::
K_{i,j} = e^{-\|\bf{r}_i-\bf{r}_j\|^2/(2\sigma^2)},

where :math:\bf{r}_i are the coordinates of point :math:i, :math:\sigma is a kernel
parameter, and :math:\|\cdot\| denotes a choice of norm. The RBF kernel is positive
semidefinite when the Euclidean norm is used, as is the case for the provided :func:rbf_kernel
function.
"""

import numpy as np
import scipy

[docs]def rbf_kernel(R: np.ndarray, sigma: float) -> np.ndarray:
r"""Calculate the RBF kernel matrix from a set of input points.

The kernel parameter :math:\sigma is used to define the kernel scale. Points that are much
further than :math:\sigma from each other lead to small entries of the kernel
matrix, whereas points much closer than :math:\sigma generate large entries.

The Euclidean norm is used to measure distance in this function, resulting in a
positive-semidefinite kernel.

**Example usage:**

>>> R = np.array([[0, 1], [1, 0], [0, 0], [1, 1]])
>>> rbf_kernel (R, 1.0)
array([[1., 0.36787944, 0.60653066, 0.60653066],
[0.36787944, 1., 0.60653066, 0.60653066],
[0.60653066, 0.60653066, 1., 0.36787944],
[0.60653066, 0.60653066, 0.36787944, 1.,]])

Args:
R (array): Coordinate matrix. Rows of this array are the coordinates of the points.
sigma (float): kernel parameter

Returns:
K (array): the RBF kernel matrix
"""
return np.exp(-((scipy.spatial.distance.cdist(R, R)) ** 2) / 2 / sigma ** 2)

[docs]def sample(K: np.ndarray, n_mean: float, n_samples: int) -> list:
"""Sample subsets of points using the permanental point process.

Points can be encoded through a radial basis function kernel, provided in :func:rbf_kernel.
Subsets of points are sampled with probabilities that are proportional to the permanent of the
submatrix of the kernel selected by those points.

This permanental point process is likely to sample points that are clustered together
:cite:jahangiri2019point. It can be realized using a variant of Gaussian boson sampling
with thermal states as input.

**Example usage:**

>>> K = np.array([[1., 0.36787944, 0.60653066, 0.60653066],
>>>               [0.36787944, 1., 0.60653066, 0.60653066],
>>>               [0.60653066, 0.60653066, 1., 0.36787944],
>>>               [0.60653066, 0.60653066, 0.36787944, 1.]])
>>> sample(K, 1.0, 10)
[[0, 1, 1, 1],
[0, 0, 0, 0],
[1, 0, 0, 0],
[0, 0, 0, 1],
[0, 1, 1, 0],
[2, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 1, 1],
[0, 0, 0, 0]]

Args:
K (array): the positive semidefinite kernel matrix
n_mean (float): average number of points per sample
n_samples (int): number of samples to be generated

Returns:
samples (list[list[int]]): samples generated by the point process
"""