Hybrid Variational Framework for Quantum Control

• The paper introduces a hybrid methodology that maps continuous quantum dynamics onto digital circuits using variational parameterization and classical optimization to achieve high-fidelity state transfer.
• The framework contrasts global versus local control strategies, showing that global control provides greater noise robustness while local control offers higher expressivity in noiseless conditions.
• Numerical benchmarks on spin-chain models demonstrate a trade-off between expressivity, stability, and computational cost, making the approach scalable for NISQ hardware.

A hybrid variational framework for quantum optimal control synthesizes quantum and classical resources to optimize control landscapes for quantum systems, leveraging variational circuit parameterizations, Trotterized dynamics, and closed-loop classical optimization. This approach directly compiles continuous-time dynamics into digital quantum circuits with tunable control parameters, which are then optimized via classical algorithms to achieve desired quantum-state objectives—such as high-fidelity state transfer—within hardware constraints and noise budgets. The framework is particularly representative in the context of spin-chain state-transfer problems, where global and local control parameterization strategies yield distinct trade-offs in expressivity, stability, and robustness.

1. Formalism and Physical Model

Consider an $N$-site, one-dimensional XXZ spin chain where the total Hamiltonian is decomposed into a static drift part ($H_d$) and time-dependent control fields ($H_c(t)$):

$H(t) = H_d + H_c(t),$

with the drift given by anisotropic Heisenberg couplings

$$H_d = \sum_{k=1}^{N-1}\left(J_x X_k X_{k+1} + J_y Y_k Y_{k+1} + J_z Z_k Z_{k+1}\right),$$

and control

$$H_c(t) = \begin{cases} \sum_{j=1}^N u(C(t), d(t), j) Z_j & \text{(global control)}, \ \sum_{j=1}^N u_j(t) Z_j & \text{(local control)}, \end{cases}$$

where $u(C,d,j)$ follows a spatially harmonic profile $u(C,d,j) = \tfrac{1}{2} C (j - d)^2$.

Control parameters are bounded, with $d(t)$ and $C(t)$ anchored at endpoints for global control; sitewise $u_j(t)$ is bounded for local control. The boundary conditions are open chain, and typical coupling choices in numerical studies are $J_x = J_y = 1$, $J_z = 0.2$.

2. Circuit Mapping and Trotterization

The continuous evolution is discretized onto a uniform time grid $t_\ell$, $\ell = 1\ldots L$, with interval $\Delta t = T/L$. For each slice, the propagator is Suzuki–Trotter decomposed:

$$U^{(\ell)} = \exp(-i H_d \Delta t) \exp(-i H_c(\boldsymbol{\theta}^{(\ell)}) \Delta t) + \mathcal{O}(\Delta t^2).$$

The drift exponentials factor into three two-qubit gates per link on each layer: XX, YY, and ZZ. The control exponential is compiled as $N$ $R_Z$ rotations per layer. The layer structure is diagrammatically straightforward, facilitating hardware execution.

3. Control Parameterization (Ansatz Design)

Two parameter-sharing schemes are featured:

• Global Control: For each time slice, only two real parameters $C_\ell$, $d_\ell$ define the control across all spins. After fixing boundary conditions, the number of variational parameters is $d_{\text{global}} = 2L-2$.
• Local Control: Sitewise distinct control amplitudes $\theta_{\ell,j}$ at each slice; $d_{\text{local}} = NL$.

Each Trotter layer requires $3(N-1)$ two-qubit gates and $N$ single-qubit gates, with linear scaling in $L$. The global ansatz offers a low-dimensional, correlated parameterization; the local ansatz provides fine-grained, high-expressivity control at the cost of increased dimensionality.

4. Optimization Procedure

The fidelity functional to be maximized is

$$F(\boldsymbol{\theta}) = |\langle \psi_f | U(\boldsymbol{\theta}) | \psi_0 \rangle|^2,$$

where $\ket{\psi_0}$ and $\ket{\psi_f}$ are the initial and target states. The cost function minimized is the infidelity

$$J(\boldsymbol{\theta}) = 1 - F(\boldsymbol{\theta}).$$

The hybrid loop alternates:

• Quantum subroutine: Simulate the circuit for proposed $\boldsymbol{\theta}$, estimate $F$ via overlap measurement.
• Classical update: Sequential Least Squares Quadratic Programming (SLSQP) optimizes $\boldsymbol{\theta}$ subject to parameter bounds. Gradients are computed by finite differences unless analytic Jacobians are available.

Each SLSQP iteration entails $\mathcal{O}(d \cdot C_{\text{sim}})$ computational cost ($d$: number of parameters).

5. Expressivity–Stability Trade-off and Noise Analysis

Global and local parameterizations present distinct trade-offs:

• Local control achieves lower infidelity and faster convergence in the noiseless regime (final $J \sim 2 \times 10^{-4}$) but is highly susceptible to noise. Under depolarizing error rate $p=10^{-3}$, achieved fidelities reduce to $F \sim 0.45$.
• Global control, despite reduced expressivity in noise-free simulations (final $J \sim 1.1 \times 10^{-3}$), maintains high-fidelity state transfer under identical error rates ($F \sim 0.989$). This robustness is attributed to the reduced parameter count and spatially correlated control profile, which regularize the landscape and mitigate overfitting to stochastic fluctuations.

The primary conclusion is an explicit expressivity-stability trade-off, with the global ansatz providing scalable robustness on NISQ hardware.

6. Numerical Benchmarks and Scalability

Simulations on $N=3$–$10$ spin chains with $L=8$–$20$ slices reveal:

• In the absence of noise, local control achieves the infidelity threshold ($J< 10^{-2}$) in $\sim 200$ function calls, global control requires $\sim 300$.
• Site population dynamics and optimized control pulses exhibit spatial and temporal structure characteristic of the chosen ansatz.
• Under realistic depolarizing noise, global control outperforms local by a factor of $\sim 2$ in robustness ratio.
• The global scheme, scaling as $\mathcal{O}(L)$ in parameters and circuit depth, is recommended for NISQ platforms with coherence times compatible with $L \sim 10$–$20$ and error rates $p \sim 10^{-3}$.

7. General Framework and Context

This hybrid variational methodology fits within a broader class of quantum optimal control frameworks—such as those leveraging digital quantum simulation and VQAs (Huang et al., 29 May 2025, Huang et al., 2022), pulse-level and gate-pulse hybrids (Liang et al., 2022), or parametric oracle-based methods (Li et al., 2016). The unifying principle is the mapping of continuous control tasks to parameterized quantum circuits, the deployment of classical optimizers to close the quantum-classical loop, and the use of hardware-efficient Trotterization to exploit near-term devices. The approach elucidates design principles for control landscape regularization, noise-aware ansatz choice, and expressivity constraints, with practical application to state transfer, gate synthesis, and scalable multi-qubit control.

The hybrid variational framework for quantum optimal control thus constitutes a fully specified recipe encompassing Hamiltonian models, circuit construction, control ansatz selection, classical optimization, and benchmarked performance metrics, enabling high-fidelity synthesis and transfer protocols on current-generation quantum hardware (Dehaghani et al., 12 Nov 2025).