# tGate

T gate

Since R2023a

Installation Required: This functionality requires MATLAB Support Package for Quantum Computing.

## Syntax

``g = tGate(targetQubit)``

## Description

example

````g = tGate(targetQubit)` applies a T gate to a single target qubit and returns a `quantum.gate.SimpleGate` object.If `targetQubit` is a vector of qubit indices, `tGate` returns a column vector of gates, where `g(i)` represents a T gate applied to a qubit with index `targetQubit(i)`.Applying this gate is equivalent to applying the R1 gate with a rotation angle of π/4, meaning that `tGate(targetQubit)` is equivalent to `r1Gate(targetQubit,pi/4)`.```

## Examples

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Create a T gate that acts on a single qubit.

`g = tGate(1)`
```g = SimpleGate with properties: Type: "t" ControlQubits: [1×0 double] TargetQubits: 1 Angles: [1×0 double]```

Get the matrix representation of the gate.

`M = getMatrix(g)`
```M = 1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.7071 + 0.7071i```

Create an array of T gates that act on qubits with indices 1 to 4.

`g = tGate(1:4)`
```g = 4×1 SimpleGate array with gates: Id Gate Control Target 1 t 1 2 t 2 3 t 3 4 t 4 ```

## Input Arguments

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Target qubit of the gate, specified as a positive integer scalar index or vector of qubit indices.

Example: `1`

Example: `3:5`

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### Matrix Representation of T Gate

The matrix representation of a T gate applied to a single qubit is

`$\left[\begin{array}{cc}1& 0\\ 0& \mathrm{exp}\left(i\text{\hspace{0.17em}}\frac{\pi }{4}\right)\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& \frac{1+i}{\sqrt{2}}\end{array}\right].$`

Applying this gate is equivalent to applying an R1 gate with a rotation angle of π/4. This gate is also known as the fourth root of Pauli Z gate because applying the T gate four times is equivalent to applying the Pauli Z gate.

## Version History

Introduced in R2023a