Fig. 2. Roles of three layers in M-cells and schematic of dynamics of two Turing machines.

[Top]: In the first part, a Turing machine, TM1, decodes a bit sequence u on the first layer through the estimation of the number of $∣1⟩$ in the second layer (step (i)). The relative frequency of 1 in the second layer is set to β, whose binary expansion is equal to u. The number of qubits TM1 should read is determined by the leftmost cell with 1 in the third layer. Here, we draw only M-cells and omit A-cells for visibility. [Bottom]: In the second part, a universal reversible Turing machine, TM2, runs with the input u (step (ii)). If TM2 halts, then TM3 starts to flip the state in the A-cells from a₁ to a₂ (step (iii)). If TM2 does not halt, the states in A-cells are not flipped. Note that we have not drawn the second and third layers of M-cells for visibility. In these figures, q_j, q_u, and r are examples of internal states of TM1, TM2, and TM3, respectively.