Extracting Dynamical Equations from Experimental Data is NP Hard
Abstract
The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a byproduct of this work, we give complexitytheoretic answers to both the quantum and classical embedding problems, two longstanding open problems in mathematics (the classical problem, in particular, dating back over 70 years).
 Publication:

Physical Review Letters
 Pub Date:
 March 2012
 DOI:
 10.1103/PhysRevLett.108.120503
 arXiv:
 arXiv:1005.0005
 Bibcode:
 2012PhRvL.108l0503C
 Keywords:

 03.67.Ac;
 03.65.Ud;
 03.65.Yz;
 89.70.Eg;
 Quantum algorithms protocols and simulations;
 Entanglement and quantum nonlocality;
 Decoherence;
 open systems;
 quantum statistical methods;
 Computational complexity;
 Quantum Physics
 EPrint:
 For mathematical details, see arXiv:0908.2128[mathph]. v2: final version, accepted in Phys. Rev. Lett