optimization

QAOA is a heuristic algorithm, meaning that it is not guaranteed to find the optimal solution, but it can often find very good solutions.

QAOA works by encoding the problem into a quantum state, and then using a series of unitary operations to evolve the state. The final state is then measured, and the result is the solution to the problem.

There are a few different ways to encode the problem into a quantum state. One common way is to use a Hamiltonian that encodes the constraints of the problem. For example, if the problem is to find the shortest path between two points, the Hamiltonian would encode the constraint that the path must be a certain length.

Once the Hamiltonian is encoded, the QAOA algorithm proceeds in two steps. In the first step, called the "preparation step", a unitary operation is applied to the state that creates a superposition of all the possible solutions. In the second step, called the "evolution step", a series of unitary operations are applied that depend on the Hamiltonian. These operations cause the state to evolve in such a way that the solutions that are "closer" to the optimum are more likely to be measured.

Finally, the state is measured, and the result is the solution to the problem.

Maxcut is a problem in graph theory that seeks to find the maximum number of edges that can be cut from a given graph. It is a NP-hard problem, meaning that it is believed to be computationally intractable. However, recent advances in quantum computing have led to the development of algorithms that can solve Maxcut on a quantum computer in polynomial time.

The algorithm that we will use to solve Maxcut on a quantum computer is called the Quantum Approximate Optimization Algorithm (QAOA). QAOA is a variationally algorithm that uses a quantum computer to find the approximate solution to an optimization problem. In order to use QAOA to solve Maxcut, we first need to encode the graph into a quantum state. This can be done using the well-known Ising model.

The Ising model is a model of a system of spins that interact with each other via the Ising interaction. In our case, the spins will represent the vertices of the graph, and the Ising interaction will represent the edges of the graph. We can then use QAOA to find the maximum number of edges that can be cut from the graph, by finding the configuration of spins that minimizes the Ising interaction.

It should be noted that QAOA is not a perfect algorithm, and it will not always find the optimal solution to the Maxcut problem. However, it is a very powerful algorithm that can find very good solutions to Maxcut in polynomial time.

In general, the optimization problem can be expressed as follows:

min x f ( x )                                        

subject to:

g i ( x ) = 0 , i = 1 , 2 , … , m 

h j ( x ) ≤ 0 , j = 1 , 2 , … , p 

where x is the decision vector to be optimized, f ( x ) is the objective function, and g i ( x ) and h j ( x ) are the constraint functions.

The quantum algorithm for solving optimization problems is to encode the objective function and the constraint functions into a quantum state, and then use a quantum circuit to search for the optimal solution through quantum interference. The quantum state encoding objective function and constraint functions is usually expressed as follows:

| ψ ⟩ = a | x ⟩ + ∑ i b i | g i ( x ) ⟩ + ∑ j c j | h j ( x ) ⟩ 

where the superposition | x 〉 of all potential solutions is usually the leading term to be optimized, and | g i ( x ) 〉 and | h j ( x ) 〉 are the superposition of all solutions that violate the constraint conditions. The coefficient a is set to 1 to ensure that all solutions are encoded in the quantum state, and the coefficients b i and c j can be set to 0 or 1. In the quantum optimization algorithm, the quantum state is evolved by a quantum circuit, which is composed of a unitary operator U and a measurement operator M. The operation of the quantum circuit is as follows:

U | ψ ⟩ → | ψ ′ ⟩ = U | ψ ⟩

M | ψ ′ ⟩ → | ψ ′ ′ ⟩ = M | ψ ⟩ 

U is a unitary operator composed of many basic gates, and M is a general measurement operator. The general unitary operator U can be expressed as follows:

U = e − i α X Δ t e − i β Z Δ t e − i γ Y Δ t 

where X, Y, and Z are the Pauli operators, and X and Y correspond to the constraint functions g i and h j . The quantum state | ψ ′ ⟩ after evolution by the quantum circuit U is given by:

| ψ ′ ⟩ = a | x ⟩ + b × e − i Δ t ( β | 0 ⟩ + γ | 1 ⟩ ) + c × e − i Δ t ( β | 1 ⟩ − γ | 0 ⟩ )

From Equation (8), we can see that, for the quantum state | ψ ′ ⟩ , the leading term | x 〉 is only evolved by the unitary operator U, and all other terms are evolved by the unitary operator U multiplied by a phase factor. The different phases of the terms cause interference between the terms, and the terms that are less beneficial to the optimization are partially canceled out. Therefore, after the quantum state | ψ ′ ⟩ is evolved by the quantum circuit, the probability of measuring the quantum state | ψ ′ ⟩ is proportional to the objective function f ( x ) , which is expressed as:

P ( x ) = | ‖ ψ ′ ⟩ | 2 ∝ f ( x )

The above equation shows that the probability of measuring the quantum state | ψ ′ ⟩ is proportional to the objective function f ( x ) . Therefore, the objective function can be optimized by repeatedly measuring the quantum state | ψ ′ ⟩