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Solving the Traveling Salesman Problem via Different Quantum Computing Architectures

Venkat Padmasola, Zhaotong Li, Rupak Chatterjee, Wesley Dyk

TL;DR

Solving the Traveling Salesman Problem (TSP) remains challenging as it is NP-hard, and this paper surveys a diverse set of emerging hardware approaches. It compares gate-based quantum computing (QAOA and QPE), quantum annealing via QUBO-Ising mappings, and optical/optoelectronic Ising machines, assessing each platform's scalability, noise sensitivity, and feasibility. The work provides concrete formulations and experimental results up to 18-node instances, showing Ising-type processors offer notable time advantages and scalability while gate-based methods are currently restricted to small instances on noisy devices. Taken together, the results outline a practical roadmap for leveraging heterogeneous hardware to tackle large-scale TSPs and establish benchmark expectations across these cutting-edge technologies.

Abstract

We study the application of emerging photonic and quantum computing architectures to solving the Traveling Salesman Problem (TSP), a well-known NP-hard optimization problem. We investigate several approaches: Simulated Annealing (SA), Quadratic Unconstrained Binary Optimization (QUBO-Ising) methods implemented on quantum annealers and Optical Coherent Ising Machines, as well as the Quantum Approximate Optimization Algorithm (QAOA) and the Quantum Phase Estimation (QPE) algorithm on gate-based quantum computers. QAOA and QPE were tested on the IBM Quantum platform. The QUBO-Ising method was explored using the D-Wave quantum annealer, which operates on superconducting Josephson junctions, and the Quantum Computing Inc (QCi) Dirac-1 entropy quantum optimization machine. Gate-based quantum computers demonstrated accurate results for small TSP instances in simulation. However, real quantum devices are hindered by noise and limited scalability. Circuit complexity grows with problem size, restricting performance to TSP instances with a maximum of 6 nodes. In contrast, Ising-based architectures show improved scalability for larger problem sizes. SQUID-based Ising machines can handle TSP instances with up to 12 nodes, while entropy computing implemented in hybrid optoelectronic components extend this capability to 18 nodes. Nevertheless, the solutions tend to be suboptimal due to hardware limitations and challenges in achieving ground state convergence as the problem size increases. Despite these limitations, Ising machines demonstrate significant time advantages over classical methods, making them a promising candidate for solving larger-scale TSPs efficiently.

Solving the Traveling Salesman Problem via Different Quantum Computing Architectures

TL;DR

Solving the Traveling Salesman Problem (TSP) remains challenging as it is NP-hard, and this paper surveys a diverse set of emerging hardware approaches. It compares gate-based quantum computing (QAOA and QPE), quantum annealing via QUBO-Ising mappings, and optical/optoelectronic Ising machines, assessing each platform's scalability, noise sensitivity, and feasibility. The work provides concrete formulations and experimental results up to 18-node instances, showing Ising-type processors offer notable time advantages and scalability while gate-based methods are currently restricted to small instances on noisy devices. Taken together, the results outline a practical roadmap for leveraging heterogeneous hardware to tackle large-scale TSPs and establish benchmark expectations across these cutting-edge technologies.

Abstract

We study the application of emerging photonic and quantum computing architectures to solving the Traveling Salesman Problem (TSP), a well-known NP-hard optimization problem. We investigate several approaches: Simulated Annealing (SA), Quadratic Unconstrained Binary Optimization (QUBO-Ising) methods implemented on quantum annealers and Optical Coherent Ising Machines, as well as the Quantum Approximate Optimization Algorithm (QAOA) and the Quantum Phase Estimation (QPE) algorithm on gate-based quantum computers. QAOA and QPE were tested on the IBM Quantum platform. The QUBO-Ising method was explored using the D-Wave quantum annealer, which operates on superconducting Josephson junctions, and the Quantum Computing Inc (QCi) Dirac-1 entropy quantum optimization machine. Gate-based quantum computers demonstrated accurate results for small TSP instances in simulation. However, real quantum devices are hindered by noise and limited scalability. Circuit complexity grows with problem size, restricting performance to TSP instances with a maximum of 6 nodes. In contrast, Ising-based architectures show improved scalability for larger problem sizes. SQUID-based Ising machines can handle TSP instances with up to 12 nodes, while entropy computing implemented in hybrid optoelectronic components extend this capability to 18 nodes. Nevertheless, the solutions tend to be suboptimal due to hardware limitations and challenges in achieving ground state convergence as the problem size increases. Despite these limitations, Ising machines demonstrate significant time advantages over classical methods, making them a promising candidate for solving larger-scale TSPs efficiently.

Paper Structure

This paper contains 32 sections, 45 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The TSP Formulation for Gate based Quantum Computing
  3. QAOA Formulation
  4. Theoretical formulation for the QAOA
  5. The QAOA algorithm step sequence
  6. Simulation Results on the IBMQ Platform for the QAOA Algorithm
  7. The QPE Formulation
  8. Overview of the Quantum Phase Estimation Algorithm
  9. Constructing the Unitary Operator $U$
  10. Phase Kickback and Superposition State
  11. Applying the Inverse QFT
  12. Encoding Hamiltonian Cycles
  13. Simulation and experimental results for the QPE algorithm on IBMQ platform
  14. The TSP Formulation for Ising type simulators and hardware
  15. Ising and QUBO Models
  16. ...and 17 more sections

Figures (15)

  • Figure 1: A single layer of the six-layer QAOA quantum circuit.
  • Figure 2: Initial set of angles before optimization.
  • Figure 3: Final set of angles obtained after optimization.
  • Figure 4: Convergence to the optimal set of angles to determine the ground state of the QAOA Hamiltonian.
  • Figure 5: Circuit for Quantum Phase Estimation(QPE)
  • ...and 10 more figures