In recent years, there has been an increase in research focusing on methods to utilize quantum computers in computational dynamics simulations. A quantum computer, unlike a classical computer, utilizes quantum states governed by quantum mechanics to perform computations. Theoretical studies have demonstrated that gate-type quantum computers can outperform classical computers to solve linear equations and time evolution simulations. However, there are several issues that must be resolved to ensure the practical applicability of quantum computers in performing practical calculations. One of the most important of these is the input of classical data to the quantum computer. In mechanical simulations on a quantum computer, the parameters of the target systems, such as the material distribution and structural information, must be reflected in the quantum state. However, it is known that writing classical information one by one into a quantum state requires a number of quantum gates in exponential order relative to the required number of qubits, which would negate the advantage of a quantum computer against the classical one. Consequently, a method to input information in polynomial time is imperative, at least for the number of qubits. In this study, we demonstrate that a quantum state with an analytic functional representation can be implemented on a number of quantum gates in logarithmic order with respect to both the inverse of the error and the number of elements in the state. This is proved by our proposing method of constructing the tensor train (TT) of the quantum state with a logarithmically small maximum rank. Notably, we have successfully constructed a TT with logarithmic rank in a direct and concrete manner, without use of indirect TT construction methods such as TT-rounding and TT-cross. As a practical illustration of the proposed method, we present TT representations of various types of analytic functions.