arXiv subjects

Cristian S. Calude

Publications and source records attributed to Cristian S. Calude.

At least 19 recordsLinked to original sources

arXiv2026

ND-Photonic QRNGs in a Noisy Environment

Standard pseudo-random generators have weaknesses that have led to the development of quantum random number generators (QRNGs). However, common QRNG validation methods, whether based on quantum indeterminism or statistical tests, are insufficient to guarantee high-quality randomness. In contrast, a mathematical theory based on the Located Kochen-Specker Theorem proves that 3D QRNGs produce maximally unpredictable outputs without using entanglement, and both theory and experiments have supported their security. The paper focuses on a practical photonic implementation of a 3D QRNG that is easier to deploy than cryogenic superconducting implementations. As any physical implementation is subject to various measurement errors, it is important to study theoretically and experimentally the type and role of errors in 3D QRNGs. In this paper, we will model the photonic 3D QRNG as an open quantum system, constructed as an arrangement of imperfect beam-splitters with a range of losses based on the fidelity of its components, and we will show that under certain conditions, the process remains within the scope of the Kochen-Specker Theorem, which guarantees maximum unpredictability.

quant-ph↗

arXiv2026

Testing the 3D QRNG by Undoing

We propose a method to test whether a photonic 3D QRNG works according to the underlying theory, thereby generating highly incomputable/unpredictable sequences of random digits. The test relies on undoing the unitary evolution realized by the 3D QRNG. The test verifies the unitarity, the magnitude of the noise, and other potential errors, such as photon loss or systematic and reproducible fabrication errors. Therefore, the test can confirm the theoretically proven features of the 3D QRNG, such as strong incomputability and unpredictability, or how one has to correct it, if necessary. In addition, the test ensures that the QRNG is not affected by limits of quantum measurement accuracy, as those described in the Wigner-Araki-Yanase Theorem. The test can be easily incorporated into the QRNG and used as a means of experimental certification.

quant-ph↗

arXiv2024

How Real is Incomputability in Physics?

A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this case, if the laws of the system can be coded into a computer program, then given the initial conditions of the system, one can compute the system's evolution. Are there incomputable physical systems? This question has been theoretically studied in the last 30-40 years. In this paper, we experimentally show for the first time the strong incomputability of a quantum experiment, namely the outputs of a quantum random number generator. Moreover, the experimental results are robust and statistically significant.

quant-ph↗

arXiv2024

Binary Quantum Random Number Generator Based on Value Indefinite Observables

All quantum random number generators based on measuring value indefinite observables are at least three-dimensional because the Kochen-Specker Theorem and the Located Kochen-Specker Theorem are false in dimension two. In this article, we construct quantum random number generators based on measuring a three-dimensional value indefinite observable that generates binary quantum random outputs with the same randomness qualities as the ternary ones: the outputs are maximally unpredictable.

quant-ph↗

arXiv2020

A New Quantum Random Number Generator Certified by Value Indefiniteness

In this paper we propose a new ternary QRNG based on measuring located value indefinite observables with probabilities $1/4, 1/2,1/4$ and prove that every sequence generated is maximally unpredictable, 3-bi-immune (a stronger form of bi-immunity), and its prefixes are Borel normal. The ternary quantum random digits produced by the QRNG are algorithmically transformed into quantum random bits using an alphabetic morphism which preserves all the above properties.

quant-ph↗

arXiv2020

Von Neumann Normalisation of a Quantum Random Number Generator

In this paper we study von Neumann un-biasing normalisation for ideal and real quantum random number generators, operating on finite strings or infinite bit sequences. In the ideal cases one can obtain the desired un-biasing. This relies critically on the independence of the source, a notion we rigorously define for our model. In real cases, affected by imperfections in measurement and hardware, one cannot achieve a true un-biasing, but, if the bias "drifts sufficiently slowly", the result can be arbitrarily close to un-biasing. For infinite sequences, normalisation can both increase or decrease the (algorithmic) randomness of the generated sequences. A successful application of von Neumann normalisation---in fact, any un-biasing transformation---does exactly what it promises, un-biasing, one (among infinitely many) symptoms of randomness; it will not produce "true" randomness.

cs.IT↗

arXiv2019

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.

quant-ph↗

arXiv2019

A Hybrid Quantum-Classical Paradigm to Mitigate Embedding Costs in Quantum Annealing

Despite rapid recent progress towards the development of quantum computers capable of providing computational advantages over classical computers, it seems likely that such computers will, initially at least, be required to run in a hybrid quantum-classical regime. This realisation has led to interest in hybrid quantum-classical algorithms allowing, for example, quantum computers to solve large problems despite having very limited numbers of qubits. Here we propose a hybrid paradigm for quantum annealers with the goal of mitigating a different limitation of such devices: the need to embed problem instances within the (often highly restricted) connectivity graph of the annealer. This embedding process can be costly to perform and may destroy any computational speedup. In order to solve many practical problems, it is moreover necessary to perform many, often related, such embeddings. We will show how, for such problems, a raw speedup that is negated by the embedding time can nonetheless be exploited to give a real speedup. As a proof-of-concept example we present an in-depth case study of a simple problem based on the maximum weight independent set problem. Although we do not observe a quantum speedup experimentally, the advantage of the hybrid approach is robustly verified, showing how a potential quantum speedup may be exploited and encouraging further efforts to apply the approach to problems of more practical interest.

cs.DS↗

arXiv2019

Spurious, Emergent Laws in Number Worlds

We study some aspects of the emergence of logos from chaos on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious, emerging laws surfaces. The spurious, emergent laws abound, they can be found almost everywhere. In accord with the ancient Greek theogony one could say that logos, the Gods and the laws of the universe, originate from "the void," or from chaos, a picture which supports the unresolvable/irreducible lawless hypothesis. The analysis presented in this paper suggests that the "laws" discovered in science correspond merely to syntactical correlations, are local and not universal.

physics.gen-ph↗

arXiv2018

Experimentally Probing the Algorithmic Randomness and Incomputability of Quantum Randomness

The advantages of quantum random number generators (QRNGs) over pseudo-random number generators (PRNGs) are normally attributed to the nature of quantum measurements. This is often seen as implying the superiority of the sequences of bits themselves generated by QRNGs, despite the absence of empirical tests supporting this. Nonetheless, one may expect sequences of bits generated by QRNGs to have properties that pseudo-random sequences do not; indeed, pseudo-random sequences are necessarily computable, a highly nontypical property of sequences. In this paper, we discuss the differences between QRNGs and PRNGs and the challenges involved in certifying the quality of QRNGs theoretically and testing their output experimentally. While QRNGs are often tested with standard suites of statistical tests, such tests are designed for PRNGs and only verify statistical properties of a QRNG, but are insensitive to many supposed advantages of QRNGs. We discuss the ability to test the incomputability and algorithmic complexity of QRNGs. While such properties cannot be directly verified with certainty, we show how one can construct indirect tests that may provide evidence for the incomputability of QRNGs. We use these tests to compare various PRNGs to a QRNG, based on superconducting transmon qutrits and certified by the Kochen-Specker Theorem, to see whether such evidence can be found in practice. While our tests fail to observe a strong advantage of the quantum random sequences due to algorithmic properties, the results are nonetheless informative: some of the test results are ambiguous and require further study, while others highlight difficulties that can guide the development of future tests of algorithmic randomness and incomputability.

quant-ph↗

arXiv2018

Quassical Computing

We present a class of hybrid classical systems using quantum co-processors and point out that unlike purely quantum computers, such hybrids can be both universal and Turing complete; we introduce such quantum-classical hybrids as "quassical." We discuss the benefits of quassical architectures from a theoretical point of view: for some classes of problems they achieve computational supremacy. From a practical point of view, quassical architectures can also reduce the overhead burden imposed by most error correction schemes and minimize the challenges of interconnecting qubits in a usefully large connection graph. All quantum computing systems are cyber-physical machines and thus quassical to at least a trivial degree but only the more profoundly quassical hybrids can exhibit an optimum problem-solving capability for the amount of quantum resources deployed. Most significantly, quassical architectures advance our thinking past that of seeing quantum machines as simply quantum embodiments of classical ones and can enliven whole new fields of analytical thinking that takes us beyond quantum information science per se into a deeper understanding of the duality between quantum information and fundamental thermodynamics, possibly suggesting unexpectedly useful new technologies.

quant-ph↗

arXiv2018

A Hybrid Quantum-Classical Paradigm to Mitigate Embedding Costs in Quantum Annealing---Abridged Version

Quantum annealing has shown significant potential as an approach to near-term quantum computing. Despite promising progress towards obtaining a quantum speedup, quantum annealers are limited by the need to embed problem instances within the (often highly restricted) connectivity graph of the annealer. This embedding can be costly to perform and may destroy any computational speedup. Here we present a hybrid quantum-classical paradigm to help mitigate this limitation, and show how a raw speedup that is negated by the embedding time can nonetheless be exploited in certain circumstances. We illustrate this approach with initial results on a proof-of-concept implementation of an algorithm for the dynamically weighted maximum independent set problem.

cs.DS↗

arXiv2015

A variant of the Kochen-Specker theorem localising value indefiniteness

The Kochen-Specker theorem proves the inability to assign, simultaneously, noncontextual definite values to all (of a finite set of) quantum mechanical observables in a consistent manner. If one assumes that any definite values behave noncontextually, one can nonetheless only conclude that some observables (in this set) are value indefinite. In this paper we prove a variant of the Kochen-Specker theorem showing that, under the same assumption of noncontextuality, if a single one-dimensional projection observable is assigned the definite value 1, then no one-dimensional projection observable that is incompatible (i.e., non-commuting) with this one can be assigned consistently a definite value. Unlike standard proofs of the Kochen-Specker theorem, in order to localise and show the extent of value indefiniteness this result requires a constructive method of reduction between Kochen-Specker sets. If a system is prepared in a pure state $|ψ\rangle$, then it is reasonable to assume that any value assignment (i.e., hidden variable model) for this system assigns the value 1 to the observable projecting onto the one-dimensional linear subspace spanned by $|ψ\rangle$, and the value 0 to those projecting onto linear subspaces orthogonal to it. Our result can be interpreted, under this assumption, as showing that the outcome of a measurement of any other incompatible one-dimensional projection observable cannot be determined in advance, thus formalising a notion of quantum randomness.

quant-ph↗

arXiv2015

A Non-Probabilistic Model of Relativised Predictability in Physics

Little effort has been devoted to studying generalised notions or models of (un)predictability, yet is an important concept throughout physics and plays a central role in quantum information theory, where key results rely on the supposed inherent unpredictability of measurement outcomes. In this paper we continue the programme started in [1] developing a general, non-probabilistic model of (un)predictability in physics. We present a more refined model that is capable of studying different degrees of "relativised" unpredictability. This model is based on the ability for an agent, acting via uniform, effective means, to predict correctly and reproducibly the outcome of an experiment using finite information extracted from the environment. We use this model to study further the degree of unpredictability certified by different quantum phenomena, showing that quantum complementarity guarantees a form of relativised unpredictability that is weaker than that guaranteed by Kochen-Specker-type value indefiniteness. We exemplify further the difference between certification by complementarity and value indefiniteness by showing that, unlike value indefiniteness, complementarity is compatible with the production of computable sequences of bits.

quant-ph↗

arXiv2015

On the unpredictability of individual quantum measurement outcomes

We develop a general, non-probabilistic model of prediction which is suitable for assessing the (un)predictability of individual physical events. We use this model to provide, for the first time, a rigorous proof of the unpredictability of a class of individual quantum measurement outcomes, a well-known quantum attribute postulated or claimed for a long time. We prove that quantum indeterminism - formally modelled as value indefiniteness - is incompatible with the supposition of predictability: measurements of value indefinite observables are unpredictable. The proof makes essential use of a strengthened form of the Kochen-Specker theorem proven previously to identify value indefinite observables. This form of quantum unpredictability, like the Kochen-Specker theorem, relies on three assumptions: compatibility with quantum mechanical predictions, non-contextuality, and the value definiteness of observables corresponding to the preparation basis of a quantum state. We explore the relation between unpredictability and incomputability and show that the unpredictability of individual measurements of a value indefinite quantum observable complements, and is independent of, the global strong incomputability of any sequence of outcomes of this particular quantum experiment. Finally, we discuss a real model of hypercomputation whose computational power has yet to be determined, as well as further open problems.

quant-ph↗

arXiv2015

Universality and Almost Decidability

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result - namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable - solves an open problem in [9]. We conclude with some open problems.

cs.CC↗

arXiv2014

Value-indefinite observables are almost everywhere

Kochen-Specker theorems assure the breakdown of certain types of non-contextual hidden variable theories through the non-existence of global, holistic frame functions; alas they do not allow us to identify where this breakdown occurs, nor the extent of it. It was recently shown [Phys. Rev. A 86, 062109 (2012)] that this breakdown does not occur everywhere; here we show that it is maximal in that it occurs almost everywhere, and thus prove that quantum indeterminacy--often referred to as contextuality or value indefiniteness--is a global property as is often assumed. In contrast to the Kochen-Specker theorem, we only assume the weaker non-contextuality condition that any potential value assignments that may exist are locally non-contextual. Under this assumption, we prove that once a single arbitrary observable is fixed to occur with certainty, almost (i.e. with Lebesgue measure one) all remaining observables are indeterminate.

quant-ph↗

arXiv2014

Kochen-Specker Theorem: Two Geometric Proofs

We present two geometric proofs of the Kochen-Specker theorem. A quite similar argument has been used by Cooke, Keane, and Moran, as well as by Kalmbach in her book to derive the Gleason theorem.

quant-ph↗