--- title: CNOTDihedral (v1.1) description: API reference for qiskit.quantum_info.CNOTDihedral in qiskit v1.1 source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral --- # CNOTDihedral *class* `qiskit.quantum_info.CNOTDihedral(data=None, num_qubits=None, validate=True)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L31-L506) Bases: `BaseOperator`, `AdjointMixin` An N-qubit operator from the CNOT-Dihedral group. > The CNOT-Dihedral group is generated by the quantum gates, [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate"), [`TGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.TGate "qiskit.circuit.library.TGate"), and [`XGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.XGate "qiskit.circuit.library.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. > > ```python > 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) > ``` ```python 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`](/docs/api/qiskit/1.1/qiskit.circuit.library.IGate "qiskit.circuit.library.IGate"), [`XGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.XGate "qiskit.circuit.library.XGate"), [`YGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.YGate "qiskit.circuit.library.YGate"), [`ZGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.ZGate "qiskit.circuit.library.ZGate"), [`TGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.TGate "qiskit.circuit.library.TGate"), [`TdgGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.TdgGate "qiskit.circuit.library.TdgGate") [`SGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SGate "qiskit.circuit.library.SGate"), [`SdgGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SdgGate "qiskit.circuit.library.SdgGate"), [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate"), [`CZGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CZGate "qiskit.circuit.library.CZGate"), [`CSGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CSGate "qiskit.circuit.library.CSGate"), [`CSdgGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CSdgGate "qiskit.circuit.library.CSdgGate"), [`SwapGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SwapGate "qiskit.circuit.library.SwapGate"), [`CCZGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CCZGate "qiskit.circuit.library.CCZGate"). They can be converted back into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit"), or [`Gate`](/docs/api/qiskit/1.1/qiskit.circuit.Gate "qiskit.circuit.Gate") object using the [`to_circuit()`](#qiskit.quantum_info.CNOTDihedral.to_circuit "qiskit.quantum_info.CNOTDihedral.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`](/docs/api/qiskit/1.1/qiskit.quantum_info.Operator "qiskit.quantum_info.Operator") objects using the [`to_operator()`](#qiskit.quantum_info.CNOTDihedral.to_operator "qiskit.quantum_info.CNOTDihedral.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](https://quantum-journal.org/papers/q-2020-12-07-369/) > 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*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") *or*[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit") *or*[*Instruction*](/docs/api/qiskit/1.1/qiskit.circuit.Instruction "qiskit.circuit.Instruction")) – Optional, operator to initialize. - **num\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int)) – Optional, initialize an empty CNOTDihedral operator. - **validate** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – if True, validates the CNOTDihedral element. **Raises** - [**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if the type is invalid. - [**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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 `adjoint()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L442-L445) Return the adjoint of the Operator. ### compose `compose(other, qargs=None, front=False)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L374-L386) Return the operator composition with another CNOTDihedral. **Parameters** - **other** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral object. - **qargs** ([*list*](https://docs.python.org/3/library/stdtypes.html#list) *or None*) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None). - **front** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – If True compose using right operator multiplication, instead of left multiplication \[default: False]. **Returns** The composed CNOTDihedral. **Return type** [CNOTDihedral](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") **Raises** [**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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()`](#qiskit.quantum_info.CNOTDihedral.dot "qiskit.quantum_info.CNOTDihedral.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()`](#qiskit.quantum_info.CNOTDihedral.dot "qiskit.quantum_info.CNOTDihedral.dot") method `A.dot(B) == A.compose(B, front=True)`. ### conjugate `conjugate()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L447-L472) Return the conjugate of the CNOTDihedral. ### copy `copy()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/base_operator.py#L143-L145) Make a deep copy of current operator. ### dot `dot(other, qargs=None)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/mixins/group.py#L133-L149) Return the right multiplied operator self \* other. **Parameters** - **other** ([*Operator*](/docs/api/qiskit/1.1/qiskit.quantum_info.Operator "qiskit.quantum_info.Operator")) – an operator object. - **qargs** ([*list*](https://docs.python.org/3/library/stdtypes.html#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](/docs/api/qiskit/1.1/qiskit.quantum_info.Operator "qiskit.quantum_info.Operator") > **Note** > > The dot product can be obtained using the `@` binary operator. Hence `a.dot(b)` is equivalent to `a @ b`. ### expand `expand(other)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L439-L440) Return the reverse-order tensor product with another CNOTDihedral. **Parameters** **other** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.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](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") ### input\_dims `input_dims(qargs=None)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/base_operator.py#L135-L137) Return tuple of input dimension for specified subsystems. ### output\_dims `output_dims(qargs=None)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/base_operator.py#L139-L141) Return tuple of output dimension for specified subsystems. ### power `power(n)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/mixins/group.py#L151-L171) Return the compose of a operator with itself n times. **Parameters** **n** ([*int*](https://docs.python.org/3/library/functions.html#int)) – the number of times to compose with self (n>0). **Returns** the n-times composed operator. **Return type** [Clifford](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford") **Raises** [**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/base_operator.py#L106-L133) Return a shallow copy with reshaped input and output subsystem dimensions. **Parameters** - **input\_dims** (*None or* [*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple)) – new subsystem input dimensions. If None the original input dims will be preserved \[Default: None]. - **output\_dims** (*None or* [*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple)) – new subsystem output dimensions. If None the original output dims will be preserved \[Default: None]. - **num\_qubits** (*None or* [*int*](https://docs.python.org/3/library/functions.html#int)) – reshape to an N-qubit operator \[Default: None]. **Returns** returns self with reshaped input and output dimensions. **Return type** BaseOperator **Raises** [**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if combined size of all subsystem input dimension or subsystem output dimensions is not constant. ### tensor `tensor(other)` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L436-L437) Return the tensor product with another CNOTDihedral. **Parameters** **other** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.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](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") > **Note** > > The tensor product can be obtained using the `^` binary operator. Hence `a.tensor(b)` is equivalent to `a ^ b`. ### to\_circuit `to_circuit()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L317-L334) Return a QuantumCircuit implementing the CNOT-Dihedral element. **Returns** a circuit implementation of the CNOTDihedral object. **Return type** [QuantumCircuit](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.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](https://quantum-journal.org/papers/q-2020-12-07-369/) 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 `to_instruction()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L336-L338) Return a Gate instruction implementing the CNOTDihedral object. ### to\_matrix `to_matrix()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L366-L368) Convert operator to Numpy matrix. ### to\_operator `to_operator()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L370-L372) Convert to an Operator object. **Return type** [*Operator*](/docs/api/qiskit/1.1/qiskit.quantum_info.Operator "qiskit.quantum_info.operators.operator.Operator") ### transpose `transpose()` [GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/quantum_info/operators/dihedral/dihedral.py#L474-L477) Return the transpose of the CNOTDihedral.