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# CNOTDihedral

class CNOTDihedral(data=None, num_qubits=None, validate=True)

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Bases: qiskit.quantum_info.operators.base_operator.BaseOperator, qiskit.quantum_info.operators.mixins.adjoint.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:

1. Shelly Garion and Andrew W. Cross, Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates, Quantum 4(369), 2020(opens in a new tab)
2. 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.

## Methods

CNOTDihedral.adjoint()

Return the adjoint of the Operator.

### compose

CNOTDihedral.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

CNOTDihedral

Raises

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

Note

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

CNOTDihedral.conjugate()

Return the conjugate of the CNOTDihedral.

### copy

CNOTDihedral.copy()

Make a deep copy of current operator.

### dot

CNOTDihedral.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

Operator

Note

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

### expand

CNOTDihedral.expand(other)

Return the reverse-order tensor product with another CNOTDihedral.

Parameters

other (CNOTDihedral) – a CNOTDihedral object.

Returns

the tensor product $b \otimes a$, where $a$

is the current CNOTDihedral, and $b$ is the other CNOTDihedral.

Return type

CNOTDihedral

### input_dims

CNOTDihedral.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

### output_dims

CNOTDihedral.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

### power

CNOTDihedral.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

Pauli

Raises

QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.

### reshape

CNOTDihedral.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

CNOTDihedral.tensor(other)

Return the tensor product with another CNOTDihedral.

Parameters

other (CNOTDihedral) – a CNOTDihedral object.

Returns

the tensor product $a \otimes b$, where $a$

is the current CNOTDihedral, and $b$ is the other CNOTDihedral.

Return type

CNOTDihedral

Note

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

### to_circuit

CNOTDihedral.to_circuit()

Return a QuantumCircuit implementing the CNOT-Dihedral element.

Returns

a circuit implementation of the CNOTDihedral object.

Return type

QuantumCircuit

References

1. Shelly Garion and Andrew W. Cross, Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates, Quantum 4(369), 2020(opens in a new tab)
2. 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

CNOTDihedral.to_instruction()

Return a Gate instruction implementing the CNOTDihedral object.

### to_matrix

CNOTDihedral.to_matrix()

Convert operator to Numpy matrix.

### to_operator

CNOTDihedral.to_operator()

Convert to an Operator object.

### transpose

CNOTDihedral.transpose()

Return the transpose of the CNOTDihedral.

## 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.