Each step that results in a bit of information being ``forgotten'' 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 study the energy expenditure associated with the computations of deterministic and quantum finite automata. We prove that zero-error quantum finite automata have no advantage over their classical deterministic counterparts in terms of the maximum obligatory thermodynamic cost associated by any step during the recognition of different regular languages. We also demonstrate languages for which ``error can be traded for energy'', i.e. whose zero-error recognition is associated with computation steps having provably bigger obligatory energy cost when compared to their bounded-error recognition by real-time finite-memory quantum devices. We show that regular languages can be classified according to the intrinsic energy requirements on the recognizing automaton as a function of input length, and prove upper and lower bounds.

翻译：每一步导致计算设备“遗忘”一个比特信息的过程都伴随固有的能量消耗。尽管任何图灵机都可以在不改变所识别语言的前提下重写为热力学可逆形式，但以“实时”方式限制为单次扫描输入的有限自动机只能以这种可逆方式识别正则语言类的一个真子集。我们研究了确定性和量子有限自动机计算过程中的能量消耗。我们证明，在识别不同正则语言的任何步骤中，零错误量子有限自动机在最大必然热力学成本方面相对于经典确定性有限自动机并无优势。我们还展示了“误差可换取能量”的语言，即这些语言的零错误识别所涉及的计算步骤，其必然能量成本明显高于实时有限记忆量子设备进行有界误差识别时的成本。我们指出，正则语言可根据识别自动机（作为输入长度的函数）的固有能量需求进行分类，并证明了其上下界。