Coins, Quantum Measurements, and Turing's Barrier

TL;DR

This paper explores the theoretical possibility of quantum computing to solve uncomputable problems like the Halting Problem by constructing a mathematical quantum device, challenging traditional views on the limits of computation.

Contribution

It presents a mathematical model of a quantum device capable of deciding the Halting Problem, suggesting quantum computing might surpass classical undecidability barriers.

Findings

Constructs a quantum device that can decide halting with high probability.

Shows the measure of indistinguishable vectors tends to zero as time increases.

Proves the mathematical possibility of building a 'halting machine'.

Abstract

Is there any hope for quantum computing to challenge the Turing barrier, i.e. to solve an undecidable problem, to compute an uncomputable function? According to Feynman's '82 argument, the answer is {\it negative}. This paper re-opens the case: we will discuss solutions to a few simple problems which suggest that {\it quantum computing is {\it theoretically} capable of computing uncomputable functions}. In this paper a mathematical quantum "device" (with sensitivity $\epsilon$) is constructed to solve the Halting Problem. The "device" works on a randomly chosen test-vector for $T$ units of time. If the "device" produces a click, then the program halts. If it does not produce a click, then either the program does not halt or the test-vector has been chosen from an {\it undistinguishable set of vectors} ${\IF}_{\epsilon, T}$. The last case is not dangerous as our main result proves: {\it the Wiener measure of} ${\IF}_{\epsilon, T}$ {\it constructively tends to zero when} $T$ {\it tends to infinity}. The "device", working in time $T$, appropriately computed, will determine with a pre-established precision whether an arbitrary program halts or not. {\it Building the "halting machine" is mathematically possible.}