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This code is part of Qiskit.","#","# (C) Copyright IBM 2017, 2021.","#","# This code is licensed under the Apache License, Version 2.0. You may","# obtain a copy of this license in the LICENSE.txt file in the root directory","# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.","#","# Any modifications or derivative works of this code must retain this","# copyright notice, and modified files need to carry a notice indicating","# that they have been altered from the originals.","","\"\"\"A class implementing a (piecewise-) linear function on qubit amplitudes.\"\"\"","","from __future__ import annotations","import numpy as np","from qiskit.circuit import QuantumCircuit","","from .piecewise_linear_pauli_rotations import PiecewiseLinearPauliRotations","","","class LinearAmplitudeFunction(QuantumCircuit):"," r\"\"\"A circuit implementing a (piecewise) linear function on qubit amplitudes.",""," An amplitude function :math:`F` of a function :math:`f` is a mapping",""," .. math::",""," F|x\\rangle|0\\rangle = \\sqrt{1 - \\hat{f}(x)} |x\\rangle|0\\rangle + \\sqrt{\\hat{f}(x)}"," |x\\rangle|1\\rangle.",""," for a function :math:`\\hat{f}: \\{ 0, ..., 2^n - 1 \\} \\rightarrow [0, 1]`, where"," :math:`|x\\rangle` is a :math:`n` qubit state.",""," This circuit implements :math:`F` for piecewise linear functions :math:`\\hat{f}`."," In this case, the mapping :math:`F` can be approximately implemented using a Taylor expansion"," and linearly controlled Pauli-Y rotations, see [1, 2] for more detail. This approximation"," uses a ``rescaling_factor`` to determine the accuracy of the Taylor expansion.",""," In general, the function of interest :math:`f` is defined from some interval :math:`[a,b]`,"," the ``domain`` to :math:`[c,d]`, the ``image``, instead of :math:`\\{ 1, ..., N \\}` to"," :math:`[0, 1]`. Using an affine transformation we can rescale :math:`f` to :math:`\\hat{f}`:",""," .. math::",""," \\hat{f}(x) = \\frac{f(\\phi(x)) - c}{d - c}",""," with",""," .. math::",""," \\phi(x) = a + \\frac{b - a}{2^n - 1} x.",""," If :math:`f` is a piecewise linear function on :math:`m` intervals"," :math:`[p_{i-1}, p_i], i \\in \\{1, ..., m\\}` with slopes :math:`\\alpha_i` and"," offsets :math:`\\beta_i` it can be written as",""," .. math::",""," f(x) = \\sum_{i=1}^m 1_{[p_{i-1}, p_i]}(x) (\\alpha_i x + \\beta_i)",""," where :math:`1_{[a, b]}` is an indication function that is 1 if the argument is in the interval"," :math:`[a, b]` and otherwise 0. The breakpoints :math:`p_i` can be specified by the"," ``breakpoints`` argument.",""," References:",""," [1]: Woerner, S., & Egger, D. J. (2018)."," Quantum Risk Analysis."," `arXiv:1806.06893 `_",""," [2]: Gacon, J., Zoufal, C., & Woerner, S. (2020)."," Quantum-Enhanced Simulation-Based Optimization."," `arXiv:2005.10780 `_"," \"\"\"",""," def __init__("," self,"," num_state_qubits: int,"," slope: float | list[float],"," offset: float | list[float],"," domain: tuple[float, float],"," image: tuple[float, float],"," rescaling_factor: float = 1,"," breakpoints: list[float] | None = None,"," name: str = \"F\","," ) -> None:"," r\"\"\""," Args:"," num_state_qubits: The number of qubits used to encode the variable :math:`x`."," slope: The slope of the linear function. Can be a list of slopes if it is a piecewise"," linear function."," offset: The offset of the linear function. Can be a list of offsets if it is a piecewise"," linear function."," domain: The domain of the function as tuple :math:`(x_\\min{}, x_\\max{})`."," image: The image of the function as tuple :math:`(f_\\min{}, f_\\max{})`."," rescaling_factor: The rescaling factor to adjust the accuracy in the Taylor"," approximation."," breakpoints: The breakpoints if the function is piecewise linear. If None, the function"," is not piecewise."," name: Name of the circuit."," \"\"\""," if not hasattr(slope, \"__len__\"):"," slope = [slope]"," if not hasattr(offset, \"__len__\"):"," offset = [offset]",""," # ensure that the breakpoints include the first point of the domain"," if breakpoints is None:"," breakpoints = [domain[0]]"," else:"," if not np.isclose(breakpoints[0], domain[0]):"," breakpoints = [domain[0]] + breakpoints",""," _check_sizes_match(slope, offset, breakpoints)"," _check_sorted_and_in_range(breakpoints, domain)",""," self._domain = domain"," self._image = image"," self._rescaling_factor = rescaling_factor",""," # do rescalings"," a, b = domain"," c, d = image",""," mapped_breakpoints = []"," mapped_slope = []"," mapped_offset = []"," for i, point in enumerate(breakpoints):"," mapped_breakpoint = (point - a) / (b - a) * (2**num_state_qubits - 1)"," mapped_breakpoints += [mapped_breakpoint]",""," # factor (upper - lower) / (2^n - 1) is for the scaling of x to [l,u]"," # note that the +l for mapping to [l,u] is already included in"," # the offsets given as parameters"," mapped_slope += [slope[i] * (b - a) / (2**num_state_qubits - 1)]"," mapped_offset += [offset[i]]",""," # approximate linear behavior by scaling and contracting around pi/4"," slope_angles = np.zeros(len(breakpoints))"," offset_angles = np.pi / 4 * (1 - rescaling_factor) * np.ones(len(breakpoints))"," for i in range(len(breakpoints)):"," slope_angles[i] = np.pi * rescaling_factor * mapped_slope[i] / 2 / (d - c)"," offset_angles[i] += np.pi * rescaling_factor * (mapped_offset[i] - c) / 2 / (d - c)",""," # use PWLPauliRotations to implement the function"," pwl_pauli_rotation = PiecewiseLinearPauliRotations("," num_state_qubits, mapped_breakpoints, 2 * slope_angles, 2 * offset_angles, name=name"," )",""," super().__init__(*pwl_pauli_rotation.qregs, name=name)"," self.append(pwl_pauli_rotation.to_gate(), self.qubits)",""," def post_processing(self, scaled_value: float) -> float:"," r\"\"\"Map the function value of the approximated :math:`\\hat{f}` to :math:`f`.",""," Args:"," scaled_value: A function value from the Taylor expansion of :math:`\\hat{f}(x)`.",""," Returns:"," The ``scaled_value`` mapped back to the domain of :math:`f`, by first inverting"," the transformation used for the Taylor approximation and then mapping back from"," :math:`[0, 1]` to the original domain."," \"\"\""," # revert the mapping applied in the Taylor approximation"," value = scaled_value - 1 / 2 + np.pi / 4 * self._rescaling_factor"," value *= 2 / np.pi / self._rescaling_factor",""," # map the value from [0, 1] back to the original domain"," value *= self._image[1] - self._image[0]"," value += self._image[0]",""," return value","","","def _check_sorted_and_in_range(breakpoints, domain):"," if breakpoints is None:"," return",""," # check if sorted"," if not np.all(np.diff(breakpoints) > 0):"," raise ValueError(\"Breakpoints must be unique and sorted.\")",""," if breakpoints[0] < 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