Synergetic quantum error mitigation by randomized compiling and zero-noise extrapolation for the variational quantum eigensolver
1Quantum Laboratory, Fujitsu Research, Fujitsu Limited. 10-1 Morinosato-wakamiya, Atsugi, Kanagawa, Japan 243-0197
2Keysight Technologies Canada, 137 Glasgow St, Kitchener, ON, Canada, N2G 4X8
| Published: | 2023-11-20, volume 7, page 1184 |
| Eprint: | arXiv:2212.11198v3 |
| Doi: | https://doi.org/10.22331/q-2023-11-20-1184 |
| Citation: | Quantum 7, 1184 (2023). |
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Abstract
We propose a quantum error mitigation strategy for the variational quantum eigensolver (VQE) algorithm. We find, via numerical simulation, that very small amounts of coherent noise in VQE can cause substantially large errors that are difficult to suppress by conventional mitigation methods, and yet our proposed mitigation strategy is able to significantly reduce these errors. The proposed strategy is a combination of previously reported techniques, namely randomized compiling (RC) and zero-noise extrapolation (ZNE). Intuitively, randomized compiling turns coherent errors in the circuit into stochastic Pauli errors, which facilitates extrapolation to the zero-noise limit when evaluating the cost function. Our numerical simulation of VQE for small molecules shows that the proposed strategy can mitigate energy errors induced by various types of coherent noise by up to two orders of magnitude.
Popular summary
In this work, we propose an error mitigation technique that effectively reduces errors induced by coherent noise. This technique utilizes synergetic effect of randomized compiling (RC) and zero-noise extrapolation (ZNE). RC converts coherent noise into stochastic Pauli noise, which can be effectively mitigated using ZNE. Our numerical simulations on variational quantum eigensolver algorithms demonstrate that our proposed mitigation technique exhibits a significant error-suppressing effect against coherent noise.
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[1] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C Benjamin, and Xiao Yuan. ``Quantum computational chemistry''. Reviews of Modern Physics 92, 015003 (2020).
https://doi.org/10.1103/RevModPhys.92.015003
[2] Hari P Paudel, Madhava Syamlal, Scott E Crawford, Yueh-Lin Lee, Roman A Shugayev, Ping Lu, Paul R Ohodnicki, Darren Mollot, and Yuhua Duan. ``Quantum Computing and Simulations for Energy Applications: Review and Perspective''. ACS Engineering Au 2, 151–196 (2022).
https://doi.org/10.1021/acsengineeringau.1c00033
[3] Julia E Rice, Tanvi P Gujarati, Mario Motta, Tyler Y Takeshita, Eunseok Lee, Joseph A Latone, and Jeannette M Garcia. ``Quantum computation of dominant products in lithium–sulfur batteries''. The Journal of Chemical Physics 154, 134115 (2021).
https://doi.org/10.1063/5.0044068
[4] Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Physical Review A 86, 032324 (2012).
https://doi.org/10.1103/PhysRevA.86.032324
[5] 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).
https://doi.org/10.1038/ncomms5213
[6] 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).
https://doi.org/10.1088/1367-2630/18/2/023023
[7] Peter JJ O’Malley, Ryan Babbush, Ian D Kivlichan, Jonathan Romero, Jarrod R McClean, Rami Barends, Julian Kelly, Pedram Roushan, Andrew Tranter, Nan Ding, et al. ``Scalable Quantum Simulation of Molecular Energies''. Physical Review X 6, 031007 (2016).
https://doi.org/10.1103/PhysRevX.6.031007
[8] 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).
https://doi.org/10.1038/nature23879
[9] James I Colless, Vinay V Ramasesh, Dar Dahlen, Machiel S Blok, Mollie E Kimchi-Schwartz, Jarrod R McClean, Jonathan Carter, Wibe A de Jong, and Irfan Siddiqi. ``Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm''. Physical Review X 8, 011021 (2018).
https://doi.org/10.1103/PhysRevX.8.011021
[10] Abhinav Kandala, Kristan Temme, Antonio D Córcoles, Antonio Mezzacapo, Jerry M Chow, and Jay M Gambetta. ``Error mitigation extends the computational reach of a noisy quantum processor''. Nature 567, 491–495 (2019).
https://doi.org/10.1038/s41586-019-1040-7
[11] Yangchao Shen, Xiang Zhang, Shuaining Zhang, Jing-Ning Zhang, Man-Hong Yung, and Kihwan Kim. ``Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure''. Physical Review A 95, 020501 (2017).
https://doi.org/10.1103/PhysRevA.95.020501
[12] Yunseong Nam, Jwo-Sy Chen, Neal C Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R Brown, Stewart Allen, Jason M Amini, Joel Apisdorf, et al. ``Ground-state energy estimation of the water molecule on a trapped-ion quantum computer''. npj Quantum Information 6, 33 (2020).
https://doi.org/10.1038/s41534-020-0259-3
[13] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes''. Nature communications 9, 4812 (2018).
https://doi.org/10.1038/s41467-018-07090-4
[14] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H Booth, et al. ``The Variational Quantum Eigensolver: A review of methods and best practices''. Physics Reports 986, 1–128 (2022).
https://doi.org/10.1016/j.physrep.2022.08.003
[15] Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. ``Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation''. Journal of the Physical Society of Japan 90, 032001 (2021).
https://doi.org/10.7566/JPSJ.90.032001
[16] Ying Li and Simon C Benjamin. ``Efficient Variational Quantum Simulator Incorporating Active Error Minimization''. Physical Review X 7, 021050 (2017).
https://doi.org/10.1103/PhysRevX.7.021050
[17] Kristan Temme, Sergey Bravyi, and Jay M Gambetta. ``Error Mitigation for Short-Depth Quantum Circuits''. Physical review letters 119, 180509 (2017).
https://doi.org/10.1103/PhysRevLett.119.180509
[18] 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, 012426 (2020).
https://doi.org/10.1103/PhysRevA.102.012426
[19] Shuaining Zhang, Yao Lu, Kuan Zhang, Wentao Chen, Ying Li, Jing-Ning Zhang, and Kihwan Kim. ``Error-mitigated quantum gates exceeding physical fidelities in a trapped-ion system''. Nature communications 11, 587 (2020).
https://doi.org/10.1038/s41467-020-14376-z
[20] Jarrod R McClean, Mollie E Kimchi-Schwartz, Jonathan Carter, and Wibe A De Jong. ``Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states''. Physical Review A 95, 042308 (2017).
https://doi.org/10.1103/PhysRevA.95.042308
[21] Joel J Wallman and Joseph Emerson. ``Noise tailoring for scalable quantum computation via randomized compiling''. Physical Review A 94, 052325 (2016).
https://doi.org/10.1103/PhysRevA.94.052325
[22] Akel Hashim, Ravi K Naik, Alexis Morvan, Jean-Loup Ville, Bradley Mitchell, John Mark Kreikebaum, Marc Davis, Ethan Smith, Costin Iancu, Kevin P O’Brien, et al. ``Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor''. Physical Review X 11, 041039 (2021).
https://doi.org/10.1103/PhysRevX.11.041039
[23] 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, et al. ``Leveraging randomized compiling for the quantum imaginary-time-evolution algorithm''. Physical Review Research 4, 033140 (2021).
https://doi.org/10.1103/PhysRevResearch.4.033140
[24] 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''. Nature Physics 19, 752–759 (2023).
https://doi.org/10.1038/s41567-022-01914-3
[25] Chao Song, Jing Cui, H Wang, J Hao, H Feng, and Ying Li. ``Quantum computation with universal error mitigation on a superconducting quantum processor''. Science advances 5, eaaw5686 (2019).
https://doi.org/10.1126/sciadv.aaw5686
[26] Matthew Ware, Guilhem Ribeill, Diego Riste, Colm A Ryan, Blake Johnson, and Marcus P Da Silva. ``Experimental Pauli-frame randomization on a superconducting qubit''. Physical Review A 103, 042604 (2021).
https://doi.org/10.1103/PhysRevA.103.042604
[27] Samuele Ferracin, Akel Hashim, Jean-Loup Ville, Ravi Naik, Arnaud Carignan-Dugas, Hammam Qassim, Alexis Morvan, David I Santiago, Irfan Siddiqi, and Joel J Wallman. ``Efficiently improving the performance of noisy quantum computers'' (2022). arXiv:2201.10672.
arXiv:2201.10672
[28] Nick S Blunt, Laura Caune, Róbert Izsák, Earl T Campbell, and Nicole Holzmann. ``Statistical phase estimation and error mitigation on a superconducting quantum processor'' (2023). arXiv:2304.05126.
arXiv:2304.05126
[29] Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. ``Noise-induced barren plateaus in variational quantum algorithms''. Nature communications 12, 6961 (2021).
https://doi.org/10.1038/s41467-021-27045-6
[30] Michael A Nielsen and Isaac Chuang. ``Quantum Computation and Quantum Information''. Cambridge University Press. (2002).
https://doi.org/10.1017/CBO9780511976667
[31] Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, et al. ``Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry''. Nature communications 14, 1952 (2023).
https://doi.org/10.1038/s41467-023-37587-6
[32] Jérôme F Gonthier, Maxwell D Radin, Corneliu Buda, Eric J Doskocil, Clena M Abuan, and Jhonathan Romero. ``Measurements as a roadblock to near-term practical quantum advantage in chemistry: Resource analysis''. Physical Review Research 4, 033154 (2022).
https://doi.org/10.1103/PhysRevResearch.4.033154
[33] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. ``Efficient quantum measurement of Pauli operators in the presence of finite sampling error''. Quantum 5, 385 (2021).
https://doi.org/10.22331/q-2021-01-20-385
[34] Tomochika Kurita, Mikio Morita, Hirotaka Oshima, and Shintaro Sato. ``Pauli String Partitioning Algorithm with the Ising Model for Simultaneous Measurement''. The Journal of Physical Chemistry A 127, 1068–1080 (2023).
https://doi.org/10.1021/acs.jpca.2c06453
[35] Stefanie J. Beale, Arnaud Carignan-Dugas, Dar Dahlen, Joseph Emerson, Ian Hincks, Pavithran Iyer, Aditya Jain, David Hufnagel, Egor Ospadov, Hammam Qassim, et al. ``True-Q software. Keysight Technologies''. url: trueq.quantumbenchmark.com.
https://trueq.quantumbenchmark.com
[36] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, et al. ``SciPy 1.0: Fundamental algorithms for scientific computing in Python''. Nature Methods 17, 261–272 (2020).
https://doi.org/10.1038/s41592-019-0686-2
[37] Michael JD Powell. ``The BOBYQA algorithm for bound constrained optimization without derivatives''. Technical report. University of Cambridge, Cambridge (2009). url: www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf.
https://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf
[38] Jarrod R. McClean, Ian D. Kivlichan, Damian S. Steiger, Yudong Cao, E. Schuyler Fried, Craig Gidney, Thomas Häner, Vojtĕch Havlíček, Zhang Jiang, Matthew Neeley, et al. ``OpenFermion: The Electronic Structure Package for Quantum Computers'' (2017). arXiv:1710.07629.
arXiv:1710.07629
[39] Ewout van den Berg, Zlatko K Minev, Abhinav Kandala, and Kristan Temme. ``Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors''. Nature Physics 19, 1116–1121 (2023).
https://doi.org/10.1038/s41567-023-02042-2
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