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The quantum ising model for perfect matching and solving it with variational quantum eigensolver

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Science China(Information Sciences) 2024年第9期67卷 306-320页

作者： Qilin ZHENG Pingyu ZHU Chao WU Miaomiao YU Weihong LUO Ping XU Institute for Quantum Information and State Key Laboratory of High Performance Computing College of Computer Science and Technology National University of Defense Technology Hefei National Laboratory

Obtaining all perfect matchings of a graph is a tough problem in graph theory, and its complexity belongs to the #P-Complete class. The problem is closely related to combinatorics, marriage matching problems, dense subgraphs, the Gaussian boson sampling, chemical molecular structures, and dimer *** this paper, we propose a quadratic unconstrained binary optimization formula of the perfect matching problem and translate it into the quantum ising model. We can obtain all perfect matchings by mapping them to the ground state of the quantum ising Hamiltonian and solving it with the variational quantum eigensolver. Adjusting the model's parameters can also achieve the maximum or minimum weighted perfect matching. The experimental results on a superconducting quantum computer of the Origin Quantum Computing Technology Company show that our model can encode 2~n dimensional optimization space with only O（n） qubits consumption and achieve a high success probability of the ground state corresponding to all perfect matchings. In addition, the further simulation results show that the model can support a scale of more than 14 qubits, effectively resist the adverse effects of noise, and obtain a high success probability at a shallow variational depth. This method can be extended to other combinatorial optimization problems.

关键词： perfect matching ising model quantum Hamiltonian variational quantum eigensolver quadratic unconstrained binary optimization

ising model on evolution networks and its application on opinion formation

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Chinese Physics B 2014年第6期23卷 613-620页

作者：朱小龙张海天桑建平黄胜友邹宪武 School of Physics and Information Engineering Jianghan University School of Physics and Technology Wuhan University Hubei Air Traffic Management Sub-Bureau of CAAC

Many phenomena show that in a favorable circumstance an agent still has an updating possibility, and in an unfavor- able circumstance an agent also has a possibility of holding its own state and reselecting its neighbors. To describe this kind of phenomena an ising model on evolution networks was presented and used for consensus formation and separation of opinion groups in human population. In this model the state-holding probability p and selection-rewiring probability q were introduced. The influence of this mixed dynamics of spin flips and network rewiring on the ordering behavior of the model was investigated, p hinders ordering of opinion networks and q accelerates the dynamical process of networks. Influence of q on the ordering and separating stems from its effect on average path length of networks.

关键词： opinion formation ising model evolution networks

ising model on an Infinite Ladder Lattice

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Communications in Theoretical Physics 2007年第3X期48卷 553-562页

作者： GAO Xing-Ru CCAST (World Laboratory) P.O. Box 8730 Beijing 100080 China Department of Physics and Institute of Theoretical Physics Beijing Normal University Beijing 100875 China Fundamental Teaching Department Beijing Union University Beijing 100101 China

In this paper we propose an ising model on an infinite ladder lattice, which is made of two infinite ising spin chains with interactions. It is essentially a quasi-one-dimessional ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.

关键词： ising model general transfer matrix phase transition Glauber dynamics critical slow down

Mathematical structure of the three-dimensional(3D) ising model

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Chinese Physics B 2013年第3期22卷 25-39页

作者：张志东 Shenyang National Laboratory for Materials Science Institute of Metal ResearchChinese Academy of Sciences

An overview of the mathematical structure of the three-dimensional（3D） ising model is given from the points of view of topology,algebra,and *** analyzing the relationships among transfer matrices of the 3D ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D ising model.1） The complex quaternion basis constructed for the 3D ising model naturally represents the rotation in a（3＋1）-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2） A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D ising model.3） A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4） The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φ*** relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in *** conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are *** analyticity in β=1/（kBT） of both the hard-core and the ising models has been proved only for β〉0,not for β=*** the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D ising model.

关键词： ising model topology algebra geometry

Critical behaviour of the ferromagnetic ising model on a triangular lattice

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Chinese Physics B 2009年第7期18卷 2696-2702页

作者：罗志环 Loan Mushtaq 刘岩林健荣 Department of Applied Physics South China Agricultural University Department of Physics Sun Yet-Sen University International School Jinan University Engineering College South China Agricultural University

We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L=8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at Тc=3.6403（2）. A convincing finite-size scaling analysis of the model yields ν=0.9995（21）, β/ν=0.12400（17）, γ/v=1.75223（22）, γ^1/ν=1.7555（22）, α/ν=0.00077（420）（scaling） and α/ν=0.0010（42）（hyperscaling）. The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.

关键词： ising model triangular lattice Monte Carlo method

Dynamic phase transition of ferroelectric nanotube described by a spin-1/2 transverse ising model

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Chinese Physics B 2021年第2期30卷 227-231页

作者： Chundong Wang Ying Wu Yulin Cao Xinying Xue College of Life Science Tarim UniversityAlaer 843300China School of Optical and Electronic Information Wuhan National Laboratory for OptoelectronicsHuazhong University of Science and TechnologyWuhan 430074China Physics Laboratory Industrial Training CenterShenzhen PolytechnicShenzhenChina Department of Physics College of ScienceShihezi UniversityXinjiang 832003China

The dynamic phase transition properties for ferroelectric nanotube under a spin-1/2 transverse ising model are studied under the effective field theory(EFT)with *** temperature effects on the pseudo-spin systems are unveiled in three-dimensional(3-D)and two-dimensional(2-D)phase ***,the dynamic behaviors of exchange interactions on the 3-D and 2-D phase transitions under high temperature are *** results present that it is hard to obtain pure ferroelectric phase under high temperature;that is,the vibration of orderly pseudo-spins cannot be eliminated completely.

关键词： ferroelectric nanotube three-dimensional(3-D)phase diagram ising model dynamic phase transitions

An extended chain ising model and its Glauber dynamics

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Chinese Physics B 2012年第2期21卷 504-508页

作者： Zhao Xing-Yu Huang Xin-Ru Fan Xiao-Hui Huang Yi-Neng Physical College and Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matters Yili Normal UniversityYining 835000China Physical College and National Laboratory of Solid State Microstructures Nanjing UniversityNanjing 210093China Department of Physics Shihezi UniversityShihezi 832003China

It was first proposed that an extended chain ising （ECI） model contains the ising chain model, single spin doublewell potentials and a pure phonon heat bath of a specific energy exchange with the spins. The extension method is easy to apply to high dimensional cases. Then the single spin-flip probability （rate） of the ECI model is deduced based on the Boltzmann principle and general statistical principles of independent events and the model is simplified to an extended chain Glauber-ising （ECGI） model. Moreover, the relaxation dynamics of the ECGI model were simulated by the Monte Carlo method and a comparison with the predictions of the special chain Clauber-ising （SCGI） model was presented. It was found that the results of the two models are consistent with each other when the ising chain length is large enough and temperature is relative low, which is the most valuable case of the model applications. These show that the ECI model will provide a firm physical base for the widely used single spin-flip rate proposed by Glauber and a possible route to obtain the single spin-flip rate of other form and even the multi-spin-flip rate.

关键词： ising model Glauber dynamics single spin-flip probability

Comment on 'Mathematical structure of the three-dimensional (3D) ising model'

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Chinese Physics B 2013年第8期22卷 356-360页

作者： Jacques H. H. Perk Department of Physics Oklahoma State UniversityStillwater OK 74078-3072 USA

The review paper by Zhang Zhi-Dong （Zhang Z D 2013 Chin. Phys. B 22 030513, arXiv：1305.2956） contains many errors and is based on several earlier works that are equally wrong.

关键词： ising model Lie algebra series analysis thermodynamic limit

Exact Calculation of Antiferromagnetic ising model on an Inhomogeneous Surface Recursive Lattice to Investigate Thermodynamics and Glass Transition on Surface/Thin Film

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Communications in Theoretical Physics 2017年第1期67卷 111-126页

作者：黄然 Purushottam D.Gujrati State Key Laboratory of Microbial Metabolism and School of Life Sciences and Biotechnology Shanghai Jiao Tong University Department of Physics University of Akron

An inhomogeneous 2-dimensional recursive lattice formed by planar elements has been designed to investigate the thermodynamics of ising spin system on the surface/thin film. The lattice is constructed as a hybrid of partial Husimi square lattice representing the bulk and 1D single bonds representing the surface. Exact calculations can be achieved with the recursive property of the lattice. The model has an anti-ferromagnetic interaction to give rise to an ordered phase identified as crystal, and a solution with higher energy to represent the amorphous/metastable *** energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function. By analyzing the free energies and entropies of the crystal and supercooled liquid state,we are able to identify the melting and ideal glass transition on the surface. The results show that due to the variation of coordination number, the transition temperatures on the surface decrease significantly compared to the bulk *** calculation qualitatively agrees with both experimental and simulation works on the thermodynamics of surfaces and thin films conducted by others. Interactions between particles farther than the nearest neighbor distance are taken into consideration, and their effects are investigated.

关键词： critical temperature ideal glass transition ising model Husimi lattice surface thin film