Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers

V. O. Shkolnikov1,2, Nicholas J. Mayhall3,2, Sophia E. Economou1,2, and Edwin Barnes1,2

1Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA
2Virginia Tech Center for Quantum Information Science and Engineering, Blacksburg, VA 24061, USA
3Department of Chemistry, Virginia Tech, Blacksburg, VA 24061, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number $n$ of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size $2n-2$ can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such "complete" pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.

Simulation of strongly correlated systems is one of the key applications envisioned for near-term quantum computers. Realizing this application requires reducing quantum and classical resources as much as possible. One of the leading approaches exploits the variational principle of quantum mechanics, but its feasibility depends crucially on finding suitable trial wavefunctions.

While adaptive algorithms that construct trial wavefunctions on the fly in a problem-tailored fashion seem particularly promising, they may come with an extra measurement cost compared to other variational algorithms. We prove that this extra cost can be reduced to being only linear in the number of qubits, and we provide explicit recipes for achieving this. We also show that it is important that these recipes must account for any symmetries in the system being simulated in order for them to work well.

► BibTeX data

► References

[1] John Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018).

[2] 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, 4213 (2014).

[3] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms''. New Journal of Physics 18, 023023 (2016).

[4] Jonathan Romero, Ryan Babbush, Jarrod R McClean, Cornelius Hempel, Peter J Love, and Alán Aspuru-Guzik. ``Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz''. Quantum Science and Technology 4, 014008 (2018).

[5] Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley. ``Generalized unitary coupled cluster wave functions for quantum computation''. Journal of Chemical Theory and Computation 15, 311–324 (2019).

[6] 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, 625–644 (2021).

[7] 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''. Rev. Mod. Phys. 94, 015004 (2022).

[8] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall. ``An adaptive variational algorithm for exact molecular simulations on a quantum computer''. Nature Communications 10, 3007 (2019).

[9] Ho Lun Tang, V.O. Shkolnikov, George S. Barron, Harper R. Grimsley, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou. ``Qubit-adapt-vqe: An adaptive algorithm for constructing hardware-efficient ansätze on a quantum processor''. PRX Quantum 2, 020310 (2021).

[10] Markus Reiher, Nathan Wiebe, Krysta M. Svore, Dave Wecker, and Matthias Troyer. ``Elucidating reaction mechanisms on quantum computers''. Proceedings of the National Academy of Sciences 114, 7555–7560 (2017).

[11] 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, 043002 (2021).

[12] P. Hohenberg and W. Kohn. ``Inhomogeneous electron gas''. Phys. Rev. 136, B864–B871 (1964).

[13] W. Kohn and L. J. Sham. ``Self-consistent equations including exchange and correlation effects''. Phys. Rev. 140, A1133–A1138 (1965).

[14] Andrew G. Taube and Rodney J. Bartlett. ``New perspectives on unitary coupled-cluster theory''. International Journal of Quantum Chemistry 106, 3393–3401 (2006).

[15] Werner Kutzelnigg. ``Error analysis and improvements of coupled-cluster theory''. Theoretica chimica acta 80, 349–386 (1991).

[16] Rodney J. Bartlett, Stanislaw A. Kucharski, and Jozef Noga. ``Alternative coupled-cluster ansätze ii. the unitary coupled-cluster method''. Chemical Physics Letters 155, 133–140 (1989).

[17] Barbara M. Terhal. ``Quantum error correction for quantum memories''. Rev. Mod. Phys. 87, 307–346 (2015).

[18] A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. ``Classical and quantum computation''. American Mathematical Society. USA (2002). url: https:/​/​dx.doi.org/​10.1090/​gsm/​047.

[19] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''. Cambridge University Press. USA (2011). 10th edition. url: https:/​/​doi.org/​10.1017/​CBO9780511976667.

[20] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. ``Quantum computational chemistry''. Rev. Mod. Phys. 92, 015003 (2020).

[21] Francesco A. Evangelista, Garnet Kin-Lic Chan, and Gustavo E. Scuseria. ``Exact parameterization of fermionic wave functions via unitary coupled cluster theory''. The Journal of Chemical Physics 151, 244112 (2019).

[22] Yordan S. Yordanov, David R. M. Arvidsson-Shukur, and Crispin H. W. Barnes. ``Efficient quantum circuits for quantum computational chemistry''. Phys. Rev. A 102, 062612 (2020).

[23] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets''. Nature 549, 242–246 (2017).

[24] 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, 1791 (2021).

[25] Ilya G. Ryabinkin, Tzu-Ching Yen, Scott N. Genin, and Artur F. Izmaylov. ``Qubit coupled cluster method: A systematic approach to quantum chemistry on a quantum computer''. Journal of Chemical Theory and Computation 14, 6317–6326 (2018).

[26] Arthur G. Rattew, Shaohan Hu, Marco Pistoia, Richard Chen, and Steve Wood. ``A domain-agnostic, noise-resistant, hardware-efficient evolutionary variational quantum eigensolver'' (2020). arXiv:1910.09694.

[27] D. Chivilikhin, A. Samarin, V. Ulyantsev, I. Iorsh, A. R. Oganov, and O. Kyriienko. ``Mog-vqe: Multiobjective genetic variational quantum eigensolver'' (2020). arXiv:2007.04424.

[28] Niladri Gomes, Anirban Mukherjee, Feng Zhang, Thomas Iadecola, Cai-Zhuang Wang, Kai-Ming Ho, Peter P. Orth, and Yong-Xin Yao. ``Adaptive variational quantum imaginary time evolution approach for ground state preparation'' (2021). arXiv:2102.01544.

[29] Yordan S. Yordanov, V. Armaos, Crispin H. W. Barnes, and David R. M. Arvidsson-Shukur. ``Qubit-excitation-based adaptive variational quantum eigensolver''. Communications Physics 4, 228 (2021).

[30] Jie Liu, Zhenyu Li, and Jinlong Yang. ``An efficient adaptive variational quantum solver of the schrödinger equation based on reduced density matrices''. The Journal of Chemical Physics 154, 244112 (2021). arXiv:https:/​/​doi.org/​10.1063/​5.0054822.

[31] Ilya G. Ryabinkin, Robert A. Lang, Scott N. Genin, and Artur F. Izmaylov. ``Iterative qubit coupled cluster approach with efficient screening of generators''. Journal of Chemical Theory and Computation 16, 1055–1063 (2020).

[32] Michael. Tinkham. ``Group theory and quantum mechanics''. New York: McGraw-Hill. (1964).

[33] V.O. Shkolnikov and Harper R Grimsley. ``Code used to generate minimal complete pool and find the ground state of Lithium hydrid''. https:/​/​github.com/​VladShkolnikov/​LiH_dissociation_curve (2020).

[34] V.O. Shkolnikov and Harper R Grimsley. ``Code used to generate minimal complete pool and find the ground state of Beryllium hydrid''. https:/​/​github.com/​VladShkolnikov/​BeH2_dissociation_curve (2020).

Cited by

[1] John S. Van Dyke, Karunya Shirali, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Scaling adaptive quantum simulation algorithms via operator pool tiling", Physical Review Research 6 1, L012030 (2024).

[2] David Linteau, Stefano Barison, Netanel H. Lindner, and Giuseppe Carleo, "Adaptive projected variational quantum dynamics", Physical Review Research 6 2, 023130 (2024).

[3] Hakon Volkmann, Raamamurthy Sathyanarayanan, Alejandro Saenz, Karl Jansen, and Stefan Kühn, "Chemically Accurate Potential Curves for H2 Molecules Using Explicitly Correlated Qubit-ADAPT", Journal of Chemical Theory and Computation 20 3, 1244 (2024).

[4] Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, "Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits", PRX Quantum 5 2, 020315 (2024).

[5] Hugh G. A. Burton, Daniel Marti-Dafcik, David P. Tew, and David J. Wales, "Exact electronic states with shallow quantum circuits from global optimisation", npj Quantum Information 9 1, 75 (2023).

[6] Alicia B. Magann, Sophia E. Economou, and Christian Arenz, "Randomized adaptive quantum state preparation", Physical Review Research 5 3, 033227 (2023).

[7] Panagiotis G. Anastasiou, Yanzhu Chen, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "TETRIS-ADAPT-VQE: An adaptive algorithm that yields shallower, denser circuit Ansätze", Physical Review Research 6 1, 013254 (2024).

[8] Huo Chen, Niladri Gomes, Siyuan Niu, and Wibe Albert de Jong, "Adaptive variational simulation for open quantum systems", Quantum 8, 1252 (2024).

[9] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[10] Panagiotis G. Anastasiou, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "How to really measure operator gradients in ADAPT-VQE", arXiv:2306.03227, (2023).

[11] Anirban Mukherjee, Noah F. Berthusen, João C. Getelina, Peter P. Orth, and Yong-Xin Yao, "Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models", Communications Physics 6 1, 4 (2023).

[12] Ada Warren, Linghua Zhu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Adaptive variational algorithms for quantum Gibbs state preparation", arXiv:2203.12757, (2022).

[13] Tatiana A. Bespalova and Oleksandr Kyriienko, "Quantum simulation and ground state preparation for the honeycomb Kitaev model", arXiv:2109.13883, (2021).

[14] Iman Marvian, "Theory of Quantum Circuits with Abelian Symmetries", arXiv:2302.12466, (2023).

[15] Mariia D. Sapova and Aleksey K. Fedorov, "Variational quantum eigensolver techniques for simulating carbon monoxide oxidation", Communications Physics 5 1, 199 (2022).

[16] Takashi Tsuchimochi, Masaki Taii, Taisei Nishimaki, and Seiichiro L. Ten-no, "Adaptive construction of shallower quantum circuits with quantum spin projection for fermionic systems", Physical Review Research 4 3, 033100 (2022).

[17] Dmitry A. Fedorov, Yuri Alexeev, Stephen K. Gray, and Matthew Otten, "Unitary Selective Coupled-Cluster Method", Quantum 6, 703 (2022).

[18] Mohammad Haidar, Marko J. Rančić, Thomas Ayral, Yvon Maday, and Jean-Philip Piquemal, "Open Source Variational Quantum Eigensolver Extension of the Quantum Learning Machine (QLM) for Quantum Chemistry", arXiv:2206.08798, (2022).

[19] Scott E. Smart and David A. Mazziotti, "Many-fermion simulation from the contracted quantum eigensolver without fermionic encoding of the wave function", Physical Review A 105 6, 062424 (2022).

[20] Yordan S. Yordanov, Crispin H. W. Barnes, and David R. M. Arvidsson-Shukur, "Molecular-excited-state calculations with the qubit-excitation-based adaptive variational quantum eigensolver protocol", Physical Review A 106 3, 032434 (2022).

[21] Luke W. Bertels, Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Symmetry breaking slows convergence of the ADAPT Variational Quantum Eigensolver", arXiv:2207.03063, (2022).

[22] Ge Bai and Iman Marvian, "Synthesis of Energy-Conserving Quantum Circuits with XY interaction", arXiv:2309.11051, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-21 23:38:01) and SAO/NASA ADS (last updated successfully 2024-06-21 23:38:02). The list may be incomplete as not all publishers provide suitable and complete citation data.