Quantum complexity theory exists and it is one of the fastest growing fields in computer science. The answer to how this occurs will come out of an elaboration on your first question, the question of how do we treat the ontological status of probability amplitudes.
In quantum mechanics we deal with what are called probability amplitudes. Probability amplitudes differ from probability distributions; amplitudes can be negative and complex numbers while distributions cannot. The reason this works is because the absolute value of the amplitude is squared before it is treated like a normal probability; this assures us that we will never end up with a negative probability. All of the much discussed "weird" or "counterintuitive" aspects of quantum mechanics are due to the fact that probability amplitudes can cancel each other out. This is just how the formalism works, it's a property of complex numbers.
The question you're asking is one of the most central questions in the philosophy of physics; it is the most central question in the philosophy of quantum mechanics without a doubt. The general discussion of the question "how should we understand the ontological status of probability amplitudes" falls under the label of interpretations of quantum mechanics.
From the Wikipedia article on interpretations of quantum mechanics:
More or less, all interpretations of quantum mechanics share two qualities:
- They interpret a formalism—a set of equations and principles to generate predictions via input of initial conditions
- They interpret a phenomenology—a set of observations, including those obtained by empirical research and those obtained informally, such as humans' experience of an unequivocal world
Two qualities vary among interpretations:
- Ontology—claims about what things, such as categories and entities, exist in the world
- Epistemology—claims about the possibility, scope, and means toward relevant knowledge of the world
The reason interpretations of quantum mechanics are more often associated with philosophy rather than science (truly, some scientists think it is all just metaphysical questions with no verifiable answer so it's nonsense to talk about them) is due to the fact that the interpretations focus on discussing the ontological and epistemological status on quantum mechanics, not the formalism. Science is the cycle of collecting data, analyzing the data, coming up with predictive models and formulas that allow us to make accurate predictions, and then preforming experiments that verify or contradict those predictions. Quantum mechanics, so far, has provided the most well tested physical theory we have ever achieved and it has done so without settling the ontological status of wave function collapses, so many scientists think the answer to this question genuinely does not matter.
There are many different interpretations but there are two that stand out as the most popular today. One is referred to as the Copenhagen interpretation and the other is the many-worlds interpretation. From the Stanford Encyclopedia of Philosophy:
Today the Copenhagen interpretation is mostly regarded as synonymous with indeterminism, Bohr's correspondence principle, Born's statistical interpretation of the wave function, and Bohr's complementarity interpretation of certain atomic phenomena.
The Copenhagen interpretation treats quantum mechanical objects as purely probabilistic objects. It treats the wave function (the collection of amplitudes) as a truly probabilistic object. Generally, the Copenhagen school of thought is described as being a system that accepts nondeterminism in physical theories. They say we have absolutely no idea what the outcome of an experiment will be and neither does the particle until the experiment happens and it then instantly makes a "choice" based off of probability.
The fundamental idea of the MWI, going back to Everett 1957, is that there are myriads of worlds in the Universe in addition to the world we are aware of. In particular, every time a quantum experiment with different possible outcomes is performed, all outcomes are obtained, each in a different world, even if we are only aware of the world with the outcome we have seen.
The many-worlds interpretation says that there is no randomness, per se, of the outcome of wave function collapse. It says that there are many parallel universes that exist and each time a measurement is made the outcomes are all experienced by a different world. The reason we do not experience all of the outcomes is because our conscious experience is limited to one world, the world in which we exist.
The answer as to "how should we interpret the probability amplitude" is not yet agreed upon. There exist many different interpretations and all of them provide different answers. The issue is that, for the most part it is believed, there is absolutely no experiment we can do that will prove one or the other. We are not able to access the parallel worlds from the many-world interpretation so we will never be able to verify that the collapses are not actually random. This means that there will be no way for us to assure that the ontological probabilities of the Copenhagen interpretation are incorrect. Much of the literature of the philosophy of physics is devoted to these questions.
A Snapshot of Foundational Attitudes Toward Quantum Mechanics is a very interesting paper describing the current consensus and divides on this question in the current physics community. The results of their polling shows that the majority of modern physicists believe that randomness is inherent in physics and that the Copenhagen is the most believed interpretation. It is conjectured that this is due to that interpretation being the "traditional" or "canonical" interpretation and thus most physicists learned it as the default idea from their professors. That being said, most of the most prominent physicists push for the many-worlds interpretations (Lenny Susskind has a somewhat technical paper about why they might both be correct).
Now, on to complexity. Scott Aaronson is generally regarded as the most authoritative voice on quantum computational complexity, at least as far as theory goes. He has also written extensively on the philosophical side of quantum computation and what philosophers, physicists, and computer scientists should all focus on in regards to these questions. His work, along with many, many others, has shown that quantum computation is theoretically a well founded idea.
However, we have not yet built fully functional quantum computers. So far, there have been no physical results that prove that quantum computation, and therefore everything predicted by quantum complexity theory, cannot be physically realized. The largest problem we have faced is the problem of decoherence, which is the tendency for quantum objects to interact destructively with their environment. Decoherence is one of the main research topics in quantum complexity theory and it is the main source of trouble for our current design of actual quantum computers.
The way that computer scientists and physicists deal docoherence is by what are called quantum error correction. This is a process whereby quantum objects are allowed to remain coherent by introducing new objects into the environment that take the decoherence onto themselves and then are removed. It is an open question as to whether or not nature experiences quantum error correct by itself, however many specialists believe that it does.
So, from a philosophical perspective, some people remain skeptical that we will ever be able to have fully functional quantum computers. Others, however, are fully committed to the idea and believe that they are just around the corner. Further study into the true ontological nature of probability amplitudes will surely shed light on some of the questions the field faces, such as the true nature of a measurement. Where we stand, however, is that quantum complexity surely exists epistemologically. We have (almost) all of the math formulated. What remains are the actual machines that prove the mathematics is correct.
As for the Doctor Who question, it seems unlikely that a being such as that would exist, because if it was a being that had consciousness it would be able to experience itself and therefore would always be able to exist. The thought process behind the show is probably more sympathetic to interpretations that rely on the human mind playing a conscious role in quantum mechanics. This interpretation is generally disfavored and discredited by most working physicists and a lot of philosophers, however it does have its proponents.