No-Collapse Accurate Quantum Feedback Control via Conditional State Tomography

Open Access

No-Collapse Accurate Quantum Feedback Control via Conditional State Tomography

Sangkha Borah and Bijita Sarma

Phys. Rev. Lett. 131, 210803 – Published 21 November 2023

Abstract

The effectiveness of measurement-based feedback control protocols is hampered by the presence of measurement noise, which affects the ability to accurately infer the underlying dynamics of a quantum system from noisy continuous measurement records to determine an accurate control strategy. To circumvent such limitations, this Letter explores a real-time stochastic state estimation approach that enables noise-free monitoring of the conditional dynamics including the full density matrix of the quantum system using noisy measurement records within a single quantum trajectory—a method we name as “conditional state tomography.” This, in turn, enables the development of precise measurement-based feedback control strategies that lead to effective control of quantum systems by essentially mitigating the constraints imposed by measurement noise and has potential applications in various feedback quantum control scenarios. This approach is particularly useful for reinforcement-learning-(RL) based control, where the RL-agent can be trained with arbitrary conditional averages of observables, and/or the full density matrix as input (observation), to quickly and accurately learn control strategies.

Received 17 January 2023
Accepted 30 October 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.210803

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. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Machine learning Quantum control Quantum feedback Quantum tomography

Quantum Information, Science & Technology

Authors & Affiliations

Sangkha Borah ^1,2,3,* and Bijita Sarma ^2,3,†

¹Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
²Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
³Okinawa Institute of Science and Technology, Okinawa 904-0495, Japan

^*sangkha.borah@mpl.mpg.de
^†bijita.sarma@fau.de

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Issue

Vol. 131, Iss. 21 — 24 November 2023

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Images

Figure 1
The schematic of the proposed protocol. Top: the estimation stage. A physical quantum system (left) described by Hamiltonian ${\hat{H}}_{0}$ is continuously monitored to probe the observable $\hat{A}$ , and the noisy measurement outcomes are fed to the estimator (right)—a simulator based on the mathematical model of the real physical system on a classical processor, e.g., a field programmable gate array. The state of the physical system (estimator) at time $t$ is described by $ρ (t)$ [ $ρ^{e} (t)$ ] which becomes equal at $t \geq t_{c}$ . Bottom: the control stage. A controller is used to apply accurate feedback $\hat{F} (t)$ to both the physical as well as the estimator systems as a function of the estimated noiseless conditional signal $⟨ \hat{A} (t) ⟩_{c}$ obtained through the estimator.
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Figure 2
Control of a linear quantum harmonic oscillator using the protocol. (a) In the estimation phase, the fidelity $F (t)$ between the physical system and the estimator steadily converges. The inset displays the conditional means of the observable $\hat{x}$ . (b) Subsequently, a state-based controller is applied, swiftly guiding the particle’s motion around the center $⟨ \hat{x} (t) ⟩_{c} = 0$ . The instantaneous fidelity $F_{g} (t)$ (depicted in black) quantifies the closeness between the physical system or estimator state and the target state, i.e., the ground state of the oscillator. The conditional mean population $⟨ {\hat{a}}^{†} \hat{a} (t) ⟩_{c}$ in the oscillator is shown in red.
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Figure 3
The protocol is applied to control a particle’s motion in a nonlinear quartic potential to cool it to its dynamic ground state using RL-based control. The training process is shown in a black colored line as the average fidelity over each episode $N$ with respect to the target state (ground state) $\bar{F} (N)$ , which is maximized through training. Note that the sudden drop at $N \sim 100$ is due to the exploration of the RL agent. The performance of the trained agent is shown in the red line.
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Figure 4
Demonstration of the proposed MBFC protocol for the preparation of symmetric, $ρ_{s}$ , and antisymmetric, $ρ_{a}$ , entangled states between two qubits as an example for when it is possible to derive control laws based on conditional moments within stochastic dynamics. Control laws $u_{1}$ and $u_{2}$ are selected depending on the conditional value of $ρ (t) ρ_{μ}$ , where $μ \in {s, a}$ (symmetric and antisymmetric) are in the three regimes, conveniently demonstrated in (a), and the arrows represent the direction of the entrance boundary of $ρ (t)$ to the middle section. $γ$ is the damping parameter, the measurement rate $κ$ is assumed to be 0.1, and the efficiency $η = 0.5$ for this simulation. After the estimation stage (not shown), these control laws are applied on conditional mean data (density matrices to compute instantaneous fidelity), which leads to convergence to the target states ( $ρ_{a}$ : black and $ρ_{s}$ : red), shown in (b). In the absence of such laws, RL can be used—the performance is shown in the inset of figure (b) with similar color settings.
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