When I programmed Q# I know it was possible to set the probability of a qubit. So for example if you set it to 0.3 the probability of it being 1 if read is 30%.
I wonder if the probability could be set to a number which has 100 decimal places. For example:
0.175028375018462834...
Of course the used programming language can have the limits of a double or float data type.
But I am interested in the physical qubits.
Do the (hardware/physical) qubits have any limits?
Quantum mechanics imposes no limit on the precision of $\theta$ in the state $\cos(\theta)|0\rangle + \sin(\theta)|1\rangle$.
In practice, because of the limitations of current quantum computing hardware, we currently have no way to prepare the state $\cos(\theta)|0\rangle + \sin(\theta)|1\rangle$ to arbitrary fidelity. So we can't check this unlimited precision claim, yet. But once we get quantum error correction working and scaling, we should be able to achieve arbitrary precision on $\theta$ in a logical qubit, at the cost of adding more physical qubits to increase the code distance.
I think it's interesting that the only known way to potentially access arbitrarily precise values of $\theta$ in practice requires using more and more physical resources to store the relevant state. Maybe there is some deep underlying reason that this is unavoidable, and you could "associate" the increasing precision as being stored in the increasing number of elements in the system, sort of like how adding digits to a counter allows it to be more precise. Or maybe when we actually start making such large states we will find that there is a limit to the precision, meaning quantum mechanics is wrong!
In practice, there is a limit on the precision with which you can set the ratio of $|0\rangle$ to $|1\rangle$ and it usually comes from the electronics used to control the physical qubit.
For example, the $|0\rangle \leftrightarrow |1\rangle$ transition in NV centers is driven by microwaves somewhere around 2-3GHz with $\pi$-pulse times of around 100ns. There are nontrivial rise and fall times at the start and end of the pulse due to switching in the electronics, so the pulses aren't perfectly square. Also, the electronics like AWGs that generate MW pulses typically have maximum modulation rates of 1's-10's of GHz. Due to these effects, pulse lengths are typically specified to 0.1-1ns precision since our MW hardware isn't realistically going to give us more precision than that.
The limit is physically induced by the hardware’s precision. Theoretically, one can manipulate a quantum state to an any arbitrary superposition (with arbitrary amount of digits after the decimal point) with an equipment precise enough. But of course any hardware isn’t perfectly precise.
