Minimum energy dissipation required for a logically irreversible operation

According to Landauer's principle, the minimum heat emission required for computing is linked to logical entropy, or logical reversibility. The validity of Landauer's principle has been investigated for several decades and was finally demonstrated in recent experiments by showing that the minimum heat emission is associated with the reduction in logical entropy during a logically irreversible operation. Although the relationship between minimum heat emission and logical reversibility is being revealed, it is not clear how much free energy is required to be dissipated for a logically irreversible operation. In the present study, in order to reveal the connection between logical reversibility and free energy dissipation, we numerically demonstrated logically irreversible protocols using adiabatic superconductor logic. The calculation results of work during the protocol showed that, while the minimum heat emission conforms to Landauer's principle, the free energy dissipation can be arbitrarily reduced by performing the protocol quasistatically. The above results show that logical reversibility is not associated with thermodynamic reversibility, and that heat is not only emitted from logic devices but also absorbed by logic devices. We also formulated the heat emission from adiabatic superconductor logic during a logically irreversible operation at a finite operation speed.

Figure 1

Schematic of the AQFP circuit used for numerical calculation. $L_1=L_2=6.58\mathrm{pH},L_{\mathrm{q}}=26.3\mathrm{pH}$. The critical currents of $J_1$ and $J_2$ are $I_{\mathrm{c}}=10\mu \mathrm{A}$. Since typical AQFP gates are designed symmetrically, $L_1=L_2=L,L_{\mathrm{x}1}=L_{\mathrm{x}2}=L_{\mathrm{x}}$, and $L_{\mathrm{d}1}=L_{\mathrm{d}2}=L_{\mathrm{d}}.I_{\mathrm{d}}$ decides the initial potential energy shape. $I_{\mathrm{x}}$ evolves the potential energy shape between a single well and a double well. $I_{\mathrm{in}}$ determines to which logic state the gate switches.

Figure 2

RT1 protocol. (a) Evolution of the potential energy of the AQFP gate. (b) Waveforms of the AQFP gate. At the initial stage (A), the logical entropy of the AQFP gate, H, is ln2, because the probability of being in logic 1 is equal to that of being in logic 0. Through forming a single-well potential in stage B, $H$ reduces to 0 in the final stage (C). This protocol is logically irreversible due to the reduction in logical entropy.

Figure 3

Numerical calculation results of the average heat emission, $-Q$, during the RT1 protocol. $-Q$ was calculated over 2000 iterations of the protocol at three different temperatures: 4.2, 2, and 1 K. The fitting curves are $-Q=2.89\times {10}^{-32}/{\tau }_{\mathrm{rf}}+k_{\mathrm{B}}T\ln 2$. As the rise and fall time of the excitation and input currents, ${\tau }_{\mathrm{rf}}$, increases, the average heat emission decreases and asymptotically approaches $k_{\mathrm{B}}T\ln 2$ in accordance with Landauer's principle.

Figure 4

Modified RT1 protocol. (a) Evolution of the potential energy of the AQFP gate. (b) Waveforms of the AQFP gate. The initial state of the AQFP gate at stage A is the same as the final state at stage E. An RT1 operation is performed between stages B and D.

Figure 5

Numerical calculation results of the free energy dissipation, $F_{\mathrm{diss}}$, during the modified RT1 protocol. $F_{\mathrm{diss}}$ was calculated over 2000 iterations of the modified protocol at three different temperatures: 4.2, 2, and 1 K. The fitting curve is $F_{\mathrm{diss}}=6.25\times {10}^{-32}/{\tau }_{\mathrm{rf}}$. As ${\tau }_{\mathrm{rf}}$ increases, the energy dissipation decreases without energy bounds.