Searcharxiv⌕ Search

arXiv subjects

Cristian S. Calude

Publications and source records attributed to Cristian S. Calude.

At least 37 records · Page 2Linked to original sources
arXiv2014

Embedding Quantum Universes into Classical Ones

Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Re-phrased, how far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result by Kochen and Specker answers the above question in the negative. However, this answer is just one among different possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical ones, more specifically, in terms of embeddings of quantum logics into classical logics; depending upon the type of restrictions imposed on embeddings the question may get negative or positive answers.

quant-ph↗
arXiv2013

Spectral Representation of Some Computably Enumerable Sets With an Application to Quantum Provability

We propose a new type of quantum computer which is used to prove a spectral representation for a class F of computable sets. When S in F codes the theorems of a formal system, the quantum computer produces through measurement all theorems and proofs of the formal system. We conjecture that the spectral representation is valid for all computably enumerable sets. The conjecture implies that the theorems of a general formal system, like Peano Arithmetic or ZFC, can be produced through measurement; however, it is unlikely that the quantum computer can produce the proofs as well, as in the particular case of F. The analysis suggests that showing the provability of a statement is different from writing up the proof of the statement.

quant-ph↗
arXiv2012

Strong Kochen-Specker theorem and incomputability of quantum randomness

The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require non-contextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefinite. Various technical and subtle aspects relating to this formal proof and its connection to quantum mechanics are discussed. This result is then utilized for the proposal and certification of a dichotomic quantum random number generator operating in a three-dimensional Hilbert space.

quant-ph↗
arXiv2012

Is Feasibility in Physics Limited by Fantasy Alone?

Although various limits on the predicability of physical phenomena as well as on physical knowables are commonly established and accepted, we challenge their ultimate validity. More precisely, we claim that fundamental limits arise only from our limited imagination and fantasy. To illustrate this thesis we give evidence that the well-known Turing incomputability barrier can be trespassed via quantum indeterminacy. From this algorithmic viewpoint, the "fine tuning" of physical phenomena amounts to a "(re)programming" of the universe.

physics.hist-ph↗
arXiv2011

A Nuclear Magnetic Resonance Implementation of a Classical Deutsch-Jozsa Algorithm

Nuclear magnetic resonance (NMR) has been widely used as a demonstrative medium for showcasing the ability for quantum computations to outperform classical ones. A large number of such experiments performed have been implementations of the Deutsch-Jozsa algorithm. It is known, however, that in some cases the Deutsch-Jozsa problem can be solved classically using as many queries to the black-box as in the quantum solution. In this paper we describe experiments in which we take the contrasting approach of using NMR as a classical computing medium, treating the nuclear spin vectors classically and utilising an alternative embedding of bits into the physical medium. This allows us to determine the actual Boolean function computed by the black-box for the n=1,2 cases, as opposed to only the nature (balanced or constant) as conventional quantum algorithms do. Discussion of these experiments leads to some clarification of the complications surrounding the comparison of different quantum algorithms, particularly black-box type algorithms.

cs.ET↗
arXiv2010

A Quantum Random Number Generator Certified by Value Indefiniteness

In this paper we propose a quantum random number generator (QRNG) which utilizes an entangled photon pair in a Bell singlet state, and is certified explicitly by value indefiniteness. While "true randomness" is a mathematical impossibility, the certification by value indefiniteness ensures the quantum random bits are incomputable in the strongest sense. This is the first QRNG setup in which a physical principle (Kochen-Specker value indefiniteness) guarantees that no single quantum bit produced can be classically computed (reproduced and validated), the mathematical form of bitwise physical unpredictability. The effects of various experimental imperfections are discussed in detail, particularly those related to detector efficiencies, context alignment and temporal correlations between bits. The analysis is to a large extent relevant for the construction of any QRNG based on beam-splitters. By measuring the two entangled photons in maximally misaligned contexts and utilizing the fact that two rather than one bitstring are obtained, more efficient and robust unbiasing techniques can be applied. A robust and efficient procedure based on XORing the bitstrings together---essentially using one as a one-time-pad for the other---is proposed to extract random bits in the presence of experimental imperfections, as well as a more efficient modification of the von Neumann procedure for the same task. Some open problems are also discussed.

quant-ph↗
arXiv2010

The Complexity of Proving Chaoticity and the Church-Turing Thesis

Proving the chaoticity of some dynamical systems is equivalent to solving the hardest problems in mathematics. Conversely, one argues that it is not unconceivable that classical physical systems may "compute the hard or even the incomputable" by measuring observables which correspond to computationally hard or even incomputable problems.

nlin.CD↗
arXiv2010

Experimental Evidence of Quantum Randomness Incomputability

In contrast with software-generated randomness (called pseudo-randomness), quantum randomness is provable incomputable, i.e.\ it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability --- an asymptotic property --- of quantum randomness by performing finite tests of randomness inspired by algorithmic information theory.

quant-ph↗
arXiv2010

Understanding the Quantum Computational Speed-up via De-quantisation

While it seems possible that quantum computers may allow for algorithms offering a computational speed-up over classical algorithms for some problems, the issue is poorly understood. We explore this computational speed-up by investigating the ability to de-quantise quantum algorithms into classical simulations of the algorithms which are as efficient in both time and space as the original quantum algorithms. The process of de-quantisation helps formulate conditions to determine if a quantum algorithm provides a real speed-up over classical algorithms. These conditions can be used to develop new quantum algorithms more effectively (by avoiding features that could allow the algorithm to be efficiently classically simulated), as well as providing the potential to create new classical algorithms (by using features which have proved valuable for quantum algorithms). Results on many different methods of de-quantisations are presented, as well as a general formal definition of de-quantisation. De-quantisations employing higher-dimensional classical bits, as well as those using matrix-simulations, put emphasis on entanglement in quantum algorithms; a key result is that any algorithm in which the entanglement is bounded is de-quantisable. These methods are contrasted with the stabiliser formalism de-quantisations due to the Gottesman-Knill Theorem, as well as those which take advantage of the topology of the circuit for a quantum algorithm. The benefits of the different methods are contrasted, and the importance of a range of techniques is emphasised. We further discuss some features of quantum algorithms which current de-quantisation methods do not cover.

quant-ph↗
arXiv2009

How Random Is Quantum Randomness? An Experimental Approach

Our aim is to experimentally study the possibility of distinguishing between quantum sources of randomness--recently proved to be theoretically incomputable--and some well-known computable sources of pseudo-randomness. Incomputability is a necessary, but not sufficient "symptom" of "true randomness". We base our experimental approach on algorithmic information theory which provides characterizations of algorithmic random sequences in terms of the degrees of incompressibility of their finite prefixes. Algorithmic random sequences are incomputable, but the converse implication is false. We have performed tests of randomness on pseudo-random strings (finite sequences) of length $2^{32}$ generated with software (Mathematica, Maple), which are cyclic (so, strongly computable), the bits of $π$, which is computable, but not cyclic, and strings produced by quantum measurements (with the commercial device Quantis and by the Vienna IQOQI group). Our empirical tests indicate quantitative differences, some statistically significant, between computable and incomputable sources of "randomness".

quant-ph↗
arXiv2009

Simplicity via Provability for Universal Prefix-free Turing Machines

Universality is one of the most important ideas in computability theory. There are various criteria of simplicity for universal Turing machines. Probably the most popular one is to count the number of states/symbols. This criterion is more complex than it may appear at a first glance. In this note we review recent results in Algorithmic Information Theory and propose three new criteria of simplicity for universal prefix-free Turing machines. These criteria refer to the possibility of proving various natural properties of such a machine (its universality, for example) in a formal theory, PA or ZFC. In all cases some, but not all, machines are simple.

cs.IT↗
arXiv2009

Every Computably Enumerable Random Real Is Provably Computably Enumerable Random

We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine" can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine $U$ such that, based on $U$, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the Kraft-Chaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle.

cs.CC↗
arXiv2007

Most Programs Stop Quickly or Never Halt

Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect non-halting programs. Our approach is to have the probability space extend over both space and time and to consider the probability that a random $N$-bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k>0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2^{-k}. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that ``long'' runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2^{N+constant} has effectively zero density.

cs.IT↗
arXiv2006

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 $f: \{0,1\} \to \{0,1\}$ and suppose that we have a (classical) black box to compute it. The problem asks whether $f$ is constant (that is, $f(0) = f(1)$) or balanced ($f(0) \not= f(1)$). Classically, to solve the problem seems to require the computation of $f(0)$ and $ f(1)$, and then the comparison of results. Is it possible to solve the problem with {\em only one} query on $f$? 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.

quant-ph↗
arXiv2006

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.

cs.CC↗
arXiv2005

Randomness Relative to Cantor Expansions

Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. When such a sequence can be called random? In this paper we offer a solution to the above question using the approach to randomness proposed by Algorithmic Information Theory.

nlin.CD↗
arXiv2004

Is Complexity a Source of Incompleteness?

In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is invariant under the change of the Gödel numbering. For this measure, the theorems of a finitely-specified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length $n$ is provable in the theory tends to zero when $n$ tends to infinity, while the probability that a sentence of length $n$ is true is strictly positive.

math.LO↗