# A Shortcut to Finite-time Memory Erasure

Geng Li  Graduate School of China Academy of Engineering Physics, Beijing
100193, China  School of Systems Science, Beijing Normal University, Beijing
100875, China  Hui Dong<EMAIL_ADDRESS>Graduate School of China Academy
of Engineering Physics, Beijing 100193, China

###### Abstract

To achieve fast computation, it is crucial to reset the memory to a desired
state within a limited time. However, the inherent delay in the system’s
response often prevents reaching the desired state once the control process is
completed in finite time. To address this challenge, we propose a shortcut
strategy that incorporates an auxiliary control to guide the system towards an
equilibrium state that corresponds to the intended control, thus enabling
accurate memory reset. Through the application of thermodynamic geometry, we
derive an optimal shortcut protocol for erasure processes that minimizes the
energy cost. This research provides an effective design principle for
realizing the finite-time erasure process while simultaneously reducing the
energy cost, thereby alleviating the burden of heat dissipation.

_Introduction._ – Memory erasure is an essential step in computations with an
unavoidable energy cost. Landauer’s principle posts a fundamental lower bound
on the energy cost of erasing a one-bit memory carried out infinitely slowly
(Landauer, 1961; Bennett, 1973; Parrondo  _et al._ , 2015). However
accelerating computing processes typically requires to complete memory erasure
in finite time. There is an inevitable trade-off between erasure speed and
accuracy when rapidly initializing the memory system. In such a finite-time
thermodynamic process, a system often does not immediately respond to external
perturbation, resulting in a lag between the final state and the desired
equilibrium state (Pearlman and Kollman, 1989; Wood, 1991). This systematic
state lag poses a significant challenge in initializing the system to the
target state promptly (Dillenschneider and Lutz, 2009; Bérut  _et al._ , 2012;
Jun  _et al._ , 2014; Bérut  _et al._ , 2015; Proesmans  _et al._ , 2020a, b;
Boyd  _et al._ , 2022), and also leads to a substantial increase in energy
costs when the erasure time is reduced (Kawai  _et al._ , 2007; Gomez-Marin
_et al._ , 2008; Vaikuntanathan and Jarzynski, 2009; Horowitz and Jarzynski,
2009; Diana  _et al._ , 2013; Zulkowski and DeWeese, 2014, 2015; Dago  _et
al._ , 2021; Vu and Saito, 2022; Ma  _et al._ , 2022; Rolandi and Perarnau-
Llobet, 2022, 2023). The quest to achieve rapid erasure with minimal energy
costs drives us to design alternative memory erasure strategies.

Much effort has been devoted to reduce the state lag developed in
nonequilibrium driving processes (Miller and Reinhardt, 2000; Jarzynski, 2002;
Vaikuntanathan and Jarzynski, 2008; Minh, 2009; Martínez  _et al._ , 2016;
Patra and Jarzynski, 2017; Li  _et al._ , 2017). As a promising candidate,
shortcut to isothermality was proposed as a finite-time driving strategy to
escort the system evolving along a series of instantaneous equilibrium states
(Li  _et al._ , 2017) with the application to realize fast transitions between
equilibrium states (Albay  _et al._ , 2019, 2020), improve free energy
calculations (Li and Tu, 2019, 2021), and design Brownian heat engines
(Martínez  _et al._ , 2017; Plata  _et al._ , 2020; Chen, 2022; Zhao  _et al._
, 2022). As shown in Fig. 1, we adopt such shortcut strategy to design a
finite-time erasure protocol to reset a classical memory. And the
thermodynamic geometric approach (Li  _et al._ , 2022, 2023; Chittari and Lu,
2023) is used to design the optimal protocol with the minimal energy cost.

Figure 1: Schematic of finite-time memory erasure. The one-bit memory is
modeled by a physical double-well system. In the initial state, the system
stores the state’s information $``0"$ or $``1"$. The task of memory erasure is
to reset the system into the blank state $``1"$. (a) Conventional memory
erasure scheme with the Hamiltonian $H_{o}$. The double-well potential is fast
switched to fulfill classical memory reset in finite time $\tau$. A systematic
state lag is accumulated between the system’s current state and its
corresponding equilibrium state. Therefore, a slow relaxation process is
needed to reach the desired erasure accuracy. (b) Shortcut to memory erasure
with the Hamiltonian $H=H_{o}+H_{a}$. In the step I, an auxiliary Hamiltonian
$H_{a}$ is added to escort the system evolving along the instantaneous
equilibrium path of the original Hamiltonian $H_{o}$ within finite time. The
system is finally expelled into the blank state $``1"$ without accumulation of
the state lag. In the step II, the potential is quenched to the double-well
form for later computations.

_Shortcuts to memory erasure._ – The memory as a simple binary system can be
simplified as a particle in a bistable potential well. Here, we consider a
one-bit memory system modeled by a Brownian particle in a double-well
potential $U_{o}(x,\vec{\lambda})=kx^{4}-A\lambda_{1}x^{2}-B\lambda_{2}x$,
where $x$ represents the coordinate. And $k$, $A$, and $B$ are constant
coefficients introduced to define the dimensionless variables
$\vec{\lambda}(t)\equiv(\lambda_{1},\lambda_{2})$ as time-dependent control
parameters. The evolution of the system is described by Hamiltonian
$H_{o}(x,p,\vec{\lambda})\equiv p^{2}/(2m)+U_{o}(x,\vec{\lambda})$ with $p$ as
the momentum and $m$ as the mass of the particle. The memory is encoded by
mapping the microstate $x$ into two macrostates. If the particle is in the
left well $(x<0)$, the system is in macrostate “$0$”. Conversely, if the
particle is in the right well $(x>0)$, the system is in macrostate “$1$”,
which serves as the blank state for storing memory.

The system is in contact with a thermal reservoir with a constant temperature
$T$. The barrier separating the double well is assumed to be much larger than
the thermal fluctuation so that the memory can be considered stable
(Dillenschneider and Lutz, 2009; Jun  _et al._ , 2014).

The particle is initially in an equilibrium microstate with the control
parameters $\vec{\lambda}(0)=(1,0)$ and has the same probability $1/2$ to stay
in each macrostate ($0$ or $1$). The entropy is $S=k_{B}\ln 2$ for the initial
microstate of the particle with $k_{B}$ being the Boltzmann constant. During
the Landauer’s quasi-static erasure process (Landauer, 1961; Bennett, 1973;
Parrondo  _et al._ , 2015), the particle is expelled into macrostate “$1$” by
changing control parameters $\vec{\lambda}$ slowly to maintain the equilibrium
state $P_{\mathrm{eq}}=\exp[\beta(F-H_{o})]$, where
$F\equiv-\beta^{-1}\ln[\iint dxdp\exp(-\beta H_{o})]$ is the free energy with
the inverse temperature $\beta\equiv 1/(k_{B}T)$. And the entropy for the
final state is $S=0.$ Such reduction of the entropy makes the memory erasure a
logically irreversible process. Once the control parameters $\vec{\lambda}$
are tuned with finite rate as illustrated in Fig. 1 (a), the system is driven
to a nonequilibrium state typically with a lag, resulting in a residual
entropy $S>0$. And additional relaxation is required to achieve the desired
accuracy with the sacrifice of longer time (Maruyama  _et al._ , 2009;
Parrondo  _et al._ , 2015; Bechhoefer, 2020).

To meet the need of fast erasure and reduce the lag, we adopt the shortcut
scheme where an auxiliary Hamiltonian $H_{a}(x,p,t)$ is supplemented to escort
the system evolving along the instantaneous equilibrium state
$P_{\mathrm{eq}}$ during the finite-time erasure process with boundary
conditions $H_{a}(0)=H_{a}(\tau)=0$. The probability distribution $P(x,p,t)$
of the microstate follows the Kramers equation,

$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial x}(\frac{\partial
H}{\partial p}P)+\frac{\partial}{\partial p}(\frac{\partial H}{\partial
x}P+\gamma\frac{\partial H}{\partial
p}P)+\frac{\gamma}{\beta}\frac{\partial^{2}P}{\partial p^{2}}],$ (1)

where $H\equiv H_{o}+H_{a}$ is the total Hamiltonian and $\gamma$ is the
dissipation coefficient. Our operation of memory erasure, illustrated in Fig.
1 (b), consists two steps as follows.

Step I, Shortcut Erasure. The particle is expelled to the right well $(x>0)$
in the finite-time interval $t\in[0,\tau]$, by raising the left well,
illustrated as the step I in Fig. 1 (b). The control parameters are tuned from
$\vec{\lambda}(0)$ to $\vec{\lambda}(\tau)=(0,1)$. With the strategy of
shortcuts to isothermality (Li  _et al._ , 2017), the system evolves along the
path of instantaneous equilibrium states, and reaches the final equilibrium
state $P=P_{\mathrm{eq}}(\tau)$ with the control parameters
$\vec{\lambda}(\tau)$. The requirement for the auxiliary Hamiltonian $H_{a}$
follows as

$\frac{\gamma}{\beta}\frac{\partial^{2}H_{a}}{\partial
p^{2}}-\text{$\frac{\gamma p}{m}$}\frac{\partial H_{a}}{\partial
p}+\frac{\partial H_{a}}{\partial p}\frac{\partial H_{o}}{\partial
x}-\frac{p}{m}\frac{\partial H_{a}}{\partial x}=\frac{dF}{dt}-\frac{\partial
H_{o}}{\partial t}.$ (2)

The auxiliary Hamiltonian is proved to have the form
$H_{a}(x,p,t)=\dot{\vec{\lambda}}\cdot\vec{f}(x,p,\vec{\lambda})$ with the
boundary condition $\dot{\vec{\lambda}}(0)=\dot{\vec{\lambda}}(\tau)=0$.

Step II, Potential Quench. For later computation purpose, the potential is
reset to the double-well form. Such operation is realized by quenching the
control parameters from $\vec{\lambda}(\tau^{-})=(0,1)$ to the initial value
$\vec{\lambda}(\tau^{+})=(1,0)$ at the time $t=\tau$ with an instantaneous
change of the system Hamiltonian $H_{o}(\tau^{-})\to H_{o}(\tau^{+})$. Here,
$\tau^{-}$ and $\tau^{+}$ denote the time before and after the potential
quench respectively.

After these two steps, the memory is erased to the blank state and the system
is reset within finite time $\tau$ to allow the later usage.

_Geometric erasure protocol._ – Energy cost is inevitable in a finite-time
erasure process. We employ the geometric approach (Li  _et al._ , 2022, 2023)
to derive the optimal erasure protocol with minimal energy costs. In the
shortcut scheme with the total Hamiltonian $H=H_{o}+H_{a}$, the work performed
in the erasure process with duration $\tau$ is
$W\equiv\langle\int_{0}^{\tau}\partial H/\partial tdt\rangle_{\mathrm{eq}},$
which is explicitly obtained as

$W_{s}=\Delta F+\gamma\int_{0}^{\tau}dt\langle(\frac{\partial
H_{\mathrm{a}}}{\partial p})^{2}\rangle_{\mathrm{eq}},$ (3)

where $\langle\cdot\rangle_{\mathrm{eq}}\equiv\iint
dxdp[\cdot]P_{\mathrm{eq}}$. Since the potential quench in the Step II is
realized instantaneously, the system distribution remains unchanged. And the
work done in this process follows as $W_{q}=\iint
dxdp(H_{o}(\tau^{+})-H_{o}(\tau^{-}))P_{\mathrm{eq}}(\tau)$.

With the explicit form of the auxiliary Hamiltonian
$H_{a}=\dot{\vec{\lambda}}\cdot\vec{f}$, the irreversible energy cost of the
erasure process is written as

$\displaystyle W_{\mathrm{irr}}$ $\displaystyle\equiv$ $\displaystyle W-\Delta
F-W_{q}$ (4) $\displaystyle=$
$\displaystyle\gamma\sum_{\mu\nu}\int_{0}^{\tau}dt\dot{\lambda}_{\mu}\dot{\lambda}_{\nu}\langle\frac{\partial
f_{\mu}}{\partial p}\frac{\partial f_{\nu}}{\partial p}\rangle_{\mathrm{eq}},$

with the total work $W=W_{s}+W_{q}.$ Here $W_{q}$ is excluded from the
irreversible energy cost $W_{\mathrm{irr}}$ in Eq. (4) since the potential
quench in the Step II is a reversible process. In the space of the control
parameters $\vec{\lambda}$, a semi-positive metric can be defined as
$g_{\mu\nu}\equiv\gamma\langle\partial f_{\mu}/\partial p\partial
f_{\nu}/\partial p\rangle_{\mathrm{eq}}$ on a Riemannian manifold. The
shortest curve connecting two given endpoints in this parametric space is the
geodesic line with the distance described by the thermodynamic length (Salamon
and Berry, 1983; Crooks, 2007; Sivak and Crooks, 2012; Chen  _et al._ , 2021)
$\mathcal{L}\equiv\int_{0}^{\tau}dt\sqrt{\sum_{\mu\nu}\dot{\lambda}_{\mu}\dot{\lambda}_{\nu}g_{\mu\nu}}$,
which provides a lower bound for the irreversible energy cost
$W_{\mathrm{irr}}\geq\mathcal{L}^{2}/\tau.$ Therefore, the geodesic line
serves as the optimal erasure protocol with minimal energy costs.

In the shortcut scheme, the auxiliary Hamiltonian $H_{a}$ can be solved from
Eq. (2) with the given original Hamiltonian $H_{o}$. However, the form of the
auxiliary Hamiltonian $H_{a}$ generally depends on the particle’s momentum (Li
_et al._ , 2017; Li and Tu, 2021; Li  _et al._ , 2023). The demand of
constantly monitoring the particle’s velocity makes it difficult for
implementation of momentum-dependent terms in experiment (Guéry-Odelin  _et
al._ , 2019, 2023). Here we design a variational auxiliary control
$H_{a}^{*}=\dot{\vec{\lambda}}\cdot\vec{f}^{*}(x,p,\vec{\lambda})$ that is
used to replace the exact auxiliary Hamiltonian $H_{a}$, where $\vec{f}^{*}$
represents an approximation to the function $\vec{f}$ to ensure the
minimization of a variational functional as

$\displaystyle\mathcal{G}(H_{a}^{*})$ $\displaystyle=$ $\displaystyle\int
dxdp(\frac{\gamma}{\beta}\frac{\partial^{2}H_{a}^{*}}{\partial
p^{2}}-\frac{\gamma p}{m}\frac{\partial H_{a}^{*}}{\partial p}+\frac{\partial
H_{o}}{\partial x}\frac{\partial H_{a}^{*}}{\partial p}$ (5)
$\displaystyle-\frac{p}{m}\frac{\partial H_{a}^{*}}{\partial x}+\frac{\partial
H_{o}}{\partial t}-\frac{\mathrm{d}F}{\mathrm{d}t})^{2}\mathrm{e}^{-\beta
H_{o}}.$

Such variational functional is defined to ensure the minimum discrepancy
between evolutions governed by the approximate Hamitonian $H_{a}^{*}$ and the
exact Hamitonian $H_{a}$. And the variational auxiliary Hamiltonian for the
Brownian particle takes the form
$H_{\mathrm{a}}^{*}=\text{$\sum_{\mu=1}^{2}\dot{\lambda}_{\mu}f_{\mu}^{*}(x,p,\vec{\lambda})$}$
with $f_{1}^{*}=a_{4}xp+a_{3}p+a_{2}x^{2}+a_{1}x,$ and
$f_{2}^{*}=b_{4}xp+b_{3}p+b_{2}x^{2}+b_{1}x$. Here $a_{n}\equiv
a_{n}(\vec{\lambda})$ and $b_{n}\equiv b_{n}(\vec{\lambda})$ with $n=1,2,3,4$
are functions to be determined through the variational procedure. With the
adoption of the shortcut scheme under the total Hamiltonian
$H=H_{o}+H_{a}^{*}$, the system can be escorted along a series of near-
equilibrium state $P_{\mathrm{eq}}^{*}\approx P_{\mathrm{eq}}$.

With the operation of a gauge transformation $X=x$ and $P=p+m\partial
H_{a}^{*}/\partial p$, we obtain an equivalent process

$\dot{X}=\frac{P}{m},\ \dot{P}=-\frac{\partial U_{o}}{\partial
X}-\frac{\partial U_{a}}{\partial X}-\gamma\dot{X}+\xi(t),$ (6)

where $\xi(t)$ is the Gaussian white noise and the auxiliary potential
$U_{a}(X,t)=C_{2}(t)X^{2}+C_{1}(t)X$ is momentum-independent. Detailed
derivations of the equivalent process in Eq. (6) and the lengthy expressions
of $C_{1}(t)$ and $C_{2}(t)$ are presented in Supplemental Material (Sup, ).
In this equivalent process, the system’s distribution follows a fixed pattern

$P_{f}(X,P,t)=\exp\left\\{\beta[F-\frac{1}{2m}(P-m\partial H_{a}^{*}/\partial
P)^{2}-U_{o}]\right\\}.$ (7)

With the boundary conditions
$\dot{\vec{\lambda}}(0)=\dot{\vec{\lambda}}(\tau)=0$, the extra term
$m\partial H_{a}^{*}/\partial P$ in Eq. (7) vanishes and the system’s
distribution $P_{f}$ returns to the instantaneous equilibrium distribution
$P_{\mathrm{eq}}^{*}$ at the beginning $t=0$ and end $t=\tau$ of the driving
process.

Figure 2: The erasure accuracy $\epsilon$ of the shortcut scheme (red
triangles) and the conventional erasure scheme (blue circles). The erasure
accuracy is defined as the probability of particles ending in the right well
$\epsilon\equiv\int_{0}^{\infty}P(x,\tau)dx$. During the erasure process, the
shortcut scheme adopts the Hamiltonian $H=H_{o}+U_{a}$ while the conventional
scheme takes the Hamiltonian $H^{\prime}=H_{o}$. In the simulation, we choose
the parameters as $k=4$, $A=8$, $B=16$, $k_{B}T=1$, $\gamma=1$, and $m=0.01$
and the straightforward control protocol as
$\lambda_{1}^{s}(t)=0.5+0.5\cos(\pi t/\tau)$ and
$\lambda_{2}^{s}(t)=0.5-0.5\cos(\pi t/\tau)$. The shortcut scheme always
ensures high erasure accuracy while the conventional erasure scheme only
achieves considerable erasure accuracy at long erasure duration.

_Reset errors._ – The conventional approach of memory erasure is to tune the
system Hamiltonian $H_{o}$ via control parameters $\vec{\lambda}$. The memory
state evolves accordingly yet with a lag, which induces reset errors
especially for the case with short erasure time $\tau$. The shortcut scheme
with the Hamiltonian $H=H_{o}+U_{a}$ effectively fulfills the demands for both
erasure speed and accuracy, without introducing any additional freedom of
control.

We numerically obtain the shortcut scheme with the Hamiltonian $H=H_{o}+U_{a}$
and compare it with the conventional scheme with the Hamiltonian
$H^{\prime}=H_{o}$ in a one-bit memory erasure process. In simulations, we
choose the parameters as $k=4$, $A=8$, $B=16$, $k_{B}T=1$, $\gamma=1$, and
$m=0.01$. See Supplemental Material for more details about the simulation
(Sup, ). A straightforward control protocol is selected as
$\lambda_{1}^{s}(t)=0.5+0.5\cos(\pi t/\tau)$ and
$\lambda_{2}^{s}(t)=0.5-0.5\cos(\pi t/\tau)$ to continuously meet the boundary
conditions. The erasure accuracy is defined as the relative number of
particles reaching the right well at the end of the erasure process
$\epsilon\equiv\int_{0}^{\infty}P(x,\tau)dx$. Figure 2 shows the erasure
accuracy $\epsilon$ for different erasure durations $\tau$. The accuracy of
the shortcut scheme is superior to that of the conventional scheme over
different durations. Especially in short erasure time, the conventional scheme
almost fails while the accuracy of the shortcut scheme still maintains high
level.

Figure 3: (a) Geodesic protocols for the memory erasure process with the
shortcut scheme. The control parameters vary form the initial value
$\vec{\lambda}(0)=(1,0)$ to the final value $\vec{\lambda}(\tau)=(0,1)$. The
dashed lines represent the straightforward protocol while the solid lines
represent the geodesic protocol. The geodesic protocol is obtained by solving
the geodesic equation corresponding to the metric $g_{\mu\nu}$. (b) The
irreversible energy cost of the straightforward protocol (red triangles) and
the geodesic protocol (green squares). The energy cost of erasure processes
given by the geodesic protocol is lower than that from the straightforward
protocol.

_Energy cost optimization._ – In the shortcut scheme, the auxiliary
Hamiltonian $H_{a}^{*}$ is obtained once the control protocol
$\vec{\lambda}(t)$ with $t\in[0,\tau]$ is specified for the conventional
erasure process. Among these protocols, one with the minimum energy cost can
be found with our geometric methods. We test the geometric approach for
minimizing the energy cost of the erasure process. The geodesic protocol is
obtained by numerically solving the geodesic equation in the parametric space
$\vec{\lambda}$ with the metric $g_{\mu\nu}$. The details on the form of the
metric and the geodesic equation are presented in the Supplementary Materials
(Sup, ). Figure 3 (a) shows the difference between the geodesic protocol and
the above straightforward protocol $\vec{\lambda}^{s}(t)$. In Fig. 3 (b), we
compare the irreversible energy cost of the geodesic protocol (green squares)
with that of the straightforward protocol (red triangles). The irreversible
energy cost $W_{\mathrm{irr}}$ given by the geodesic protocol is lower than
that from the straightforward protocol. Such observation proves the geometric
approach for the shortcut scheme to be effective. With the application of
tools from Riemannian geometry (Berger, 2007), the procedure of searching the
optimal erasure protocol is largely simplified with our two-step processes.

_Conclusions._ – In summary, we have developed a shortcut strategy to realize
a finite-time one-bit memory erasure with minimal energy costs. In the
shortcut scheme, an auxiliary control have been introduced to steer the system
evolving along the path of instantaneous equilibrium states. We have employed
the variational procedure to remove the momentum-dependent terms and derive an
accessible auxiliary control. The irreversible energy cost of the erasure
process have been minimized by adopting the geometric method that connects the
optimal erasure protocol with the geodesic line in a Riemannian manifold. This
property helps us to solve the optimal protocol by using methods developed in
geometry. Numerical results have verified that the shortcut strategy can
largely improve the accuracy of finite-time memory erasure without additional
control means. Our strategy shall provide an effective design principle for
finite-time memory erasure with low energy costs.

Recently, much effort has been devoted to realize the one-bit memory erasure
with quasi-static control strategies (Bérut  _et al._ , 2012; Jun  _et al._ ,
2014; Bérut  _et al._ , 2015; Gavrilov and Bechhoefer, 2016; Dago  _et al._ ,
2021). The state lag accumulated in a finite-rate operation hinders the
development of finite-time erasure schemes. Our strategy offers an operable
approach to eliminate the nonequilibrium lag and realize the finite-time
memory erasure with high accuracy and low energy costs. Besides, the
Landauer’s bound have been approached in an underdamped micromechanical
oscillator (Dago  _et al._ , 2021; Dago and Bellon, 2022). The fast
equilibrium recovery strategy have also been achieved with a levitated
particle in the underdamped regime (Raynal  _et al._ , 2023). The driving
force in our shortcut scheme only depends on the particle’s position.
Therefore, it is promising to realize our finite-time memory erasure strategy
with current experimental platforms.

_Acknowledgment._ –This work is supported by the National Natural Science
Foundation of China (NSFC) (Grants No. 12088101, No. 11534002, No. 11875049,
No. U1930402, No. U1930403 and No. 12047549) and the National Basic Research
Program of China (Grant No. 2016YFA0301201).

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