Solving The Travelling Salesman Problem Using A Single Qubit (2407.17207v2)

Abstract: The travelling salesman problem (TSP) is a popular NP-hard-combinatorial optimization problem that requires finding the optimal way for a salesman to travel through different cities once and return to the initial city. The existing methods of solving TSPs on quantum systems are either gate-based or binary variable-based encoding. Both approaches are resource-expensive in terms of the number of qubits while performing worse compared to existing classical algorithms even for small-size problems. We present an algorithm that solves an arbitrary TSP using a single qubit by invoking the principle of quantum parallelism. The cities are represented as quantum states on the Bloch sphere while the preparation of superposition states allows us to traverse multiple paths at once. The underlying framework of our algorithm is a quantum version of the classical Brachistochrone approach. Optimal control methods are employed to create a selective superposition of the quantum states to find the shortest route of a given TSP. The numerical simulations solve a sample of four to nine cities for which exact solutions are obtained. The algorithm can be implemented on any quantum platform capable of efficiently rotating a qubit and allowing state tomography measurements. For the TSP problem sizes considered in this work, our algorithm is more resource-efficient and accurate than existing quantum algorithms with the potential for scalability. A potential speed-up of polynomial time over classical algorithms is discussed.

• The paper presents a novel quantum algorithm that maps TSP cities to quantum states on the Bloch sphere, solving the NP-hard problem with a single qubit.
• It utilizes quantum parallelism through superposition and tailored rotation operators to explore multiple paths simultaneously.
• Resource efficiency and high accuracy, with approximation ratios near 1 for over 90% of tested instances, indicate strong potential for broader combinatorial optimization.

Solving the Travelling Salesman Problem Using A Single Qubit

This paper introduces a novel approach for solving the Travelling Salesman Problem (TSP) utilizing a single qubit by leveraging quantum parallelism. Notably, TSP is an NP-hard combinatorial optimization problem, where solving it efficiently using classical algorithms becomes impractical as the number of cities increases. The proposed method aims to mitigate resource intensiveness and enhance the accuracy of existing quantum solutions by employing superposition principles on a Bloch sphere.

Summary of the Approach

The authors present an algorithm that encodes the cities of TSP as quantum states on the Bloch sphere. Traditional quantum methods often employ gate-based or QUBO-based strategies, which demand numerous qubits and are not competitive with classical algorithms for even small-size TSPs. The innovative aspect of this work is the representation of cities as quantum states on the Bloch sphere and the preparation of superposition states for traversing multiple paths simultaneously.

The algorithm draws upon the classical Brachistochrone problem, where the goal is to find the shortest path that a particle travels under gravity, but maps it into a quantum context. Cities are encoded as quantum states, and the problem solution involves optimally steering the qubit's state to achieve a selective population distribution across these states.

Detailed Procedure

1. Mapping and Initialization: Each city in the TSP is mapped onto quantum states on the Bloch sphere. The initial quantum state is placed arbitrarily on the equator.
2. Superposition and Rotations: By employing rotation operators, the qubit is driven from one state to another, effectively exploring multiple paths through quantum parallelism. Specific rotation operators are used:
   • $U^u_{ii-ij}$ (upward rotations) steer the qubit towards an intermediate state
   • $U^d_{ij-ii}$ (downward rotations) return the qubit to the equatorial plane
3. Measurement: Optimizing these rotations to create a superposition state at each layer of the algorithm and performing quantum state tomography to measure the penultimate state. The measurement information is processed to calculate the TSP cost function, $D$.

Results and Numerical Simulations

The authors implement their algorithm on a single qubit and demonstrate solving TSP instances with city sizes ranging from 4 to 9. For all problem sizes considered, the algorithm proves more resource-efficient and accurate than existing quantum algorithms. Specifically, the paper reports an approximation ratio close to 1 for more than 90% of the problems solved, indicating high accuracy.

For a 5-city TSP, the optimal route is identified, and a one-to-one correspondence between the classical cost $D$ and the quantum time functional $T$ is observed. This correspondence provides proof of the validity and accuracy of the proposed encoding scheme on the Bloch sphere.

Implications and Future Work

Practically, this work suggests a robust method for solving TSP by significantly reducing the quantum resource overhead, which is crucial in the NISQ era. This encoding on a Bloch sphere could be extended to other combinatorial optimization problems, offering a new avenue for resource-efficient quantum algorithms.

Theoretically, this method aligns with the ideas of achieving polynomial speed-ups in quantum computing by exploiting quantum superposition. Further work could focus on rigorously establishing the polynomial quantum advantage and extending the control to multi-qubit systems to handle larger instances of TSP or even more complex optimization problems.

Moreover, geometric visualization of the quantum problem-solving process presented in this paper, through the routing charts and Bloch sphere mappings, provides unique insights. Such visualization can act as a stepping stone for future development of more intuitive and efficient quantum algorithms.

In conclusion, the proposed algorithm represents a significant advancement in solving combinatorial optimization problems using quantum systems. It opens up potential pathways for leveraging quantum computing in practical, high-dimensional optimization tasks with limited quantum resources.