Estimating vibrational entropy is a significant challenge in thermodynamics and statistical mechanics due to its reliance on quantum mechanical properties. This paper introduces a quantum algorithm designed to estimate vibrational entropy via energy derivatives. Our approach block encodes the exact expression for the second derivative of the energy and uses quantum linear systems algorithms to deal with the reciprocal powers of the gaps that appear in the expression. We further show that if prior knowledge about the values of the second derivative is used then our algorithm can $ε$-approximate the entropy using a number of queries that scales with the condition number $κ$, the temperature $T$, error tolerance $ε$ and an analogue of the partition function $\mathcal{Z}$, as $\widetilde{O}\left(\frac{\mathcal{Z}κ^2 }{εT}\right)$. We show that if sufficient prior knowledge is given about the second derivative then the query scales quadratically better than these results. This shows that, under reasonable assumptions of the temperature and a quantum computer can be used to compute the vibrational contributions to the entropy faster than analogous classical algorithms would be capable of. Our findings highlight the potential of quantum algorithms to enhance the prediction of thermodynamic properties, paving the way for advancements in fields such as material science, molecular biology, and chemical engineering.

翻译：振动熵的估计是热力学与统计力学中的一个重要挑战，因其依赖于量子力学特性。本文提出一种通过能量导数估计振动熵的量子算法。我们的方法对能量的二阶导数精确表达式进行块编码，并利用量子线性系统算法处理表达式中出现的能隙倒数幂次。进一步证明，若利用关于二阶导数值的先验知识，算法可通过查询次数以条件数κ、温度T、误差容限ε及配分函数类比量Z为参数，按Õ(Zκ²/(εT))的复杂度实现对熵的ε近似估计。研究表明，若对二阶导数具备充分先验知识，查询复杂度可较上述结果实现二次方级改进。这表明在温度合理性假设下，量子计算机计算熵的振动贡献可超越经典算法的计算速度。本研究成果凸显了量子算法在提升热力学性质预测能力方面的潜力，为材料科学、分子生物学及化学工程等领域的进展开辟了新路径。