To connect a PS4 controller to the Xbox One, you'll need to use the CronusMAX PLUS adapter, which is one of the more reliable ways to pair the controller and console.

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Set up the CronusMAX PLUS
Plug the CronusMAX PLUS into your internet-connected PC and it will automatically install the drivers.
After the CronusMAX PLUS is loaded onto the PC, load the Cronus PRO application and select Tools > Options > Device.
Set the output protocol to Xbox One, shown as XB1 in the drop-down list at the top of the menu.
Make sure the Enable Dualshock 4/Wiimote Bluetooth searching box is checked.
Remove the CronusMAX PLUS from your PC and plug it into one of your Xbox One via the USB port. Once you've done so, the CronusMAX PLUS will begin to show "AU" on its LED display for "Authentication."
Remove the batteries from your Xbox One controller. If any Xbox One controllers are connected to the console, their signal can overwrite the PS4 controller.
Connect your Xbox One controller to the CronusMAX PLUS using a USB cable. The display on the adapter should switch from "AU" to "0."
Once the number appears on the CronusMAX PLUS, disconnect the Xbox One controller and look for the standby animation. This will look like two halves of a circle spinning around the display.

Connect a Bluetooth 4.0 USB adapter to the CronusMAX PLUS.
Hold down the SHARE and PS buttons on your PS4 controller.
The controller will begin to flash white, indicating it's searching for a connection; it should find the CronusMAX PLUS. After a few seconds of this, the light bar should change to a solid color.
Look for the number on the CronusMAX PLUS to change to "0" again. Once this happens, the controller should be connected and ready to use.
How to Disable Bluetooth Searching
If you intend to use your PS4 controller with your Xbox One from now on and don't want to go through this process again in the future, you can disable Bluetooth searching so the device stays connected.

Disabling Bluetooth searching makes subsequent connections easier so you can start gaming faster. Here's how to disable it:

Plug the CronusMAX PLUS into your PC and access Cronus PRO.
Make sure the Enable Dualshock 4/Wiimote Bluetooth searching box is not checked.
Repeat steps 5-9 from the above section.
Press the PS button on the PS4 controller.
Follow steps 11 and 12. At this point, your PlayStation 4 controller should connect to the Xbox One.