Turing-equivalent automata using a fixed-size quantum memory

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.

💡 Research Summary

This paper introduces two novel quantum computational models that augment classical space‑bounded frameworks with a constant‑size quantum register. The first model, called quantum Arthur‑Merlin (qAM), is a public proof system derived from the classical Arthur‑Merlin protocol, while the second, quantum alternating Turing machine (qATM), extends the classical alternating Turing machine (ATM) with the same quantum resource. The authors focus on space‑bounded computation and obtain several striking results that demonstrate the extraordinary power of even a tiny quantum memory.

The first set of results concerns “weak” qAM systems, where the verifier is required to accept members of a language with high probability but is not required to reject non‑members with comparable confidence. The authors prove that for every Turing‑recognizable language L there exists a constant‑space weak‑qAM protocol that decides L. The construction uses a single qubit (or a constant number of qubits) as a quantum register. The prover encodes a computation trace into a sequence of unitary rotations applied to the register; the verifier interleaves these rotations with measurements that check consistency. Because quantum superposition and interference can compress arbitrarily long computation histories into a constant‑size quantum state, the verifier can simulate an unbounded classical computation while using only O(1) classical work tape cells.

The second result upgrades the model to a “strong” qAM, where both members and non‑members must be distinguished with high probability. The authors show that any space‑bounded classical private‑coin AM protocol can be simulated by a public qAM with exactly the same space bound. The simulation maps the private random bits of the prover onto the quantum register and lets the verifier measure them, thereby reproducing the same error probabilities as the original private protocol. Consequently, public quantum proofs are at least as powerful as their classical private counterparts, and in some cases strictly stronger because the quantum part can generate interference patterns that no classical bounded‑space protocol can emulate.

Turning to qATM, the paper proves that a constant‑space quantum alternating machine with a one‑way input head can recognize every Turing‑recognizable language. The key idea is to use the constant‑size quantum register as a “quantum counter” that can be incremented or decremented by unitary operations during state transitions. This counter enables the machine to encode an unbounded amount of configuration information despite having only O(1) classical cells. As a result, the one‑way qATM can simulate the full power of a classical ATM, which normally requires unbounded work tape space and a two‑way input head.

A further refinement introduces “strong” qATMs that are required to halt on every computation path (analogous to private ATMs). The authors demonstrate that such machines can simulate deterministic space‑S(n) computations using only O(log S(n)) quantum bits while still using O(1) classical space. This yields a precise shift of the deterministic space hierarchy by one level: any language that can be decided in deterministic space S(n) can be decided by a strong qATM in space O(log S(n)) plus a constant‑size quantum register. The proof constructs a simulation where the deterministic configuration is encoded into the amplitudes of the quantum register; each transition of the deterministic machine corresponds to a fixed unitary operation, and halting is guaranteed because the quantum state space is finite.

The technical core of all these results is a new public quantum protocol that cleverly exploits the fixed‑size quantum register. The protocol forces the prover to apply specific unitaries that generate interference patterns impossible to reproduce with any space‑bounded classical protocol. The authors formalize this impossibility by showing that any classical simulation would require more than constant space, thereby establishing a genuine quantum advantage in the space‑bounded regime.

In summary, the paper establishes that a constant‑size quantum memory dramatically enhances the computational power of space‑restricted models. It shows that both proof systems and alternating machines become Turing‑complete with only O(1) classical workspace, that strong quantum versions can simulate classical space‑bounded protocols without extra space, and that deterministic space hierarchies can be shifted exactly by one level using strong qATMs. These findings open new avenues for research on quantum complexity with limited memory, suggesting that even minimal quantum hardware could provide substantial computational benefits in memory‑constrained environments.