The anomalously slow dynamics of inhomogeneous quantum annealing (2502.03535v2)

Abstract: Inhomogeneous quantum annealing (IQA), in which transverse fields are turned off one by one rather than simultaneously, has been proposed as an effective way to avoid the first-order phase transitions that impede conventional quantum annealing (QA). Here we explicitly study the dynamics of IQA, rather than merely the thermodynamics, and find that it is appreciably slower than the phase diagram would suggest. Interestingly, this slowdown manifests both when IQA succeeds in circumventing phase transitions and when it fails. Even in the absence of transitions, such as for the mean-field models that have been analyzed previously, IQA is slower than expected by a factor of the number of spins $N$. More significantly, we show that in non-mean-field models, first-order transitions are likely to be quite common, and the gap at such transitions is not merely exponential in $N$ but exactly zero. Thus IQA cannot reach the ground state on any timescale. Both of these results can be understood through the simple observation that a spin's magnetization becomes conserved once its field is turned off during the IQA protocol.

• The paper demonstrates that inhomogeneous quantum annealing avoids first-order transitions in mean-field models but suffers from linearly scaling time-to-threshold.
• It employs a comparative analysis of mean-field (p-spin) and non-mean-field (Sherrington-Kirkpatrick) systems to reveal inherent dynamic limitations.
• The study suggests that introducing noise could help counteract qubit freezing and improve overall quantum annealing performance.

Anomalously Slow Dynamics in Inhomogeneous Quantum Annealing

The paper "The anomalously slow dynamics of inhomogeneous quantum annealing" provides a detailed examination of the effectiveness and dynamics of inhomogeneous quantum annealing (IQA) compared to conventional quantum annealing (QA). The research particularly highlights the unexpected dynamic behavior of IQA when applied to both mean-field and non-mean-field quantum systems, unveiling insights into hardware and algorithmic challenges faced by quantum computing.

Inhomogeneous quantum annealing has been suggested as an alternative approach aimed at circumventing first-order phase transitions seen in traditional quantum annealing. Unlike conventional methods where transverse fields are adjusted simultaneously across all qubits, IQA turns off these fields one-by-one. Theoretically, this staggered approach indicates improved performance for specific models, potentially allowing systems to sidestep some quantum mechanical hurdles present in conventional QA.

The research begins with an analysis of mean-field systems, particularly focusing on the ferromagnetic $p$-spin model. In conventional QA, this model traditionally exhibits first-order phase transitions, resulting in an exponential scaling of annealing time with the number of qubits, $N$. However, when studying the thermodynamics of IQA, the phase diagram suggests the potential to avoid these first-order transitions entirely. Despite these promising thermodynamic facets, the paper reveals that IQA remains dynamics limited. The mean-field model, even in the absence of transitions, exhibits dynamics where the expected time-to-threshold increases linearly with $N$. This indicates that each qubit's transverse field must be decreased individually, adding complexities in time management that were not previously apparent.

To explore more realistic conditions, the paper extends to non-mean-field systems, particularly examining instances of the Sherrington-Kirkpatrick model. It is here that IQA seemingly struggles more profoundly. Due to the nature of IQA, once a qubit's transverse field is turned off, its magnetization tends to become conserved, leading to exact energy level crossings rather than mere gaps. This conservation implies that spins can freeze into incorrect states, thereby making the quantum system unable to reach its ground state within any time scale, even infinite. The paper reports that such level crossings become increasingly common as $N$ increases, suggesting an increasingly limited applicability of IQA for large, complex systems.

The research asserts a need for careful consideration when implementing IQA — despite its ability to theoretically avoid phase transitions, performance improvements are metastable. Significant practical challenges lie in efficiently managing the dynamics, which rely heavily on the precise timing of field adjustments. This poses an additional burden for hardware performance requirements and may limit architectural strategies for embedding such algorithms.

The research advocates for a deeper investigation into the potential of noise and other external perturbations to improve IQA performance since the presence of noise could potentially help qubits avoid getting stuck in incorrect states. This counterintuitive notion that imperfections might lead to improved outcomes under IQA highlights a fascinating avenue for further exploration and experimental verification.

In sum, while IQA offers theoretically interesting properties, its performance is undermined by anomalous dynamics that extend annealing times and severely limit its capability to reach correct states, especially in complex, non-mean-field quantum systems. The insights will fuel ongoing research into optimizing annealing protocols and may influence architectural strategies in the design of future quantum hardware.