If a quantum state is an eigenstate of the measurement apparatus, the measurement result is deterministic. That means, if you want deterministic answers, you need to design an algorithm where the desired answer is an eigenstate of the measurement apparatus. In effect, quantum interference is used to ensure that all other possibilities destructively interfere, leaving the desired output.
Think of each bit as being an arrow in three dimensional space. The quantum computer rotates those arrows around according to the algorithm. The aim is that in the final state, the arrows point either up or down. These are the two deterministic qubit directions. That means, to get a deterministic answer, you need to find an algorithm that ensures that all the rotations are precisely aligned at the output.
This requirement severely limits the design of quantum algorithms, and that is why there are still relatively few. The notable examples being Shorâ€™s factoring algorithm, Grover's search algorithm, and a few others.
It would be futile if Shor's algorithm only found prime factors probabilistically. How would you sort the correct answer from all the incorrect answers? In practice, with error-prone operation, you could find that nine times out of ten, you get the correct answer. Then that's just a signal-to-noise problem. The algorithm is designed to give the correct factors in the absence of noise. In fact, if you are interested in learning about quantum computing, I suggest looking at the details of Shor's algorithm.
The most natural problems to run on a quantum computer are actually quantum simulations. There have been quite a number of those that have been successfully run to date. Furthermore, these applications are in an area where classical computers struggle. These are applications applicable to materials, chemical, and pharmacological research. In other words, these are big money applications if they can be realised on a large scale quantum computer. That's why people are pouring money into this area.
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