Quantum-to-Quantum Bernoulli Factory

TL;DR

This paper explores the quantum analogue of the Bernoulli factory problem, characterizing which quantum states can be exactly generated from a given quantum coin and revealing surprising differences from the classical case.

Contribution

It provides a complete characterization of quantum state transformations from a quantum coin and introduces an algorithm for such transformations, highlighting differences from classical scenarios.

Findings

Characterization of simulable quantum states from a quantum coin

Development of an algorithm for quantum state transformation

Discovery that quantum and classical simulable sets are not subsets of each other

Abstract

Given a coin with unknown bias $p\in [0,1]$, can we exactly simulate another coin with bias $f(p)$? The exact set of simulable functions has been well characterized 20 years ago. In this paper, we ask the quantum counterpart of this question: Given the quantum coin $|p\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, can we exactly simulate another quantum coin $|f(p)\rangle=\sqrt{f(p)}|0\rangle+\sqrt{1-f(p)}|1\rangle$? We give the full characterization of simulable quantum state $k_0(p)|0\rangle+k_1(p)|1\rangle$ from quantum coin $|p\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, and present an algorithm to transform it. Surprisingly, we show that simulable sets in the quantum-to-quantum case and classical-to-classical case have no inclusion relationship with each other.