# Source code for pennylane.templates.subroutines.approx_time_evolution

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r"""
Contains the ApproxTimeEvolution template.
"""
# pylint: disable-msg=too-many-branches,too-many-arguments,protected-access
import pennylane as qml
from pennylane.operation import Operation, AnyWires
from pennylane.ops import PauliRot

[docs]class ApproxTimeEvolution(Operation): r"""Applies the Trotterized time-evolution operator for an arbitrary Hamiltonian, expressed in terms of Pauli gates. The general time-evolution operator for a time-independent Hamiltonian is given by .. math:: U(t) \ = \ e^{-i H t}, for some Hamiltonian of the form: .. math:: H \ = \ \displaystyle\sum_{j} H_j. Implementing this unitary with a set of quantum gates is difficult, as the terms :math:H_j don't necessarily commute with one another. However, we are able to exploit the Trotter-Suzuki decomposition formula, .. math:: e^{A \ + \ B} \ = \ \lim_{n \to \infty} \Big[ e^{A/n} e^{B/n} \Big]^n, to implement an approximation of the time-evolution operator as .. math:: U \ \approx \ \displaystyle\prod_{k \ = \ 1}^{n} \displaystyle\prod_{j} e^{-i H_j t / n}, with the approximation becoming better for larger :math:n. The circuit implementing this unitary is of the form: .. figure:: ../../_static/templates/subroutines/approx_time_evolution.png :align: center :width: 60% :target: javascript:void(0); It is also important to note that this decomposition is exact for any value of :math:n when each term of the Hamiltonian commutes with every other term. .. note:: This template uses the :class:~.PauliRot operation in order to implement exponentiated terms of the input Hamiltonian. This operation only takes terms that are explicitly written in terms of products of Pauli matrices (:class:~.PauliX, :class:~.PauliY, :class:~.PauliZ, and :class:~.Identity). Thus, each term in the Hamiltonian must be expressed this way upon input, or else an error will be raised. Args: hamiltonian (.Hamiltonian): The Hamiltonian defining the time-evolution operator. The Hamiltonian must be explicitly written in terms of products of Pauli gates (:class:~.PauliX, :class:~.PauliY, :class:~.PauliZ, and :class:~.Identity). time (int or float): The time of evolution, namely the parameter :math:t in :math:e^{- i H t}. n (int): The number of Trotter steps used when approximating the time-evolution operator. .. details:: :title: Usage Details The template is used inside a qnode: .. code-block:: python import pennylane as qml from pennylane.templates import ApproxTimeEvolution n_wires = 2 wires = range(n_wires) dev = qml.device('default.qubit', wires=n_wires) coeffs = [1, 1] obs = [qml.PauliX(0), qml.PauliX(1)] hamiltonian = qml.Hamiltonian(coeffs, obs) @qml.qnode(dev) def circuit(time): ApproxTimeEvolution(hamiltonian, time, 1) return [qml.expval(qml.PauliZ(wires=i)) for i in wires] >>> circuit(1) tensor([-0.41614684 -0.41614684], requires_grad=True) """ num_wires = AnyWires grad_method = None def __init__(self, hamiltonian, time, n, do_queue=True, id=None): if not isinstance(hamiltonian, qml.Hamiltonian): raise ValueError( f"hamiltonian must be of type pennylane.Hamiltonian, got {type(hamiltonian).__name__}" ) # extract the wires that the op acts on wire_list = [term.wires for term in hamiltonian.ops] wires = qml.wires.Wires.all_wires(wire_list) self._hyperparameters = {"hamiltonian": hamiltonian, "n": n} # trainable parameters are passed to the base init method super().__init__(*hamiltonian.data, time, wires=wires, do_queue=do_queue, id=id)
[docs] @staticmethod def compute_decomposition( *coeffs_and_time, wires, hamiltonian, n ): # pylint: disable=arguments-differ,unused-argument r"""Representation of the operator as a product of other operators. .. math:: O = O_1 O_2 \dots O_n. .. seealso:: :meth:~.ApproxTimeEvolution.decomposition. Args: coeffs_and_time (list[tensor_like or float]): list of coefficients of the Hamiltonian, appended by the time variable wires (Any or Iterable[Any]): wires that the operator acts on hamiltonian (.Hamiltonian): The Hamiltonian defining the time-evolution operator. The Hamiltonian must be explicitly written in terms of products of Pauli gates (:class:~.PauliX, :class:~.PauliY, :class:~.PauliZ, and :class:~.Identity). n (int): The number of Trotter steps used when approximating the time-evolution operator. Returns: list[.Operator]: decomposition of the operator """ pauli = {"Identity": "I", "PauliX": "X", "PauliY": "Y", "PauliZ": "Z"} theta = [] pauli_words = [] wires = [] coeffs = coeffs_and_time[:-1] time = coeffs_and_time[-1] for i, term in enumerate(hamiltonian.ops): word = "" try: if isinstance(term.name, str): word = pauli[term.name] if isinstance(term.name, list): word = "".join(pauli[j] for j in term.name) except KeyError as error: raise ValueError( f"hamiltonian must be written in terms of Pauli matrices, got {error}" ) from error # skips terms composed solely of identities if word.count("I") != len(word): theta.append((2 * time * coeffs[i]) / n) pauli_words.append(word) wires.append(term.wires) op_list = [] for i in range(n): for j, term in enumerate(pauli_words): op_list.append(PauliRot(theta[j], term, wires=wires[j])) return op_list

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