Tight bounds on quantum sample complexity and quantum query complexity have been known for various computational problems in the literature, whereas tight bounds on quantum time complexity (i.e., the size of quantum circuits) remain unresolved. In this paper, we provide a framework to establish lower bounds on the quantum time complexity for testing permutation-invariant properties of quantum states, via a reduction from quantum sample complexity. As an application, we obtain a series of matching lower bounds when given sample access to the input quantum states, including: 1. The SWAP test due to Buhrman, Cleve, Watrous, and de Wolf (Phys. Rev. Lett. 2001) is time-optimal to estimate the purity $\operatorname{tr}(ρ^2)$ and the inner product $\operatorname{tr}(ρσ)$. 2. The Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki, and Kwek (Phys. Rev. Lett. 2002) is time-optimal to estimate the high-order functionals $\operatorname{tr}(ρ^k)$. 3. The productness tester for multipartite pure states due to Harrow and Montanaro (J. ACM 2013) is time-optimal. 4. The LMR protocol due to Lloyd, Mohseni, and Rebentrost (Nat. Phys. 2014) is time-optimal to implement the reflection operator about a pure state. 5. The samplizer due to Wang and Zhang (IEEE Trans. Inf. Theory 2025) is time-optimal for pure states. 6. The estimator for pure-state trace distance and fidelity due to Wang and Zhang (ICALP 2026) is time-optimal. To the best of our knowledge, this is the first method that allows us to systematically establish tight lower bounds on quantum time complexity.

翻译：量子样本复杂度和量子查询复杂度的紧致下界在相关文献中已有多种计算问题的已知结果，而量子时间复杂度（即量子电路规模）的紧致下界仍悬而未决。本文提出一个框架，通过从量子样本复杂度归约，为检验量子态置换不变性质建立量子时间复杂度的下界。作为应用，当对输入量子态具有样本访问权限时，我们获得一系列匹配下界，包括：1. Buhrman、Cleve、Watrous 与 de Wolf（Phys. Rev. Lett. 2001）提出的SWAP检验在估计纯度 $\operatorname{tr}(ρ^2)$ 和内积 $\operatorname{tr}(ρσ)$ 时具有时间最优性；2. Ekert、Alves、Oi、Horodecki、Horodecki 与 Kwek（Phys. Rev. Lett. 2002）提出的Shift检验在估计高阶泛函 $\operatorname{tr}(ρ^k)$ 时具有时间最优性；3. Harrow 与 Montanaro（J. ACM 2013）提出的多体纯态乘积性检验器具有时间最优性；4. Lloyd、Mohseni 与 Rebentrost（Nat. Phys. 2014）提出的LMR协议在实现关于纯态的反射算子时具有时间最优性；5. Wang 与 Zhang（IEEE Trans. Inf. Theory 2025）提出的采样器（samplizer）对纯态具有时间最优性；6. Wang 与 Zhang（ICALP 2026）提出的纯态迹距离与保真度估计器具有时间最优性。据我们所知，这是首个能系统性地建立量子时间复杂度紧致下界的方法。