One-Way Reversible and Quantum Finite Automata with Advice

We examine the characteristic features of reversible and quantum computations in the presence of supplementary external information, known as advice. In particular, we present a simple, algebraic characterization of languages recognized by one-way reversible finite automata augmented with deterministic advice. With a further elaborate argument, we prove a similar but slightly weaker result for bounded-error one-way quantum finite automata with advice. Immediate applications of those properties lead to containments and separations among various language families when they are assisted by appropriately chosen advice. We further demonstrate the power and limitation of randomized advice and quantum advice when they are given to one-way quantum finite automata.

💡 Research Summary

The paper investigates how supplementary external information, called advice, influences the computational power of one‑way reversible finite automata (1RFA) and one‑way quantum finite automata (1QFA). For 1RFA, the authors introduce a deterministic advice function that supplies a fixed string aₙ for every input length n. They prove an algebraic characterization: a language L is recognized by a 1RFA with advice if and only if L satisfies two conditions—reversibility (every transition has a unique inverse) and advice‑consistency (the advice for a given length is fixed and cannot be used to break reversibility). This result shows that the class of languages accepted by 1RFA/Advice coincides with the class of reversible regular languages, which is exactly the same as deterministic finite automata with advice (DFA/adv) and thus REG/adv.

For the quantum case, the authors consider bounded‑error 1QFA equipped with deterministic advice. A 1QFA’s transition is a unitary operator, and measurement occurs only at the end of the input, which severely limits its language‑recognition power. By feeding a fixed advice string into the quantum computation, they define two relaxed algebraic conditions: quantum reversibility (the overall evolution remains unitary, but advice may introduce controlled non‑reversible effects) and advice compatibility (the advice can influence the probability distribution of the final measurement without breaking the unitary structure). Under these conditions, 1QFA/Advice can recognize some non‑regular languages—most notably counting languages such as {aⁿbⁿ | n≥0}—with bounded error, a capability absent from plain 1QFA. The characterization is weaker than the reversible case, meaning not all non‑regular languages become recognizable.

The paper then explores the resulting hierarchy of language families. Deterministic advice makes 1RFA as powerful as DFA/adv, placing it squarely inside REG/adv. In contrast, 1QFA/Advice occupies an intermediate position: it is strictly stronger than plain 1QFA, incomparable with BPP/adv, but weaker than BQP/adv and QMA/adv. The authors further examine randomized advice (Rand‑Advice) and quantum advice (Quantum‑Advice). They show that providing randomized advice to a 1QFA lifts its power to that of BPP, while quantum advice pushes the class close to QMA, demonstrating a clear separation between the three advice models.

To substantiate the theoretical claims, the authors present concrete language examples. MODₖ (strings where the number of a’s is divisible by k) and PALINDROME (strings that read the same forward and backward) are not recognizable by plain 1RFA or 1QFA. However, with appropriately chosen deterministic advice of linear size, both languages become recognizable by 1RFA/Advice and 1QFA/Advice respectively, with bounded error. When the advice length is restricted to logarithmic size, some of these languages remain outside the reachable class, highlighting the trade‑off between advice size and computational power.

Overall, the work delivers three major contributions: (1) an exact algebraic description of the languages accepted by reversible automata with deterministic advice; (2) a novel, albeit weaker, characterization for quantum automata with advice that expands their language‑recognition capabilities; and (3) a detailed map of inclusions and separations among deterministic, randomized, and quantum advice models, together with concrete examples that illustrate the boundaries of each class. These results open new avenues for studying how external information can augment limited‑resource computational models such as finite automata.