[Paper Review] Improved constructions of quantum automata

Andris Ambainis, Nikolajs Nahimovs|ArXiv.org|May 12, 2008

Quantum Computing Algorithms and Architecture7 references6 citations

TL;DR

This paper presents a simplified and more efficient quantum finite automaton (QFA) construction that achieves an exponential space advantage over classical finite automata. By combining multiple small quantum automata with randomized parameters, the proposed QFA recognizes the language $L_p = \{a^i \mid i \text{ divisible by } p\}$ using $\frac{4}{\epsilon}\log 2p + O(1)$ states, significantly improving the constant factor over prior work while maintaining high accuracy for non-memberships.

ABSTRACT

We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use \frac{4}ε \log 2p + O(1) states to recognize a language that requires p states classically. The construction is both substantially simpler and achieves a better constant in the front of \log p than the previously known construction of Ambainis and Freivalds (quant-ph/9802062). Similarly to Ambainis and Freivalds, our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction.

Motivation & Objective

• To develop a simpler and more efficient quantum finite automaton (QFA) that achieves an exponential space advantage over classical finite automata.
• To reduce the constant factor in the $O(\log p)$ state complexity compared to the prior construction by Ambainis and Freivalds (AF).
• To explore derandomization of the probabilistic construction while preserving correctness and efficiency.
• To provide provably correct, non-probabilistic QFA constructions for the same language, including one with $O(\log^{2+\epsilon}p)$ states.

Proposed method

• Construct a base QFA $U_k$ with two states that applies a rotation by angle $\frac{2\pi k}{p}$ for each input symbol $a$, using unitary transformations.
• Combine $d = \frac{4}{\epsilon}\log 2p$ such automata in parallel, each with a randomly chosen $k_i \in \{1,\dots,p-1\}$, to form a composite QFA.
• Use a left endmarker to initialize the system into a uniform superposition over the $d$ initial states $|q_{i,0}\rangle$, enabling interference across components.
• Apply the right endmarker to collapse the superposition to the first component $|q_{1,0}\rangle$, which is the only accepting state.
• Leverage quantum interference: acceptance probability for non-multiples of $p$ depends on the squared sum of cosines of $\frac{2\pi k_i j}{p}$, which concentrates near zero for random $k_i$.
• Provide two derandomized constructions: one numerically validated with $O(\log p)$ states, and one provably correct with $O(\log^{2+\epsilon}p)$ states.

Experimental results

Research questions

• RQ1 Can a simpler and more efficient quantum finite automaton be constructed that achieves an exponential space advantage over classical automata for the language $L_p$?
• RQ2 What is the minimal constant factor in front of $\log p$ for a QFA recognizing $L_p$ with high success probability?
• RQ3 Is it possible to derandomize the probabilistic construction while maintaining correctness and near-optimal state complexity?
• RQ4 Can a provably correct, non-probabilistic QFA be constructed for $L_p$ with $O(\log^{2+\epsilon}p)$ states?

Key findings

• The proposed QFA recognizes $L_p = \{a^i \mid i \text{ divisible by } p\}$ with $\frac{4}{\epsilon}\log 2p + O(1)$ states, achieving a better constant factor than the prior $16\log p$ construction.
• For inputs not in $L_p$, the acceptance probability is bounded by $\frac{1}{d^2}\left(\sum_{i=1}^d \cos\left(\frac{2\pi k_i j}{p}\right)\right)^2$, which is small with high probability due to random phase cancellation.
• The construction is significantly simpler than the prior probabilistic method of Ambainis and Freivalds, with a cleaner analysis and reduced complexity.
• A derandomized construction with $O(\log p)$ states is proposed, though its correctness is supported only by numerical evidence.
• A provably correct derandomized construction exists with $O(\log^{2+\epsilon}p)$ states for any $\epsilon > 0$, offering a trade-off between state count and determinism.
• The paper confirms that quantum automata can achieve an exponential space advantage over classical automata, with the new construction improving both efficiency and simplicity.