Singularity-free dynamical invariants-based quantum control

TL;DR

This work develops a singularity-free invariant-based framework for robust single-qubit state preparation under non-Markovian noise. By splitting trajectories into subsegments and parameterizing dynamical invariants, it constructs a family of bounded control pulses that guarantee physical, smooth pulses in the closed system. An explicit two-stage approach then selects the pulse that minimizes noise impact, using either known-noise whitebox optimization (via Dyson expansion) or unknown-noise graybox ML modeling, enabling high-fidelity state preparation across diverse targets. Numerically, the method yields substantial fidelity gains over baseline pulses, with graybox performance closely approaching whitebox results, and it significantly improves state purity under non-Markovian noise. Overall, the framework extends invariant-based control to realistic open-system regimes and offers a scalable, robust route to quantum state engineering on NISQ hardware and beyond.

Abstract

State preparation is a cornerstone of quantum technologies, underpinning applications in computation, communication, and sensing. Its importance becomes even more pronounced in non-Markovian open quantum systems, where environmental memory and model uncertainties pose significant challenges to achieving high-fidelity control. Invariant-based inverse engineering provides a principled framework for synthesizing analytic control fields, yet existing parameterizations often lead to experimentally infeasible, singular pulses and are limited to simplified noise models such as those of Lindblad form. Here, we introduce a generalized invariant-based protocol for single-qubit state preparation under arbitrary noise conditions. The control proceeds in two-stages: first, we construct a family of bounded pulses that achieve perfect state preparation in a closed system; second, we identify the optimal member of this family that minimizes the effect of noise. The framework accommodates both (i) characterized noise, enabling noise-aware control synthesis, and (ii) uncharacterized noise, where a noise-agnostic variant preserves robustness without requiring a master-equation description. Numerical simulations demonstrate high-fidelity state preparation across diverse targets while producing smooth, hardware-feasible control fields. This singularity-free framework extends invariant-based control to realistic open-system regimes, providing a versatile route toward robust quantum state engineering on NISQ hardware and other platforms exhibiting non-Markovian dynamics.

Figures (6)

• Figure 1: The workflow of the proposed method. The first step in the protocol is splitting the target trajectory into sub-trajectories to avoid singularities in control pulses. Next, for each subtrajectory, the invariant is defined at the boundary points, followed by finding a parameterized functional form of the invariant, and bounding the parameters to avoid complex-valued pulses. The corresponding family of control pulses can then be computed, achieving the target evolution in the absence of noise. The next step, is finding the optimal control pulse from the constructed family to mitigate the noise effects. If the noise model is known, a control cost function is optimized directly. Otherwise, a machine learning stage is introduced where an ML model is designed and trained on a dataset. The trained ML model can then be integrated into the cost function to find the optimal control.
• Figure 2: Trajectory splitting to avoid singular (unbounded) pulses. Given the initial and target control Hamiltonians, $H_{\text{ctrl}}(0)=h_z(0)\sigma_z/2$ and $H_{\text{ctrl}}(T)=(h_x(T)\sigma_x+h_y(T)\sigma_y + h_z(T)\sigma_z)/2$, respectively, we split the trajectory into a number of subtrajectories, based on the location of the target state. This step of the protocol is designed to avoid the control pulse amplitude from growing to infinity. For each subtrajectory, a reference axis is chosen alongside the corresponding intermediate points.
• Figure 3: System setting for the numerical simulations. a) The simulation model is a generic two-level system with RTN noise acting along X- and Z-axis, and control applied along all three axes. b) The Bloch sphere representation of the target states to be prepared. The corresponding control Hamiltonians are shown in Table \ref{['table:target']}.
• Figure 4: Family of invariants for target (vi). This case requires 3 subtrajectories. Different examples from the invariant family are plotted as function of time. The parameters are chosen to be of linear scaling of $\vec{\Theta}_0$ that corresponds to the extreme case where $f_3(t)\to 0$ at exactly 1 point. The dotted curve represents degree 3 solution, i.e. the minimal polynomial that satisfies the boundary conditions.
• Figure 5: Purity of the state of the system for different pulses sequences. The purity of the state is plotted as a function of time for the worst-case and average-case pulse from the dataset, and the optimized pulse (known and unknown noise), in comparison to the case where the system is subject purely to the noise (i.e. $H_{\text{ctrl}}(t)=0$). Plots a)-f) correspond to the targets (i)-(vi), respectively.

Theorems & Definitions (2)

• Theorem 1
• proof