Quantum geometric protocols for fast high-fidelity adiabatic state transfer

TL;DR

This work develops a geometric framework, the geometric fast-QUAD, that leverages the quantum metric tensor to map control parameter trajectories to geodesics, enabling fast and high-fidelity adiabatic state transfer in multi-level quantum systems with dense spectra. By minimizing energy fluctuations along geodesics, the method suppresses diabatic transitions without introducing extra control fields, and it scales with the number of control parameters rather than the Hilbert-space size. The authors apply the approach to a double quantum dot system, comparing full, truncated, and reduced models, and demonstrate superior performance over linear pulses under both unitary and non-unitary dynamics, including robustness to 1/f detuning noise and pulse miscalibration. These results indicate a practical, platform-flexible route to fast, high-fidelity initialization/readout in semiconductor spin qubits and potentially other quantum architectures, with extensions to mixed states and non-Abelian quantum geometry left for future work.

Abstract

Efficient control schemes that enable fast, high-fidelity operations are essential for any practical quantum computation. However, current optimization protocols are intractable due to stringent requirements imposed by the microscopic systems encoding the qubit, including dense energy level spectra and cross talk, and generally require a trade-off between speed and fidelity of the operation. Here, we address these challenges by developing a general framework for optimal control based on the quantum metric tensor. This framework allows for fast and high-fidelity adiabatic pulses, even for a dense energy spectrum, based solely on the Hamiltonian of the system instead of the full time evolution propagator and independent of the size of the underlying Hilbert space. Furthermore, the framework suppresses diabatic transitions and state-dependent crosstalk effects without the need for additional control fields. As an example, we study the adiabatic charge transfer in a double quantum dot to find optimal control pulses with improved performance. We show that for the geometric protocol, the transfer fidelites are lower bounded $F>99\%$ for ultrafast 20 ns pulses, regardless of the size of the anti-crossing.

Figures (12)

• Figure 1: Schematic representation of the geometric fast-QUAD approach with the example of a qubit Hamiltonian. First, one defines the system Hamiltonian and the respective parameters $x^\mu=(x^1,x^2,x^3)$ one wants to optimally operate. The number of parameters defines the dimension of the quantum metric tensor $g_{\mu \nu}$. For a qubit Hamiltonian, the underlying parameter manifold is the Bloch sphere. The geodesic on the sphere (red line) gives the optimal path connecting two states. Examples of non-optimal state transfer protocols ($x^\mu_1,x^\mu_2$) are drawn as blue and purple, dotted lines.
• Figure 2: (a) Coherent transfer probability $p(t_\text{f})=|\braket{\psi_0(t_\text{f})}{\psi(t_\text{f})}|^2$ (inset shows energy levels as a function of $\rho$) and (b) transfer error $1-p(t_\text{f})$ via quantum geometric protocols for a single qubit system (\ref{['eqn: Pauli qubit Hamiltonian']}) as a function of pulse time $t_\text{f}$. As the quantum metric tensor (\ref{['eqn: Pauli quantum geometric tensor']}) is independent of $\phi$ we can choose a protocol exclusively for $\rho(t)$ or for the ratio $\theta(t)=\text{arctan2}(\rho(t),z(t))$. Using the quantum adiabatic protocol (\ref{['eqn: adiabatic quantum geo protocol']}) we find analytically that $\theta_\text{geo}(t)=(\theta_\text{f}-\theta_\text{0})\,t/t_\text{f}+\theta_\text{0}$, where the adiabaticity is given by the ramp rate $\delta=(\theta_\text{f}-\theta_\text{0})/t_\text{f}$. We also simulate the pulse numerically, as indicated by $\rho_\text{geo}(t)$. Remarkably, the two methods align even up to the minimum transfer error. Throughout these simulations the values of $\phi=0$, $\rho_\text{0}=-10$ GHz, $\rho_\text{f}=10$ GHz, $z=0.1$ GHz have been used. Note that the precision of the numerical simulations determines the singular behavior (gray area) in the transfer error.
• Figure 3: (a) Transfer errors as a function of the pulse time $t_\text{f}$ are plotted for the linear and geometric protocols following Eq. \ref{['eqn: 6x6 Hamiltonian']} without incoherent errors. (b) Illustration of the 6x6 energy levels as a function of detuning, where we only focus on the two lower levels. The $ST_-$ anti-crossing, which induces diabatic transitions in the initialization process, is given by the size of $\Delta E_X= \unit[0.1]{GHz}$, similar to saez-mollejoMicrowaveDrivenSinglettriplet2024a for out-of-plane magnetic fields. After around $\unit[150]{ns}$ pulse time, we can expect an initialization fidelity of $\unit[99.99]{\%}$ for the $\ket{\downarrow\downarrow}$ state. The parameters used in the simulation are: $\Tilde{U}=\unit[100]{GHz}$, $E_Z=\unit[10]{GHz}$, $\Omega=\unit[10]{GHz}$, $\Delta E_Z=\unit[1]{GHz}$, $\varepsilon_\text{f}=\unit[10]{GHz}$, and $\varepsilon_0=3\Tilde{U}/2$.
• Figure 4: (a) Illustration of the spin-to-charge conversion protocol of spin qubits. The particle on the right side of the DQD potential can only be transferred to an antisymmetric spin-state on the left side due to the Pauli exclusion principle. (b) Plot of the energy spectrum $E_n$ of the full model \ref{['eqn: 6x6 Hamiltonian']} as a function of the detuning $\varepsilon$ (tilt of the potential wells). The colored lines correspond to the energy levels of the reduced three-level model \ref{['eqn: 3x3 Hamiltonian']}. (c) Detailed energy spectrum for the three-level Hamiltonian \ref{['eqn: 3x3 Hamiltonian']}. The dashed line represents the energy of the $\ket{S(2,0)}$ state, which for large detuning $\varepsilon\gg \Tilde{U}, \Omega, \Delta E_Z$ becomes the eigenstate of the DQD Hamiltonian.
• Figure 5: Comparison of initialization pulses using the linear (a), (c), (d) and geometric fast-QUAD (b), (d), (f) protocols with $\varepsilon_\text{0}=2\Tilde{U}=200$ GHz and $\varepsilon_\text{f}=0$. (a), (b) Illustrates the effect of both the tunnel coupling and the pulse time on the transfer error. We note that the geometric fast-QUAD achieves a transfer fidelity of $\unit[99.99]{\%}$ after $\unit[20]{ns}$, which defines a lower bound. Transfer errors as low as $10^{-8}$ may be reached even for very small tunnel couplings. (c), (d) is a contour plot sweeping the Zeeman splitting difference and the pulse time $(\Delta E_Z, t_\text{f})$ at fixed tunnel coupling $\Omega=\unit[1]{GHz}$. Here the clear advantage of the geometric fast-QUAD comes through, as the size of the anti-crossing hinders the coherent charge transfer in the linear case. A similar trend can be seen in (e), (f). Remarkably, in (f) the charge transfer fidelity is lower bounded at $\unit[99]{\%}$, independent of the combination $(\Delta E_Z,\Omega)$ for an ultrafast pulse with pulse time $t_\text{f}=\unit[20]{ns}$. In addition, due to the conservation of energy along adiabatic paths, we observe no difference between the initialization and readout protocols.