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Computational difficulty of global variations in the density matrix renormalization group.
1QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom.
Physical Review Letters
|February 7, 2007
Summary
The density matrix renormalization group (DMRG) method, used for quantum spin chains, can be computationally difficult. Optimizing matrix-product states globally can lead to NP-hard problems, impacting quantum many-body system descriptions.
Area of Science:
- Quantum Physics
- Computational Physics
- Condensed Matter Theory
Background:
- The density matrix renormalization group (DMRG) is a leading numerical method for finding ground states of quantum spin chains.
- DMRG iteratively optimizes matrix-product states to approximate the true ground state.
- A formal proof of convergence and complexity assessment for DMRG are currently lacking.
Purpose of the Study:
- To establish a result on the computational complexity of approximation with matrix-product states in DMRG.
- To investigate the worst-case complexity when globally optimizing over multiple sites.
Main Methods:
- Analysis of the computational complexity of approximating ground states using matrix-product states.
- Relating the optimization problem to binary quadratic programming.
Main Results:
- The global optimization of matrix-product states for local Hamiltonians can be an NP-hard problem in the worst case.
- This difficulty persists even for approximation, indicating inherent computational challenges.
Conclusions:
- The study reveals that approximating quantum many-body systems using DMRG can be computationally intractable.
- The findings have significant ramifications for the efficiency and limitations of describing complex quantum systems.
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