Can I programmatically move the steps of a mat-horizontal-stepper in Angular / Angular Material

Yes. It is possible to jump to a specific stepper by using selectedIndex property of the MatStepper. Also, MatStepper exposes public methods next() and previous(). You can use them to move back and forth.

In your template:

Add an id to your stepper e.g. #stepper. Then in your goBack() and goForward() methods, pass the stepper id:

<mat-horizontal-stepper #stepper>
    <!-- Steps -->
</mat-horizontal-stepper>    
<!-- second option -->
<div>
   <button (click)="goBack(stepper)" type="button">Back</button>
   <button (click)="goForward(stepper)" type="button">Next</button>
</div>

.. and in your typescript:

import { MatStepper } from '@angular/material/stepper';

goBack(stepper: MatStepper){
    stepper.previous();
}

goForward(stepper: MatStepper){
    stepper.next();
}

Link to stackblitz demo.

You can also use ViewChild to get a reference to the stepper component in your TypeScript as shown below:

@ViewChild('stepper') private myStepper: MatStepper;

goBack(){
    this.myStepper.previous();
}

goForward(){
    this.myStepper.next();
}

In this case, you don't have to pass the stepper reference in the method in your component's html. Link to Demo with ViewChild

You can enable/disable the Back and Next buttons by using the following:

<button (click)="goBack(stepper)" type="button" 
        [disabled]="stepper.selectedIndex === 0">Back</button>
<button (click)="goForward(stepper)" type="button" 
        [disabled]="stepper.selectedIndex === stepper._steps.length-1">Next</button>

In addition to @Faisal's answer, this is my take on making the MatStepper jump without needing to pass the stepper in the arguments.

This is helpful when you need more flexibility in manipulating the stepper e.g. from a Service or something else.

HTML:

<div fxLayout="row" fxLayoutAlign="center center" fxLayoutGap="6px">
  <button (click)="move(0)">1st</button>
  <button (click)="move(1)">2nd</button>
  <button (click)="move(2)">3rd</button>
  <button (click)="move(3)">4th</button>
</div>

TS File:

move(index: number) {
    this.stepper.selectedIndex = index;
}

Here's the stackblitz demo.