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

TL;DR

This paper characterizes the complexity of quantum-to-quantum Bernoulli factories, providing bounds and variants, which are essential for quantum randomness manipulation in algorithms like Bayesian inference and encryption.

Contribution

It offers a formal analysis of the complexity bounds and introduces variants of quantum Bernoulli factories, advancing understanding of their capabilities and limitations.

Findings

Lower bound on qubits needed for implementation

Upper bound on success probability

Quantum circuit that saturates the bounds

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.