Tikhonov-regularised projected gradient flow for equality-constrained bilinear quantum control

Abstract: We study a projection-type gradient flow for equality-constrained maximisation of a smooth bilinear control objective on $\mathcal{H}=L^2(0,T;\mathbb{R})$, eliminating Lagrange multipliers through an $(M{+}1)\times(M{+}1)$ moving Gram matrix $Γ(s){\ell\ell'}=\int_0^T S(t)\,c\ell(s,t)\,c_{\ell'}(s,t)\,\mathrm{d}t$. The flow generates monotonic ascent in continuous time but becomes unstable on discretisation; existing implementations rely on heuristic step-size safeguards lacking rigorous justification. We close this gap by replacing $Γ$ with $Γ{\varepsilon}:=Γ+\varepsilon^{2}I$ and prove: (i) an exact spectral identity giving $κ(Γ{\varepsilon})=(σ{\max}^{2}+\varepsilon^{2})/(σ{\min}^{2}+\varepsilon^{2})$; (ii) objective monotonicity $\mathrm{d}J/\mathrm{d}s\ge 0$ for all $\varepsilon\ge 0$; (iii) constraint drift $|h_{m}-C_{m}|=\mathcal{O}(\varepsilon^{2})$ with a computable prefactor; (iv) convergence of the regularised trajectory to the unregularised one in $L^{2}(0,T)$ at rate $\mathcal{O}(\varepsilon^{2})$ under uniform invertibility of $Γ$; and (v) a discrete CFL criterion $Δs\,G\,|Γ_{\varepsilon}^{-1}|\leα<2$ guaranteeing objective monotonicity of the forward-Euler scheme up to $\mathcal{O}(Δs^{2})$ local truncation error. The theory is validated on a three-level bilinear benchmark for all-optical Bell-state preparation, where $κ(Γ)\in[10^{9},10^{11}]$, the predicted $\varepsilon^{2}$ rate is confirmed over eight decades, and moderate regularisation eliminates step rejections and reduces constraint drift by more than an order of magnitude at unchanged final fidelity.

• The paper presents a novel Tikhonov‐regularised projected gradient flow that stabilizes equality‐constrained quantum control by addressing ill-conditioning in the Gram matrix.
• It rigorously establishes theoretical guarantees including quadratic L2 convergence, exact conditioning identities, and discrete stability via CFL-type bounds.
• Empirical benchmarks on a three-level system confirm the method's effectiveness in reducing fluence drift and step rejections while achieving near-perfect state fidelity.

Tikhonov-Regularised Projected Gradient Flow for Equality-Constrained Bilinear Quantum Control

Problem Formulation and Motivation

The paper addresses the problem of constrained quantum optimal control where a smooth bilinear objective must be maximized under several integral equality constraints in the control field. The state of a finite-dimensional quantum system, evolving according to a time-dependent Schrödinger equation perturbed by a control field, must be transferred to a target state with maximal fidelity. However, physical and engineering considerations introduce crucial constraints, including zero pulse area, fixed fluence, and fixed area against a reference oscillation. These physically motivated constraints make the unconstrained maximization of the control objective fundamentally ill-posed for practical implementation.

A key issue with classical projection-type gradient flows is the emergence of ill-conditioning due to near-linear dependencies among constraint gradients in the $L^2$ inner product space. This ill-conditioning manifests as Gram matrices with persistently large condition numbers, leading to instability and loss of constraint preservation upon discretisation. Previous implementations have relied on heuristic step-size adaptation and lack quantitative guarantees on stability and constraint enforcement.

Regularised Flow Formulation and Theoretical Results

To overcome the instability in numerical implementations, the paper introduces a Tikhonov-regularised projected gradient flow, replacing the moving Gram matrix $\Gamma$ with $\Gamma + \epsilon^2 I$, with $\epsilon$ a regularisation parameter. This regularisation systematically stabilises the inversion required in the projected flow and offers transparent analytical control over the step size in explicit integrators, yielding five rigorous results:

• Exact Conditioning Identity: The spectrum of the regularised Gram matrix is shifted by $\epsilon^2$, so the condition number becomes $$( \sigma_0^2 + \epsilon^2 ) / ( \sigma_M^2 + \epsilon^2 )$$, where $\sigma_k$ are the singular values of the constraint assembly operator (Theorem 1).
• Monotonic Ascent: Objective monotonicity of the flow, $dJ/ds \geq 0$, is preserved for all $\epsilon \geq 0$ (Theorem 2).
• Constraint Drift Quantification: The deviation from constraint satisfaction due to regularisation is exactly quadratic in $\epsilon$, with explicit computable prefactors (Theorem 3).
• $\Gamma$0 Convergence: The regularised flow converges to the unregularised flow at rate $\Gamma$1 under appropriate invertibility assumptions (Theorem 4).
• Discrete Stability (CFL) Condition: In the forward-Euler discretisation, a Courant–Friedrichs–Lewy (CFL) type step-size bound ensures discrete monotonicity through explicit dependence on the regularised Gram matrix norm, supplanting heuristic step-halving (Theorem 5).

The structure of the regularised algorithm is summarised in an explicit update in Algorithm 1 of the paper.

Numerical Verification and Empirical Phenomenology

A central contribution is the empirical investigation of the regularised flow on a three-level quantum control benchmark relevant to the all-optical generation of Bell states in coupled atoms, subject to zero-area, fixed-fluence, and fixed-resonant-area constraints. All key theoretical predictions are robustly validated:

• The unregularised flow ($\Gamma$2) suffers from extreme Gram matrix condition numbers $\Gamma$3–$\Gamma$4, leading to significant fluence constraint drift ($\Gamma$5–$\Gamma$6), while affine constraints are preserved only up to solver precision.

  Figure 1: Baseline unregularised flow ($\Gamma$7). (a) Objective $\Gamma$8, (b) relative fluence drift, (c) absolute pulse area, (d) Gram-matrix condition number $\Gamma$9. Gram-matrix ill-conditioning is persistent and severe for all pulse durations.

• For the regularised flow, the empirical $\Gamma + \epsilon^2 I$0-distance to the unregularised solution exhibits precise $\Gamma + \epsilon^2 I$1 scaling over eight orders of magnitude in $\Gamma + \epsilon^2 I$2, saturating only as predicted for larger regularisations.

  Figure 2: Empirical verification of quadratic convergence: relative $\Gamma + \epsilon^2 I$3-distance between regularised and unregularised solutions scales as $\Gamma + \epsilon^2 I$4 across eight decades.

• The condition number of $\Gamma + \epsilon^2 I$5 follows theoretical scaling, sharply decreasing as $\Gamma + \epsilon^2 I$6 increases and entering a regime where discretisation becomes robust, with significant reductions in both step rejections and constraint drift.

  Figure 3: (a) Maximum condition number $\Gamma + \epsilon^2 I$7 along the trajectory. (b) Relative fluence drift at final iteration, reflecting the regularisation-induced trade-off.

• In aggressive step-size regimes, moderate regularisation ($\Gamma + \epsilon^2 I$8 in $\Gamma + \epsilon^2 I$9–$\epsilon$0, non-dimensionalised) eliminates step rejections and lowers fluence drift by an order of magnitude without penalising final objective fidelity.

  Figure 4: Practical impact of regularisation at $\epsilon$1. (a) Required iterations, (b) number of step-halving rejections, (c) relative fluence drift. Moderate $\epsilon$2 significantly improves all metrics.

• The $\epsilon$3 distances between final optimal controls at different $\epsilon$4 match theoretical predictions, and all regularised procedures achieve $\epsilon$5.

  Figure 5: Initial control (dashed) and final controls for $\epsilon$6, $\epsilon$7, $\epsilon$8. Final state fidelity and inter-control distances are consistent with analytic bounds.

Implications and Outlook

The introduction and analysis of Tikhonov regularisation in projected gradient flows for constrained quantum control offers robust theoretical and practical remedies for numerical ill-conditioning endemic to equality-constrained bilinear problems. It provides:

• A principled calibration of the regularisation parameter, balancing the trade-off between constraint preservation and numerical stability.
• Quantitative, predictive guarantees on constraint drift and convergence rates, enabling algorithmic transparency and reproducibility.
• A mathematically-founded alternative to step-size tuning heuristics, directly applicable to a range of scenarios beyond the quantum control setting.

The mathematical structure and regularisation strategy is independent of the details of the bilinear system and readily generalises to more complex scenarios, including non-bilinear dynamics, higher-dimensional constraints, and even PDE-constrained quantum or classical control. Future directions include adaptive regularisation (e.g., Tikhonov–Morozov principles), advanced integrators, and multi-control extensions, as well as direct applications to the design of quantum gates for neutral-atom and superconducting quantum technologies, where the constraint-ill-conditioning challenge is structurally pervasive.

Conclusion

The work provides an integrated analytical and empirical treatment of Tikhonov-regularised projected gradient flows for equality-constrained bilinear quantum control, quantifying how moderate regularisation yields both robust discrete-time convergence and precise control over constraint satisfaction, validated on physically relevant benchmarks. The framework represents a significant step towards reliable and reproducible implementation of constraint-sensitive quantum optimal control algorithms (2604.26625).