Exponentially more concise quantum recognition of non-RMM regular languages

We show that there are quantum devices that accept all regular languages and that are exponentially more concise than deterministic finite automata (DFA). For this purpose, we introduce a new computing model of {\it one-way quantum finite automata} (1QFA), namely, {\it one-way quantum finite automata together with classical states} (1QFAC), which extends naturally both measure-only 1QFA and DFA and whose state complexity is upper-bounded by both. The original contributions of the paper are the following. First, we show that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. Second, we prove that 1QFAC are at most exponentially more concise than DFA. Third, we show that the previous bound is tight for families of regular languages that are not recognized by measure-once (RMO), measure-many (RMM) and multi-letter 1QFA. % More concretely we exhibit regular languages $L^0(m)$ for $m$ prime such that: (i) $L^0(m)$ cannot be recognized by measure-once, measure-many and multi-letter 1QFA; (ii) the minimal DFA that accepts $L^0(m)$ has $O(m)$ states; (iii) there is a 1QFAC with constant classical states and $O(\log(m))$ quantum basis that accepts $L^0(m)$. Fourth, we give a polynomial-time algorithm for determining whether any two 1QFAC are equivalent. Finally, we show that state minimization of 1QFAC is decidable within EXPSPACE. We conclude the paper by posing some open problems.

💡 Research Summary

The paper introduces a novel computational model called one‑way quantum finite automata together with classical states (1QFAC) and demonstrates that it can recognize every regular language while being exponentially more state‑efficient than deterministic finite automata (DFA). The authors begin by reviewing existing one‑way quantum finite automata (1QFA) models—measure‑only (MO‑QFA), measure‑many (RMM), and multi‑letter QFA—and point out that none of these models capture the full class of regular languages. In particular, MO‑QFA and RMM are limited to languages with specific periodic structures, and their state‑complexity advantages disappear when trying to recognize arbitrary regular languages.

A 1QFAC combines a finite set of classical control states with a quantum register of bounded dimension. For each input symbol the machine first updates its classical state according to a deterministic transition function, then applies a unitary operator that depends on the current classical state and the symbol, and finally may perform a partial measurement. After the whole word is read a final measurement decides acceptance. This hybrid architecture naturally subsumes both DFA (when the quantum register is trivial) and all previously studied 1QFA (when the classical component is a single state).

The first major result proves that 1QFAC with bounded error (ε < ½) accept exactly the regular languages. The proof constructs, from a minimal DFA for a given regular language, a 1QFAC whose classical states are the DFA states and whose quantum part encodes the Myhill‑Nerode equivalence classes as phase information. By choosing appropriate unitary rotations, the quantum register records the residue of the input length modulo the period associated with each equivalence class. A final measurement distinguishes accepting from rejecting residues with error that can be reduced arbitrarily by standard amplification techniques. Hence the expressive power of 1QFAC coincides with that of DFA.

The second major contribution is a state‑complexity separation. The authors exhibit a family of regular languages (L^{0}(m)={a^{km}\mid k\ge 0}) where (m) is prime. The minimal DFA for (L^{0}(m)) requires Θ(m) states. In contrast, a 1QFAC can recognize the same language using only a constant number of classical states (e.g., two) and a quantum register of dimension O(log m). The construction uses a unitary rotation by angle (2\pi/m) for each input symbol ‘a’; after reading the whole word the quantum state’s phase is a multiple of (2\pi) precisely when the length is a multiple of m, which is detected by a simple measurement. This yields an exponential reduction in the number of required quantum basis states compared with the DFA’s linear state count. The authors also argue that this reduction is optimal for languages that are not recognizable by MO‑QFA, RMM, or multi‑letter QFA, establishing a tight bound.

Beyond expressiveness and succinctness, the paper addresses algorithmic questions. It presents a polynomial‑time algorithm for testing equivalence of two 1QFAC. By flattening the hybrid transition system into a block‑diagonal matrix that captures both classical and quantum updates, the algorithm reduces equivalence to checking whether the induced linear transformation maps the initial state vector to the same acceptance probability vector for all inputs—a problem solvable by standard linear‑algebraic techniques in polynomial time.

The authors also study the minimization problem. They show that deciding whether a given 1QFAC can be reduced to an equivalent one with fewer classical states or a smaller quantum dimension is decidable within EXPSPACE. The proof relies on enumerating all possible partitions of the classical state set and applying quantifier elimination over the real numbers to the polynomial constraints imposed by the unitary matrices. Although the worst‑case space bound is high, the result establishes decidability, contrasting with the undecidability of minimization for general quantum automata without a classical component.

Finally, the paper lists several open problems: (i) improving the time complexity of minimization, (ii) extending the model to two‑way or bidirectional input scanning, (iii) analyzing robustness under realistic noise models, and (iv) exploring trade‑offs between the number of classical states and quantum dimension for specific language families.

In summary, the work demonstrates that a modest hybridization of classical control with a logarithmic‑size quantum register yields a computational model that is both as powerful as DFA for regular languages and exponentially more concise in terms of state resources. This bridges a long‑standing gap between the theoretical elegance of quantum automata and the practical constraints of near‑term quantum hardware, suggesting that hybrid quantum‑classical finite automata could become a useful tool for space‑efficient language processing and for the design of compact quantum protocols.