We introduce a new hybrid quantum-classical algorithm inspired by an advanced control strategy known as model predictive control (MPC). This algorithm unifies the optimization-based design of variational quantum algorithms (VQAs) with the feedback-based design of feedback-based quantum algorithms (FQAs). Variational circuit parameters are optimized using a layer-wise receding horizon strategy, where observable measurements after every layer initialize a classically simulated dynamic model used to predict quantum state evolution and optimize over future parameterized gates. This hybrid algorithm can be used for applications such as ground state preparation and approximate combinatorial optimization, and presents an ideal use case for the integration of quantum computers with high-performance computing, where the latter resource can be used to increase the scale and efficiency of the predictions critical to MPC. We show through mathematical proof and numerical evidence that the MPC-based algorithm can be guaranteed to at least match the performance of FQAs. Through simulations on Max-Cut problems and a two-dimensional transverse-field Ising model, we demonstrate that relaxed implementations of the MPC-based algorithm can also provide improved performance in practice compared to an FQA.
We introduce a new hybrid quantum-classical algorithm inspired by an advanced control strategy known as model predictive control (MPC). This algorithm unifies the optimization-based design of variational quantum algorithms (VQAs) with the feedback-based design of feedback-based quantum algorithms (FQAs). Variational circuit parameters are optimized using a layer-wise receding horizon strategy, where observable measurements after every layer initialize a classically simulated dynamic model used to predict quantum state evolution and optimize over future parameterized gates. This hybrid algorithm can be used for applications such as ground state preparation and approximate combinatorial optimization, and presents an ideal use case for the integration of quantum computers with high-performance computing, where the latter resource can be used to increase the scale and efficiency of the predictions critical to MPC. We show through mathematical proof and numerical evidence that the MPC-based algorithm can be guaranteed to at least match the performance of FQAs. Through simulations on Max-Cut problems and a two-dimensional transverse-field Ising model, we demonstrate that relaxed implementations of the MPC-based algorithm can also provide improved performance in practice compared to an FQA.
The potential for greater algorithmic efficiency for some problems on quantum computers compared to classical computers is appealing in many fields including, for example, the process systems engineering field. While quantum algorithms have been studied for a variety of applications related to optimization, molecular modeling, and machine learning, there remain many applications in process systems engineering, including process control, where it is not clear how quantum computing algorithms would be beneficial. One idea for attempting to understand when a quantum algorithm might provide benefits for control is to start with algorithms that would be expected to benefit “similar” problems (e.g., optimization problems) and to see if controllers can be implemented within those algorithmic frameworks. Therefore, in this work, we study the use of a quantum computing algorithm related to Grover’s algorithm, which is an amplitude amplification strategy that can search an unordered list with improved efficiency compared to a classical algorithm for the task. It has been extended to perform a search for optimal paths over a graph. Given its potential utility for search and optimization, this is an example of an algorithm where we might wonder if it could be adjusted or used to provide speed-ups for large control problems if the controller could function within this algorithmic framework. This work provides the first steps toward attempting to address this question by investigating how optimization-based control problems would fit into this framework. A process described by ẋ=x+u is considered as a test case. The modified Grover’s algorithm requires the optimization problem to be mapped into quantum gates. We discuss ideas for attempting to represent an optimization-based controller known as model predictive control (MPC) in the modified Grover’s algorithm framework. We test how various parameters of the control and quantum algorithm designs, including fundamental parameters in MPC such as the number of sampling periods and length of the sampling periods, impact the success of using the quantum algorithm for the MPC. We provide analyses regarding why the results are what they are to give perspective on how quantum computing algorithms work and intersect with engineering problems.
In this paper, we present Quantum-Inspired Model Predictive Control (QIMPC), an approach that uses Variational Quantum Circuits (VQCs) to learn control polices in MPC problems. The viability of the approach is tested in five experiments: A target-tracking control strategy, energy-efficient building climate control, autonomous vehicular dynamics, the simple pendulum, and the compound pendulum. Three safety guarantees were established for the approach, and the experiments gave the motivation for two important theoretical results that, in essence, identify systems for which the approach works best.
The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024
Feedback-based quantum algorithms (FQAs) operate by iteratively growing a quantum circuit to optimize a given task. At each step, feedback from qubit measurements is used to inform the next quantum circuit update. In practice, the sampling cost associated with these measurements can be significant. Here, we ask whether FQA parameter sequences can be predicted using classical machine learning, obviating the need for qubit measurements altogether. To this end, we train a teacher-student model to map problem instances to associated FQA parameter curves in a single classical inference step. We first study weighted MaxCut problem instances, where numerical experiments show that this machine learning model can accurately predict FQA parameter curves across a range of problem sizes, including sizes not seen during model training. We then extend the analysis to transverse-field Ising models on graphs, and analyze how this learning strategy performs and generalizes beyond the original MaxCut setting. To evaluate performance, we compare the predicted parameter curves in simulation against FQA reference curves and linear quantum annealing schedules. We observe similar results to the former and performance improvements over the latter. These results suggest that machine learning can offer a heuristic, practical path to reducing sampling costs and resource overheads in quantum algorithms.
A critical engineering challenge in quantum technology is the accurate control of quantum dynamics. Model-based methods for optimal control have been shown to be highly effective when theory and experiment closely match. Consequently, realizing high-fidelity quantum processes with model-based control requires careful device characterization. In quantum processors based on cold atoms, the Hamiltonian can be well-characterized. For superconducting qubits operating at milli-Kelvin temperatures, the Hamiltonian is not as well-characterized. Unaccounted for physics (i.e., mode discrepancy), coherent disturbances, and increased noise compromise traditional model-based control. This work introduces model predictive control (MPC) for quantum control applications. MPC is a closed-loop optimization framework that (i) inherits a natural degree of disturbance rejection by incorporating measurement feedback, (ii) utilizes finite-horizon model-based optimizations to control complex multi-input, multi-output dynamical systems under state and input constraints, and (iii) is flexible enough to develop synergistically alongside other modern control strategies. We show how MPC can be used to generate practical optimized control sequences in representative examples of quantum state preparation. Specifically, we demonstrate for a qubit, a weakly-anharmonic qubit, and a system undergoing crosstalk, that MPC can realize successful model-based control even when the model is inadequate. These examples showcase why MPC is an important addition to the quantum engineering control suite.
Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.