The battle of clean and dirty qubits in the era of partial error correction

Daniel Bultrini1,2, Samson Wang1,3, Piotr Czarnik1,4, Max Hunter Gordon1,5, M. Cerezo6,7, Patrick J. Coles1,7, and Lukasz Cincio1,7

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany
3Imperial College London, London, UK
4Institute of Theoretical Physics, Jagiellonian University, Krakow, Poland.
5Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid 28049, Spain
6Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
7Quantum Science Center, Oak Ridge, TN 37931, USA

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When error correction becomes possible it will be necessary to dedicate a large number of physical qubits to each logical qubit. Error correction allows for deeper circuits to be run, but each additional physical qubit can potentially contribute an exponential increase in computational space, so there is a trade-off between using qubits for error correction or using them as noisy qubits. In this work we look at the effects of using noisy qubits in conjunction with noiseless qubits (an idealized model for error-corrected qubits), which we call the "clean and dirty" setup. We employ analytical models and numerical simulations to characterize this setup. Numerically we show the appearance of Noise-Induced Barren Plateaus (NIBPs), i.e., an exponential concentration of observables caused by noise, in an Ising model Hamiltonian variational ansatz circuit. We observe this even if only a single qubit is noisy and given a deep enough circuit, suggesting that NIBPs cannot be fully overcome simply by error-correcting a subset of the qubits. On the positive side, we find that for every noiseless qubit in the circuit, there is an exponential suppression in concentration of gradient observables, showing the benefit of partial error correction. Finally, our analytical models corroborate these findings by showing that observables concentrate with a scaling in the exponent related to the ratio of dirty-to-total qubits.

In a future with fault-tolerant quantum computers, a whole new world of quantum algorithms will open up which may offer advantage over many classical algorithms. This will not come without some sacrifice – the number of qubits required to encode an error corrected (or logical) qubit will be large. Adding a single qubit to a system doubles the machine's available computational space, so in this paper we ask the question: can you combine error-corrected qubits with physical qubits? Since noise greatly impedes quantum algorithms, perhaps combining the benefits of error-correction with the additional Hilbert space afforded by non-error-corrected physical qubits may be beneficial for some classes of algorithms. We approach this question using an approximation where noiseless qubits take the place of error-corrected qubits, which we call clean; and they are coupled to noisy physical qubits, which we call dirty. We show analytically and numerically that errors in the measurement of expectation values are exponentially suppressed for each noisy qubit that is replaced with a clean qubit, and that this behavior closely follows what the machine would do had you reduced the error rate of a uniformly noisy machine by the ratio of dirty qubits to total qubits.

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► References

[1] Richard P. Feynman. ``Simulating physics with computers''. International Journal of Theoretical Physics 21, 467–488 (1982).

[2] Laird Egan, Dripto M Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R Brown, Marko Cetina, et al. ``Fault-tolerant control of an error-corrected qubit''. Nature 598, 281–286 (2021).

[3] Peter W Shor. ``Algorithms for quantum computation: discrete logarithms and factoring''. In Proceedings 35th annual symposium on foundations of computer science. Pages 124–134. Ieee (1994).

[4] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. ``Quantum algorithm for linear systems of equations''. Physical Review Letters 103, 150502 (2009).

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

[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, et al. ``Noisy intermediate-scale quantum algorithms''. Reviews of Modern Physics 94, 015004 (2022).

[8] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. ``Quantum machine learning''. Nature 549, 195–202 (2017).

[9] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''. Cambridge University Press. Cambridge (2000).

[10] Dorit Aharonov, Michael Ben-Or, Russell Impagliazzo, and Noam Nisan. ``Limitations of noisy reversible computation'' (1996). url: https:/​/​​10.48550/​arXiv.1106.6189.

[11] Michael Ben-Or, Daniel Gottesman, and Avinatan Hassidim. ``Quantum refrigerator'' (2013). url: https:/​/​​10.48550/​arXiv.1301.1995.

[12] Daniel Stilck França and Raul Garcia-Patron. ``Limitations of optimization algorithms on noisy quantum devices''. Nature Physics 17, 1221–1227 (2021).

[13] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. ``Noise-induced barren plateaus in variational quantum algorithms''. Nature Communications 12, 1–11 (2021).

[14] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes''. Nature Communications 9, 1–6 (2018).

[15] 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, 1–12 (2021).

[16] Andrew Arrasmith, Zoë Holmes, Marco Cerezo, and Patrick J Coles. ``Equivalence of quantum barren plateaus to cost concentration and narrow gorges''. Quantum Science and Technology 7, 045015 (2022).

[17] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J Coles. ``Effect of barren plateaus on gradient-free optimization''. Quantum 5, 558 (2021).

[18] M. Cerezo and Patrick J Coles. ``Higher order derivatives of quantum neural networks with barren plateaus''. Quantum Science and Technology 6, 035006 (2021).

[19] Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. ``Entanglement-induced barren plateaus''. PRX Quantum 2, 040316 (2021).

[20] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo. ``Diagnosing Barren Plateaus with Tools from Quantum Optimal Control''. Quantum 6, 824 (2022).

[21] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J Coles. ``Connecting ansatz expressibility to gradient magnitudes and barren plateaus''. PRX Quantum 3, 010313 (2022).

[22] Supanut Thanasilp, Samson Wang, Nhat A Nghiem, Patrick J. Coles, and M. Cerezo. ``Subtleties in the trainability of quantum machine learning models'' (2021). url: https:/​/​​abs/​2110.14753.

[23] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J Coles. ``Can error mitigation improve trainability of noisy variational quantum algorithms?'' (2021). url: https:/​/​​10.48550/​arXiv.2109.01051.

[24] Ningping Cao, Junan Lin, David Kribs, Yiu-Tung Poon, Bei Zeng, and Raymond Laflamme. ``NISQ: Error correction, mitigation, and noise simulation'' (2021). url: https:/​/​​10.48550/​arXiv.2111.02345.

[25] Adam Holmes, Mohammad Reza Jokar, Ghasem Pasandi, Yongshan Ding, Massoud Pedram, and Frederic T Chong. ``NISQ+: Boosting quantum computing power by approximating quantum error correction''. In 2020 ACM/​IEEE 47th Annual International Symposium on Computer Architecture (ISCA). Pages 556–569. IEEE (2020). url: https:/​/​​10.1109/​ISCA45697.2020.00053.

[26] 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, 010345 (2022).

[27] Emanuel Knill and Raymond Laflamme. ``Power of one bit of quantum information''. Physical Review Letters 81, 5672 (1998).

[28] Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. ``Power of Quantum Computation with Few Clean Qubits''. 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) 55, 13:1–13:14 (2016).

[29] Tomoyuki Morimae, Keisuke Fujii, and Harumichi Nishimura. ``Power of one nonclean qubit''. Physical Review A 95, 042336 (2017).

[30] Craig Gidney. ``Factoring with n+2 clean qubits and n-1 dirty qubits'' (2017). url: https:/​/​​10.48550/​arXiv.1706.07884.

[31] Anirban N. Chowdhury, Rolando D. Somma, and Yiğit Subaşı. ``Computing partition functions in the one-clean-qubit model''. Physical Review A 103, 032422 (2021).

[32] Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. ``Impossibility of classically simulating one-clean-qubit model with multiplicative error''. Physical Review Letters 120, 200502 (2018).

[33] Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. ``Perfect quantum error correcting code''. Phys. Rev. Lett. 77, 198–201 (1996).

[34] Daniel Gottesman. ``An introduction to quantum error correction and fault-tolerant quantum computation''. Quantum information science and its contributions to mathematics, Proceedings of Symposia in Applied Mathematics 63, 13–58 (2010).

[35] 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).

[36] A Yu Kitaev. ``Quantum computations: algorithms and error correction''. Russian Mathematical Surveys 52, 1191 (1997).

[37] Chris N Self, Marcello Benedetti, and David Amaro. ``Protecting expressive circuits with a quantum error detection code'' (2022). url: https:/​/​​10.48550/​arXiv.2211.06703.

[38] Rolando D Somma. ``Quantum eigenvalue estimation via time series analysis''. New Journal of Physics 21, 123025 (2019).

[39] Vojtěch Havlíček, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta. ``Supervised learning with quantum-enhanced feature spaces''. Nature 567, 209–212 (2019).

[40] Andrew G Taube and Rodney J Bartlett. ``New perspectives on unitary coupled-cluster theory''. International journal of quantum chemistry 106, 3393–3401 (2006).

[41] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T Sornborger, and Patrick J Coles. ``Quantum-assisted quantum compiling''. Quantum 3, 140 (2019).

[42] Colin J Trout, Muyuan Li, Mauricio Gutiérrez, Yukai Wu, Sheng-Tao Wang, Luming Duan, and Kenneth R Brown. ``Simulating the performance of a distance-3 surface code in a linear ion trap''. New Journal of Physics 20, 043038 (2018).

[43] Lukasz Cincio, Yiğit Subaşı, Andrew T Sornborger, and Patrick J Coles. ``Learning the quantum algorithm for state overlap''. New Journal of Physics 20, 113022 (2018).

[44] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. ``A quantum approximate optimization algorithm'' (2014). url: https:/​/​​10.48550/​arXiv.1411.4028.

[45] Stuart Hadfield, Zhihui Wang, Bryan O'Gorman, Eleanor G Rieffel, Davide Venturelli, and Rupak Biswas. ``From the quantum approximate optimization algorithm to a quantum alternating operator ansatz''. Algorithms 12, 34 (2019).

[46] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. ``Evaluating analytic gradients on quantum hardware''. Physical Review A 99, 032331 (2019).

[47] Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar, and Patrick J. Coles. ``Machine learning of noise-resilient quantum circuits''. PRX Quantum 2, 010324 (2021).

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

[49] Sergey Danilin, Nicholas Nugent, and Martin Weides. ``Quantum sensing with tuneable superconducting qubits: optimization and speed-up'' (2022). url: https:/​/​​abs/​2211.08344.

[50] Nikolai Lauk, Neil Sinclair, Shabir Barzanjeh, Jacob P Covey, Mark Saffman, Maria Spiropulu, and Christoph Simon. ``Perspectives on quantum transduction''. Quantum Science and Technology 5, 020501 (2020).

[51] Bernhard Baumgartner. ``An inequality for the trace of matrix products, using absolute values'' (2011). url: https:/​/​​10.48550/​arXiv.1106.6189.

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[2] Duc Tuan Hoang, Friederike Metz, Andreas Thomasen, Tran Duong Anh-Tai, Thomas Busch, and Thomás Fogarty, "Variational quantum algorithm for ergotropy estimation in quantum many-body batteries", Physical Review Research 6 1, 013038 (2024).

[3] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and opportunities in quantum machine learning", Nature Computational Science 2 9, 567 (2022).

[4] Linshu Chen, Tao Li, Yuxiang Chen, Xiaoyan Chen, Marcin Wozniak, Neal Xiong, and Wei Liang, "Design and analysis of quantum machine learning: a survey", Connection Science 36 1, 2312121 (2024).

[5] 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).

[6] Abdullah Ash Saki, Amara Katabarwa, Salonik Resch, and George Umbrarescu, "Hypothesis Testing for Error Mitigation: How to Evaluate Error Mitigation", arXiv:2301.02690, (2023).

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

[8] Nikolaos Koukoulekidis, Samson Wang, Tom O'Leary, Daniel Bultrini, Lukasz Cincio, and Piotr Czarnik, "A framework of partial error correction for intermediate-scale quantum computers", arXiv:2306.15531, (2023).

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