Krylov's State Complexity and Information Geometry in Qubit Dynamics

TL;DR

This work investigates Krylov's state complexity $\mathcal{K}(t)$ alongside a quantum information geometry (IG) complexity $C(t_A,t_B)$ for qubit dynamics on the Bloch sphere, comparing geodesic and nongeodesic evolutions under stationary and nonstationary Hamiltonians.It reformulates Krylov complexity in geometric terms using Bloch vectors and the Lanczos Krylov basis, and defines IG complexity via FS-volume-based accessible and maximum volumes, highlighting their distinct geometric meanings: spread along a Krylov chain versus explored volume on the Bloch sphere.Across stationary and nonstationary, geodesic and nongeodesic cases, the two measures generally behave differently: Krylov complexity tracks distance moved from the initial state along a preferred direction, while IG complexity tracks how much of the Bloch-sphere region is explored, revealing complementary insights into quantum evolution.The results show that phase motion can alter trajectory length without necessarily increasing Krylov spread, underscoring the nontrivial separation between length-based and volume-based complexity notions and suggesting a broader, geometrically unified view of quantum complexity.

Abstract

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.

Figures (8)

• Figure 1: Sketch of an energy level diagram for a two-level system. The Hamiltonian H of the system has a spectral decomposition given by H$=E_{\_}\left\vert E_{\_}\right\rangle \left\langle E_{\_}\right\vert +E_{+}\left\vert E_{+}\right\rangle \left\langle E_{+}\right\vert$, with energy gap $\Delta E\overset{\text{def}}{=}E_{+}-E_{\_}\geq0$. In general, $\left\vert E_{+}\right\rangle =\left\vert e\right\rangle =\left\vert \uparrow \right\rangle$ denotes the excited state or spin up state (maximum eigenvalue), while $\left\vert E_{-}\right\rangle =\left\vert g\right\rangle =\left\vert \downarrow\right\rangle$ denotes the ground state or spin down state (minimum eigenvalue).
• Figure 2: Sketch of a variety of magnetic field vector configurations. In (a), $\mathbf{B}\overset{\text{def}}{=}B_{0}\hat{B}_{0}$. In (b), $\mathbf{B}\overset{\text{def}}{=}B\left( t\right) \hat{B}_{0}$. In (c), $\mathbf{B}\overset{\text{def}}{=}B_{0}\hat{B}\left( t\right)$. Finally, in (d), $\mathbf{B}\overset{\text{def}}{=}B\left( t\right) \hat{B}\left( t\right)$. We note that $\mathbf{B}$ is the magnetic field vector, $B_{0}$ denotes a constant intensity, and $\hat{B}_{0}$ is a constant unit vector.
• Figure 3: Illustrations of low (a) and high (b) scenarios of Krylov state complexity $\mathcal{K}\left( t\right)$. In both (a) and (b), $n$ represents the Krylov index, while the dashed vertical lines indicate the probabilities $p_{n}(t)\overset{\text{def}}{=}\left\vert \left\langle K_{n}\left\vert \psi\left( t\right) \right. \right\rangle \right\vert ^{2}$. The Krylov state complexity $\mathcal{K}\left( t\right)$ is defined as the average position $\left\langle n\right\rangle$ of this probability distribution $p_{n}(t)$ with $0\leq p_{n}(t)\leq1$ and $\sum_{n}p_{n}(t)=1$. In scenario (a), the wavefunction exhibits minimal spreading, resulting in a slow increase in Krylov's state complexity. Generally, a low complexity scenario is characterized by a narrow distribution concentrated around $n=0$. Conversely, in scenario (b), the state investigates numerous Krylov directions, leading to a rapid increase in Krylov's state complexity. Typically, a high energy scenario is marked by a wide distribution with its center significantly distant from $n=0$.
• Figure 4: A visual representation that aids in understanding the concept of complexity C$\left( t_{A}\text{, }t_{B}\right)$ with $t_{A}\leq t\leq t_{B}$. In (a), we illustrate a complex situation where the accessed volume $\overline {\mathrm{V}}_{1}$ (the area enclosed by the dashed circular line) constitutes a minor portion of the total accessible volume V$_{\max\text{-}1}$ (the area enclosed by the thick solid circular line). Conversely, in (b), we depict a scenario with a reduced level of complexity, as $\overline {\mathrm{V}}_{2}$ (the area enclosed by the dashed circular line) represents a larger portion of V$_{\max\text{-}2}$ (the area enclosed by the thick solid circular line). The length $s$ of a path serves as an estimate for the maximum number $n_{s}$ of statistically distinguishable states along that path. Generally, the higher complexity scenario in (a) corresponds to a longer quantum path of length $s_{1}$, characterized by a small fraction ($\overline{\mathrm{V}}_{1}/\mathrm{V}_{\max\text{-}1}$) of the total accessible volume V$_{\max\text{-}1}$. In contrast, the lower complexity scenario in (b) typically corresponds to a shorter quantum path of length $s_{2}<s_{1}$, described by a larger fraction ($\overline{\mathrm{V}}_{2}/\mathrm{V}_{\max\text{-}2}$, with $\overline{\mathrm{V}}_{2}/\mathrm{V}_{\max\text{-}2}>\overline{\mathrm{V}}_{1}/\mathrm{V}_{\max \text{-}1}$) of the total accessible volume V$_{\max\text{-}2}$. The number $n_{\mathrm{L}_{\mathrm{C}}}$ of points represented in an accessed volume $\overline{\mathrm{V}}$ is proportional to the square of the length $s^{2}$ of the path, enhanced by a factor that is the reciprocal of $\overline{\mathrm{V}}/\mathrm{V}_{\max}$, where $0\leq$$\overline {\mathrm{V}}/\mathrm{V}_{\max}\leq1$. These two scenarios in (a) and (b) are accurately characterized by the concept of complexity length scale L$_{\mathrm{C}}$, with $n_{\mathrm{L}_{\mathrm{C}}}\overset {\text{def}}{=}$L$_{\mathrm{C}}^{2}=s^{2}\cdot(\overline{\mathrm{V}}/\mathrm{V}_{\max})^{-1}$leo25emma25.
• Figure 5: Geometric sketch for Rodrigues' rotation formula, $\mathbf{a}_{t}=\mathcal{R}_{\mathbf{n}}\left( \theta_{t}\right) \mathbf{a}_{0}$ with $\mathbf{a}_{0}\overset{\text{def}}{=}\mathbf{a}_{0}^{\parallel}+\mathbf{a}_{0}^{\perp}$ and $\mathbf{a}_{t}\overset{\text{def}}{=}\mathbf{a}_{t}^{\parallel}+\mathbf{a}_{t}^{\perp}$. We stress that $\mathcal{R}_{\mathbf{n}}\left( \theta_{t}\right)$ denotes a rotation about the axis specified by means of the unit vector $\mathbf{n}$ by an angle $\theta_{t}$.