# CUGate¶

class CUGate(theta, phi, lam, gamma, label=None, ctrl_state=None)[source]

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 $$e^{i\gamma}$$ of the U gate.

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

Circuit symbol:

q_0: ──────■──────
┌─────┴──────┐
q_1: ┤ U(ϴ,φ,λ,γ) ├
└────────────┘


Matrix representation:

\begin{align}\begin{aligned}\newcommand{\th}{\frac{\theta}{2}}\\\begin{split}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(\th) & 0 & -e^{i(\gamma + \lambda)}\sin(\th) \\ 0 & 0 & 1 & 0 \\ 0 & e^{i(\gamma+\phi)}\sin(\th) & 0 & e^{i(\gamma+\phi+\lambda)}\cos(\th) \end{pmatrix}\end{split}\end{aligned}\end{align}

নোট

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_0: ┤ U(ϴ,φ,λ,γ) ├
└─────┬──────┘
q_1: ──────■───────

$\begin{split}CU(\theta, \phi, \lambda, \gamma)\ q_1, q_0 = |0\rangle\langle 0| \otimes I + e^{i\gamma}|1\rangle\langle 1| \otimes U(\theta,\phi,\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{i\gamma} \cos(\th) & -e^{i(\gamma + \lambda)}\sin(\th) \\ 0 & 0 & e^{i(\gamma + \phi)}\sin(\th) & e^{i(\gamma + \phi+\lambda)}\cos(\th) \end{pmatrix}\end{split}$

Create new CU gate.

Methods Defined Here

 inverse Return inverted CU gate.

Attributes

condition_bits

Get Clbits in condition.

রিটার্ন টাইপ

List[Clbit]

ctrl_state

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

রিটার্ন টাইপ

int

decompositions

Get the decompositions of the instruction from the SessionEquivalenceLibrary.

definition

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.

রিটার্ন টাইপ

List

duration

Get the duration.

label

Return instruction label

রিটার্ন টাইপ

str

name

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

<original_name_o<ctrl_state>

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.

রিটার্ন টাইপ

str

num_clbits

Return the number of clbits.

num_ctrl_qubits

Get number of control qubits.

রিটার্নস

The number of control qubits for the gate.

রিটার্ন টাইপ

int

num_qubits

Return the number of qubits.

params

Get parameters from base_gate.

রিটার্নস

List of gate parameters.

রিটার্ন টাইপ

list

রেইজেস

CircuitError -- Controlled gate does not define a base gate

unit

Get the time unit of duration.