Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Grover Adaptive Search with Problem-Specific State Preparation

Maximilian Hess, Lilly Palackal, Abhishek Awasthi, Peter J. Eder, Manuel Schnaus, Laurin Demmler, Karen Wintersperger, Joseph Doetsch

TL;DR

This paper tackles applying Grover-based quantum search to combinatorial optimization by introducing Grover Adaptive Search (GAS) for the Traveling Salesperson Problem (TSP) with a Lin-Kernighan–inspired, problem-specific state preparation. The approach biases the quantum superposition toward promising tours using a neighborhood-exchange mechanism and Dicke-state encodings, aiming for a quadratic speedup when combined with amplitude amplification. Through classical benchmarking that simulates GAS on 15 TSP instances with $n$ in {8,10,12}, the authors compare local-search state preparation against various GAS termination and iteration strategies, observing that fixed-interval termination is often suboptimal and that incremental strategies can incur higher iteration counts, while a well-tuned GAS setup achieves better trade-offs and polynomial-like scaling. The results emphasize the critical role of tailored state preparation for quantum optimization and outline future directions for larger instances, circuit-level implementations, and more efficient oracles to evaluate practical quantum advantage.

Abstract

Grover's search algorithm is one of the basic building block in the world of quantum algorithms. Successfully applying it to combinatorial optimization problems is a subtle challenge. As a quadratic speedup is not enough to naively search an exponentially large space, the search has to be complemented with a state preparation routine which increases the amplitudes of promising states by exploiting the problem structure. In this paper, we build upon previous work by Baertschi and Eidenbenz to construct heuristic state preparation routines for the Traveling Salesperson Problem (TSP), mimicking the well-known classical Lin-Kernighan heuristic. With our heuristic, we aim to achieve a reasonable approximation ratio with only a polynomial number of Grover iterations. Further, we compare several algorithmic settings relating to termination criteria and the choice of Grover iterations when the number of marked solutions is unknown.

Grover Adaptive Search with Problem-Specific State Preparation

TL;DR

This paper tackles applying Grover-based quantum search to combinatorial optimization by introducing Grover Adaptive Search (GAS) for the Traveling Salesperson Problem (TSP) with a Lin-Kernighan–inspired, problem-specific state preparation. The approach biases the quantum superposition toward promising tours using a neighborhood-exchange mechanism and Dicke-state encodings, aiming for a quadratic speedup when combined with amplitude amplification. Through classical benchmarking that simulates GAS on 15 TSP instances with in {8,10,12}, the authors compare local-search state preparation against various GAS termination and iteration strategies, observing that fixed-interval termination is often suboptimal and that incremental strategies can incur higher iteration counts, while a well-tuned GAS setup achieves better trade-offs and polynomial-like scaling. The results emphasize the critical role of tailored state preparation for quantum optimization and outline future directions for larger instances, circuit-level implementations, and more efficient oracles to evaluate practical quantum advantage.

Abstract

Grover's search algorithm is one of the basic building block in the world of quantum algorithms. Successfully applying it to combinatorial optimization problems is a subtle challenge. As a quadratic speedup is not enough to naively search an exponentially large space, the search has to be complemented with a state preparation routine which increases the amplitudes of promising states by exploiting the problem structure. In this paper, we build upon previous work by Baertschi and Eidenbenz to construct heuristic state preparation routines for the Traveling Salesperson Problem (TSP), mimicking the well-known classical Lin-Kernighan heuristic. With our heuristic, we aim to achieve a reasonable approximation ratio with only a polynomial number of Grover iterations. Further, we compare several algorithmic settings relating to termination criteria and the choice of Grover iterations when the number of marked solutions is unknown.
Paper Structure (11 sections, 7 equations, 4 figures, 3 algorithms)

This paper contains 11 sections, 7 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Methods and Related Work
  3. Grover Adaptive Search for the TSP
  4. TSP: problem definition
  5. Initial state preparation
  6. Further algorithm settings
  7. Parameters in GAS
  8. Termination criterion
  9. Number of Grover iterations for an unknown number of marked states
  10. Results and Discussion
  11. Conclusion and Outlook

Figures (4)

  • Figure 1: Example for sampling a new state from the reference on the left. Here, each color corresponds to a different node in the TSP. The exchange chain starts at index $0$ and has a length of $2$. This means the first two nodes in the route are swapped out for different ones.
  • Figure 2: Approximation ratio and iteration count of the local search strategy inspired by the Lin-Kernighan algorithm.
  • Figure 3: Comparison of the three variations of GAS outlined in \ref{['alg:Grover Adaptive Search highlighted']}. We show the approximation ratio for three different choices for the number of Grover iterations, cf. \ref{['sec: Number of Grover iterations for an unknown number of marked states']}. Additionally, we vary the termination criteria, where the upper bound for the number of Grover iterations is set by $\lambda^R$ and $R$ is the number of rounds within each improvement step.
  • Figure 4: Comparison of the three variations of GAS outlined in \ref{['alg:Grover Adaptive Search highlighted']}. We show the total number of iterations for three different choices for the number of Grover iterations, cf. \ref{['sec: Number of Grover iterations for an unknown number of marked states']}. The termination criteria depend on the upper bound for the number of Grover iterations $\lambda^R$, where $R$ is the number of rounds within each improvement step.