Quantum optimal control is a technique for controlling the evolution of a quantum system and has been applied to a wide range of problems in quantum physics. We study a binary quantum control optimization problem, where control decisions are binary-valued and the problem is solved in diverse quantum algorithms. In this paper, we utilize classical optimization and computing techniques to develop an algorithmic framework that sequentially optimizes the number of control switches and the duration of each control interval on a continuous time horizon. Specifically, we first solve the continuous relaxation of the binary control problem based on time discretization and then use a heuristic to obtain a controller sequence with a penalty on the number of switches. Then, we formulate a switching time optimization model and apply sequential least-squares programming with accelerated time-evolution simulation to solve the model. We demonstrate that our computational framework can obtain binary controls with high-quality performance and also reduce computational time via solving a family of quantum control instances in various quantum physics applications.
Quantum optimal control is a technique for controlling the evolution of a quantum system and has been applied to a wide range of problems in quantum physics. We study a binary quantum control optimization problem, where control decisions are binary-valued and the problem is solved in diverse quantum algorithms. In this paper, we utilize classical optimization and computing techniques to develop an algorithmic framework that sequentially optimizes the number of control switches and the duration of each control interval on a continuous time horizon. Specifically, we first solve the continuous relaxation of the binary control problem based on time discretization and then use a heuristic to obtain a controller sequence with a penalty on the number of switches. Then, we formulate a switching time optimization model and apply sequential least-squares programming with accelerated time-evolution simulation to solve the model. We demonstrate that our computational framework can obtain binary controls with high-quality performance and also reduce computational time via solving a family of quantum control instances in various quantum physics applications.
Quantum control aims to manipulate quantum systems toward specific quantum states or desired operations. Designing highly accurate and effective control steps is vitally important to various quantum applications, including energy minimization and circuit compilation. In this paper we focus on discrete binary quantum control problems and apply different optimization algorithms and techniques to improve computational efficiency and solution quality. Specifically, we develop a generic model and extend it in several ways. We introduce a squared L2 -penalty function to handle additional side constraints, to model requirements such as allowing at most one control to be active. We introduce a total variation (TV) regularizer to reduce the number of switches in the control. We modify the popular gradient ascent pulse engineering (GRAPE) algorithm, develop a new alternating direction method of multipliers (ADMM) algorithm to solve the continuous relaxation of the penalized model, and then apply rounding techniques to obtain binary control solutions. We propose a modified trust-region method to further improve the solutions. Our algorithms can obtain high-quality control results, as demonstrated by numerical studies on diverse quantum control examples.
Applying sophisticated optimization methods to control problems in various systems offers a structured approach for achieving system efficiency and reliability, while ensuring the feasibility of physics-based constraints. This dissertation primarily focuses on developing optimization algorithms for control problems in complex systems with mixed-integer variables and parameter uncertainties. In Chapter 1, we provide a detailed introduction about the problem motivation and summarize our problems and contributions. Specifically, we develop new optimization models and algorithms for network-based traffic control and binary control in quantum systems, detailed as follows. Traffic signal control is an effective way of mitigating traffic congestion, where its complexity escalates in large cities due to numerous intersections and varying traffic conditions. Advancements in communication technologies within transportation infrastructures have made distributed traffic signal control at the network level possible and worth investigating. In Chapter 2, we build a stochastic optimization model for network-level traffic signal control under traffic demand uncertainty and solve it using decentralized algorithms. We compare the results with state-of-the-art traffic control methods via testing instances of real-world traffic networks and data. Quantum computing and quantum systems provide a novel way of significantly accelerating computation and their operations largely rely on quantum pulse control optimization to attain desired states or status. The optimization of binary quantum pulse control can potentially enhance the performance of classical quantum variational techniques and improve solution quality. In Chapter 3, we study a discrete-valued binary quantum control problem and propose an algorithmic framework to solve the problem with penalty on switches. In Chapter 4, we explore the binary quantum control problem in a continuous time horizon by optimizing both control functions and control time intervals. Moreover, the time-varying noise in quantum systems and the wide use of inhomogeneous quantum ensembles highlight the need for quantum controls considering uncertainties. In Chapter 5, we develop a stochastic optimization model for the binary quantum control problem with uncertain Hamiltonian controllers and solve it using gradient-based methods with rounding algorithms. We test the performance of our approaches and benchmark with state-of-the-art control methods in all three chapters. Lastly, in Chapter 6, we conclude the dissertation and state future research directions.
We develop quantum algorithms for optimal control over a prescribed finite time interval (finite horizon), focusing on continuous-time linear-quadratic regulation (LQR). This foundational control problem often requires solving a nonlinear matrix Riccati equation, which can be costly for large systems and long control durations on conventional classical solvers. We instead work directly with the state-costate optimality equations, reformulating the LQR problem as a multiple-shooting linear system. Importantly, we prove a condition-number bound for the system that is independent of the horizon, under stabilizability and detectability assumptions. By incorporating the initial and final conditions into a single linear system, we can solve the subsequent linear system more efficiently with a quantum linear systems subroutine. For a given initial state, our quantum algorithms return classical estimates of the optimal cost and the optimal control inputs at a selected time, without constructing a Riccati solution operator. Given sparse-matrix block-encodings, an initial-state preparation oracle and a fixed number of control inputs, the query and gate complexities scale polylogarithmically with system size and control duration, provided that sparsity, relevant norms, and stability parameters are bounded. The dependence on the target error is inverse-linear up to logarithmic factors. We establish the required bounds uniformly in system size and duration for a family of damped mass-spring chains with two endpoint controls. Numerical experiments support the analysis.
Optimizing the controls of quantum systems plays a crucial role in advancing quantum technologies. The time-varying noises in quantum systems and the widespread use of inhomogeneous quantum ensembles raise the need for high-quality quantum controls under uncertainties. In this paper, we consider a stochastic discrete optimization formulation of a discretized binary optimal quantum control problem involving Hamiltonians with predictable uncertainties. We propose a sample-based reformulation that optimizes both risk-neutral and risk-averse measurements of control policies, and solve these with two gradient-based algorithms using sum-up-rounding approaches. Furthermore, we discuss the differentiability of the objective function and prove upper bounds of the gaps between the optimal solutions to binary control problems and their continuous relaxations. We conduct numerical simulations on various sized problem instances based on two applications of quantum pulse optimization; we evaluate different strategies to mitigate the impact of uncertainties in quantum systems. We demonstrate that the controls of our stochastic optimization model achieve significantly higher quality and robustness compared with the controls of a deterministic model. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This work was supported by the US Department of Energy, Advanced Scientific Computing Research [Grants DE-AC02-06CH11357, DE-SC0018018]; Defense Sciences Office, DARPA [Grant IAA-8839-annex-130]; the US National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2041745]; and the US National Aeronautics and Space Administration (NASA) Ames Research Center [Grant 80ARC020D0010]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0560 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0560 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
Quantum computing has developed rapidly in recent years and has shown potential advantages over classical computing in various domains. In this paper, we study whether quantum computing can accelerate the convergence rate of discrete simulation optimization in the fixed-budget setting. We first formulate the quantum discrete simulation optimization problem by defining the quantum simulation oracle, the algorithmic performance measure, and the corresponding optimality guarantee. Based on this formulation, we establish a minimax lower bound on the expected optimality gap for any quantum algorithm. The lower bound analysis reduces the discrete simulation optimization problem to a sequential quantum phase-testing problem, and the resulting lower-bound argument may be of independent interest. By comparing the quantum lower bound with its classical counterpart, we show that the optimal convergence rate in the quantum setting exhibits a quadratic improvement in its dependence on the simulation budget, at the cost of a worse dependence on the number of decisions. We then propose the first fixed-budget quantum algorithm, called \textsc{QUEST}, for discrete simulation optimization. The algorithm combines a quantum Monte Carlo subroutine with a sequential elimination framework and uses a slow-measurement mechanism to help preserve quantum superposition during the optimization process. We prove that \textsc{QUEST} achieves the optimal convergence rate up to logarithmic factors. Numerical experiments further show that \textsc{QUEST} outperforms its classical counterpart, which provides empirical evidence for the quantum advantage in improving the convergence rate of discrete simulation optimization algorithms.
This paper presents a novel approach for solving finite-horizon linear optimal control based on block-Toeplitz least-squares problem, by recasting the latter as Quadratic Unconstrained Binary Optimization (QUBO) suitable for adiabatic quantum computing (AQC). Classical block-Toeplitz leastsquares problem, scales asO((Nm)3)for horizon lengthNand control dimensionm. We demonstrate that this Toeplitzstructured least-squares cost can be transformed into a QUBO formulation by introducing binary precision vectors that encode each continuous control parameter into finite number of bits. We establish the general case in mathematically rigorous manner wherein the total number of binary decision variables remain dependent onN. To respect current quantum annealer capabilities, we propose a basis-function parametrization that approximates the full control sequence with a small set of basis coefficients, reducing the total number of binary-variables and rendering the latter independent ofN. Through simulation study, we show that, for large datasets, the QUBO pipeline, comprising of hardware-constant anneal step outperforms classical least square based solvers by several factors whilst demonstrating acceptable accuracy.