Hybrid quantum-classical approach for coupled-cluster Green’s function theory

Trevor Keen1, Bo Peng2, Karol Kowalski2, Pavel Lougovski3, and Steven Johnston1,4

1Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, United States of America
2Physical Sciences and Computational Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States of America
3Quantum Information Science Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States of America
4Institute for Advanced Materials and Manufacturing, University of Tennessee, Knoxville, Tennessee 37996, United States of America

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The three key elements of a quantum simulation are state preparation, time evolution, and measurement. While the complexity scaling of time evolution and measurements are well known, many state preparation methods are strongly system-dependent and require prior knowledge of the system's eigenvalue spectrum. Here, we report on a quantum-classical implementation of the coupled-cluster Green's function (CCGF) method, which replaces explicit ground state preparation with the task of applying unitary operators to a simple product state. While our approach is broadly applicable to many models, we demonstrate it here for the Anderson impurity model (AIM). The method requires a number of $T$ gates that grows as $ \mathcal{O} \left(N^5 \right)$ per time step to calculate the impurity Green's function in the time domain, where $N$ is the total number of energy levels in the AIM. Since the number of $T$ gates is analogous to the computational time complexity of a classical simulation, we achieve an order of magnitude improvement over a classical CCGF calculation of the same order, which requires $ \mathcal{O} \left(N^6 \right)$ computational resources per time step.

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[1] Gadi Aleksandrowicz, Thomas Alexander, Panagiotis Barkoutsos, Luciano Bello, Yael Ben-Haim, David Bucher, Francisco Jose Cabrera-Hernández, Jorge Carballo-Franquis, Adrian Chen, Chun-Fu Chen, Jerry M. Chow, Antonio D. Córcoles-Gonzales, Abigail J. Cross, Andrew Cross, Juan Cruz-Benito, Chris Culver, Salvador De La Puente González, Enrique De La Torre, Delton Ding, Eugene Dumitrescu, Ivan Duran, Pieter Eendebak, Mark Everitt, Ismael Faro Sertage, Albert Frisch, Andreas Fuhrer, Jay Gambetta, Borja Godoy Gago, Juan Gomez-Mosquera, Donny Greenberg, Ikko Hamamura, Vojtech Havlicek, Joe Hellmers, Łukasz Herok, Hiroshi Horii, Shaohan Hu, Takashi Imamichi, Toshinari Itoko, Ali Javadi-Abhari, Naoki Kanazawa, Anton Karazeev, Kevin Krsulich, Peng Liu, Yang Luh, Yunho Maeng, Manoel Marques, Francisco Jose Martín-Fernández, Douglas T. McClure, David McKay, Srujan Meesala, Antonio Mezzacapo, Nikolaj Moll, Diego Moreda Rodríguez, Giacomo Nannicini, Paul Nation, Pauline Ollitrault, Lee James O'Riordan, Hanhee Paik, Jesús Pérez, Anna Phan, Marco Pistoia, Viktor Prutyanov, Max Reuter, Julia Rice, Abdón Rodríguez Davila, Raymond Harry Putra Rudy, Mingi Ryu, Ninad Sathaye, Chris Schnabel, Eddie Schoute, Kanav Setia, Yunong Shi, Adenilton Silva, Yukio Siraichi, Seyon Sivarajah, John A. Smolin, Mathias Soeken, Hitomi Takahashi, Ivano Tavernelli, Charles Taylor, Pete Taylour, Kenso Trabing, Matthew Treinish, Wes Turner, Desiree Vogt-Lee, Christophe Vuillot, Jonathan A. Wildstrom, Jessica Wilson, Erick Winston, Christopher Wood, Stephen Wood, Stefan Wörner, Ismail Yunus Akhalwaya, and Christa Zoufal. Qiskit: An Open-source Framework for Quantum Computing, January 2019. URL https:/​/​doi.org/​10.5281/​zenodo.2562111.

[2] P. W. Anderson. Localized magnetic states in metals. Phys. Rev., 124: 41–53, Oct 1961. 10.1103/​PhysRev.124.41.

[3] Rodney J. Bartlett and Monika Musiał. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys., 79: 291–352, Feb 2007. 10.1103/​RevModPhys.79.291.

[4] Bela Bauer, Dave Wecker, Andrew J. Millis, Matthew B. Hastings, and Matthias Troyer. Hybrid quantum-classical approach to correlated materials. Physical Review X, 6 (3): 1–11, 2016. ISSN 21603308. 10.1103/​PhysRevX.6.031045.

[5] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Simulating Hamiltonian Dynamics with a Truncated Taylor Series. Phys. Rev. Lett., 114 (9): 90502, mar 2015. 10.1103/​PhysRevLett.114.090502.

[6] Kiran Bhaskaran-Nair, Karol Kowalski, and William A. Shelton. Coupled cluster green function: Model involving single and double excitations. J. Chem. Phys., 144 (14): 144101, 2016. http:/​/​dx.doi.org/​10.1063/​1.4944960.

[7] Andrew M. Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information and Computation, 12 (11-12): 901–924, 2012. ISSN 15337146. 10.26421/​QIC12.11-12.

[8] J. Čížek­. On the correlation problem in atomic and molecular systems. calculation of wavefunction components in Ursell-type expansion using quantum-field theoretical methods. J. Chem. Phys., 45 (11): 4256–4266, 1966. http:/​/​dx.doi.org/​10.1063/​1.1727484.

[9] F. Coester. Bound states of a many-particle system. Nucl. Phys., 7: 421–424, 1958. ISSN 0029-5582. http:/​/​dx.doi.org/​10.1016/​0029-5582(58)90280-3.

[10] F. Coester and H. Kümmel. Short-range correlations in nuclear wave functions. Nucl. Phys., 17: 477–485, 1960. ISSN 0029-5582. http:/​/​dx.doi.org/​10.1016/​0029-5582(60)90140-1.

[11] Suguru Endo, Iori Kurata, and Yuya O. Nakagawa. Calculation of the Green's function on near-term quantum computers. Phys. Rev. Research, 2: 033281, Aug 2020a. 10.1103/​PhysRevResearch.2.033281.

[12] Suguru Endo, Iori Kurata, and Yuya O. Nakagawa. Calculation of the Green's function on near-term quantum computers. Phys. Rev. Research, 2: 033281, Aug 2020b. 10.1103/​PhysRevResearch.2.033281.

[13] Antoine Georges and Gabriel Kotliar. Hubbard model in infinite dimensions. Phys. Rev. B, 45: 6479–6483, Mar 1992. 10.1103/​PhysRevB.45.6479.

[14] Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Reviews of Modern Physics, 68 (1): 13–125, 1996. ISSN 1526498X. 10.1103/​RevModPhys.68.13.

[15] A.C. Hewson. The Kondo Problem to Heavy Fermions. Cambridge Studies in Magnetism, Vol. 2. Cambridge University Press, Cambridge, 1993.

[16] Ian D. Kivlichan, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, Hartmut Neven, and Ryan Babbush. Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization. Quantum, 4: 296, 2020. ISSN 2521-327X. 10.22331/​q-2020-07-16-296.

[17] Taichi Kosugi and Yu-ichiro Matsushita. Construction of Green's functions on a quantum computer: Quasiparticle spectra of molecules. Phys. Rev. A, 101: 012330, Jan 2020. 10.1103/​PhysRevA.101.012330.

[18] Lin Lin and Yu Tong. Near-optimal ground state preparation. Quantum, 4: 372, December 2020. ISSN 2521-327X. 10.22331/​q-2020-12-14-372.

[19] Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Qubitization. Quantum, 3: 163, 2019. ISSN 2521-327X. 10.22331/​q-2019-07-12-163.

[20] G.D. Mahan. Many-Particle Physics. Physics of Solids and Liquids. Springer US, 1990. ISBN 9780306434235.

[21] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition, 2011. ISBN 1107002176, 9781107002173.

[22] J. Noga and R. J. Bartlett. Erratum: The full ccsdt model for molecular electronic structure [j. chem. phys. 86, 7041 (1987)]. J. Chem. Phys., 89 (5): 3401–3401, 1988. 10.1063/​1.455742.

[23] Jozef Noga and Rodney J. Bartlett. The full ccsdt model for molecular electronic structure. J. Chem. Phys., 86 (12): 7041–7050, 1987. 10.1063/​1.452353.

[24] Marcel Nooijen and Jaap G. Snijders. Coupled cluster approach to the single-particle Green's function. Int. J. Quantum Chem., 44 (S26): 55–83, 1992. ISSN 1097-461X. 10.1002/​qua.560440808.

[25] Marcel Nooijen and Jaap G. Snijders. Coupled cluster Green's function method: Working equations and applications. Int. J. Quantum Chem., 48 (1): 15–48, 1993. ISSN 1097-461X. 10.1002/​qua.560480103.

[26] Marcel Nooijen and Jaap G. Snijders. Second order many-body perturbation approximations to the coupled cluster Green's function. J. Chem. Phys., 102 (4): 1681–1688, 1995. http:/​/​dx.doi.org/​10.1063/​1.468900.

[27] J. Paldus, J. Čížek, and I. Shavitt. Correlation problems in atomic and molecular systems. IV. Extended coupled-pair many-electron theory and its application to the B${\mathrm{H}}_{3}$ molecule. Phys. Rev. A, 5: 50–67, Jan 1972. 10.1103/​PhysRevA.5.50.

[28] Bo Peng and Karol Kowalski. Green's function coupled-cluster approach: Simulating photoelectron spectra for realistic molecular systems. J. Chem. Theory Comput., 14 (8): 4335–4352, 2018. 10.1021/​acs.jctc.8b00313.

[29] Bo Peng, Roel Van Beeumen, David B. Williams-Young, Karol Kowalski, and Chao Yang. Approximate Green’s function coupled cluster method employing effective dimension reduction. Journal of Chemical Theory and Computation, 15 (5): 3185–3196, 2019. 10.1021/​acs.jctc.9b00172. PMID: 30951302.

[30] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man Hong Yung, Xiao Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (May), 2014. ISSN 20411723. 10.1038/​ncomms5213.

[31] G. Purvis and R. Bartlett. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples. J. Chem. Phys., 76 (4): 1910–1918, 1982. http:/​/​dx.doi.org/​10.1063/​1.443164.

[32] J. R. Schrieffer and P. A. Wolff. Relation between the Anderson and Kondo Hamiltonians. Phys. Rev., 149: 491–492, Sep 1966. 10.1103/​PhysRev.149.491.

[33] Gustavo E. Scuseria and Henry F. Schaefer. A new implementation of the full ccsdt model for molecular electronic structure. Chem. Phys. Lett., 152 (4): 382–386, 1988. ISSN 0009-2614. https:/​/​doi.org/​10.1016/​0009-2614(88)80110-6.

[34] Avijit Shee and Dominika Zgid. Coupled cluster as an impurity solver for Green’s function embedding methods. Journal of Chemical Theory and Computation, 15 (11): 6010–6024, 2019. 10.1021/​acs.jctc.9b00603. PMID: 31518129.

[35] F. D. Vila, J. J. Rehr, J. J. Kas, K. Kowalski, and B. Peng. Real-time coupled-cluster approach for the cumulant Green’s function. Journal of Chemical Theory and Computation, 16 (11): 6983–6992, 2020. 10.1021/​acs.jctc.0c00639. PMID: 33108872.

[36] Kianna Wan. Exponentially faster implementations of select(h) for fermionic hamiltonians. Quantum, 5: 380, Jan 2021. ISSN 2521-327X. 10.22331/​q-2021-01-12-380.

[37] Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A - Atomic, Molecular, and Optical Physics, 92 (4): 1–11, 2015. ISSN 10941622. 10.1103/​PhysRevA.92.042303.

[38] Tianyu Zhu, Carlos A. Jiménez-Hoyos, James McClain, Timothy C. Berkelbach, and Garnet Kin-Lic Chan. Coupled-cluster impurity solvers for dynamical mean-field theory. Phys. Rev. B, 100: 115154, Sep 2019. 10.1103/​PhysRevB.100.115154.

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