Source code for pennylane.optimize.lie_algebra

# Copyright 2018-2021 Xanadu Quantum Technologies Inc.

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"""Lie algebra optimizer"""
import warnings

import numpy as np
from scipy.sparse.linalg import expm
import pennylane as qml


@qml.qfunc_transform
def append_time_evolution(tape, riemannian_gradient, t, n, exact=False):
    r"""Append an approximate time evolution, corresponding to a Riemannian
    gradient on the Lie group, to an existing circuit.

    We want to implement the time evolution generated by an operator of the form

    .. math::

        \text{grad} f(U) = sum_i c_i O_i,

    where :math:`O_i` are Pauli words and :math:`c_t \in \mathbb{R}`.
    If ``exact`` is ``False``, we Trotterize this operator and apply the unitary

    .. math::

        U' = \prod_{n=1}^{N_{Trot.}} \left(\prod_i \exp{-it / N_{Trot.} O_i}\right),

    which is then appended to the current circuit.

    If ``exact`` is ``True``, we calculate the exact time evolution for the Riemannian gradient
     by way of the matrix exponential.

    .. math:

        U' = \exp{-it \text{grad} f(U)}

    and append this unitary.

    Args:
        tape (QuantumTape or .QNode): circuit to transform.
        riemannian_gradient (.Hamiltonian): Hamiltonian object representing the Riemannian gradient.
        t (float): time evolution parameter.
        n (int): number of Trotter steps.

    """
    for obj in tape.operations:
        qml.apply(obj)
    if exact:
        qml.QubitUnitary(
            expm(-1j * t * qml.utils.sparse_hamiltonian(riemannian_gradient).toarray()),
            wires=range(len(riemannian_gradient.wires)),
        )
    else:
        qml.templates.ApproxTimeEvolution(riemannian_gradient, t, n)

    for obj in tape.measurements:
        qml.apply(obj)


@qml.batch_transform
def algebra_commutator(tape, observables, lie_algebra_basis_names, nqubits):
    """Calculate the Riemannian gradient in the Lie algebra with the parameter shift rule
    (see :meth:`LieAlgebraOptimizer.get_omegas`).

    Args:
        tape (.QuantumTape or .QNode): input circuit
        observables (list[.Observable]): list of observables to be measured. Can be grouped.
        lie_algebra_basis_names (list[str]): List of strings corresponding to valid Pauli words.
        nqubits (int): the number of qubits.

    Returns:
         func: Function which accepts the same arguments as the QNode. When called, this
         function will return the Lie algebra commutator.

    """
    tapes_plus_total = []
    tapes_min_total = []
    for obs in observables:
        for o in obs:
            # create a list of tapes for the plus and minus shifted circuits
            tapes_plus = [qml.tape.QuantumTape(name=f"{p}_p") for p in lie_algebra_basis_names]
            tapes_min = [qml.tape.QuantumTape(name=f"{p}_m") for p in lie_algebra_basis_names]

            # loop through all operations on the input tape
            for op in tape.operations:
                for t in tapes_plus + tapes_min:
                    with t:
                        qml.apply(op)
            for i, t in enumerate(tapes_plus):
                with t:
                    qml.PauliRot(
                        np.pi / 2,
                        lie_algebra_basis_names[i],
                        wires=list(range(nqubits)),
                    )
                    qml.expval(o)
            for i, t in enumerate(tapes_min):
                with t:
                    qml.PauliRot(
                        -np.pi / 2,
                        lie_algebra_basis_names[i],
                        wires=list(range(nqubits)),
                    )
                    qml.expval(o)
            tapes_plus_total.extend(tapes_plus)
            tapes_min_total.extend(tapes_min)
    return tapes_plus_total + tapes_min_total, None


[docs]class LieAlgebraOptimizer: r"""Lie algebra optimizer. Riemannian gradient descent algorithms can be used to optimize a function directly on a Lie group as opposed to on a Euclidean parameter space. Consider the function :math:`f(U) = \text{Tr}(U \rho_0 U^\dagger H)` for a given Hamiltonian :math:`H`, unitary :math:`U\in \text{SU}(2^N)` and initial state :math:`\rho_0`. One can show that this function is minimized by the flow equation .. math:: \dot{U} = \text{grad}f(U) where :math:`\text{grad}` is the Riemannian gradient operator on :math:`\text{SU}(2^N)`. By discretizing the flow above, we see that a step of the Lie algebra optimizer iterates the Riemannian gradient flow on :math:`\text{SU}(2^N)` as .. math:: U^{(t+1)} = \exp\left\{\epsilon\: \text{grad}f(U^{(t)}) U^{(t)}\right\}, where :math:`\epsilon` is a user-defined hyperparameter corresponding to the step size. The gradient in the Lie algebra is given by .. math:: \text{grad}f(U^{(t)}) = -\left[U \rho U^\dagger, H\right] . Hence we see that subsequent steps of this optimizer will append the unitary generated by the Lie gradient and grow the circuit. The exact Riemannian gradient flow on :math:`\text{SU}(2^N)` has desirable optimization properties that can guarantee convergence to global minima under mild assumptions. However, this comes at a cost. Since :math:`\text{dim}(\text{SU}(2^N)) = 4^N-1`, we need an exponential number of parameters to calculate the gradient. This will not be problematic for small systems (:math:`N<5`), but will quickly get out of control as the number of qubits increases. To resolve this issue, we can restrict the Riemannian gradient to a subspace of the Lie algebra and calculate an approximate Riemannian gradient flow. The choice of restriction will affect the optimization behavior and quality of the final solution. For more information on Riemannian gradient flows on Lie groups see `T. Schulte-Herbrueggen et. al. (2008) <https://arxiv.org/abs/0802.4195>`_ and the application to quantum circuits `Wiersema and Killoran (2022) <https://arxiv.org/abs/2202.06976>`_. Args: circuit (.QNode): a user defined circuit that does not take any arguments and returns the expectation value of a ``qml.Hamiltonian``. stepsize (float): the user-defined hyperparameter :math:`\epsilon`. restriction (.Hamiltonian): Restrict the Lie algebra to a corresponding subspace of the full Lie algebra. This restriction should be passed in the form of a ``qml.Hamiltonian`` that consists only of Pauli words. exact (bool): Flag that indicates wether we approximate the Riemannian gradient with a Trotterization or calculate the exact evolution via a matrix exponential. The latter is not hardware friendly and can only be done in simulation. **Examples** Define a Hamiltonian cost function to minimize: >>> coeffs = [-1., -1., -1.] >>> observables = [qml.PauliX(0), qml.PauliZ(1), qml.PauliY(0) @ qml.PauliX(1)] >>> hamiltonian = qml.Hamiltonian(coeffs, observables) Create an initial state and return the expectation value of the Hamiltonian: >>> @qml.qnode(qml.device("default.qubit", wires=2)) ... def quant_fun(): ... qml.RX(0.1, wires=[0]) ... qml.RY(0.5, wires=[1]) ... qml.CNOT(wires=[0,1]) ... qml.RY(0.6, wires=[0]) ... return qml.expval(hamiltonian) Instantiate the optimizer with the initial circuit and the cost function and set the stepsize accordingly: >>> opt = qml.LieAlgebraOptimizer(circuit=quant_fun, stepsize=0.1) Applying 5 steps gets us close the ground state of :math:`E\approx-2.23`: >>> for step in range(6): ... circuit, cost = opt.step_and_cost() ... print(f"Step {step} - cost {cost}") Step 0 - cost -1.3351865007304005 Step 1 - cost -1.9937887238935206 Step 2 - cost -2.1524234485729834 Step 3 - cost -2.1955105378898487 Step 4 - cost -2.2137628169764256 Step 5 - cost -2.2234364822091575 The optimized circuit is returned at each step, and can be used as any other QNode: >>> circuit() -2.2283086057521713 """ # pylint: disable=too-many-arguments # pylint: disable=too-many-instance-attributes def __init__(self, circuit, stepsize=0.01, restriction=None, exact=False, trottersteps=1): if not isinstance(circuit, qml.QNode): raise TypeError(f"circuit must be a QNode, received {type(circuit)}") self.circuit = circuit self.circuit.construct([], {}) self.hamiltonian = circuit.func().obs if not isinstance(self.hamiltonian, qml.Hamiltonian): raise TypeError( f"circuit must return the expectation value of a Hamiltonian," f"received {type(circuit.func().obs)}" ) self.nqubits = len(circuit.device.wires) if self.nqubits > 4: warnings.warn( "The exact Riemannian gradient is exponentially expensive in the number of qubits, " f"optimizing a {self.nqubits} qubit circuit may be slow.", UserWarning, ) if restriction is not None and not isinstance(restriction, qml.Hamiltonian): raise TypeError(f"restriction must be a Hamiltonian, received {type(restriction)}") ( self.lie_algebra_basis_ops, self.lie_algebra_basis_names, ) = self.get_su_n_operators(restriction) self.exact = exact self.trottersteps = trottersteps self.coeffs, self.observables = self.hamiltonian.terms() self.stepsize = stepsize
[docs] def step(self): r"""Update the circuit with one step of the optimizer. Returns: float: the optimized circuit and the objective function output prior to the step. """ return self.step_and_cost()[0]
[docs] def step_and_cost(self): r"""Update the circuit with one step of the optimizer and return the corresponding objective function value prior to the step. Returns: tuple[.QNode, float]: the optimized circuit and the objective function output prior to the step. """ # pylint: disable=not-callable cost = self.circuit() omegas = self.get_omegas() non_zero_omegas = -omegas[omegas != 0] nonzero_idx = np.nonzero(omegas)[0] non_zero_lie_algebra_elements = [self.lie_algebra_basis_names[i] for i in nonzero_idx] lie_gradient = qml.Hamiltonian( non_zero_omegas, [qml.pauli.string_to_pauli_word(ps) for ps in non_zero_lie_algebra_elements], ) new_circuit = append_time_evolution( lie_gradient, self.stepsize, self.trottersteps, self.exact )(self.circuit.func) # we can set diff_method=None because the gradient of the QNode is computed # directly in this optimizer self.circuit = qml.QNode(new_circuit, self.circuit.device, diff_method=None) return self.circuit, cost
[docs] def get_su_n_operators(self, restriction): r"""Get the SU(N) operators. The dimension of the group is :math:`N^2-1`. Args: restriction (.Hamiltonian): Restrict the Riemannian gradient to a subspace. Returns: tuple[list[array[complex]], list[str]]: list of :math:`N^2 \times N^2` NumPy complex arrays and corresponding Pauli words. """ operators = [] names = [] # construct the corresponding pennylane observables wire_map = dict(zip(range(self.nqubits), range(self.nqubits))) if restriction is None: for ps in qml.pauli.pauli_group(self.nqubits): operators.append(ps) names.append(qml.pauli.pauli_word_to_string(ps, wire_map=wire_map)) else: for ps in set(restriction.ops): operators.append(ps) names.append(qml.pauli.pauli_word_to_string(ps, wire_map=wire_map)) return operators, names
[docs] def get_omegas(self): r"""Measure the coefficients of the Riemannian gradient with respect to a Pauli word basis. We want to calculate the components of the Riemannian gradient in the Lie algebra with respect to a Pauli word basis. For a Hamiltonian of the form :math:`H = \sum_i c_i O_i`, where :math:`c_i\in\mathbb{R}`, this can be achieved by calculating .. math:: \omega_{i,j} = \text{Tr}(c_i[\rho, O_i] P_j) where :math:`P_j` is a Pauli word in the set of Pauli monomials on :math:`N` qubits. Via the parameter shift rule, the commutator can be calculated as .. math:: [\rho, O_i] = \frac{1}{2}(V(\pi/2) \rho V^\dagger(\pi/2) - V(-\pi/2) \rho V^\dagger(-\pi/2)) where :math:`V` is the unitary generated by the Pauli word :math:`V(\theta) = \exp\{-i\theta P_j\}`. Returns: array: array of omegas for each direction in the Lie algebra. """ obs_groupings, _ = qml.pauli.group_observables(self.observables, self.coeffs) # get all circuits we need to calculate the coefficients circuits = algebra_commutator( self.circuit.qtape, obs_groupings, self.lie_algebra_basis_names, self.nqubits, )[0] circuits = qml.execute(circuits, self.circuit.device, gradient_fn=None) circuits_plus = np.array(circuits[: len(circuits) // 2]).reshape( len(self.coeffs), len(self.lie_algebra_basis_names) ) circuits_min = np.array(circuits[len(circuits) // 2 :]).reshape( len(self.coeffs), len(self.lie_algebra_basis_names) ) # For each observable O_i in the Hamiltonian, we have to calculate all Lie coefficients omegas = circuits_plus - circuits_min return np.dot(self.coeffs, omegas)