I can't understand how should be possible that a brain can integrate with logical gates.
But now, even newest working chips still operate with 0 and 1 and logical gates
So do neurons. You might think your wetware is analogue, but nerves don't actually work that way.
The magic phrase you need to know is action potential, possibly along with voltage-gated ion channel. Nerve signals are in fact trains of pulses, with reasonably well-defined on and off states. They're not nice square waves like digital logic would like to be, and nor do they use nice user-friendly communication protocols.
A wetware-hardware interface will mostly consist of a whole load of signal analysis and signal generation logic to interpret the messy organic stuff you work with and translate it into something machine-friendly and vice-versa. There's been a fair bit of work done on this sort of thing, especially with regards to artificial vision. Obviously we aren't in the realms of direct neural interfaces yet, but you've got to start somewhere.
Would the brain "feel" a qubit interface maybe?
I think trying to get a quantum-coherent communications channel from a computer into a brain is going to be harder than making an actual functional cyborg. There's some hint that brains do make use of interesting and (comparatively) long-lived quantum states internally, but they are unlikely to propagate far because human bodies are a hot mess and coherency is fragile. Stick with digital stuff. It'll work fine.
What it an implant would actually feel like would obviously depend on where you hooked it up. Remember that your brain doesn't have vacant expansion slots you can just plug new toys in to... every bit is used for something. If you're replacing something (eg. adding an artificial eye or limb) then you hook it up to the appropriate bits of the visual or motor cortices and hopefully it will feel and work just like the original squidgy version. If you're wedging in new functionality, then it'll feel like whatever the old bit of functionality did that you're overriding.