Reducing Problem Size for Quantum Optimization
In the quest for leveraging quantum computing to solve complex combinatorial optimization problems, our recent study has delved into the knapsack problem—a classic challenge that finds applications across various fields from cryptography to resource management. While the theoretical potential of quantum computing promises significant advancements, practical implementation on current hardware remains a hurdle due to resource constraints. Here, we focus on our innovative approaches to reduce problem size, making quantum optimization more feasible with today’s quantum computers.
Background: The Knapsack Problem and Quantum Computing
The knapsack problem involves selecting items with given weights and values to maximize total value without exceeding a weight limit. Formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, it can be mapped to quantum computing hardware like neutral atom quantum computers. However, as the number of items increases, the required quantum resources grow quadratically, presenting a significant challenge for current quantum systems.
Innovative Methods to Reduce Problem Size
To address these challenges, we developed several classical algorithms aimed at reducing the size of the QUBO formulation of the knapsack problem:
- Matrix Sparsity Enhancement
- Euclidean Scaling
- Coarse Grouping
These methods enhance the feasibility of quantum optimization by decreasing the computational complexity and resource requirements, thus bringing us closer to practical quantum advantage.
Matrix Sparsity Enhancement
Increasing matrix sparsity involves pruning variables or interactions with minimal impact on the objective function. This hybrid heuristic approach systematically identifies and removes less significant elements, thereby simplifying the problem.
Example: For an intricate instance where the decision between items is challenging due to closely matched value-to-weight ratios and tight weight limits, we applied the following steps:
- Calculated value-to-weight ratios for all items.
- Determined the upper bound (UB) and lower bound (LB) for optimal solutions.
- Pruned items with UB less than or equal to LB, reducing the problem size significantly while retaining solution accuracy.
Euclidean Scaling
This method reduces the maximum weight capacity of the knapsack while maintaining the problem’s landscape. By scaling down the weights and values using their greatest common divisors, we preserve the relative differences, allowing the scaled-down problem to yield equivalent solutions with fewer resources.
Example: We applied Euclidean Scaling to our knapsack instance:
- Calculated the greatest common divisor for weights and values.
- Scaled down the knapsack capacity and item weights.
- Adjusted the problem iteratively, maintaining the optimal value-to-weight ratio.
Coarse Grouping
Grouping similar items together and treating them as a single entity can drastically reduce the problem size. This approach is particularly useful when items have closely matched characteristics, allowing us to simplify the problem without losing significant detail.
Results and Future Directions
Our tests with these methods on various data instances have shown promising results. By effectively managing computational complexity, we achieved optimal or near-optimal solutions in all cases with significantly reduced problem sizes. These optimizations enhance the practicality of quantum algorithms, making them more accessible with current hardware capabilities.
Moving forward, our focus will shift towards identifying other combinatorial problems that can benefit from similar size-reducing techniques. Problems like portfolio optimization, which naturally map to sparse QUBO formulations, hold promise for future explorations. By continuing to refine these algorithms and exploring new applications, we aim to pave the way for realizing the full potential of quantum optimization.
At Decision Lab, we are helping make quantum computing a reality for practical use. By partnering with us, you can leverage our expertise in reducing problem size and optimizing computational efficiency. Whatever challenges you face, our tailored approaches can help. Contact us to discover how our solutions can help you on your path to giving your projects a quantum advantage and driving your success. Contact us today!
Author: Dr. Joshua Liu, Decision Lab Principal Technology Strategist
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