Efficiency, Curvature, and Complexity of Quantum Evolutions for Qubits in Nonstationary Magnetic Fields

TL;DR

This work develops a geometric framework for nonstationary qubit evolutions driven by time-dependent magnetic fields, deriving an exact analytical curvature expression and a Bloch-sphere–based complexity measure. By expressing curvature in terms of the Bloch vector $\mathbf{a}(t)$ and magnetic field $\mathbf{h}(t)$, the authors connect trajectory bending to geodesic and speed efficiencies, and analyze five phase-growth profiles to contrast path length, curvature, and complexity. The key contributions are the closed-form curvature for qubit evolutions, a concrete complexity metric based on accessed and accessible Bloch-sphere volumes, and a detailed comparative study showing that efficient evolutions generally exhibit lower complexity, though longer, highly curved paths can reduce complexity relative to shorter, less curved ones. The results offer guidance for designing time-dependent driving protocols that achieve fast, energy-efficient state transfers with minimal trajectory complexity, and they lay the groundwork for extensions to higher-dimensional systems and Krylov-based complexity analyses.

Abstract

In optimal quantum-mechanical evolutions, motion can take place along paths of minimal length within an optimal time frame. Alternatively, optimal evolutions may occur along established paths without any waste of energy resources and achieving 100% speed efficiency. Unfortunately, realistic physical scenarios often lead to less-than-ideal evolutions that demonstrate suboptimal efficiency, nonzero curvature, and a high level of complexity. In this paper, we provide an exact analytical expression for the curvature of a quantum evolution pertaining to a two-level quantum system subjected to various time-dependent magnetic fields. Specifically, we examine the dynamics produced by a two-parameter nonstationary Hermitian Hamiltonian with unit speed efficiency. To enhance our understanding of the physical implications of the curvature coefficient, we analyze the curvature behavior in relation to geodesic efficiency, speed efficiency, and the complexity of the quantum evolution (as described by the ratio of the difference between accessible and accessed Bloch-sphere volumes for the evolution from initial to final state to the accessible volume for the given quantum evolution). Our findings indicate that, generally, efficient quantum evolutions exhibit lower complexity compared to inefficient ones. However, we also note that complexity transcends mere length. In fact, longer paths that are sufficiently curved can demonstrate a complexity that is less than that of shorter paths with a lower curvature coefficient.

Figures (4)

• Figure 1:
• Figure 2: Illustrative depiction of the nongeodesic evolution path (thick solid line) on the Bloch sphere generated by the nonstationary Hamiltonian H$\left( t\right)$ associated to a phase $\beta\left( t\right)$ that exhibits exponential growth. The evolution occurs from $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 0\right\rangle$ to $\left\vert B\right\rangle \overset{\text{def}}{=}\left\vert 1\right\rangle$. For simplicity, we set $\nu_{0}=\omega_{0}=1$ and $0\leq t\leq\pi/2$. Physical units are chosen using $\hslash=1$.
• Figure 3: Temporal behavior of the curvature coefficient $\kappa_{\mathrm{AC}}^{2}(t)$ of the quantum evolution when the phase $\beta\left( t\right)$ exhibits linear growth (a), quadratic growth (b), exponential growth (c) and, finally, exponential decay (d). In all plots, we set $\nu_{0}=\omega_{0}=1$ and $0\leq t\leq\pi/2$. Physical units are chosen using $\hslash=1$.
• Figure 4: Behavior of the complexity of the quantum evolution specified by the nonstationary Hamiltonian H$\left( t\right)$ corresponding to the phase $\beta\left( t\right)$ that grows exponentially (dashed line) or decays exponentially (thick solid line). The complexity Cis plotted versus the ratio $\nu_{0}/\omega_{0}$. When $\nu_{0}/\omega_{0}\gg1$ and $\beta\left( t\right)$ grows exponentially, the complexity asymptotically approaches its maximum value $1$. Alternatively, when $\nu_{0}/\omega_{0}\gg1$ and $\beta\left( t\right)$ decays exponentially, the complexity asymptotically approaches the limiting value of $1/2$, that is to say the complexity value that characterizes the geodesic evolution on the Bloch sphere between initial and final states $\left\vert A\right\rangle \overset{\text{def}}{=}\left\vert 0\right\rangle$ and $\left\vert B\right\rangle \overset{\text{def}}{=}\left\vert 1\right\rangle$, respectively.