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class qiskit.algorithms.PVQD(fidelity, ansatz, initial_parameters, estimator=None, optimizer=None, num_timesteps=None, evolution=None, use_parameter_shift=True, initial_guess=None)

GitHub(opens in a new tab)

Bases: RealTimeEvolver

The projected Variational Quantum Dynamics (p-VQD) Algorithm.

In each timestep, this algorithm computes the next state with a Trotter formula (specified by the evolution argument) and projects the timestep onto a variational form (ansatz). The projection is determined by maximizing the fidelity of the Trotter-evolved state and the ansatz, using a classical optimization routine. See Ref. [1] for details.

The following attributes can be set via the initializer but can also be read and updated once the PVQD object has been constructed.


The parameterized circuit representing the time-evolved state.




The parameters of the ansatz at time 0.




The classical optimization routine used to maximize the fidelity of the Trotter step and ansatz.


Optional[Union[Optimizer, Minimizer]]


The number of timesteps to take. If None, it is automatically selected to achieve a timestep of approximately 0.01.


Optional[int(opens in a new tab)]


The method to perform the Trotter step. Defaults to first-order Lie-Trotter evolution.




If True, use the parameter shift rule for loss function gradients (if the ansatz supports).


bool(opens in a new tab)


The starting point for the first classical optimization run, at time 0. Defaults to random values in [0.01,0.01][-0.01, 0.01].




This snippet computes the real time evolution of a quantum Ising model on two neighboring sites and keeps track of the magnetization.

import numpy as np
from qiskit.algorithms.state_fidelities import ComputeUncompute
from qiskit.algorithms.time_evolvers import TimeEvolutionProblem, PVQD
from qiskit.primitives import Estimator, Sampler
from qiskit.circuit.library import EfficientSU2
from qiskit.quantum_info import SparsePauliOp, Pauli
from qiskit.algorithms.optimizers import L_BFGS_B
sampler = Sampler()
fidelity = ComputeUncompute(sampler)
estimator = Estimator()
hamiltonian = 0.1 * SparsePauliOp(["ZZ", "IX", "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(
# specify the evolution problem
problem = TimeEvolutionProblem(
    hamiltonian, time, aux_operators=[hamiltonian, observable]
# and evolve!
result = pvqd.evolve(problem)


[1] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo (2021), An efficient

quantum algorithm for the time evolution of parameterized circuits, Quantum 5, 512(opens in a new tab).


  • fidelity (BaseStateFidelity) – A fidelity primitive used by the algorithm.
  • ansatz (QuantumCircuit) – A parameterized circuit preparing the variational ansatz to model the time evolved quantum state.
  • initial_parameters (np.ndarray) – The initial parameters for the ansatz. Together with the ansatz, these define the initial state of the time evolution.
  • estimator (BaseEstimator | None) – An estimator primitive used for calculating expected values of auxiliary operators (if provided via the problem).
  • optimizer (Optimizer |Minimizer | None) – The classical optimizers used to minimize the overlap between Trotterization and ansatz. Can be either a Optimizer or a callable using the Minimizer protocol. This argument is optional since it is not required for get_loss(), but it has to be set before evolve() is called.
  • num_timesteps (int(opens in a new tab) | None) – The number of time steps. If None it will be set such that the timestep is close to 0.01.
  • evolution (EvolutionSynthesis | None) – The evolution synthesis to use for the construction of the Trotter step. Defaults to first-order Lie-Trotter decomposition, see also evolution for different options.
  • use_parameter_shift (bool(opens in a new tab)) – If True, use the parameter shift rule to compute gradients. If False, the optimizer will not be passed a gradient callable. In that case, Qiskit optimizers will use a finite difference rule to approximate the gradients.
  • initial_guess (np.ndarray | None) – The initial guess for the first VQE optimization. Afterwards the previous iteration result is used as initial guess. If None, this is set to a random vector with elements in the interval [0.01,0.01][-0.01, 0.01].




Perform real time evolution exp(itH)Ψ\exp(-i t H)|\Psi\rangle.

Evolves an initial state Ψ|\Psi\rangle for a time tt under a Hamiltonian HH, as provided in the evolution_problem.


evolution_problem (TimeEvolutionProblem) – The evolution problem containing the hamiltonian, total evolution time and observables to evaluate.


A result object containing the evolution information and evaluated observables.


Return type



get_loss(hamiltonian, ansatz, dt, current_parameters)

Get a function to evaluate the infidelity between Trotter step and ansatz.


  • hamiltonian (BaseOperator | PauliSumOp) – The Hamiltonian under which to evolve.
  • ansatz (QuantumCircuit) – The parameterized quantum circuit which attempts to approximate the time-evolved state.
  • dt (float(opens in a new tab)) – The time step.
  • current_parameters (np.ndarray) – The current parameters.


A callable to evaluate the infidelity and, if gradients are supported and required,

a second callable to evaluate the gradient of the infidelity.

Return type

tuple(opens in a new tab)[Callable[[np.ndarray], float(opens in a new tab)], Callable[[np.ndarray], np.ndarray]] | None


step(hamiltonian, ansatz, theta, dt, initial_guess)

Perform a single time step.


  • hamiltonian (BaseOperator | PauliSumOp) – The Hamiltonian under which to evolve.
  • ansatz (QuantumCircuit) – The parameterized quantum circuit which attempts to approximate the time-evolved state.
  • theta (np.ndarray) – The current parameters.
  • dt (float(opens in a new tab)) – The time step.
  • initial_guess (np.ndarray) – The initial guess for the classical optimization of the fidelity between the next variational state and the Trotter-evolved last state. If None, this is set to a random vector with elements in the interval [0.01,0.01][-0.01, 0.01].


A tuple consisting of the next parameters and the fidelity of the optimization.

Return type

tuple(opens in a new tab)[np.ndarray, float(opens in a new tab)]

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