Efficient QAOA Architecture for Solving Multi-Constrained Optimization Problems

TL;DR

The paper tackles constrained combinatorial optimization by integrating constraint-preserving QAOA components: XY-Mixers enforce one-hot constraints and IF-based oracles handle inequality constraints, specifically applied to Multi-Knapsack and Prosumer problems. The authors develop a modular circuit-generation pipeline and a memory-efficient state-tensor simulator that exploits the reduced feasible space, enabling simulations up to 88 binary variables. Empirical results show IF+XY consistently outperforms standard QUBO and other constrained variants in RAAR, optimal-solution probability, and Time-to-Solution, with notable speedups over baselines and favorable scaling trends. These findings highlight a practical, scalable path toward more capable quantum optimization of real-world constrained problems and point to hardware experiments as a next step.

Abstract

This paper proposes a novel combination of constraint encoding methods for the Quantum Approximate Optimization Ansatz (QAOA). Real-world optimization problems typically consist of multiple types of constraints. To solve these optimization problems with quantum methods, normally, all constraints are added as quadratic penalty terms to the objective, which expands the search space and increases problem complexity. This work proposes a general workflow that extracts and encodes specific constraints directly into the circuit of QAOA: One-hot constraints are enforced through $XY$-mixers that restrict the search space to the feasible sub-space naturally. Inequality constraints are implemented through oracle-based Indicator Functions (IF). This paper focuses on the numerical benchmarks of the combined approach for solving the Multi-Knapsack (MKS) and the Prosumer Problem (PP), a modification of the MKS in the domain of electricity optimization. To this end, we introduce computational techniques that efficiently simulate the two presented constraint architectures. Since $XY$-mixers restrict the search space, specific state vector entries are always zero and can be omitted from the simulation, saving valuable memory and computing resources. We benchmark the combined method against the established QUBO formulation, yielding a better solution quality and probability of sampling the optimal solution. Despite more complex circuits, the time-to-solution is more than an order of magnitude faster compared to the baseline methods and exhibits more favorable scaling properties.

Figures (6)

• Figure 1: Panel (a) shows the circuit diagram of the combined architecture, implementing both $XY$-mixers for one-hot constraints and IF penalties for inequality constraint satisfaction. Panel (b) depicts the simulation loop for a mixture of qubits and qudits. The mixer layer is split into two sections: First, the $R_X$ mixer is applied on all qubits by reshaping the state tensor into a matrix and applying batched gate multiplication. Afterwards, the qudit legs are expanded and the $U_{XY}^\text{Ring}$ matrices are contracted onto the legs. Both operations rely on NVIDIA's cuQuantum SDK.
• Figure 2: Algorithm for efficient brute-forcing based on bit keys and masks. Symbols $\mathbin{\&}$, $\mathbin{|}$ and $\mathbin{\ll}$ refer to bitwise operations.
• Figure 3: State vector size required for simulation depending on the PP instance and encoding method. The dashed line indicates the 28-qubit equivalent, which marks the maximum for our numerical simulations.
• Figure 4: $P^*$ and $P_{90}$ of the combined IF+XY (at $p=5,10$) in comparison to the most promising TAE (slack and no-slack at $p=20$) approach from Ref. hess2024. Multi-Knapsack instances start at scenario number 10. All smaller instance numbers are single Knapsack instances (no one-hot constraints and $XY$-Mixer necessary).
• Figure 5: Results showing the RAAR (upper panel) and $P^*$ (lower panel) for the four investigated pipelines at maximum QAOA layer depth $p=12$. The dashed and dotted lines in the lower plot mark probability when random sampling in the $N$ bits or the one-hot constrained subspace, respectively.