CNOTDihedral#

class qiskit.quantum_info.CNOTDihedral(data=None, num_qubits=None, validate=True)[fuente]#

Bases: BaseOperator, AdjointMixin

An N-qubit operator from the CNOT-Dihedral group.

The CNOT-Dihedral group is generated by the quantum gates, CXGate, TGate, and XGate.

Representation

An \(N\)-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 \(N \times N\) 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, CSGate, CSdgGate, SwapGate, CCZGate. They can be converted back into a QuantumCircuit, or Gate object using the to_circuit() or 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 Operator objects using the to_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.

Parámetros:

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.

Muestra:

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()[fuente]#: Return the adjoint of the Operator.

compose(other, qargs=None, front=False)[fuente]#

Return the operator composition with another CNOTDihedral.

Parámetros:

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].

Devuelve:

The composed CNOTDihedral.

Tipo del valor devuelto:

CNOTDihedral

Muestra:

QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.

Nota

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()[fuente]#: Return the conjugate of the CNOTDihedral.

copy()#: Make a deep copy of current operator.

dot(other, qargs=None)#

Return the right multiplied operator self * other.

Parámetros:

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).

Devuelve:

The right matrix multiplied Operator.

Tipo del valor devuelto:

Operator

Nota

The dot product can be obtained using the @ binary operator. Hence a.dot(b) is equivalent to a @ b.

expand(other)[fuente]#

Return the reverse-order tensor product with another CNOTDihedral.

Parámetros:

other (CNOTDihedral) – a CNOTDihedral object.

Devuelve:

the tensor product \(b \otimes a\), where \(a\): is the current CNOTDihedral, and \(b\) is the other CNOTDihedral.

Tipo del valor devuelto:

CNOTDihedral

input_dims(qargs=None)#: Return tuple of input dimension for specified subsystems.

output_dims(qargs=None)#: Return tuple of output dimension for specified subsystems.

power(n)#

Return the compose of a operator with itself n times.

Parámetros:: n (int) – the number of times to compose with self (n>0).
Devuelve:: the n-times composed operator.
Tipo del valor devuelto:: Pauli
Muestra:: QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.

reshape(input_dims=None, output_dims=None, num_qubits=None)#

Return a shallow copy with reshaped input and output subsystem dimensions.

Parámetros:

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].

Devuelve:

returns self with reshaped input and output dimensions.

Tipo del valor devuelto:

BaseOperator

Muestra:

QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.

tensor(other)[fuente]#

Return the tensor product with another CNOTDihedral.

Parámetros:

other (CNOTDihedral) – a CNOTDihedral object.

Devuelve:

the tensor product \(a \otimes b\), where \(a\): is the current CNOTDihedral, and \(b\) is the other CNOTDihedral.

Tipo del valor devuelto:

CNOTDihedral

Nota

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

to_circuit()[fuente]#

Return a QuantumCircuit implementing the CNOT-Dihedral element.

Devuelve:: a circuit implementation of the CNOTDihedral object.
Tipo del valor devuelto:: QuantumCircuit

Referencias

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).

to_instruction()[fuente]#: Return a Gate instruction implementing the CNOTDihedral object.

to_matrix()[fuente]#: Convert operator to Numpy matrix.

to_operator()[fuente]#

Convert to an Operator object.

Tipo del valor devuelto:: Operator

transpose()[fuente]#: Return the transpose of the CNOTDihedral.