Krylov complexity as a proxy for circuit or holographic complexity (especially for subsystems)

Determine whether Krylov complexity provides a reliable proxy for circuit complexity and for holographic complexity in quantum field theories, particularly for subsystem complexity, given that Krylov complexity’s behavior depends sensitively on its ultraviolet definition.

Background

Beyond circuit complexity, other measures such as Krylov complexity have been widely investigated as potential diagnostics of operator/state growth. However, rigorous connections between such measures and circuit or holographic complexity are not established.

The authors note that in QFT the behavior of Krylov complexity depends on UV regularization choices, and they highlight uncertainty about whether it tracks circuit or holographic complexity—especially in the context of subsystems—thereby posing a clear open question for future work.

References

However, the behavior of Krylov complexity in quantum field theory depends on the precise UV definition, and whether it is a good proxy for circuit or holographic complexity, particularly in relation to subsystems, remains an open question.

— Sharp Transitions for Subsystem Complexity  (2510.18832 - Fan et al., 21 Oct 2025) in Section 5 (Discussion)

More broadly, a general, regulator-controlled field-theoretic formulation for interacting higher-dimensional QFTs remains open.

— Quantum Chaos and Spread of States in Krylov Subspace: A Topical Review  (2608.26825 - Jeong et al., 27 Aug 2026) in Section 7, Conclusions and outlook

Secondly, to the best of our knowledge, the holographic prescription to connect the spread complexity rate to the proper momentum of an infalling test particle, is not proven in general, except for the analytically controllable AdS${}_3$/CFT${}_2$ case scrutinized in .

— Holographic Krylov complexity in confining backgrounds: particle, string, and frames  (2610.13555 - Nongmaithem et al., 9 Oct 2026) in Section Conclusion and future perspectives