Quantum Algorithm Offers Efficient Solution to Traveling Salesman Problem, Paving Way for Quantum Supremacy

The Quantum Approximate Optimization Algorithm (QAOA) is a variational algorithm that transforms the discrete solution space of Combinatorial Optimization Problems (COPs) into a continuous parameter space of a quantum circuit. It is particularly useful for solving the Traveling Salesman Problem (TSP), a classic COP. Traditional algorithms struggle with the tradeoff between accuracy and computational speed, but quantum computation offers a more efficient solution. Two main methods, Variational Quantum Algorithms and Quantum Annealing, have emerged. However, current quantum hardware limitations present challenges. A new QAOA-based algorithm reduces the number of Binary Variables, demonstrating an efficient quantum algorithm for contemporary quantum computers and contributing to quantum supremacy.

What is the Quantum Approximate Optimization Algorithm (QAOA) and How Does it Address the Traveling Salesman Problem (TSP)?

The Quantum Approximate Optimization Algorithm (QAOA) is a variational algorithm that transforms the discrete solution space of Combinatorial Optimization Problems (COPs) into a continuous parameter space of a quantum circuit. This algorithm is particularly relevant for addressing COPs, which form a critical class of challenges where the goal is to find the best solution from a finite set of discrete possibilities. These problems have applications in various fields including logistics, scheduling, and network design.

The Traveling Salesman Problem (TSP) is a quintessential example within the realm of COPs. In the TSP, the objective is to determine the most efficient route that visits each of the given cities exactly once and returns to the starting city. This problem encapsulates the challenge of optimizing a permutation of cities and as the number of cities increases, the number of possible routes grows factorially.

Classical algorithms such as the greedy algorithm and genetic algorithm often face a tradeoff between the accuracy of the solution and the computational speed. Achieving high accuracy in finding optimal solutions typically involves intensive computation, limiting the applicability of these methods to real-time decision-making scenarios. The best-known exact algorithm for solving the TSP is the Held-Karp algorithm, also known as the dynamic programming approach. This algorithm computes the shortest tour that visits each city exactly once by considering all possible subsets of cities and storing intermediate results to avoid redundant calculations.

How Does Quantum Computation Offer a Paradigm Shift from Classical Approaches?

Quantum computation offers promising avenues for solving COPs more efficiently, presenting a paradigm shift from classical approaches. Among the various quantum algorithms, two main methods have emerged as particularly relevant for addressing COPs: Variational Quantum Algorithms and Quantum Annealing.

Variational Quantum Algorithms, like the QAOA, transform the discrete solution space of COPs into a continuous parameter space of a quantum circuit. These algorithms leverage a hybrid method wherein a classical computer explores the continuous parameter space while a quantum computer performs calculations to evaluate the cost function. This synergistic approach allows for more efficient exploration of potential solutions.

Quantum Annealing is another prominent method that directly seeks solutions to problems without requiring a classical counterpart during the computation. In both QAOA and quantum annealing, the sought-after solution corresponds to the ground state of the Hamiltonian, a fundamental concept in quantum mechanics. However, the methodologies differ: QAOA achieves this by navigating through the parameter space while quantum annealing exploits the quantum adiabatic process to directly obtain the ground state of the Hamiltonian using suitable quantum hardware.

What are the Limitations and Advancements in Quantum Computing for COPs?

While quantum annealing presents a compelling approach due to its inherent simplicity, bypassing the need to search through a parameterized space, it faces a notable challenge in the constraints imposed by current quantum hardware. Quantum computers, particularly those designed for quantum annealing, currently exhibit limitations. The hardware constraints include the ability to handle only quadratic Hamiltonians, which essentially means that the Hamiltonian can be at most quadratic in the binary variables representing the problem.

Traditionally, for a TSP with 𝑛 cities, the standard formulation involves the introduction of 𝑛^2 Binary Variables (BVs), leading to a quadratic Hamiltonian in BVs. However, a novel approach rooted in the QAOA framework strategically reduces the requisite qubit count from 𝑛^2 to 𝑛log2𝑛. This reduction in the number of BVs is a substantial advancement, demonstrating an efficient quantum algorithm tailored for contemporary quantum computers.

How Does the New QAOA-based Algorithm Contribute to Quantum Supremacy?

The new QAOA-based algorithm not only contributes to the ongoing discourse on qubit efficiency but also demonstrates improved performance based on established metrics. This underscores its potential for achieving Noisy Intermediate-Scale Quantum (NISQ) era supremacy in solving real-world optimization challenges.

The NISQ era of quantum technology presents both opportunities and challenges. As researchers strive to achieve quantum supremacy, where quantum computers surpass classical counterparts in specific tasks, explorations have primarily centered around sampling problems. However, the practical implications of these achievements, particularly in domains like combinatorial optimization, remain a critical aspect to address.

The proposed Hamiltonian departs from the conventional quadratic structure associated with TSP formulations. This research introduces a novel approach rooted in the QAOA framework to address the TSP, demonstrating an efficient quantum algorithm tailored for contemporary quantum computers. This not only contributes to the ongoing discourse on qubit efficiency but also underscores its potential for achieving NISQ era supremacy in solving real-world optimization challenges.