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
Piecewisepolynomiallycontrolled 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 \([0, 2^n1]\), 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 piecewiselinear function. Defaults to[0]
.coeffs (
Optional
[List
[List
[float
]]]) – The coefficients of the polynomials for different segments of thefunction. coeffs[j][i] is the coefficient of the ith power of x (piecewiselinear) –
the jth 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
Classically evaluate the piecewise polynomial rotation.
Attributes

ancillas
¶ Returns a list of ancilla bits in the order that the registers were added.
 Return type
List
[AncillaQubit
]

basis
¶ The kind of Pauli rotation to be used.
Set the basis to ‘X’, ‘Y’ or ‘Z’ for controlledX, 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 is2**(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}}
 Return type
dict

clbits
¶ Returns a list of classical bits in the order that the registers were added.
 Return type
List
[Clbit
]

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.
 Return type
Union
[ParameterExpression
,float
]

header
= 'OPENQASM 2.0;'¶

instances
= 9¶

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. Return type
dict

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.
 Return type
int

num_clbits
¶ Return number of classical bits.
 Return type
int

num_parameters
¶  Return type
int

num_qubits
¶ Return number of qubits.
 Return type
int

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.
 Return type
List
[Qubit
]