Kyle J. Ray and James P. Crutchfield |

**ABSTRACT: **
We introduce a fast and highly-efficient physically-realizable bit
swap. Employing readily available and scalable Josephson junction
microtechnology, the design implements the recently introduced
paradigm of momentum computing. Its nanosecond speeds and
sub-Landauer thermodynamic efficiency arise from dynamically
storing memory in momentum degrees of freedom. As such, during the
swap, the microstate distribution is never near equilibrium and
the memory-state dynamics fall far outside of stochastic
thermodynamics that assumes detailed-balanced Markovian dynamics.
The device implements a bit-swap operation—a fundamental
operation necessary to build reversible universal computing.
Extensive, physically-calibrated simulations demonstrate that
device performance is robust and that momentum computing can
support thermodynamically-efficient, high-speed, large-scale
general-purpose computing.

Kyle J. Ray and James P. Crutchfield, “Gigahertz Sub-Landauer Momentum Computing”, Physical Review Applied

doi:10.1103/PhysRevApplied.19.014049.

[pdf]

arxiv.org:2202.07122 [cond-mat.stat-mech].

**Selected animations of continuous-time thermodynamically-free bit swap:**

See Supplemental Material at [URL will be provided by publisher] for a suite of animations that showcase the action of the proposed flux qubit implementation of bit swap on the underlying state space.

Bit swap animations of 1000 trajectories sampled from the equilibrium distribution of the storage potential from four different projections. Color encodes the initial informational state of each trajectory, with red meaning '1' and orange meaning '0'.

- Top:
(left) The position space projection shows how the degrees of
freedom react to the potential energy surface as it transitions
between the computational and storage potential parameters,
depicted by the underlying contours. The energy imparted by the
change in potential is manifest in nonequilibrium oscillations of
the conjugate momenta (right). This momentum space projection most
clearly shows the departure and eventual return to equilibrium.
Note that the transition to the storage potential imparts large
oscillations in the ɸ
_{dc}dimension, because the double-well (V^{store}) and single-well (V^{comp}) profiles are centered about different ɸ_{dc}values. However, the bulk of these oscillations are re-absorbed when the storage potential is reinstated. - Bottom:
Marginal phase space projections in both the (left) computational
ɸ and (right) computationally passive ɸ
_{dc}dimensions. The left plot shows how the \varphi momenta serves as a transient memory by allowing the informational states to avoid crossing in phase space. On the right, we see that the ɸ_{dc}dimension is informationally mute—trajectories corresponding to both the '0' and '1' informational states lying on top of eachother in the marginal phase space. The oscillations in this informationally mute dimension are not necessary for the computation, but are an inevitable result of the control protocol when switches between the two potential energy landscapes. - Both: The device and protocol simulated for these animations do not represent a particularly efficient device. Instead, for illustrations purposes, a device was selected so that both the swap and the return to equilibrium could be easily seen at relatively similar timescales. For efficient devices, the particles end closer to an equilibrium distribution and the equilibration takes a couple orders of magnitude longer. Even when simulating this higher dissipation device, the time is sped up once the swap is accomplished to avoid an overly long video—an indicator that the dynamics of the swap really must happen much faster than thermal equilibration.