qml.labs.trotter_error.effective_hamiltonian

effective_hamiltonian(product_formula, fragments, order, timestep=1.0)

Compute the effective Hamiltonian \(\hat{H}_{eff} = \hat{H} + \hat{\epsilon}\) that corresponds to a given product formula.

Parameters:

• product_formula (ProductFormula) – A product formula used to approximate the time-evolution operator for a Hamiltonian.

• fragments (Dict[Hashable, Fragment) – The fragments that sum to the Hamiltonian. The keys in the dictionary must match the labels used to build the ProductFormula object.

• order (int) – The order of the approximatation.

• timestep (float) – The timestep for simulation.

Example

>>> import numpy as np
>>> from pennylane.labs.trotter_error.fragments import vibrational_fragments
>>> from pennylane.labs.trotter_error.product_formulas import ProductFormula, effective_hamiltonian
>>>
>>> n_modes = 4
>>> r_state = np.random.RandomState(42)
>>> freqs = r_state.random(4)
>>> taylor_coeffs = [
>>>     np.array(0),
>>>     r_state.random(size=(n_modes, )),
>>>     r_state.random(size=(n_modes, n_modes)),
>>>     r_state.random(size=(n_modes, n_modes, n_modes))
>>> ]
>>>
>>> delta = 0.001
>>> frag_labels = [0, 1, 1, 0]
>>> frag_coeffs = [1/2, 1/2, 1/2, 1/2]
>>>
>>> pf = ProductFormula(frag_labels, coeffs=frag_coeffs)
>>> frags = dict(enumerate(vibrational_fragments(n_modes, freqs, taylor_coeffs)))
>>> type(effective_hamiltonian(pf, frags, order=5, timestep=delta))
<class 'pennylane.labs.trotter_error.realspace.realspace_operator.RealspaceSum'>

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