squlearn.kernel.lowlevel_kernel.FidelityKernel

class squlearn.kernel.lowlevel_kernel.FidelityKernel(encoding_circuit: EncodingCircuitBase, executor: Executor, evaluate_duplicates: str = 'off_diagonal', mit_depol_noise: str | None = None, initial_parameters: ndarray | None = None, parameter_seed: int | None = 0, regularization: str | None = None, use_expectation: bool = False, caching: bool = False)

Fidelity Quantum Kernel.

The Fidelity Quantum Kernel is a based on the overlap of the quantum states. These quantum states can be defined by a parameterized quantum circuit. The Fidelity Quantum Kernel is defined as:

\[K(x,y) = |\langle \phi(x) | \phi(y) \rangle|^2\]

This class wraps to the respective Quantum Kernel implementations from Qiskit Machine Learning. Depending on the choice of the backend and the choice of trainable parameters, the appropriate Quantum Kernel implementation is chosen.

Parameters:

  • encoding_circuit (EncodingCircuitBase) – PQC encoding circuit.

  • executor (Executor) – Executor object.

  • evaluate_duplicates (str) – Option for evaluating duplicates (‘all’, ‘off_diagonal’, ‘none’).

  • mit_depol_noise (Union[str, None]) – Option for mitigating depolarizing noise ("msplit" or "mmean") after Ref. [4]. Only meaningful for FQKs computed on a real backend.

  • initial_parameters (Union[np.ndarray, None], default=None) – Initial parameters for the encoding circuit.

  • parameter_seed (Union[int, None], default=0) – Seed for the random number generator for the parameter initialization, if initial_parameters is None.

  • regularization (Union[str, None], default=None) – Option for choosing different regularization techniques ("thresholding" or "tikhonov") after Ref. [4] for the training kernel matrix, prior to solving the linear system in the fit()-procedure.

  • use_expectation (bool, default=False) – Option for using the expectation value of a QNN circuit that computes the fidelity as the expectation value of the observable \(P_0 = |0\rangle\langle0|^{\otimes n}\). This is not very efficient in simulation, but allows the computation of derivatives of the kernel.

References

[1]: Havlicek et al., Supervised learning with quantum-enhanced feature spaces, Nature 567, 209-212 (2019).

[2]: Schuld et al., Quantum Machine Learning in Feature Hilbert Spaces, Phys. Rev. Lett. 122, 040504 (2019).

[3]: Schuld et al., Quantum Machine Learning Models are Kernel Methods: Noise-Enhanced Quantum Embeddings, arXiv:2105.02276 (2021).

[4]: T. Hubregtsen et al., “Training Quantum Embedding Kernels on Near-Term Quantum Computers”, arXiv:2105.02276v1 (2021)

assign_parameters(parameters)

Fix the training parameters of the encoding circuit to numerical values

Parameters:

parameters (np.ndarray) – Array containing numerical values to be assigned to the trainable parameters of the encoding circuit

evaluate(x: ndarray, y: ndarray | None = None) → ndarray

Evaluates the fidelity kernel matrix.

Parameters:

  • x (np.ndarray) – Vector of training or test data for which the kernel matrix is evaluated

  • y (np.ndarray, default=None) – Vector of training or test data for which the kernel matrix is evaluated

Returns:

Returns the quantum kernel matrix as 2D numpy array.

evaluate_derivatives(x: ndarray, y: ndarray = None, values: str | tuple = 'dKdx') → dict

Evaluates the Fidelity Kernel and its derivatives for the given data points x and y.

Parameters:

  • x (np.ndarray) – Data points x

  • y (np.ndarray) – Data points y, if None y = x is used

  • values (Union[str, tuple]) – Values to evaluate. Can be a string or a tuple of strings. Possible values are: dKdx, dKdy, dKdxdx, dKdydy, dKdxdy, dKdydx, dKdp and jacobian.

Returns:

Dictionary with the evaluated values

evaluate_pairwise(x: ndarray, y: ndarray = None) → float

Computes the quantum kernel matrix.

Parameters:

  • x (np.ndarray) – Vector of training or test data for which the kernel matrix is evaluated

  • y (np.ndarray, default=None) – Vector of training or test data for which the kernel matrix is evaluated

evaluate_with_parameters(x: ndarray, y: ndarray, parameters: ndarray) → ndarray

Computes the quantum kernel matrix with assigned parameters

Parameters:

  • x (np.ndarray) – Vector of training or test data for which the kernel matrix is evaluated

  • y (np.ndarray) – Vector of training or test data for which the kernel matrix is evaluated

  • parameters (np.ndarray) – Array containing numerical values to be assigned to the trainable parameters of the encoding circuit

get_params(deep: bool = True) → dict

Returns hyper-parameters and their values of the fidelity kernel.

Parameters:

deep (bool) – If True, also the parameters for contained objects are returned (default=True).

Returns:

Dictionary with hyper-parameters and values.

set_params(**params)

Sets value of the fidelity kernel hyper-parameters.

Parameters:

params – Hyper-parameters and their values, e.g. num_qubits=2