328 lines
16 KiB
Plaintext
328 lines
16 KiB
Plaintext
---
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title: CNOTDihedral
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description: API reference for qiskit.quantum_info.CNOTDihedral
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in_page_toc_min_heading_level: 1
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python_api_type: class
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python_api_name: qiskit.quantum_info.CNOTDihedral
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---
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# CNOTDihedral
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<Class id="qiskit.quantum_info.CNOTDihedral" isDedicatedPage={true} github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L31-L505" signature="qiskit.quantum_info.CNOTDihedral(data=None, num_qubits=None, validate=True)" modifiers="class">
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Bases: `BaseOperator`, `AdjointMixin`
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An N-qubit operator from the CNOT-Dihedral group.
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> The CNOT-Dihedral group is generated by the quantum gates, [`CXGate`](qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate"), [`TGate`](qiskit.circuit.library.TGate "qiskit.circuit.library.TGate"), and [`XGate`](qiskit.circuit.library.XGate "qiskit.circuit.library.XGate").
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>
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> **Representation**
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>
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> An $N$-qubit CNOT-Dihedral operator is stored as an affine function and a phase polynomial, based on the convention in references \[1, 2].
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>
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> The affine function consists of an $N \times N$ invertible binary matrix, and an $N$ binary vector.
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>
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> 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.
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>
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> ```python
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> from qiskit import QuantumCircuit
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> from qiskit.quantum_info import CNOTDihedral
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>
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> circ = QuantumCircuit(3)
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> circ.cx(0, 1)
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> circ.x(2)
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> circ.t(1)
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> circ.t(1)
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> circ.t(1)
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> elem = CNOTDihedral(circ)
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>
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> # Print the CNOTDihedral element
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> print(elem)
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> ```
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```python
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phase polynomial =
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0 + 3*x_0 + 3*x_1 + 2*x_0*x_1
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affine function =
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(x_0,x_0 + x_1,x_2 + 1)
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```
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**Circuit Conversion**
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> CNOTDihedral operators can be initialized from circuits containing *only* the following gates: [`IGate`](qiskit.circuit.library.IGate "qiskit.circuit.library.IGate"), [`XGate`](qiskit.circuit.library.XGate "qiskit.circuit.library.XGate"), [`YGate`](qiskit.circuit.library.YGate "qiskit.circuit.library.YGate"), [`ZGate`](qiskit.circuit.library.ZGate "qiskit.circuit.library.ZGate"), [`TGate`](qiskit.circuit.library.TGate "qiskit.circuit.library.TGate"), [`TdgGate`](qiskit.circuit.library.TdgGate "qiskit.circuit.library.TdgGate") [`SGate`](qiskit.circuit.library.SGate "qiskit.circuit.library.SGate"), [`SdgGate`](qiskit.circuit.library.SdgGate "qiskit.circuit.library.SdgGate"), [`CXGate`](qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate"), [`CZGate`](qiskit.circuit.library.CZGate "qiskit.circuit.library.CZGate"), [`CSGate`](qiskit.circuit.library.CSGate "qiskit.circuit.library.CSGate"), [`CSdgGate`](qiskit.circuit.library.CSdgGate "qiskit.circuit.library.CSdgGate"), [`SwapGate`](qiskit.circuit.library.SwapGate "qiskit.circuit.library.SwapGate"), [`CCZGate`](qiskit.circuit.library.CCZGate "qiskit.circuit.library.CCZGate"). They can be converted back into a [`QuantumCircuit`](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit"), or [`Gate`](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.
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>
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> CNOTDihedral operators can also be converted to [`Operator`](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.
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>
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> **References:**
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>
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> 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/)
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> 2. Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta, *Scalable randomized benchmarking of non-Clifford gates*, npj Quantum Inf 2, 16012 (2016).
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Initialize a CNOTDihedral operator object.
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**Parameters**
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* **data** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") *or*[*QuantumCircuit*](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit") *or*[*Instruction*](qiskit.circuit.Instruction "qiskit.circuit.Instruction")) – Optional, operator to initialize.
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* **num\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")) – Optional, initialize an empty CNOTDihedral operator.
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* **validate** ([*bool*](https://docs.python.org/3/library/functions.html#bool "(in Python v3.12)")) – if True, validates the CNOTDihedral element.
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**Raises**
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* [**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if the type is invalid.
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* [**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if validate=True and the CNOTDihedral element is invalid.
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## Attributes
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### dim
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<Attribute id="qiskit.quantum_info.CNOTDihedral.dim">
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Return tuple (input\_shape, output\_shape).
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</Attribute>
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### name
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<Attribute id="qiskit.quantum_info.CNOTDihedral.name">
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Unique string identifier for operation type.
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</Attribute>
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### num\_clbits
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<Attribute id="qiskit.quantum_info.CNOTDihedral.num_clbits">
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Number of classical bits.
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</Attribute>
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### num\_qubits
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<Attribute id="qiskit.quantum_info.CNOTDihedral.num_qubits">
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Return the number of qubits if a N-qubit operator or None otherwise.
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</Attribute>
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### qargs
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<Attribute id="qiskit.quantum_info.CNOTDihedral.qargs">
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Return the qargs for the operator.
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</Attribute>
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## Methods
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### adjoint
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<Function id="qiskit.quantum_info.CNOTDihedral.adjoint" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L442-L445" signature="adjoint()">
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Return the adjoint of the Operator.
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</Function>
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### compose
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<Function id="qiskit.quantum_info.CNOTDihedral.compose" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L374-L386" signature="compose(other, qargs=None, front=False)">
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Return the operator composition with another CNOTDihedral.
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**Parameters**
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* **other** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral object.
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* **qargs** ([*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)") *or None*) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).
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* **front** ([*bool*](https://docs.python.org/3/library/functions.html#bool "(in Python v3.12)")) – If True compose using right operator multiplication, instead of left multiplication \[default: False].
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**Returns**
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The composed CNOTDihedral.
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**Return type**
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[CNOTDihedral](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")
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**Raises**
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[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.
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<Admonition title="Note" type="note">
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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.
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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)`.
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</Admonition>
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</Function>
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### conjugate
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<Function id="qiskit.quantum_info.CNOTDihedral.conjugate" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L447-L471" signature="conjugate()">
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Return the conjugate of the CNOTDihedral.
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</Function>
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### copy
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<Function id="qiskit.quantum_info.CNOTDihedral.copy" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/base_operator.py#L143-L145" signature="copy()">
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Make a deep copy of current operator.
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</Function>
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### dot
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<Function id="qiskit.quantum_info.CNOTDihedral.dot" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/mixins/group.py#L133-L149" signature="dot(other, qargs=None)">
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Return the right multiplied operator self \* other.
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**Parameters**
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* **other** ([*Operator*](qiskit.quantum_info.Operator "qiskit.quantum_info.Operator")) – an operator object.
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* **qargs** ([*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)") *or None*) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).
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**Returns**
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The right matrix multiplied Operator.
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**Return type**
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[Operator](qiskit.quantum_info.Operator "qiskit.quantum_info.Operator")
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<Admonition title="Note" type="note">
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The dot product can be obtained using the `@` binary operator. Hence `a.dot(b)` is equivalent to `a @ b`.
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</Admonition>
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</Function>
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### expand
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<Function id="qiskit.quantum_info.CNOTDihedral.expand" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L439-L440" signature="expand(other)">
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Return the reverse-order tensor product with another CNOTDihedral.
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**Parameters**
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**other** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral object.
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**Returns**
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**the tensor product $b \otimes a$, where $a$**
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is the current CNOTDihedral, and $b$ is the other CNOTDihedral.
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**Return type**
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[CNOTDihedral](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")
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</Function>
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### input\_dims
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<Function id="qiskit.quantum_info.CNOTDihedral.input_dims" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/base_operator.py#L135-L137" signature="input_dims(qargs=None)">
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Return tuple of input dimension for specified subsystems.
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</Function>
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### output\_dims
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<Function id="qiskit.quantum_info.CNOTDihedral.output_dims" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/base_operator.py#L139-L141" signature="output_dims(qargs=None)">
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Return tuple of output dimension for specified subsystems.
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</Function>
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### power
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<Function id="qiskit.quantum_info.CNOTDihedral.power" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/mixins/group.py#L151-L171" signature="power(n)">
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Return the compose of a operator with itself n times.
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**Parameters**
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**n** ([*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")) – the number of times to compose with self (n>0).
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**Returns**
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the n-times composed operator.
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**Return type**
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[Clifford](qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")
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**Raises**
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[**QiskitError**](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.
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</Function>
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### reshape
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<Function id="qiskit.quantum_info.CNOTDihedral.reshape" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/base_operator.py#L106-L133" signature="reshape(input_dims=None, output_dims=None, num_qubits=None)">
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Return a shallow copy with reshaped input and output subsystem dimensions.
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**Parameters**
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* **input\_dims** (*None or* [*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple "(in Python v3.12)")) – new subsystem input dimensions. If None the original input dims will be preserved \[Default: None].
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* **output\_dims** (*None or* [*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple "(in Python v3.12)")) – new subsystem output dimensions. If None the original output dims will be preserved \[Default: None].
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* **num\_qubits** (*None or* [*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")) – reshape to an N-qubit operator \[Default: None].
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**Returns**
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returns self with reshaped input and output dimensions.
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**Return type**
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BaseOperator
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**Raises**
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[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.
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</Function>
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### tensor
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<Function id="qiskit.quantum_info.CNOTDihedral.tensor" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L436-L437" signature="tensor(other)">
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Return the tensor product with another CNOTDihedral.
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**Parameters**
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**other** ([*CNOTDihedral*](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral object.
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**Returns**
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**the tensor product $a \otimes b$, where $a$**
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is the current CNOTDihedral, and $b$ is the other CNOTDihedral.
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**Return type**
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[CNOTDihedral](#qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")
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<Admonition title="Note" type="note">
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The tensor product can be obtained using the `^` binary operator. Hence `a.tensor(b)` is equivalent to `a ^ b`.
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</Admonition>
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</Function>
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### to\_circuit
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<Function id="qiskit.quantum_info.CNOTDihedral.to_circuit" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L317-L334" signature="to_circuit()">
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Return a QuantumCircuit implementing the CNOT-Dihedral element.
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**Returns**
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a circuit implementation of the CNOTDihedral object.
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**Return type**
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[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
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**References**
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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/)
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2. Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta, *Scalable randomized benchmarking of non-Clifford gates*, npj Quantum Inf 2, 16012 (2016).
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</Function>
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### to\_instruction
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<Function id="qiskit.quantum_info.CNOTDihedral.to_instruction" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L336-L338" signature="to_instruction()">
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Return a Gate instruction implementing the CNOTDihedral object.
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</Function>
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### to\_matrix
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<Function id="qiskit.quantum_info.CNOTDihedral.to_matrix" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L366-L368" signature="to_matrix()">
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Convert operator to Numpy matrix.
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</Function>
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### to\_operator
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<Function id="qiskit.quantum_info.CNOTDihedral.to_operator" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L370-L372" signature="to_operator()">
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Convert to an Operator object.
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**Return type**
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[*Operator*](qiskit.quantum_info.Operator "qiskit.quantum_info.operators.operator.Operator")
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</Function>
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### transpose
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<Function id="qiskit.quantum_info.CNOTDihedral.transpose" github="https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/quantum_info/operators/dihedral/dihedral.py#L473-L476" signature="transpose()">
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Return the transpose of the CNOTDihedral.
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</Function>
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</Class>
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