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

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.

💡 Research Summary

This paper presents a comprehensive analysis of the Quantum-to-Quantum Bernoulli Factory (QQBF), a protocol that transforms multiple copies of an input qubit state |z⟩ into an output qubit state |f(z)⟩ corresponding to a target complex rational function f(z). The work goes beyond merely identifying the class of simulable functions, delivering a detailed characterization of the protocol’s complexity in terms of quantum resources and success probability, while also introducing and analyzing novel multi-functional extensions.

The core contribution begins by demonstrating that any QQBF circuit can be mapped to a canonical form: it uses n copies of the input |z⟩ state and m ancillary |0⟩ qubits, applies a global unitary operation U, and then measures all but the first qubit in the computational basis, accepting the output only if all measurement results are 0. Analyzing this general structure reveals that the amplitudes of the final output state are ratios of polynomials in z of degree at most n. This reconfirms that the set of simulable functions is precisely the set of complex rational functions, f(z)=P(z)/Q(z). Crucially, it establishes a direct resource lower bound: implementing a function of degree d requires at least n ≥ d input qubits.

The paper then proves this lower bound is tight and constructively defines the optimal circuit. For any rational function of degree n, an explicit method is provided to construct a unitary operator U that implements the function using exactly n input qubits and, for n≥2, requires no ancillary qubits. This optimal construction involves defining specific row vectors of U based on the coefficients of the polynomials P(z) and Q(z), with a free parameter w. The analysis further derives the success probability of the protocol and shows that it is a decreasing function of |w|^2. Therefore, setting |w|^2=0 not only yields the minimal circuit but also maximizes the success probability, confirming the optimality of the proposed design.

Beyond characterizing the standard QQBF, the paper formalizes two significant generalizations. First, the Multivariate QQBF extends the framework to handle multiple input quantum coins with different complex biases z1, z2, …, zk, aiming to produce an output state encoding a multivariate rational function s(z1,…,zk). Second, the Multifunctional QQBF enables a single quantum circuit to implement one function from a predefined set {f_i(z)}. The choice of which function is executed is determined by the state of internal control parameters (e.g., specific ancillary qubits), offering a versatile architecture for programmable randomness processing.

An application section illustrates the theory by constructing optimal QQBF circuits for elementary operations like addition and scalar multiplication. In conclusion, this work provides a complete framework for understanding and constructing QQBFs. It establishes fundamental limits on qubit count and success probability, delivers a recipe for optimal circuit synthesis, and expands the scope of Bernoulli factories through its novel multivariate and multifunctional variants. These results lay a solid foundation for employing QQBFs as efficient subroutines in quantum algorithms for Bayesian inference, Monte Carlo methods, and blind quantum computation.