Trotterized Variational Quantum Control for Spin-Chain State Transfer

TL;DR

The paper addresses high-fidelity state transfer in spin chains under hardware constraints by mapping finite-horizon quantum control to a Trotterized parameterized quantum circuit. It introduces two control parameterizations, global and local, and optimizes circuit parameters with a hybrid loop using SLSQP to maximize terminal fidelity $F$ via $J=1-F$. The key contributions are (i) a clean bilinear Hamiltonian to circuit mapping, (ii) explicit parameter-counts showing an expressivity–stability trade-off, (iii) a practical VQC loop with finite-difference gradients, and (iv) empirical evidence that global control offers noise robustness while local control can yield marginal gains in noiseless settings. This work provides a scalable, NISQ-friendly route to quantum control synthesis for spin-chain state transfer and informs design choices under realistic noise models.

Abstract

We present a hybrid variational framework for quantum optimal control aimed at high-fidelity state transfer in spin chains. The system dynamics are discretized and compiled into a parameterized circuit, where deterministic two-qubit blocks implement the drift interactions, while trainable on-site RZ rotations encode the control inputs. We study two parameterizations: a compact global scheme with a small number of shared parameters per slice, and a local scheme with site-wise angles. Using a Sequential Least Squares Quadratic Programming (SLSQP) optimization to minimize infidelity, simulations on XXZ spin chains show that both parameterizations can achieve near-unit fidelities in the noiseless regime. Under depolarizing noise, the global scheme provides improved robustness for comparable circuit depth and iteration budgets. The results make explicit an expressivity-stability trade-off and suggest a scalable route to Noisy Intermediate-Scale Quantum (NISQ) compatible control synthesis.

Figures (7)

• Figure 1: Hybrid variational quantum control loop. The time-discretized Hamiltonian is mapped to a parameterized circuit; the simulator returns $F(\theta)$ and $J(\theta)=1-F(\theta)$ to SLSQP, which updates $\theta$ and hence $\{u_k(t_j)\}$. Repeat $L$ times per pass until convergence.
• Figure 2: One Trotter layer $t$ for a 3-qubit chain used in the simulations. Nearest-neighbor drift via $R_{XX},R_{YY},R_{ZZ}$ followed by on-site $R_Z$ controls. Local control: $\phi_{t,j}=\Delta t\,\theta_{j,t}$. Global control: $\phi_{t,j}=\Delta t\,\tfrac{1}{2} C_t(j-d_t)^2$. Stacking $L$ layers realizes the full evolution.
• Figure 3: Optimization trajectories (semilog scale) for global and local control across 10 realizations ($N=3$, $T=2$, $L=8$). Solid lines: mean loss $J(\theta)$; shaded bands: $\pm1\sigma$. The dashed line marks $J=10^{-2}$. Local control converges faster and attains lower final loss (mean $\sim 2\!\times\!10^{-4}$) than global (mean $\sim 1.1\!\times\!10^{-3}$).
• Figure 4: Average fidelity evolution for state transfer in a $N=3$ XXZ chain ($J_x=J_y=1,\,J_z=0.2$) over a total time $T=2.0$ discretized into $L=8$ layers ($\Delta t=0.25$). Solid blue: mean fidelity to the target state; dashed orange: mean fidelity to the initial state. Error bars denote $\pm 1\sigma$ across $R$ independent realizations (different random initializations of the control parameters).
• Figure 5: Optimized piecewise-constant local controls for the best realization (highest terminal fidelity) in an $N=3$ XXZ chain with $J_x{=}J_y{=}1$, $J_z{=}0.2$, total time $T{=}2.0$, and $L{=}8$ layers ($\Delta t{=}0.25$). Each panel shows the on-site control $u_j(t)=\theta_{\ell,j}$ held constant on $[t_\ell,t_{\ell+1})$ for site $j\in\{0,1,2\}$. Angles are the optimized amplitudes used in the $R_Z(2\,\theta_{\ell,j}\Delta t)$ gates of our Trotterized circuit.

Theorems & Definitions (3)

• Remark 1: Ansatz design and expressivity
• Remark 2: On assumptions
• Remark 3: Parameter counts and anchoring