Turing-equivalent automata using a fixed-size quantum memory

TL;DR

This paper introduces quantum automata with fixed-size quantum memory, demonstrating they are Turing-equivalent under constant space and can simulate classical protocols, revealing new quantum computational capabilities.

Contribution

It presents the first quantum alternating Turing machine and quantum Arthur-Merlin proof system with fixed-size quantum memory, establishing their Turing-equivalence and superior simulation power.

Findings

Constant-space quantum systems recognize all Turing-recognizable languages.

Quantum protocols can simulate classical space-bounded protocols.

Quantum automata can exponentially reduce space requirements for certain computations.

Abstract

In this paper, we introduce a new public quantum interactive proof system and the first quantum alternating Turing machine: qAM proof system and qATM, respectively. Both are obtained from their classical counterparts (Arthur-Merlin proof system and alternating Turing machine, respectively,) by augmenting them with a fixed-size quantum register. We focus on space-bounded computation, and obtain the following surprising results: Both of them with constant-space are Turing-equivalent. More specifically, we show that for any Turing-recognizable language, there exists a constant-space weak-qAM system, (the nonmembers do not need to be rejected with high probability), and we show that any Turing-recognizable language can be recognized by a constant-space qATM even with one-way input head. For strong proof systems, where the nonmembers must be rejected with high probability, we show that the known space-bounded classical private protocols can also be simulated by our public qAM system with the same space bound. Besides, we introduce a strong version of qATM: The qATM that must halt in every computation path. Then, we show that strong qATMs (similar to private ATMs) can simulate deterministic space with exponentially less space. This leads to shifting the deterministic space hierarchy exactly by one-level. The method behind the main results is a new public protocol cleverly using its fixed-size quantum register. Interestingly, the quantum part of this public protocol cannot be simulated by any space-bounded classical protocol in some cases.