Reliable Computation by Formulas of Noisy AND Gates and Noiseless XOR Gates, with Applications to Quantum Mechanics
Noah Shutty, Mary K. Wootters, Patrick M. Hayden · arXiv (Cornell University) · 2018
It has long been known that the existence of certain superquantum nonlocal correlations would cause complexity to collapse. The absurdity of a world in which any nonlocal binary function could be evaluated with a constant amount of in turn provides a tantalizing way to distinguish quantum mechanics from incorrect theories of physics; the statement communication complexity is nontrivial has even been conjectured to be a concise information-theoretic axiom for characterizing quantum mechanics. We directly address the viability of that perspective with two results. First, we exhibit a nonlocal game such that complexity collapses in any physical theory whose maximal winning probability exceeds the quantum value. Second, we consider the venerable CHSH game that initiated this line of inquiry. In that case, the quantum value is about 0.85 but it is known that a winning probability of approximately 0.91 would collapse complexity. We show that the 0.91 result is the best possible using a large class of proof strategies, suggesting that the complexity axiom is insufficient for characterizing CHSH correlations. Both results build on new insights about reliable classical computation. The first exploits our formalization of an equivalence between amplification and reliable computation, while the second follows from a rigorous determination of the threshold for reliable computation with formulas of noise-free XOR gates and $\varepsilon$-noisy AND gates.