Pseudo-deterministic Quantum Algorithms

TL;DR

This paper systematically studies pseudo-deterministic quantum algorithms, revealing complexity separations and limitations, and identifying classes of problems that can be made pseudo-deterministic efficiently.

Contribution

It introduces new complexity separations in the query model and characterizes the potential advantage and limitations of pseudo-deterministic quantum algorithms.

Findings

Classical randomized query complexity is O(1) for AOES but Omega(N) for pseudo-deterministic quantum algorithms.

Exponential speed-up of pseudo-deterministic quantum algorithms over classical for QL-Estimation.

Pseudo-deterministic quantum algorithms have at most a quintic advantage over deterministic algorithms.

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 ($\Omega(N)$ 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. $\Theta(\sqrt{N})$), 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.