Project: s41467-025-59198-z | Function: single_data_point_generation_TriMKernel (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Generates feature representations for quantum circuit parameters using trigonometric kernel mapping. Maps input parameters to frequency coefficients using cosine and sine functions, then computes products across frequency sets to create the Phi_w(theta) feature vectors for kernel-based learning.

**KEY TERMS:** {
  "map_freq_coefficients": "frequency coefficient matrix",
  "frequence_list": "frequency set W",
  "data_feature": "feature vector Phi_w",
  "f1": "cosine function",
  "f2": "sine function"
}

**CONTEXT CODE:** '''This file collects the code of predicting the two-point correlation of 60-qubit GHZ states on any pair of qubit'''

import numpy as np
try:
    import h5py
except ImportError:
    # Handle missing h5py for testing environment
    pass
try:
    from pennylane import classical_shadow, shadow_expval, ClassicalShadow
    import pennylane as qml
except ImportError:
    # Handle missing pennylane for testing environment
    # Create dummy qml for code execution
    class qml:
        @staticmethod
        def PauliZ(x):
            return None
        @staticmethod
        def PauliY(x):
            return None
import itertools
import os

seed = np.random.seed(123)
dim_feature = 3
observable = None
test_data = np.random.uniform(-np.pi, np.pi, dim_feature) # a particular input for prediction
print('The test point is {}'.format(test_data))


### Load training data, Trigo representations, and (estimated) label with the specified backend

# Generate the frequency set with the specified truncations
frequence_index_single = [0, 1, -1]
frequence_index_all_qubits = []
[frequence_index_all_qubits.append(frequence_index_single) for i in range(3)] # non-truncation by setting 3
frequence_list = list(itertools.product(*frequence_index_all_qubits))


# precomputing the length of each frequency in the frequency set
f1 = lambda x: np.cos(x)
f2 = lambda x: np.sin(x)

lentgh_freq_list = []
for index_list in frequence_list:
    index_list = np.asarray(index_list)
    lentgh_freq_list.append(np.count_nonzero(index_list))

def single_data_point_generation_TriMKernel(params):
    """
    Generates feature representations for quantum circuit parameters using trigonometric kernel mapping.

    This function implements the trigonometric monomial expansion described in the paper "Efficient
    learning for linear properties of bounded-gate quantum circuits" (Nature Communications, s41467-025-59198-z)
    for kernel-based learning algorithm.

    Maps input parameters to frequency coefficients using cosine and sine functions, then computes
    products across frequency sets to create the Phi_w(theta) feature vectors for efficient classical
    representations of quantum circuits.

    Paper reference: "Then, the learner applies shadow estimation and the trigonometric monomial
    expansion to the collected training dataset to obtain efficient classical representations,
    where any new input of the explored quantum circuits can be efficiently predicted offline"

    Formula: Phi_w(theta) = Product_i[f_{w_i}(theta_i)] where f_0=1, f_1=cos, f_{-1}=sin

    Args:
        params: Input quantum circuit parameters

    Returns:
        list: Feature vector Phi_w for kernel-based learning
    """
    # MASKED: Trigonometric kernel mapping for quantum circuit feature generation
    raise NotImplementedError(
        "This function is masked for testing purposes. "
        "Original function generates feature representations for quantum circuit parameters using "
        "trigonometric kernel mapping for kernel-based learning as described in Nature Communications "
        "article s41467-025-59198-z. Key operations: frequency coefficient matrix computation, "
        "trigonometric function application (cosine/sine), product computation across frequency sets "
        "to create Phi_w(theta) feature vectors for efficient quantum circuit prediction."
    )

# load data and calculate shadow estimations for each training point
file_dir = 'dataset/data_60_qubits/shots1000/'
num_data = 500
n_qubits = 60
X_train = np.zeros(shape=(num_data, 3**dim_feature)) # num_data x dim of classical control
trace_est = np.zeros(shape=(num_data, 1)) # record the shadow estimation
two_point_est_array = np.zeros(shape=(n_qubits, n_qubits, num_data))
try:
    obs = qml.PauliZ(0) @ qml.PauliY(59)
except (TypeError, AttributeError):
    # Handle missing pennylane operations for testing
    obs = None


if __name__ == '__main__':
    for i in range(num_data):
        index_ = i + 1
        f = open(file_dir + 'angles_' + str(index_) + '.txt', 'r')
        data_tempt = f.read().split()
        data_tempt = np.reshape(data_tempt, newshape = (dim_feature))
        data_tempt = data_tempt.astype(float)
        X_train[i] = single_data_point_generation_TriMKernel(data_tempt)
        hf = h5py.File(file_dir + 'qst_circuit_'+ str(index_) + '.h5', 'r')
        GHZ_shadow_bases = hf.get('bases')[:]
        GHZ_shadow_bases = np.transpose(GHZ_shadow_bases)
        dic_map_basis = {b'X':0, b'Y':1, b'Z':2} # Map bases to the format acceptable by Pennylane
        GHZ_recipes = np.vectorize(dic_map_basis.get)(GHZ_shadow_bases)
        GHZ_shadow_res = hf.get('outcomes')[:]
        GHZ_shadow_res = np.transpose(GHZ_shadow_res)
        shadow = ClassicalShadow(GHZ_shadow_res, GHZ_recipes)
        for qubit_i in range(n_qubits):
            for qubit_j in range(n_qubits):
                obs_two_point = (qml.PauliZ(qubit_i) @ qml.PauliZ(qubit_j) + qml.PauliY(qubit_i) @ qml.PauliY(qubit_j) + qml.PauliX(qubit_i) @ qml.PauliX(qubit_j))/3
                two_point_est_array[qubit_i, qubit_j, i] = shadow.expval(obs_two_point)

    print('data loading is finished')

# Compute classical representations
def Tri_geo_kernel_pre_compute(train_data_, y_train_, truncation_length):
    r'''
     :param train_data: data feature of train set
     :param y_train: label of training dataset
     :return: a representation vector obtained from traiing data
     '''
    n_train = len(y_train_)
    representation_train_data = np.zeros(shape=(3** dim_feature))
    for i in range(n_train):
        # compute the kernel W(x,xl)=sum_w 2^|w| Phi_w(x) Phi_w(x')
        for j in range(3 ** (dim_feature)):
            if lentgh_freq_list[j] <= truncation_length:
                representation_train_data[j] += (2 ** lentgh_freq_list[j]) * train_data_[i][j] * y_train_[i]
    representation_train_data = representation_train_data / n_train
    return representation_train_data

trunc_length = 3
prediction_array = np.zeros(shape=(n_qubits, n_qubits))
if __name__ == '__main__':
    for qubit_i in range(n_qubits):
        for qubit_j in range(n_qubits):
            representation_train_correlator_ij = Tri_geo_kernel_pre_compute(X_train,  two_point_est_array[qubit_i, qubit_j, :], truncation_length= trunc_length)
            rep_test_data = single_data_point_generation_TriMKernel(test_data)
            prediction_array[qubit_i, qubit_j] = np.inner(representation_train_correlator_ij, rep_test_data)

    np.save('prediction_two_point_correlation', prediction_array)
    print(prediction_array)
    ### start to prediction


Complete the function implementation replacing "raise NotImplementedError".