Resource-Efficient Hadamard Test Circuits for Nonlinear Dynamics on a Trapped-Ion Quantum Computer

Resource-efficient, low-depth implementations of quantum circuits remain a promising strategy for achieving reliable and scalable computation on quantum hardware, as they reduce gate resources and limit the accumulation of noisy operations. Here, we propose a low-depth implementation of a class of Hadamard test circuits, complemented by the development of a parameterized quantum ansatz specifically tailored for variational algorithms that exploit the underlying Hadamard test framework. Our findings demonstrate a significant reduction in single- and two-qubit gate counts, suggesting a reliable circuit architecture for noisy intermediate-scale quantum (NISQ) devices. Building on this foundation, we tested our low-depth scheme to investigate the expressive capacity of the proposed parameterized ansatz in simulating nonlinear Burgers' dynamics. The resulting variational quantum states faithfully capture the shockwave feature of the turbulent regime and maintain high overlaps with classical benchmarks, underscoring the practical effectiveness of our framework. Furthermore, we evaluate the effect of hardware noise by modeling the error properties of real quantum processors and by executing the variational algorithm on a trapped-ion-based IBEX Q1 device. The outcomes of our demonstrations highlight the resilience of our low-depth scheme in the turbulent regime, consistently preparing high-fidelity variational states that exhibit strong agreement with classical benchmarks. Our work contributes to the advancement of resource-efficient strategies for quantum computation, offering a robust framework for tackling a range of computationally intensive problems across numerous applications.