Jafari and Colleagues Model Defect Density Scaling for Non-Hermitian Quantum Annealing

H. Najafzadeh and colleagues at Sharif University of Technology, Institute for Advanced Studies in Basic Sciences (IASBS) and Institute for Research in Fundamental Sciences (IPM) show that non-Hermitian dynamics, unlike Hermitian systems, display complex defect density behaviours linked to dissipation strength. Their analytical and numerical results reveal examples of standard Kibble-Zurek scaling, alongside both anti-Kibble-Zurek behaviour and a suppression of defect density greater than Kibble-Zurek predictions. The findings clarify how underlying excitation probabilities dictate these deviations, offering key insight into optimising quantum annealing performance and understanding the impact of dissipation on quantum systems.

Non-Hermitian annealing expands defect control beyond gap-closing dynamics

Defect density, a critical measure of errors in quantum annealing processes, now demonstrates a capacity to decrease at a rate exceeding previous Kibble-Zurek predictions, according to researchers at Sharif University of Technology, Zanjan and the Institute for Research in Fundamental Sciences. Quantum annealing, a metaheuristic for finding the global minimum of a given objective function over a set of candidate solutions, relies on the principle of quantum fluctuations to explore the solution space. Defects, representing suboptimal solutions, hinder the process and limit the accuracy of the final result. Earlier methods were largely limited to approximating dynamics near the point where energy levels converge, a phenomenon known as the ‘gap-closing point’. This point represents a critical slowing down in the annealing process, and traditional Kibble-Zurek theory predicts a specific scaling relationship between the annealing time and the resulting defect density. Non-Hermitian quantum annealing, however, achieves enhanced suppression of these defects, overcoming the limitations of Hermitian systems that previously focused calculations solely on energy levels immediately surrounding this convergence point. The work details that the rate of defect suppression can vary sharply depending on the strength of dissipation within the system, with certain levels leading to standard Kibble-Zurek scaling where defect density decreases proportionally to annealing time raised to a power. Conversely, other conditions induce ‘anti-Kibble-Zurek’ behaviour, causing defect density to increase with longer annealing times, a previously observed, yet unexplained, phenomenon. Originating from a vanishing excitation probability across a range of annealing times and all allowed modes, even those not immediately near the gap-closing point, defect density can fall at a rate exceeding standard Kibble-Zurek predictions. This suggests a more nuanced relationship between annealing parameters and defect formation than previously understood.

The research focuses on the non-Hermitian transverse-field Ising model, a theoretical framework used to simulate quantum annealing. In this model, the system’s Hamiltonian, which describes the total energy, is non-Hermitian, meaning it does not satisfy the mathematical condition of being equal to its conjugate transpose. This non-Hermiticity introduces dissipation, representing energy loss from the system, and fundamentally alters the dynamics of the annealing process. The team employed both analytical calculations and numerical simulations to investigate the behaviour of defects under varying dissipation strengths. Analytical solutions provide a theoretical understanding of the underlying physics, while numerical simulations allow for the exploration of more complex scenarios and validation of the analytical results. The simulations were performed across a range of system sizes and annealing schedules to ensure the robustness of the findings. The observed deviations from standard Kibble-Zurek scaling have significant implications for the design and optimisation of quantum annealers, potentially enabling the development of more efficient and accurate algorithms.

Excitation probabilities and their correlation with quantum annealing efficacy

Minimising defects, errors arising during computation, is crucial for optimising quantum annealing, and a team from Sharif University of Technology, Zanjan and the Institute for Research in Fundamental Sciences has demonstrated a striking level of control over their formation. This ability highlights a key gap in current understanding. The analysis relies heavily on calculating excitation probabilities, the likelihood of a system jumping between energy states, and practical application demands a direct link between these probabilities and tangible improvements in annealing performance. Revealing how non-Hermitian dynamics, a complex area of quantum mechanics dealing with systems that don’t follow standard energy conservation rules, influence the process and lead to deviations from expected scaling behaviour linked to excitation probabilities across different energy states, this research offers valuable insight into the fundamental behaviour of quantum annealers. Understanding these excitation probabilities is paramount, as they directly determine the number of defects created during the annealing process. A higher excitation probability implies a greater likelihood of the system transitioning to an incorrect, higher-energy state, thus increasing the defect density.

Unlike previous assumptions, the examination reveals that defect density is not solely determined by energy levels close to a critical point. Instead, momentum sectors, a broader range of energy states characterised by their momentum, significantly influence defect formation, leading to behaviours beyond standard predictions. Momentum sectors represent different spatial distributions of the quantum fluctuations, and their contribution to defect formation was previously underestimated. Depending on the strength of energy loss within the system, the rate at which these defects appear can either follow, oppose, or even fall below established theoretical limits linked to Kibble-Zurek scaling. Specifically, the researchers found that strong dissipation can suppress defect formation by effectively damping out the quantum fluctuations that drive the system away from the optimal solution. Conversely, weak dissipation can lead to an increase in defect density, as the system remains susceptible to fluctuations for a longer period. The observed anti-Kibble-Zurek behaviour, where defect density increases with annealing time, is attributed to a specific interplay between dissipation and excitation probabilities, resulting in a sustained generation of defects even as the system approaches its ground state. The ability to manipulate dissipation strength, therefore, offers a novel pathway for controlling defect density and improving the performance of quantum annealers. Further research is needed to explore the practical implications of these findings and develop strategies for implementing dissipation control in real-world quantum annealing devices.

The findings have implications beyond quantum annealing, potentially informing our understanding of other non-Hermitian quantum systems and their behaviour in various physical contexts. The detailed analysis of excitation probabilities and their correlation with defect density provides a valuable framework for studying the dynamics of open quantum systems, where interactions with the environment lead to dissipation and decoherence. This work contributes to a growing body of research exploring the unique properties of non-Hermitian physics and its potential applications in quantum technologies.

The research determined that defect density in the non-Hermitian transverse-field Ising model does not always follow standard Kibble-Zurek scaling, instead exhibiting both faster suppression and anti-Kibble-Zurek behaviour depending on dissipation strength. This matters because it demonstrates that broad momentum sectors significantly contribute to defect formation, unlike previous understanding which focused on modes near gap-closing points. Researchers found that defect density is linked to excitation probabilities and can be controlled by manipulating dissipation. The authors suggest further investigation is needed to explore the practical implications of these findings for quantum annealing devices.