Constraint-oriented biased quantum search for linear constrained combinatorial optimization problems

TL;DR

This work extends Grover-based quantum search to linear-constrained combinatorial optimization by developing Constraint-oriented biased quantum search (CBQS), which uses biased state preparation and amplitude amplification to favor feasible, high-quality solutions. It provides a unified framework for single and multiple linear constraints, plus improvement techniques such as advanced biasing, look-ahead, and item ordering, and demonstrates benchmarking strategies that reveal potential quantum advantages over classical solvers on large instances. The results show that CBQS can outperform certain classical methods in finding incumbent solutions early, with the potential for significant speedups when implemented on error-corrected quantum hardware. Overall, CBQS represents a practical pathway toward exploiting quantum speedups in structured constrained optimization problems like knapsack variants.

Abstract

In this paper, we extend a previously presented Grover-based heuristic to tackle general combinatorial optimization problems with linear constraints. We further describe the introduced method as a framework that enables performance improvements through circuit optimization and machine learning techniques. Comparisons with state-of-the-art classical solvers further demonstrate the algorithm's potential to achieve a quantum advantage in terms of speed, given appropriate quantum hardware.

Figures (7)

• Figure 1: Circuit for conditional branching and constraint updating on a single variable.
• Figure 2: State preparation quantum circuit example for a two-variable gapped knapsack problem with constraints $w^T x \leq c$ and $-w^T x \leq -(c-\varepsilon)$. Rather than checking whether $w_j$ fits the remaining constraint, we can subtract its absolute value and check if the result is not negative by looking at the uppermost sign qubit in each of the lower two registers. Then we only have to do a controlled uncomputation after assigning the variable.
• Figure 3: Properties of a 70 variable instance investigated in this work for the \ref{['eq:gkp']}.
• Figure 4: (a) Behavior of the objective value over time of the exact against the sampling-based benchmarking method. (b) Probability of finding a feasible solution when incorporating both or just one of the two constraints in the state preparation, tested on a non-trivial 70-variable instance with random ordering of items. (c) Number of oracle calls of the exact and sampling-based heuristic benchmarking method to find the optimal solution of a 25-variable instance. We observe that approximate benchmarking provides an empirical lower bound on the number of oracles and that significantly fewer Grover iterations are required to find a feasible state when incorporating both constraints.
• Figure 5: Comparison of our benchmarking strategy (red) and the actual amplitude amplification protocol suggested in this work (blue). We compare both success probability and expected number of iterations and find that the benchmarking protocol behaves very similarly in both metrics and can thus be used to produce meaningful benchmarks for the suggested quantum algorithm.

Theorems & Definitions (11)

• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3
• Definition 3.4
• Lemma 3.5
• proof
• Theorem 3.6
• proof
• Lemma 3.7