def cross_entropy(predictions: np.ndarray,
targets: np.ndarray,
epsilon: float = 1e-15) -> float:
"""
Cross entropy calculation between :py:attr:`targets` (encoded as one-hot vectors)
and :py:attr:`predictions`. Predictions are normalized to sum up to `1.0`.
.. note::
The implementation of this function is based on the discussion on
`StackOverflow <https://stackoverflow.com/a/47398312/10138546>`_.
Due to ArrayBoxes that are required for automatic differentiation, we currently
use this implementation instead of implementations provided by sklearn for
example.
:param predictions: Predictions in same order as targets. In case predictions for
several samples are given, the weighted cross entropy is returned.
:param targets: Ground truth labels for supplied samples.
:param epsilon: Amount to clip predictions as log is not defined for `0` and `1`.
"""
assert (
predictions.shape == targets.shape
), f"Shape of predictions {predictions.shape} must match targets {targets.shape}"
current_sum = np.sum(predictions, axis=predictions.ndim - 1)
if predictions.ndim == 1:
sample_count = 1
predictions = predictions / current_sum
else:
sample_count = predictions.shape[0]
predictions = predictions / current_sum[:, np.newaxis]
predictions = np.clip(predictions, epsilon, 1.0 - epsilon)
return -np.sum(targets * np.log(predictions)) / sample_count
def mitigate_depolarizing_noise(K, num_wires, method, use_entries=None):
r"""Estimate depolarizing noise rate(s) using on the diagonal entries of a kernel
matrix and mitigate the noise, assuming a global depolarizing noise model.
Args:
K (array[float]): Noisy kernel matrix.
num_wires (int): Number of wires/qubits of the quantum embedding kernel.
method (``'single'`` | ``'average'`` | ``'split_channel'``): Strategy for mitigation
* ``'single'``: An alias for ``'average'`` with ``len(use_entries)=1``.
* ``'average'``: Estimate a global noise rate based on the average of the diagonal
entries in ``use_entries``, which need to be measured on the quantum computer.
* ``'split_channel'``: Estimate individual noise rates per embedding, requiring
all diagonal entries to be measured on the quantum computer.
use_entries (array[int]): Diagonal entries to use if method in ``['single', 'average']``.
If ``None``, defaults to ``[0]`` (``'single'``) or ``range(len(K))`` (``'average'``).
Returns:
array[float]: Mitigated kernel matrix.
Reference:
This method is introduced in Section V in
`arXiv:2105.02276 <https://arxiv.org/abs/2105.02276>`_.
**Example:**
For an example usage of ``mitigate_depolarizing_noise`` please refer to the
`PennyLane demo on the kernel module <https://github.com/PennyLaneAI/qml/tree/master/demonstrations/tutorial_kernel_module.py>`_ or `the postprocessing demo for arXiv:2105.02276 <https://github.com/thubregtsen/qhack/blob/master/paper/post_processing_demo.py>`_.
"""
dim = 2**num_wires
if method == "single":
if use_entries is None:
use_entries = (0, )
diagonal_element = K[use_entries[0], use_entries[0]]
noise_rate = (1 - diagonal_element) * dim / (dim - 1)
mitigated_matrix = (K - noise_rate / dim) / (1 - noise_rate)
elif method == "average":
if use_entries is None:
diagonal_elements = np.diag(K)
else:
diagonal_elements = np.diag(K)[np.array(use_entries)]
noise_rates = (1 - diagonal_elements) * dim / (dim - 1)
mean_noise_rate = np.mean(noise_rates)
mitigated_matrix = (K - mean_noise_rate / dim) / (1 - mean_noise_rate)
elif method == "split_channel":
eff_noise_rates = np.clip((1 - np.diag(K)) * dim / (dim - 1), 0.0, 1.0)
noise_rates = 1 - np.sqrt(1 - eff_noise_rates)
inverse_noise = (-np.outer(noise_rates, noise_rates) +
noise_rates.reshape(
(1, len(K))) + noise_rates.reshape((len(K), 1)))
mitigated_matrix = (K - inverse_noise / dim) / (1 - inverse_noise)
return mitigated_matrix
def imshow(inp, title=None):
"""Display image from tensor."""
inp = inp.numpy().transpose((1, 2, 0))
# Inverse of the initial normalization operation.
mean = np.array([0.485, 0.456, 0.406])
std = np.array([0.229, 0.224, 0.225])
inp = std * inp + mean
inp = np.clip(inp, 0, 1)
plt.imshow(inp)
if title is not None:
plt.title(title)
def threshold_matrix(K):
r"""Remove negative eigenvalues from the given kernel matrix.
This method yields the closest positive semi-definite matrix in
any unitarily invariant norm, e.g. the Frobenius norm.
Args:
K (array[float]): Kernel matrix, assumed to be symmetric.
Returns:
array[float]: Kernel matrix with cropped negative eigenvalues.
**Example:**
Consider a symmetric matrix with both positive and negative eigenvalues:
.. code-block :: pycon
>>> K = np.array([[0, 1, 0], [1, 0, 0], [0, 0, 2]])
>>> np.linalg.eigvalsh(K)
array([-1., 1., 2.])
We then can threshold/truncate the eigenvalues of the matrix via
.. code-block :: pycon
>>> K_thresh = qml.kernels.threshold_matrix(K)
>>> np.linalg.eigvalsh(K_thresh)
array([0., 1., 2.])
If the input matrix does not have negative eigenvalues, ``threshold_matrix``
does not have any effect.
"""
w, v = np.linalg.eigh(K)
if w[0] < 0:
# Transform spectrum: Threshold/clip at 0.
w0 = np.clip(w, 0, None)
return (v * w0) @ np.transpose(v)
return K
def step(self, objective_fn, *args, **kwargs):
"""Update trainable arguments with one step of the optimizer.
Args:
objective_fn (function): the objective function for optimization
*args: variable length argument list for objective function
**kwargs: variable length of keyword arguments for the objective function
Returns:
list[array]: The new variable values :math:`x^{(t+1)}`.
If single arg is provided, list[array] is replaced by array.
"""
self.trainable_args = set()
for index, arg in enumerate(args):
if getattr(arg, "requires_grad", True):
self.trainable_args |= {index}
if self.s is None:
# Number of shots per parameter
self.s = [
np.zeros_like(a, dtype=np.int64) + self.min_shots
for i, a in enumerate(args)
if i in self.trainable_args
]
# keep track of the number of shots run
s = np.concatenate([i.flatten() for i in self.s])
self.max_shots = max(s)
self.shots_used = int(2 * np.sum(s))
self.total_shots_used += self.shots_used
# compute the gradient, as well as the variance in the gradient,
# using the number of shots determined by the array s.
grads, grad_variances = self.compute_grad(objective_fn, args, kwargs)
new_args = self.apply_grad(grads, args)
if self.xi is None:
self.chi = [np.zeros_like(g, dtype=np.float64) for g in grads]
self.xi = [np.zeros_like(g, dtype=np.float64) for g in grads]
# running average of the gradient
self.chi = [self.mu * c + (1 - self.mu) * g for c, g in zip(self.chi, grads)]
# running average of the gradient variance
self.xi = [self.mu * x + (1 - self.mu) * v for x, v in zip(self.xi, grad_variances)]
for idx, (c, x) in enumerate(zip(self.chi, self.xi)):
xi = x / (1 - self.mu ** (self.k + 1))
chi = c / (1 - self.mu ** (self.k + 1))
# determine the new optimum shots distribution for the next
# iteration of the optimizer
s = np.ceil(
(2 * self.lipschitz * self.stepsize * xi)
/ ((2 - self.lipschitz * self.stepsize) * (chi ** 2 + self.b * (self.mu ** self.k)))
)
# apply an upper and lower bound on the new shot distributions,
# to avoid the number of shots reducing below min(2, min_shots),
# or growing too significantly.
gamma = (
(self.stepsize - self.lipschitz * self.stepsize ** 2 / 2) * chi ** 2
- xi * self.lipschitz * self.stepsize ** 2 / (2 * s)
) / s
argmax_gamma = np.unravel_index(np.argmax(gamma), gamma.shape)
smax = max(s[argmax_gamma], 2)
self.s[idx] = np.squeeze(np.int64(np.clip(s, min(2, self.min_shots), smax)))
self.k += 1
# unwrap from list if one argument, cleaner return
if len(new_args) == 1:
return new_args[0]
return new_args
