Adaptive time Compressed QITE (ACQ) and its geometrical interpretation

Imaginary Time Evolution (QITE) approximates this evolution on quantum hardware but suffers from high circuit depth and numerous measurements. In this work we introduce Adaptive-time Compressed QITE (ACQ), a novel algorithm that reduces resource-cost by combining adaptive time steps with circuit compression. This approach leverages geometric insights by characterizing its relationship to geodesic trajectories with a measure that distinguishes trajectories in $\mathbb{CP}^N$. Recalling that ITE is a gradient flow on the complex projective plane $\mathbb{CP}^N$, such trajectory measures allow one to measure the deviation from geodesicity of said flow. For rank-2 Hamiltonians, ITE and QITE exactly trace geodesics, this fact motivates an adaptive strategy for higher rank systems where QITE unitaries are reused until an energy increase signals departure from the ITE path. This is implemented via a line search for energy minimization. Circuit compression is achieved by approximating the sequence of QITE unitaries with a single element of a one-parameter group. Numerical simulations on the Transverse Field Ising Model demonstrate that ACQ achieves comparable fidelity to standard QITE while significantly reducing the number of QITE optimizations and maintaining fixed circuit depth during propagation. Gate-count estimates and an analysis of the fidelity scaling with truncation parameters are provided.