We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism: - We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is $O(1)$ but is maximally hard for pseudo-deterministic quantum algorithms ($Ω(N)$ query complexity). - We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms ($O(\log(N))$ vs. $Θ(\sqrt{N})$), while the randomized query complexity is $O(1)$. Complementing these separations, we show that for any total problem $R$, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., $D(R) = \tilde O(psQ(R)^5)$. On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, $k$-sum, and graph collision.

翻译：我们系统性地研究了伪确定性量子算法。这类算法对于任意输入，都能以高概率输出一个规范解。聚焦于查询复杂度模型，我们的主要贡献包括以下复杂度分离结果，这些结果需要专门针对伪确定性设计的新下界技术：- 我们提出了一个名为"避免单加密字符串"的问题，其经典随机化查询复杂度为$O(1)$，但对伪确定性量子算法具有最大难度（查询复杂度为$Ω(N)$）。- 我们提出了量子锁定估计问题，其伪确定性量子算法相比经典伪确定性算法具有指数级加速优势（$O(\log(N))$对比$Θ(\sqrt{N})$），而随机化查询复杂度仅为$O(1)$。作为这些分离结果的补充，我们证明对于任何完全问题$R$，伪确定性量子算法相比确定性算法至多具有五次方优势，即$D(R) = \tilde O(psQ(R)^5)$。在算法层面，我们识别出一类可通过较小开销实现伪确定性的量子搜索问题，包括Grover搜索、元素互异性、三角形查找、$k$-和以及图碰撞问题。