Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3592110.22363/2658-4670-2023-31-3-247-259Research ArticleHamiltonian simulation in the Pauli basis of multi-qubit clusters for condensed matter physicsAndréEduardo L.<p>PhD student, Department of Applied Physics, Tver State University</p>lumonansoni@gmail.comhttps://orcid.org/0000-0002-0697-1639TsirulevAlexander N.<p>Doctor of Sciences in Physics and Mathematics, Professor of the Department of General Mathematics and Mathematical Physics</p>tsirulev.an@tversu.ruhttps://orcid.org/0000-0003-4168-3613Agostinho Neto UniversityTver State University1209202331324725912092023Copyright © 2023, André E.L., Tsirulev A.N.2023<p style="text-align: justify;">We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a <span class="math inline">\(n\)</span>-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval <span class="math inline">\(0.1\!-\!2\:\!\text{K}\)</span>, which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.</p>Hamiltonian simulationcluster of qubitsoperator exponentialthermal environmentGibbs statethermodynamic propertiesмоделирование квантовых гамильтониановкластер кубитовоператорная экспонентатермостатсостояние Гиббсатермодинамические свойства[J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum, vol. 2, p. 79, 2018. DOI: 10.22331/q-2018-08-06-79.][S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, “Quantum computational chemistry,” Reviews of Modern Physics, vol. 92, no. 1, 2020. DOI: 10.1103/revmodphys.92.015003.][G. H. Low and I. L. Chuang, “Hamiltonian simulation by qubitization,” Quantum, vol. 3, p. 163, 2019. DOI: 10.22331/q-2019-07-12-163.][B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter. Springer New York, 2019. DOI: 10.1007/9781-4939-9084-9.][L. Bassman, M. Urbanek, M. Metcalf, J. Carter, A. F. Kemper, and W. A. de Jong, “Simulating quantum materials with digital quantum computers,” Quantum Science and Technology, vol. 6, no. 4, p. 043002, 2021. DOI: 10.1088/2058-9565/ac1ca6.][D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, “Simulating Hamiltonian dynamics with a truncated Taylor series,” Physical Review Letters, vol. 114, p. 090502, 9 2015. DOI: 10.1103/ PhysRevLett.114.090502.][D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, “Exponential improvement in precision for simulating sparse Hamiltonians,” in Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, New York, NY, USA, 2014, pp. 283-292. DOI: 10.1145/2591796.2591854.][I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Reviews of Modern Physics, vol. 86, pp. 153-185, 1 2014. DOI: 10.1103/RevModPhys.86.153.][G. L. Deçordi and A. Vidiella-Barranco, “Two coupled qubits interacting with a thermal bath: A comparative study of different models,” Optics Communications, vol. 387, pp. 366-376, 2017. DOI: 10.1016/j.optcom. 2016.10.017.][C. Boudreault, H. Eleuch, M. Hilke, and R. MacKenzie, “Universal quantum computation with symmetric qubit clusters coupled to an environment,” Physical Review A, vol. 106, no. 6, p. 062610, 2022. DOI: 10.1103/PhysRevA.106.062610.][T. Menke et al., “Demonstration of tunable three-body interactions between superconducting qubits,” Physical Review Letters, vol. 129, no. 22, p. 220501, 2022. DOI: 10.1103/PhysRevLett.129.220501.][V. Verma and M. Sisodia, “Two-way quantum communication using four-qubit cluster state: Mutual exchange of quantum information,” Modern Physics Letters A, vol. 37, no. 04, p. 2250020, 2022. DOI: 10.1142/S0217732322500201.][J. Von Neumann, “Proof of the ergodic theorem and the H-theorem in quantum mechanics: translation in English,” European Physical Journal H, vol. 35, pp. 201-235, 2010.][A. Kossakowski, A. Frigerio, V. Gorini, and M. Verri, “Quantum detailed balance and KMS condition,” Communications in Mathematical Physics, vol. 57, no. 2, pp. 97-110, 1977.][G. Bulnes Cuetara, M. Esposito, and G. Schaller, “Quantum thermodynamics with degenerate eigenstate coherences,” Entropy, vol. 18, no. 12, 2016. DOI: 10.3390/e18120447.][A. Short and T. Farrelly, “Quantum equilibration in finite time,” New Journal of Physics, vol. 14, no. 1, p. 013063, 2012. DOI: 10.1088/13672630/14/1/013063.][M. A. Novotny, F. Jin, S. Yuan, S. Miyashita, H. De Raedt, and K. Michielsen, “Quantum decoherence and thermalization at finite temperature within the canonical-thermal-state ensemble,” Phys. Rev. A, vol. 93, no. 3, p. 032110, 2016. DOI: 10.1103/PhysRevA.93.032110]