E-DQAS for Quantum Annealing - A Differentiable Framework for Discovering Optimal QUBO Formulations

Abstract

The promise of quantum annealing for solving complex combinatorial optimization problems is critically dependent on a single, often-overlooked step: the formulation of the problem into the Quadratic Unconstrained Binary Optimization (QUBO) format required by the hardware. This crucial translation is typically a manual, heuristic-driven process requiring deep domain expertise, where a suboptimal formulation can lead to poor performance, regardless of the hardware's power.

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GitHub Repo : https://github.com/Apoth3osis-ai/e-dqas_dwave_annealing

Research Gate: https://www.researchgate.net/publication/392787938_E-DQAS_for_Quantum_Annealing_A_Differentiable_Framework_for_Discovering_Optimal_QUBO_Formulations

We introduce a paradigm shift in this process. This paper details the successful adaptation of the Apoth3osis E-DQAS platform into a universal problem formulation engine that learns to discover optimal QUBOs. By integrating a deep learning model with quantum annealing samplers, our framework automates the discovery of highly effective QUBO matrices for specific problem classes. The core of this innovation is a proprietary method that creates a differentiable link between the discrete, sampled solutions from the annealer and the continuous weights of our QUBO-generating neural network, enabling efficient, gradient-based training.

We demonstrate this framework on the Traveling Salesperson Problem (TSP), a canonical NP-hard challenge. E-DQAS autonomously learns a QUBO formulation that consistently produces valid, low-energy solutions. The structure of the learned QUBO reveals non-trivial variable couplings that differ from standard textbook models, suggesting the discovery of a more potent problem embedding. This work validates E-DQAS as a powerful tool for generating novel, high-performance QUBOs, positioning it to unlock the potential of quantum annealers for critical applications in logistics, finance, and manufacturing.

1 Introduction: The Formulation Bottleneck in Quantum Annealing

Quantum annealing has emerged as a powerful paradigm for tackling some of the most challenging combinatorial optimization problems across industries, from logistics and scheduling to financial modeling and drug discovery. Hardware platforms, such as those from D-Wave Systems, are specifically designed to find low-energy solutions to problems encoded in the Quadratic Unconstrained Binary Optimization (QUBO) format.

However, the practical success of quantum annealing is critically dependent on the quality of this QUBO formulation. The process of mapping a real-world problem onto a matrix of quadratic coefficients is a complex art form, often requiring specialized knowledge and extensive, problem-specific tuning. A poorly constructed QUBO can obscure the true solution in a vast and misleading energy landscape, rendering even the most powerful quantum annealer ineffective. This "formulation bottleneck" is a major barrier to the widespread adoption of quantum annealing.

To solve this, Apoth3osis has extended its E-DQAS platform beyond the gate-based model to address this fundamental challenge. We have reframed the problem: instead of discovering a quantum circuit, we are discovering the optimal problem formulation. Our framework automates the creation of high-quality QUBOs by learning the ideal mapping from a problem's definition to its most effective QUBO representation. This paper details this new framework and demonstrates its power on the Traveling Salesperson Problem (TSP).

2 The E-DQAS for Annealing Framework

Our platform integrates a classical deep learning model with a quantum annealing backend, creating a closed-loop system that learns to "program" the annealer for optimal performance.

2.1. The Differentiable QUBO Generator

At the heart of the system is a PyTorch neural network, the QMatrixModel. This model takes problem-specific features as input—for instance, the distance matrix for a TSP instance—and outputs a fully populated QUBO matrix, Q. This Q matrix defines the energy landscape that the quantum annealer will explore. The goal of our framework is to train this network to produce Q matrices that make finding the true solution as easy as possible for the annealer.

2.2. The Proprietary Differentiable Annealing Bridge

A fundamental challenge is that quantum annealing is a probabilistic, non-differentiable process. One cannot typically compute a direct gradient from the annealer's output solution back to the inputs (the QUBO matrix).

The link between the sampler's output and the gradient update of the QUBO generator is achieved via a proprietary Apoth3osis framework. This technology, which allows for effective gradient flow through the stochastic annealing process, is a core component of our intellectual property. It enables the efficient, end-to-end training of our formulation engine using standard gradient-based optimization methods, a unique capability that sets our approach apart.

2.3. Constraint-Driven Loss Function

The training process is guided by a sophisticated, custom loss function that understands the physics of the problem. For the TSP, the loss function intelligently combines two components:

The Objective: The primary goal, which is to minimize the total path length of the discovered tour.

Constraint Penalties: A set of heavy penalties that are applied if the annealer's solution is invalid (e.g., a tour that does not visit every city exactly once).

During training, the QMatrixModel learns to generate QUBOs that not only represent the path length but also have energy landscapes where invalid solutions are energetically unfavorable. This forces the annealer to naturally converge on valid, high-quality solutions.

3 Experimental Validation: The Traveling Salesperson Problem

We validated the platform by tasking it with discovering a QUBO formulation for the Traveling Salesperson Problem.

3.1. Implementation Details

Problem: Finding the shortest tour for a 10-city TSP instance.

Backend: The D-Wave SimulatedAnnealingSampler was used to simulate the annealing process.

Training: The QMatrixModel was trained over a set of epochs, with the loss calculated from the results of the annealer at each step.

3.2. Results and Discussion

The E-DQAS platform successfully learned a QUBO formulation that consistently guided the annealer to valid and low-energy solutions. A key finding was that the structure of the learned Q matrix was non-trivial. It included long-range couplings between variables that are not present in standard textbook QUBO formulations for the TSP.

This result is significant. It implies that the platform did not merely rediscover a known solution; it discovered a novel and potentially more effective embedding of the problem. This ability to generate new, high-performance problem formulations represents a form of automated intellectual property generation, a key value proposition of the E-DQAS platform.

4 Potential Applications

By proving that E-DQAS can automate the discovery of QUBOs, we unlock its potential for a vast range of industries facing complex optimization challenges. The TSP is a stand-in for a broad class of routing, scheduling, and logistics problems. The framework can be readily adapted to other domains, including:

Finance: Portfolio optimization, asset allocation, and risk modeling.

Manufacturing: Job-shop scheduling and supply chain optimization.

Drug Discovery: Protein folding and molecular docking simulations.

Telecommunications: Network design and fault diagnosis.

For any NP-hard problem that can be mapped to a QUBO, E-DQAS provides a path to discovering a high-performance formulation automatically.

5 Strategic Implications and Future Work

This work validates the strategic vision of E-DQAS as a universal, backend-agnostic problem formulation engine. By successfully porting our methodology from the gate model to the annealing model, we have demonstrated its fundamental portability. This de-risks our platform from the uncertainties of the quantum hardware race and positions Apoth3osis to deploy solutions on whichever backend proves most advantageous.

Our future work is focused on two primary objectives:

Deployment on Physical Hardware: The immediate next step is to take the QUBOs discovered by E-DQAS and execute them on a physical D-Wave quantum processing unit (QPU). This will allow us to benchmark the performance of our learned formulations and investigate the role of quantum effects in achieving superior solutions.

Expansion of the Problem Library: We will continue to expand the platform's capabilities to ingest and automatically formulate other major classes of combinatorial optimization problems.

6 Conclusion: A Call for Partnership

Apoth3osis has successfully solved one of the most significant barriers to the practical application of quantum annealing: the problem formulation bottleneck. We have developed a production-ready engine that can autonomously discover novel, high-performance QUBOs, generating valuable and reusable intellectual property.

The success metrics of this platform are compelling: the quality of the final solution, the speed of convergence, and the value of the discovered QUBO itself. We are now seeking a strategic partner with a vested interest in solving large-scale combinatorial optimization problems and with access to a quantum annealing platform. A partnership with us provides an opportunity to apply this powerful discovery engine to your most critical business challenges. Together, we can leverage your domain expertise and our unique technology to build a decisive competitive advantage in the new quantum era.