Exploiting the features of quantum computers, such as the ability to simultaneously represent and process different states of information thanks to qubits, to generate sets of random numbers more efficiently is a prerequisite for beneficial applications in the context of computational simulation of physical systems such as cryptography. This is the claim made in two studies carried out within the framework of ICSC - National Research Centre in High Performance Computing, Big Data and Quantum Computing and the European projects 'PHOQUSING - PHotonic QUantum SamplING Machine' and 'QU-BOSS' by the Quantum Lab group of Sapienza University of Rome, in collaboration with the International Iberian Nanotechnology Labs (INL) and the Institute of Photonics and Nanotechnology - National Research Council (IFN-CNR). The results of the two papers, which recently appeared in the prestigious scientific journals Nature Photonics and Science Advances, demonstrate how a properly designed and controlled quantum photonics platform can implement the Bernoulli Factory algorithm. The latter is a well-known algorithm used for generating sets of random variables, thus defining a new technique for manipulating random variables using quantum mechanics called the 'quantum-to-quantum Bernoulli Factory'.

Using the example of the series of results from a coin toss, the Bernoulli factory algorithm, which plays a central role in numerical integration and Monte Carlo methods used in probabilistic calculations, makes it possible, from a known probability distribution of tosses used as input, to generate coin tosses with a different distribution as output.

"If we have, for example, the objective of creating a new coin that shows heads with a different probability from the known probability revealed by the tosses performed", says Fabio Sciarrino, head of the Quantum Lab group of Sapienza University and head of the photonics platform for Spoke 10 "Quantum Computing" at ICSC, "the Bernoulli factory algorithm cunningly allows us to toss the original coin several times and exploit the various results to simulate the tosses of a new coin with the desired probability distribution. Within the framework of quantum mechanics, this procedure was translated by encoding probability distributions as quantum states in both input and output. Hence the name 'quantum-to-quantum Bernoulli factory".

The unique characteristics of the Bernoulli factories have therefore prompted the research groups that authored the two studies to explore various methods of implementing these 'randomness factories' by constructing optical platforms within which, by modifying the configuration of the circuits, it has been possible to make the dynamics of photons and the quantum information states they carry evolve in the desired manner. Given the statistical behaviour that characterises them, the evolution of photons within these devices can therefore more effectively generate a distribution of random results than can a simulation performed by a classical computer.

"To implement the Bernoulli factory algorithms", Sciarrino continues, "we have developed two platforms that manipulate distinct degrees of freedom of single-photon states. The first, developed in collaboration with INL and IFN-CNR which led to the article in Nature Photonics, works with so-called path-encoded qubits, in which information is written in the path of each photon. This was made possible thanks to the high control and precision achievable in the programming of the glass-integrated photonic circuits made by CNR-IFN, the reproducibility of which is guaranteed by the automated systems adopted, which also facilitate the use of these devices for the implementation of algorithms with greater complexity. In the second platform, developed in collaboration with INL, the qubits are instead encoded in the polarisation states of single photons. With both platforms, we could demonstrate all the necessary steps to realise genuine Bernoulli Factory quantum algorithms.'

These advances represent significant steps forward in research aimed at understanding how to process information by exploiting the quantum properties of light. The quantum-to-quantum Bernoulli Factories also represent further proof of the advantages that quantum devices can provide over their classical counterparts. Indeed, by exploiting the unique properties of quantum light, researchers will be able to explore new possibilities for efficient computation and sophisticated manipulation of random variables, paving the way for innovative applications in fields as diverse as cryptography, computation and simulation.

"On the one hand, the architecture used for path-encoding qubits, which are manipulated through integrated optics devices, is an ideal solution for the implementation of Bernoulli Factories in the context of quantum computation, as it can, for instance, be used as a component of more complex quantum photonic hardware", says Francesco Hoch, post-doc and first author of the article in Nature Photonics. "On the other hand, the second platform operating on the polarisation states of photons, realised through optical elements operating in both air and fibre, is particularly suitable for interfacing the device with quantum networks and, more generally, with existing complex communication and quantum cryptography protocols", concludes Giovanni Rodari, PhD student and first author of the article in Science Advances.

F. Hoch, et al., Modular quantum-to-quantum Bernoulli factory in an integrated photonic processor Nature Photonics (2024). https://www.nature.com/articles/s41566-024-01526-8

G. Rodari, et al. Polarization-encoded photonic quantum-to-quantum Bernoulli factory based on a quantum dot source, Science Advances 10, 30 (2024). https://www.science.org/doi/10.1126/sciadv.ado6244