qSHIFT: An Adaptive Sampling Protocol for Higher-Order Quantum Simulation

Quantum simulation is a cornerstone application for quantum computing, yet standard methods face a trade-off between circuit depth and accuracy: Trotterization depth scales with the number of Hamiltonian terms L, while sampling-based qDRIFT is restricted to O\(t²\) error scaling. Here, We introduce qSHIFT, an adaptive sampling protocol that overcomes these limitations. By adaptively updating sampling distributions, qSHIFT maintains L-independent gate complexity while achieving an improved error scaling of O\(t^1+r\) for an adjustable parameter r. This performance is enabled by a classical subroutine solving L^r linear equations per sampling round. Numerical demonstrations confirm the O\(t^1+r\) scaling, showcasing qSHIFT as a resource-efficient framework for high-precision quantum simulation. Furthermore, the protocol's reduced circuit depth enhances its compatibility with physical error mitigation, making it a promising candidate for implementation on near-term quantum devices. In addition to its role as a standalone algorithm, qSHIFT can provide a high-precision foundation for modular quantum frameworks such as qSWIFT or Krylov quantum diagonalization.