Quantum computing does not have capability of physically implementing algorithm-output extraction: a nonconstructive proof for the nonexistence of the process solution T(t) satisfying the equivalence between bit-output probability and outcome probability
Authors: Liu Bowen 
Published: February 2023

Quantum computing consists of algorithm and algorithm-output extraction. There are two major philosophical bubbles of quantum computing: the output-extraction model has low mathematical coverage closing to zero; the real mathematized goal of output-extraction, the existence of process solution, is hidden. Piercing the two bubbles, we reduce the mathematical structure of quantum computing to the two barest mathematical problems: to prove the existence of the program solution for given input and output state in algorithm; to prove the existence of process solution for given initial and final probability in output-extraction. The mathematical theme of algorithm-output extraction is two probability binaries (PA(λn), PB(λn) and (PB(λn), PBorn(λn)). For (PA(λn), PB(λn)), PA(λn) is the outcome probability obtained by the researchers by means of the eigenvalue-to-outcome transmission, PB(λn) is the probability within the state space based on the average of the results obtained from a finite discrete set of trials. The researchers assert PA(λn) = PB(λn), and assume that the mathematical basis for PA(λn) = PB(λn) is the existence of process solution satisfying PA(λn) = PB(λn). For (PB(λn), PBorn(λn), PBorn(λn) ≡|Ψ|2is the theoretical probability for single-bit output specified by quantum mechanical principles. The researchers assert PB(λn) = PBorn(λn), and assume that the mathematical basis for PB(λn) = PBorn(λn) is the so-called Law (QC) of Large Numbers. Thus, there are two major math-bugs of quantum computing: the existence of process solution satisfying PA(λn) = PB(λn) is unproven, the Law (QC) of Large Numbers guaranteeing PB(λn) = PBorn(λn) is unproven. For the first math-bug, we introduce a completely new proof method, nonconstructive proof based on path classification. We reformulate the problem of the existence of process solution as the existence problem of path classes. Physicists rarely have the opportunity to use non-constructive proofs, which are very effective for the desired existence problem. For our nonconstructive proof, it is not necessary to provide an example of a path and a mathematical algorithm that produces an example of the path. By means of non-constructive proof, we have shown the nonexistence of process solution satisfying PA(λn) = PB(λn). For the second math-bug, we show that the law (QC) of large numbers is not a corollary of any known law of large numbers in probability theory. For the qubit-output probability of quantum computing experiments there is no reliable basis of probability theory. We provide the three examples to prove that all known minimized quantum factorization experiments are invalid. We conclude that Quantum computing cannot survive the two major math-bugs. quantum computing theory is wrong because the mathematics of the output-extraction theory is wrong; the output-extraction theory, which assumes PA(λn) = PB(λn) = PBorn(λn), is wrong because PA(λn) = PB(λn) is wrong and PB(λn) = PBorn(λn) is unproven. Our study shows that two major math-bugs of quantum computing are very serious and unfixable, because they challenge mathematicians' understanding of the underlying mathematical rules.