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arXiv:2512.10810 · Physical Review Research

Complexity and multi-functional variants of the Quantum-to-Quantum Bernoulli Factories

Francesco Hoch, Taira Giordani, Gonzalo Carvacho, Nicolò Spagnolo, Fabio Sciarrino

quant-phSubmitted Dec 11, 2025• Announced Dec 12, 2025• 0 citations
arXivHTMLPDF

Abstract

A Bernoulli factory is a model for randomness manipulation that transforms an initial Bernoulli random variable into another Bernoulli variable by applying a predetermined function relating the output bias to the input one. In literature, quantum-to-quantum Bernoulli factory schemes have been proposed, which encode both the input and output variables using qubit amplitudes. This fundamental concept can serve as a subroutine for quantum algorithms that involve Bayesian inference and Monte Carlo methods, or that require data encryption, like in blind quantum computation. In this work, we present a characterisation of the complexity of the quantum-to-quantum Bernoulli factory by providing a lower bound on the required number of qubits needed to implement the protocol, an upper bound on the success probability and the quantum circuit that saturates the bounds. We also formalise and analyse two different variants of the original problem that address the possibility of increasing the number of input biases or the number of functions implemented by the quantum-to-quantum Bernoulli factory. The obtained results can be used as a framework for randomness manipulation via such an approach.

Journal reference
Phys. Rev. Research 8, 033003 (2026)
DOI
10.1103/k8kd-58h8

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A Bernoulli factory is a model for randomness manipulation that transforms an initial Bernoulli random variable into another Bernoulli variable by applying a predetermined function relating the output bias to the input one. In literature, quantum-to-quantum Bernoulli factory schemes have been proposed, which encode both the input and output variables using qubit amplitudes. This fundamental concept can serve as a subroutine for quantum algorithms that involve Bayesian inference and Monte Carlo methods, or that require data encryption, like in blind quantum computation. In this work, we present a characterisation of the complexity of the quantum-to-quantum Bernoulli factory by providing a lower bound on the required number of qubits needed to implement the protocol, an upper bound on the success probability and the quantum circuit that saturates the bounds. We also formalise and analyse two different variants of the original problem that address the possibility of increasing the number of input biases or the number of functions implemented by the quantum-to-quantum Bernoulli factory. The obtained results can be used as a framework for randomness manipulation via such an approach.
A Bernoulli factory is a model of randomness processing. Recently, the quantum generalization of a Bernoulli factory with a quantum input and classical output offers a clear advantage over classical means in simulating classical randomness. A more ``quantum'' version of a Bernoulli factory with a quantum input and quantum output has been proposed and shows applications in simulating quantum randomness using quantum resources, the so-called quantum-to-quantum Bernoulli factory, which simulates probability amplitudes rather than probability. In this paper, we report an experimental implementation of a quantum-to-quantum Bernoulli factory with linear optics. We implement three elements of a quantum-to-quantum Bernoulli factory, including inversion, multiplication, and addition, and demonstrate how to compare quantum states via a quantum-to-quantum Bernoulli factory. Our results provide a thorough understanding of the Bernoulli factory problem in the quantum world and will stimulate the quantum advantages in simulating a wider range of classically infeasible random processes.
A Bernoulli factory is a randomness manipulation routine that takes as input a Bernoulli random variable, outputting another Bernoulli variable whose bias is a function of the input bias. Recently proposed quantum-to-quantum Bernoulli factory schemes encode both input and output variables in qubit amplitudes. This primitive could be used as a subroutine for more complex quantum algorithms involving Bayesian inference and Monte Carlo methods. Here, we report an experimental implementation of a polarization-encoded photonic quantum-to-quantum Bernoulli factory. We present and test three interferometric setups implementing the basic operations of an algebraic field (inversion, multiplication, and addition), which, chained together, allow for the implementation of a generic quantum-to-quantum Bernoulli factory. These in-bulk schemes are validated using a quantum dot-based single-photon source featuring high brightness and indistinguishability, paired with a time-to-spatial demultiplexing setup to prepare input resources of up to three single-photon states.
Generation and manipulation of randomness is a relevant task for several applications of information technology. It has been shown that quantum mechanics offers some advantages for this type of task. A promising model for randomness manipulation is provided by Bernoulli factories—protocols capable of changing the bias of Bernoulli random processes in a controlled way. At first, this framework was proposed and investigated in a fully classical regime. Recent extensions of this model to the quantum case showed the possibility of implementing a wider class of randomness manipulation functions. We propose a Bernoulli factory scheme with quantum states as the input and output, using a photonic-path-encoding approach. Our scheme is modular and universal and its functioning is truly oblivious of the input bias—characteristics that were missing in earlier work. We report on experimental implementations using an integrated and fully programmable photonic platform, thereby demonstrating the viability of our approach. These results open new paths for randomness manipulation with integrated quantum technologies. A quantum-to-quantum Bernoulli factory is demonstrated by using a reconfigurable Clements’s squared unitary circuit in an integrated quantum photonic platform. Three interferometer designs are proposed for the basic operations of a field on qubit states.
Randomness generation and manipulation algorithms are a cornerstone of information theory and in cryptography. In this context, the inherently random nature of quantum mechanics has proved to be a useful resource, leading to the realisation of several quantum information protocols. Within this framework, there is an increasing search for new techniques for generating and manipulating quantum randomness that can be used as sub-routines for quantum protocols. In recent years, there have been various proposals for quantum generalisations of a family of classical algorithms known as Bernoulli factories [1] which, starting from extractions of a Bernoulli distribution with unknown parameterppp, allow one to simulate exactly a new Bernoulli distribution with parameterf(p)f(p)f(p)for a given functionf(x):D⊆[0,1]→[0,1]f(x):\mathscr{D}\subseteq[0,1]\rightarrow[0,1]f(x):D⊆[0,1]→[0,1]. In particular, it was shown that the use of quantum resources results in an advantage in terms of implementable functions [2], [3] and the average number of samples required by the algorithm.
Quantum computing is rapidly becoming one of the largest research areas in quantum physics and yet we are still unable to prove the superiority of quantum computing over classical with the exception of a few contrived and unrealistic scenarios. Understanding the areas in which quantum computing provides an advantage and, more importantly, where that advantage comes from is the key to making progress in the field. We certainly do not solve this mystery in this thesis but do explore some unconventional routes through which answers may be found in the future. To this end we explore quantum advantage in a different computational setting. Rather than working with Turing Machines as most research does, we focus on the Bernoulli Factory: a randomness processing scheme studied in the fields of statistics and computer science. The Bernoulli Factory takes an infinite string of random bits generated with a fixed unknown bias as input and outputs a single random bit, the bias of which is a function of the input bias. This can be seen more intuitively as generating one biased coin from another, given a desired relationship between the two biases. The Bernoulli Factory is inherently probabilistic, like quantum mechanics, and so, it leads us to questions about the fundamental nature of quantum randomness and quantum sampling. We find, as we hoped, that the Bernoulli Factory is more tractable than the Turing Machine and that we are able to prove quantum superiority in both resource efficiency and classes of computable functions. We first give the quantum analogue of the classical Keane-O'’Brien theorem which establishes the increased class of computable functions for the quantum case. We then adjust the problem in a variety of ways, such as introducing multiple coins, errors in state preparation, phases and restriction on the bias or operations allowed. These lead to additional results which lend insight into the problems faced in quantum computing as well as the nature of quantum randomness.
There has been a concerted effort to identify problems computable with quantum technology, which are intractable with classical technology or require far fewer resources to compute. Recently, randomness processing in a Bernoulli factory has been identified as one such task. Here, we report two quantum photonic implementations of a Bernoulli factory, one using quantum coherence and single-qubit measurements and the other one using quantum coherence and entangling measurements of two qubits. We show that the former consumes three orders of magnitude fewer resources than the best-known classical method, while entanglement offers a further fivefold reduction. These concepts may provide a means for quantum-enhanced performance in the simulation of stochastic processes and sampling tasks.