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# This code is part of Qiskit.
#
# (C) Copyright IBM 2018, 2022.
#
# obtain a copy of this license in the LICENSE.txt file in the root directory
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.

"""
"""

from typing import Union, Tuple
import itertools as it
from functools import reduce
import numpy as np
from qiskit.utils import algorithm_globals, optionals

from qiskit_machine_learning.datasets.dataset_helper import (
features_and_labels_transform,
)

[docs]def ad_hoc_data( training_size: int, test_size: int, n: int, gap: int, plot_data: bool = False, one_hot: bool = True, include_sample_total: bool = False, ) -> Union[ Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray], ]: r"""Generates a toy dataset that can be fully separated with qiskit.circuit.library.ZZ_Feature_Map according to the procedure outlined in [1]. To construct the dataset, we first sample uniformly distributed vectors :math:\vec{x} \in (0, 2\pi]^{n} and apply the feature map .. math:: |\Phi(\vec{x})\rangle = U_{{\Phi} (\vec{x})} H^{\otimes n} U_{{\Phi} (\vec{x})} H^{\otimes n} |0^{\otimes n} \rangle where .. math:: U_{{\Phi} (\vec{x})} = \exp \left( i \sum_{S \subseteq [n] } \phi_S(\vec{x}) \prod_{i \in S} Z_i \right) and .. math:: \begin{cases} \phi_{\{i, j\}} = (\pi - x_i)(\pi - x_j) \\ \phi_{\{i\}} = x_i \end{cases} We then attribute labels to the vectors according to the rule .. math:: m(\vec{x}) = \begin{cases} 1 & \langle \Phi(\vec{x}) | V^\dagger \prod_i Z_i V | \Phi(\vec{x}) \rangle > \Delta \\ -1 & \langle \Phi(\vec{x}) | V^\dagger \prod_i Z_i V | \Phi(\vec{x}) \rangle < -\Delta \end{cases} where :math:\Delta is the separation gap, and :math:V\in \mathrm{SU}(4) is a random unitary. The current implementation only works with n = 2 or 3. **References:** [1] Havlíček V, Córcoles AD, Temme K, Harrow AW, Kandala A, Chow JM, Gambetta JM. Supervised learning with quantum-enhanced feature spaces. Nature. 2019 Mar;567(7747):209-12. arXiv:1804.11326 <https://arxiv.org/abs/1804.11326>_ Args: training_size: the number of training samples. test_size: the number of testing samples. n: number of qubits (dimension of the feature space). Must be 2 or 3. gap: separation gap (:math:\Delta). plot_data: whether to plot the data. Requires matplotlib. one_hot: if True, return the data in one-hot format. include_sample_total: if True, return all points in the uniform grid in addition to training and testing samples. Returns: Training and testing samples. Raises: ValueError: if n is not 2 or 3. """ class_labels = [r"A", r"B"] count = 0 if n == 2: count = 100 elif n == 3: count = 20 # coarseness of data separation else: raise ValueError(f"Supported values of 'n' are 2 and 3 only, but {n} is provided.") # Define auxiliary matrices and initial state z = np.diag([1, -1]) i_2 = np.eye(2) h_2 = np.array([[1, 1], [1, -1]]) / np.sqrt(2) h_n = reduce(np.kron, [h_2] * n) psi_0 = np.ones(2**n) / np.sqrt(2**n) # Generate Z matrices acting on each qubits z_i = np.array([reduce(np.kron, [i_2] * i + [z] + [i_2] * (n - i - 1)) for i in range(n)]) # Construct the parity operator bitstrings = ["".join(bstring) for bstring in it.product(*[["0", "1"]] * n)] if n == 2: bitstring_parity = [bstr.count("1") % 2 for bstr in bitstrings] d_m = np.diag((-1) ** np.array(bitstring_parity)) elif n == 3: bitstring_majority = [0 if bstr.count("0") > 1 else 1 for bstr in bitstrings] d_m = np.diag((-1) ** np.array(bitstring_majority)) # Generate a random unitary operator by collecting eigenvectors of a # random hermitian operator basis = algorithm_globals.random.random( (2**n, 2**n) ) + 1j * algorithm_globals.random.random((2**n, 2**n)) basis = np.array(basis).conj().T @ np.array(basis) eigvals, eigvecs = np.linalg.eig(basis) idx = eigvals.argsort()[::-1] eigvecs = eigvecs[:, idx] m_m = eigvecs.conj().T @ d_m @ eigvecs # Generate a grid of points in the feature space and compute the # expectation value of the parity xvals = np.linspace(0, 2 * np.pi, count, endpoint=False) ind_pairs = list(it.combinations(range(n), 2)) _sample_total = [] for x in it.product(*[xvals] * n): x = np.array(x) phi = np.sum(x[:, None, None] * z_i, axis=0) phi += sum(((np.pi - x[i1]) * (np.pi - x[i2]) * z_i[i1] @ z_i[i2] for i1, i2 in ind_pairs)) # u_u was actually scipy.linalg.expm(1j * phi), but this method is # faster because phi is always a diagonal matrix. # We first extract the diagonal elements, then do exponentiation, then # construct a diagonal matrix from them. u_u = np.diag(np.exp(1j * np.diag(phi))) psi = u_u @ h_n @ u_u @ psi_0 exp_val = np.real(psi.conj().T @ m_m @ psi) if np.abs(exp_val) > gap: _sample_total.append(np.sign(exp_val)) else: _sample_total.append(0) sample_total = np.array(_sample_total).reshape(*[count] * n) # Extract training and testing samples from grid x_sample, y_sample = _sample_ad_hoc_data(sample_total, xvals, training_size + test_size, n) if plot_data: _plot_ad_hoc_data(x_sample, y_sample, training_size) training_input = { key: (x_sample[y_sample == k, :])[:training_size] for k, key in enumerate(class_labels) } test_input = { key: (x_sample[y_sample == k, :])[training_size : (training_size + test_size)] for k, key in enumerate(class_labels) } training_feature_array, training_label_array = features_and_labels_transform( training_input, class_labels, one_hot ) test_feature_array, test_label_array = features_and_labels_transform( test_input, class_labels, one_hot ) if include_sample_total: return ( training_feature_array, training_label_array, test_feature_array, test_label_array, sample_total, ) else: return ( training_feature_array, training_label_array, test_feature_array, test_label_array, )
def _sample_ad_hoc_data(sample_total, xvals, num_samples, n): count = sample_total.shape[0] sample_a, sample_b = [], [] for i, sample_list in enumerate([sample_a, sample_b]): label = 1 if i == 0 else -1 while len(sample_list) < num_samples: draws = tuple(algorithm_globals.random.choice(count) for i in range(n)) if sample_total[draws] == label: sample_list.append([xvals[d] for d in draws]) labels = np.array([0] * num_samples + [1] * num_samples) samples = [sample_a, sample_b] samples = np.reshape(samples, (2 * num_samples, n)) return samples, labels @optionals.HAS_MATPLOTLIB.require_in_call def _plot_ad_hoc_data(x_total, y_total, training_size): import matplotlib.pyplot as plt n = x_total.shape[1] fig = plt.figure() projection = "3d" if n == 3 else None ax1 = fig.add_subplot(1, 1, 1, projection=projection) for k in range(0, 2): ax1.scatter(*x_total[y_total == k][:training_size].T) ax1.set_title("Ad-hoc Data") plt.show()