diff --git a/qiskit/circuit/library/arithmetic/piecewise_chebyshev.py b/qiskit/circuit/library/arithmetic/piecewise_chebyshev.py
new file mode 100644
index 000000000000..65cdc4a444a5
--- /dev/null
+++ b/qiskit/circuit/library/arithmetic/piecewise_chebyshev.py
@@ -0,0 +1,327 @@
+# 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 typing import Callable, List, Optional
+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
+
+
+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].
+
+    Examples:
+
+        .. jupyer-execute::
+
+            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)
+            print(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>`_
+    """
+
+    def __init__(self,
+                 f_x: Callable[[int], float],
+                 degree: Optional[int] = None,
+                 breakpoints: Optional[List[int]] = None,
+                 num_state_qubits: Optional[int] = None,
+                 name: str = 'pw_cheb') -> None:
+        r"""
+        Args:
+            f_x: the function to be approximated.
+            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 = None
+
+        self.num_state_qubits = num_state_qubits
+
+    def _check_configuration(self, raise_on_failure: bool = True) -> bool:
+        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) -> 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: Optional[Callable[[int], float]]) -> 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: Optional[int]) -> 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: Optional[List[int]]) -> 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
+
+            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.
+        """
+        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)
+        num_intervals = len(self._breakpoints)
+
+        # Calculate the polynomials
+        polynomials = []
+        for i in range(0, num_intervals - 1):
+            # Calculate the polynomial approximating the function on the current interval
+            poly = Chebyshev.interpolate(self._f_x, self._degree,
+                                         domain=[self._breakpoints[i],
+                                                 self._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())
+
+        # If the last breakpoint is < 2 ** num_qubits, add the identity polynomial
+        if self._breakpoints[-1] < 2 ** self.num_state_qubits:
+            polynomials = polynomials + [[2 * np.arcsin(1)]]
+
+        # If the first breakpoint is > 0, add the identity polynomial
+        if self._breakpoints[0] > 0:
+            polynomials = [[2 * np.arcsin(1)]] + polynomials
+
+        return polynomials
+
+    @polynomials.setter
+    def polynomials(self, polynomials: Optional[List[List[float]]]) -> 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: Optional[int]) -> 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: Optional[int]) -> None:
+        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]
+            self._ancillas = []
+            self._qubits = qr_state[:] + qr_target[:]
+
+            num_ancillas = num_state_qubits
+            if num_ancillas > 0:
+                qr_ancilla = AncillaRegister(num_ancillas)
+                self.add_register(qr_ancilla)
+
+        else:
+            self.qregs = []
+            self._qubits = []
+            self._ancillas = []
+
+    def _build(self):
+        """Build the circuit. The operation is considered successful when q_objective is :math:`|1>`
+        """
+        # do not build the circuit if _data is already populated
+        if self._data is not None:
+            return
+
+        self._data = []
+
+        # check whether the configuration is valid
+        self._check_configuration()
+
+        poly_r = PiecewisePolynomialPauliRotations(self.num_state_qubits,
+                                                   self.breakpoints, self.polynomials)
+
+        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_instruction(), qr_state[:] + qr_target + qr_ancillas[:])
diff --git a/qiskit/circuit/library/arithmetic/piecewise_polynomial_pauli_rotations.py b/qiskit/circuit/library/arithmetic/piecewise_polynomial_pauli_rotations.py
index bfe8726b6a5b..ff576c4da80a 100644
--- a/qiskit/circuit/library/arithmetic/piecewise_polynomial_pauli_rotations.py
+++ b/qiskit/circuit/library/arithmetic/piecewise_polynomial_pauli_rotations.py
@@ -127,6 +127,10 @@ def breakpoints(self) -> List[int]:
         Returns:
             The list of breakpoints.
         """
+        if self.num_state_qubits is not None and len(self._breakpoints) == len(self.coeffs) and\
+                self._breakpoints[-1] < 2 ** self.num_state_qubits:
+            return self._breakpoints + [2 ** self.num_state_qubits]
+
         return self._breakpoints
 
     @breakpoints.setter
@@ -246,20 +250,27 @@ def _reset_registers(self, num_state_qubits: Optional[int]) -> None:
             if self.contains_zero_breakpoint:
                 num_ancillas -= 1
             if num_ancillas > 0:
-                self._ancillas = []
                 qr_ancilla = AncillaRegister(num_ancillas)
                 self.add_register(qr_ancilla)
+            else:
+                qr_ancilla = []
+
+            self._qubits = qr_state[:] + qr_target[:] + qr_ancilla[:]
+            self._ancillas = qr_ancilla[:]
         else:
             self.qregs = []
+            self._qubits = []
+            self._ancillas = []
 
     def _build(self):
-        # Add the last breakpoint if necessary
-        if self.num_state_qubits is not None and len(self._breakpoints) == len(self._coeffs) and\
-                self._breakpoints[-1] < 2 ** self.num_state_qubits:
-            self.breakpoints = self._breakpoints + [2 ** self.num_state_qubits]
+        # do not build the circuit if _data is already populated
+        if self._data is not None:
+            return
+
+        self._data = []
 
-        # The number of ancilla might have changed, so reset registers
-        super()._build()
+        # check whether the configuration is valid
+        self._check_configuration()
 
         qr_state = self.qubits[:self.num_state_qubits]
         qr_target = [self.qubits[self.num_state_qubits]]
diff --git a/qiskit/circuit/library/arithmetic/polynomial_pauli_rotations.py b/qiskit/circuit/library/arithmetic/polynomial_pauli_rotations.py
index c43767a68b66..3c093a40e25e 100644
--- a/qiskit/circuit/library/arithmetic/polynomial_pauli_rotations.py
+++ b/qiskit/circuit/library/arithmetic/polynomial_pauli_rotations.py
@@ -303,7 +303,14 @@ def _get_rotation_coefficients(self) -> Dict[Sequence[int], float]:
         return rotation_coeffs
 
     def _build(self):
-        super()._build()
+        # do not build the circuit if _data is already populated
+        if self._data is not None:
+            return
+
+        self._data = []
+
+        # check whether the configuration is valid
+        self._check_configuration()
 
         qr_state = self.qubits[:self.num_state_qubits]
         qr_target = self.qubits[self.num_state_qubits]
diff --git a/qiskit/circuit/library/blueprintcircuit.py b/qiskit/circuit/library/blueprintcircuit.py
index bf26cbcff327..2131e477b6e1 100644
--- a/qiskit/circuit/library/blueprintcircuit.py
+++ b/qiskit/circuit/library/blueprintcircuit.py
@@ -59,7 +59,7 @@ def _check_configuration(self, raise_on_failure: bool = True) -> bool:
     def _build(self) -> None:
         """Build the circuit."""
         # do not build the circuit if _data is already populated
-        if self._data:
+        if self._data is not None:
             return
 
         self._data = []
diff --git a/releasenotes/notes/add-piecewise-polynomial-chebyshev-66c82ffe1d749dd6.yaml b/releasenotes/notes/add-piecewise-polynomial-chebyshev-66c82ffe1d749dd6.yaml
new file mode 100644
index 000000000000..3bb21dc3f530
--- /dev/null
+++ b/releasenotes/notes/add-piecewise-polynomial-chebyshev-66c82ffe1d749dd6.yaml
@@ -0,0 +1,6 @@
+---
+features:
+  - |
+    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`.
diff --git a/test/python/circuit/library/test_piecewise_chebyshev.py b/test/python/circuit/library/test_piecewise_chebyshev.py
new file mode 100644
index 000000000000..78940f91b7b0
--- /dev/null
+++ b/test/python/circuit/library/test_piecewise_chebyshev.py
@@ -0,0 +1,127 @@
+# 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.
+
+"""Test the piecewise Chebyshev approximation."""
+
+import unittest
+from collections import defaultdict
+import numpy as np
+from ddt import ddt, data, unpack
+
+from qiskit.test.base import QiskitTestCase
+from qiskit import BasicAer, execute
+from qiskit.circuit import QuantumCircuit
+from qiskit.circuit.library.arithmetic.piecewise_chebyshev import PiecewiseChebyshev
+
+
+@ddt
+class TestPiecewiseChebyshev(QiskitTestCase):
+    """Test the piecewise Chebyshev approximation."""
+
+    def assertFunctionIsCorrect(self, function_circuit, reference):
+        """Assert that ``function_circuit`` implements the reference function ``reference``."""
+        function_circuit._build()
+        num_state_qubits = function_circuit.num_state_qubits
+        num_ancilla_qubits = function_circuit.num_ancillas
+        circuit = QuantumCircuit(num_state_qubits + 1 + num_ancilla_qubits)
+        circuit.h(list(range(num_state_qubits)))
+        circuit.append(function_circuit.to_instruction(), list(range(circuit.num_qubits)))
+
+        backend = BasicAer.get_backend('statevector_simulator')
+        statevector = execute(circuit, backend).result().get_statevector()
+
+        probabilities = defaultdict(float)
+        for i, statevector_amplitude in enumerate(statevector):
+            i = bin(i)[2:].zfill(circuit.num_qubits)[num_ancilla_qubits:]
+            probabilities[i] += np.real(np.abs(statevector_amplitude) ** 2)
+
+        unrolled_probabilities = []
+        unrolled_expectations = []
+        for i, probability in probabilities.items():
+            x, last_qubit = int(i[1:], 2), i[0]
+            if last_qubit == '0':
+                expected_amplitude = np.cos(reference(x)) / np.sqrt(2 ** num_state_qubits)
+            else:
+                expected_amplitude = np.sin(reference(x)) / np.sqrt(2 ** num_state_qubits)
+
+            unrolled_probabilities += [probability]
+            unrolled_expectations += [np.real(np.abs(expected_amplitude) ** 2)]
+
+        np.testing.assert_array_almost_equal(unrolled_probabilities, unrolled_expectations,
+                                             decimal=3)
+
+    @ data((lambda x: np.arcsin(1 / x), 2, [2, 4], 2),
+           (lambda x: x / 8, 1, [1, 8], 3)
+           )
+    @ unpack
+    def test_piecewise_chebyshev(self, f_x, degree, breakpoints, num_state_qubits):
+        """Test the piecewise Chebyshev approximation."""
+
+        def pw_poly(x):
+            if breakpoints[0] <= x < breakpoints[-1]:
+                return f_x(x)
+            return np.arcsin(1)
+
+        pw_approximation = PiecewiseChebyshev(f_x, degree, breakpoints, num_state_qubits)
+
+        self.assertFunctionIsCorrect(pw_approximation, pw_poly)
+
+    def test_piecewise_chebyshev_mutability(self):
+        """Test the mutability of the piecewise Chebyshev approximation."""
+
+        def pw_poly(x, f_x):
+            if breakpoints[0] <= x < breakpoints[-1]:
+                return f_x(x)
+            return np.arcsin(1)
+
+        def f_x_1(x):
+            return x / 2
+
+        pw_approximation = PiecewiseChebyshev(f_x_1)
+
+        with self.subTest(msg='missing number of state qubits'):
+            with self.assertRaises(AttributeError):  # no state qubits set
+                print(pw_approximation.draw())
+
+        with self.subTest(msg='default setup, just setting number of state qubits'):
+            pw_approximation.num_state_qubits = 2
+            pw_approximation.f_x = f_x_1
+            # set to the default breakpoints for pw_poly
+            breakpoints = [0, 4]
+            pw_approximation.breakpoints = breakpoints
+            self.assertFunctionIsCorrect(pw_approximation, lambda x: pw_poly(x, f_x_1))
+
+        def f_x_2(x):
+            return x / 4
+        with self.subTest(msg='setting non-default values'):
+            breakpoints = [0, 2]
+            degree = 2
+            pw_approximation.breakpoints = breakpoints
+            pw_approximation.degree = degree
+            pw_approximation.f_x = f_x_2
+            self.assertFunctionIsCorrect(pw_approximation, lambda x: pw_poly(x, f_x_2))
+
+        def f_x_3(x):
+            return x ** 2
+
+        with self.subTest(msg='changing all values'):
+            pw_approximation.num_state_qubits = 4
+            breakpoints = [1, 3, 6]
+            degree = 3
+            pw_approximation.breakpoints = breakpoints
+            pw_approximation.degree = degree
+            pw_approximation.f_x = f_x_3
+            self.assertFunctionIsCorrect(pw_approximation, lambda x: pw_poly(x, f_x_3))
+
+
+if __name__ == '__main__':
+    unittest.main()