Stochastic optimal control of open quantum systems

TL;DR

This work recasts open-quantum-system state preparation as a stochastic optimal control problem and solves it via path integral control, obviating gradient computations. By leveraging SSE unravelings and a gauge-invariant transformation to anti-Hermitian operators, the authors derive closed-form, trajectory-based expressions for optimal pulses and implement adaptive importance sampling (QDC) to estimate them efficiently. They extend QDC to function as a diffusion-based annealer to tackle unitary dynamics and demonstrate strong performance on single-qubit, multi-qubit, and NMR-inspired GHZ tasks, with notable fidelity gains and reduced variance compared to opensource gradient-based baselines. The framework is inherently parallelizable and opens a practical path toward hardware-aware quantum control, including on-hardware implementations and pulse-level control. Overall, QDC offers a scalable, gradient-free approach for optimal control in noisy quantum environments, with broad implications for quantum information processing and quantum technologies.

Abstract

We address the generic problem of optimal quantum state preparation for open quantum systems. It is well known that open quantum systems can be simulated by quantum trajectories described by a stochastic Schrödinger equation. In this context, the state preparation becomes a stochastic optimal control (SOC) problem. The latter requires the solution of the Hamilton-Jacobi-Bellman equation, which is, in general, challenging to solve. A notable exception are the so-called path integral (PI) control problems, for which one can estimate the optimal control solution by direct sampling of the cost objective. In this work, we derive a class of quantum state preparation problems that are amenable to PI control techniques, and propose a corresponding algorithm, which we call Quantum Diffusion Control (QDC). Unlike conventional quantum control algorithms, QDC avoids computing gradients of the cost function to determine the optimal control. Instead, it employs adaptive importance sampling, a technique where the controls are iteratively improved based on global averages over quantum trajectories. We also demonstrate that QDC, used as an annealer in the environmental coupling strength, finds high accuracy solutions for unitary (noiseless) quantum control problems. We further discuss the implementation of this technique on quantum hardware. We illustrate the effectiveness of our approach through examples of open-loop control for single- and multi-qubit systems.

Figures (8)

• Figure 1: Control of a noise qubit from $X\to Y$. ( A) Average fidelity $F_\text{avg}$, minimal fidelity $F_\text{min}$, effective sample size ESS and cost $C$ in \ref{['eq:stoch_cost']} versus importance sampling iterations $p$. The converged average fidelity is $F^* = 0.9759 \pm 0.0006$. ( B) Optimal control solution $u_{x,y}(t)$ after convergence of the algorithm. The optimal solution is two-fold degenerate. The two solutions are related by a global sign $u_{x, y} \rightarrow -u_{x, y}$. ( C). Dependence of the quality of the optimal control, measured by the ESS, on the number of pulses $K$. ( D). Optimally controlled trajectories on the Bloch sphere. Parameters: final time $T=1$, noise coupling $D=0.005$, control weight $R= 1$, fidelity weight $Q=10$, number of pulses $K=128$, number of trajectories $N_{traj}=400$ per IS step, maximum number of IS steps $n_{IS}=1000$ and time discretization $dt=T/N_T$ with $N_T=128$. IS smoothing window $w=40$ (see Section \ref{['sec:IS']}).
• Figure 2: Histograms of optimal control costs obtained with Open GRAPE (main panel) and QDC (inset) on $505$ runs. For Open GRAPE, the average cost, fidelity and fluence (and standard deviations) are $\left\langle C \right\rangle = -4 \pm 1$, $\left\langle F \right\rangle = 0.8 \pm 0.2$ and $\left\langle U \right\rangle = 3 \pm 2$, respectively. The minimum cost achieved is $C_{min} = -4.07$. For QDC, the corresponding average values (yellow dot) are $\left\langle C \right\rangle = -4.171 \pm 0.003$, $\left\langle F \right\rangle = 0.9759 \pm 0.0006$ and $\left\langle U \right\rangle = 1.4170 \pm 0.0009$. In the main panels, the dashed vertical lines at the QDC average are wider than the actual standard deviations of the QDC distributions. This illustrates the large difference in variance between the two methods. All QDC runs converge to solutions with the same cost, fidelity, fluence and effective sample size up to small statistical fluctuations due to the stochastic nature of QDC algorithm. Parameter setting: $D=0.005$, $T=1$, $R=1$, $Q=10, K=128$, $w=20$.
• Figure 3: ( A) Examples of Open GRAPE solutions with diverse levels of regularity for different seeds. ( B) 2-D t-SNE dimension reduction representation followed by a K-means decomposition of Open GRAPE solutions (red, blue and black dots) and the two average QDC solutions (yellow dots). The cluster of black dots correspond to local minima of relatively high cost $C$. ( C) Average Open GRAPE solutions for cluster 1 (red) and cluster 2 (blue). In the bottom panels, the QDC solutions corresponding to each cluster. ( D) Robustness $\Delta C = C(u+\xi)-C(u)$ versus $\sigma$ with $u$ the high-fidelity high-ESS QDC solution (red) and the high-fidelity low-ESS Open Grape solution (black).
• Figure 4: ( A) Single qubit, $X \rightarrow Y$. In the left y-axis, fidelities $F_{open}$ and $F_{closed}$ vs $D$; in the right y-axis, normalized fluence $U/U_{max}$, vs $D$, with $U_{max}=1.86$. Parameters: $T=1$, $K=50$, $R=0.5$, $N_T=100$, $N_{traj}=2000$ and $Q$ iterates over $5, 50$ and $100$, and stopping when the fidelity threshold condition $F_{avg} > 0.98$ is met. ( B) $X \rightarrow Haar$. Distribution of $D_{max}$ for $100$ random Haar target states. Average $\left\langle D_{max} \right\rangle=1 \pm 0.3$.
• Figure 5: 4-NMR control. ( A) Infidelity $1-F$ vs $K$. Curve open corresponds to the open system case, with fixed noise coupling $D=1/T_1$. Curves anneal1, anneal2 and anneal3 correspond are the infidelities using annealing control. The rest of parameters are: $N_T = 500$, $n_{IS}=1000$. We show the control solutions for the open system case for $K=5$ pulses ( B) and $K=500$ pulses ( C). In panel ( D) we show an example of annealing training for $K=500$ pulses and logarithmic end cost. We plot the average and the worst case infidelities as function of the IS step, and in panel ( E) we plot the corresponding final solutions. These solutions achieve a infidelity of $6.0e-11$.

Theorems & Definitions (18)

• Proposition 1
• Theorem 1
• proof
• proof
• Lemma 1
• proof
• Corollary 2
• Theorem 3
• proof
• Corollary 4