Knapsack Problem variants of QAOA for battery revenue optimisation (1908.02210v2)

Abstract: We implement two Quantum Approximate Optimisation Algorithm (QAOA) variants for a battery revenue optimisation problem, equivalent to the weakly NP-hard Knapsack Problem. Both approaches investigate how to tackle constrained problems with QAOA. A first 'constrained' approach introduces a quadratic penalty to enforce the constraint to be respected strictly and reformulates the problem into an Ising Problem. However, simulations on IBM's simulator highlight non-convergent results for intermediate depth ($ p\leq 50$). A second 'relaxed' approach applies the QAOA with a non-Ising target function to compute a linear penalty, running in time $O(p(\log_2 n)^3)$ and needing $O(n \log n)$ qubits. Simulations reveal an exponential improvement over the number of depth levels and obtain approximations about $0.95$ of the optimum with shallow depth ($p \leq 10$).

• The paper introduces two novel QAOA variants—constrained and relaxed—for optimizing battery revenue within a knapsack problem framework.
• The constrained approach integrates a quadratic penalty to mimic an Ising model but shows non-convergence at moderate circuit depths.
• The relaxed method employs a linear penalty, achieving up to 95% of the optimal result with shallow-depth circuits (p ≤ 10), demonstrating practical potential.

An Examination of QAOA Variants for Optimizing Battery Revenue within Knapsack Problem Frameworks

The paper "Knapsack Problem Variants of QAOA for Battery Revenue Optimization" investigates two variations of the Quantum Approximate Optimization Algorithm (QAOA) to tackle a specific battery revenue optimization problem, which can be seen as an extension of the Knapsack Problem—a combinatorial problem known to be weakly NP-hard. The primary focus is to explore both constrained and relaxed approaches of applying QAOA in the context of managing energy storage systems efficiently.

The authors implement a constrained approach that incorporates a quadratic penalty directly into the optimization function, reformulating the problem into an Ising model. However, simulated results using IBM's quantum simulators exhibit non-convergence for intermediate circuit depths (specifically, depths $p \leq 50$). In contrast, the relaxed approach uses a non-Ising target function where the constraints are imposed through a linear penalty. This variant demonstrates exponential improvements in performance with respect to the number of depths, achieving approximations of up to 95% of the optimum with shallow-depth circuits ($p \leq 10$).

Key Results and Contrasts

Constrained Approach

• Uses a quadratic penalty to enforce constraints, aligning the problem with the Ising model.
• Simulations indicate non-convergence at intermediate depths, suggesting inefficacy at those levels.

Relaxed Approach

• Employs a linear penalty for constraint violation, significantly reducing the complexity compared to the constrained variant.
• Provides strong numerical results, obtaining an approximation ratio of 0.95 at low depths, which highlights its viability for quantum circuit execution with existing quantum hardware constraints.

Relevance and Implications

The implications of this paper are noteworthy for both theoretical advancement and practical applications in quantum computing and energy management:

1. Theoretical Contribution: The paper expands the application potential of QAOA beyond traditional Ising model-centric problems like MAX-CUT, introducing novel methodologies for non-Ising objectives. This extension is crucial for broadening the selection of practical problems that can benefit from quantum optimization.
2. Practical Relevance: With the increasing penetration of renewable energy sources into power grids, efficient battery management becomes vital. The research discusses quantum-inspired solutions to optimize the economic returns of battery systems, which bear significant implications for real-world electricity market operations and renewable integration challenges.
3. Potential for Future Research: The promising results from the relaxed approach demonstrate not just theoretical appeal but also provide a viable path for developing more fine-tuned hybrid quantum-classical algorithms in energy system optimization and beyond.

Future Prospects

Beyond what is directly studied, this work opens potential paths for future research. Investigating the impact of different penalty coefficients and optimization of parameters $(\beta, \gamma)$ could refine these algorithms further. Furthermore, exploring the applicability of these QAOA variants to other domains beyond energy, particularly those involving constrained optimization problems, presents a fertile area for advancing quantum algorithm development.

In conclusion, while the constrained approach revealed limitations at certain depths, the relaxed QAOA variant shows substantial promise for practical applications. As quantum computing hardware continues to evolve, such innovative approaches provide a cornerstone for catalyzing computational advancements in combinatorial optimization, meshing quantum theoretical exploration with tangible industrial applicability.