Adiabatic Quantum Optimization

What is Adiabatic Quantum Optimization

Adiabatic Quantum Optimization (AQO) is a quantum computing technique based on the principles of adiabatic quantum computation (AQC), which leverages the adiabatic theorem of quantum mechanics to find solutions to complex optimization problems. Unlike gate-based quantum computing, which relies on the application of discrete quantum gates to manipulate qubits, AQO employs a continuous evolution of a quantum system from an initial Hamiltonian to a final Hamiltonian.

The goal of AQO is to prepare a quantum system in the ground state of an initial, easily-preparable Hamiltonian, and then evolve the system slowly enough such that it remains in the ground state throughout the process. The final Hamiltonian is designed such that its ground state encodes the solution to the optimization problem of interest. AQO is particularly useful for combinatorial optimization problems, where classical methods may struggle due to the exponential growth of possible solutions.

Principles of Hamiltonian Evolution

The underlying mechanism of AQO is grounded in the adiabatic theorem, which states that a quantum system will remain in its instantaneous ground state if the Hamiltonian governing the system is varied sufficiently slowly compared to the energy gap between the ground state and the first excited state. The Hamiltonian evolution can be described using this formula: 

• H₀ is the initial Hamiltonian, whose ground state is easily prepared.
• Hp is the problem Hamiltonian, whose ground state encodes the solution to the problem.
• T is a function of time dependent on the gap in energy eigenvalues of the Hamiltonian. 

The system begins in the ground state of H₀, and by slowly evolving the Hamiltonian toward Hp, the system ideally ends up in the ground state of Hp. This gradual evolution allows the system to "explore" the problem space and converge upon the optimal solution. The formula describes an adiabatic “schedule”, which provides the path of evolution from H₀ to Hp, and guarantees that the final state has a large overlap with the desired solution ground state. If the spectral gap (energy eigenvalue) is small, the Hamiltonian has to be evolved slowly, causing T to be greater. 

Relationship Between Adiabatic Quantum Optimization (AQO) and Quantum Annealing (QA) Algorithms

AQO is closely related to Quantum Annealing (QA), a heuristic algorithm that uses quantum fluctuations to sample the solution space. While AQO relies on maintaining the system in the ground state throughout the evolution process, Quantum annealing algorithms allow for the possibility of non-adiabatic transitions, allowing for the calculation of approximate solutions even if the system does not remain strictly in the ground state, making it more robust to certain types of noise and imperfections.

The primary difference between AQO and QA is that QA permits some diabatic transitions and can exploit thermal relaxation. This difference makes QA potentially faster in practice, though it may sacrifice solution accuracy. While AQO guarantees finding the ground state if evolution is slow enough, QA may find less-accurate approximate solutions more quickly.

Additionally, AQO can be considered a special case of Quantum Annealing where the process is carried out slowly enough to ensure adiabaticity. Quantum Annealing has gained commercial attention with systems such as D-Wave’s quantum annealers, which are specifically designed for solving optimization problems.

Applications of Adiabatic Quantum Computing

Adiabatic quantum computing is particularly suited for solving unstructured optimization problems where the solution space is discrete and vast. AQO provides a natural framework for solving NP-hard problems by finding the minimum cost of a function encoded in the ground states. Some key applications include:

• Combinatorial Optimization: Problems such as the Traveling Salesman Problem, Max-Cut, and Graph Partitioning can be formulated as energy minimization problems suitable for AQO.
• Machine Learning: Quantum optimization techniques are being explored for training machine learning models, particularly in the context of minimizing loss functions or finding optimal network architectures.
• Material Science: Ground-state problems in quantum chemistry and condensed matter physics can be tackled using AQO to identify low-energy configurations of complex systems. See this work by D-Wave and Google covering promising results in quantum simulation. 
• Financial Modeling: Optimization problems related to portfolio optimization, risk analysis, and derivative pricing can benefit from the enhanced sampling capabilities of AQO.

It’s been proven (Aharonov, et al, 2004) that adiabatic quantum computation is polynomially equivalent to the circuit model. However, while equivalence exists in terms of computational capability, there is no guarantee that an algorithm optimized for one model will be efficient when translated to the other, leading to questions about the real-world usability of this method. 

On gate-based machines, a different family of algorithms: Quantum Approximate Optimization Algorithms (QAOA) are being developed today. Inspired by adiabatic evolution, this cutting-edge family of algorithms utilizes the strengths of a hybrid quantum-classical system to approximate the adiabotic process. Case studies show promising data for the future of this technique. 

Advantages and Limitations of Adiabatic Quantum Optimization

Advantages:

• Robustness to Decoherence: Since AQO relies on slow, continuous evolution, it is somewhat more resilient to certain types of noise compared to gate-based quantum computing. Longer coherence times allows for more complex algorithms to be performed. 
• Error Correction: The continuous nature of the evolution process reduces the need for some complex error correction protocols, like high fidelity 2-qubit gates, which are a major hurdle for gate-based quantum computers. However, dicoherence and thermalization still remain significant issues that need error correction. 
• Applicability to Optimization Problems: AQO is particularly well-suited to discrete optimization problems, providing a potential quantum advantage over classical algorithms.

Limitations:

• Slow Evolution Requirement: The requirement of slow evolution to maintain adiabaticity can make AQO inefficient for large-scale problems.
• Energy Gap Sensitivity: If the energy gap between the ground state and excited states is small, maintaining adiabaticity becomes increasingly challenging, particularly for complex or large-scale problems. This issue majorly limits the scalability of solutions. 
• Limited Generalization: Unlike gate-based quantum computing, AQO is not yet practically applicable past optimization problems. While it is theoretically universal given sufficient control over the Hamiltonian, gate-based solutions are far more universal today. 
• Algorithms: Solutions discovered today show advantage in small data sets, which could disappear or become minimal at scale. More work is needed to develop new approaches aswell as to explore the performance of algorithms at scale. 

Despite these limitations, AQO remains a promising approach for near-term quantum devices, especially when applied to optimization problems where classical algorithms struggle. As hardware continues to improve, AQO and related quantum annealing techniques are expected to play an increasingly important role in the broader landscape of quantum computing. See this arXiv paper for a deep dive into AQO.