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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Photonic ternary quantum random number generators

José Manuel Agüero Trejo, Cristian S. Calude

Published May 1, 2023• 5 citations
Publisher

Abstract

We construct a class of three-dimensional photonic quantum random number generators (QRNGs) and prove that each of them generates maximally unpredictable digits via measurements that are robust to errors. This shows that every sequence generated is strongly incomputable; hence its quality is provably better than that of every pseudo-random sequence. These results suggest that incomputability in physics is real and practically applicable. Finally, we present photonic implementations of three-dimensional QRNGs and discuss device independence.

DOI
10.1098/rspa.2022.0543

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We construct a class of three-dimensional photonic quantum random number generators (QRNGs) and prove that each of them generates maximally unpredictable digits via measurements that are robust to errors. This shows that every sequence generated is strongly incomputable; hence its quality is provably better than that of every pseudo-random sequence. These results suggest that incomputability in physics is real and practically applicable. Finally, we present photonic implementations of three-dimensional QRNGs and discuss device independence.
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.
In this thesis we study the generation of randomness from quantum mechanical sources: quantum random number generators (QRNGs). Such devices are thought to provide a better quality of randomness than is possible with classical devices, but the issue is in need of more rigorous study. In Chapter 1 we provide the necessary background for the thesis from both computability and probability theory. We then present and extend some recent results providing a more theoretical grounding for the indeterminism in quantum mechanics. In particular, we show that sequences of quantum random bits are incomputable in the strongest sense: no bit can be provably computed in advance. In Chapter 2 we study in detail the use of von Neumann normalisation for QRNGs producing both finite strings and infinite sequences of bits; such normalisation methods are necessary in order to counter for experimental imperfections. We show that, in the case of a constantly biased source, this normalisation gives the desired uniform distribution. The effect of this normalisation on the (in)computability of the generated sequences is studied, and it is shown that algorithmic randomness and Borel normality are preserved, but e-randomness (and thus incomputability) is, in general, not. Experimental bounds for the extent of departure from the uniform distribution in the non-ideal case of a slowly drifting bias are derived. In Chapter 3 we propose a new QRNG which uses entangled photon pairs and is certified to produce incomputable bits by value indefiniteness; this is the first QRNG with explicit certification by a physical principle. The effects of various experimental imperfections are discussed in detail, and care is taken to make the proposed QRNG as robust as is possible to these. A robust method based on XORing the produced bitstrings together is proposed to further improve the quality of the distribution produced by the QRNG. Various improvements and optimisations to this scheme are discussed.
Random numbers are essential resources in science and engineering, with indispensable applications in simulation, cybersecurity, and finance. Quantum random number generators (QRNGs), based on the principles of quantum mechanics, ensure genuine randomness and unpredictability. Silicon photonics enables the large-scale deployment of integrated QRNGs due to its low cost, miniaturization, and compatibility with CMOS technology. However, current integrated QRNGs are typically based on perfect or partially perfect device models, deviating from real-world devices, which compromises the unpredictability of quantum random numbers. In this study, we implemented a silicon-based QRNG that makes no assumptions about the source and only uses trusted but uncharacterized measurement devices. In experimental demonstration, we show that our setup can generate secure random numbers with different choices of intensities of laser light, and achieve an optimized random number generation rate of up to 4.04 Mbps. Our work significantly advances the security, practicality, and commercial development of QRNGs by employing imperfect devices.
In this Thesis, we develop the only class of QRNGs, to date, that has been proved to be better than any PRNG. This type of QRNG is based on a located form of the Kochen-Specker Theorem. This certification is unique because i) the assumptions used have been experimentally validated, ii) the robustness of measurements was proved theoretically, iii) the quality of quantum random digits generated with this class of QRNG was experimentally shown to be better than that of the best PRNGs, demonstrating for the first time the strong effects of incomputability on the outputs of a quantum random number generator. Finally, we propose a method using high-quality randomness for inverting a class of one-way functions in a pre-given time.