Criticality, the area law, and the computational power of PEPS

2006/01/27 by Frank Verstraete, F. Verstraete, M. M. Wolf +3 · 1 voice · 65 citations
Computer Science · Physics and Astronomy · #Quantum Computing Algorithms and Architecture #Quantum and electron transport phenomena #Quantum many-body systems #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevlett.96.220601

published as Phys. Rev. Lett. 96, 220601 (2006).

arxiv created 2006/01/27 · openalex publication_date 2006/06/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit an enormously rich structure including states with critical and topological properties as well as resonating valence bond states. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS within the same class as the cluster state, which can serve as computational resources for the solution of NP-hard problems.

Citations

• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Fault-tolerant quantum computation by anyons

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Discussions

• In the paper Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States , Verstraete et al give the following definition of the PEPS of a ground state of the toric code: [stackexchange, 2 points]

Related

• Fermionic projected entangled pair states
• 2D Hamiltonians with exotic bipartite and topological entanglement
• PEPS as unique ground states of local Hamiltonians
• Entanglement Entropy and Entanglement Spectrum for Two-Dimensional Classical Spin Configuration
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