Computational Difficulty of Global Variations in the Density Matrix Renormalization Group
Abstract
The density matrix renormalization group approach is arguably the most successful method to numerically find ground states of quantum spin chains. It amounts to iteratively locally optimizing matrixproduct states, aiming at better and better approximating the true ground state. To date, both a proof of convergence to the globally best approximation and an assessment of its complexity are lacking. Here we establish a result on the computational complexity of an approximation with matrixproduct states: The surprising result is that when one globally optimizes over several sites of local Hamiltonians, avoiding local optima, one encounters in the worst case a computationally difficult NPhard problem (hard even in approximation). The proof exploits a novel way of relating it to binary quadratic programming. We discuss intriguing ramifications on the difficulty of describing quantum manybody systems.
 Publication:

Physical Review Letters
 Pub Date:
 December 2006
 DOI:
 10.1103/PhysRevLett.97.260501
 arXiv:
 arXiv:quantph/0609051
 Bibcode:
 2006PhRvL..97z0501E
 Keywords:

 03.67.Lx;
 02.60.Pn;
 64.60.Ak;
 75.10.Pq;
 Quantum computation;
 Numerical optimization;
 Renormalizationgroup fractal and percolation studies of phase transitions;
 Spin chain models;
 Quantum Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 5 pages, 1 figure, RevTeX, final version