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Computational complexity theory contains a corpus of theorems and conjecturesregarding the time a Turing machine will need to solve certain types ofproblems as a function of the input size. Nature {\em need not} be a Turingmachine and, thus, these theorems do not apply directly to it. But {\emclassical simulations} of physical processes are programs running on Turingmachines and, as such, are subject to them. In this work, computationalcomplexity theory is applied to classical simulations of systems performing anadiabatic quantum computation (AQC), based on an annealed extension of thedensity matrix renormalization group (DMRG). We conjecture that thecomputational time required for those classical simulations is controlledsolely by the {\em maximal entanglement} found during the process. Thus, lowerbounds on the growth of entanglement with the system size can be provided. Insome cases, quantum phase transitions can be predicted to take place in certaininhomogeneous systems. Concretely, physical conclusions are drawn from theassumption that the complexity classes {\bf P} and {\bf NP} differ. As aby-product, an alternative measure of entanglement is proposed which, viaChebyshev's inequality, allows to establish strict bounds on the requiredcomputational time.

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