Counterdiabaticity and the quantum approximate optimization algorithm

Jonathan Wurtz; Peter J. Love

doi:10.22331/q-2022-01-27-635

Counterdiabaticity and the quantum approximate optimization algorithm

Jonathan Wurtz and Peter J. Love

Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

Published:	2022-01-27, volume 6, page 635
Eprint:	arXiv:2106.15645v3
Doi:	https://doi.org/10.22331/q-2022-01-27-635
Citation:	Quantum 6, 635 (2022).

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

Abstract

The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the $p\to\infty$ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of $p$ large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" terms to approximate adiabatic gauge potentials which suppress diabatic excitations arising from finite ramp speed. In our construction, these "error" terms are helpful, not detrimental, to QAOA. Using this matching to link QAOA with quantum adiabatic algorithms (QAA), we show that the approximation ratio converges to one at least as $1-C(p)\sim 1/p^{\mu}$. We show that transfer of parameters between graphs, and interpolating angles for $p+1$ given $p$ are both natural byproducts of CD-QAOA matching. Optimization of CD-QAOA angles is equivalent to optimizing a continuous adiabatic schedule. Finally, we show that, using a property of variational adiabatic gauge potentials, QAOA is at least counterdiabatic, not just adiabatic, and has better performance than finite time adiabatic evolution. We demonstrate the method on three examples: a 2 level system, an Ising chain, and the MaxCut problem.

Featured image: Schematic diagram of the CD-QAOA angle matching procedure. As input, the procedure receives the counterdiabatic annealing protocol (Black line), effective Hamiltonian H, and number of QAOA steps p. For each step of the CD-QAOA evolution (pink), the effective unitaries over the interval are matched by minimizing the difference of their generators (right). This procedure sets the value of each angle γ, β in the QAOA (blue). By best matching along each step, the procedure fixes the angles {γ, β} and total adiabatic annealing time T.

► BibTeX data

@article{Wurtz2022counterdiabaticity,
  doi = {10.22331/q-2022-01-27-635},
  url = {https://doi.org/10.22331/q-2022-01-27-635},
  title = {Counterdiabaticity and the quantum approximate optimization algorithm},
  author = {Wurtz, Jonathan and Love, Peter J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {635},
  month = jan,
  year = {2022}
}

► References

[1] E. Farhi, J. Goldstone, and S. Gutmann, (2014), arXiv:1411.4028 [quant-ph].
arXiv:1411.4028

[2] J. Preskill, Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[3] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, (2021), arXiv:2101.08448 [quant-ph].
arXiv:2101.08448

[4] A. Lucas, Frontiers in Physics 2 (2014), 10.3389/fphy.2014.00005.
https://doi.org/10.3389/fphy.2014.00005

[5] T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018).
https://doi.org/10.1103/RevModPhys.90.015002

[6] M. H. S. Amin, Phys. Rev. Lett. 102, 220401 (2009).
https://doi.org/10.1103/PhysRevLett.102.220401

[7] S. H. Sack and M. Serbyn, (2021), arXiv:2101.05742 [quant-ph].
https://doi.org/10.22331/q-2021-07-01-491
arXiv:2101.05742

[8] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Rev. Mod. Phys. 91, 045001 (2019).
https://doi.org/10.1103/RevModPhys.91.045001

[9] G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508 (2008).
https://doi.org/10.1103/PhysRevA.78.052508

[10] S. Bachmann, W. De Roeck, and M. Fraas, Phys. Rev. Lett. 119, 060201 (2017).
https://doi.org/10.1103/PhysRevLett.119.060201

[11] A. del Campo and W. H. Zurek, International Journal of Modern Physics A 29, 1430018 (2014).
https://doi.org/10.1142/s0217751x1430018x

[12] J. I. Latorre and R. Orús, Phys. Rev. A 69, 062302 (2004).
https://doi.org/10.1103/PhysRevA.69.062302

[13] R. Barankov and A. Polkovnikov, Physical Review Letters 101 (2008), 10.1103/physrevlett.101.076801.
https://doi.org/10.1103/physrevlett.101.076801

[14] B. F. Schiffer, J. Tura, and J. I. Cirac, (2021), arXiv:2103.01226 [quant-ph].
arXiv:2103.01226

[15] L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Physical Review X 10 (2020), 10.1103/physrevx.10.021067.
https://doi.org/10.1103/physrevx.10.021067

[16] F. G. S. L. Brandao, M. Broughton, E. Farhi, S. Gutmann, and H. Neven, (2018), arXiv:1812.04170 [quant-ph].
arXiv:1812.04170

[17] Z.-C. Yang, A. Rahmani, A. Shabani, H. Neven, and C. Chamon, Physical Review X 7 (2017), 10.1103/physrevx.7.021027.
https://doi.org/10.1103/physrevx.7.021027

[18] L. T. Brady, C. L. Baldwin, A. Bapat, Y. Kharkov, and A. V. Gorshkov, Phys. Rev. Lett. 126, 070505 (2021a).
https://doi.org/10.1103/PhysRevLett.126.070505

[19] M. Streif and M. Leib, Quantum Science and Technology 5, 034008 (2020).
https://doi.org/10.1088/2058-9565/ab8c2b

[20] M. Born and V. Fock, Zeitschrift für Physik 51, 165 (1928).
https://doi.org/10.1007/BF01343193

[21] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, (2000), arXiv:quant-ph/0001106 [quant-ph].
arXiv:quant-ph/0001106

[22] A. T. Rezakhani, W.-J. Kuo, A. Hamma, D. A. Lidar, and P. Zanardi, Phys. Rev. Lett. 103, 080502 (2009).
https://doi.org/10.1103/PhysRevLett.103.080502

[23] C. Brif, M. D. Grace, M. Sarovar, and K. C. Young, New Journal of Physics 16, 065013 (2014).
https://doi.org/10.1088/1367-2630/16/6/065013

[24] L. T. Brady, L. Kocia, P. Bienias, A. Bapat, Y. Kharkov, and A. V. Gorshkov, (2021b), arXiv:2107.01218 [quant-ph].
arXiv:2107.01218

[25] S. Bao, S. Kleer, R. Wang, and A. Rahmani, Phys. Rev. A 97, 062343 (2018).
https://doi.org/10.1103/PhysRevA.97.062343

[26] T. Chasseur, L. S. Theis, Y. R. Sanders, D. J. Egger, and F. K. Wilhelm, Phys. Rev. A 91, 043421 (2015).
https://doi.org/10.1103/PhysRevA.91.043421

[27] P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Phys. Rev. Lett. 123, 090602 (2019).
https://doi.org/10.1103/PhysRevLett.123.090602

[28] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011).

[29] H. Nishimori and K. Takada, Frontiers in ICT 4, 2 (2017).
https://doi.org/10.3389/fict.2017.00002

[30] M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Nature 473, 194 (2011).
https://doi.org/10.1038/nature10012

[31] S. Sugiura, P. W. Claeys, A. Dymarsky, and A. Polkovnikov, Phys. Rev. Research 3, 013102 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013102

[32] M. V. Berry, Journal of Physics A: Mathematical and Theoretical 42, 365303 (2009).
https://doi.org/10.1088/1751-8113/42/36/365303

[33] M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Physics Reports 697, 1 (2017).
https://doi.org/10.1016/j.physrep.2017.07.001

[34] M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Phys. Rev. X 10, 041017 (2020).
https://doi.org/10.1103/PhysRevX.10.041017

[35] D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences 114, E3909 (2017).
https://doi.org/10.1073/pnas.1619826114

[36] M. B. Hastings and X.-G. Wen, Phys. Rev. B 72, 045141 (2005).
https://doi.org/10.1103/PhysRevB.72.045141

[37] J. Wurtz and A. Polkovnikov, Phys. Rev. B 101, 195138 (2020).
https://doi.org/10.1103/PhysRevB.101.195138

[38] N. Hatano and M. Suzuki, Lecture Notes in Physics , 37–68 (2005).
https://doi.org/10.1007/11526216_2

[39] S. Blanes, F. Casas, J. Oteo, and J. Ros, Physics Reports 470, 151–238 (2009).
https://doi.org/10.1016/j.physrep.2008.11.001

[40] X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010).
https://doi.org/10.1103/PhysRevLett.105.123003

[41] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
https://doi.org/10.1103/PhysRevLett.95.245701

[42] M. Kolodrubetz, B. K. Clark, and D. A. Huse, Physical Review Letters 109 (2012), 10.1103/physrevlett.109.015701.
https://doi.org/10.1103/physrevlett.109.015701

[43] A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, (2015), arXiv:1012.0653 [cond-mat.stat-mech].
arXiv:1012.0653

[44] D. J. Egger, J. Mareček, and S. Woerner, Quantum 5, 479 (2021).
https://doi.org/10.22331/q-2021-06-17-479

[45] A. G. R. Day, M. Bukov, P. Weinberg, P. Mehta, and D. Sels, Phys. Rev. Lett. 122, 020601 (2019).
https://doi.org/10.1103/PhysRevLett.122.020601

[46] G. B. Mbeng, R. Fazio, and G. Santoro, (2019), arXiv:1906.08948 [quant-ph].
arXiv:1906.08948

[47] V. Viswanath and G. Müller, The Recursion Method (Springer US, 2008).

[48] J. Wurtz and P. Love, Phys. Rev. A 103, 042612 (2021).
https://doi.org/10.1103/PhysRevA.103.042612

Cited by

[1] Akel Hashim, Rich Rines, Victory Omole, Ravi K. Naik, John Mark Kreikebaum, David I. Santiago, Frederic T. Chong, Irfan Siddiqi, and Pranav Gokhale, "Optimized SWAP networks with equivalent circuit averaging for QAOA", Physical Review Research 4 3, 033028 (2022).

[2] Zhiqiang Fan, Jinchen Xu, Guoqiang Shu, Xiaodong Ding, Hang Lian, and Zheng Shan, "Solving the Shortest Path Problem with QAOA", SPIN 13 01, 2350002 (2023).

[3] Danylo Lykov, Jonathan Wurtz, Cody Poole, Mark Saffman, Tom Noel, and Yuri Alexeev, "Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm", npj Quantum Information 9 1, 73 (2023).

[4] Lucas K. Kovalsky, Fernando A. Calderon-Vargas, Matthew D. Grace, Alicia B. Magann, James B. Larsen, Andrew D. Baczewski, and Mohan Sarovar, "Self-Healing of Trotter Error in Digital Adiabatic State Preparation", Physical Review Letters 131 6, 060602 (2023).

[5] Stefan H. Sack, Raimel A. Medina, Richard Kueng, and Maksym Serbyn, "Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement", Physical Review A 107 6, 062404 (2023).

[6] Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon, and Robert Joynt, "Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm", Physical Review Research 5 2, 023147 (2023).

[7] Michelle Chalupnik, Hans Melo, Yuri Alexeev, and Alexey Galda, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 97 (2022) ISBN:978-1-6654-9113-6.

[8] Zheng-Hang Sun, Yong-Yi Wang, Jian Cui, and Heng Fan, "Improving the performance of quantum approximate optimization for preparing non-trivial quantum states without translational symmetry", New Journal of Physics 25 1, 013015 (2023).

[9] Luis S. Yagüe Bosch, Tim Ehret, Francesco Petiziol, Ennio Arimondo, and Sandro Wimberger, "Shortcut‐to‐Adiabatic Controlled‐Phase Gate in Rydberg Atoms", Annalen der Physik 2300275 (2023).

[10] John Golden, Andreas Bärtschi, Daniel O'Malley, and Stephan Eidenbenz, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 307 (2023) ISBN:979-8-3503-4323-6.

[11] Antonio A. Mele, Glen B. Mbeng, Giuseppe E. Santoro, Mario Collura, and Pietro Torta, "Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz", Physical Review A 106 6, L060401 (2022).

[12] Phillip C. Lotshaw, George Siopsis, James Ostrowski, Rebekah Herrman, Rizwanul Alam, Sarah Powers, and Travis S. Humble, "Approximate Boltzmann distributions in quantum approximate optimization", Physical Review A 108 4, 042411 (2023).

[13] P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and Xi Chen, "Digitized-counterdiabatic quantum approximate optimization algorithm", Physical Review Research 4 1, 013141 (2022).

[14] Ieva Čepaitė, Anatoli Polkovnikov, Andrew J. Daley, and Callum W. Duncan, "Counterdiabatic Optimized Local Driving", PRX Quantum 4 1, 010312 (2023).

[15] Nicolas PD Sawaya, Albert T Schmitz, and Stuart Hadfield, "Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems", Quantum 7, 1111 (2023).

[16] Dan Sun, Pranav Chandarana, Zi-Hua Xin, and Xi Chen, "Optimizing counterdiabaticity by variational quantum circuits", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 2239, 20210282 (2022).

[17] Stuart Hadfield, Tad Hogg, and Eleanor G Rieffel, "Analytical framework for quantum alternating operator ansätze", Quantum Science and Technology 8 1, 015017 (2023).

[18] Zeqiao Zhou, Yuxuan Du, Xinmei Tian, and Dacheng Tao, "QAOA-in-QAOA: Solving Large-Scale MaxCut Problems on Small Quantum Machines", Physical Review Applied 19 2, 024027 (2023).

[19] Pranav Chandarana, Pablo Suárez Vieites, Narendra N Hegade, Enrique Solano, Yue Ban, and Xi Chen, "Meta-learning digitized-counterdiabatic quantum optimization", Quantum Science and Technology 8 4, 045007 (2023).

[20] N. N. Hegade, P. Chandarana, K. Paul, Xi Chen, F. Albarrán-Arriagada, and E. Solano, "Portfolio optimization with digitized counterdiabatic quantum algorithms", Physical Review Research 4 4, 043204 (2022).

[21] Wenyang Qian, Robert A. M. Basili, Mary Mehrnoosh Eshaghian-Wilner, Ashfaq Khokhar, Glenn Luecke, and James P. Vary, "Comparative Study of Variations in Quantum Approximate Optimization Algorithms for the Traveling Salesman Problem", Entropy 25 8, 1238 (2023).

[22] Pranav Chandarana, Narendra N. Hegade, Iraitz Montalban, Enrique Solano, and Xi Chen, "Digitized Counterdiabatic Quantum Algorithm for Protein Folding", Physical Review Applied 20 1, 014024 (2023).

[23] Qing Xie, Kazuhiro Seki, and Seiji Yunoki, "Variational counterdiabatic driving of the Hubbard model for ground-state preparation", Physical Review B 106 15, 155153 (2022).

[24] Pietro Torta, Glen B. Mbeng, Carlo Baldassi, Riccardo Zecchina, and Giuseppe E. Santoro, "Quantum approximate optimization algorithm applied to the binary perceptron", Physical Review B 107 9, 094202 (2023).

[25] Johannes Weidenfeller, Lucia C. Valor, Julien Gacon, Caroline Tornow, Luciano Bello, Stefan Woerner, and Daniel J. Egger, "Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware", Quantum 6, 870 (2022).

[26] Phillip C. Lotshaw, Kevin D. Battles, Bryan Gard, Gilles Buchs, Travis S. Humble, and Creston D. Herold, "Modeling noise in global Mølmer-Sørensen interactions applied to quantum approximate optimization", Physical Review A 107 6, 062406 (2023).

[27] Abhinav Suresh, Vishal Varma, Priya Batra, and T. S. Mahesh, "Counterdiabatic driving for long-lived singlet state preparation", The Journal of Chemical Physics 159 2, 024202 (2023).

[28] G. Wendin, Encyclopedia of Condensed Matter Physics 246 (2024) ISBN:9780323914086.

[29] Yahui Chai, Yong-Jian Han, Yu-Chun Wu, Ye Li, Menghan Dou, and Guo-Ping Guo, "Shortcuts to the quantum approximate optimization algorithm", Physical Review A 105 4, 042415 (2022).

[30] Phillip C. Lotshaw, Thien Nguyen, Anthony Santana, Alexander McCaskey, Rebekah Herrman, James Ostrowski, George Siopsis, and Travis S. Humble, "Scaling quantum approximate optimization on near-term hardware", Scientific Reports 12 1, 12388 (2022).

[31] Narendra N. Hegade, Xi Chen, and Enrique Solano, "Digitized counterdiabatic quantum optimization", Physical Review Research 4 4, L042030 (2022).

[32] Camille Grange, Michael Poss, and Eric Bourreau, "An introduction to variational quantum algorithms for combinatorial optimization problems", 4OR 21 3, 363 (2023).

[33] Vladimir Kremenetski, Tad Hogg, Stuart Hadfield, Stephen J. Cotton, and Norm M. Tubman, "Quantum Alternating Operator Ansatz (QAOA) Phase Diagrams and Applications for Quantum Chemistry", arXiv:2108.13056, (2021).

[34] Jiahao Yao, Lin Lin, and Marin Bukov, "Reinforcement Learning for Many-Body Ground-State Preparation Inspired by Counterdiabatic Driving", Physical Review X 11 3, 031070 (2021).

[35] James Sud, Stuart Hadfield, Eleanor Rieffel, Norm Tubman, and Tad Hogg, "A Parameter Setting Heuristic for the Quantum Alternating Operator Ansatz", arXiv:2211.09270, (2022).

[36] Ruslan Shaydulin, Phillip C. Lotshaw, Jeffrey Larson, James Ostrowski, and Travis S. Humble, "Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut", arXiv:2201.11785, (2022).

[37] Jiahao Yao, Haoya Li, Marin Bukov, Lin Lin, and Lexing Ying, "Monte Carlo Tree Search based Hybrid Optimization of Variational Quantum Circuits", arXiv:2203.16707, (2022).

[38] Lucas T. Brady, Lucas Kocia, Przemyslaw Bienias, Aniruddha Bapat, Yaroslav Kharkov, and Alexey V. Gorshkov, "Behavior of Analog Quantum Algorithms", arXiv:2107.01218, (2021).

[39] Jonathan Wurtz and Danylo Lykov, "Fixed-angle conjectures for the quantum approximate optimization algorithm on regular MaxCut graphs", Physical Review A 104 5, 052419 (2021).

[40] Changhao Yi, "Success of digital adiabatic simulation with large Trotter step", Physical Review A 104 5, 052603 (2021).

[41] Jonathan Wurtz and Danylo Lykov, "The fixed angle conjecture for QAOA on regular MaxCut graphs", arXiv:2107.00677, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2023-12-07 04:42:21) and SAO/NASA ADS (last updated successfully 2023-12-07 04:42:24). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.