1. How does QAOA solve combinatorial optimization problems?
The Quantum Approximate Optimization Algorithm (QAOA) solves combinatorial optimization problems by encoding them into Ising formulations. It performs a coarsely trotterized adiabatic time evolution, minimizing an Ising Hamiltonian. This algorithm is suitable for near-term quantum devices with limited circuit depth. QAOA is an expected candidate for demonstrating quantum supremacy. In this paper, QAOA is used to solve the Vehicle Routing Problem (VRP), an NP-hard combinatorial optimization problem. The paper discusses the structure of VRP, its Ising formulation, QAOA's details, simulation results, and performance limitations.
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2. What is the Vehicle Routing Problem (VRP)?
The Vehicle Routing Problem (VRP) is an NP-hard combinatorial optimization problem. It involves k vehicles and n-1 locations, with each vehicle starting and ending at the depot D. The goal is to find routes where each location is visited exactly once, minimizing the total distance traveled by the vehicles. VRP is a generalization of the Travelling Salesman Problem, where a group of salesmen serve n-1 locations. In real-world applications, VRP often includes constraints like vehicle capacity and limited coverage time. However, this explanation focuses on the basic version of VRP without additional constraints.
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3. How can the Ising formulation of VRP be represented using QAOA?
To solve a problem instance (n, k) of VRP using QAOA, we first need to map it to the minimization of an Ising Hamiltonian H c 9. This is done by finding the H c, which encodes the given problem instance. Decision variables x ij, where x ij is 1 if there exists an edge from i to j with weight w ij > 0, and 0 otherwise, represent the solution to the VRP problem. The VRP can be formulated as a set of equations (2-5) with constraints (4-5). The energy functional H V RP of the problem can be written using these equations and a constant A > 0. Using the decision variables x, we define vectors z S[i] and z T[i] for each node i. The QUBO problem for a graph G = (N, V) can be defined as f(x) = min x{0,1} (N xV) x T Qx + g T x + c. By substituting Eqs. (13-14) into Eq. (6), we can find the coefficients Q, g, and c. The Ising Hamiltonian for VRP (n, k) can be constructed by expanding Eq. (15) using Eqs. (17-19) and replacing x ij with spin variables s ij. This gives the quantum mechanical description of HISING, which can be implemented on a quantum computer.
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4. What is the role of Hamiltonians in Adiabatic Quantum Computation (AQC)?
In Adiabatic Quantum Computation (AQC), two Hamiltonians, Hmixer and Hcost, play crucial roles. Hmixer's ground state should be easily preparable, such as |+ N, while Hcost's ground state encodes the solution to the problem. Both Hamiltonians should be local, involving interactions between a constant number of particles. The instantaneous Hamiltonian H(t) is defined as EQUATION, where E represents the energy difference between the ground state and the first excited state of H(t). This energy difference bounds the step size for following the adiabatic pathway. However, the computation time to solve problems increases exponentially as E becomes infinitesimally small, limiting AQC's capability to solve certain instances of hard optimization problems. In contrast, Quantum Approximate Optimization Algorithm (QAOA) eliminates this restriction by discretizing the adiabatic pathway into p steps, represented by the parameters {b, g}, as shown in EQUATION. These parameters are provided by a classical processor and optimized using a classical optimization routine based on energy measurement results. QAOA is a hybrid quantum-classical variational algorithm.
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