---
title: TwoQubitControlledUDecomposer (latest version)
description: API reference for qiskit.synthesis.TwoQubitControlledUDecomposer in the latest version of qiskit
source: https://quantum.cloud.ibm.com/docs/en/api/qiskit/qiskit.synthesis.TwoQubitControlledUDecomposer
---

# TwoQubitControlledUDecomposer

*class* `qiskit.synthesis.TwoQubitControlledUDecomposer(rxx_equivalent_gate, euler_basis='ZXZ')`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.5/qiskit/synthesis/two_qubit/two_qubit_decompose.py#L267-L399)

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

Decompose a general two-qubit unitary in terms of a target two-qubit gate, that is locally equivalent to an [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate").

**Synthesis algorithm**

Any two-qubit unitary $U$ can be written, through its canonical (Weyl) decomposition (see [`TwoQubitWeylDecomposition`](/docs/api/qiskit/qiskit.synthesis.TwoQubitWeylDecomposition "qiskit.synthesis.TwoQubitWeylDecomposition")), as a Weyl gate $U_d(a, b, c)$ surrounded by four single-qubit unitary gates:

```text
     ┌─────┐┌───────┐┌─────┐
q_0: ┤ c2r ├┤0      ├┤ c1r ├
     ├─────┤│  Weyl │├─────┤
q_1: ┤ c2l ├┤1      ├┤ c1l ├
     └─────┘└───────┘└─────┘
```

The Weyl gate factorizes into a product of three two-qubit rotations, $U_d(a, b, c) = R_{XX}(a)\, R_{YY}(b)\, R_{ZZ}(c)$:

```text
     ┌─────────┐┌─────────┐
q_0: ┤0        ├┤0        ├─■──────
     │  Rxx(a) ││  Ryy(b) │ │ZZ(c)
q_1: ┤1        ├┤1        ├─■──────
     └─────────┘└─────────┘
```

The $R_{YY}$ and $R_{ZZ}$ rotations are then mapped onto $R_{XX}$ rotations using single-qubit basis changes. With $R_{YY}(b) = (S^\dagger \otimes S^\dagger)\, R_{XX}(b)\, (S \otimes S)$:

```text
     ┌─────┐┌─────────┐┌───┐
q_0: ┤ Sdg ├┤0        ├┤ S ├
     ├─────┤│  Rxx(b) │├───┤
q_1: ┤ Sdg ├┤1        ├┤ S ├
     └─────┘└─────────┘└───┘
```

and $R_{ZZ}(c) = (H \otimes H)\, R_{XX}(c)\, (H \otimes H)$:

```text
     ┌───┐┌─────────┐┌───┐
q_0: ┤ H ├┤0        ├┤ H ├
     ├───┤│  Rxx(c) │├───┤
q_1: ┤ H ├┤1        ├┤ H ├
     └───┘└─────────┘└───┘
```

Finally, each $R_{XX}$ rotation is realized with the user-supplied gate that is locally equivalent to [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate") (the `rxx_equivalent_gate`), wrapped by the single-qubit gates that account for the local equivalence and for any scaling of the rotation angle. After every rotation is expanded, all single-qubit gates that fall between two consecutive two-qubit gates are multiplied together and consolidated, so the synthesized circuit uses at most three applications of `rxx_equivalent_gate` and at most eight single-qubit unitary gates:

```text
     ┌─────┐┌───────────┐┌─────┐┌───────────┐┌─────┐┌───────────┐┌─────┐
q_0: ┤ d2r ├┤0          ├┤ d1r ├┤0          ├┤ e1r ├┤0          ├┤ f1r ├
     ├─────┤│  Equiv(a) │├─────┤│  Equiv(b) │├─────┤│  Equiv(c) │├─────┤
q_1: ┤ d2l ├┤1          ├┤ d1l ├┤1          ├┤ e1l ├┤1          ├┤ f1l ├
     └─────┘└───────────┘└─────┘└───────────┘└─────┘└───────────┘└─────┘
```

Here `Equiv(a)`, `Equiv(b)` and `Equiv(c)` are the user-supplied `rxx_equivalent_gate` (the gate locally equivalent to [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate")) realizing the $R_{XX}(a)$, $R_{XX}(b)$ and $R_{XX}(c)$ rotations, and the remaining boxes are the consolidated single-qubit unitary gates.

The number of two-qubit gates actually emitted depends on the Weyl parameters of the target: rotations with a vanishing angle are dropped, so unitaries that are closer to a single or two instances of [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate") use one or two applications of `rxx_equivalent_gate` respectively instead of three. A target close to the identity will use no applications of it.

**Parameters**

- **rxx\_equivalent\_gate** ([*type*](/docs/api/qiskit/qiskit.dagcircuit.DAGDepNode#type "qiskit.dagcircuit.DAGDepNode.type")*\[*[*Gate*](/docs/api/qiskit/qiskit.circuit.Gate "qiskit.circuit.Gate")*]*) – Gate that is locally equivalent to an [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate"): $U \sim U_d(\alpha, 0, 0) \sim \text{Ctrl-U}$ gate. Valid options are \[[`RZZGate`](/docs/api/qiskit/qiskit.circuit.library.RZZGate "qiskit.circuit.library.RZZGate"), [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate"), [`RYYGate`](/docs/api/qiskit/qiskit.circuit.library.RYYGate "qiskit.circuit.library.RYYGate"), [`RZXGate`](/docs/api/qiskit/qiskit.circuit.library.RZXGate "qiskit.circuit.library.RZXGate"), [`CPhaseGate`](/docs/api/qiskit/qiskit.circuit.library.CPhaseGate "qiskit.circuit.library.CPhaseGate"), [`CRXGate`](/docs/api/qiskit/qiskit.circuit.library.CRXGate "qiskit.circuit.library.CRXGate"), [`CRYGate`](/docs/api/qiskit/qiskit.circuit.library.CRYGate "qiskit.circuit.library.CRYGate"), [`CRZGate`](/docs/api/qiskit/qiskit.circuit.library.CRZGate "qiskit.circuit.library.CRZGate")].
- **euler\_basis** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – Basis string to be provided to [`OneQubitEulerDecomposer`](/docs/api/qiskit/qiskit.synthesis.OneQubitEulerDecomposer "qiskit.synthesis.OneQubitEulerDecomposer") for 1Q synthesis. Valid options are \[`'ZXZ'`, `'ZYZ'`, `'XYX'`, `'XZX'`, `'U'`, `'U3'`, `'U321'`, `'U1X'`, `'PSX'`, `'ZSX'`, `'ZSXX'`, `'RR'`].

**Raises**

[**QiskitError**](/docs/api/qiskit/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – If the gate is not locally equivalent to an [`RXXGate`](/docs/api/qiskit/qiskit.circuit.library.RXXGate "qiskit.circuit.library.RXXGate").

### \_\_call\_\_

`__call__(unitary, approximate=False, use_dag=False, *, atol=1e-12)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.5/qiskit/synthesis/two_qubit/two_qubit_decompose.py#L379-L399)

Decompose a two-qubit `unitary` using the [`TwoQubitControlledUDecomposer`](#qiskit.synthesis.TwoQubitControlledUDecomposer "qiskit.synthesis.TwoQubitControlledUDecomposer").

**Parameters**

- **unitary** ([*Operator*](/docs/api/qiskit/qiskit.quantum_info.Operator "qiskit.quantum_info.operators.operator.Operator")  *|*[*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray)) – $4 \times 4$ unitary to synthesize.
- **approximate** – Currently not used by this decomposer; accepted for signature compatibility with the other two-qubit decomposers. Reserved for future use.
- **use\_dag** – Currently not used by this decomposer; accepted for signature compatibility with the other two-qubit decomposers. Reserved for future use.
- **atol** – Absolute tolerance for checking angles of the single-qubit unitaries when simplifying the returned circuit \[Default: 1e-12].

**Returns**

Synthesized quantum circuit.

**Return type**

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

Note: atol is passed to [`OneQubitEulerDecomposer`](/docs/api/qiskit/qiskit.synthesis.OneQubitEulerDecomposer "qiskit.synthesis.OneQubitEulerDecomposer").