Multi-level spectral navigation with geometric diabatic-adiabatic control

TL;DR

This work introduces a geometric diabatic-adiabatic (di-ad) framework for high-fidelity state transfer in multi-level quantum systems, interpolating between adiabatic and diabatic dynamics via the di-ad tensor $\mathcal{G}_{\text{di-ad}}$ and transition matrix $\xi_{mn}$. A single control parameter reduces the optimization to a first-order ODE, enabling fast, hardware-friendly pulse generation that can traverse energy landscapes beyond strict adiabatic limits. The framework is demonstrated on a double quantum dot spin-qubit platform for state initialization and shuttling, achieving fidelities above 99% and flexible parity-sector access, with runtime and optimization strategies enabling practical deployment. The method is system-agnostic, combining geometric control with automated parameter optimization to adapt pulses to experimental constraints in scalable quantum devices.

Abstract

We introduce a geometric framework for efficient few-parameter pulse optimization in multi-level quantum systems, enabling high-fidelity state transfer beyond the adiabatic limit. Our method interpolates smoothly between adiabatic and diabatic dynamics to minimize unwanted excitations and maximize desired transitions even within a multi-level structure. Crucially, for single-parameter pulse control, the optimization reduces to solving a first-order ordinary differential equation. We showcase the flexibility of our diabatic-adiabatic protocols through two examples in spin-based quantum information processing: state initialization and qubit state transfer.

Figures (7)

• Figure 1: Application of the di-ad protocol (a), here, spin qubit shuttling, and the emergent dense energy spectrum as a function of some control parameter (b). The di-ad protocols allow for flexible and high-fidelity state transfer beyond the extrema of adiabatic (red, dashed) and diabatic (blue, dashed) limits by selectively operating between both regimes (purple).
• Figure 2: Coherent state transfer for the adiabatic ($\ket{\psi_0}\mapsto \ket{\psi_0}$, dashed lines) and the diabatic ($\ket{\psi_0}\mapsto \ket{\psi_1}$, solid lines) evolution in the Landau-Zener model. We show the energy spectrum (a), the di-ad pulse (b), originating from the corresponding di-ad metric (c), and the resulting infidelities $1{-}\mathcal{F}$(d) with $\mathcal{F}{=}|\braket{\psi_0(t_\text{f})}{U(t_\text{f})\psi_0(0)}|^2$ for adiabatic and $\mathcal{F}{=}|\braket{\psi_1(t_\text{f})}{U(t_\text{f})\psi_0(0)}|^2$ for diabatic evolution, where $U(t_\text{f})$ describes the unitary evolution given the pulse $z(t)/x$. The Hamiltonian parameters used are $z(0)/x{=}{-}z(t_\text{f})/x{=}{-}10$. The colored pulses refer to different $n_+$ with the appropriate sign for adiabatic (dashed) and diabatic (solid) protocols: $|n_+|=0,2,3$ corresponds to black, red, and blue, respectively.
• Figure 3: Initialization of a spin qubit in a DQD. The energy spectrum is shown (a). The optimal infidelity and corresponding pulse times for the adiabatic protocol ($\ket{\uparrow\downarrow,\cdot}\mapsto \ket{\downarrow,\downarrow}$) are illustrated in (b),(c). Similarly, for the di-ad protocol ($\ket{\uparrow\downarrow,\cdot}\mapsto \ket{\uparrow,\downarrow}$), we showcase the infidelities and pulse times in (d),(e). The DQD parameters read $\tilde{U}/t_c=10, \,E_Z/t_c=0.9,\, \Delta\!E_Z/t_c=0.1, \,\Delta\!E_X/t_c=0.01$, where we sweep the detuning $\varepsilon(0)/t_c=15$ to $\varepsilon(t_\text{f})/t_c=0$. For the di-ad protocol, we fix the adiabatic parameters to $(\alpha,\beta){=}(2,2)$. For the protocol, we sweep possible pulse times $t_\text{f}t_c\in [0,1000]$.
• Figure 4: On-the-go operations using shuttling-induced excitations. The energy spectrum of the Hamiltonian \ref{['eqn: 4x4 Ham']}, where we marked in orange the undesired leakage states, is plotted in (a). In addition, we plot the minimum infidelities $1-\tilde{\mathcal{F}}$ for the state transfers $\ket{\psi_0}{\to}\ket{\psi_1}$(b) and $\ket{\psi_1}{\to}\ket{\psi_0}$ (inset), for possible pulse times $\Delta_Lt_\text{f}\in [0,500]$. Using the Horodecki identity, in (c), we analyze the effective X-gate average gate fidelity $F_\text{gate}=(1+\sum_{j=0,1} |\braket{\psi_j(0)}{\psi_{1-j}(t_\text{f})}|^2)/3$ for the computational subspace spanned by the lower two eigenstates as a function of possible pulse times $\Delta_L\,t_\text{f}$ for the indicated circle, square and star parameters in (b). The parameters used are $t_c/\Delta_L=.1,\Delta_R/\Delta_L=5, \phi_L=0.1, \phi_R=\pi/2$. The detuning is swept from $\varepsilon/\Delta_L\in[-10,10]$.
• Figure 5: Pulse shapes for highest-fidelity DQD initialization using the di-ad protocol (blue) and fully adiabatic (orange) in Figure \ref{['fig: dqd init']}.