PiecewisePolynomialPauliRotations¶

class PiecewisePolynomialPauliRotations(num_state_qubits=None, breakpoints=None, coeffs=None, basis='Y', name='pw_poly')[source]¶

Bases: qiskit.circuit.library.arithmetic.functional_pauli_rotations.FunctionalPauliRotations

Piecewise-polynomially-controlled Pauli rotations.

This class implements a piecewise polynomial (not necessarily continuous) function, \(f(x)\), on qubit amplitudes, which is defined through breakpoints and coefficients as follows. Suppose the breakpoints \((x_0, ..., x_J)\) are a subset of \([0, 2^n-1]\), where \(n\) is the number of state qubits. Further on, denote the corresponding coefficients by \([a_{j,1},...,a_{j,d}]\), where \(d\) is the highest degree among all polynomials.

Then \(f(x)\) is defined as:

\[\begin{split}f(x) = \begin{cases} 0, x < x_0 \\ \sum_{i=0}^{i=d}a_{j,i}/2 x^i, x_j \leq x < x_{j+1} \end{cases}\end{split}\]

where if given the same number of breakpoints as polynomials, we implicitly assume \(x_{J+1} = 2^n\).

Note

Note the \(1/2\) factor in the coefficients of \(f(x)\), this is consistent with Qiskit’s Pauli rotations.

Examples

>>> from qiskit import QuantumCircuit
>>> from qiskit.circuit.library.arithmetic.piecewise_polynomial_pauli_rotations import\
... PiecewisePolynomialPauliRotations
>>> qubits, breakpoints, coeffs = (2, [0, 2], [[0, -1.2],[-1, 1, 3]])
>>> poly_r = PiecewisePolynomialPauliRotations(num_state_qubits=qubits,
...breakpoints=breakpoints, coeffs=coeffs)
>>>
>>> qc = QuantumCircuit(poly_r.num_qubits)
>>> qc.h(list(range(qubits)));
>>> qc.append(poly_r.to_instruction(), list(range(qc.num_qubits)));
>>> qc.draw()
     ┌───┐┌──────────┐
q_0: ┤ H ├┤0         ├
     ├───┤│          │
q_1: ┤ H ├┤1         ├
     └───┘│          │
q_2: ─────┤2         ├
          │  pw_poly │
q_3: ─────┤3         ├
          │          │
q_4: ─────┤4         ├
          │          │
q_5: ─────┤5         ├
          └──────────┘

References

[1]: Haener, T., Roetteler, M., & Svore, K. M. (2018).: Optimizing Quantum Circuits for Arithmetic. arXiv:1805.12445
[2]: Carrera Vazquez, A., Hiptmair, R., & Woerner, S. (2020).: Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation. arXiv:2009.04484

Parameters

num_state_qubits (Optional[int]) – The number of qubits representing the state.
breakpoints (Optional[List[int]]) – The breakpoints to define the piecewise-linear function. Defaults to [0].
coeffs (Optional[List[List[float]]]) – The coefficients of the polynomials for different segments of the
function. coeffs[j][i] is the coefficient of the i-th power of x (piecewise-linear) –
the j-th polynomial. (for) – Defaults to linear: [[1]].
basis (str) – The type of Pauli rotation ('X', 'Y', 'Z').
name (str) – The name of the circuit.

Methods Defined Here

evaluate

Classically evaluate the piecewise polynomial rotation.

Attributes

ancillas¶: Returns a list of ancilla bits in the order that the registers were added.

basis¶

The kind of Pauli rotation to be used.

Set the basis to ‘X’, ‘Y’ or ‘Z’ for controlled-X, -Y, or -Z rotations respectively.

Return type: str
Returns: The kind of Pauli rotation used in controlled rotation.

breakpoints¶

The breakpoints of the piecewise polynomial function.

The function is polynomial in the intervals [point_i, point_{i+1}] where the last point implicitly is 2**(num_state_qubits + 1).

Return type: List[int]
Returns: The list of breakpoints.

calibrations¶

Return calibration dictionary.

The custom pulse definition of a given gate is of the form: {‘gate_name’: {(qubits, params): schedule}}

clbits¶: Returns a list of classical bits in the order that the registers were added.

coeffs¶

The coefficients of the polynomials.

Return type: List[List[float]]
Returns: The polynomial coefficients per interval as nested lists.

contains_zero_breakpoint¶

Whether 0 is the first breakpoint.

Return type: bool
Returns: True, if 0 is the first breakpoint, otherwise False.

data¶

extension_lib = 'include "qelib1.inc";'¶

global_phase¶: Return the global phase of the circuit in radians.

header = 'OPENQASM 2.0;'¶

instances = 16¶

mapped_coeffs¶

The coefficients mapped to the internal representation, since we only compare x>=breakpoint.

Return type: List[List[float]]
Returns: The mapped coefficients.

metadata¶

The user provided metadata associated with the circuit

The metadata for the circuit is a user provided dict of metadata for the circuit. It will not be used to influence the execution or operation of the circuit, but it is expected to be passed between all transforms of the circuit (ie transpilation) and that providers will associate any circuit metadata with the results it returns from execution of that circuit.

num_ancilla_qubits¶

The minimum number of ancilla qubits in the circuit.

Return type: int
Returns: The minimal number of ancillas required.

num_ancillas¶: Return the number of ancilla qubits.

num_clbits¶: Return number of classical bits.

num_parameters¶

Return type: int

num_qubits¶: Return number of qubits.

num_state_qubits¶

The number of state qubits representing the state \(|x\rangle\).

Return type: int
Returns: The number of state qubits.

parameters¶

Return type: ParameterView

prefix = 'circuit'¶

qregs¶: A list of the quantum registers associated with the circuit.

qubits¶: Returns a list of quantum bits in the order that the registers were added.