# CU3Gate¶

class CU3Gate(theta, phi, lam, label=None, ctrl_state=None)[source]

Controlled-U3 gate (3-parameter two-qubit gate).

This is a controlled version of the U3 gate (generic single qubit rotation). It is restricted to 3 parameters, and so cannot cover generic two-qubit controlled gates).

Circuit symbol:

q_0: ──────■──────
┌─────┴─────┐
q_1: ┤ U3(ϴ,φ,λ) ├
└───────────┘


Matrix representation:

\begin{align}\begin{aligned}\newcommand{\th}{\frac{\theta}{2}}\\\begin{split}CU3(\theta, \phi, \lambda)\ q_0, q_1 = I \otimes |0\rangle\langle 0| + U3(\theta,\phi,\lambda) \otimes |1\rangle\langle 1| = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\th) & 0 & e^{-i\lambda}\sin(\th) \\ 0 & 0 & 1 & 0 \\ 0 & e^{i\phi}\sin(\th) & 0 & e^{i(\phi+\lambda)\cos(\th)} \end{pmatrix}\end{split}\end{aligned}\end{align}

Note

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: ┤ U3(ϴ,φ,λ) ├
└─────┬─────┘
q_1: ──────■──────

$\begin{split}CU3(\theta, \phi, \lambda)\ q_1, q_0 = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes U3(\theta,\phi,\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\th) & e^{-i\lambda}\sin(\th) \\ 0 & 0 & e^{i\phi}\sin(\th) & e^{i(\phi+\lambda)\cos(\th)} \end{pmatrix}\end{split}$

Create new CU3 gate.

Attributes

 CU3Gate.ctrl_state Return the control state of the gate as a decimal integer. CU3Gate.decompositions Get the decompositions of the instruction from the SessionEquivalenceLibrary. CU3Gate.definition Return definition in terms of other basic gates. CU3Gate.label Return gate label CU3Gate.params return instruction params.

Methods

 CU3Gate.add_decomposition(decomposition) Add a decomposition of the instruction to the SessionEquivalenceLibrary. Assemble a QasmQobjInstruction CU3Gate.broadcast_arguments(qargs, cargs) Validation and handling of the arguments and its relationship. CU3Gate.c_if(classical, val) Add classical condition on register classical and value val. CU3Gate.control([num_ctrl_qubits, label, …]) Return controlled version of gate. CU3Gate.copy([name]) Copy of the instruction. Return inverted CU3 gate. Return True .IFF. For a composite instruction, reverse the order of sub-gates. CU3Gate.power(exponent) Creates a unitary gate as gate^exponent. Return a default OpenQASM string for the instruction. Creates an instruction with gate repeated n amount of times. Return a Numpy.array for the gate unitary matrix.