features:
  - |
    Added the :class:`~.PVQD` class to the time evolution framework in :mod:`qiskit.algorithms`.
    This class implements the projected Variational Quantum Dynamics (p-VQD) algorithm
    `Barison et al. <https://quantum-journal.org/papers/q-2021-07-28-512/>`_.

    In each timestep this algorithm computes the next state with a Trotter formula and projects it
    onto a variational form. The projection is determined by maximizing the fidelity of the
    Trotter-evolved state and the ansatz, using a classical optimization routine.

    .. code-block:: python

        import numpy as np

        from qiskit.algorithms.state_fidelities import ComputeUncompute
        from qiskit.algorithms.evolvers import EvolutionProblem
        from qiskit.algorithms.time_evolvers.pvqd import PVQD
        from qiskit.primitives import Estimator, Sampler
        from qiskit import BasicAer
        from qiskit.circuit.library import EfficientSU2
        from qiskit.quantum_info import Pauli, SparsePauliOp
        from qiskit.algorithms.optimizers import L_BFGS_B

        sampler = Sampler()
        fidelity = ComputeUncompute(sampler)
        estimator = Estimator()
        hamiltonian = 0.1 * SparsePauliOp([Pauli("ZZ"), Pauli("IX"), Pauli("XI")])
        observable = Pauli("ZZ")
        ansatz = EfficientSU2(2, reps=1)
        initial_parameters = np.zeros(ansatz.num_parameters)

        time = 1
        optimizer = L_BFGS_B()

        # setup the algorithm
        pvqd = PVQD(
            fidelity,
            ansatz,
            initial_parameters,
            estimator,
            num_timesteps=100,
            optimizer=optimizer,
        )

        # specify the evolution problem
        problem = EvolutionProblem(
            hamiltonian, time, aux_operators=[hamiltonian, observable]
        )

        # and evolve!
        result = pvqd.evolve(problem)