Constrained Quantum Optimization at Utility Scale: Application to the Knapsack Problem

TL;DR

This work presents the largest successful demonstration of the knapsack problem on IBM Quantum hardware using up to 150 qubits, and more generally, the largest demonstration of constrained combinatorial optimization where constraints are enforced via shallow mixers.

Abstract

Constrained combinatorial optimization problems are challenging for quantum computing, particularly at utility-relevant scales and on near-term hardware. At the same time, these problems are of practical significance in industry; for example, the Unit Commitment (UC) problem in energy systems involves complex operational constraints. To address this challenge, we apply copula-QAOA (cop-QAOA), a hardware-efficient approach for constrained optimization to a single-period UC that can be reduced to a one-dimensional knapsack. Cop-QAOA biases the quantum state toward feasible solutions using constant-depth mixers and appropriately biased initial states. We implement our benchmark on problem instances that are confirmed to be hard for classical solvers such as Gurobi. Our results show that cop-QAOA often finds solutions better than a lazy greedy baseline and very close to, and in some instances surpasses, those obtained by Gurobi, with only a few QAOA rounds. This work presents the largest successful demonstration of the knapsack problem on IBM Quantum hardware using up to 150 qubits, and more generally, the largest demonstration of constrained combinatorial optimization where constraints are enforced via shallow mixers.

Figures (5)

• Figure 1: Impact of the parameter $\mathcal{D}$ on the quality of the solution find when mapping the UC problem to a Knapsack like problem. Figure shows the UC cost function (eq. \ref{['eq:UC_cost']}) as a value of $\mathcal{D}$ when solving problems with different values of load factor. The load factor is the ratio $L / \sum_i p_{max, i}$. Problem instances are generated for 100 units with randomized parameters drawn from the following ranges: $A \in [10, 50]$, $B \in [0.5, 1.5]$, $C \in [0.01, 0.2]$, $p_{\min} \in [10, 20]$, and $p_{\max} \in [50, 100]$.
• Figure 2: Results on 100-item instance: Cost distribution sampled from lazy greedy and cop-QAOA (a) simulator, $p=5$; (b) hardware, $p=1$; (c) hardware, $p=5$; (d) the ratio between the best value found by cop-QAOA at each depth $p$ and the solutions obtained by Gurobi and by the lazy greedy; (e) Ratio of valid solutions versus $p$ on simulator and hardware. The horizontal gray dashed line shows the ratio for lazy greedy; (f) Approximation ratio (Ar) versus $p$ on simulator and hardware. The gray horizontal dashed line shows the approximation ration of lazy greedy. The instance is generated based on jooken2022new. Gurobi reports a solution with an optimality gap of $4 \cdot 10^{-2}$. A random sampler fails to produce even a single valid solution for this instance. The number of shots for both the lazy greedy baseline and cop-QAOA is $10^{5}$. The hardware experiments are executed on the ibm_kingston quantum processor using the Q-CTRL Performance Management Qiskit function fire_opalqiskit-function. The two-qubit gate depths for $p=1$ through $p=5$ are 5,7,7,5,6. The parameter $k$ for lazy greedy distribution (see Eq. \ref{['EQ:proba']}) is set to 8.
• Figure 3: Results on 150-item instance: Cost distribution sampled from lazy greedy and cop-QAOA (a) simulator, $p=1$; (b) hardware, $p=1$. This instance is generated based on Inversely Strongly Correlated Distribution introduced in pisinger2005hard. For this instance the solution found by Gurobi is optimal. A random sampler fails to produce even a single valid solution for this instance. The number of shots for both the lazy greedy baseline and cop-QAOA is $10^{5}$. The 2-qubit gate depth here is 28. The hardware experiments are executed on the ibm_kingston quantum processor using the Q-CTRL Performance Management Qiskit function fire_opalqiskit-function. The parameter $k$ for lazy greedy distribution (see Eq. \ref{['EQ:proba']}) is set to 10.
• Figure 4: Best value and approximation ratio on simulator. The plot addresses two instances generated based on jooken2022new, whose difficulty for Gurobi is discussed in the main text. (a) shows the ratio of the best value found by cop-QAOA to the solutions obtained by Gurobi and by the lazy greedy algorithm, as a function of $p$. (b) shows the approximation ratio as a function of $p$. Horizontal dashed lines show the approximation ratio for lazy greedy.
• Figure 5: Cost value as a function of $\gamma$ and $\beta$ for four problem instances with $P=1$. Panels (a) and (c) correspond to the same instances shown in Fig. \ref{['fig:fig2']} and Fig. \ref{['fig:fig3']}, respectively. The remaining two instances are generated using the Inversely Strongly Correlated Distribution. Small red dots indicate all $(\gamma, \beta)$ pairs for which the obtained cost exceeds the cost at $(\gamma=0, \beta=0)$. The star marks the best-performing parameter setting, while the large red circles highlight the three best solutions. The similarity of the parameter landscapes—particularly in the regions corresponding to near-optimal values—across different problem instances highlights the potential for parameter transferability and suggests that effective performance may be achieved without instance-specific training.