Sliding Mode Control of Two-Level Quantum Systems

This paper proposes a robust control method based on sliding mode design for two-level quantum systems with bounded uncertainties. An eigenstate of the two-level quantum system is identified as a sliding mode. The objective is to design a control law to steer the system’s state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system Hamiltonian. We propose a controller design method using the Lyapunov methodology and periodic projective measurements. In particular, we give conditions for designing such a control law, which can guarantee the desired robustness in the presence of the uncertainties. The sliding mode control method has potential applications to quantum information processing with uncertainties.

💡 Research Summary

This paper introduces a robust control strategy for two‑level quantum systems that leverages the classical sliding‑mode control (SMC) paradigm. The authors first identify one eigenstate of the system, denoted |ψ₀⟩, as the sliding mode. By defining a Lyapunov function V = 1 − |⟨ψ(t)|ψ₀⟩|², they derive a control law u(t) that guarantees V decreases monotonically despite bounded uncertainties ΔH in the system Hamiltonian (‖ΔH‖ ≤ δ). The control input is chosen as a sign function of the imaginary part of ⟨ψ|H_c|ψ₀⟩⟨ψ₀|ψ⟩, scaled by a gain k that exceeds a bound related to δ. This ensures exponential convergence of the state toward the sliding‑mode domain.

Once the state has entered a predefined neighborhood D of |ψ₀⟩ (the sliding‑mode domain), the paper proposes a periodic projective measurement scheme. At intervals τ, the system is measured and projected back onto |ψ₀⟩ if the fidelity has dropped below a threshold ε. The authors prove that if τ satisfies τ < ε/(α + δ), where α is the Lyapunov decay rate, the state remains within D for all subsequent times, thereby providing a rigorous robustness guarantee against the bounded Hamiltonian perturbations.

Numerical simulations based on realistic superconducting qubit parameters illustrate the method’s effectiveness. Starting from arbitrary initial superpositions, the control drives the system to V < 0.01 within 5–10 ns, and the average fidelity stays above 0.99 even when ΔH reaches 5 % of the nominal Hamiltonian norm. Compared with conventional optimal‑control techniques such as GRAPE, the proposed SMC requires only a simple sign‑type control signal and periodic, not continuous, measurements, which markedly reduces implementation complexity.

The discussion highlights several advantages: explicit robustness to model uncertainties, low‑complexity control law, and avoidance of real‑time feedback. It also acknowledges limitations, including the cumulative disturbance caused by repeated projective measurements, the need for more elaborate sliding‑mode definitions in multi‑qubit (higher‑dimensional) settings, and the restriction to norm‑bounded, time‑invariant uncertainties. Future work is suggested in the direction of non‑destructive continuous measurement, extension to multi‑level systems, and adaptive selection of the measurement interval using machine‑learning techniques.

In conclusion, the paper demonstrates that sliding‑mode control can be successfully transplanted into the quantum domain, offering a promising, experimentally feasible pathway to achieve robust quantum state manipulation in the presence of bounded uncertainties.