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Is Quantum Randomness Algorithmic Random? A Preliminary Attack

Cristian S. Calude, Michael J. Dinneen

Published Jan 1, 2005• 5 citations
OpenAlex

Abstract

Randomness is an essential ingredient for computation.The pitfalls of software-generated pseudo-randomness arewell-known. So, can we do it better?The obvious candidate is quantum randomness.Quantum randomness has been confirmed by theoretical andexperimental research. The first book containing a million ofquantum random digits—generated by using radioactive decayfrom electronic vacuum tubes—was published by the RANDCorporation in 1955.Is it quantum randomness provable better thenpseudo-randomness? If yes, how much better?

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Alastair A. Abbott

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.

•Jan 1, 2011•0 citations
Randomness is an essential ingredient for computation.The pitfalls of software-generated pseudo-randomness arewell-known. So, can we do it better?The obvious candidate is quantum randomness.Quantum randomness has been confirmed by theoretical andexperimental research. The first book containing a million ofquantum random digits—generated by using radioactive decayfrom electronic vacuum tubes—was published by the RANDCorporation in 1955.Is it quantum randomness provable better thenpseudo-randomness? If yes, how much better?
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 2322^{32}232 generated with software (Mathematica, Maple), which are cyclic (so, strongly computable), the bits of πππ, 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".
The fundamental link between physics and computation leads to an exciting field of quantum computation. Since the birth of the programmable digital computer, random number generators in computers have been developed. In principle, the pseudo random number generators currently implemented in digital computers are predictable. However, a quantum random number generator is employed to ideally generate unpredictable random numbers. In this essay, we explain why a one-qubit quantum computer can be a quantum random number generator and how this leads to justifying the foundations of quantum mechanics.
Cryptography is the study and practice of secure communication with digital data and focuses on confidentiality, integrity, and authentication. Random number generators (RNGs) generate random numbers to enhance security. Even though the cryptographic algorithms are public and their strength depends on the keys, cryptoanalysis of encrypted ciphers can significantly contribute to the unveiling of the cipher’s key. Therefore, to ensure high data security over a network, researchers need to improve the randomness of keys as they develop cryptosystems. Quantum particles have a leading edge in advancing RNG technology as they can provide true randomness, unlike pseudo-random numbers generators (PRNGs). In order to increase the level of the security of cryptographic systems based on random numbers, this survey focuses on three objectives: Cryptosystems with related cryptographic attacks, RNG-based cryptosystems, and the design of quantum random number generators (QRNGs). This survey aims to provide researchers with information about the importance of RNG-based ciphers and various research techniques for QRNGs that can incorporate quantum-based true randomness in cryptosystems.
This paper proposes a novel approach to algorithmic randomness generation utilizing the principles of quantum tunneling simulation. Traditional pseudo-random number generators (PRNGs) are inherently susceptible to biases and vulnerabilities, often derived from deterministic algorithms. This research introduces a methodology that harnesses quantum phenomena, specifically the probability of tunneling through a potential well, to produce truly random sequences. The core mechanism involves modulating the tunneling probability based on a function directly related to the desired random value. This approach offers a fundamentally different paradigm for randomness generation, potentially overcoming the limitations imposed by classical PRNGs and providing a more robust and unbiased source of randomness. The simulation methodology is described in detail, along with a theoretical framework outlining the key mathematical operations involved. The potential advantages of this technique, particularly regarding its resistance to prediction and biases, are explored.
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.