CNOTDihedral¶

class
CNOTDihedral
(data=None, num_qubits=None, validate=True)[source]¶ Bases:
qiskit.quantum_info.operators.base_operator.BaseOperator
,qiskit.quantum_info.operators.mixins.adjoint.AdjointMixin
An Nqubit operator from the CNOTDihedral group.
The CNOTDihedral group is generated by the quantum gates,
CXGate
,TGate
, andXGate
.Representation
An \(N\)qubit CNOTDihedral 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 \(N \times N\) invertible binary matrix, and an \(N\) binary vector.
The phase polynomial is a polynomial of degree at most 3, in \(N\) variables, whose coefficients are in the ring Z_8 with 8 elements.
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)
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:
IGate
,XGate
,YGate
,ZGate
,TGate
,TdgGate
SGate
,SdgGate
,CXGate
,CZGate
,SwapGate
. They can be converted back into aQuantumCircuit
, orGate
object using theto_circuit()
orto_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
Operator
objects using theto_operator()
method. This is done via decomposing to a circuit, and then simulating the circuit as a unitary operator. References:
Shelly Garion and Andrew W. Cross, Synthesis of CNOTDihedral circuits with optimal number of two qubit gates, Quantum 4(369), 2020
Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta, Scalable randomised benchmarking of nonClifford gates, npj Quantum Inf 2, 16012 (2016).
Initialize a CNOTDihedral operator object.
 Parameters
data (CNOTDihedral or QuantumCircuit or 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.
Methods
Return the adjoint of the Operator.
Return the operator composition with another CNOTDihedral.
Return the conjugate of the CNOTDihedral.
Make a deep copy of current operator.
Return the right multiplied operator self * other.
Return the reverseorder tensor product with another CNOTDihedral.
Return tuple of input dimension for specified subsystems.
Return tuple of output dimension for specified subsystems.
Return the compose of a operator with itself n times.
Return a shallow copy with reshaped input and output subsystem dimensions.
Return the tensor product with another CNOTDihedral.
Return a QuantumCircuit implementing the CNOTDihedral element.
Return a Gate instruction implementing the CNOTDihedral object.
Convert operator to Numpy matrix.
Convert to an Operator object.
Return the transpose of the CNOTDihedral.
Attributes

dim
¶ Return tuple (input_shape, output_shape).

num_qubits
¶ Return the number of qubits if a Nqubit operator or None otherwise.

qargs
¶ Return the qargs for the operator.

settings
¶ Return operator settings.