Abstract
The erasure of each bit of information by a computing device has an intrinsic energy cost. Although any Turing machine can be rewritten to be thermodynamically reversible without changing the recognized language, finite automata that are restricted to scan their input once in “real-time” fashion can only recognize the members of a proper subset of the class of regular languages in this reversible manner. We use a general quantum finite automaton model to study the thermodynamic cost per step associated with the recognition of different regular languages. We show that zero-error quantum finite automata have no energy cost advantage over their classical deterministic counterparts, and prove an upper bound for the cost that holds for all regular languages. We also demonstrate languages for which “error can be traded for energy”, i.e. whose zero-error recognition is associated with provably bigger energy cost per step when compared to their bounded-error recognition by real-time finite-memory quantum devices.
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Notes
- 1.
- 2.
“Zero-energy” QFAs with a single operation element in each superoperator correspond to the earliest definition in [3, 12], and can recognize all and only the group languages (a proper subclass of the class of regular languages, whose DFAs have the property that one again obtains a DFA if one reverses all their transitions) with bounded error [3, 4].
- 3.
The left end-marker is inconsequential in DFA simulations, and its superoperator is not shown.
- 4.
Since none of the three states with \(\texttt {b}\)-transitions into state 3 is more likely to be the source than the others, this information amounts to \(\log _2 3\) bits.
- 5.
The fact that zero-error QFAs have no advantage over equivalent DFAs in terms of the number of machine states was first proven by Klauck [6], using Holevo’s theorem and communication complexity arguments.
- 6.
At this point, one may be tempted to declare that the set of subspaces already provides the state set of the DFA we wish to construct. After all, each matrix of the form \(U_\sigma \) that we saw in Sect. 2 “maps” a vector in \(S_i\) to one or more vectors in \(S_j\) if and only if D switches from state i to state j upon consuming \(\sigma \). However, this simple construction does not guarantee our aim of keeping the maximum number of incoming transitions with the same label to any state in the machine to a minimum.
References
Ambainis, A., Yakaryılmaz, A.: Automata and quantum computing (2018). arXiv:1507.01988v2
Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)
Bertoni, A., Carpentieri, M.: Regular languages accepted by quantum automata. Inf. Comput. 165(2), 174–182 (2001)
Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata. SIAM J. Comput. 31, 1456–1478 (2002)
Hirvensalo, M.: Quantum automata with open time evolution. Int. J. Nat. Comput. 1, 70–85 (2010)
Klauck, H.: On quantum and probabilistic communication: Las Vegas and one-way protocols. In: 32th ACM Symposium on Theory of Computing, pp. 644–651 (2000)
Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proceedings 38th Symposium on Foundations of Computer Science, pp. 66–75 (1997)
Kutrib, M.: Aspects of reversibility for classical automata. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Computing with New Resources. LNCS, vol. 8808, pp. 83–98. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13350-8_7
Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961)
Leff, H.S., Rex, A.F. (eds.): Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing. CRC Press, Boca Raton (2002)
Li, L., Qiu, D., Zou, X., Li, L., Wu, L., Mateus, P.: Characterizations of one-way general quantum finite automata. Theoret. Comput. Sci. 419, 73–91 (2012)
Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theoret. Comput. Sci. 237(1), 275–306 (2000)
Pin, J.-E.: On reversible automata. In: Simon, I. (ed.) LATIN 1992. LNCS, vol. 583, pp. 401–416. Springer, Heidelberg (1992). https://doi.org/10.1007/BFb0023844
Say, A.C.C., Yakaryılmaz, A.: Quantum finite automata: A modern introduction. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Computing with New Resources. LNCS, vol. 8808, pp. 208–222. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13350-8_16
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Yılmaz, Ö., Kıyak, F., Üngör, M., Say, A.C.C. (2022). Energy Complexity of Regular Language Recognition. In: Caron, P., Mignot, L. (eds) Implementation and Application of Automata. CIAA 2022. Lecture Notes in Computer Science, vol 13266. Springer, Cham. https://doi.org/10.1007/978-3-031-07469-1_16
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