Dual Correspondence Between Classical Spin Models and Quantum States
This workshop delves into the intriguing connection between classical spin models and quantum CSS states. Explore topics like computational complexity, quantum computing, and quantum information theory meeting statistical physics. Discover the applications in measurement-based quantum computing and the classical simulatability of MBQC on quantum entangled states.
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Dual correspondence between classical spin models and quantum CSS states Mohammad Hossein Zarei Shiraz University Iran Workshop on quantum science and technology, ICTP, 15 September 2017
Classical spin models in statistical mechanics: = ???? + ???????+ ??????????+... Computational complexity Spin glasses Critical behavior m NP P 2D Ising model T T C
Quantum computing: Quantum computing Quantum measurement memory Unitary evolution ? QC CC Different computational models and an ocean of open problems
Quantum information theory meets statistical physics: Quantum information theory Statistical mechanics A new field of physics with an ocean of open problems A well-stablished field of physics
partition function of classical spin models entangled states = H JS S i j 1 i j JS S i j = kT Z e , { } S i ??(??? |0 + ? ?? |?? = ?? |??+ ?? ;??= 1?? |? = |1 ) ?? M. Van den Nest, W. Dur, H. J. Briegel, Phys.Rev. Lett. 98, 117207 (2007)
= ( ) J ( ) J Z G G An entangled state A product state = Encoding coupling constants Encoding interaction pattern
Applications: Measurement-based quantum computing and completeness of classical spin models M. Van den Nest, W. Dur, H. J. Briegel, Phys. Rev. Lett. 100, 110501 (2008) V. Karimipour, M. H. Zarei, Phys. Rev. A 85, 032316 (2012) V. Karimipour, M. H. Zarei, Phys. Rev. A 86, 052303 (2012). Classical simulatability of MBQC on quantum entangled states S. Bravyi, R. Raussendorf, Phys. Rev. A 76, 022304 (2007) H. Bombin, M. A. Martin-Delgado, Phys. Rev. A77, 042322 (2008) New complexity classes of classical spin models G. De las Cuevas, W. Dr, M. Van den Nest and M. A.Martin-Delgado, New J. Phys. 13:093021 (2011) Error threshold of topological quantum codes and phase transition of spin glasses E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002).
Which classical model a quantum state is mapped to? It is not a simple problem. Corresponding quantum state Classical model GHZ states 1D Ising model 1D cluster states 1D Ising model in a magnetic field 2D Kitaev s toric codes 2D Ising model 2D Topological color code Ising-like model with three-body interactions 2D cluster states 2D Ising model in a magnetic field
Hypergraphs and a dual correspondence Graphs are useful for encoding two-body interactions. Hypergraphs are a better candidate for encoding many-body interactions. Graph(?) Hypergraph(?)
Dual correspondence: Mapping all Calderbank-Shor-Steane states to hypergraphs: An entangled state which is stabilized by X-type and Z-type Pauli operators Mapping all classical spin models to hypergraphs ZH(J) = (J)| CSS H A quantum CSS state on the ? Partition function of a classical model on the ?
What we can do: Considering different well-known models by mapping to hypergraphs: To find new complete models To study simulatability of MBQC
What about more general concepts: Criticality ZH(J) = (J)| CSS H ? Probability of robustness against noise Partition function diverges at a critical temperature critical robustness
Critical robustness and topological codes: Corresponding quantum state Classical model Kitaev s toric code on an arbitrary graph Ising model on the same graph Color code D-colexes Ising-like model on D-simplicial lattices
Summary: An explicit correspondence between classical spin models and quantum CSS states Critical robustness for topological CSS states M. H. Zarei, A. Montakhab, Dual correspondence between classical spin models and quantum CSS states , submitted to PRL Quantum CSS code on hypergraphs M. H. Zarei, Strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems , submitted to PRB
Thanks for your attention Thanks for your attention
