CNOTDihedral
class qiskit.quantum_info.CNOTDihedral(data=None, num_qubits=None, validate=True)
Bases: BaseOperator, AdjointMixin
An N-qubit operator from the CNOT-Dihedral group.
The CNOT-Dihedral group is generated by the quantum gates,
CXGate,TGate, andXGate.Representation
An -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 invertible binary matrix, and an binary vector.
The phase polynomial is a polynomial of degree at most 3, in 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,TdgGateSGate,SdgGate,CXGate,CZGate,CSGate,CSdgGate,SwapGate,CCZGate. They can be converted back into aQuantumCircuit, orGateobject 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
Operatorobjects 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 CNOT-Dihedral 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 non-Clifford gates, npj Quantum Inf 2, 16012 (2016).
Initialize a CNOTDihedral operator object.
Parameters
- data (CNOTDihedral orQuantumCircuit orInstruction) – 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.
Attributes
dim
Return tuple (input_shape, output_shape).
name
Unique string identifier for operation type.
num_clbits
Number of classical bits.
num_qubits
Return the number of qubits if a N-qubit operator or None otherwise.
qargs
Return the qargs for the operator.
Methods
adjoint
compose
compose(other, qargs=None, front=False)
Return the operator composition with another CNOTDihedral.
Parameters
- other (CNOTDihedral) – a CNOTDihedral object.
- qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).
- front (bool) – If True compose using right operator multiplication, instead of left multiplication [default: False].
Returns
The composed CNOTDihedral.
Return type
Raises
QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.
Composition (&) by default is defined as left matrix multiplication for matrix operators, while @ (equivalent to dot()) is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B @ A == B.dot(A) when A and B are of the same type.
Setting the front=True kwarg changes this to right matrix multiplication and is equivalent to the dot() method A.dot(B) == A.compose(B, front=True).
conjugate
copy
dot
dot(other, qargs=None)
Return the right multiplied operator self * other.
Parameters
- other (Operator) – an operator object.
- qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).
Returns
The right matrix multiplied Operator.
Return type
The dot product can be obtained using the @ binary operator. Hence a.dot(b) is equivalent to a @ b.
expand
expand(other)
Return the reverse-order tensor product with another CNOTDihedral.
Parameters
other (CNOTDihedral) – a CNOTDihedral object.
Returns
the tensor product , where
is the current CNOTDihedral, and is the other CNOTDihedral.
Return type
input_dims
output_dims
power
power(n)
Return the compose of a operator with itself n times.
Parameters
n (int) – the number of times to compose with self (n>0).
Returns
the n-times composed operator.
Return type
Raises
QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.
reshape
reshape(input_dims=None, output_dims=None, num_qubits=None)
Return a shallow copy with reshaped input and output subsystem dimensions.
Parameters
- input_dims (None or tuple) – new subsystem input dimensions. If None the original input dims will be preserved [Default: None].
- output_dims (None or tuple) – new subsystem output dimensions. If None the original output dims will be preserved [Default: None].
- num_qubits (None or int) – reshape to an N-qubit operator [Default: None].
Returns
returns self with reshaped input and output dimensions.
Return type
BaseOperator
Raises
QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.
tensor
tensor(other)
Return the tensor product with another CNOTDihedral.
Parameters
other (CNOTDihedral) – a CNOTDihedral object.
Returns
the tensor product , where
is the current CNOTDihedral, and is the other CNOTDihedral.
Return type
The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.
to_circuit
to_circuit()
Return a QuantumCircuit implementing the CNOT-Dihedral element.
Returns
a circuit implementation of the CNOTDihedral object.
Return type
References
- Shelly Garion and Andrew W. Cross, Synthesis of CNOT-Dihedral 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 non-Clifford gates, npj Quantum Inf 2, 16012 (2016).