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CNOTDihedral operator class.
) annotationsN) QiskitError) BaseOperator)Operator)Pauli)ScalarOp)generate_apidocs AdjointMixin)QuantumCircuit Instruction )_append_circuit)SpecialPolynomialc ^ \ rS rSrSr S S U 4S jjjr\S 5 r\S 5 rS r S r
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rS rS rS rS rS S jrS rS!S jr S" S#S jjrS$S jrS%S jrS%S jrS r S r S r S r S r U =r!$ )& CNOTDihedral a An N-qubit operator from the CNOT-Dihedral group.
The CNOT-Dihedral group is generated by the quantum gates,
:class:`~qiskit.circuit.library.CXGate`, :class:`~qiskit.circuit.library.TGate`,
and :class:`~qiskit.circuit.library.XGate`.
**Representation**
An :math:`N`-qubit CNOT-Dihedral operator is stored as an affine function and a
phase polynomial, based on the convention in references [1, 2].
The affine function consists of an :math:`N \times N` invertible binary matrix,
and an :math:`N` binary vector.
The phase polynomial is a polynomial of degree at most 3,
in :math:`N` variables, whose coefficients are in the ring Z_8 with 8 elements.
.. plot::
:include-source:
:nofigs:
from qiskit import QuantumCircuit
from qiskit.quantum_info import CNOTDihedral
circ = QuantumCircuit(3)
circ.cx(0, 1)
circ.x(2)
circ.t(1)
circ.t(1)
circ.t(1)
elem = CNOTDihedral(circ)
# Print the CNOTDihedral element
print(elem)
.. code-block:: text
phase polynomial =
0 + 3*x_0 + 3*x_1 + 2*x_0*x_1
affine function =
(x_0,x_0 + x_1,x_2 + 1)
**Circuit Conversion**
CNOTDihedral operators can be initialized from circuits containing *only* the
following gates: :class:`~qiskit.circuit.library.IGate`,
:class:`~qiskit.circuit.library.XGate`, :class:`~qiskit.circuit.library.YGate`,
:class:`~qiskit.circuit.library.ZGate`,
:class:`~qiskit.circuit.library.TGate`, :class:`~qiskit.circuit.library.TdgGate`
:class:`~qiskit.circuit.library.SGate`, :class:`~qiskit.circuit.library.SdgGate`,
:class:`~qiskit.circuit.library.CXGate`, :class:`~qiskit.circuit.library.CZGate`,
:class:`~qiskit.circuit.library.CSGate`, :class:`~qiskit.circuit.library.CSdgGate`,
:class:`~qiskit.circuit.library.SwapGate`, :class:`~qiskit.circuit.library.CCZGate`.
They can be converted back into a :class:`~qiskit.circuit.QuantumCircuit`,
or :class:`~qiskit.circuit.Gate` object using the :meth:`~CNOTDihedral.to_circuit`
or :meth:`~CNOTDihderal.to_instruction` methods respectively. Note that this
decomposition is not necessarily optimal in terms of number of gates
if the number of qubits is more than two.
CNOTDihedral operators can also be converted to
:class:`~qiskit.quantum_info.Operator` objects using the
:meth:`to_operator` method. This is done via decomposing to a circuit,
and then simulating the circuit as a unitary operator.
References:
1. Shelly Garion and Andrew W. Cross, *Synthesis of CNOT-Dihedral circuits
with optimal number of two qubit gates*, `Quantum 4(369), 2020
`_
2. Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta,
*Scalable randomized benchmarking of non-Clifford gates*,
npj Quantum Inf 2, 16012 (2016).
c > U( a X l [ U R 5 U l [ R " U R [ R
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S9U l GO+[ U[ 5 ( a5 UR U l UR U l UR U l GO[ U[ 5 ( a UR 5 ( a' [ UR 5 S1:w d
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Args:
data (CNOTDihedral or QuantumCircuit or ~qiskit.circuit.Instruction):
Optional, operator to initialize.
num_qubits (int): Optional, initialize an empty CNOTDihedral operator.
validate (bool): if True, validates the CNOTDihedral element.
Raises:
QiskitError: if the type is invalid.
QiskitError: if validate=True and the CNOTDihedral element is invalid.
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*Scalable randomized benchmarking of non-Clifford gates*,
npj Quantum Inf 2, 16012 (2016).
r )synth_cnotdihedral_full) qiskit.synthesis.cnotdihedralr )r* r s r/
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circuit (QuantumCircuit or ~qiskit.circuit.Instruction):
instruction to initialize.
Returns:
CNOTDihedral: the CNOTDihedral object for the circuit.
Raises:
QiskitError: if the input instruction is not CNOTDihedral or contains
classical register instruction.
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