QUANTUM COMPUTING NOTES

This brief note is dedicated to all quantum computer scientists, engineers, and bystanders. The goal is to reduce the amount of unnecessary hype and considerably lower expectations.

1) If noise is present, the state of a real quantum system cannot be determined with a probability of 1. This is a fundamental law of nature.

Assume a quantum system with two quantum states, ∣Ψ0⟩ and ∣Ψ1⟩, a virtual state ∣Ψv⟩, and a single noise state ∣Ψn⟩. The superposition of those states would be described by the following expression

∣Ψ⟩ = α∣Ψ0⟩ + β∣Ψ1⟩ + ν∣Ψv⟩ + n(t)∣Ψn⟩,  (1)

where α, β, ν, and n(t) are complex coefficients. In addition, n(t) is a function of time and, therefore, not reproducible.

From Born formula

∣α∣^2+ ∣β∣^2 +∣ν∣^2 + ∣n(t)∣^2 = 1.  (2)

There are too many unknowns to solve those two equations (1-2), four unknowns, and only two equations. In general, ∣Ψn⟩ may be in a superposition of several noise states with additional unknown coefficients. Both noise n(t) coefficient and ∣Ψn⟩ states are random and change with time. Repeating experiments multiple times will produce constantly changing α, β, ν, and n(t) coefficients.

The situation with a real quantum system is rather difficult. In virtually all quantum mechanics textbooks, the virtual states are ignored, so ν=0 and the noise part is also ignored, so n(t)=0. Now α and β can be determined, and the time dependence has been eliminated. The coefficient ν alone can be eliminated if the particle is surrounded by a high potential, but this approach may not always be practical.

2) Noise-free real quantum systems do not exist. The probabilistic nature of everything surrounding us dictates that noise is never going away. At least not in our Universe!

3) Quantum superposition, which the quantum computer relies on, is a linear combination of quantum states. The quantum system's state will be undetermined if there is at least a single noise source. This obviously is a significant issue for quantum computer designs based on the superposition of quantum states.

4) A particle's probability vs. energy curve inside a quantum system is always continuous, between zero and infinity. There are no gaps in the probability, and you cannot call a particular part of the curve a 0 and another part of the curve a 1.

4) Now, another issue that was already discussed in the literature. Any quantum approach will need help to be able to represent integers. This means that integers 0, 1, 2, etc., can only appear with certain probabilities, always less than 1. Natural integers always appear with probabilities of 1, which is their fundamental property.

5) As an exercise, try this search program: 1) apple, 2) pear. How will you find an apple if neither 1 nor 2 can be represented correctly by the quantum machine? Just don't tell me; it's the first one.

6) As a verification, you can check that no quantum computer can handle integer factorization satisfactorily. Shor's algorithm can run only on a noise-free quantum computer that cannot be built. Such a brilliant idea, damn it! The same is true for most famous quantum algorithms. To all future quantum algorithm developers, ensure you don't ignore the noise.

7) Quantum computers generate objects with strange properties, and there are no algorithms to handle those properties. Can they even exist?

8) An electron or photon must be surrounded by a potential barrier. For example, Cooper pairs in superconducting qubits, electrons in semiconducting qubits, photons in ion-based qubits, photons in atom-based qubits, or photons in photonic qubits. That potential barrier will scatter electrons or photons, thus preventing them from traveling over the barrier with a probability equal to 1. This is always true. If an electron is in one place and needs to be moved to another, this can only be done with p<1. You need to find out if the intended operation happened, and you need to know if the electron was in the first place with p=1. If you want to perform a measurement again, the electron may arrive with p<1. The measured result will always be very unreliable.

9) Quantum tunneling into the potential barrier is still another issue, and there is no way around it. Build infinitely tall energy barriers, then particles cannot escape the confinement, and nothing about their state can ever be learned; that's not a good solution.

10) To verify that those remarks are correct, test X = ∣0⟩ with a probability of 1 on a quantum computer. Repeat this experiment a large number of times. Then analyze the distribution of X; it should be a straight vertical line, and other distributions are not acceptable.

In slightly more detail: a) x=0, b) Determine x, c) Save x, don't overwrite previous values, d) Repeat steps a-c a large number of times, e) Plot distribution of x.

11) Or, try to determine 1 + 1 with a probability of 1 on a quantum computer. If the above does not work, then this has no chance either. Then what exactly do you do with a computer that cannot add numbers?

12) An abacus can handle integers very well but is relatively slow. A binary computer can handle integers with high precision and speed. A quantum computer of today is a real dud! Quoted error rates are about 2%, 10 times better with an error correction. Speed is also an issue in some designs. Regarding accuracy, regular binary computers are roughly 13 orders of magnitude ahead of quantum computers.

13) A binary computer has only two logical levels, 0 and 1. Both levels can handle a fair amount of noise, and those levels never overlap. There is a gap between 0 and 1. Someone who invented it was an absolute genius! 0 means 0, and 1 means 1, such a simple idea. One can build almost anything with three logical gates: AND, OR, and NOT. High-speed and accurate supercomputers verify that the binary approach is working well, which is not valid for the existing quantum approaches.

14) Quantum algorithms can be extensive but must be relatively shallow. The noise keeps accumulating, and after only several steps, the simulation no longer has any meaning.

In an analog setup, a source will contribute noise and the first stage if the power gain of that stage is sufficiently high. So the amplified signal can be processed by other stages without significant distortions. In a quantum computer, the first qubit generates noise and sends it to the second qubit, and the second qubit adds its own noise and sends it to the third qubit. After several steps, there is nothing but noise coming out of it.

15) Quantum error correction can only work to some small degree. After all, the arrangement that is supposed to reduce noise is adding its own noise. How could this work?

16) Obviously, something is wrong with all existing quantum ideas, algorithms, etc. We are underlining some issues; this is only a partial review of cases.

17) Those remarks deal with potential sources of noise. However, Heisenberg's uncertainty principle is another problem to consider and can worsen things.

18) Please note that the issue of simulation time has yet to be raised even once. Because if the machine generates noise, who cares how fast it is?

19) Quantum computers are the thing of the future and will always remain that way.