We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g., energy) and include states with diverging correlation length and unbounded multiparticle entanglement. As a demonstration, we apply our approach to the Ising model on 1D, 2D, and 3D square lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g., energy) and include states with diverging correlation length and unbounded multiparticle entanglement. As a demonstration, we apply our approach to the Ising model on 1D, 2D, and 3D square lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.

arXiv:quant-ph/0602230Physical Review LettersSep 8, 200665 citations

Extension of ground state variational methods

H. G. Miller, R. M. Dreizler

Nuclear Physics AMar 1, 197920 citations

Efficient algorithm for approximating one-dimensional ground states

Dorit Aharonov, Itai Arad, Sandy Irani

The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well-defined conditions. More precisely, our algorithm finds a matrix product state of bond dimension $D$ whose energy approximates the minimal energy such states can achieve. The running time is exponential in $D$ , and so the algorithm can be considered tractable even for $D$ , which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well-defined conditions. More precisely, our algorithm finds a matrix product state of bond dimension $D$ whose energy approximates the minimal energy such states can achieve. The running time is exponential in $D$, and so the algorithm can be considered tractable even for $D$, which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.

arXiv:0910.5055Physical Review AJul 16, 201022 citations

Variational Methods for Quantum Systems

Philipp O. J. Scherer

Graduate texts in physicsJan 1, 20171 citation

A numerical method for finding the ground states of one-dimensional systems

Kevin Hood

Journal of Computational PhysicsJul 1, 19904 citations

Polynomial-time degenerate ground state approximation of gapped 1D Hamiltonians.

Christopher T. Chubb, Steven T. Flammia

In spite of a plethora of no-go results that prevent classically efficient simulation of quantum systems in general, heuristic methods[1] have seen great success, proving typically efficient in practice. The downside of these heuristic methods is that we lack a rigorous understanding of how, and for which systems, they fail[2]. Provably efficient algorithms – even if practically surpassed by heuristic methods – are of key importance for understanding the true worst-case complexity of simulation. The cutting edge for provably efficient simulation is the ground state approximation of 1D spin systems. Under the assumption of a gap, there exists a provably efficient algorithm[3, 4] for approximation of a unique ground state of such systems; we present ongoing work to extend this algorithm to work for degenerate systems. This result comes in the wake of a recent degenerate extension of the area law[5], an important structural bound on the complexity of ground states. In practice this algorithm is no match for heuristic analogues such as the density matrix renormalisation group. However, it does add to the increasing understanding of the systems for which provably efficient simulation is attainable, and might hopefully lead to a future algorithm that is both provably and practically efficient.In spite of a plethora of no-go results that prevent classically efficient simulation of quantum systems in general, heuristic methods[1] have seen great success, proving typically efficient in practice. The downside of these heuristic methods is that we lack a rigorous understanding of how, and for which systems, they fail[2]. Provably efficient algorithms – even if practically surpassed by heuristic methods – are of key importance for understanding the true worst-case complexity of simulation. The cutting edge for provably efficient simulation is the ground state approximation of 1D spin systems. Under the assumption of a gap, there exists a provably efficient algorithm[3, 4] for approximation of a unique ground state of such systems; we present ongoing work to extend this algorithm to work for degenerate systems. This result comes in the wake of a recent degenerate extension of the area law[5], an important structural bound on the complexity of ground states. In practice this algorithm is no match for heuristic analogues such as the density matrix renormalisation group. However, it does add to the increasing understanding of the systems for which provably efficient simulation is attainable, and might hopefully lead to a future algorithm that is both provably and practically efficient.

Jan 1, 20140 citations

D

whose energy approximates the minimal energy such states can achieve. The running time is exponential in

D

, and so the algorithm can be considered tractable even for

D

, which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.