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qiskit.circuit.library.CUGate(theta, phi, lam, gamma, label=None, ctrl_state=None, *, duration=None, unit='dt', _base_label=None) GitHub(opens in a new tab)

Bases: ControlledGate

Controlled-U gate (4-parameter two-qubit gate).

This is a controlled version of the U gate (generic single qubit rotation), including a possible global phase eiγe^{i\gamma} of the U gate.

Can be applied to a QuantumCircuit with the cu() method.

Circuit symbol:

q_0: ──────■──────

Matrix representation:

CU(θ,ϕ,λ,γ) q0,q1=I00+eiγU(θ,ϕ,λ)11=(10000eiγcos(θ2)0ei(γ+λ)sin(θ2)00100ei(γ+ϕ)sin(θ2)0ei(γ+ϕ+λ)cos(θ2))\providecommand{\rotationangle}{\frac{\theta}{2}} CU(\theta, \phi, \lambda, \gamma)\ q_0, q_1 = I \otimes |0\rangle\langle 0| + e^{i\gamma} U(\theta,\phi,\lambda) \otimes |1\rangle\langle 1| = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & e^{i\gamma}\cos(\rotationangle) & 0 & -e^{i(\gamma + \lambda)}\sin(\rotationangle) \\ 0 & 0 & 1 & 0 \\ 0 & e^{i(\gamma+\phi)}\sin(\rotationangle) & 0 & e^{i(\gamma+\phi+\lambda)}\cos(\rotationangle) \end{pmatrix}

In Qiskit’s convention, higher qubit indices are more significant (little endian convention). In many textbooks, controlled gates are presented with the assumption of more significant qubits as control, which in our case would be q_1. Thus a textbook matrix for this gate will be:

q_1: ──────■───────
CU(θ,ϕ,λ,γ) q1,q0=00I+eiγ11U(θ,ϕ,λ)=(1000010000eiγcos(θ2)ei(γ+λ)sin(θ2)00ei(γ+ϕ)sin(θ2)ei(γ+ϕ+λ)cos(θ2))\providecommand{\rotationangle}{\frac{\theta}{2}} CU(\theta, \phi, \lambda, \gamma)\ q_1, q_0 = |0\rangle\langle 0| \otimes I + e^{i\gamma}|1\rangle\langle 1| \otimes U(\theta,\phi,\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{i\gamma} \cos(\rotationangle) & -e^{i(\gamma + \lambda)}\sin(\rotationangle) \\ 0 & 0 & e^{i(\gamma + \phi)}\sin(\rotationangle) & e^{i(\gamma + \phi+\lambda)}\cos(\rotationangle) \end{pmatrix}

Create new CU gate.



Get the base class of this instruction. This is guaranteed to be in the inheritance tree of self.

The “base class” of an instruction is the lowest class in its inheritance tree that the object should be considered entirely compatible with for _all_ circuit applications. This typically means that the subclass is defined purely to offer some sort of programmer convenience over the base class, and the base class is the “true” class for a behavioural perspective. In particular, you should not override base_class if you are defining a custom version of an instruction that will be implemented differently by hardware, such as an alternative measurement strategy, or a version of a parametrised gate with a particular set of parameters for the purposes of distinguishing it in a Target from the full parametrised gate.

This is often exactly equivalent to type(obj), except in the case of singleton instances of standard-library instructions. These singleton instances are special subclasses of their base class, and this property will return that base. For example:

>>> isinstance(XGate(), XGate)
>>> type(XGate()) is XGate
>>> XGate().base_class is XGate

In general, you should not rely on the precise class of an instruction; within a given circuit, it is expected that should be a more suitable discriminator in most situations.


The classical condition on the instruction.


Get Clbits in condition.


Return the control state of the gate as a decimal integer.


Get the decompositions of the instruction from the SessionEquivalenceLibrary.


Return definition in terms of other basic gates. If the gate has open controls, as determined from self.ctrl_state, the returned definition is conjugated with X without changing the internal _definition.


Get the duration.


Return instruction label


Is this instance is a mutable unique instance or not.

If this attribute is False the gate instance is a shared singleton and is not mutable.


Get name of gate. If the gate has open controls the gate name will become:


where <original_name> is the gate name for the default case of closed control qubits and <ctrl_state> is the integer value of the control state for the gate.


Return the number of clbits.


Get number of control qubits.


The number of control qubits for the gate.

Return type

int(opens in a new tab)


Return the number of qubits.



Get the time unit of duration.




Return inverted CU gate.

CU(θ,ϕ,λ,γ)=CU(θ,ϕ,λ,γ))CU(\theta,\phi,\lambda,\gamma)^{\dagger} = CU(-\theta,-\phi,-\lambda,-\gamma))


annotated (bool(opens in a new tab)) – when set to True, this is typically used to return an AnnotatedOperation with an inverse modifier set instead of a concrete Gate. However, for this class this argument is ignored as the inverse of this gate is always a CUGate with inverse parameter values.


inverse gate.

Return type


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