Pseudo-deterministic Quantum Algorithms

2026-02-19

AI Summary

This work investigates pseudodeterministic quantum algorithms—quantum procedures that, on any input, output a unique solution with high probability—and delineates their capabilities within the query complexity model. By introducing novel problems such as AOES and QL-Estimation and developing lower-bound techniques tailored to the pseudodeterministic setting, the study establishes the first provable separations between this model and classical randomized or deterministic algorithms. Leveraging quantum primitives like Grover search and element distinctness, the authors present a general method for efficiently transforming standard quantum search problems into pseudodeterministic form. The results demonstrate that, for certain problems, pseudodeterministic quantum algorithms achieve exponential speedups over their classical pseudodeterministic counterparts; however, for any total function, the quantum advantage is limited to at most a fifth-power polynomial improvement.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Privacy-enhancing technologies

Abstract

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.

Problem

• pseudo-deterministic quantum algorithms
• query complexity
• complexity separations
• quantum computing
• canonical solution

Innovation

• pseudo-deterministic quantum algorithms
• query complexity
• quantum-classical separation
• lower bound techniques
• Grover search