Graph-controlled Permutation Mixers in QAOA for the Flexible Job-Shop Problem

One of the most promising attempts towards solving optimization problems with quantum computers in the noisy intermediate scale era of quantum computing are variational quantum algorithms. The Quantum Alternating Operator Ansatz provides an algorithmic framework for constrained, combinatorial optimization problems. As opposed to the better known standard QAOA protocol, the constraints of the optimization problem are built into the mixing layers of the ansatz circuit, thereby limiting the search to the much smaller Hilbert space of feasible solutions. In this work we develop mixing operators for a wide range of scheduling problems including the flexible job shop problem. These mixing operators are based on a special control scheme defined by a constraint graph model. After describing an explicit construction of those mixing operators, they are proven to be feasibility preserving, as well as exploring the feasible subspace

Efficient learning of Sparse Pauli Lindblad models for fully connected qubit topology

The challenge to achieve practical quantum computing considering current hardware size and gate fidelity is the sensitivity to errors and noise. Recent work has shown that by learning the underlying noise model capturing qubit cross-talk, error mitigation can push the boundary of practical quantum computing. This has been accomplished using Sparse Pauli-Lindblad models only on devices with a linear topology connectivity (i.e. superconducting qubit devices). In this work we extend the theoretical requirement for learning such noise models on hardware with full connectivity (i.e. ion trap devices).Comment: 6 pages, 3 figure

Quantum-Assisted Solution Paths for the Capacitated Vehicle Routing Problem

Many relevant problems in industrial settings result in NP-hard optimization problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced variant, the Travelling Salesperson Problem (TSP). Even with today's most powerful classical algorithms, the CVRP is challenging to solve classically. Quantum computing may offer a way to improve the time to solution, although the question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ) devices can achieve a practical advantage compared to classical heuristics. The most prominent algorithms proposed to solve combinatorial optimization problems in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and the more general Variational Quantum Eigensolver (VQE). However, implementing them in a way that reliably provides high-quality solutions is challenging, even for toy examples. In this work, we discuss decomposition and formulation aspects of the CVRP and propose an application-driven way to measure solution quality. Considering current hardware constraints, we reduce the CVRP to a clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA and VQE and investigate the influence of various hyperparameters, such as the classical optimizer choice and strength of constraint penalization. Results of QAOA are generally of limited quality because the algorithm does not reach the energy threshold for feasible TSP solutions, even when considering various extensions such as recursive, warm-start and constraint-preserving mixer QAOA. On the other hand, the VQE reaches the energy threshold and shows a better performance. Our work outlines the obstacles to quantum-assisted solutions for real-world optimization problems and proposes perspectives on how to overcome them.Comment: Submitted to the IEEE for possible publicatio

Quantum Computing Techniques for Multi-Knapsack Problems

Optimization problems are ubiquitous in various industrial settings, and multi-knapsack optimization is one recurrent task faced daily by several industries. The advent of quantum computing has opened a new paradigm for computationally intensive tasks, with promises of delivering better and faster solutions for specific classes of problems. This work presents a comprehensive study of quantum computing approaches for multi-knapsack problems, by investigating some of the most prominent and state-of-the-art quantum algorithms using different quantum software and hardware tools. The performance of the quantum approaches is compared for varying hyperparameters. We consider several gate-based quantum algorithms, such as QAOA and VQE, as well as quantum annealing, and present an exhaustive study of the solutions and the estimation of runtimes. Additionally, we analyze the impact of warm-starting QAOA to understand the reasons for the better performance of this approach. We discuss the implications of our results in view of utilizing quantum optimization for industrial applications in the future. In addition to the high demand for better quantum hardware, our results also emphasize the necessity of more and better quantum optimization algorithms, especially for multi-knapsack problems.Comment: 20 page