Sliding Mode Control of Two-Level Quantum Systems

I. INTRODUCTION

• The paper frames robustness as a key issue for practical quantum technology and contrasts its goal with earlier Lyapunov designs requiring perfect initial-state and Hamiltonian knowledge.The introduction also notes earlier quantum robust-control work for linear quantum stochastic systems and prior sliding-mode work that did not consider robustness.Source: §I · pp. 2–3
• The paper formally presents sliding-mode control for two-level quantum systems with bounded uncertainties in the system Hamiltonian.It develops a variable-structure control approach to enhance robustness.Source: §I · p. 2
• The method identifies a free-Hamiltonian eigenstate as the sliding mode and uses Lyapunov control to steer the state into that domain.Periodic projective measurements then maintain the state in the domain when Hamiltonian uncertainties exist.Source: §I · p. 3
• The approach addresses the difficulty that direct feedback is not applicable when acquiring state information would destroy the quantum state.The authors describe the Lyapunov method as tolerating small drifts and say this is demonstrated by simulation.Source: §I · p. 3
• Periodic measurement design has two approaches based on uncertainty information, including the uncertainty bound and type, and the allowed probability of failure.The design objective is to make the measurement period as large as possible while retaining the stated control performance.Source: §I · pp. 3–4

A. Quantum Control Model

• The model specifies a free Hamiltonian with eigenstates and introduces a control Hamiltonian formed from time-independent Hamiltonians weighted by controls.The controls are real-valued functions of time.Source: A. Quantum Control Model · p. 4
• The control problem is to find controls that drive a two-level system from a given initial state to a target state.The model focuses on pure-state quantum systems.Source: A. Quantum Control Model · pp. 4–5
• Pure states of a two-level system are represented on the surface of the Bloch sphere, with coordinates expressed using θ and ϕ.The global phase is not considered because it has no observable physical effect.Source: A. Quantum Control Model · pp. 4–5

B. Sliding Modes

• A sliding mode is represented by a condition S(|ψ⟩,H)=0, and an eigenstate of the free Hamiltonian can be selected to define that surface.The paper defines S as one minus the squared overlap between the system state and the selected eigenstate.Source: B. Sliding Modes · p. 5
• When the initial state is on the sliding surface, free-Hamiltonian evolution keeps the system state on that surface.The paper thereby treats a free-Hamiltonian eigenstate as a sliding mode.Source: B. Sliding Modes · pp. 5–6
• The paper identifies |0⟩ as the sliding mode for its two-level system.It states that either |0⟩ or |1⟩ may be selected.Source: B. Sliding Modes · p. 6

C. Problem Formulation

• The selected eigenstate |0⟩ is identified as a sliding mode, with the state maintained on the sliding surface after reaching it.This identifies the target mode used in the control problem.Source: C. Problem Formulation · p. 6
• The uncertainties are modeled as bounded perturbations to the Hamiltonian, including components along Ix, Iy, and Iz.The stated class includes unitary errors and uncertainties associated with one-qubit gates; noisy magnetic fields and unwanted spin coupling are cited as possible sources.Source: C. Problem Formulation · p. 6
• The paper introduces sliding-mode control to address Hamiltonian uncertainty, with bounded uncertainty taken as the nontrivial case.The formulation assumes ε > 0; ε = 0 corresponds to no uncertainty.Source: C. Problem Formulation · p. 6
• The formulation assumes the uncertainty-free system is completely controllable and can generate arbitrary unitary operations.For a two-level system, the stated condition can be met by realizing arbitrary rotations along the z-axis and either the x- or y-axis.Source: C. Problem Formulation · p. 6
• The control law must drive the state into D, maintain it under bounded Hamiltonian uncertainty, and return it to D after a measurement takes it out.The formulation divides the task into reaching, maintaining, and recovery stages.Source: C. Problem Formulation · pp. 6–7
• D contains states whose overlap probability with |0⟩ is at least 1−p0, while a measurement can collapse the state out of D with probability at most p0.The formulation expects the state to remain in D except for such measurement-induced departures.Source: C. Problem Formulation · pp. 6–7
• The problem assumes the initial state is known and that no uncertainties act during the reaching and recovery processes.The maintenance stage is the part explicitly considered in the presence of uncertainties.Source: C. Problem Formulation · p. 7

A. General Method

• Existing Lyapunov-based results do not address robustness to uncertainty, and measurement can destroy the measured state, motivating a design that combines Lyapunov control and projective measurement.Prior approaches described here use feedback design to construct an open-loop control.Source: A. General Method · pp. 7–8
• The method selects eigenstate |0⟩ of H0 as sliding mode, constructs a Lyapunov function and control period T0 for known |ψ0⟩ at failure probability p0, and similarly designs T1 for initial |1⟩.It designs period T for periodic projective measurements according to p0 and ε.Source: A. General Method · p. 8
• The selected Lyapunov function uses Hilbert-Schmidt distance from the sliding-mode state, and the control law is chosen to make its derivative nonpositive.The gain Kk adjusts control amplitude, while f(·) is required to satisfy xf(x) ≥ 0.Source: A. General Method · pp. 8–9
• Under the Lyapunov control law, the quantum state converges asymptotically to the sliding mode |0⟩.The convergence is asymptotic, motivating a projective measurement after a designed control duration.Source: A. General Method · p. 9
• A projective σz measurement after Lyapunov control for T0 or T1 can project the state into |0⟩.The corresponding control duration depends on whether the initial condition is the known state or |1⟩.Source: A. General Method · p. 10
• Periodic measurements can maintain the state despite uncertainty, but very frequent measurements are difficult and costly, so the design seeks the largest suitable period.The period is estimated from the uncertainty bound and allowed failure probability.Source: A. General Method · p. 10

B. The Design of the Measurement Period T

• The period T(1) is designed for bounded transverse uncertainties, while a z-axis perturbation is ignored because it does not drive a state already in the sliding mode away from it.The z-axis case is treated separately from the transverse x/y uncertainty design.Source: B. The Design of the Measurement… · pp. 10–11
• For bounded x- and y-component Hamiltonian uncertainties, the state remains in D and the probability of measurement failure is no greater than p0.The stated guarantee applies to the system initialized in |0⟩ and evolving under the specified Hamiltonian.Source: B. The Design of the Measurement… · p. 11
• For a single bounded x- or y-axis uncertainty, the theorem likewise guarantees that the state remains in D and the measurement failure probability is at most p0.This is the restricted uncertainty setting used for the second period design.Source: B. The Design of the Measurement… · p. 11
• When the uncertainty has only one x- or y-axis component and p0 and ε satisfy the stated condition, T(2) exceeds T(1) and is an optimal measurement period.The paper states that either designed period can guarantee the required robustness under its applicable conditions.Source: B. The Design of the Measurement… · p. 12
• For fixed ε, both designed periods approach zero as p0 approaches zero; for fixed p0, T(1) decreases as the uncertainty bound ε increases.The paper concludes that stringent failure bounds or very large uncertainties require frequent measurements.Source: B. The Design of the Measurement… · p. 12

C. An Illustrative Example

• The example’s Lyapunov design takes T1=0.060, while the cited bang-bang time-optimal strategy for this task takes 0.030.The paper notes that complete time-optimal solutions are difficult to find for high-dimensional quantum systems.Source: C. An Illustrative Example · pp. 13–14
• Simulations suggest that small uncertainties can also be tolerated while driving the system into the sliding mode domain.The authors note that changing the Lyapunov control terminal condition or using a bigger K for the same T1 may help ensure entry into that domain.Source: C. An Illustrative Example · pp. 14–15
• Final |0⟩ probability is (99.00 ± 0.01)% for ε(t)Ix (|ε(t)| ≤ ε, ε=0.02 or 0.2), (99.00 ± 0.02)% for ε(t)Iy (|ε(t)| ≤ 0.02), and (99.00 ± 0.13)% for ε(t)Iy (|ε(t)| ≤ 0.2).The values are reported for simulations with bounded uncertainties in the Hamiltonian.Source: C. An Illustrative Example · p. 14
• Fig. 4 compares |0⟩ probability with and without noise; the noisy-case probabilities are very close to the no-uncertainty probability.The example uses uniform noise on [−0.02, 0.02] and applies the control shown in Fig. 3.Source: C. An Illustrative Example · p. 14, Figure 4 · p. 15

IV. PROOF OF THEOREMS

• The proof presents Theorem 2 before Theorem 1 because some arguments in the former are useful for proving the latter.Source: §IV · pp. 15–16
• Theorem 1’s proof compares failure probabilities across Hamiltonians with constant and time-varying uncertainty components, then uses those comparisons to estimate the measurement period.The listed comparisons include uncertainty along Ix and combined Ix and Iy perturbations.Source: §IV · p. 15
• Theorem 2’s proof formulates the worst case as an optimal-control problem, solves nonsingular cases, excludes singular cases, and uses the worst case to estimate the measurement period.Source: §IV · pp. 15–16

A. Proof of Theorem 2

• The worst-case analysis provides an upper comparison for the failure probability under the other considered uncertainty Hamiltonian.Source: A. Proof of Theorem 2 · pp. 16, 18
• For the worst-case control problem, the optimal uncertainty control is bang-bang, with ε*(t)=+ε or −ε.The proof states this strategy is optimal over the interval considered.Source: A. Proof of Theorem 2 · pp. 17–18
• The proof rules out singular cases by showing the switching quantity does not change sign and that no singular condition can exist.Source: A. Proof of Theorem 2 · p. 18
• The measurement period is designed from the worst-case Hamiltonian; the proof states the same conclusion for uncertainty along Iy as for Ix.Source: A. Proof of Theorem 2 · p. 19

B. Proof of Theorem 1

• The proof establishes Lemmas 3 and 5 for a two-level system initialized at |0⟩, comparing evolution under the stated constant and time-varying Hamiltonian cases.Source: B. Proof of Theorem 1 · pp. 19, 21
• The proof uses monotonicity of f(t) and F(t), with min_t F(t)=F(0)=0, to establish the stated comparison result.Source: B. Proof of Theorem 1 · pp. 20–21
• For the performance index J(ε)=zf, the proof states that the optimal control has a bang-bang form, so it analyzes a bang-bang strategy.Source: B. Proof of Theorem 1 · p. 22
• The analysis extends to time-varying uncertainty components εx(t) and εy(t) in the Hamiltonian Iz+εx(t)Ix+εy(t)Iy.Source: B. Proof of Theorem 1 · p. 23

• The proof reuses Lemma 5's argument for arbitrary t.Source: §2 · p. 24
• The stated comparison bounds one failure probability by the other: p_A^t is not greater than p_B^t.Source: §2 · p. 24
• The maximum measurement period is designed using the H_B case.Source: §2 · p. 24

V. CONCLUSIONS

• The proposed variable-structure sliding-mode scheme targets robust control of two-level quantum systems, with an eigenstate identified as the sliding mode.Source: §V · p. 24
• The design uses Lyapunov methodology and periodic projective measurements to drive and maintain the state in the sliding-mode domain.The control problem is recast as designing Lyapunov functions and the measurement period.Source: §V · pp. 24–25
• Two measurement-period design approaches address different system-Hamiltonian uncertainty situations and guarantee robust control performance.The paper reports using simulation to drive the state into the sliding-mode domain.Source: §V · p. 25
• The proposed strategy is presented as a quantum-engineering approach with potential applications in state preparation, decoherence control, and quantum error correction.The paper also identifies protection of encoded quantum information as a potential application.Source: §V · p. 25
• Physical implementation remains future work, with NMR spin systems proposed as candidates for testing the approach.Source: §V · p. 25
• The sliding-mode idea can extend to multi-level systems, but obtaining an analytical solution is more difficult.A numerical result for a three-level system is noted in prior work.Source: §V · p. 25
• For dissipative or stochastic quantum systems, new methods are needed to drive the state into the sliding-mode domain because the Lyapunov-based approach usually does not work.Source: §V · p. 25
• Practical applications of the proposed approaches remain an area for further exploration.Source: §V · p. 25

APPENDIX: PROOF OF T (2) ≥T (1)

• The proof uses that eG(x) is continuous on [-1, 1] and vanishes only at x = ±1.Source: APPENDIX: PROOF OF T (2) ≥T (1) · p. 26
• Since eG(0) > 0, the proof concludes that eG(x) is nonnegative on [-1, 1].Source: APPENDIX: PROOF OF T (2) ≥T (1) · p. 26
• For ε > 0, T(2)(p′) is at least T(1)(p′).Source: APPENDIX: PROOF OF T (2) ≥T (1) · p. 26