Quantum Annealing (QA) is a heuristic for solving optimization problems by leveraging quantum mechanical effects. In particular, quantum tunneling is used. Here is a deeper dive into this:
Basics of Quantum Annealing
Quantum Tunneling - Unlike in classical optimization where a system might get stuck in a local minimum due to barriers, quantum tunneling allows the system to pass through these barriers. Picture a ball rolling down a landscape but instead of being restricted by hills, it can “tunnel” through them to potentially find a deeper valley (global minimum).
Adiabatic Quantum Computation - Quantum Annealing is closely related to adiabatic quantum computation. The system starts in a prepared ground state (initial state) of a simple Hamiltonian (mathematical description of the system’s energy). Then, the Hamiltonian is slowly changed (annealed) to another one which encodes the problem into the state. If this change is slow enough, the system will remain in its ground state, which hopefully will correspond to the solution of the problem.
Adiabatic Theorem - States that if a change is slow enough, the system will remain in the ground state of the Hamiltonian throughout the transition. This is critical for Quantum Annealing because the final ground state should be the solution to the problem.
Transverse Field - In practical implementations, a transverse field is applied to the qubits. This field induces quantum fluctuations. This allows the system to explore different configurations simultaneously. This is similar to searching all possible solutions at the same time due to superposition.
How Quantum Annealing Works
Problem Encoding - The optimization problem is mapped to an Ising model or a QUBO (Quadratic Unconstrained Binary Optimization) problem. In the Ising model, each qubit represents a spin that can be either up or down. The goal is to find the configuration of spins that minimizes the energy of the system.
Initial State - Start with a quantum superposition of all possible solutions. This is often facilitated by a transverse field.
Annealing Process
Decrease Transverse Field - Gradually reduce the strength of the transverse field. This causes high quantum fluctuations allowing for tunneling.
Increase Problem Hamiltonian - The problem Hamiltonian, which represents the actual optimization problem, is increased in influence.
Measurement - Once the annealing process is complete, the system is measured. This measurement collapses the quantum state to one classical configuration. This would hopefully be the optimal solution.
Applications:
Optimization Problems - QA is particularly good for NP-hard problems like: Traveling Salesperson Problem - Finding the shortest route visiting each location once and returning to the start.
Scheduling - Assigning tasks to resources over time.
Portfolio Optimization - Selecting a set of investments that minimizes risk for a given return.
Machine Learning - For tasks like clustering or feature selection where optimization plays a key role.
Cryptography - There is an interest in using QA for breaking certain encryption schemes. Although this is not as universally applicable as gate-based quantum computers for this purpose.
Challenges:
Noise and Decoherence - Maintaining quantum coherence over the annealing process is challenging.
Problem Mapping - Not all problems are easily or efficiently mapped to the form suitable for QA machines.
Verification - Ensuring the solution found by QA is indeed the global minimum rather than a local one.
Quantum annealing represents a different paradigm from gate-based quantum computing. QA focuses more on specific types of optimization problems rather than general-purpose computation.
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