We analyze the properties of Krylov state complexity in qubit dynamics, considering a single qubit and a qubit pair. A geometrical picture of the Krylov complexity is discussed for the single-qubit case, whereas it becomes non-trivial for the two-qubit case. Considering the particular case of interacting Rydberg two-level atoms, we show that the Krylov basis obtained using an effective Hamiltonian minimizes the time-averaged spread complexity compared to that which is obtained from the original Hamiltonian. We further generalize the latter property to an arbitrary Hamiltonian in which the entire Hilbert space comprises of two subspaces provided a weak coupling between them.
We analyze the properties of Krylov state complexity in qubit dynamics, considering a single qubit and a qubit pair. A geometrical picture of the Krylov complexity is discussed for the single-qubit case, whereas it becomes non-trivial for the two-qubit case. Considering the particular case of interacting Rydberg two-level atoms, we show that the Krylov basis obtained using an effective Hamiltonian minimizes the time-averaged spread complexity compared to that which is obtained from the original Hamiltonian. We further generalize the latter property to an arbitrary Hamiltonian in which the entire Hilbert space comprises of two subspaces provided a weak coupling between them.
We compare Krylov's state complexity with an information-geometric (IG) measure of complexity for the quantum evolution of two-level systems. Focusing on qubit dynamics on the Bloch sphere, we analyze evolutions generated by stationary and nonstationary Hamiltonians, corresponding to geodesic and nongeodesic trajectories. We formulate Krylov complexity in geometric terms, both instantaneously and in a time-averaged sense, and contrast it with an IG complexity of quantum evolutions characterized in terms of efficiency and curvature. We show that the two measures reflect fundamentally different aspects of quantum dynamics: Krylov's state complexity quantifies the directional spread of the evolving state relative to the initial state, whereas the IG complexity captures the effective volume explored along the trajectory on the Bloch sphere. This geometric distinction explains their inequivalent behavior and highlights the complementary nature of state-based and information-geometric notions of complexity in quantum systems.
Krylov complexity provides a powerful framework for characterizing the dynamical evolution of quantum systems through the spreading of states in Krylov space. The motivation for this is rooted in the optimality of the Krylov basis for the analyzed cost function. In this work, we reinterpret the motivation for the Krylov basis from a dynamical perspective and show that it corresponds to a first-order approximation of the time-evolution operator. We extend this framework to higher-order generators and analytically disprove the optimality assumption by showing that an infinite-order generator can be constructed to exhibit smaller spread for arbitrary times. We propose a natural time scale for the construction of these higher-order generators and discuss results for matrices sampled from Gaussian Unitary Ensembles, demonstrating smaller Krylov complexity at all higher orders. These results extend the framework of Krylov complexity beyond the conventional Krylov basis by disproving the widely held assumption of optimality, extending the construction to higher-order generators, and introducing a physically motivated method for their construction. Our findings therefore suggest that previous statements and results concerning Krylov complexity may need to be reconsidered.
A recently developed notion, known as spread complexity, captures how a quantum state spreads in Krylov space under unitary evolution. Related Krylov-space approaches are increasingly being investigated for applications in quantum control, simulation, and metrology, as well as are being considered for implementation in near-term quantum platforms. However, its connection to fundamental quantum resources such as entanglement and quantum coherence remains unclear. We show that the dynamics of entanglement is constrained by the extent of spreading in Krylov space. For multipartite systems, we relate state delocalization in the Krylov basis, quantified by the inverse participation ratio, to geometric measures of multipartite quantum correlations. Furthermore, we derive analytical relations between the quantum coherence of the initial state in the energy eigenbasis and spread complexity for qubit and qutrit systems. Our results make progress towards understanding the subtle interplay between the dynamical spreading in Krylov space and fundamental quantum resources.
Quantum simulation of complex many-body systems beyond classical computational capabilities provides a promising route toward understanding novel quantum phases and their transitions. In particular, analog quantum simulators with global control fields have attracted considerable attention due to their potential to simulate arbitrary Hamiltonians and perform quantum computing tasks. However, a clear, quantitative measure for the complexity of implementing specific quantum operations in such systems is still lacking. In this Letter, we address this challenge by introducing generalized Krylov complexity, a concept originating from operator growth dynamics, as a direct diagnosis for this synthesis complexity. We construct the block Krylov basis generated by a set of Hamiltonians, which naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators. By analyzing representative systems including Rydberg atom arrays, we demonstrate that the generalized Krylov complexity of a target operation serves as a strong predictor of the minimum time required for its realization. Our results establish Krylov complexity as an intuitive and predictive tool for designing efficient control protocols in analog quantum simulators.
Investigating the time evolution of complexity in quantum systems entails evaluating the spreading of the system's state across a defined basis in its corresponding Hilbert space. Recently, the Krylov basis has been identified as the one that minimizes this spreading. In this study, we develop a numerical exploration of the Krylov complexity in quantum states following a quench in the Lipkin-Meshkov-Glick model. Our results reveal that the long-term averaged Krylov complexity acts as an order parameter for this model. It effectively discriminates between the two dynamic phases induced by the quench, sharing a critical point with the conventional order parameter. Additionally, we examine the inverse participation ratio and the Shannon entropy in both the Krylov basis and the energy basis. A matching dynamic behavior is observed in both bases when the initial state possesses a specific symmetry. This behavior is analytically explained by establishing the equivalence between the Krylov basis and the prequench energy eigenbasis.
A bstract Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ( C K ) and Spread complexity ( C S ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators . After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between C K and 2 C S . Furthermore, for maximally entangled pure states, we find that the moment-generating function of C K becomes the Spectral Form Factor and, at late-times, C K is simply related to NC S for N ≥ 2 within the N -dimensional Hilbert space. Notably, we confirm that C K = 2 C S holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.