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Europhysics Letters (EPL)

Exploiting automatic differentiation for quantum optimal control of driven dissipative two-level systems

Hamza Jirari, HERSCHEL A. RABITZ

Published Aug 28, 2024• 0 citations
Publisher

Abstract

Abstract In principle one can use Pontryagin's minimum principle to treat an optimal control problem and derive the gradient for the cost functional. However, for the driven spin-boson model studied here, the response of the system to the variation of the control is determined by the master equation and the equation of motion for the propagator of the coherent system dynamics. As a result, the application of Pontryagin's minimum principle is less straightforward. An alternative to this approach is the technique of automatic differentiation which in principle amounts to doing calculus on the fully discretized form of the optimal control problem. Automatic differentiation tools can be viewed as black boxes taking as input a program computing the cost functional and giving as output another program computing its gradient. First we derive for the polaron transformed Hamiltonian a Born-Markov master equation using the Bloch-Redfield formalism. By combining the latter with automatic differentiation we are able to implement Z -gate and coherent destruction of tunnelling with high fidelity. Optimization of a dissipative quantum gate poses a more complex numerical problem, since it should occur independent of the input state. To overcome this difficulty, we apply optimal control to the concept of resolvent of the master equation which is the generalisation of quantum unitary evolution operator in the case of dissipative dynamics.

DOI
10.1209/0295-5075/ad749b

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Abstract In principle one can use Pontryagin's minimum principle to treat an optimal control problem and derive the gradient for the cost functional. However, for the driven spin-boson model studied here, the response of the system to the variation of the control is determined by the master equation and the equation of motion for the propagator of the coherent system dynamics. As a result, the application of Pontryagin's minimum principle is less straightforward. An alternative to this approach is the technique of automatic differentiation which in principle amounts to doing calculus on the fully discretized form of the optimal control problem. Automatic differentiation tools can be viewed as black boxes taking as input a program computing the cost functional and giving as output another program computing its gradient. First we derive for the polaron transformed Hamiltonian a Born-Markov master equation using the Bloch-Redfield formalism. By combining the latter with automatic differentiation we are able to implement Z -gate and coherent destruction of tunnelling with high fidelity. Optimization of a dissipative quantum gate poses a more complex numerical problem, since it should occur independent of the input state. To overcome this difficulty, we apply optimal control to the concept of resolvent of the master equation which is the generalisation of quantum unitary evolution operator in the case of dissipative dynamics.
This paper investigates the dynamics of a two-level system coupled to a super-Ohmic boson bath, which is the so-called driven spin-boson model. Using the polaron transformation we derive the Bloch vector of a master equation where the bar tunneling parameter is used as the basis for a perturbation expansion. Our method is based on the canonical quantization provided by the Bloch-Redfield formalism. We apply quantum optimal control theory to the realization of a quantum ZZZ gate and also the coherent destruction of tunneling. To solve our optimal control problem, we use the techniques of automatic differentiation as an alternative to Pontryagin's minimum principle. The latter is less straightforward since the master equation is complex.
In this paper we investigate the application of optimal control of a single qubit coupled to an ohmic heat bath. For the weak bath coupling regime, we derive a Bloch-Redfield master equation describing the evolution of the qubit state parametrized by vectors in the Bloch sphere. By use of the optimal control methodology we determine the field that generates an arbitrary single qubit rotation in minimum time. In particular we develop an efficient method for solving the time-optimal control problem, namely the implementation of XXX gate, which consists of a rotation about the $\stackrel{P\vec}{ox}$ axis through angle $\ensuremath{\pi}$. The time-optimal control problem requires a bounded control. If upper and lower bounds are imposed for the external control, the optimal solution is of bang-bang type and switches from the upper to the lower values of the control bounds. We compare our results using the techniques of automatic differentiation to compute the gradient for the cost functional with some known results using Pontryagin's minimum principle. We use an alternative numerical approach for the optimization strategy.
Optimal control in open quantum systems is considered, with a focus on a two-level logical qubit system whose time evolution is governed by Markovian dynamics described within the Lindblad (GKSL) formalism. The objective is to minimize the net heat transferred to the environment during the transition while explicitly accounting for the work performed by the external control field. In this context, we formulate a variational optimal control framework for minimizing dissipated heat under fidelity constraints, which establishes existence under standard assumptions of bounded controls and lower semi-continuity of at least one bounded control that minimizes the environmental heat dissipation functional while maintaining the final-state fidelity above a designated threshold. The analysis combines the direct method of the calculus of variations with the Pontryagin Maximum Principle. The direct method in the calculus of variations (in L ∞ spaces) guarantees existence by demonstrating the compactness of admissible controls and the lower semi-continuity of the cost functional. Second, the Pontryagin Maximum Principle for open quantum systems establishes essential optimality conditions, resulting in coupled state–costate equations and switching functions that define optimal controls. We establish a physical foundation for the optimization problem by defining internal energy, heat flow, and work within the Lindblad framework, resulting in a thermodynamically coherent expression for dissipated energy. This formulation allows for both analytical understanding together with a practical numerical approximation based on the MDR-GRAPE algorithm. To demonstrate numerical consistency, we present MDR-GRAPE, a modified gradient-based algorithm that extends GRAPE to dissipative environments, integrating positivity-preserving techniques for density matrices. Numerical experiments demonstrate reproducible convergence across the five independent initializations considered in this study, providing empirical evidence of consistent convergence toward feasible low-dissipation solutions.
We develop a framework of "semi-automatic differentiation" that combines existing gradient-based methods of quantum optimal control with automatic differentiation. The approach allows to optimize practically any computable functional and is implemented in two open source Julia packages, GRAPE.jl and Krotov.jl , part of the QuantumControl.jl framework. Our method is based on formally rewriting the optimization functional in terms of propagated states, overlaps with target states, or quantum gates. An analytical application of the chain rule then allows to separate the time propagation and the evaluation of the functional when calculating the gradient. The former can be evaluated with great efficiency via a modified GRAPE scheme. The latter is evaluated with automatic differentiation, but with a profoundly reduced complexity compared to the time propagation. Thus, our approach eliminates the prohibitive memory and runtime overhead normally associated with automatic differentiation and facilitates further advancement in quantum control by enabling the direct optimization of non-analytic functionals for quantum information and quantum metrology, especially in open quantum systems. We illustrate and benchmark the use of semi-automatic differentiation for the optimization of perfectly entangling quantum gates on superconducting qubits coupled via a shared transmission line. This includes the first direct optimization of the non-analytic gate concurrence.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.
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.