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 $\Gamma (s{)}_{\ell {\ell }^{\prime }}={\int }_0^{T}S(t)c_{\ell }(s,t)c_{{\ell }^{\prime }}(s,t)\mathrm{d}t$. 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 $\Gamma$ with ${\Gamma }_{\epsilon }:=\Gamma +{\epsilon }^{2}I$ and prove: (i) an exact spectral identity giving $\kappa ({\Gamma }_{\epsilon })=({\sigma }_{\max }^2+{\epsilon }^2)/({\sigma }_{\min }^2+{\epsilon }^2)$; (ii) objective monotonicity $\mathrm{d}J/\mathrm{d}s\ge 0$ for all $\epsilon \ge 0$; (iii) constraint drift $|h_m-C_m|=\mathcal{O}({\epsilon }^2)$ with a computable prefactor; (iv) convergence of the regularised trajectory to the unregularised one in $L^2(0,T)$ at rate $\mathcal{O}({\epsilon }^2)$ under uniform invertibility of $\Gamma$; and (v) a discrete CFL criterion $\Delta sG\Vert {\Gamma }_{\epsilon }^{-1}\Vert \le \alpha <2$ guaranteeing objective monotonicity of the forward-Euler scheme up to $\mathcal{O}(\Delta s^2)$ local truncation error. The theory is validated on a three-level bilinear benchmark for all-optical Bell-state preparation, where $\kappa (\Gamma )\in [10^9,10^{11}]$, the predicted ${\epsilon }^2$ 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.