Physical consequences of P$\neq$NP and the DMRG-annealing conjecture

TL;DR

This paper explores the implications of P≠NP on classical simulations of quantum systems, proposing that entanglement growth bounds computational complexity and predicting quantum phase transitions.

Contribution

It introduces a conjecture linking classical simulation time to maximal entanglement and applies complexity theory to quantum phase transition predictions.

Findings

Lower bounds on entanglement growth can predict simulation difficulty.

Quantum phase transitions can be inferred from complexity assumptions.

An alternative entanglement measure provides bounds on computational time.

Abstract

Computational complexity theory contains a corpus of theorems and conjectures regarding the time a Turing machine will need to solve certain types of problems as a function of the input size. Nature {\em need not} be a Turing machine and, thus, these theorems do not apply directly to it. But {\em classical simulations} of physical processes are programs running on Turing machines and, as such, are subject to them. In this work, computational complexity theory is applied to classical simulations of systems performing an adiabatic quantum computation (AQC), based on an annealed extension of the density matrix renormalization group (DMRG). We conjecture that the computational time required for those classical simulations is controlled solely by the {\em maximal entanglement} found during the process. Thus, lower bounds on the growth of entanglement with the system size can be provided. In some cases, quantum phase transitions can be predicted to take place in certain inhomogeneous systems. Concretely, physical conclusions are drawn from the assumption that the complexity classes {\bf P} and {\bf NP} differ. As a by-product, an alternative measure of entanglement is proposed which, via Chebyshev's inequality, allows to establish strict bounds on the required computational time.