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Please, be patient with my question

I already read that there is a heuristic that makes superposition a fact of reality. In addition, this superposition, when observed, it has a state of 0 or 1. Furthermore, quantum programming is done on our workstation and executed on remote quantum hardware. Finally, the use of Hadamard gates was able to create a superposition. Therefore, I'm curious to know if the superposition really is real or is it an assumption?

Edit note: I started this question with Toffoli. However, my question is about Hadamard gate.

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  • $\begingroup$ As explained here, Toffoli gates cannot create superposition. So I'm not quite sure what you mean by "Finally, the use of Toffoli gates was able to create a superposition". Either your observation of the experimental results was wrong or it was just noise. $\endgroup$ – Sanchayan Dutta Dec 1 '19 at 5:22
  • $\begingroup$ Hey @igormonteirovieira Check this experiment: physics.stackexchange.com/q/104657/236559 $\endgroup$ – walid Dec 1 '19 at 8:49
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Whether or not quantum superposition is a "truth", is a philosophical question. Quantum theory is simply an axiomatic mathematical model of the universe that happens to give correct experimental predictions (at least, for a fairly broad range) for several physical phenomena. It certainly might be possible to come up with a different mathematical model of the universe (though, the prospects look bleak) that would not require quantum superposition to explain such physical phenomena. In that sense, you can definitely call it an assumption. Every axiomatic system begins with assumptions (i.e., the axioms) after all!

If you're wondering which basic axiom of quantum mechanics implies quantum superposition, it's this:

Each physical system is associated with a (topologically) separable complex Hilbert space $H$ with inner product $\langle \phi|\psi\rangle$. Rays (that is, subspaces of complex dimension 1) in $H$ are associated with quantum states of the system.

If you've taken some introductory courses in linear algebra you should be aware that the Hilbert space is a vector space whose elements can be written as a linear superposition of its basis elements. Quantum superposition is just that; a quantum state can be written in terms of linear superpositions of the other (basis) quantum states.

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  • $\begingroup$ I know there is a mathematical theory and practice behind the superposition. But I am curious if I can state that it is possible to have a state of superposition, rather than just binary 0 or 1. Please give me a practical example and be patient in the answer. $\endgroup$ – haxkkroto Dec 1 '19 at 1:47
  • $\begingroup$ It really boils down to whether you believe quantum mechanics is a valid theory. Within the framework of the theory, it certainly is possible to have a state of superposition. For instance, an electron is considered to be in a superposition of up and down spin states. $\endgroup$ – Sanchayan Dutta Dec 1 '19 at 5:12
  • $\begingroup$ The superposition is supported by the observations of, as example, the spin. The fact that it has "only" two measurable values that have a random appearance and this is supported by the superposition makes the super position a useful tool. Quantum computing takes advantage of the successful prediction of the model and implementations of means that realize the theory of the Quantum Functions (gates and algorithms). The fact that simulations is not perfect will result some applications that will be useful and some that will require refining. The answer is a good one - deserve approval. $\endgroup$ – Moti Dec 1 '19 at 6:02
  • $\begingroup$ @SanchayanDutta I understand Heisenberg’s uncertainty principle said the level of particles do not behave like numbers. This is very weird, sometimes difficult to digest. However, it seems to be the fundamental truth behind quantum physics. ...And I am curious to know if superposition is a truth or an assumption. $\endgroup$ – haxkkroto Dec 1 '19 at 14:11
  • $\begingroup$ In general, Can I do an analogy that a superposition is just a high speed interpretation between one state and another? $\endgroup$ – haxkkroto Dec 2 '19 at 0:08
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[Can't comment and so, writing an answer]

As @Sanchayan aptly pointed the theoretical/mathematical arguments which could indicate that superposition is not just an assumption.

A couple more thoughts:

  1. You're right in observing that the superposition collapses to 0 or 1 when observed and that's expected due to the collapse of the wave function. A quick way to check if superposition exists could be to see if it's symptoms are observable - interference (destructive/constructive) as an example. You could verify this by running Grover's search algorithm. You'd see how select amplitudes are amplified while others are diminished. We could make a case here that this extends support for superposition being real.

  2. It's unclear what you mean by "quantum programming is done on our workstation and executed on remote quantum hardware". When we write a quantum program on our workstation, it's just a blueprint of an "experiment" that we would like the quantum hardware to perform. Once that information is received by the hardware, it sets up the experiment (initializing qubits), manipulates the qubits (using gates - which might result in superposition, entanglement and/or interference in due course) and finally measures the outcome (collapse of the superposition) and then reports it to our workstation.

  3. When you mention, "the use of Toffoli gates was able to create a superposition", is that a question? Technically, any state which is not a pure basis state (|0> and |1>) could be termed as a superposition since then it would be a free point on the bloch sphere apart from the poles.

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  • $\begingroup$ 2. The IBM QX, for example, allows us to interface with IBM's remote quantum hardware and quantum simulation devices. We can execute code to simulate a quantum computer on our local machines, but it is good to have the option to run on IBM's simulation architecture or IBM's quantum computing hardware as we so choose. To do this, we will need an API key. Find $\endgroup$ – haxkkroto Dec 1 '19 at 10:04

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