Experimental Quantum Bernoulli Factories via Bell-Basis Measurements

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.

💡 Research Summary

The paper presents a compact experimental realization of a quantum Bernoulli factory (QBF) using Bell‑basis measurements on two identical “p‑quoins” prepared on an IBM superconducting quantum processor. A p‑quoin is the quantum encoding of a classical biased coin with unknown probability p, namely |p⟩ = √(1‑p) |0⟩ + √p |1⟩. By preparing two copies of this state, applying a CNOT followed by a Hadamard gate, and measuring in the computational basis, the joint state is projected onto the four Bell states |Φ⁺⟩, |Φ⁻⟩, |Ψ⁺⟩, |Ψ⁻⟩. The probabilities of these outcomes are simple polynomials in p: P(|Φ⁺⟩)=P(|Ψ⁻⟩)=½, P(|Ψ⁺⟩)=2p(1‑p), and P(|Φ⁻⟩)=½‑2p(1‑p).

Exploiting these statistics, the authors implement three target functions without any external source of randomness:

1. Exact fair coin (f(p)=½). By mapping |Φ⁺⟩ and |Ψ⁻⟩ to “heads” and the other two outcomes to “tails”, a perfectly unbiased coin is obtained from a single Bell measurement, consuming exactly two p‑quoins regardless of p.

2. Quadratic function (f(p)=4p(1‑p)). Conditioning on the event that the measurement yields either |Ψ⁺⟩ or |Φ⁻⟩ (which occurs with probability ½), and assigning heads to |Ψ⁺⟩ and tails to |Φ⁻⟩, yields a coin whose bias is precisely 4p(1‑p). On average four p‑quoins are required, again independent of the unknown bias. This function is classically inconstructible because it attains the values 0 and 1 inside the domain.

3. Bernoulli‑doubling (f(p)=2p). The probability of observing |Φ⁻⟩ alone is (1‑2p)². Taking the square‑root of this quantity (i.e., |1‑2p|) can be achieved by a classical Bernoulli‑factory subroutine that uses the fair coins generated in step 1. After a “switch” operation that flips 0↔1 when necessary, the resulting bias equals 2p for p∈