134 research outputs found
De-Quantising the Solution of Deutsch's Problem
Probably the simplest and most frequently used way to illustrate the power of
quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a
Boolean function and suppose that we have a
(classical) black box to compute it. The problem asks whether is constant
(that is, ) or balanced (). Classically, to solve
the problem seems to require the computation of and , and then
the comparison of results. Is it possible to solve the problem with {\em only
one} query on ? In a famous paper published in 1985, Deutsch posed the
problem and obtained a ``quantum'' {\em partial affirmative answer}. In 1998 a
complete, probability-one solution was presented by Cleve, Ekert, Macchiavello,
and Mosca. Here we will show that the quantum solution can be {\it
de-quantised} to a deterministic simpler solution which is as efficient as the
quantum one. The use of ``superposition'', a key ingredient of quantum
algorithm, is--in this specific case--classically available.Comment: 8 page
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
We introduce the zeta number, natural halting probability and natural
complexity of a Turing machine and we relate them to Chaitin's Omega number,
halting probability, and program-size complexity. A classification of Turing
machines according to their zeta numbers is proposed: divergent, convergent and
tuatara. We prove the existence of universal convergent and tuatara machines.
Various results on (algorithmic) randomness and partial randomness are proved.
For example, we show that the zeta number of a universal tuatara machine is
c.e. and random. A new type of partial randomness, asymptotic randomness, is
introduced. Finally we show that in contrast to classical (algorithmic)
randomness--which cannot be naturally characterised in terms of plain
complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin
Computing A Glimpse of Randomness
A Chaitin Omega number is the halting probability of a universal Chaitin
(self-delimiting Turing) machine. Every Omega number is both computably
enumerable (the limit of a computable, increasing, converging sequence of
rationals) and random (its binary expansion is an algorithmic random sequence).
In particular, every Omega number is strongly non-computable. The aim of this
paper is to describe a procedure, which combines Java programming and
mathematical proofs, for computing the exact values of the first 64 bits of a
Chaitin Omega:
0000001000000100000110001000011010001111110010111011101000010000. Full
description of programs and proofs will be given elsewhere.Comment: 16 pages; Experimental Mathematics (accepted
From Heisenberg to Goedel via Chaitin
In 1927 Heisenberg discovered that the ``more precisely the position is
determined, the less precisely the momentum is known in this instant, and vice
versa''. Four years later G\"odel showed that a finitely specified, consistent
formal system which is large enough to include arithmetic is incomplete. As
both results express some kind of impossibility it is natural to ask whether
there is any relation between them, and, indeed, this question has been
repeatedly asked for a long time. The main interest seems to have been in
possible implications of incompleteness to physics. In this note we will take
interest in the {\it converse} implication and will offer a positive answer to
the question: Does uncertainty imply incompleteness? We will show that
algorithmic randomness is equivalent to a ``formal uncertainty principle''
which implies Chaitin's information-theoretic incompleteness. We also show that
the derived uncertainty relation, for many computers, is physical. In fact, the
formal uncertainty principle applies to {\it all} systems governed by the wave
equation, not just quantum waves. This fact supports the conjecture that
uncertainty implies randomness not only in mathematics, but also in physics.Comment: Small change
An Observer-Based De-Quantisation of Deutschâs Algorithm
Deutschâs problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutschâs quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various dequantised solutions for Deutschâs quantum algorithmâclassical solutions which are as efficient as the quantum oneâhave been proposed in the literature. These solutions are based on the possibility of classically simulating âsuperpositionsâ, a key ingredient of quantum algorithms, in particular, Deutschâs algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous dequantisations of Deutschâs quantum algorithm, the resulting dequantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their âobservational powerâ
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
A note on the differences of computably enumerable reals
We show that given any non-computable left-c.e. real α there exists a left-c.e. real ÎČ such that αâ ÎČ+Îł for all left-c.e. reals and all right-c.e. reals Îł. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Loef random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact
How much contextuality?
The amount of contextuality is quantified in terms of the probability of the
necessary violations of noncontextual assignments to counterfactual elements of
physical reality.Comment: 5 pages, 3 figure
- âŠ