A Variational Surrogate Approach to Finite-Horizon Quantum Control via Hardware-Efficient Ansatz

Technical Summary: A Variational Surrogate Approach to Finite-Horizon Quantum Control via Hardware-Efficient Ansatz

Problem Statement
The paper addresses the finite-horizon quantum optimal control problem, specifically the task of steering a closed quantum system from a given initial state $|\psi_{in}\rangle$ to a desired target state $|\psi_{tar}\rangle$ within a fixed time $T$. Traditional approaches to this problem typically rely on explicitly parameterizing time-dependent control fields $u(t)$ and solving for the optimal trajectory using gradient-based methods or Pontryagin's Minimum Principle. However, these classical methods can become computationally prohibitive for high-dimensional systems and often require detailed knowledge of the system's drift and control Hamiltonians, limiting their direct compatibility with near-term quantum hardware.

Methodology
The authors propose a variational quantum framework that reformulates the control objective as a variational optimization problem. Instead of synthesizing time-dependent control fields or enforcing Hamiltonian reachability constraints, the method treats the terminal evolution operator as a trainable parameterized quantum circuit (PQC).

• Surrogate Parameterization: The core innovation is the use of a hardware-efficient ansatz as a surrogate for the terminal evolution. The circuit $U(\theta)$ maps the initial state to the terminal state $|\psi(\theta)\rangle = U(\theta)|\psi_{in}\rangle$.
• Circuit Structure: The ansatz consists of alternating layers of single-qubit rotations ($R_z$ and $R_x$) and nearest-neighbor entangling gates (CNOT). This structure is designed for compatibility with current Noisy Intermediate-Scale Quantum (NISQ) devices, avoiding the need for physics-inspired constructions like Trotterized Hamiltonian evolutions.
• Optimization Loop: The control task is reduced to minimizing a terminal cost function defined by the infidelity: $J(\theta) = 1 - |\langle \psi_{tar} | U(\theta) | \psi_{in} \rangle|^2$. This is solved via a hybrid quantum-classical loop where a classical optimizer (SLSQP) updates the circuit parameters $\theta$ based on fidelity evaluations performed on a quantum simulator.
• Theoretical Basis: The paper establishes that the variational formulation admits a global minimizer on compact parameter domains and that any unitary generated by the ansatz corresponds to a valid finite-horizon Hamiltonian evolution, ensuring the surrogate is physically realizable in principle.

Key Contributions

1. Control-Theoretic Formulation: The authors present a variational formulation of finite-horizon quantum control that utilizes hardware-efficient ansätze, shifting the search space from control functions to circuit parameters.
2. Surrogate Control Strategy: The work introduces a hardware-efficient surrogate control formulation where trainable circuit parameters replace explicit time-dependent control fields. This distinguishes the approach from Hamiltonian-inspired variational methods, offering a flexible, implementation-friendly alternative that does not require problem-specific ansätze.
3. Numerical Investigation: The paper provides a numerical analysis of the framework's convergence, expressivity, and robustness, specifically examining the trade-offs between circuit depth, system size, and optimization complexity.

Results
Numerical experiments were conducted on multi-qubit state-transfer benchmarks (transferring a single excitation from one end of a register to the other) using noiseless state-vector simulators and various Azure Quantum backends.

• System Size and Convergence: For small systems ($N=3$), the method achieved high fidelities ($>0.999$) with stable convergence. As system size increased ($N=5, 7$), optimization became more challenging, with final fidelities decreasing to approximately $0.996$ and $0.955$ respectively, reflecting the growth of the Hilbert space and the emergence of barren plateau phenomena.
• Circuit Depth vs. Expressivity: Increasing the circuit depth $L$ generally improved the achievable fidelity, particularly for larger systems ($N=7$), where deeper circuits ($L \ge 8$) were necessary to recover high-fidelity solutions. However, deeper circuits also increased the number of trainable parameters and the sensitivity to random initialization.
• Robustness and Initialization: The optimization showed stability for small and moderate system sizes. For larger systems ($N=7$), the solution distribution broadened, indicating multiple local minima and increased sensitivity to initialization, consistent with theoretical expectations for large-scale variational circuits.
• Hardware Compatibility: The optimized circuits were successfully executed on Azure Quantum simulator backends (IonQ, Rigetti, and Quantinuum H2 emulator). The results demonstrated that the optimized parameters transfer effectively across platforms, achieving target-state probabilities above $0.99$ on ideal simulators and $0.94$ on a noise-inclusive emulator.

Significance and Claims
The paper claims to provide a flexible and implementation-friendly approach to quantum control that is naturally compatible with near-term quantum devices. By avoiding reliance on problem-specific or physics-inspired ansätze, the method offers a generic framework for finite-horizon state transfer. The authors emphasize that while the approach successfully demonstrates high-fidelity state transfer for small-to-moderate multi-qubit systems, it highlights the inherent trade-off between ansatz expressivity and optimization complexity. The work is presented as an open-loop control framework; the authors modestly note that extending this to closed-loop control with intermediate measurements remains a direction for future research. The significance lies in demonstrating that hardware-efficient circuits can serve as effective variational surrogates for terminal evolution, bypassing the need for explicit control field synthesis in the optimization phase.