# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# 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.
"""Piecewise polynomial Chebyshev approximation to a given f(x)."""
from __future__ import annotations
from typing import Callable
import numpy as np
from numpy.polynomial.chebyshev import Chebyshev
from qiskit.circuit import QuantumRegister, AncillaRegister
from qiskit.circuit.library.blueprintcircuit import BlueprintCircuit
from qiskit.circuit.exceptions import CircuitError
from .piecewise_polynomial_pauli_rotations import PiecewisePolynomialPauliRotations
[docs]class PiecewiseChebyshev(BlueprintCircuit):
r"""Piecewise Chebyshev approximation to an input function.
For a given function :math:`f(x)` and degree :math:`d`, this class implements a piecewise
polynomial Chebyshev approximation on :math:`n` qubits to :math:`f(x)` on the given intervals.
All the polynomials in the approximation are of degree :math:`d`.
The values of the parameters are calculated according to [1] and see [2] for a more
detailed explanation of the circuit construction and how it acts on the qubits.
Examples:
.. plot::
:include-source:
import numpy as np
from qiskit import QuantumCircuit
from qiskit.circuit.library.arithmetic.piecewise_chebyshev import PiecewiseChebyshev
f_x, degree, breakpoints, num_state_qubits = lambda x: np.arcsin(1 / x), 2, [2, 4], 2
pw_approximation = PiecewiseChebyshev(f_x, degree, breakpoints, num_state_qubits)
pw_approximation._build()
qc = QuantumCircuit(pw_approximation.num_qubits)
qc.h(list(range(num_state_qubits)))
qc.append(pw_approximation.to_instruction(), qc.qubits)
qc.draw(output='mpl')
References:
[1]: Haener, T., Roetteler, M., & Svore, K. M. (2018).
Optimizing Quantum Circuits for Arithmetic.
`arXiv:1805.12445 <http://arxiv.org/abs/1805.12445>`_
[2]: Carrera Vazquez, A., Hiptmair, H., & Woerner, S. (2022).
Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation.
`ACM Transactions on Quantum Computing 3, 1, Article 2 <https://doi.org/10.1145/3490631>`_
"""
def __init__(
self,
f_x: float | Callable[[int], float],
degree: int | None = None,
breakpoints: list[int] | None = None,
num_state_qubits: int | None = None,
name: str = "pw_cheb",
) -> None:
r"""
Args:
f_x: the function to be approximated. Constant functions should be specified
as f_x = constant.
degree: the degree of the polynomials.
Defaults to ``1``.
breakpoints: the breakpoints to define the piecewise-linear function.
Defaults to the full interval.
num_state_qubits: number of qubits representing the state.
name: The name of the circuit object.
"""
super().__init__(name=name)
# define internal parameters
self._num_state_qubits = None
# Store parameters
self._f_x = f_x
self._degree = degree if degree is not None else 1
self._breakpoints = breakpoints if breakpoints is not None else [0]
self._polynomials: list[list[float]] | None = None
self.num_state_qubits = num_state_qubits
def _check_configuration(self, raise_on_failure: bool = True) -> bool:
"""Check if the current configuration is valid."""
valid = True
if self._f_x is None:
valid = False
if raise_on_failure:
raise AttributeError("The function to be approximated has not been set.")
if self._degree is None:
valid = False
if raise_on_failure:
raise AttributeError("The degree of the polynomials has not been set.")
if self._breakpoints is None:
valid = False
if raise_on_failure:
raise AttributeError("The breakpoints have not been set.")
if self.num_state_qubits is None:
valid = False
if raise_on_failure:
raise AttributeError("The number of qubits has not been set.")
if self.num_qubits < self.num_state_qubits + 1:
valid = False
if raise_on_failure:
raise CircuitError(
"Not enough qubits in the circuit, need at least "
"{}.".format(self.num_state_qubits + 1)
)
return valid
@property
def f_x(self) -> float | Callable[[int], float]:
"""The function to be approximated.
Returns:
The function to be approximated.
"""
return self._f_x
@f_x.setter
def f_x(self, f_x: float | Callable[[int], float] | None) -> None:
"""Set the function to be approximated.
Note that this may change the underlying quantum register, if the number of state qubits
changes.
Args:
f_x: The new function to be approximated.
"""
if self._f_x is None or f_x != self._f_x:
self._invalidate()
self._f_x = f_x
self._reset_registers(self.num_state_qubits)
@property
def degree(self) -> int:
"""The degree of the polynomials.
Returns:
The degree of the polynomials.
"""
return self._degree
@degree.setter
def degree(self, degree: int | None) -> None:
"""Set the error tolerance.
Note that this may change the underlying quantum register, if the number of state qubits
changes.
Args:
degree: The new degree.
"""
if self._degree is None or degree != self._degree:
self._invalidate()
self._degree = degree
self._reset_registers(self.num_state_qubits)
@property
def breakpoints(self) -> list[int]:
"""The breakpoints for the piecewise approximation.
Returns:
The breakpoints for the piecewise approximation.
"""
breakpoints = self._breakpoints
# it the state qubits are set ensure that the breakpoints match beginning and end
if self.num_state_qubits is not None:
num_states = 2**self.num_state_qubits
# If the last breakpoint is < num_states, add the identity polynomial
if breakpoints[-1] < num_states:
breakpoints = breakpoints + [num_states]
# If the first breakpoint is > 0, add the identity polynomial
if breakpoints[0] > 0:
breakpoints = [0] + breakpoints
return breakpoints
@breakpoints.setter
def breakpoints(self, breakpoints: list[int] | None) -> None:
"""Set the breakpoints for the piecewise approximation.
Note that this may change the underlying quantum register, if the number of state qubits
changes.
Args:
breakpoints: The new breakpoints for the piecewise approximation.
"""
if self._breakpoints is None or breakpoints != self._breakpoints:
self._invalidate()
self._breakpoints = breakpoints if breakpoints is not None else [0]
self._reset_registers(self.num_state_qubits)
@property
def polynomials(self) -> list[list[float]]:
"""The polynomials for the piecewise approximation.
Returns:
The polynomials for the piecewise approximation.
Raises:
TypeError: If the input function is not in the correct format.
"""
if self.num_state_qubits is None:
return [[]]
# note this must be the private attribute since we handle missing breakpoints at
# 0 and 2 ^ num_qubits here (e.g. if the function we approximate is not defined at 0
# and the user takes that into account we just add an identity)
breakpoints = self._breakpoints
# Need to take into account the case in which no breakpoints were provided in first place
if breakpoints == [0]:
breakpoints = [0, 2**self.num_state_qubits]
num_intervals = len(breakpoints)
# Calculate the polynomials
polynomials = []
for i in range(0, num_intervals - 1):
# Calculate the polynomial approximating the function on the current interval
try:
# If the function is constant don't call Chebyshev (not necessary and gives errors)
if isinstance(self.f_x, (float, int)):
# Append directly to list of polynomials
polynomials.append([self.f_x])
else:
poly = Chebyshev.interpolate(
self.f_x, self.degree, domain=[breakpoints[i], breakpoints[i + 1]]
)
# Convert polynomial to the standard basis and rescale it for the rotation gates
poly = 2 * poly.convert(kind=np.polynomial.Polynomial).coef
# Convert to list and append
polynomials.append(poly.tolist())
except ValueError as err:
raise TypeError(
" <lambda>() missing 1 required positional argument: '"
+ self.f_x.__code__.co_varnames[0]
+ "'."
+ " Constant functions should be specified as 'f_x = constant'."
) from err
# If the last breakpoint is < 2 ** num_qubits, add the identity polynomial
if breakpoints[-1] < 2**self.num_state_qubits:
polynomials = polynomials + [[2 * np.arcsin(1)]]
# If the first breakpoint is > 0, add the identity polynomial
if breakpoints[0] > 0:
polynomials = [[2 * np.arcsin(1)]] + polynomials
return polynomials
@polynomials.setter
def polynomials(self, polynomials: list[list[float]] | None) -> None:
"""Set the polynomials for the piecewise approximation.
Note that this may change the underlying quantum register, if the number of state qubits
changes.
Args:
polynomials: The new breakpoints for the piecewise approximation.
"""
if self._polynomials is None or polynomials != self._polynomials:
self._invalidate()
self._polynomials = polynomials
self._reset_registers(self.num_state_qubits)
@property
def num_state_qubits(self) -> int:
r"""The number of state qubits representing the state :math:`|x\rangle`.
Returns:
The number of state qubits.
"""
return self._num_state_qubits
@num_state_qubits.setter
def num_state_qubits(self, num_state_qubits: int | None) -> None:
"""Set the number of state qubits.
Note that this may change the underlying quantum register, if the number of state qubits
changes.
Args:
num_state_qubits: The new number of qubits.
"""
if self._num_state_qubits is None or num_state_qubits != self._num_state_qubits:
self._invalidate()
self._num_state_qubits = num_state_qubits
# Set breakpoints if they haven't been set
if num_state_qubits is not None and self._breakpoints is None:
self.breakpoints = [0, 2**num_state_qubits]
self._reset_registers(num_state_qubits)
def _reset_registers(self, num_state_qubits: int | None) -> None:
"""Reset the registers."""
self.qregs = []
if num_state_qubits is not None:
qr_state = QuantumRegister(num_state_qubits, "state")
qr_target = QuantumRegister(1, "target")
self.qregs = [qr_state, qr_target]
num_ancillas = num_state_qubits
if num_ancillas > 0:
qr_ancilla = AncillaRegister(num_ancillas)
self.add_register(qr_ancilla)
def _build(self):
"""Build the circuit if not already build. The operation is considered successful
when q_objective is :math:`|1>`"""
if self._is_built:
return
super()._build()
poly_r = PiecewisePolynomialPauliRotations(
self.num_state_qubits, self.breakpoints, self.polynomials, name=self.name
)
# qr_state = self.qubits[: self.num_state_qubits]
# qr_target = [self.qubits[self.num_state_qubits]]
# qr_ancillas = self.qubits[self.num_state_qubits + 1 :]
# Apply polynomial approximation
self.append(poly_r.to_gate(), self.qubits)