Source code for pennylane.optimize.rotoselect

# Copyright 2018-2021 Xanadu Quantum Technologies Inc.

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"""Rotoselect gradient free optimizer"""

import numpy as np

import pennylane as qml
from pennylane.utils import _flatten, unflatten


[docs]class RotoselectOptimizer: r"""Rotoselect gradient-free optimizer. The Rotoselect optimizer minimizes an objective function with respect to the rotation gates and parameters of a quantum circuit without the need for calculating the gradient of the function. The algorithm works by updating the parameters :math:`\theta = \theta_1, \dots, \theta_D` and rotation gate choices :math:`R = R_1,\dots,R_D` one at a time according to a closed-form expression for the optimal value of the :math:`d^{th}` parameter :math:`\theta^*_d` when the other parameters and gate choices are fixed: .. math:: \theta^*_d = \underset{\theta_d}{\text{argmin}}\left<H\right>_{\theta_d} = -\frac{\pi}{2} - \text{arctan2}\left(2\left<H\right>_{\theta_d=0} - \left<H\right>_{\theta_d=\pi/2} - \left<H\right>_{\theta_d=-\pi/2}, \left<H\right>_{\theta_d=\pi/2} - \left<H\right>_{\theta_d=-\pi/2}\right), where :math:`\left<H\right>_{\theta_d}` is the expectation value of the objective function optimized over the parameter :math:`\theta_d`. :math:`\text{arctan2}(x, y)` computes the element-wise arc tangent of :math:`x/y` choosing the quadrant correctly, avoiding, in particular, division-by-zero when :math:`y = 0`. Which parameters and gates that should be optimized over is decided in the user-defined cost function, where :math:`R` is a list of parametrized rotation gates in a quantum circuit, along with their respective parameters :math:`\theta` for the circuit and its gates. Note that the number of generators should match the number of parameters. The algorithm is described in further detail in `Ostaszewski et al. (2021) <https://doi.org/10.22331/q-2021-01-28-391>`_. Args: possible_generators (list[~.Operation]): List containing the possible ``pennylane.ops.qubit`` operators that are allowed in the circuit. Default is the set of Pauli rotations :math:`\{R_x, R_y, R_z\}`. **Example:** Initialize the Rotoselect optimizer, set the initial values of the weights ``x``, choose the initial generators, and set the number of steps to optimize over. >>> opt = qml.optimize.RotoselectOptimizer() >>> x = [0.3, 0.7] >>> generators = [qml.RX, qml.RY] >>> n_steps = 10 Set up the PennyLane circuit using the ``default.qubit`` simulator device. >>> dev = qml.device("default.qubit", shots=None, wires=2) >>> @qml.qnode(dev) ... def circuit(params, generators=None): # generators will be passed as a keyword arg ... generators[0](params[0], wires=0) ... generators[1](params[1], wires=1) ... qml.CNOT(wires=[0, 1]) ... return qml.expval(qml.PauliZ(0)), qml.expval(qml.PauliX(1)) Define a cost function based on the above circuit. >>> def cost(x, generators): ... Z_1, X_2 = circuit(x, generators=generators) ... return 0.2 * Z_1 + 0.5 * X_2 Run the optimization step-by-step for ``n_steps`` steps. >>> cost_rotosel = [] >>> for _ in range(n_steps): ... cost_rotosel.append(cost(x, generators)) ... x, generators = opt.step(cost, x, generators) The optimized values for x should now be stored in ``x`` together with the optimal gates for the circuit, while steps-vs-cost can be seen by plotting ``cost_rotosel``. """ # pylint: disable=too-few-public-methods def __init__(self, possible_generators=None): self.possible_generators = possible_generators or [qml.RX, qml.RY, qml.RZ]
[docs] def step_and_cost(self, objective_fn, x, generators, **kwargs): """Update trainable arguments with one step of the optimizer and return the corresponding objective function value prior to the step. Args: objective_fn (function): The objective function for optimization. It must have the signature ``objective_fn(x, generators=None)`` with a sequence of the values ``x`` and a list of the gates ``generators`` as inputs, returning a single value. x (Union[Sequence[float], float]): sequence containing the initial values of the variables to be optimized over or a single float with the initial value generators (list[~.Operation]): list containing the initial ``pennylane.ops.qubit`` operators to be used in the circuit and optimized over **kwargs : variable length of keyword arguments for the objective function. Returns: tuple: the new variable values :math:`x^{(t+1)}`, the new generators, and the objective function output prior to the step """ x_new, generators = self.step(objective_fn, x, generators, **kwargs) return x_new, generators, objective_fn(x, generators, **kwargs)
[docs] def step(self, objective_fn, x, generators, **kwargs): r"""Update trainable arguments with one step of the optimizer. Args: objective_fn (function): The objective function for optimization. It must have the signature ``objective_fn(x, generators=None)`` with a sequence of the values ``x`` and a list of the gates ``generators`` as inputs, returning a single value. x (Union[Sequence[float], float]): sequence containing the initial values of the variables to be optimized over or a single float with the initial value generators (list[~.Operation]): list containing the initial ``pennylane.ops.qubit`` operators to be used in the circuit and optimized over **kwargs : variable length of keyword arguments for the objective function. Returns: array: The new variable values :math:`x^{(t+1)}` as well as the new generators. """ x_flat = np.fromiter(_flatten(x), dtype=float) # wrap the objective function so that it accepts the flattened parameter array objective_fn_flat = lambda x_flat, gen: objective_fn( unflatten(x_flat, x), generators=gen, **kwargs ) try: assert len(x_flat) == len(generators) except AssertionError as e: raise ValueError( f"Number of parameters {x} must be equal to the number of generators." ) from e for d, _ in enumerate(x_flat): x_flat[d], generators[d] = self._find_optimal_generators( objective_fn_flat, x_flat, generators, d ) return unflatten(x_flat, x), generators
def _find_optimal_generators(self, objective_fn, x, generators, d): r"""Optimizer for the generators. Optimizes for the best generator at position ``d``. Args: objective_fn (function): The objective function for optimization. It must have the signature ``objective_fn(x, generators=None)`` with a sequence of the values ``x`` and a list of the gates ``generators`` as inputs, returning a single value. x (Union[Sequence[float], float]): sequence containing the initial values of the variables to be optimized over or a single float with the initial value generators (list[~.Operation]): list containing the initial ``pennylane.ops.qubit`` operators to be used in the circuit and optimized over d (int): the position in the input sequence ``x`` containing the value to be optimized Returns: tuple: tuple containing the parameter value and generator that, at position ``d`` in ``x`` and ``generators``, optimizes the objective function """ params_opt_d = x[d] generators_opt_d = generators[d] params_opt_cost = objective_fn(x, generators) for generator in self.possible_generators: generators[d] = generator x = self._rotosolve(objective_fn, x, generators, d) params_cost = objective_fn(x, generators) # save the best paramter and generator for position d if params_cost <= params_opt_cost: params_opt_d = x[d] params_opt_cost = params_cost generators_opt_d = generator return params_opt_d, generators_opt_d @staticmethod def _rotosolve(objective_fn, x, generators, d): r"""The rotosolve step for one parameter and one set of generators. Updates the parameter :math:`\theta_d` based on Equation 1 in `Ostaszewski et al. (2021) <https://doi.org/10.22331/q-2021-01-28-391>`_. Args: objective_fn (function): The objective function for optimization. It must have the signature ``objective_fn(x, generators=None)`` with a sequence of the values ``x`` and a list of the gates ``generators`` as inputs, returning a single value. x (Union[Sequence[float], float]): sequence containing the initial values of the variables to be optimized overs or a single float with the initial value generators (list[~.Operation]): list containing the initial ``pennylane.ops.qubit`` operators to be used in the circuit and optimized over d (int): the position in the input sequence ``x`` containing the value to be optimized Returns: array: the input sequence ``x`` with the value at position ``d`` optimized """ # helper function for x[d] = theta def insert(x, d, theta): x[d] = theta return x H_0 = float(objective_fn(insert(x, d, 0), generators)) H_p = float(objective_fn(insert(x, d, np.pi / 2), generators)) H_m = float(objective_fn(insert(x, d, -np.pi / 2), generators)) a = np.arctan2(2 * H_0 - H_p - H_m, H_p - H_m) x[d] = -np.pi / 2 - a if x[d] <= -np.pi: x[d] += 2 * np.pi return x