Algebraic Bethe Circuits

Alejandro Sopena1, Max Hunter Gordon1,2, Diego García-Martín3,1, Germán Sierra1, and Esperanza López1

1Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid, Spain
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3Barcelona Supercomputing Center, Barcelona, Spain

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The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary $R$ matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-$\frac{1}{2}$ XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on $4$ sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.

The Bethe Ansatz is a tremendously successful analytical technique to solve a large class of many-body quantum systems. However, there remain properties of these systems that are inaccessible to classical methods. This motivates applying quantum computing to their study. Starting from the tensor-network structure provided by the Algebraic Bethe Ansatz (ABA), we present a quantum-classical algorithm to deterministically prepare ABA eigenstates on a quantum computer. We note that these eigenstates may also prove useful inputs to other quantum algorithms, and may be employed to benchmark quantum hardware.

The main issue encountered when translating the ABA into a quantum circuit is that the matrices that make up the tensor network are not unitary. Hence, we use classical resources to bring the ABA to unitary form by iteratively computing a QR decomposition. This leads to a set of unitary matrices that must be compiled down to quantum gates. We have thus transformed the ABA into a quantum circuit, implemented some of them in IBM quantum computers and applied advanced mitigation techniques obtaining precise results. We moreover find a novel representation of the Yang-Baxter equation in terms of unitary matrices, that may be of independent interest and have broad applicability.

Our work represents an important step in quantum simulation. We bring together techniques from statistical physics, condensed-matter physics and near-term quantum computing to provide an exact alternative approach to directly prepare quantum many-body eigenstates. This opens up a new promising area of research leveraging the power of the Bethe ansatz on quantum computers.

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► References

[1] Seth Lloyd. Universal quantum simulators. Science, 273 (5278): 1073–1078, 1996. ISSN 0036-8075. 10.1126/​science.273.5278.1073.

[2] Rafael I. Nepomechie. Bethe ansatz on a quantum computer? 2021. 10.48550/​arXiv.2010.01609.

[3] 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), Nov 2021. ISSN 2691-3399. 10.1103/​prxquantum.2.040329.

[4] John S Van Dyke, Edwin Barnes, Sophia E Economou, and Rafael I Nepomechie. Preparing exact eigenstates of the open xxz chain on a quantum computer. Journal of Physics A: Mathematical and Theoretical, 55 (5): 055301, Jan 2022. ISSN 1751-8121. 10.1088/​1751-8121/​ac4640.

[5] Wen Li, Mert Okyay, and Rafael I Nepomechie. Bethe states on a quantum computer: success probability and correlation functions. Journal of Physics A: Mathematical and Theoretical, 55 (35): 355305, September 2022. ISSN 1751-8113, 1751-8121. 10.1088/​1751-8121/​ac8255.

[6] H. Bethe. Zur theorie der metalle. Zeitschrift für Physik, 71 (3): 205–226, Mar 1931. ISSN 0044-3328. 10.1007/​BF01341708.

[7] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin. Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1993. 10.1017/​CBO9780511628832.

[8] L. D. Faddeev. How algebraic bethe ansatz works for integrable model. 1996. 10.48550/​arXiv.hep-th/​9605187.

[9] César Gómez, Martí Ruiz-Altaba, and Germán Sierra. Quantum Groups in Two-Dimensional Physics. Cambridge University Press, 1 edition, April 1996. ISBN 9780521460651 9780521020046 9780511628825. 10.1017/​CBO9780511628825.

[10] Balázs Pozsgay. Excited state correlations of the finite heisenberg chain. Journal of Physics A: Mathematical and Theoretical, 50 (7): 074006, Jan 2017. ISSN 1751-8121. 10.1088/​1751-8121/​aa5344.

[11] Wen Wei Ho and Timothy H. Hsieh. Efficient variational simulation of non-trivial quantum states. SciPost Physics, 6 (3), Mar 2019. ISSN 2542-4653. 10.21468/​scipostphys.6.3.029.

[12] Carlos Bravo-Prieto, Josep Lumbreras-Zarapico, Luca Tagliacozzo, and José I. Latorre. Scaling of variational quantum circuit depth for condensed matter systems. Quantum, 4: 272, May 2020. ISSN 2521-327X. 10.22331/​q-2020-05-28-272.

[13] Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, and Henry Yuen. Exploring entanglement and optimization within the hamiltonian variational ansatz. PRX Quantum, 1 (2), Dec 2020. ISSN 2691-3399. 10.1103/​prxquantum.1.020319.

[14] Manpreet Singh Jattana, Fengping Jin, Hans De Raedt, and Kristel Michielsen. Assessment of the Variational Quantum Eigensolver: Application to the Heisenberg Model. Frontiers in Physics, 10: 907160, June 2022. ISSN 2296-424X. 10.3389/​fphy.2022.907160.

[15] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature Communications, 9 (1), Nov 2018. ISSN 2041-1723. 10.1038/​s41467-018-07090-4.

[16] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics, 3 (9): 625–644, Aug 2021a. ISSN 2522-5820. 10.1038/​s42254-021-00348-9.

[17] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo. Diagnosing barren plateaus with tools from quantum optimal control. 2021a. 10.48550/​arXiv.2105.14377.

[18] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles. Noise-induced barren plateaus in variational quantum algorithms. Nature Communications, 12 (1), Nov 2021. ISSN 2041-1723. 10.1038/​s41467-021-27045-6.

[19] U. Schollwöck. The density-matrix renormalization group. Reviews of Modern Physics, 77 (1): 259–315, April 2005. 10.1103/​RevModPhys.77.259.

[20] B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo. Matrix product states for critical spin chains: Finite-size versus finite-entanglement scaling. Physical Review B, 86 (7): 075117, August 2012. ISSN 1098-0121, 1550-235X. 10.1103/​PhysRevB.86.075117.

[21] James Dborin, Fergus Barratt, Vinul Wimalaweera, Lewis Wright, and Andrew G Green. Matrix product state pre-training for quantum machine learning. Quantum Science and Technology, 7 (3): 035014, July 2022. ISSN 2058-9565. 10.1088/​2058-9565/​ac7073.

[22] Adam Smith, Bernhard Jobst, Andrew G. Green, and Frank Pollmann. Crossing a topological phase transition with a quantum computer. Physical Review Research, 4 (2): L022020, April 2022. ISSN 2643-1564. 10.1103/​PhysRevResearch.4.L022020.

[23] Gene H. Golub and Charles F. Van Loan. Matrix computations. Johns Hopkins studies in the mathematical sciences. Johns Hopkins University Press, Baltimore, 3rd ed edition, 1996. ISBN 9780801854132 9780801854149.

[24] Pieter W. Claeys, Jonah Herzog-Arbeitman, and Austen Lamacraft. Correlations and commuting transfer matrices in integrable unitary circuits. SciPost Physics, 12 (1), Jan 2022. ISSN 2542-4653. 10.21468/​scipostphys.12.1.007.

[25] Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush. Quantum simulation of electronic structure with linear depth and connectivity. Physical Review Letters, 120 (11), Mar 2018. ISSN 1079-7114. 10.1103/​physrevlett.120.110501.

[26] Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo. Quantum algorithms to simulate many-body physics of correlated fermions. Physical Review Applied, 9 (4), Apr 2018. ISSN 2331-7019. 10.1103/​physrevapplied.9.044036.

[27] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, et al. Observation of separated dynamics of charge and spin in the fermi-hubbard model. 2020. 10.48550/​arXiv.2010.07965.

[28] Frank Verstraete, J. Ignacio Cirac, and José I. Latorre. Quantum circuits for strongly correlated quantum systems. Physical Review A, 79 (3), Mar 2009. ISSN 1094-1622. 10.1103/​physreva.79.032316.

[29] Alba Cervera-Lierta. Exact Ising model simulation on a quantum computer. Quantum, 2: 114, December 2018. ISSN 2521-327X. 10.22331/​q-2018-12-21-114.

[30] Francisco C Alcaraz and Matheus J Lazo. The bethe ansatz as a matrix product ansatz. Journal of Physics A: Mathematical and General, 37 (1): L1–L7, Dec 2003. ISSN 1361-6447. 10.1088/​0305-4470/​37/​1/​l01.

[31] Hosho Katsura and Isao Maruyama. Derivation of the matrix product ansatz for the heisenberg chain from the algebraic bethe ansatz. Journal of Physics A: Mathematical and Theoretical, 43 (17): 175003, Apr 2010. ISSN 1751-8121. 10.1088/​1751-8113/​43/​17/​175003.

[32] V. Murg, V. E. Korepin, and F. Verstraete. Algebraic bethe ansatz and tensor networks. Physical Review B, 86 (4), Jul 2012. ISSN 1550-235X. 10.1103/​physrevb.86.045125.

[33] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. 2007. 10.48550/​arXiv.quant-ph/​0608197.

[34] Shi-Ju Ran. Encoding of matrix product states into quantum circuits of one- and two-qubit gates. Phys. Rev. A, 101: 032310, Mar 2020. 10.1103/​PhysRevA.101.032310.

[35] Sheng-Hsuan Lin, Rohit Dilip, Andrew G. Green, Adam Smith, and Frank Pollmann. Real- and imaginary-time evolution with compressed quantum circuits. PRX Quantum, 2: 010342, Mar 2021. 10.1103/​PRXQuantum.2.010342.

[36] F. Barratt, James Dborin, Matthias Bal, Vid Stojevic, Frank Pollmann, and A. G. Green. Parallel quantum simulation of large systems on small nisq computers. npj Quantum Information, 7 (1), May 2021. ISSN 2056-6387. 10.1038/​s41534-021-00420-3.

[37] Michael Foss-Feig, David Hayes, Joan M. Dreiling, Caroline Figgatt, John P. Gaebler, Steven A. Moses, Juan M. Pino, and Andrew C. Potter. Holographic quantum algorithms for simulating correlated spin systems. Physical Review Research, 3 (3), Jul 2021. ISSN 2643-1564. 10.1103/​physrevresearch.3.033002.

[38] Reza Haghshenas, Johnnie Gray, Andrew C. Potter, and Garnet Kin-Lic Chan. Variational Power of Quantum Circuit Tensor Networks. Physical Review X, 12 (1): 011047, March 2022. ISSN 2160-3308. 10.1103/​PhysRevX.12.011047.

[39] R. J. Baxter. Exactly Solved Models in Statistical Mechanics. In Integrable Systems in Statistical Mechanics, volume 1, pages 5–63. WORLD SCIENTIFIC, May 1985. ISBN 9789971978112 9789814415255. 10.1142/​9789814415255_0002.

[40] Bill Sutherland. Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems. WORLD SCIENTIFIC, June 2004. ISBN 9789812388599 9789812562142. 10.1142/​5552.

[41] Giuseppe Mussardo. Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics. Oxford University PressOxford, 2 edition, March 2020. ISBN 9780198788102 9780191830082. 10.1093/​oso/​9780198788102.001.0001.

[42] Thierry Giamarchi. Quantum Physics in One Dimension. Oxford University Press, 12 2003. ISBN 9780198525004. 10.1093/​acprof:oso/​9780198525004.001.0001.

[43] Dave Wecker, Matthew B. Hastings, Nathan Wiebe, Bryan K. Clark, Chetan Nayak, and Matthias Troyer. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A, 92: 062318, Dec 2015. 10.1103/​PhysRevA.92.062318.

[44] Diogo Cruz, Romain Fournier, Fabien Gremion, Alix Jeannerot, Kenichi Komagata, Tara Tosic, Jarla Thiesbrummel, Chun Lam Chan, Nicolas Macris, Marc‐André Dupertuis, and et al. Efficient quantum algorithms for ghz and w states, and implementation on the ibm quantum computer. Advanced Quantum Technologies, 2 (5-6): 1900015, Apr 2019. ISSN 2511-9044. 10.1002/​qute.201900015.

[45] Richard Jozsa and Akimasa Miyake. Matchgates and classical simulation of quantum circuits. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464 (2100): 3089–3106, December 2008. 10.1098/​rspa.2008.0189.

[46] G. Vidal and C. M. Dawson. Universal quantum circuit for two-qubit transformations with three controlled-not gates. Phys. Rev. A, 69: 010301, Jan 2004. 10.1103/​PhysRevA.69.010301.

[47] Ed Younis, Ethan Smith, Mathias Weiden, GoodwillComputingLab, Marc Davis, and Tirthak Patel. Bqskit/​bqskit: 0.5.2. April 2022. 10.5281/​zenodo.6499836.

[48] Efekan Kökcü, Thomas Steckmann, Yan Wang, J. K. Freericks, Eugene F. Dumitrescu, and Alexander F. Kemper. Fixed Depth Hamiltonian Simulation via Cartan Decomposition. Physical Review Letters, 129 (7): 070501, August 2022a. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.129.070501.

[49] Alexander B Zamolodchikov and Alexey B Zamolodchikov. Factorized s-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Annals of Physics, 120 (2): 253–291, 1979. ISSN 0003-4916. https:/​/​​10.1016/​0003-4916(79)90391-9.

[50] Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego García-Martín, Artur Garcia-Saez, José Ignacio Latorre, and Stefano Carrazza. Qibo: a framework for quantum simulation with hardware acceleration. Quantum Science and Technology, 7 (1): 015018, Dec 2021a. ISSN 2058-9565. 10.1088/​2058-9565/​ac39f5.

[51] Stavros Efthymiou, Stefano Carrazza, AdrianPerezSalinas, Sergi Ramos, Carlos Bravo-Prieto, Diego García-Martín, Nicole Zattarin, Marco Lazzarin, Paul, Andrea Pasquale, Javier Serrano, and atomicprinter. qiboteam/​qibo: Qibo 0.1.7-rc1. November 2021b. 10.5281/​zenodo.5711842.

[52] Stavros Efthymiou, Stefano Carrazza, Marco Lazzarin, and Andrea Pasquale. qiboteam/​qibojit: qibojit 0.0.4. February 2022. 10.5281/​zenodo.6080210.

[53] Alejandro Sopena, Max Hunter Gordon, and Diego García-Martín. Algebraic-bethe-circuits 1.0. July 2022. 10.5281/​zenodo.6908365.

[54] Daniel Bultrini, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio. Unifying and benchmarking state-of-the-art quantum error mitigation techniques. 2021. 10.48550/​arXiv.2107.13470.

[55] Cristina Cirstoiu, Silas Dilkes, Daniel Mills, Seyon Sivarajah, and Ross Duncan. Volumetric benchmarking of error mitigation with qermit. 2022. 10.48550/​arXiv.2204.09725.

[56] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. Error mitigation for short-depth quantum circuits. Physical Review Letters, 119 (18), Nov 2017. ISSN 1079-7114. 10.1103/​physrevlett.119.180509.

[57] Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio. Error mitigation with clifford quantum-circuit data. Quantum, 5: 592, Nov 2021. ISSN 2521-327X. 10.22331/​q-2021-11-26-592.

[58] Angus Lowe, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio. Unified approach to data-driven quantum error mitigation. Physical Review Research, 3 (3), Jul 2021. ISSN 2643-1564. 10.1103/​physrevresearch.3.033098.

[59] Ryan LaRose, Andrea Mari, Sarah Kaiser, Peter J. Karalekas, Andre A. Alves, Piotr Czarnik, Mohamed El Mandouh, Max H. Gordon, Yousef Hindy, Aaron Robertson, Purva Thakre, Misty Wahl, Danny Samuel, Rahul Mistri, Maxime Tremblay, Nick Gardner, Nathaniel T. Stemen, Nathan Shammah, and William J. Zeng. Mitiq: A software package for error mitigation on noisy quantum computers. Quantum, 6: 774, August 2022. ISSN 2521-327X. 10.22331/​q-2022-08-11-774.

[60] Chao Zheng, Jun lin Li, Si yu Song, and Gui Lu Long. Direct experimental simulation of the Yang-Baxter equation. J. Opt. Soc. Am. B, 30 (6): 1688–1693, Jun 2013. 10.1364/​JOSAB.30.001688.

[61] F. Anvari Vind, A. Foerster, I. S. Oliveira, R. S. Sarthour, D. O. Soares-Pinto, A. M. Souza, and I. Roditi. Experimental realization of the Yang-Baxter Equation via NMR interferometry. Scientific Reports, 6 (1): 20789, August 2016. ISSN 2045-2322. 10.1038/​srep20789.

[62] Efekan Kökcü, Daan Camps, Lindsay Bassman, J. K. Freericks, Wibe A. de Jong, Roel Van Beeumen, and Alexander F. Kemper. Algebraic compression of quantum circuits for Hamiltonian evolution. Physical Review A, 105 (3): 032420, March 2022b. ISSN 2469-9926, 2469-9934. 10.1103/​PhysRevA.105.032420.

[63] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum, 3: 214, Dec 2019. ISSN 2521-327X. 10.22331/​q-2019-12-09-214.

[64] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communications, 12 (1), Mar 2021b. ISSN 2041-1723. 10.1038/​s41467-021-21728-w.

[65] Thomas Lubinski, Sonika Johri, Paul Varosy, Jeremiah Coleman, Luning Zhao, Jason Necaise, Charles H. Baldwin, Karl Mayer, and Timothy Proctor. Application-oriented performance benchmarks for quantum computing. 2021. 10.48550/​arXiv.2110.03137.

[66] Bobak Toussi Kiani, Seth Lloyd, and Reevu Maity. Learning unitaries by gradient descent. 2020. 10.48550/​arXiv.2001.11897.

[67] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo. Theory of overparametrization in quantum neural networks. 2021b. 10.48550/​arXiv.2109.11676.

[68] Alejandro Sopena, Max Hunter Gordon, Germán Sierra, and Esperanza López. Simulating quench dynamics on a digital quantum computer with data-driven error mitigation. Quantum Science and Technology, 6 (4): 045003, Jul 2021. ISSN 2058-9565. 10.1088/​2058-9565/​ac0e7a.

[69] Andre He, Benjamin Nachman, Wibe A. de Jong, and Christian W. Bauer. Zero-noise extrapolation for quantum-gate error mitigation with identity insertions. Physical Review A, 102 (1), Jul 2020. ISSN 2469-9934. 10.1103/​physreva.102.012426.

[70] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-state quantum systems. Physical Review A, 58 (4): 2733–2744, Oct 1998. ISSN 1094-1622. 10.1103/​physreva.58.2733.

[71] Youngseok Kim, Christopher J. Wood, Theodore J. Yoder, Seth T. Merkel, Jay M. Gambetta, Kristan Temme, and Abhinav Kandala. Scalable error mitigation for noisy quantum circuits produces competitive expectation values. 2021. 10.48550/​arXiv.2108.09197.

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[2] Yuan Miao, Vladimir Gritsev, and Denis V. Kurlov, "The Floquet Baxterisation", arXiv:2206.15142.

[3] Kazunobu Maruyoshi, Takuya Okuda, Juan William Pedersen, Ryo Suzuki, Masahito Yamazaki, and Yutaka Yoshida, "Conserved charges in the quantum simulation of integrable spin chains", arXiv:2208.00576.

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