# PiecewisePolynomialPauliRotations¶

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

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 $$x_{J+1} = 2^n$$.

Bemerkung

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

Parameter
• 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

• x (piecewise-linear function. coeffs[j][i] is the coefficient of the i-th power of) –

• polynomial. (for the j-th) – 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.

Rückgabetyp

List[AncillaQubit]

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.

Rückgabetyp

str

Rückgabe

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).

Rückgabetyp

List[int]

Rückgabe

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}}

Rückgabetyp

dict

clbits

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

Rückgabetyp

List[Clbit]

coeffs

The coefficients of the polynomials.

Rückgabetyp

List[List[float]]

Rückgabe

The polynomial coefficients per interval as nested lists.

contains_zero_breakpoint

Whether 0 is the first breakpoint.

Rückgabetyp

bool

Rückgabe

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.

Rückgabetyp

Union[ParameterExpression, float]

instances = 9
mapped_coeffs

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

Rückgabetyp

List[List[float]]

Rückgabe

The mapped coefficients.

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.

Rückgabetyp

dict

num_ancilla_qubits

The minimum number of ancilla qubits in the circuit.

Rückgabetyp

int

Rückgabe

The minimal number of ancillas required.

num_ancillas

Return the number of ancilla qubits.

Rückgabetyp

int

num_clbits

Return number of classical bits.

Rückgabetyp

int

num_parameters
Rückgabetyp

int

num_qubits

Return number of qubits.

Rückgabetyp

int

num_state_qubits

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

Rückgabetyp

int

Rückgabe

The number of state qubits.

parameters
Rückgabetyp

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.

Rückgabetyp

List[Qubit]