Exact Ising model simulation on a quantum computer

Alba Cervera-Lierta

Barcelona Supercomputing Center (BSC), Barcelona, Spain
Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain

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We present an exact simulation of a one-dimensional transverse Ising spin chain with a quantum computer. We construct an efficient quantum circuit that diagonalizes the Ising Hamiltonian and allows to obtain all eigenstates of the model by just preparing the computational basis states. With an explicit example of that circuit for $n=4$ spins, we compute the expected value of the ground state transverse magnetization, the time evolution simulation and provide a method to also simulate thermal evolution. All circuits are run in IBM and Rigetti quantum devices to test and compare them qualitatively.

In this work, it is presented a quantum circuit that diagonalizes exactly the 1D antiferromagnetic Ising Hamiltonian. With this circuit, it is possible to simulate time and temperature evolution since we have access to the whole model spectrum by just preparing a product state. As an example, it is provided an explicit circuit for four spins which is run in IBM's and Rigetti's quantum devices. As the Ising model can be solved analitically and this circuit can be extenend to higher number of qubits, it can also be used to benchmark quantum computers.

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[1] D. P. DiVincenzo, Fortschritte der Physik 48, 771 (2000).

[2] IBM Quantum Experience, https:/​/​www.research.ibm.com/​ibm-q/​.

[3] R. Smith, M. J. Curtis and W. J. Zeng, arXiv:1608.03355 [quant-ph] (2016).

[4] D. Alsina and J. I. Latorre, Phys. Rev. A 94, 012314 (2016).

[5] Y. Wang, Y. Li, Z. Yin and B. Zeng, npj Quantum Information 4, 46 (2018).

[6] J. S. Devitt, Phys. Rev. A 94, 032329 (2016).

[7] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).

[8] M. H. Kalos, Phys. Rev. 128, 1791 (1962).

[9] B.L. Hammond, W. A. Lester Jr. and P.J. Reynolds, MonteCarlo Methods in Ab Initio Quantum Chemistry, World Scientific, Singapore (1994).

[10] N. S. Blunt, T. W. Rogers, J. S. Spencer and W. M. C. Foulkes, Phys. Rev. B 89, 245124 (2014).

[11] R. Orús, Ann. Phys. 349, 117 (2014).

[12] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).

[13] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 64, 022319 (2001).

[14] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak and M. Troyer, Phys. Rev. A 92, 062318 (2015).

[15] Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy and S. Boixo, Phys. Rev. Appl. 9, 044036 (2018).

[16] B. Kraus, Phys. Rev. Lett. 107, 250503 (2011).

[17] M. Hebenstreit, D. Alsina, J. I. Latorre and B. Kraus, Phys. Rev. A 95, 052339 (2017).

[18] F. Verstraete, J. I. Cirac and J. I. Latorre, Phys. Rev. A 79, 032316 (2008).

[19] P. Schmoll and R. Orús, Phys. Rev. B 95, 045112 (2017).

[20] H. Bethe, Z. Phys. 71, 205 (1931).

[21] V. Murg, V. E. Korepin and F. Verstraete, Phys. Rev. B 86, 045125 (2012).

[22] E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407 (1961).

[23] S. Katsura, Phys. Rev. 127, 1508 (1962).

[24] P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).

[25] A. J. Ferris, Phys. Rev. Lett. 113, 010401 (2014).

[26] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge (1999).

[27] Device specifications: https:/​/​github.com/​Qiskit/​qiskit-backend-information/​tree/​master/​backends.

[28] Official announce of IBM ``Teach Me QISKit" award winnerhttps:/​/​www.ibm.com/​blogs/​research/​2018/​06/​teach-qiskit-winner/​.

[29] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin and H. Weinfurter, Phys. Rev. A 52 3457 (1995).

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[2] Sean Greenaway, Frédéric Sauvage, Kiran E. Khosla, and Florian Mintert, "Efficient assessment of process fidelity", Physical Review Research 3 3, 033031 (2021).

[3] Juan Carlos Criado, Michael Spannowsky, and Roman Kogler, "Quantum fitting framework applied to effective field theories", Physical Review D 107 1, 015023 (2023).

[4] Alejandro Sopena, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López, "Algebraic Bethe Circuits", Quantum 6, 796 (2022).

[5] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[6] Manoranjan Swain, Amit Rai, Bikash K. Behera, and Prasanta K. Panigrahi, "Experimental demonstration of the violations of Mermin’s and Svetlichny’s inequalities for W and GHZ states", Quantum Information Processing 18 7, 218 (2019).

[7] Francesco Tarantelli and Ettore Vicari, "Out-of-equilibrium dynamics arising from slow round-trip variations of Hamiltonian parameters across quantum and classical critical points", Physical Review B 105 23, 235124 (2022).

[8] Lindsay Bassman Oftelie, Katherine Klymko, Diyi Liu, Norm M. Tubman, and Wibe A. de Jong, "Computing Free Energies with Fluctuation Relations on Quantum Computers", Physical Review Letters 129 13, 130603 (2022).

[9] Erik Gustafson, Patrick Dreher, Zheyue Hang, and Yannick Meurice, "Indexed improvements for real-time trotter evolution of a (1 + 1) field theory using NISQ quantum computers", Quantum Science and Technology 6 4, 045020 (2021).

[10] Kübra Yeter-Aydeniz, George Siopsis, and Raphael C Pooser, "Scattering in the Ising model with the quantum Lanczos algorithm * ", New Journal of Physics 23 4, 043033 (2021).

[11] Guanlin Jian, Yuan Yang, Ze Liu, Zhen-Gang Zhu, and Zhengchuan Wang, "Towards simulating time evolution of specific quantum many-body system by lower counts of quantum gates", Europhysics Letters 141 1, 10003 (2023).

[12] Muhammad Asaduzzaman, Goksu Can Toga, Simon Catterall, Yannick Meurice, and Ryo Sakai, "Quantum simulation of the N -flavor Gross-Neveu model", Physical Review D 106 11, 114515 (2022).

[13] Lindsay Bassman Oftelie, Kuang Liu, Aravind Krishnamoorthy, Thomas Linker, Yifan Geng, Daniel Shebib, Shogo Fukushima, Fuyuki Shimojo, Rajiv K. Kalia, Aiichiro Nakano, and Priya Vashishta, "Towards simulation of the dynamics of materials on quantum computers", Physical Review B 101 18, 184305 (2020).

[14] Kübra Yeter-Aydeniz, Zachary Parks, Aadithya Nair Thekkiniyedath, Erik Gustafson, Alexander F. Kemper, Raphael C. Pooser, Yannick Meurice, and Patrick Dreher, "Measuring qubit stability in a gate-based NISQ hardware processor", Quantum Information Processing 22 2, 96 (2023).

[15] Nizar Ahami and Morad El Baz, "Thermal entanglement in a mixed spin Heisenberg XXX chain with DM interaction", International Journal of Quantum Information 19 05, 2150021 (2021).

[16] Fereshte Shahbeigi, Mahsa Karimi, and Vahid Karimipour, "Simulating of X-states and the two-qubit XYZ Heisenberg system on IBM quantum computer", Physica Scripta 97 2, 025101 (2022).

[17] Francesco Tacchino, Alessandro Chiesa, Stefano Carretta, and Dario Gerace, "Quantum Computers as Universal Quantum Simulators: State‐of‐the‐Art and Perspectives", Advanced Quantum Technologies 3 3, 1900052 (2020).

[18] Francesca De Franco and Ettore Vicari, "Out-of-equilibrium finite-size scaling in generalized Kibble-Zurek protocols crossing quantum phase transitions in the presence of symmetry-breaking perturbations", Physical Review B 107 11, 115175 (2023).

[19] Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Variational fast forwarding for quantum simulation beyond the coherence time", npj Quantum Information 6 1, 82 (2020).

[20] Vasily Sazonov and Mohamed Tamaazousti, "Quantum error mitigation for parametric circuits", Physical Review A 105 4, 042408 (2022).

[21] Noah F. Berthusen, Thaís V. Trevisan, Thomas Iadecola, and Peter P. Orth, "Quantum dynamics simulations beyond the coherence time on noisy intermediate-scale quantum hardware by variational Trotter compression", Physical Review Research 4 2, 023097 (2022).

[22] Steve Abel, Andrew Blance, and Michael Spannowsky, "Quantum optimization of complex systems with a quantum annealer", Physical Review A 106 4, 042607 (2022).

[23] Erik J. Gustafson, "Prospects for simulating a qudit-based model of (1+1)D scalar QED", Physical Review D 103 11, 114505 (2021).

[24] Leon Bello, Marcello Calvanese Strinati, Emanuele G. Dalla Torre, and Avi Pe’er, "Persistent Coherent Beating in Coupled Parametric Oscillators", Physical Review Letters 123 8, 083901 (2019).

[25] Alessio Franchi, Davide Rossini, and Ettore Vicari, "Critical crossover phenomena driven by symmetry-breaking defects at quantum transitions", Physical Review E 105 3, 034139 (2022).

[26] Alessio Franchi, Andrea Pelissetto, and Ettore Vicari, "Quantum critical behaviors and decoherence of weakly coupled quantum Ising models within an isolated global system", Physical Review E 107 1, 014113 (2023).

[27] Jason P. Terry, Prosper D. Akrobotu, Christian F. A. Negre, Susan M. Mniszewski, and Itay Hen, "Quantum isomer search", PLOS ONE 15 1, e0226787 (2020).

[28] Jin Ming Koh, Tommy Tai, and Ching Hua Lee, "Simulation of Interaction-Induced Chiral Topological Dynamics on a Digital Quantum Computer", Physical Review Letters 129 14, 140502 (2022).

[29] Ricardo Pérez-Castillo, Manuel A. Serrano, and Mario Piattini, "Software modernization to embrace quantum technology", Advances in Engineering Software 151, 102933 (2021).

[30] James Holehouse, Hector Pollitt, and Talib Al-Ameri, "Non-equilibrium time-dependent solution to discrete choice with social interactions", PLOS ONE 17 5, e0267083 (2022).

[31] Alba Cervera-Lierta, José Ignacio Latorre, and Dardo Goyeneche, "Quantum circuits for maximally entangled states", Physical Review A 100 2, 022342 (2019).

[32] Konrad Jałowiecki, Andrzej Więckowski, Piotr Gawron, and Bartłomiej Gardas, "Parallel in time dynamics with quantum annealers", Scientific Reports 10 1, 13534 (2020).

[33] Maxime Dupont and Joel E. Moore, "Quantum criticality using a superconducting quantum processor", Physical Review B 106 4, L041109 (2022).

[34] Bruno Murta, Pedro M. Q. Cruz, and J. Fernández-Rossier, "Preparing valence-bond-solid states on noisy intermediate-scale quantum computers", Physical Review Research 5 1, 013190 (2023).

[35] Alessandro Santini and Vittorio Vitale, "Experimental violations of Leggett-Garg inequalities on a quantum computer", Physical Review A 105 3, 032610 (2022).

[36] Marcello Calvanese Strinati, Leon Bello, Avi Pe'er, and Emanuele G. Dalla Torre, "Theory of coupled parametric oscillators beyond coupled Ising spins", Physical Review A 100 2, 023835 (2019).

[37] Alessio Franchi, Davide Rossini, and Ettore Vicari, "Decoherence and energy flow in the sunburst quantum Ising model", Journal of Statistical Mechanics: Theory and Experiment 2022 8, 083103 (2022).

[38] Kishore S. Shenoy, Dev Y. Sheth, Bikash K. Behera, and Prasanta K. Panigrahi, "Demonstration of a measurement-based adaptation protocol with quantum reinforcement learning on the IBM Q experience platform", Quantum Information Processing 19 5, 161 (2020).

[39] Alessio Franchi, Davide Rossini, and Ettore Vicari, "Quantum many-body spin rings coupled to ancillary spins: The sunburst quantum Ising model", Physical Review E 105 5, 054111 (2022).

[40] Alba Cervera-Lierta, Jakob S. Kottmann, and Alán Aspuru-Guzik, "Meta-Variational Quantum Eigensolver: Learning Energy Profiles of Parameterized Hamiltonians for Quantum Simulation", PRX Quantum 2 2, 020329 (2021).

[41] Juan Carlos Criado and Michael Spannowsky, "Qade: solving differential equations on quantum annealers", Quantum Science and Technology 8 1, 015021 (2023).

[42] Kenneth Robbins and Peter J. Love, "Benchmarking near-term quantum devices with the variational quantum eigensolver and the Lipkin-Meshkov-Glick model", Physical Review A 104 2, 022412 (2021).

[43] Steve Abel, Juan C. Criado, and Michael Spannowsky, "Completely quantum neural networks", Physical Review A 106 2, 022601 (2022).

[44] John S. Van Dyke, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Preparing Bethe Ansatz Eigenstates on a Quantum Computer", PRX Quantum 2 4, 040329 (2021).

[45] Erik Gustafson, Yingyue Zhu, Patrick Dreher, Norbert M. Linke, and Yannick Meurice, "Real-time quantum calculations of phase shifts using wave packet time delays", Physical Review D 104 5, 054507 (2021).

[46] Hailong Fu, Pengjie Wang, Zhenhai Hu, Yifan Li, and Xi Lin, "Low-temperature environments for quantum computation and quantum simulation* ", Chinese Physics B 30 2, 020702 (2021).

[47] Jean-Loup Ville, Alexis Morvan, Akel Hashim, Ravi K. Naik, Marie Lu, Bradley Mitchell, John-Mark Kreikebaum, Kevin P. O'Brien, Joel J. Wallman, Ian Hincks, Joseph Emerson, Ethan Smith, Ed Younis, Costin Iancu, David I. Santiago, and Irfan Siddiqi, "Leveraging randomized compiling for the quantum imaginary-time-evolution algorithm", Physical Review Research 4 3, 033140 (2022).

[48] Xiao Xiao, J. K. Freericks, and A. F. Kemper, "Robust measurement of wave function topology on NISQ quantum computers", Quantum 7, 987 (2023).

[49] Benedikt Fauseweh and Jian-Xin Zhu, "Digital quantum simulation of non-equilibrium quantum many-body systems", Quantum Information Processing 20 4, 138 (2021).

[50] Korbinian Kottmann, Friederike Metz, Joana Fraxanet, and Niccolò Baldelli, "Variational quantum anomaly detection: Unsupervised mapping of phase diagrams on a physical quantum computer", Physical Review Research 3 4, 043184 (2021).

[51] Robert Maxton and Yannick Meurice, "Perturbative boundaries of quantum advantage: Real-time evolution for digitized λϕ4 lattice models", Physical Review D 107 7, 074508 (2023).

[52] P M Q Cruz, G Catarina, R Gautier, and J Fernández-Rossier, "Optimizing quantum phase estimation for the simulation of Hamiltonian eigenstates", Quantum Science and Technology 5 4, 044005 (2020).

[53] Adrián Pérez-Salinas, Juan Cruz-Martinez, Abdulla A. Alhajri, and Stefano Carrazza, "Determining the proton content with a quantum computer", Physical Review D 103 3, 034027 (2021).

[54] Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planagumà, and José I. Latorre, "Quantum unary approach to option pricing", Physical Review A 103 3, 032414 (2021).

[55] Anshuman Padhi, Sudev Pradhan, Pragna Paramita Sahoo, Kalyani Suresh, Bikash K. Behera, and Prasanta K. Panigrahi, "Studying the effect of lockdown using epidemiological modelling of COVID-19 and a quantum computational approach using the Ising spin interaction", Scientific Reports 10 1, 21741 (2020).

[56] Michał Białończyk, Fernando Gómez-Ruiz, and Adolfo del Campo, "Exact thermal properties of free-fermionic spin chains", SciPost Physics 11 1, 013 (2021).

[57] Aydin Deger and Tzu-Chieh Wei, "Geometric entanglement and quantum phase transition in generalized cluster-XY models", Quantum Information Processing 18 10, 326 (2019).

[58] Joseph Vovrosh and Johannes Knolle, "Confinement and entanglement dynamics on a digital quantum computer", Scientific Reports 11 1, 11577 (2021).

[59] Yannick Meurice, Ryo Sakai, and Judah Unmuth-Yockey, "Tensor lattice field theory for renormalization and quantum computing", Reviews of Modern Physics 94 2, 025005 (2022).

[60] Yeonghun Lee, "Symmetric Trotterization in digital quantum simulation of quantum spin dynamics", Journal of the Korean Physical Society 82 5, 479 (2023).

[61] Gabriel Matos, Chris N. Self, Zlatko Papić, Konstantinos Meichanetzidis, and Henrik Dreyer, "Characterization of variational quantum algorithms using free fermions", Quantum 7, 966 (2023).

[62] Adam Smith, M. S. Kim, Frank Pollmann, and Johannes Knolle, "Simulating quantum many-body dynamics on a current digital quantum computer", npj Quantum Information 5 1, 106 (2019).

[63] Lindsay Bassman, Miroslav Urbanek, Mekena Metcalf, Jonathan Carter, Alexander F Kemper, and Wibe A de Jong, "Simulating quantum materials with digital quantum computers", Quantum Science and Technology 6 4, 043002 (2021).

[64] Martin Rodriguez-Vega, Ella Carlander, Adrian Bahri, Ze-Xun Lin, Nikolai A. Sinitsyn, and Gregory A. Fiete, "Real-time simulation of light-driven spin chains on quantum computers", Physical Review Research 4 1, 013196 (2022).

[65] Kh. P. Gnatenko and V. M. Tkachuk, "Observation of spin-1 tunneling on a quantum computer", The European Physical Journal Plus 138 4, 346 (2023).

[66] Joseph Vovrosh, Kiran E. Khosla, Sean Greenaway, Christopher Self, M. S. Kim, and Johannes Knolle, "Simple mitigation of global depolarizing errors in quantum simulations", Physical Review E 104 3, 035309 (2021).

[67] Sudev Pradhan, Amlandeep Nayak, Sritam Kumar Satpathy, Tanmaya Shree Behera, Ankita Misra, Debashis Swain, and Bikash K. Behera, "Simulating the Hamiltonian of dimer atomic spin model of one-dimensional optical lattice on quantum computers", International Journal of Quantum Information 21 01, 2350002 (2023).

[68] Ren Liao, Jingxin Sun, Pengju Zhao, Shifeng Yang, Hui Li, Xinyi Huang, Wei Xiong, Xiaoji Zhou, Dingping Li, Xiongjun Liu, and Xuzong Chen, "Simulation of exact quantum Ising models with a Mott insulator of paired atoms", Physical Review A 106 5, 053308 (2022).

[69] Lindsay Bassman Oftelie, Sahil Gulania, Connor Powers, Rongpeng Li, Thomas Linker, Kuang Liu, T K Satish Kumar, Rajiv K Kalia, Aiichiro Nakano, and Priya Vashishta, "Domain-specific compilers for dynamic simulations of quantum materials on quantum computers", Quantum Science and Technology 6 1, 014007 (2021).

[70] Santiago Higuera-Quintero, Ferney J. Rodríguez, Luis Quiroga, and Fernando J. Gómez-Ruiz, "Experimental validation of the Kibble-Zurek mechanism on a digital quantum computer", Frontiers in Quantum Science and Technology 1, 1026025 (2022).

[71] Xiao Xiao, J. K. Freericks, and A. F. Kemper, "Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware", Quantum 5, 553 (2021).

[72] Ashley Montanaro and Stasja Stanisic, "Compressed variational quantum eigensolver for the Fermi-Hubbard model", arXiv:2006.01179, (2020).

[73] Bhupesh Bishnoi, "Quantum Computation", arXiv:2006.02799, (2020).

[74] Rafael I. Nepomechie, "Bethe ansatz on a quantum computer?", arXiv:2010.01609, (2020).

[75] Bartłomiej Gardas, Marek M. Rams, and Jacek Dziarmaga, "Quantum neural networks to simulate many-body quantum systems", Physical Review B 98 18, 184304 (2018).

[76] Amandeep Singh Bhatia and Mandeep Kaur Saggi, "Implementing Entangled States on a Quantum Computer", arXiv:1811.09833, (2018).

[77] Guillermo Blázquez-Cruz and Pierre-Luc Dallaire-Demers, "Quantum supremacy regime for compressed fermionic models", arXiv:2110.09550, (2021).

[78] K. M. Anandu, Muhammad Shaharukh, Bikash K. Behera, and Prasanta K. Panigrahi, "Demonstration of teleportation-based error correction in the IBM quantum computer", arXiv:1902.01692, (2019).

[79] Alakesh Baishya, Lingraj Kumar, Bikash K. Behera, and Prasanta K. Panigrahi, "Experimental Demonstration of Force Driven Quantum Harmonic Oscillator in IBM Quantum Computer", arXiv:1906.01436, (2019).

[80] Harshavardhan Reddy Nareddula, Bikash K. Behera, and Prasanta K. Panigrahi, "Quantum Cost Efficient Scheme for Violating the Holevo Bound and Cloning in the Presence of Deutschian Closed Timelike Curves", arXiv:1901.00379, (2018).

[81] Robert Maxton and Yannick Meurice, "Perturbative boundaries of quantum computing: real-time evolution for digitized lambda phi^4 lattice models", arXiv:2210.05493, (2022).

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