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

TL;DR

This work provides a complete characterization of the quantum-to-quantum Bernoulli factory (QQBF) by proving that the set of simulable functions is exactly the complex rational functions Rat$(z)$ and that the minimum number of input qubits satisfies $n \ge \deg(f)$. It delivers a constructive circuit framework that saturates these bounds and shows no ancilla is required except in the deg$(f)=1$ case, enabling explicit circuit design for any QQBF. The authors extend the model to multivariate inputs (multivariate QQBF) and to multifunctional QQBF, where multiple functions are implemented within a single protocol, including compatibility conditions between functions. They illustrate the theory with applications to sum and product operations, analyzing resource requirements and heralded outputs, thereby positioning QQBF as a versatile subroutine for quantum algorithms and randomness manipulation in quantum information processing.

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.

Figures (4)

• Figure 1: Schematic representation of the general circuit for the Quantum Bernoulli. The circuit takes as input $n$ qubit in the state $\ket{z}$ and $m$ ancillary qubits in the state $\ket{0_c}$. The resulting state evolves through a unitary evolution $U$. All qubits are then measured in the computational basis, except the first one, which carries the output state of the Bernoulli factory. The output state $\ket{\psi_O}$ is accepted if all the measurements $q_2 \dots q_{n+m}$ return the outcome $0$. This circuit is the most general one since every circuit implementing a Quantum-to-quantum Bernoulli factory can be mapped into the one shown in the figure, as described in the main text.
• Figure 2: Average success probabilities $\langle \mathrm{Pr}_n\rangle_{U/C}$ for the function $g(z) = \eta z^2$. Representation of the success probability for the function $g(z) = \eta z^2$ for different numbers of qubits employed by the quantum circuit and for different sets of states. In a) we considered the average over the uniform distribution of all the qubits and in b) we consider the average only on the equatorial states. The background colour represents the region in which a particular probability is higher than the others. It can be deduced from the figures that the optimal number of qubits required depends strongly on both the function itself and the set of states considered.
• Figure 3: Multivariate Quantum to Quantum Bernoulli factory general circuit. The circuit takes in input $n_k$ qubits in the state $\ket{z_k}$ for a total of $n_t$ and $m$ ancillary qubits in the state $\ket{0}$. The rest of the circuit is similar to that of Fig. \ref{['fig: BF_circuit']}. As before, this circuit is the most general one implementing a Multivariate Quantum to Quantum Bernoulli factory.
• Figure 4: Multifunctional Quantum Bernoulli factory general circuit. This circuit implementing a Multifunctional Quantum Bernoulli factory is similar to the Fig. \ref{['fig: BF_circuit']}. The difference is at the measurement stage. Previously, all the measures had to return the outcome $0$. In this case, if the measure $q_2$ returns $0$, the circuit implements the function $g_0(z)$, while if the outcome is 1 $1$ the circuit implements the function $g_1(z)$. As before, this circuit is the most general one implementing a Multifunctional Quantum to Quantum Bernoulli factory.