U Gate

In Quantum Computing, U Gate is the most important and the most fundamental single Qubit Quantum gate. All single Qubit unitary Quantum gates can be considered as a special case of the U Gate. In this chapter of the OpenQASM Tutorial, you will be learning about how to apply U Gate on a Qubit.

In OpenQASM, U Gate is a built-in gate and therefore OpenQASM provides the syntax to apply U Gate on a Qubit. A U Gate has 3 parameters- θ, φ, and λ.

A U Gate with parameters θ, φ, and λ can be considered as equivalent to the following operations- RZ(φ) RY(θ) RZ(λ)

where RZ(x) is rotation about the Z-axis on bloch sphere by x radians and RY(x) is rotation about the Y-axis on the bloch sphere by x radians.

The U Gate with parameters θ, φ, and λ can be represented by the following matrix-

For example, H Gate is a special case of U Gate when the value of θ = π/2, φ = 0 , λ = π.

The value of parameters θ, φ, and λ can range between 0(included) and 4π(excluded).

U Gate

The U Gate can be applied on any Qubit.

The following syntax is used for applying the U Gate with parameters theta, phi, lambda on Qubit quantum_register[index]–

U(theta, phi, lambda) quantum_register[index]; Note: U is a built-in gate therefore u is a reserved keyword.

Example

Below is an example of applying the U Gate on first Qubit of Quantum Register qubits with parameters θ = π/2, φ = 0 , λ = π. Applying U Gate with these parameters is the equivalent of applying H Gate on the Qubit.

Diagram
OPENQASM 2.0;
include "qelib1.inc";

// Define quantum registers
qreg qubits[1];

// Applying H gate on the first Qubit of Quantum Register
u(pi/2, 0, pi) qubits[0];
Translation
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U Gate on Quantum Registers

It is also possible to apply U Gate with a particular set of parameters on all Qubits of a Quantum Register at once.

The following syntax is used for applying the U Gate with parameters theta, phi, lambda on all Qubits of a Quantum Register quantum_register at once.

u(theta, phi, lambda) quantum_register;

Example

Below is an example of applying the U Gate on all the Qubits of Quantum Register qubits with parameters θ = π/2, φ = 0 , λ = π. Effectively, we are applying H Gate on all Qubits of the Quantum Register qubits.

// Applying H gate on all Qubits of the Quantum Register
u(pi/2, 0, pi) qubits;