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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[1] 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).

[2] 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).

[3] 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).

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

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

[6] Mikel Garcia-de-Andoin, Álvaro Saiz, Pedro Pérez-Fernández, Lucas Lamata, Izaskun Oregi, and Mikel Sanz, "Digital-Analog Quantum Computation with Arbitrary Two-Body Hamiltonians", arXiv:2307.00966, (2023).

[7] 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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