Experimental Quantum Bernoulli Factories via Bell-Basis Measurements

TL;DR

The paper addresses quantum advantages in Bernoulli factories by encoding the input bias $p$ into $p$-quoins and performing Bell-basis measurements on two copies. A single Bell measurement yields an exact fair coin and a classically inconstructible $f_\frown(p)=4p(1-p)$ with constant average quoin cost, and, from the same data, the Bernoulli-doubling $f_\wedge(p)=2p$; the latter is obtained via a conditional square-root operation. The authors implement the protocol on IBM superconducting hardware, validating the predicted biases and highlighting practical limitations from readout errors. This work demonstrates a compact, resource-efficient quantum-to-classical randomness processing primitive with potential for quantum-enhanced stochastic simulation and sampling on near-term devices.

Abstract

Randomness processing in the Bernoulli factory framework provides a concrete setting in which quantum resources can outperform classical ones. We experimentally demonstrate an entanglement-assisted quantum Bernoulli factory based on Bell-basis measurements of two identical input quoins prepared on IBM superconducting hardware. Using only the measurement outcomes (and no external classical randomness source), we realize the classically inconstructible Bernoulli doubling primitive $f(p)=2p$ and, as intermediate outputs from the same Bell-measurement statistics, an exact fair coin $f(p)=1/2$ and the classically inconstructible function $f(p)=4p(1-p)$. We benchmark the measured output biases against ideal predictions and discuss the impact of device noise. Our results establish a simple, resource-efficient experimental primitive for quantum-to-classical randomness processing and support the viability of quantum Bernoulli factories for quantum-enhanced stochastic simulation and sampling tasks.

Figures (3)

• Figure 1: Fair coin generation. (a) The number of biased coins consumed to generate a fair coin using the classical von Neumann method is shown in the dashed line as a function of biasness $p$. The solid line shows that a quantum Bernoulli method needs only two quantum coin or quoins per fair coin irrespective of $p$. (b) A quantum circuit using a single qubit for generating a fair coin with $\theta = 2\sin^{-1}\sqrt{p}$ where $Y(\theta)$ represents a rotation about the $y$-axis by an amount $\theta$ and the result of the last measurement is the outcome.
• Figure 2: Experimental protocol and measurements. (a) Two quoins are prepared with identical bias $p$ by applying rotations about the y-axis by an amount $\theta=2 \sin^{-1}\sqrt{p}$. A transformation to the Bell-basis is performed by a CNOT followed by a Hadamard gate. The measured states are mapped to the corresponding Bell basis. (b) Measured counts of the four basis states as a function of bias $p$ obtained after $5\times10^4$ experimental repetitions.
• Figure 3: Experimental quantum Bernoulli factory. (a) Experimental data of a fair coin (cyan circles) and a $f_\frown(p)$ coin (green circles) as a function of bias $p$. The solid lines shows ideal values. Each fair and $f_\frown(p)$ coin needs, on average, two and four quoins, respectively. (b) Top panel shows experimental data for the Bernoulli doubling function $f_\wedge(p)$ (green circles) and the ideal values (red lines). The number of quoins used is shown in the bottom plot with a median value of 303.