Statistical mechanical models and topological color codes
Open accessAbstract
We find that the overlap of a topological quantum color-code state, representing a quantum memory, with a factorized state of qubits can be written as the partition function of a three-body classical Ising model on triangular or Union Jack lattices. This mapping allows us to test that different computational capabilities of color codes correspond to qualitatively different universality classes of their associated classical spin models. By generalizing these statistical mechanical models for arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-based quantum computation with a color-code state and we find that their classical simulability remains an open problem. We complement the measurement-based computation with the construction of a cluster state that yields the topological color code and this also gives the possibility to represent statistical models with external magnetic fields.
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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 [99]
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 [99]
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Cites
Fault-tolerant quantum computation by anyons
Kitaev, A. Y. (2003) · Annals of Physics · doi:10.1016/s0003-4916(02)00018-0 · Cited by 7,238 · see citing works
Raussendorf, R., & Briegel, H. J. (2001) · Physical Review Letters · doi:10.1103/physrevlett.86.5188 · Cited by 4,477 · see citing works
Quantum Field Theory of Many-Body Systems
Wen, X.-G. (2007) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199227259.001.0001 · Cited by 2,008 · see citing works
Persistent Entanglement in Arrays of Interacting Particles
Briegel, H. J., & Raussendorf, R. (2001) · Physical Review Letters · doi:10.1103/physrevlett.86.910 · Cited by 1,950 · see citing works
Statistics of the Two-Dimensional Ferromagnet. Part I
Kramers, H. A., & Wannier, G. H. (1941) · Physical Review · doi:10.1103/physrev.60.252 · Cited by 1,819 · see citing works
Dennis, E., Kitaev, A., Landahl, A., & Preskill, J. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1499754 · Cited by 1,768 · see citing works
Dennis, E., Kitaev, A., Landahl, A., & Preskill, J. (2001) · CERN Document Server (European Organization for Nuclear Research) · Cited by 1,485 · see citing works
String-net condensation: A physical mechanism for topological phases
Levin, M., & Wen, X.-G. (2005) · Physical Review B · doi:10.1103/physrevb.71.045110 · Cited by 1,455 · see citing works
On the computational complexity of Ising spin glass models
Barahona, F. (1982) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/15/10/028 · Cited by 1,303 · see citing works
The statistics of dimers on a lattice
Kasteleyn, P. (1961) · Physica · doi:10.1016/0031-8914(61)90063-5 · Cited by 1,281 · see citing works
Class of quantum error-correcting codes saturating the quantum Hamming bound
Gottesman, D. (1996) · Physical Review A · doi:10.1103/physreva.54.1862 · Cited by 1,227 · see citing works
Topological Quantum Distillation
Bombin, H., & Martin-Delgado, M. A. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.180501 · Cited by 700 · see citing works
Self-citationOn the Theory of the Ising Model of Ferromagnetism
Newell, G. F., & Montroll, E. W. (1953) · Reviews of Modern Physics · doi:10.1103/revmodphys.25.353 · Cited by 488 · see citing works
Exact Solution of an Ising Model with Three-Spin Interactions on a Triangular Lattice
Baxter, R. J., & Wu, F. Y. (1973) · Physical Review Letters · doi:10.1103/physrevlett.31.1294 · Cited by 319 · see citing works
Topological Computation without Braiding
Bombín, H., & Martin-Delgado, M. A. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.160502 · Cited by 191 · see citing works
Self-citationOptimal resources for topological two-dimensional stabilizer codes: Comparative study
Bombin, H., & Martin-Delgado, M. A. (2007) · Physical Review A · doi:10.1103/physreva.76.012305 · Cited by 178 · see citing works
Self-citationExact topological quantum order in D=3 and beyond: Branyons and brane-net condensates
Bombín, H., & Martin-Delgado, M. A. (2007) · Physical Review B · doi:10.1103/physrevb.75.075103 · Cited by 138 · see citing works
Self-citationIsing Model on a Triangular Lattice with Three-spin Interactions. I. The Eigenvalue Equation
Baxter, R., & Wu, F. (1974) · Australian Journal of Physics · doi:10.1071/ph740357 · Cited by 131 · see citing works
Homological error correction: Classical and quantum codes
Bombín, H., & Martin-Delgado, M. A. (2007) · Journal of Mathematical Physics · doi:10.1063/1.2731356 · Cited by 124 · see citing works
Self-citationHomological Error Correction: Classical and Quantum Codes
Bombín, H., & Martín-Delgado, M. A. (2006) · CERN Document Server (European Organization for Nuclear Research) · Cited by 74 · see citing works
Measurement-based quantum computation with the toric code states
Bravyi, S., & Raussendorf, R. (2007) · Physical Review A · doi:10.1103/physreva.76.022304 · Cited by 69 · see citing works
A self dual relation for an Ising model with triplet interactions
Wood, D. W., & Griffiths, H. P. (1972) · Journal of Physics C Solid State Physics · doi:10.1088/0022-3719/5/18/001 · Cited by 66 · see citing works
Completeness of the Classical 2D Ising Model and Universal Quantum Computation
Van den Nest, M., Dür, W., & Briegel, H. J. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.110501 · Cited by 63 · see citing works
Spin-½ Lattice System: Group Structure and Duality Relation
Merlini, D., & Gruber, C. (1972) · Journal of Mathematical Physics · doi:10.1063/1.1665913 · Cited by 58 · see citing works
Classical Spin Models and the Quantum-Stabilizer Formalism
den Nest, M. V., Dür, W., & Briegel, H. J. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.117207 · Cited by 56 · see citing works
Exact solution of a two dimensional Ising model with pure 3 spin interactions
Hintermann, A., & Merlini, D. (1972) · Physics Letters A · doi:10.1016/0375-9601(72)90261-7 · Cited by 42 · see citing works
2D multipartite valence bond states in quantum anti-ferromagnets
Rico, E., & Briegel, H. J. (2008) · Annals of Physics · doi:10.1016/j.aop.2008.03.006 · Cited by 12 · see citing works
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Eisert, J., Cramer, M., & Plenio, M. B. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.277 · Cited by 2,873 · see citing works
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Briegel, H. J., Browne, D. E., Dür, W., Raussendorf, R., & Van den Nest, M. (2009) · Nature Physics · doi:10.1038/nphys1157 · Cited by 1,090 · see citing works
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Devitt, S. J., Munro, W. J., & Nemoto, K. (2013) · Reports on Progress in Physics · doi:10.1088/0034-4885/76/7/076001 · Cited by 775 · see citing works
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de Mello, D. O., Schäffner, D., Werkmann, J., Preuschoff, T., Kohfahl, L., Schlosser, M., & Birkl, G. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.203601 · Cited by 165 · see citing works
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Ohzeki, M. (2009) · Physical Review E · doi:10.1103/physreve.80.011141 · Cited by 40 · see citing works
Khan, M. N., Teo, J. C. Y., & Hughes, T. L. (2014) · Physical Review B · doi:10.1103/physrevb.90.235149 · Cited by 36 · see citing works
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Topological color codes on Union Jack lattices: a stable implementation of the whole Clifford group
Katzgraber, H. G., Bombín, H., Andrist, R. S., & Martin-Delgado, M. A. (2010) · Physical Review A · doi:10.1103/physreva.81.012319 · Cited by 31 · see citing works
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Cuevas, G. D. L. (2013) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/0953-4075/46/24/243001 · Cited by 8 · see citing works
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Dür, W., & den Nest, M. V. (2011) · Physical Review Letters · doi:10.1103/physrevlett.107.170402 · Cited by 6 · see citing works
Classical spin systems and the quantum stabilizer formalism: General mappings and applications
Hübener, R., den Nest, M. V., Dür, W., & Briegel, H. J. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3190486 · Cited by 6 · see citing works
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Miyazaki, R. (2013) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.82.094001 · Cited by 5 · see citing works
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Pecorari, L., & Pupillo, G. (2025) · Physical Review A · doi:10.1103/mgkt-ctv8 · Cited by 4 · see citing works
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Zarei, M. H., & Ramezanpour, A. (2019) · Physical Review A · doi:10.1103/physreva.100.062313 · Cited by 4 · see citing works
Entangled multi-knot lattice model of anyon current
Si, T. (2019) · Chinese Physics B · doi:10.1088/1674-1056/28/4/040501 · Cited by 4 · see citing works
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Zhang, Y. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0801.2561 · Cited by 4 · see citing works
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den Nest, M. V., & Dür, W. (2014) · Physical Review A · doi:10.1103/physreva.89.012334 · Cited by 2 · see citing works
Frustration and Transition Point of Ising Spin Glasses
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Quantum error correction and fault tolerance: A comprehensive tutorial
Spencer, D. J., Jain, S. P., Tanggara, A., Sun, Z., Haug, T., Khu, D., & Bharti, K. (2026) · arXiv
- Authors
- Bombin, H., & Martin-Delgado, M. A.
- Affiliations
- Universidad Complutense de Madrid (ES)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2008-04-24
- Volume
- 77
- Issue
- 4
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1992610563
- Cited by
- 39
- References
- 27
- Citations by year
- 2025: 2 · 2022: 1 · 2021: 1 · 2020: 1 · 2019: 5 · 2018: 2 · 2017: 1 · 2016: 1 · 2015: 2 · 2014: 2 · 2013: 6 · 2012: 1
- FWCI
- 7.40
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (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 › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum memory, partition function, universality class, classical spin systems, measurement-based quantum computation, classical simulability, cluster states, external magnetic field, inhomogeneous coupling
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