Completeness of the Classical 2D Ising Model and Universal Quantum Computation
Open accessAbstract
We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.
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The multiparticle entangled state |ψ_L> encodes the interaction pattern and is a graph state.
“The multiparticle entangled state | ψ L 〉 95 encodes the interaction pattern and is a graph state 21 (see Fig. 1).”
From Measurement-based quantum computation (2009) · cites this paper as [95]
The Ising model on an arbitrary lattice L can be mapped to an MQC model in which the entangled resource is determined by the geometry of L.
“The connection between MQC and the Ising model is given by a mapping of the Ising model on an arbitrary lattice L to an MQC model where the entangled resource is determined by the geometry of L ,”
From Measurement-based quantum computation (2009) · cites this paper as [95]
The efficient classical simulation of MQC on stabilizer states with bounded tree-width yields a novel classical algorithm to efficiently calculate the partition function on (inhomogeneous) q-state spin models on tree-like graphs with or without magnetic fields.
“In turn, the efficient classical simulation of MQC on stabilizer states with bounded tree-width 94 , as demonstrated in 91 , yields a novel classical algorithm to efficiently calculate the partition function on (inhomogenous) q -state spin models on tree-like-graphs with or without magnetic fields 95,96 .”
From Measurement-based quantum computation (2009) · cites this paper as [95]
The universality of 2D-cluster states was used to prove that the 2D Ising model with magnetic fields is 'complete' in the sense that the partition function of any q-state spin model on an arbitrary lattice can be expressed as a special instance of the partition function of the 2D Ising model in a complex parameter regime.
“For example, in 95 the universality of the 2D-cluster states was used to prove that the 2D Ising model with magnetic fields is 'complete' in the sense that the partition function of any q -state spin model on an arbitrary lattice can be expressed as a special instance of the partition function of the 2D Ising model (in a complex parameter regime).”
From Measurement-based quantum computation (2009) · cites this paper as [95]
These connections enable one to phrase statistical physics problems naturally in a quantum mechanical setting.
“For one, these connections enable one to phrase statistical physics problems naturally in a quantum mechanical setting 87,92,95,96,99-102 .”
From Measurement-based quantum computation (2009) · cites this paper as [95]
The partition function Z_L is identified with a quantum mechanical amplitude obtained as an overlap between two quantum states.
“This expression states that the partition function Z L is identified with a quantum mechanical amplitude which is obtained as an overlap between two quantum states 92,95,96 (see also 99 ).”
From Measurement-based quantum computation (2009) · cites this paper as [95]
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References 15 · Cited by 63
Research works printed but not linked: 1
- [4]F. Verstraete et al. , Phys. Rev. Lett. 96 , 220601 (2006); R. D. Somma, C. D. Batista, G. Ortiz, quant-ph/0609216; D. A. Lidar, New J. Phys. 6 , 167 (2004); D. A. Lidar and O. Biham, Phys. Rev. E 56 , 3661 (1997); D. Aharonov et al. , quant-ph/0702008. · arXiv
- Authors
- Van den Nest, M., Dür, W., & Briegel, H. J.
- Affiliations
- Austrian Academy of Sciences (AT)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2008-03-17
- Volume
- 100
- Issue
- 11
- Pages
- 110501-110501
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1971755522
- Cited by
- 63
- References
- 15
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 3 · 2023: 3 · 2022: 2 · 2021: 2 · 2020: 4 · 2019: 5 · 2018: 5 · 2017: 3 · 2016: 2 · 2015: 1 · 2014: 7 · 2013: 3 · 2012: 2
- FWCI
- 9.51
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Keywords
- two-dimensional Ising model, completeness, partition function, classical spin systems, Potts model, complex coupling, external fields, measurement-based quantum computation, cluster states, universal quantum computation
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