What I mean is, in regular data any particular bit can only be in one of two states, "off" or "on", "0" or "1".
Not quite true. And
data is not really the program (as such... really) and we are talking about running programs here.
Those states. 1 and 0, are abstractions. What they really are are HIGH and LOW states where low states could be for example 0.2v.
In classical electronic von-neumann arch stored bit values in RAM are non-discrete. We just abstract away from that, (employ auto-correction methods) and choose to view that possible range of electrical charge ranges (amplitudes) as 1's or 0's. That abstraction allows for the binary boolean logic than underpins classical circuitry which then covered by microcode and instruction set, machine code and so on all the way up. at each level we apply a further abstraction which removes implementation details and adds expressive power.
Your phone RAM does not physically contain "0"s and "1"'s .. that is a misnomer, it contains charges which fall within ranges.
But in quantum data, that bit is in a superposition determined by probabilities. Since the probability is a wave function, it is no longer defined by a specific number of possible states, it's not in "discrete" states at all, there are literally infinite possibilities.
To simplify, imagine if you said, "Now the superposition has a 32.66869948377385% chance of being a '0'. if we perform this operation, now the superposition has a 36.34383834334343% chance of being a '0'." There's an infinite # of %'s that can define its superposition between '1' and '0', so the information that can be contained by its ambiguous state is now far more complex.
Thus its potential for holding information becomes far, far more complex. Of course, there are likely going to be practical limits to how much information you can actually contain and measure within any particular network of superpositions.
A super-positioned qubit (as is the classical case) has a range of values with the added probabilistic element. BUT then we as scientists need to build and abstract away from that in order to build a theoretical framework for basic operations and an instruction set as a set of standard system calls (or system services). and then on top of that we build the algorithms.
As part of those abstractions we view qubits logically as containing 00, 01, 10, 11 not the underlying
real numbers that you referred to.
We then add logical operations (analogues to boolean logic) as explained in this video ..
Then we build our quantum algorithms on top of that - new research field.
As this is still at its infancy the last I looked at this related to the DWAVE machines, they were used to solve decision problems which could be transformed into a hill-walking (min) problem. This is then used in classical problem terms by changing the representation of the NP-hard / EXP or harder problem that you want to solve using some polynomial time transformation, load it into DWAVE, get your solution, transform back. Then you have a polynomial time solution to the original harder problem.
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TL;DR - the speed up is related to the how that extra super-positioned "1+0 at the same time" state works against the exponential complexity explosion of classical computing problems from NP-hard, EXP and up .. the fact that the range of real (probabilistic) values in qubits is very large ("
infinite" in principle) is not at all related to this speed up. It is the superposition of multiple 0,1 states (where each qubit has 2) which is.
