Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?

Samson Wang1,2, Piotr Czarnik1,3,4, Andrew Arrasmith1,5, M. Cerezo1,5,6, Lukasz Cincio1,5, and Patrick J. Coles1,5

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Department of Physics, Imperial College London, London, SW7 2AZ, UK
3Faculty of Physics, Astronomy, and Applied Computer Science, Jagiellonian University, Kraków, Poland
4Mark Kac Center for Complex Systems Research, Jagiellonian University, Kraków, Poland
5Quantum Science Center, Oak Ridge, TN 37931, USA
6Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

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Abstract

Variational Quantum Algorithms (VQAs) are often viewed as the best hope for near-term quantum advantage. However, recent studies have shown that noise can severely limit the trainability of VQAs, e.g., by exponentially flattening the cost landscape and suppressing the magnitudes of cost gradients. Error Mitigation (EM) shows promise in reducing the impact of noise on near-term devices. Thus, it is natural to ask whether EM can improve the trainability of VQAs. In this work, we first show that, for a broad class of EM strategies, exponential cost concentration cannot be resolved without committing exponential resources elsewhere. This class of strategies includes as special cases Zero Noise Extrapolation, Virtual Distillation, Probabilistic Error Cancellation, and Clifford Data Regression. Second, we perform analytical and numerical analysis of these EM protocols, and we find that some of them (e.g., Virtual Distillation) can make it harder to resolve cost function values compared to running no EM at all. As a positive result, we do find numerical evidence that Clifford Data Regression (CDR) can aid the training process in certain settings where cost concentration is not too severe. Our results show that care should be taken in applying EM protocols as they can either worsen or not improve trainability. On the other hand, our positive results for CDR highlight the possibility of engineering error mitigation methods to improve trainability.

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Cited by

[1] Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O'Brien, "Quantum error mitigation", Reviews of Modern Physics 95 4, 045005 (2023).

[2] Ryuji Takagi, Hiroyasu Tajima, and Mile Gu, "Universal Sampling Lower Bounds for Quantum Error Mitigation", Physical Review Letters 131 21, 210602 (2023).

[3] Louis Schatzki, Andrew Arrasmith, Patrick J. Coles, and M. Cerezo, "Entangled Datasets for Quantum Machine Learning", arXiv:2109.03400, (2021).

[4] Ryuji Takagi, Suguru Endo, Shintaro Minagawa, and Mile Gu, "Fundamental limits of quantum error mitigation", npj Quantum Information 8, 114 (2022).

[5] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, "Theory of overparametrization in quantum neural networks", arXiv:2109.11676, (2021).

[6] Valentin Heyraud, Zejian Li, Kaelan Donatella, Alexandre Le Boité, and Cristiano Ciuti, "Efficient Estimation of Trainability for Variational Quantum Circuits", PRX Quantum 4 4, 040335 (2023).

[7] Yihui Quek, Daniel Stilck França, Sumeet Khatri, Johannes Jakob Meyer, and Jens Eisert, "Exponentially tighter bounds on limitations of quantum error mitigation", arXiv:2210.11505, (2022).

[8] Kento Tsubouchi, Takahiro Sagawa, and Nobuyuki Yoshioka, "Universal Cost Bound of Quantum Error Mitigation Based on Quantum Estimation Theory", Physical Review Letters 131 21, 210601 (2023).

[9] Patrick J. Coles, Collin Szczepanski, Denis Melanson, Kaelan Donatella, Antonio J. Martinez, and Faris Sbahi, "Thermodynamic AI and the fluctuation frontier", arXiv:2302.06584, (2023).

[10] R. Au-Yeung, B. Camino, O. Rathore, and V. Kendon, "Quantum algorithms for scientific applications", arXiv:2312.14904, (2023).

[11] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras", PRX Quantum 3 1, 010345 (2022).

[12] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics, and Astronomy 66 5, 250302 (2023).

[13] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum error mitigation as a universal error-minimization technique: applications from NISQ to FTQC eras", arXiv:2010.03887, (2020).

[14] Gokul Subramanian Ravi, Pranav Gokhale, Yi Ding, William M. Kirby, Kaitlin N. Smith, Jonathan M. Baker, Peter J. Love, Henry Hoffmann, Kenneth R. Brown, and Frederic T. Chong, "CAFQA: A classical simulation bootstrap for variational quantum algorithms", arXiv:2202.12924, (2022).

[15] Supanut Thanasilp, Samson Wang, M. Cerezo, and Zoë Holmes, "Exponential concentration and untrainability in quantum kernel methods", arXiv:2208.11060, (2022).

[16] Giacomo De Palma, Milad Marvian, Cambyse Rouzé, and Daniel Stilck França, "Limitations of Variational Quantum Algorithms: A Quantum Optimal Transport Approach", PRX Quantum 4 1, 010309 (2023).

[17] Abhinav Deshpande, Pradeep Niroula, Oles Shtanko, Alexey V. Gorshkov, Bill Fefferman, and Michael J. Gullans, "Tight Bounds on the Convergence of Noisy Random Circuits to the Uniform Distribution", PRX Quantum 3 4, 040329 (2022).

[18] Frédéric Sauvage, Martín Larocca, Patrick J. Coles, and M. Cerezo, "Building spatial symmetries into parameterized quantum circuits for faster training", Quantum Science and Technology 9 1, 015029 (2024).

[19] Ingo Tews, Zohreh Davoudi, Andreas Ekström, Jason D. Holt, Kevin Becker, Raúl Briceño, David J. Dean, William Detmold, Christian Drischler, Thomas Duguet, Evgeny Epelbaum, Ashot Gasparyan, Jambul Gegelia, Jeremy R. Green, Harald W. Grießhammer, Andrew D. Hanlon, Matthias Heinz, Heiko Hergert, Martin Hoferichter, Marc Illa, David Kekejian, Alejandro Kievsky, Sebastian König, Hermann Krebs, Kristina D. Launey, Dean Lee, Petr Navrátil, Amy Nicholson, Assumpta Parreño, Daniel R. Phillips, Marek Płoszajczak, Xiu-Lei Ren, Thomas R. Richardson, Caroline Robin, Grigor H. Sargsyan, Martin J. Savage, Matthias R. Schindler, Phiala E. Shanahan, Roxanne P. Springer, Alexander Tichai, Ubirajara van Kolck, Michael L. Wagman, André Walker-Loud, Chieh-Jen Yang, and Xilin Zhang, "Nuclear Forces for Precision Nuclear Physics: A Collection of Perspectives", Few-Body Systems 63 4, 67 (2022).

[20] Adam Callison and Nicholas Chancellor, "Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond", Physical Review A 106 1, 010101 (2022).

[21] C. Huerta Alderete, Max Hunter Gordon, Frédéric Sauvage, Akira Sone, Andrew T. Sornborger, Patrick J. Coles, and M. Cerezo, "Inference-Based Quantum Sensing", Physical Review Letters 129 19, 190501 (2022).

[22] Supanut Thanasilp, Samson Wang, Nhat A. Nghiem, Patrick J. Coles, and M. Cerezo, "Subtleties in the trainability of quantum machine learning models", arXiv:2110.14753, (2021).

[23] Laurin E. Fischer, Daniel Miller, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Daniel J. Egger, and Ivano Tavernelli, "Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space", Physical Review Research 4 3, 033027 (2022).

[24] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[25] Kazunobu Maruyoshi, Takuya Okuda, Juan W. Pedersen, Ryo Suzuki, Masahito Yamazaki, and Yutaka Yoshida, "Conserved charges in the quantum simulation of integrable spin chains", Journal of Physics A Mathematical General 56 16, 165301 (2023).

[26] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian G. Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", arXiv:2112.00760, (2021).

[27] Manuel S. Rudolph, Sacha Lerch, Supanut Thanasilp, Oriel Kiss, Sofia Vallecorsa, Michele Grossi, and Zoë Holmes, "Trainability barriers and opportunities in quantum generative modeling", arXiv:2305.02881, (2023).

[28] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and Opportunities in Quantum Machine Learning", arXiv:2303.09491, (2023).

[29] Zhenyu Cai, "A Practical Framework for Quantum Error Mitigation", arXiv:2110.05389, (2021).

[30] Minh C. Tran, Kunal Sharma, and Kristan Temme, "Locality and Error Mitigation of Quantum Circuits", arXiv:2303.06496, (2023).

[31] Tailong Xiao, Xinliang Zhai, Xiaoyan Wu, Jianping Fan, and Guihua Zeng, "Practical advantage of quantum machine learning in ghost imaging", Communications Physics 6 1, 171 (2023).

[32] Keita Kanno, Masaya Kohda, Ryosuke Imai, Sho Koh, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa, "Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers", arXiv:2302.11320, (2023).

[33] Marvin Bechtold, Johanna Barzen, Frank Leymann, Alexander Mandl, Julian Obst, Felix Truger, and Benjamin Weder, "Investigating the effect of circuit cutting in QAOA for the MaxCut problem on NISQ devices", Quantum Science and Technology 8 4, 045022 (2023).

[34] Cristina Cirstoiu, Silas Dilkes, Daniel Mills, Seyon Sivarajah, and Ross Duncan, "Volumetric Benchmarking of Error Mitigation with Qermit", Quantum 7, 1059 (2023).

[35] N. M. Guseynov, A. A. Zhukov, W. V. Pogosov, and A. V. Lebedev, "Depth analysis of variational quantum algorithms for the heat equation", Physical Review A 107 5, 052422 (2023).

[36] Christoph Hirche, Cambyse Rouzé, and Daniel Stilck França, "On contraction coefficients, partial orders and approximation of capacities for quantum channels", arXiv:2011.05949, (2020).

[37] Piotr Czarnik, Michael McKerns, Andrew T. Sornborger, and Lukasz Cincio, "Improving the efficiency of learning-based error mitigation", arXiv:2204.07109, (2022).

[38] Antonio Anna Mele, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck França, and Yihui Quek, "Noise-induced shallow circuits and absence of barren plateaus", arXiv:2403.13927, (2024).

[39] Daniel Bultrini, Samson Wang, Piotr Czarnik, Max Hunter Gordon, M. Cerezo, Patrick J. Coles, and Lukasz Cincio, "The battle of clean and dirty qubits in the era of partial error correction", arXiv:2205.13454, (2022).

[40] Muhammad Kashif and Saif Al-Kuwari, "The impact of cost function globality and locality in hybrid quantum neural networks on NISQ devices", Machine Learning: Science and Technology 4 1, 015004 (2023).

[41] Muhammad Kashif and Saif Al-kuwari, "ResQNets: A Residual Approach for Mitigating Barren Plateaus in Quantum Neural Networks", arXiv:2305.03527, (2023).

[42] P. Singkanipa and D. A. Lidar, "Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and fixed points", arXiv:2402.08721, (2024).

[43] Olivia Di Matteo and R. M. Woloshyn, "Quantum computing fidelity susceptibility using automatic differentiation", Physical Review A 106 5, 052429 (2022).

[44] Enrico Fontana, Ivan Rungger, Ross Duncan, and Cristina Cîrstoiu, "Spectral analysis for noise diagnostics and filter-based digital error mitigation", arXiv:2206.08811, (2022).

[45] Piotr Czarnik, Michael McKerns, Andrew T. Sornborger, and Lukasz Cincio, "Robust design under uncertainty in quantum error mitigation", arXiv:2307.05302, (2023).

[46] Matteo Robbiati, Alejandro Sopena, Andrea Papaluca, and Stefano Carrazza, "Real-time error mitigation for variational optimization on quantum hardware", arXiv:2311.05680, (2023).

[47] Wei-Bin Ewe, Dax Enshan Koh, Siong Thye Goh, Hong-Son Chu, and Ching Eng Png, "Variational Quantum-Based Simulation of Waveguide Modes", IEEE Transactions on Microwave Theory Techniques 70 5, 2517 (2022).

[48] Zichang He, Bo Peng, Yuri Alexeev, and Zheng Zhang, "Distributionally Robust Variational Quantum Algorithms with Shifted Noise", arXiv:2308.14935, (2023).

[49] Nico Meyer, Daniel D. Scherer, Axel Plinge, Christopher Mutschler, and Michael J. Hartmann, "Quantum Natural Policy Gradients: Towards Sample-Efficient Reinforcement Learning", arXiv:2304.13571, (2023).

[50] Siddharth Dangwal, Gokul Subramanian Ravi, Poulami Das, Kaitlin N. Smith, Jonathan M. Baker, and Frederic T. Chong, "VarSaw: Application-tailored Measurement Error Mitigation for Variational Quantum Algorithms", arXiv:2306.06027, (2023).

[51] Christoph Hirche, Cambyse Rouzé, and Daniel Stilck França, "On contraction coefficients, partial orders and approximation of capacities for quantum channels", Quantum 6, 862 (2022).

[52] Jessie M. Henderson, Marianna Podzorova, M. Cerezo, John K. Golden, Leonard Gleyzer, Hari S. Viswanathan, and Daniel O'Malley, "Quantum Algorithms for Geologic Fracture Networks", arXiv:2210.11685, (2022).

[53] Marco Schumann, Frank K. Wilhelm, and Alessandro Ciani, "Emergence of noise-induced barren plateaus in arbitrary layered noise models", arXiv:2310.08405, (2023).

[54] André Melo, Nathan Earnest-Noble, and Francesco Tacchino, "Pulse-efficient quantum machine learning", Quantum 7, 1130 (2023).

[55] Jessie M. Henderson, Marianna Podzorova, M. Cerezo, John K. Golden, Leonard Gleyzer, Hari S. Viswanathan, and Daniel O'Malley, "Quantum algorithms for geologic fracture networks", Scientific Reports 13, 2906 (2023).

[56] Sharu Theresa Jose and Osvaldo Simeone, "Error Mitigation-Aided Optimization of Parameterized Quantum Circuits: Convergence Analysis", arXiv:2209.11514, (2022).

[57] Kosuke Ito and Keisuke Fujii, "SantaQlaus: A resource-efficient method to leverage quantum shot-noise for optimization of variational quantum algorithms", arXiv:2312.15791, (2023).

[58] Kevin Lively, Tim Bode, Jochen Szangolies, Jian-Xin Zhu, and Benedikt Fauseweh, "Robust Experimental Signatures of Phase Transitions in the Variational Quantum Eigensolver", arXiv:2402.18953, (2024).

The above citations are from SAO/NASA ADS (last updated successfully 2024-04-12 05:30:52). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-04-12 05:30:51).