There's a lot of complex maths and polar notation. There are a couple of good simulators out there that lets you play with qubits and their probabilistic coefficients.
I'll be honest that I was never all that great at the higher maths and a lot of this taxes my brain or goes way above my head. But all these quantum computers are deterministic. The simulators can fully simulate them.
It's just that simulating several qubits requires gigs and gigs of ram. A real quantum computer can't do anything you can't do with a traditional computer, it can just do it in a more computational faster time and fewer resources.
You can run small quantum programs yourself on the IBM cloud quantum platform. They allow people to queue up programs to run, similar to old punch card systems:
Not just "both", you have to consider all quantum states.
It is more like adding an imaginary component to a real probability making it a complex probability. It is possible to calculate with the complex probabilities and only collapse to the field of R in the end.
While the alternative seems a little too far out to be true, I have to ask, how do you know?
> So the idea that a quantum computers is better than the classical computer is opposed to the idea that quantum physics is some hack to reduce computational resources.
Keep in mind, not all operations in a computer take the same amount of time. If those qubits are entangled, you are going to get the state of all of them from a single "operation". Finally, we assume deterministic and stochastic computation take the same time, but that's only true for us because we perform stochastic computations deterministically - I'm pretty sure we could squeeze a lot more performance out of our silicon if we relaxed our accuracy constraints.
Additionally, if the universe is a computing system, the probabilistic nature of quantum mechanics may be a way to work around paradoxes, i.e. Godel's incompleteness theorem.
Gödel's incompleteness theorems have in some sense much deeper reasons, they are based on the logical consistency of the entire construction. Maybe you can look at it in a similar way, a theory is an object like a signal above and the properties of being consistent and complete can not be realized at the same time. But I have a hard time imagining that this could really be similar to signals where you can trade localization in time for localization in frequency and vice versa, but how would you trade a bit of consistency for a bit of completeness?
EDIT: To be a bit more concrete, in classical mechanics you have to specify position and momentum (velocity) of a particle to specify its state, those are two independent properties that can have specific and independent values. That is not true in quantum mechanics, there position or momentum alone fully specify the state of the system. The wave function (in position space) tells you where the particle is with what probability, the frequencies of the wave function tell you what the momenta are with what probability.
And from here it is the same as above, if you force a particle into a very well localized position, i.e. make the wave function a narrow spike at some place, then the frequencies and therefore the momenta are no longer well defined. If, on the other hand, you make the wave function of the particle like a sine wave, then you get a well defined frequency and therefore momentum but the wave function becomes spread out across space and the position is therefore no longer well localized.
>In 1927 Heisenberg discovered that the ``more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa''. Four years later G\"odel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the {\it converse} implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a ``formal uncertainty principle'' which implies Chaitin's information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to {\it all} systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics.
How do we "know" that the camera collapses the wave function? Maybe WE collapse the camera by observing IT.
That's how you get things like Schrodingers cat.
That sounds ridiculous. The double slit experiment is a small system. The detector apparatus is much larger. The world outside of that, even more so. If your view is how the universe works, the number of superpositions of states that must exist between the "magical human observer" and the experimental result must be huge. Every particle in contact with every other particle. It seems ridiculous that the entire universe must be in superposition to give humans this special property.
The "observer" here is simply an outside system.
(FWIW, we only covered the basics of quantum mechanics in my chemistry undergrad. Perhaps a physicist can explain better.)
According to your argument, a situation like Schrödinger's cat, where a Cat is both dead and alive, is completely ridiculous.
Which is a fair point. But apparently a whole lot of quantum physicists believe that Schrödinger's cat is a perfectly reasonable situation.
This is a common misconception (among programmers). There's zero experimental evidence for the effect you mention, and zero theoretical derivation. Circumstances under which wave function collapses is the greatest mystery of QM.
I agree with that.
Anyway, if you have an optical system, you can assume that coherence is present while the light hits mirrors, lens and similar optical equipment. (If the difference in the optical paths are smaller than the coherence length of your laser or light source.) But as soon as the light hits a white screen or a brick wall, all the further calculations must use only the intensity of the light at each spot in the screen, forgetting about the phase angle. And all the spots are not coherent.
It's not clear what cause the wave function collapse, but if you are using photons in the visible spectrum probably a mirror will not collapse it, and a brick wall will collapse it. [Or your preferred rewrite with the multiple word interpretation, or the abstract Hilbert space calculations.) I'm guessing decoherence is the correct explanation, there is a nice comment in a reply.
For other particles, the abstract calculation is equivalent, but it's necessary to choose another system to do the experiments.
If you follow the equations of the wave packet of the particle arriving to the wall (or CCD) detector, you then need to solve the equation of the interaction of the particle + all the particles in the wall or the CCD. The challenge of the collapse is that simulating anything beyond a few dozen quantum particles is too demanding. Mathematical models that simulate millions of particles need to make assumptions (typically they are too hot, too cold, too strongly bound, so you can ignore most effects - think 1D Ising model, Bose-Einstein Condensates, Photon gases, etc). But the full description of a particle + all particles in a detector still escapes us.
Therefore the transition between: superposition of paths -> particle lands at specific points has never been truly explored. The best description currently involves decoherence. Many theorems have been proven (and experiments done) in that area. The gist is that as you add particles to a system (2, 3, 4, 5, 10, ...) the superposition effects slowly cancel each other out. Another angle is the monogamy of entanglement (the more particles are entangled, the weaker the entanglement between any 2 particles). The idea of decoherence is that as things get larger, the weird effects of quantum physics become more "dilute". However, going from double slit to macroscopic reading still has many assumptions along the way.
Take the above explanation with a grain of salt (as I have tried to make it accessible).
This is a common misconception (¿among programmers?).
Let's think about the double slit experiment. https://en.wikipedia.org/wiki/Double-slit_experiment
In a classical word, you must simulate only one path. In a quantum word, you must simulate both. You don´t need some magical conscious observer to force the collapse of the wave function. A CCD detector of a camera or a simple wall is enough to force that the "wave" collapse into a "particle" and the detector or wall gets a small spot where the "particle" hits it.
A similar experiment is possible with spin, and you can use that to get a qbit. In a classical word, a qbit is simply a bit and you only have to chose between the 0 or 1 state and simulate it. In a quantum word, you must simulate both.
But it's worst with many qbits. Let's say you have a 10 qbits computer. In a classical word you pick a value for each of them and simulate each one, so the total computation is ~10. In a quantum word, you can combine any of the states of the 10 qbits and you must simulate the 1024 states.
So the idea that a quantum computers is better than the classical computer is opposed to the idea that quantum physics is some hack to reduce computational resources.
I think that the main problem is the quantum mechanics is weird, use a lot of linear algebra, but the calculations are somewhat straightforward and well defined. But the popularization explanations try to avoid the algebra and make some simplifications, so the explanation only keeps the weird part.