Luna v1.0 & FlexQAOA bring constraint-aware quantum optimization to real-world problems

Aqarios’ platform Luna v1.0 marks a major milestone in quantum optimization. This release significantly improves usability, performance, and real-world applicability by introducing FlexQAOA, a hybrid quantum algorithm designed specifically to handle industrial constraints directly within quantum circuits.

Luna enables professionals across logistics, energy, manufacturing and other industries to easily model, solve, and interpret complex optimization problems, bringing quantum computing into practical use today.

On June 4, 2025, we launched Luna v1.0, our most significant upgrade to our quantum optimization platform yet. Quantum computing promises powerful optimization capabilities, but the complexity of real world problems, in particular handling problem constraints, still poses a major challenge in this field. Luna v1.0 tackles these challenges head-on.

What makes Luna v1.0 special? It provides an intuitive user experience, deeper integration of quantum and hybrid algorithms, and native support for FlexQAOA, our new constraint-native quantum optimization algorithm.

Luna is designed for professionals who face complex problems in decision-making. It doesn’t require prior quantum computing expertise, just real-world optimization challenges that demand robust, constraint-respecting solutions.

Quantum optimization meets constraints: Why we built FlexQAOA

Quantum computing promises significant breakthroughs in optimization, but when real-world constraints are introduced, many quantum methods falter. Industrial problems rarely exist without constraints: budgets, timeframes, capacities. Traditional quantum algorithms cannot handle these constraints natively, requiring complicated problem reformulations through penalty terms.

We saw an opportunity: Why not create a quantum algorithm built specifically around constraints from the start?

That’s FlexQAOA.

Constraints are everywhere

Constraints are fundamental to optimization problems and define the limitations of which solutions are valid in the first place. Consider the following scenarios:

• Energy grids: Balancing power generation within strict regulatory and capacity limits.
• Manufacturing: Efficiently assigning tasks to machines while respecting capacities and deadlines.
• Logistics: Optimizing vehicle routes with precise load limits and delivery windows.

Traditional quantum methods struggle here. The reformulations to work with constraints waste resources exploring infeasible solutions, complicate the optimization process by having to balance the original objective with additional penalties, and require additional slack variables, increasing the number of required qubits.

We aimed for something simpler, clearer, and more efficient.

How FlexQAOA solves constraints directly

FlexQAOA directly encodes constraints into quantum circuits, eliminating complicated penalty structures. We introduced two concepts:

• XY-Mixers: Handle decisions requiring exclusive (one-hot) selections by constraining the quantum state to feasible solutions.
• Indicator Functions: Manage inequality constraints (like budgets or capacities) by applying targeted phase shifts, efficiently encoding constraint satisfaction.

These innovations allow FlexQAOA to deliver clear, practical solutions faster.

FlexQAOA in practice: Benchmarking results

We benchmarked FlexQAOA using the multi-dimensional knapsack problem, a well-known, complex optimization challenge involving multiple constraints. Even at a few QAOA layers, FlexQAOA matches or surpasses baseline quantum methods. With more QAOA iterations, it consistently delivers superior results, clearly outperforming conventional penalty-based algorithms.

FlexQAOA achieves a probability of sampling high-quality solutions of more than 90% at just 10 QAOA layers for the investigated instances, without requiring slack variables or tuning penalty weights.

Its constraint-aware architecture enhances solution quality and improves time-to-solution, thanks to a dramatically reduced search space, making FlexQAOA a strong candidate for solving industrial-scale problems as quantum hardware continues to evolve. The reduced search space not only increases performance but also enables the simulation of problem sizes that are inaccessible to conventional methods.

Detailed results are available in our recent paper posted to the arXiv preprint server.

Real-world impact: FlexQAOA in energy optimization with E.ON

One of the first real-world applications of FlexQAOA was developed in collaboration with E.ON Digital Technology, where we addressed a key challenge in the future of energy: optimizing electricity demand from flexible appliances in prosumer households.

The goal was to coordinate smart devices like EV chargers and heat pumps in a way that minimizes electricity costs while making better use of locally generated renewable energy—all without violating grid constraints.

Using FlexQAOA, we successfully encoded the problem’s complex structure directly into a quantum circuit, enabling constraint-aware optimization that respects real-world feasibility. The results show clear potential for improving flexibility and efficiency in energy systems.

You can read the full case study here.

Luna v1.0: Quantum optimization for everyone

With Luna, users can model, benchmark and solve optimization problems intuitively using Python, while Luna provides hardware-agnostic access to various algorithms and hardware backends to choose from. It combines proprietary algorithms with automated pipelines, making the process of problem-solving more intuitive and easier than ever before. Already in active use across logistics, manufacturing, and energy systems, Luna proves that quantum optimization is now within reach for anyone looking to get started.

Our roadmap includes extending FlexQAOA for broader constraint types, enhancing performance on quantum hardware, and expanding hybrid optimization workflows. We believe this is just the beginning of the transformative potential of quantum optimization.

This story is part of Science X Dialog, where researchers can report findings from their published research articles. Visit this page for information about Science X Dialog and how to participate.

Aqarios is a Munich-based deep-tech startup, founded in 2021 as a spin-off from LMU Munich and its “Quantum Applications and Research Laboratory” (QAR-Lab). We specialize in advancing quantum computing for industrial use by transforming cutting-edge research into practical applications. We focus on quantum algorithms and quantum-enhanced machine learning, making them accessible and usable across real-world scenarios.

With a strong foundation in sectors such as aerospace, automotive, finance, energy, logistics, and manufacturing, Aqarios leverages over a decade of quantum application research to deliver innovative, user-centric software solutions. Our intuitive tools provide streamlined access to quantum applications, algorithms, and hardware—empowering everyone from novice users to seasoned experts to solve complex problems more efficiently and powerfully.