Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle (1607.06473v3)

Abstract: We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.

• The paper applies Pontryagin's minimum principle from optimal control theory to show that optimal variational quantum algorithms exhibit 'bang-bang' control characteristics.
• This demonstration that optimal VQAs follow 'bang-bang' control confirms the efficacy of protocols used in algorithms like QAOA and shows resilience to noise.
• The findings suggest efficient, system-size invariant pulse time scales for pulse-based VQA strategies, potentially influencing future quantum hardware development and algorithm deployment.

Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle

In the paper titled "Optimizing Variational Quantum Algorithms using Pontryagin’s Minimum Principle," the authors advance the understanding of variational quantum algorithms (VQAs) by applying concepts from optimal control theory, specifically leveraging Pontryagin's minimum principle. This approach reveals insights into optimizing quantum algorithms, with applications extending to combinatorial optimization problems. The paper asserts the efficacy of nonadiabatic "bang-bang" protocols and addresses the performance of VQAs in quantum systems with both closed and open dynamics.

Key Contributions

1. Application of Optimal Control Theory: The authors apply Pontryagin’s minimum principle to establish that optimal control strategies within VQAs possess "bang-bang" characteristics, where control signals switch abruptly between extremal values. The principle directs optimal quantum evolution strategies by effectively minimizing control Hamiltonians over permissible controls, reinforcing theoretical assertions that constrain evolution protocols to abrupt signal transitions.
2. Bang-Bang Protocols in VQAs: The research demonstrates that optimal VQAs exhibit "bang-bang" control forms. This is evident under circumstances wherein system dynamics, driven by Hamiltonian controls, manifest conjointly linear properties—the linearity in control parameters aligns under Pontryagin’s framework to output pulse signal optimizations that alternate swiftly between high and low states.
3. Implications for Quantum Approximate Optimization Algorithm (QAOA): The insights derived through control optimization foster substantial alignment with precedents in quantum approximate optimization. This research further supports specifically structured protocols utilized within QAOA, offering a rigorous foundation for its pulse-based ansatz. The QAOA employs discrete square pulses to navigate quantum state spaces, a method shown to have foundational merit through the principles explored.
4. Noise and Decoherence Robustness: Evaluations persist even in noise-afflicted contexts, employing the Redfield master equation for modeling thermal bath interactions and analyzing system resilience amidst external perturbations. Results indicate that optimal nonadiabatic bang-bang protocols maintain superior performance in comparison to conventional quantum annealing techniques.
5. Characteristic Time Scale: The authors derive system-size invariant pulse time scales by numerically analyzing optimizations under specific Hamiltonian instances—a critical assertion indicating the scalability and efficiency of pulse-based evolution strategies, setting a computationally feasible precedent amid quantum algorithm applications.

Practical and Theoretical Implications

The research imparts efficient methodology considerations for quantum algorithm applications, emphasizing practical parameters such as pulse duration and control switching optimization. By affirming model applicability in quantum computing endeavors, particularly QAOA, these findings entrench protocols with firm applicability and expansion potential, particularly within computational fields involving complex optimization problems.

Furthermore, these findings impart significant implications for quantum technologies, potentially influencing quantum hardware development and algorithmic deployment, which beckons further exploration into control strategies and quantum state manipulation.

Future Developments

Continued exploration may delve into improving algorithmic outer-loop efficiency, enhancing noise model accuracies, addressing large-system scale challenges, and concretizing practical implementations across broader quantum computational paradigms. Bridging control optimization theory with quantum hardware capabilities fosters prospects for innovations in quantum information science and beyond.

In summary, the paper achieves substantive milestones in embedding control theory within quantum algorithm designs, suggesting improvements in quantum computational efficacy and stability.