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

Resource-efficient quantum circuits remain a promising strategy for achieving scalable computations on quantum hardware. In this study, we propose a low-depth implementation of a class of Hadamard test circuits, and a parameterized ansatz tailored to the 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. We implement this low-depth scheme to examine the expressivity of the proposed ansatz for nonlinear Burgers' dynamics. The resulting variational quantum states faithfully capture the shockwave features of the turbulent regime while maintaining high overlaps with classical benchmarks. Furthermore, we evaluate the effect of hardware noise by executing the variational algorithm on a trapped-ion-based IBEX Q1 device. Our results highlight the resilience of the low-depth scheme, consistently producing high-fidelity variational states in strong agreement with classical benchmarks. This 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.