# Source code for pennylane.kernels.utils

# Copyright 2018-2021 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
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
"""
This file contains functionalities that simplify working with kernels.
"""
from itertools import product
import pennylane as qml

[docs]def square_kernel_matrix(X, kernel, assume_normalized_kernel=False):
r"""Computes the square matrix of pairwise kernel values for a given dataset.

Args:
X (list[datapoint]): List of datapoints
kernel ((datapoint, datapoint) -> float): Kernel function that maps
datapoints to kernel value.
assume_normalized_kernel (bool, optional): Assume that the kernel is normalized, in
which case the diagonal of the kernel matrix is set to 1, avoiding unnecessary
computations.

Returns:
array[float]: The square matrix of kernel values.

**Example:**

Consider a simple kernel function based on :class:~.templates.embeddings.AngleEmbedding:

.. code-block :: python

dev = qml.device('default.qubit')
@qml.qnode(dev)
def circuit(x1, x2):
qml.templates.AngleEmbedding(x1, wires=dev.wires)
return qml.probs(wires=dev.wires)

kernel = lambda x1, x2: circuit(x1, x2)[0]

We can then compute the kernel matrix on a set of 4 (random) feature
vectors X via

>>> X = np.random.random((4, 2))
>>> qml.kernels.square_kernel_matrix(X, kernel)
tensor([[1.        , 0.9532702 , 0.96864001, 0.90932897],
[0.9532702 , 1.        , 0.99727485, 0.95685561],
[0.96864001, 0.99727485, 1.        , 0.96605621],
[0.90932897, 0.95685561, 0.96605621, 1.        ]], requires_grad=True)
"""
N = qml.math.shape(X)[0]
if assume_normalized_kernel and N == 1:
return qml.math.eye(1, like=qml.math.get_interface(X))

matrix = [None] * N**2

# Compute all off-diagonal kernel values, using symmetry of the kernel matrix
for i in range(N):
for j in range(i + 1, N):
matrix[N * i + j] = (kernel_value := kernel(X[i], X[j]))
matrix[N * j + i] = kernel_value

if assume_normalized_kernel:
# Create a one-like entry that has the same interface and batching as the kernel output
# As we excluded the case N=1 together with assume_normalized_kernel above, matrix[1] exists
one = qml.math.ones_like(matrix[1])
for i in range(N):
matrix[N * i + i] = one
else:
# Fill the diagonal by computing the corresponding kernel values
for i in range(N):
matrix[N * i + i] = kernel(X[i], X[i])

shape = (N, N) if qml.math.ndim(matrix[0]) == 0 else (N, N, qml.math.size(matrix[0]))

return qml.math.moveaxis(qml.math.reshape(qml.math.stack(matrix), shape), -1, 0)

[docs]def kernel_matrix(X1, X2, kernel):
r"""Computes the matrix of pairwise kernel values for two given datasets.

Args:
X1 (list[datapoint]): List of datapoints (first argument)
X2 (list[datapoint]): List of datapoints (second argument)
kernel ((datapoint, datapoint) -> float): Kernel function that maps datapoints to kernel value.

Returns:
array[float]: The matrix of kernel values.

**Example:**

Consider a simple kernel function based on :class:~.templates.embeddings.AngleEmbedding:

.. code-block :: python

dev = qml.device('default.qubit')
@qml.qnode(dev)
def circuit(x1, x2):
qml.templates.AngleEmbedding(x1, wires=dev.wires)
return qml.probs(wires=dev.wires)

kernel = lambda x1, x2: circuit(x1, x2)[0]

With this method we can systematically evaluate the kernel function kernel on
pairs of datapoints, where the points stem from different datasets, like a training
and a test dataset.

>>> X_train = np.random.random((4,2))
>>> X_test = np.random.random((3,2))
>>> qml.kernels.kernel_matrix(X_train, X_test, kernel)
tensor([[0.88875298, 0.90655175, 0.89926447],
[0.93762197, 0.98163781, 0.93076383],
[0.91977339, 0.9799841 , 0.91582698],

As we can see, for :math:n and :math:m datapoints in the first and second
dataset respectively, the output matrix has the shape :math:n\times m.
"""
N = qml.math.shape(X1)[0]
M = qml.math.shape(X2)[0]

matrix = qml.math.stack([kernel(x, y) for x, y in product(X1, X2)])

if qml.math.ndim(matrix[0]) == 0:
return qml.math.reshape(matrix, (N, M))

return qml.math.moveaxis(qml.math.reshape(matrix, (N, M, qml.math.size(matrix[0]))), -1, 0)


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