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Physics-informed tracking of qubit fluctuations

Fabrizio Berritta, Jan A. Krzywda, Jacob Benestad, Joost van der Heijden, Federico Fedele, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, Evert van Nieuwenburg, Jeroen Danon, Anasua Chatterjee, and Ferdinand Kuemmeth

Phys. Rev. Applied 22, 014033 – Published 15 July 2024

Abstract

Environmental fluctuations degrade the performance of solid-state qubits but can in principle be mitigated by real-time Hamiltonian estimation down to timescales set by the estimation efficiency. We implement a physics-informed and an adaptive Bayesian estimation strategy and apply them in real time to a semiconductor spin qubit. The physics-informed strategy propagates a probability distribution inside the quantum controller according to the Fokker-Planck equation, appropriate for describing the effects of nuclear spin diffusion in gallium arsenide. Evaluating and narrowing the anticipated distribution by a predetermined qubit probe sequence enables improved dynamical tracking of the uncontrolled magnetic field gradient within the singlet-triplet qubit. The adaptive strategy replaces the probe sequence by a small number of qubit probe cycles, with each probe time conditioned on the previous measurement outcomes, thereby further increasing the estimation efficiency. The combined real-time estimation strategy efficiently tracks low-frequency nuclear spin fluctuations in solid-state qubits, and can be applied to other qubit platforms by tailoring the appropriate update equation to capture their distinct noise sources.

Received 16 April 2024
Revised 26 April 2024
Accepted 7 June 2024

DOI:https://doi.org/10.1103/PhysRevApplied.22.014033

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum control Quantum fluctuations & noise Quantum information with solid state qubits

Double quantum dots

Bayesian methods Fokker–Planck equation Machine learning Radio frequency techniques

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Fabrizio Berritta ¹, Jan A. Krzywda ², Jacob Benestad ³, Joost van der Heijden ⁴, Federico Fedele ^1,5, Saeed Fallahi ^6,7, Geoffrey C. Gardner ⁷, Michael J. Manfra^6,7,8,9, Evert van Nieuwenburg ², Jeroen Danon ³, Anasua Chatterjee ¹, and Ferdinand Kuemmeth ^1,4,*

¹Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
²Lorentz Institute and Leiden Institute of Advanced Computer Science, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands
³Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
⁴QDevil, Quantum Machines, 2750 Ballerup, Denmark
⁵Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom
⁶Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
⁷Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA
⁸Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
⁹School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA

^*Contact author: [email protected]

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Vol. 22, Iss. 1 — July 2024

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Figure 1
Qubit implementation and estimation schedule. (a) Scanning electron micrograph of a $Ga As$ double-dot device similar to the one used in this work [35] comprising a singlet-triplet qubit (black circles) next to a sensor dot (SD) used for qubit readout. Scale bar $100 nm$ . (b) Exchange coupling $J (ε)$ and Overhauser gradient $Δ B \propto h f_{B} (t)$ drive rotations of the qubit around two orthogonal axes of the Bloch sphere, providing universal qubit control if the prevailing Overhauser frequency $f_{B}$ can be estimated sufficiently efficiently. (c) Qubit schedule, alternating between periods $T_{op}$ of quantum information processing (dashed box) and short periods $T_{est}$ for efficiently learning the fluctuating environment (gray box).
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Figure 2
Tracking the Overhauser frequency by anticipating nuclear spin diffusion on the quantum controller. (a) The physics-informed estimation sequence for $f_{B}$ initializes the prior distribution $P_{0} (f_{B})$ by evolving an older final distribution $P_{final} (f_{B})$ (Fokker-Planck update). For each of the $N$ probe cycles, labeled $i$ , the quantum controller initializes the qubit to the singlet state, performs an FID for time $t_{i} = i t_{0}$ , then updates the probability distribution $P_{i} (f_{B})$ based on the measurement outcome $m_{i}$ . After $N$ probe cycles, the final distribution $P_{final} (f_{B})$ is saved. (b) Simulation of the unknown fluctuating Overhauser gradient (black) and five physics-informed estimation sequences, illustrating the tracking protocol. Every $40 ms$ , a sequence of FID probe cycles results in a final distribution with expected value $f_{B}^{f} = ⟨ f_{B} ⟩$ and error bar $2 σ_{f}$ (red markers). The simulation assumes a uniform prior distribution $P_{0} (f_{B})$ at $t = 0$ , whereas subsequent priors $P_{0} (f_{B})$ are based on the mean $μ (t)$ and standard deviation $σ (t)$ propagated by the Fokker-Planck equation over period $T_{op}$ (shaded in light red). (c) Experimental results for the nontracking reference protocol, using $P_{0} (f_{B}) \equiv P_{uniform} (f_{B})$ for each estimation sequence. (d) Experimental results for the physics-informed tracking protocol, obtained simultaneously with nontracking estimates in panel (c). The initial prior $P_{0} (f_{B})$ for each column is $P_{final} (f_{B})$ from the previous column, propagated in time according to Eq. (5a). Note the absence of multipeaked distributions $P_{final} (f_{B})$ .
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Figure 3
Efficiency of the nontracking and physics-informed protocols. (a) Estimation uncertainty as a function of the number of FID probes in the estimation sequence, for the nontracking (black) and physics-informed (red) protocols. Symbols denote the average standard deviation of 10,000 $⟨ f_{B} ⟩$ values, whereas shaded regions show their standard deviation, for different choices of operation time. (b) Uncertainty from (a) plotted as a function of the ratio $T_{est} / T_{op}$ , where the estimation time is $T_{est} = N \cdot 20 μ s$ . The dash-dotted gray line indicates the resolution limit imposed by our setup, see the main text.
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Figure 4
Adaptive Bayesian tracking by real-time choice of qubit probe times. (a) In this adaptive Bayesian estimation sequence, probe times $t_{i}$ are chosen based on the standard deviation $σ_{i - 1}$ of the previous Bayesian distribution. $P_{0} (f_{B})$ is initialized based on the FP equation. (b) Adaptive tracking obtained from short estimation sequences ( $N = 10$ ) for $T_{op} = 1 ms$ . (c) Reconstructed uncertainty in the distribution function within an estimation sequence (defined in the text) as a function of the measurement update $m_{i}$ . Squares at the end of the curves correspond to the experimental posterior distributions computed on the quantum controller. (d) Simulated uncertainty expected at the end of a short estimation sequence ( $N \leq 10$ ) for different probe time protocols, including evenly distributed $t_{i}$ (probe time spacing of 1 or 5 ns), adaptive probe times, and random probe times (see the main text). The initial prior distributions are assumed to be determined from the FP equation.
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Physical Review Applied

Physics-informed tracking of qubit fluctuations

Fabrizio Berritta, Jan A. Krzywda, Jacob Benestad, Joost van der Heijden, Federico Fedele, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, Evert van Nieuwenburg, Jeroen Danon, Anasua Chatterjee, and Ferdinand Kuemmeth

Phys. Rev. Applied 22, 014033 – Published 15 July 2024

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