A no-go theorem for sequential and retro-causal hidden-variable theories based on computational complexity

TL;DR

This paper introduces a new no-go theorem based on computational complexity, showing that certain hidden-variable theories cannot replicate quantum circuit sampling unless they are as computationally hard as quantum computing.

Contribution

It develops an independent complexity-based no-go theorem for sequential and retro-causal hidden-variable theories, expanding beyond Bell's theorem.

Findings

Shows classical sampling problems are computationally easier than quantum circuits

Proves that certain hidden-variable theories cannot model quantum correlations efficiently

Suggests some retro-causal models might evade the no-go constraints

Abstract

The celebrated Bell's no-go theorem rules out the hidden-variable theories falling in the hypothesis of locality and causality, by requiring the theory to model the quantum correlation-at-a-distance phenomena. Here I develop an independent no-go theorem, by inspecting the ability of a theory to model quantum \emph{circuits}. If a theory is compatible with quantum mechanics, then the problems of solving its mathematical models must be as hard as calculating the output of quantum circuits, i.e., as hard as quantum computing. Rigorously, I provide complexity classes capturing the idea of sampling from sequential (causal) theories and from post-selection-based (retro-causal) theories; I show that these classes fail to cover the computational complexity of sampling from quantum circuits. The result is based on widely accepted conjectures on the superiority of quantum computers over classical ones. The result represents a no-go theorem that rules out a large family of sequential and post-selection-based theories. I discuss the hypothesis of the no-go theorem and the possible ways to circumvent them. In particular, I discuss the Schulman model and its extensions, which is retro-causal and is able to model quantum correlation-at-a-distance phenomena: I provides clues suggesting that it escapes the hypothesis of the no-go theorem.