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

Sören Wilkening

TL;DR

Problem and approach: The paper tackles general constrained combinatorial optimization with arbitrary efficiently computable objective and constraint functions, proposing CBQS to bias Grover search via problem-specific state preparation. Method: It develops sum-constraint and product-constraint routines, with bounded-depth quantum circuits to compute objective values and constraint feasibility, and introduces iterative QSearch and QMaxSearch stages. Key contributions: formalized state-preparation theorems, polynomial-depth objective evaluation, and a benchmarking framework showing improved oracle efficiency and potential quantum advantage over classical solvers on large-scale instances. Findings and impact: CBQS offers a path to far-term quantum advantage for general constrained optimization, while acknowledging hardware realism limits and room for methodological improvements.

Abstract

We present a quantum algorithmic routine that extends the realm of Grover-based heuristics for tackling combinatorial optimization problems with arbitrary efficiently computable objective and constraint functions. Building on previously developed quantum methods that were primarily restricted to linear constraints, we generalize the approach to encompass a broader class of problems in discrete domains. To evaluate the potential of our algorithm, we assume the existence of sufficiently advanced logical quantum hardware. With this assumption, we demonstrate that our method has the potential to outperform state-of-the-art classical solvers and heuristics in terms of both runtime scaling and solution quality. The same may be true for more realistic implementations, as the logical quantum algorithm can achieve runtime savings of up to $10^2-10^3$.

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

TL;DR

Problem and approach: The paper tackles general constrained combinatorial optimization with arbitrary efficiently computable objective and constraint functions, proposing CBQS to bias Grover search via problem-specific state preparation. Method: It develops sum-constraint and product-constraint routines, with bounded-depth quantum circuits to compute objective values and constraint feasibility, and introduces iterative QSearch and QMaxSearch stages. Key contributions: formalized state-preparation theorems, polynomial-depth objective evaluation, and a benchmarking framework showing improved oracle efficiency and potential quantum advantage over classical solvers on large-scale instances. Findings and impact: CBQS offers a path to far-term quantum advantage for general constrained optimization, while acknowledging hardware realism limits and room for methodological improvements.

Abstract

We present a quantum algorithmic routine that extends the realm of Grover-based heuristics for tackling combinatorial optimization problems with arbitrary efficiently computable objective and constraint functions. Building on previously developed quantum methods that were primarily restricted to linear constraints, we generalize the approach to encompass a broader class of problems in discrete domains. To evaluate the potential of our algorithm, we assume the existence of sufficiently advanced logical quantum hardware. With this assumption, we demonstrate that our method has the potential to outperform state-of-the-art classical solvers and heuristics in terms of both runtime scaling and solution quality. The same may be true for more realistic implementations, as the logical quantum algorithm can achieve runtime savings of up to .

Paper Structure

This paper contains 9 sections, 6 theorems, 30 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Previous work
  3. Method
  4. The sum constraint
  5. The product constraint
  6. Additional quantum search stage
  7. Benchmarking
  8. Results
  9. Discussion

Key Result

Theorem 1

For any optimization problem with an efficiently computable objective function, there exists a quantum circuit that efficiently computes the respective objective value of any solution contained in a quantum state $\ket{\psi}$.

Figures (3)

  • Figure 1: Comparison of the number of oracle calls required by Grover's algorithm, qBnB and CBQS (above plot), and the relative objective value of Simanneal compared to CBQS. Simulated annealing is limited to using 1$0^5$ steps, leading to a vanishing success probability in finding feasible solutions from 90 variables.
  • Figure 2: Comparison of the time-to-solution of CBQS versus the classical state-of-the-art solvers Gurobi and Hexaly. The quantum gates are assumed to be executed within $6.5ns$. Our quantum algorithm potentially outperforms the classical solvers at finding various incumbent solutions.
  • Figure 3: Best found incumbent solution over time up to $\approx$7h for one 1000 variable instance. The comparison was made between Hexaly, Gurobi, CBQS, and its classical sampling equivalent. The quantum algorithm and its classical counterpart can quickly outperform Hexaly and Gurobi, which struggle to improve upon the incumbent solution by much.

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • Theorem 2
  • ...and 8 more