---
title: OneQubitEulerDecomposer (v2.2)
description: API reference for qiskit.synthesis.OneQubitEulerDecomposer in qiskit v2.2
source: https://quantum.cloud.ibm.com/docs/en/api/qiskit/2.2/qiskit.synthesis.OneQubitEulerDecomposer
---

# OneQubitEulerDecomposer

*class* `qiskit.synthesis.OneQubitEulerDecomposer(basis='U3', use_dag=False)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.2/qiskit/synthesis/one_qubit/one_qubit_decompose.py#L76-L288)

Bases: [`object`](https://docs.python.org/3/library/functions.html#object)

A class for decomposing 1-qubit unitaries into Euler angle rotations.

The resulting decomposition is parameterized by 3 Euler rotation angle parameters $(\theta, \phi, \lambda)$, and a phase parameter $\gamma$. The value of the parameters for an input unitary depends on the decomposition basis. Allowed bases and the resulting circuits are shown in the following table. Note that for the non-Euler bases ($U3$, $U1X$, $RR$), the $ZYZ$ Euler parameters are used.

| Basis  | Euler Angle Basis              | Decomposition Circuit                                                                                                    |
| ------ | ------------------------------ | ------------------------------------------------------------------------------------------------------------------------ |
| ‘ZYZ’  | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} R_Z(\phi).R_Y(\theta).R_Z(\lambda)$                                                                         |
| ‘ZXZ’  | $Z(\phi) X(\theta) Z(\lambda)$ | $e^{i\gamma} R_Z(\phi).R_X(\theta).R_Z(\lambda)$                                                                         |
| ‘XYX’  | $X(\phi) Y(\theta) X(\lambda)$ | $e^{i\gamma} R_X(\phi).R_Y(\theta).R_X(\lambda)$                                                                         |
| ‘XZX’  | $X(\phi) Z(\theta) X(\lambda)$ | $e^{i\gamma} R_X(\phi).R_Z(\theta).R_X(\lambda)$                                                                         |
| ‘U3’   | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} U_3(\theta,\phi,\lambda)$                                                                                   |
| ‘U321’ | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} U_3(\theta,\phi,\lambda)$                                                                                   |
| ‘U’    | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} U_3(\theta,\phi,\lambda)$                                                                                   |
| ‘PSX’  | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} U_1(\phi+\pi).R_X\left(\frac{\pi}{2}\right).$ $U_1(\theta+\pi).R_X\left(\frac{\pi}{2}\right).U_1(\lambda)$  |
| ‘ZSX’  | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} R_Z(\phi+\pi).\sqrt{X}.$ $R_Z(\theta+\pi).\sqrt{X}.R_Z(\lambda)$                                            |
| ‘ZSXX’ | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} R_Z(\phi+\pi).\sqrt{X}.R_Z(\theta+\pi).\sqrt{X}.R_Z(\lambda)$ or $e^{i\gamma} R_Z(\phi+\pi).X.R_Z(\lambda)$ |
| ‘U1X’  | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} U_1(\phi+\pi).R_X\left(\frac{\pi}{2}\right).$ $U_1(\theta+\pi).R_X\left(\frac{\pi}{2}\right).U_1(\lambda)$  |
| ‘RR’   | $Z(\phi) Y(\theta) Z(\lambda)$ | $e^{i\gamma} R\left(-\pi,\frac{\phi-\lambda+\pi}{2}\right).$ $R\left(\theta+\pi,\frac{\pi}{2}-\lambda\right)$            |

### \_\_call\_\_

`__call__(unitary, simplify=True, atol=1e-12)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.2/qiskit/synthesis/one_qubit/one_qubit_decompose.py#L178-L215)

Decompose single qubit gate into a circuit.

**Parameters**

- **unitary** ([*Operator*](/docs/api/qiskit/2.2/qiskit.quantum_info.Operator "qiskit.quantum_info.Operator")  *|*[*Gate*](/docs/api/qiskit/2.2/qiskit.circuit.Gate "qiskit.circuit.Gate") *| np.ndarray*) – 1-qubit unitary matrix
- **simplify** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – reduce gate count in decomposition \[Default: True].
- **atol** ([*float*](https://docs.python.org/3/library/functions.html#float)) – absolute tolerance for checking angles when simplifying returned circuit \[Default: 1e-12].

**Returns**

the decomposed single-qubit gate circuit

**Return type**

[QuantumCircuit](/docs/api/qiskit/2.2/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")

**Raises**

[**QiskitError**](/docs/api/qiskit/2.2/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if input is invalid or synthesis fails.

Initialize decomposer

Supported bases are: `'U'`, `'PSX'`, `'ZSXX'`, `'ZSX'`, `'U321'`, `'U3'`, `'U1X'`, `'RR'`, `'ZYZ'`, `'ZXZ'`, `'XYX'`, `'XZX'`.

**Parameters**

- **basis** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – the decomposition basis \[Default: `'U3'`]
- **use\_dag** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – If true the output from calls to the decomposer will be a [`DAGCircuit`](/docs/api/qiskit/2.2/qiskit.dagcircuit.DAGCircuit "qiskit.dagcircuit.DAGCircuit") object instead of [`QuantumCircuit`](/docs/api/qiskit/2.2/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit").

**Raises**

[**QiskitError**](/docs/api/qiskit/2.2/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – If input basis is not recognized.

## Attributes

### basis

The decomposition basis.

## Methods

### angles

`angles(unitary)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.2/qiskit/synthesis/one_qubit/one_qubit_decompose.py#L258-L269)

Return the Euler angles for input array.

**Parameters**

**unitary** ([*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray)) – $2\times2$ unitary matrix.

**Returns**

`(theta, phi, lambda)`.

**Return type**

[tuple](https://docs.python.org/3/library/stdtypes.html#tuple)

### angles\_and\_phase

`angles_and_phase(unitary)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.2/qiskit/synthesis/one_qubit/one_qubit_decompose.py#L271-L281)

Return the Euler angles and phase for input array.

**Parameters**

**unitary** ([*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray)) – $2\times2$

**Returns**

`(theta, phi, lambda, phase)`.

**Return type**

[tuple](https://docs.python.org/3/library/stdtypes.html#tuple)

### build\_circuit

`build_circuit(gates, global_phase)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.2/qiskit/synthesis/one_qubit/one_qubit_decompose.py#L157-L176)

Return the circuit or dag object from a list of gates.

**Return type**

[QuantumCircuit](/docs/api/qiskit/2.2/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit") | [DAGCircuit](/docs/api/qiskit/2.2/qiskit.dagcircuit.DAGCircuit "qiskit.dagcircuit.DAGCircuit")