Direct Construction of Quantized Tensor Trains With Small Rank for Laplace‐ or Gaussian‐Transformed Analytic Functions

ABSTRACT Preparing quantum states with desired amplitude distributions is a key bottleneck in the implementation of quantum linear and nonlinear dynamics solvers, including Linear Combination of Hamiltonian Simulation (LCHS) and Schrödingerization. We present a direct, closed‐form construction of Quantized Tensor Train (QTT) representations for two important classes of analytic functions—Laplace‐transformed and Gaussian‐transformed functions—which include the states required for implementing LCHS and Schrödingerization. Our approach removes the need for data‐driven QTT construction procedures, such as TT‐SVD or TT‐Cross, which often lack analytical rank‐accuracy guarantees. Instead, we utilize the analytic structure of these functions through a double‐exponential quadrature formula and truncated polynomial approximation. Using these techniques, we establish provable bounds: such Laplace‐transformed functions admit QTT representations with TT‐rank , while Gaussian‐transformed functions admit TT‐rank to achieve a target accuracy . Moreover, applying data‐driven rank minimization to our analytically constructed QTTs further reduces maximal TT‐ranks without sacrificing accuracy. The resulting pipeline provides analytically controlled, implementation‐ready QTTs, which effectively reduce the cost of state preparation in quantum algorithms and suggest a path toward closed‐form QTTs for broader classes of functions.