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arXiv:2411.19644 · Journal of Physics A Mathematical and Theoretical

Gradient projection method for constrained quantum control

Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen

Published Mar 27, 2025• 7 citations
arXivHTMLPDF

Abstract

Abstract In this work, we adopt the gradient projection method (GPM) to problems of quantum control. For general N-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals and provide the projection versions of the Pontryagin maximum principle and the GPM, all directly in terms of quantum objects such as evolution operator, Hamiltonians, density matrices, etc. Various forms of the GPM, including one- and two-step, are provided and compared. We formulate the GPM both for closed and open quantum systems, latter for the general case with simultaneous coherent and incoherent controls. The GPM is designed to perform local gradient based optimization in the case when bounds are imposed on the controls. The main advantage of the method is that it allows to exactly satisfy the bounds, in difference to other approaches such as adding constraints as weight to an objective. We apply the GPM to several examples including generation of one- and two-qubit gates and two-qubit Bell and Werner states for models of superconducting qubits under the constraint when controls are zero at the initial and final times, steering an open quantum system to a target density matrix for simulating action of the Werner–Holevo channel, etc.

DOI
10.1088/1751-8121/adbe1a

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Abstract In this work, we adopt the gradient projection method (GPM) to problems of quantum control. For general N-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals and provide the projection versions of the Pontryagin maximum principle and the GPM, all directly in terms of quantum objects such as evolution operator, Hamiltonians, density matrices, etc. Various forms of the GPM, including one- and two-step, are provided and compared. We formulate the GPM both for closed and open quantum systems, latter for the general case with simultaneous coherent and incoherent controls. The GPM is designed to perform local gradient based optimization in the case when bounds are imposed on the controls. The main advantage of the method is that it allows to exactly satisfy the bounds, in difference to other approaches such as adding constraints as weight to an objective. We apply the GPM to several examples including generation of one- and two-qubit gates and two-qubit Bell and Werner states for models of superconducting qubits under the constraint when controls are zero at the initial and final times, steering an open quantum system to a target density matrix for simulating action of the Werner–Holevo channel, etc.
In this work, we consider the problem of the analysis of quantum control landscapes in both situations when the controls during optimization are unconstrained and when they should satisfy a given constraint, for example their amplitude should be bounded by some constant value. For the analysis of unconstrained quantum control landscapes, a special approach was developed in [Pechen and Tannor, Israel J. of Chemistry 52 (5), 467–472 (2012)] based on a fixed step gradient ascent pulse engineering (GRAPE) algorithm. Here we also adopt a GRAPE approach based on the Pontryagin maximum principle (PMP). For the analysis of the constrained quantum control landscapes, we adopt the gradient projection method which was developed for quantum systems in [A. Oza et al., J. Phys. A: Math. Theor. 42, 205305 (2009)] (GPM; for our purpose the most suitable is a one-step GPM with a fixed step), both for closed and open quantum systems. As an explicit example, we apply these approaches to study quantum control landscape for optimization of an observable mean value for a three-level degenerate Ξ\XiΞ -system system having a seventh-order trap.
This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.
This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls. Bibliography: 72 titles.
Methods of optimization and control of quantum systems are widely used for various applications in quantum technologies including laser control of chemical reactions, nuclear magnetic resonance, gate generation for quantum computing, describing quantum transport in photosynthesis, etc. Among many topics in quantum control, we briefly outline Riemannian optimization and gradient projection method (GPM) on complex Stiefel manifolds and their relation to quantum control, as well as various related optimization tools such as incoherent GRAPE (InGRAPE) modification of the gradient ascent pulse engineering (GRAPE) algorithm, GPM in the dynamic picture for constrained optimization, and demonstrate operation of GRAPE and GPM for an investigation of the optimization efficiency over unconstrained and constrained quantum control landscapes for an example of a three-level Ξ\Xi Ξ -system having a seventh-order trap.1
We study a projection-type gradient flow for equality-constrained maximisation of a smooth bilinear control objective on H=L2(0,T;R)\mathcal{H}=L^2(0,T;\mathbb{R})H=L2(0,T;R), eliminating Lagrange multipliers through an (M+1)×(M+1)(M{+}1)\times(M{+}1)(M+1)×(M+1) moving Gram matrix Γ(s)ℓℓ′=∫0TS(t) cℓ(s,t) cℓ′(s,t) dtΓ(s)_{\ell\ell'}=\int_0^T S(t)\,c_\ell(s,t)\,c_{\ell'}(s,t)\,\mathrm{d}tΓ(s)ℓℓ′​=∫0T​S(t)cℓ​(s,t)cℓ′​(s,t)dt. 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 ΓΓΓ with Γε:=Γ+ε2IΓ_{\varepsilon}:=Γ+\varepsilon^{2}IΓε​:=Γ+ε2I and prove: (i) an exact spectral identity giving κ(Γε)=(σmax⁡2+ε2)/(σmin⁡2+ε2)κ(Γ_{\varepsilon})=(σ_{\max}^{2}+\varepsilon^{2})/(σ_{\min}^{2}+\varepsilon^{2})κ(Γε​)=(σmax2​+ε2)/(σmin2​+ε2); (ii) objective monotonicity dJ/ds≥0\mathrm{d}J/\mathrm{d}s\ge 0dJ/ds≥0 for all ε≥0\varepsilon\ge 0ε≥0; (iii) constraint drift ∣hm−Cm∣=O(ε2)|h_{m}-C_{m}|=\mathcal{O}(\varepsilon^{2})∣hm​−Cm​∣=O(ε2) with a computable prefactor; (iv) convergence of the regularised trajectory to the unregularised one in L2(0,T)L^{2}(0,T)L2(0,T) at rate O(ε2)\mathcal{O}(\varepsilon^{2})O(ε2) under uniform invertibility of ΓΓΓ; and (v) a discrete CFL criterion Δs G ∥Γε−1∥≤α<2Δs\,G\,\|Γ_{\varepsilon}^{-1}\|\leα<2ΔsG∥Γε−1​∥≤α<2 guaranteeing objective monotonicity of the forward-Euler scheme up to O(Δs2)\mathcal{O}(Δs^{2})O(Δs2) local truncation error. The theory is validated on a three-level bilinear benchmark for all-optical Bell-state preparation, where κ(Γ)∈[109,1011]κ(Γ)\in[10^{9},10^{11}]κ(Γ)∈[109,1011], the predicted ε2\varepsilon^{2}ε2 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 article considers a two-level open quantum system evolving under the action of coherent and incoherent controls using the general control method proposed in Phys. Rev. A. 73, 062102 (2006). Coherent control determines the Hamiltonian aspects of the dynamics whereas incoherent control determines the dissipative aspects. The goal is to find controls which steer the initial density matrix into a state which maximizes overlap with a predefined target density matrix. The controlled dynamics is represented as evolution in the Bloch ball and is analyzed analytically using Pontryagin maximum principle and numerically using optimization either in the functional space of controls (with the conditional and projected gradient methods) or through reduction to a finite-dimensional control space.