The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm designed to solve combinatorial optimization problems. Some examples of popular combinatorial optimization problems include finding the shortest path on a map, optimizing job scheduling, and optimizing the loading of a truck.
How QAOA Works:
- Problem Formulation:
The optimization problem that is to be solved must first be formulated to find the extrema of some objective function. This is usually represented as a Hamiltonian* in quantum terms.
- Quantum Circuit Design:
Initialization:
This step involves starting with a quantum state that is easy to prepare. One common approach is to start with all qubits in the |0⟩ state.
Alternating Layers:
Mixing Hamiltonian - This is applied to mix the quantum state and can be accomplished by implementing a set of rotations to each qubits based on the structure of the objective function.
Problem Hamiltonian - This encodes the problem’s cost function to approach the objective functions extrema. This is done by adjusting the phase of the quantum states based on the desired solution of the objective function.
Rapid Repetition:
This process involves rapidly repeating these two Hamiltonians in succession. Parameters in the Hamiltonians are adjusted to optimize the solution as this occurs. The number of times this is repeated is called the “depth” of the quantum circuit.
- Measurement and Optimization:
After the application of the quantum circuit the system must be measured to yield a classical result. This is usually represented with a classical bit string which corresponds to a potential solution to the given problem.
The result will now be evaluated for its efficacy and then the parameters of the quantum circuit should be adjusted using classical optimization techniques to attempt to improve the next iteration of the quantum circuit.
Challenges:
Parameter Tuning - Tuning the parameters to optimize the quantum circuit requires careful consideration. This can take a large amount of computational power.
Quantum Noise - Noise introduced from the outside systems can interfere and degrade the performance of the algorithm.
Scalability - As the problem grows in complexity the depth of the circuit required may exceed current quantum hardware capabilities.
*What is a Hamiltonian?
The Hamiltonian is a function that represents the total energy of a system. This helps to demonstrate how a given system evolves from one state to another. Understanding this is key in enabling quantum computers to solve complex problems.
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