An anatomy of a quantum adiabatic algorithm that transcends the Turing computability

We give an update on a quantum adiabatic algorithm for the Turing noncomputable Hilbert's tenth problem, and briefly go over some relevant issues and misleading objections to the algorithm.Comment: 7 pages, no figure. Submitted to the Proceedings of the conference "Foundations of Quantum Information" (April 2004, Camerino, Italy

A reformulation of Hilbert's tenth problem through Quantum Mechanics

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shr\"odinger propagator with some appropriate kernel. Either way, Mathematics and Physics could be combined for Hilbert's tenth problem and for the notion of effective computability

Quantum adiabatic algorithm for Hilbert's tenth problem: I. The algorithm

We review the proposal of a quantum algorithm for Hilbert's tenth problem and provide further arguments towards the proof that: (i) the algorithm terminates after a finite time for any input of Diophantine equation; (ii) the final ground state which contains the answer for the Diophantine equation can be identified as the component state having better-than-even probability to be found by measurement at the end time--even though probability for the final ground state in a quantum adiabatic process need not monotonically increase towards one in general. Presented finally are the reasons why our algorithm is outside the jurisdiction of no-go arguments previously employed to show that Hilbert's tenth problem is recursively non-computable.Comment: Typos fixed, substantial results added in Section III, new reference and footnotes added. Now 22 pages, one figur

Hypercomputability of quantum adiabatic processes: Fact versus Prejudices

We give an overview of a quantum adiabatic algorithm for Hilbert's tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementation.Comment: 25 pages, 4 figures. Invited paper for a special issue of the Journal of Applied Mathematics and Computatio

Reply to ``The quantum algorithm of Kieu does not solve the Hilbert's tenth problem"

The arguments employed in quant-ph/0111009, to claim that the quantum algorithm in quant-ph/0110136 does not work, are so general that were they true then the adiabatic theorem itself would have been wrong. As a matter of fact, those arguments are only valid for the sudden approximation, not the adiabatic process.Comment: 3 page

What's Wrong with Anomalous Chiral Gauge Theory?

It is argued on general ground and demonstrated in the particular example of the Chiral Schwinger Model that there is nothing wrong with apparently anomalous chiral gauge theory. If quantised correctly, there should be no gauge anomaly and chiral gauge theory should be renormalisable and unitary, even in higher dimensions and with non-abelian gauge groups. Furthermore, mass terms for gauge bosons and chiral fermions can be generated without spoiling the gauge invariance.Comment: 9 A4 pages in LATEX, talk given at the First Symposium on Symmetries in Subatomic Physics, Taipei, 16-18 May, 1994, UM-P-94/58 and RCHEP-94/1

Mathematical computability questions for some classes of linear and non-linear differential equations originated from Hilbert's tenth problem

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger equations with some appropriate time-dependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are non-trivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the Church-Turing thesis.Comment: 8 pages; submitted to the Second International Conference on Computability and Complexity in Analysis, August 25-29, 2005, Kyoto, Japa

Chiral gauge theory in four dimensions

A formulation of abelian and non-abelian chiral gauge theories is presented together with arguments for the unitarity and renormalisability in four dimensions. IASSNS-HEP-94/70, UM-P-94/96, and RCHEP-94/26.Comment: Latex, 9 pages; IASSNS-HEP-94/70, UM-P-94/96, and RCHEP-94/2

On the identification of the ground state based on occupation probabilities: An investigation of Smith's apparent counterexamples

We study a set of truncated matrices, given by Smith~\cite{Smith2005}, in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert's tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, should be applicable provided certain conditions are satisfied. We also point out that, apart from this one, other criteria serving the same identification purpose may also be available.Comment: This is an edited version, with updated references, of an earlier reply to Smith in July, 2005. 9 pages and 12 figure

Quantum Principles and Mathematical Computability

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical "algorithm" for one of the insoluble problems of mathematics, the Hilbert's tenth and equivalently the Turing halting problem. The key element of this algorithm is the {\em computability} and {\em measurability} of both the values of physical observables and of the quantum-mechanical probability distributions for these values.Comment: 9 pages in A4 size and 10pt fonts, 3 figures. Modified with a new reference added for submission to QS200