Origami-based tunable truss structures \\ for non-volatile mechanical memory operation
Hiromi Yasuda$^1$
Tomohiro Tachi$^2$
Mia Lee$^1$
and Jinkyu Yang$^1$
%====================================================================== {Origami has recently received significant interest from the scientific community as a building block for constructing metamaterials~\cite{Schenk, Lv, Cheung, Yasuda,Waitukaitis,Silverberg,Hawkes,Overvelde,Filipov. However, the primary focus has been placed on their kinematic applications, such as deployable space structures~\cite{Miura1985,Tsuda, Zirbel and sandwich core materials~\cite{Schenk2, by leveraging the compactness and auxeticity of planar origami platforms. Here, we present volumetric origami cells -- specifically triangulated cylindrical origami (TCO)~\cite{Miura,Kresling,Hunt,Zhao,Jianguo1,Guest1,Ishida -- with tunable stability and stiffness, and demonstrate their feasibility as non-volatile mechanical memory storage devices. We show that a pair of origami cells can develop a double-well potential to store bit information without the need of residual forces. What makes this origami-based approach more appealing is the realization of two-bit mechanical memory, in which two pairs of TCO cells are interconnected and one pair acts as a control for the other pair. Using TCO-based truss structures, we present an experimental demonstration of purely mechanical one- and two-bit memory storage mechanisms. %We envision that the versatile and controllable nature of the volumetric origami can open a new way to building a rugged mechanical memory management system
OrigamiMetamaterials_NatMater-rev6.tex
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\begin{document}
%\title{\textsf{Triangulated cylindrical origami-based mechanical metamtarials with tunable stability and application to a mechanical computing device}}
%\title{\textsf{Origami-based truss structures \\ for mechanical memory cells and logic gates}}
%\title{\textsf{Origami-based truss structures with tunable stability \\ enabling bit storage in mechanical memory and two-bit coupling}}
%\title{\textsf{Origami-based tunable truss structures \\ for two-bit mechanical memory management}}
\title{\textsf{Origami-based tunable truss structures \\ for non-volatile mechanical memory operation}}
\author{Hiromi Yasuda$^1$, Tomohiro Tachi$^2$, Mia Lee$^1$, and Jinkyu Yang$^1$}
\affiliation{$^1$Department of Aeronautics \& Astronautics, University of Washington, Seattle, WA 98195-2400, USA\\ $^2$Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902, Japan}

\begin{abstract}
%======================================================================
{Origami has recently received significant interest from the scientific community as a building block for constructing metamaterials~\cite{Schenk, Lv, Cheung, Yasuda,Waitukaitis,Silverberg,Hawkes,Overvelde,Filipov}. However, the primary focus has been placed on their kinematic applications, such as deployable space structures~\cite{Miura1985,Tsuda, Zirbel} and sandwich core materials~\cite{Schenk2}, by leveraging the compactness and auxeticity of planar origami platforms. Here, we present volumetric origami cells -- specifically triangulated cylindrical origami (TCO)~\cite{Miura,Kresling,Hunt,Zhao,Jianguo1,Guest1,Ishida} -- with tunable stability and stiffness, and demonstrate their feasibility as non-volatile mechanical memory storage devices. We show that a pair of origami cells can develop a double-well potential to store bit information without the need of residual forces. What makes this origami-based approach more appealing is the realization of two-bit mechanical memory, in which two pairs of TCO cells are interconnected and one pair acts as a control for the other pair. Using TCO-based truss structures, we present an experimental demonstration of purely mechanical one- and two-bit memory storage mechanisms. %We envision that the versatile and controllable nature of the volumetric origami can open a new way to building a rugged mechanical memory management system
}
\end{abstract}

\maketitle


%\section{Introduction}
%\hl{(Previous introduction)}
% Origami can be seen as an effective approach of building 2D or 3D structures from sheets of paper.
% Depending upon the crease patterns of paper sheets, origami shows various shapes and folding behaviors, which can be exploited to construct unique structures with tunable mechanical properties.
% For example, origami offers a way to designing deployable space structures such as solar sails~\cite{Tsuda} and solar arrays~\cite{Zirbel}. 
% Also, with the recent emergence of mechanical metamaterials, unique properties of origami also has been reported, e.g., negative Poisson's ratio~\cite{Schenk, Lv, Yasuda}, tunable stability~\cite{Waitukaitis,Silverberg}, reconfigurable shape change~\cite{Hawkes,Overvelde}, and variable stiffness~\cite{Overvelde,Filipov}.
% 
% In particular, for 3D origami structures, one of the well-known origami crease patterns is the Yoshimura pattern which enables to enhance bending stiffness~\cite{Miura}.
% This Yoshimura pattern consists of an identical isosceles triangle facet, and it can be formed as a post-buckling pattern when a thin-walled cylinder is subjected to axial compression, and if twist is applied as well as axial compression at the same time, it induces a derivative of the Yoshimura pattern~\cite{Kresling,Hunt,Zhao,Jianguo1,Jianguo2}, triangulated cylindrical origami (TCO), which is composed non-isosceles triangle surfaces.
% A variation of these triangulated cylindrical structures was studied ~\cite{Guest1,Guest2,Guest3}. 
% Recently, numerical analysis on the TCO-based structure, a truss-like structure, shows bistable behavior under axial compression~\cite{Jianguo1,Jianguo2}.
% However, one of the major issues in application of origami for engineering devices is deformation in the facets of origami, which tends to lead to plastic deformation.
% Rigid origami (e.g., Miura-ori) is one of the solutions for this because rigid origami shows folding motion only along its crease lines without having surface bending, but instead the structure possesses limited degrees of freedom (DOF). 
% To extract useful folding behavior of origami without surface bending, we develop a linear spring network model to characterize the folding behavior and the potential energy analysis is applied to it. 
%
% In this study, we investigate the folding behavior of the TCO-based structure by employing the total potential energy analysis. 
% We model a unit cell of the TCO as a 2 DOF structure, and this multi-DOF feature leads to highly tunable mechanical behaviors such as monostability/bistability, zero-stiffness, and bifurcation behavior.
% We show analytically and experimentally these unique behaviors by introducing a truss-like structure to accommodate deformations of the facets.
% In addition, our potential energy analysis to multi-unit cell structures indicates multistability, especially symmetric double-well potential from the analysis on a two-unit cell structure, and utilizing this behavior, we propose a mechanical computing unit, speficially a mechanical memory device which does not require external power source to maintain bit information and a logic gate operation which is a controlled NOT gate.
% This mechanical computing unit can be highly robust against external noises and perturbations unlike other electronic devices.
%By leveraging their unique and tunable mechanical behaviors, the TCO-based mechanical metamaterials have great potential for engineering applications, especially in aerospace structures.
%
%\hl{(New introduction)}
% Origami is a way to connect flat sheet materials to desired 2D or 3D shapes through folding motion.
% By folding sheets of paper along crease lines, various kinds of origami shapes are created.
% This shape transformation feature can be useful for designing deployable space structures, such as solar sails and solar arrays.
% Although research on origami tends to focus on initial and folded shapes, transition state between those two configurations (i.e., folding/unfolding motions) is also of great interest because structural deformations can be well predictable by introducing origami crease patterns. 
% This predictable structural deformation, together with unique mechanical properties (e.g., negative Poisson's ratio~\cite{Schenk, Lv, Yasuda}, tunable stability~\cite{Waitukaitis,Silverberg}, reconfigurable shape change~\cite{Hawkes,Overvelde}, and variable stiffness~\cite{Overvelde,Filipov}) can be exploited to construct multi-functional structures, e.g., a device which not only exhibits unique mechanical responses to external load/displacement, but also stores information based on repeatable structural deformations.
% 
% To achieve multi-functional devices, we show that tunable behaviors of origami play an important role.
% One of the major issues in application of origami for engineering devices is deformation in the facets of origami, which tends to lead to plastic deformation.
% Rigid origami (e.g., Miura-ori) is one of the solutions for this issue because rigid origami shows folding motion only along its crease lines without having surface bending, however, the structure possesses limited degrees of freedom (DOF) instead. 
% Here, starting from simple origami unit cell used as a building block, we study origami-based mechanical metamaterials, particularly triangulated cylindrical origami (TCO) (Fig.~\ref{fig:TCO_geometry}a) ~\cite{Miura,Kresling,Hunt,Zhao,Jianguo1,Jianguo2,Guest1,Guest2,Guest3} by employing the total potential energy analysis.
% The TCO can exhibits 2 DOF motions, i.e., axial and rotational motions, and this additional DOF leads to unique tunable behaviors, which are not only the bistable behavior of the TCO-based structure, a truss-like structure~\cite{Jianguo1,Jianguo2}, but also other distinctive behaviors; transition between mono- and bistable behaviors, inifitesimaly zero-stiffness, and bifurcations 
% In addition to unique behaviors of a single unit cell, we build multi-unit cell structures, and our potential energy analysis to multi-unit cell structures indicates multistability, especially symmetric double-well potential from the analysis on a two-unit cell structure.
% Taking advantage of this behavior, we propose a mechanical computing unit, speficially a mechanical memory device which does not require external power source to maintain bit information and a logic gate operation which is a controlled NOT gate.
% This mechanical computing unit can be highly robust against external noises and perturbations unlike other electronic devices.
%By leveraging their unique and tunable mechanical behaviors, the TCO-based mechanical metamaterials have great potential for multi-functional engineering devices.

% Previous introduciton ==================================================
%Mechanical logic gates have been a long-standing subject of research, not only to mimic electronic/optical computing devices, but also to control the flow of mechanical energy for sound isolation and heat insulation purposes~\cite{Liang, BaowenLi, Maldovan2013, Cummer2014, Fleury2014}.
% These mechanical devices can be highly useful in harsh environments, such as space and nuclear power plants, where extreme thermal, mechanical, and radiation conditions can hinder the operation of electronic devices. % \hl{While previous studies showed the possibility of mechanical logic gates by using XXX, they mostly adopted systems that cannot be easily expanded to two- or three-dimensional architectures. [ref].
%% Also, those computing units tend to be operated independently without crosstalking to neighboring units. Was there any previous work on realizing a CNOT gate in a mechanical setting?} 
% Such robustness of mechanical systems, together with nanoelectromechanical technologies, has indicated the possibility and effectiveness of mechanical computing devices~\cite{Roukes, Pott,Lee}. In previous studies, however, the operations of mechanical logic gates are limited to an individual cell level, and the possibility of cross-linked operations across the neighboring cells has not been fully explored. This is partly due to the lack of the expandability of the proof-of-concept prototypes employed by the previous studies to two- or three-dimensions. Here we study how we can realize mechanical computing units -- particularly memory storage and controlled NOT (CNOT) gate -- by using origami cells. These modular units can function in a hierarchical way, potentially stacked in multi-dimensions, and they can interact with each other to demonstrate the CNOT operation.


% New introduction
% Mechanical devices can be seen as a system which receives input and gives output accordingly in the form of force, torque, mechanical energy, etc, and mechanical computing units are the striking example to understand the input-output relationship of the mechanical system.
 Mechanical memory operations can be highly useful not only to mimic electronic/optical memory devices, but also to manage the flow of mechanical energy for sound isolation, heat insulation, and energy harvesting purposes~\cite{Liang, BaowenLi, Maldovan2013, Cummer2014, Fleury2014}. These mechanical devices can be highly useful in harsh environments, such as space and nuclear power plants, where extreme thermal, mechanical, and radiation conditions can hinder the operation of electronic devices. The robustness
of mechanical systems, together with nanoelectromechanical technologies, has indicated the possibility and effectiveness of mechanical memory storage and computing devices~\cite{Pott,Lee}. In previous studies, however, the operations of mechanical memory devices are mostly limited to an individual one-bit memory level (few attempts on multi-bit memories, e.g., ~\cite{Guo}), and the possibility of cross-linked operations across the neighboring bits has not been fully explored. 
 
Here, we study how we can realize a two-bit mechanical memory operation by using origami cells.  These origami units can function in a modular way, and they can interact with each other to demonstrate hierarchical, multi-bit memory operations. For this purpose, tunability of unit cells plays an important role to enable the coupling and bit-flipping behavior between the adjacent cells. We show that origami-based structures provide an excellent platform to manipulate their tunable mechanical characteristics, such as stability and stiffness, in a controllable manner. 

% \OLD{
%Recently, origami has been a popular building block to construct mechanical metamaterials~\cite{Schenk, Lv, Yasuda,Waitukaitis,Silverberg,Hawkes,Overvelde,Filipov}.
%% These structures take advantage of outstanding flexibility, expandability, and tailorability of mostly 2D planar origami elements by altering their folding patterns.
%% In such planar elements, rigid foldability can be widely applied.
%% Thus, engineering structures based on 2D origami patterns (e.g., Miura-ori pattern) can ideally offer a single degree of freedom (DOF), in which the actuation in one corner of the plane can govern the motion of the whole structure.
% In particular, a rigid foldable origami pattern  -- so called rigid origami represented by planar Miura-ori pattern~\cite{Miura1985} -- has been studied extensively, because its deformation takes place only along crease lines without relying on the elasticity of materials.
% Engineering structures based on rigid origami can ideally offer a single degree of freedom (DOF), in which the actuation in one corner of the plane can govern the motion of the whole structure.
% This can be beneficial for the control of deployable planar structures, such as solar panels and sails~\cite{Tsuda, Zirbel}.
% %
% In contrast to the rigid planar origami, volumetric origami generally inherits a more enhanced DOF (e.g., accordion that can be compressed and twisted simultaneously), often at the sacrifice of the deformation of origami facets. This can result in versatile kinematic and dynamic motions of volumetric origami. We here investigate the mechanics of volumetric origami, specifically triangulated cylindrical origami (TCO), which can develop coupled dynamics of axial and rotational motions during folding (Fig.~\ref{fig:TCO_geometry}a).
% Despite its simplicity, we find a rich tunability and expandability it offers for the construction of mechanical memory storage units.
%% The tunability of the TCO can be achieved in two different ways, 1) altering its initial geometry and 2) applying pre-compression, and we demonstrate the tunable behavior from single unit cell level to multi-cell level.
%% and also for the durability of origami based structures due to the prevention of warping of the origami facets.
%}

Origami has been a popular building block to construct mechanical metamaterials~\cite{Schenk, Lv, Cheung, Yasuda, Waitukaitis,Silverberg,Hawkes,Overvelde,Filipov}.
In particular, an overconstrained quadrangular mesh origami such as Miura-ori pattern~\cite{Miura1985} has been studied extensively because it offers a single degree of freedom (DOF) mechanism of folding without relying on the elasticity of materials. Such a structure is called rigid foldable origami, or rigid origami, in which the kinematics of folding governs the behavior of the whole structure. This can be beneficial for the control of deployable planar structures, such as solar panels and sails~\cite{Tsuda, Zirbel}.


In this paper, we feature volumetric origami as a building block of tunable metamaterials that can store information. In contrast to the rigid planar origami, volumetric origami generally inherits a highly nonlinear elastic behavior, at the sacrifice of small deformation of panels. 
The coupled behavior of folding and elastic deformation can result in versatile kinematic and dynamic motions. We here investigate the mechanics of volumetric origami, specifically triangulated cylindrical origami (TCO), which can develop coupled dynamics of axial and rotational motions during folding (Fig.~\ref{fig:TCO_geometry}a).
 Despite its simplicity, we find a rich tunability in this structure to enable the design of monostable/bistable, zero-stiffness, and bifurcation structures from one-parameter family of the initial geometry. By assembling multiple origami modules with designed initial conditions, we demonstrate the feasibility of mechanical memory storage units with non-volatile, bit-flipping behavior.

%======================================================================
% Geometry
%======================================================================

%\section{Geometry}
% First, we examine the geometry of the TCO.
 The TCO consists of repeating triangular arrays, which are characterized by valley crease lines (length $a$) and mountain crease lines (length $b$) as shown in Fig~\ref{fig:TCO_geometry}b. Top and bottom surfaces of the TCO unit cell are $n$-sided polygons (e.g., $n=5$ in Fig.~\ref{fig:TCO_geometry}) with side length $c$. Since the TCO is not a rigid foldable origami, folding/unfolding motions cause the warping of each facet, which may result in surface fatigue and damage under repeated usage. To overcome this issue while preserving the key characteristics of the TCO, we replace its surfaces with purely elastic truss members, which support tension/compression by using linear springs (Fig.~\ref{fig:TCO_geometry}c; See Supplementary Movie 1 for the comparison between the paper- and truss-based TCO models). If we assume that the top and bottom surfaces always share the same rotational axis during folding/unfolding, we can characterize the shape of the unit cell by defining its height ($h$), relative angle between the top and bottom polygons ($\theta$), and radius of the circle circumscribing the polygon ($R$). Note that for the sake of mathematical simplicity, $\theta$ is defined as an angle between $OB$ and the perpendicular bisector of $A_aA_b$ as shown in the top view of Fig~\ref{fig:TCO_geometry}c.  Letting $h_0$ and $\theta_0$ be the initial height and relative angle respectively, we can express deformations of the structure by axial displacement $u=-(h-h_0)$ where compression is defined to be positive, and rotational angle $\varphi=\theta-\theta_0$.
 
In this truss model, two crease lines $a$ and $b$ are intersecting at a vertex of the polygon (e.g., $B$ in Fig.~\ref{fig:TCO_geometry}c). For the fabrication of this truss model, we need to secure space for mechanical joints. Thus, we modify the geometry of the TCO model, such that the two crease lines avoid intersecting (Fig.~\ref{fig:TCO_geometry}d). The difference between the original (Fig.~\ref{fig:TCO_geometry}c) and modified (Fig.~\ref{fig:TCO_geometry}d) models is characterized by the correction of the relative angle ($\theta_{\text{cal}}$ in Fig.~\ref{fig:TCO_geometry}d). By adopting this modified model, we fabricate, test, and analyze four different types of the TCO structures in various combinations of $h_0$ and $\theta_0$. Figure~\ref{fig:TCO_geometry}e-h show the graphical illustration of these four original models (top row) and the digital images of their modified physical prototypes (bottom row): $(h_0,\theta_0)$ = (90 mm, 46$^\circ$), (150 mm, 40$^\circ$), (140 mm, 92$^\circ$), and (119 mm, 0$^\circ$). In these models, we use $R$ = 90 mm and $\theta_{cal}$ = 9.7$^\circ$. See Methods and Supplementary Information for more details on the experimental configuration. 

 
%======================================================================
% Compression test
%======================================================================
%\section{Compression test on single unit cells}
 To understand the folding behavior of the TCO-based structure, we calculate the total elastic energy ($U$) stored in the TCO cell as a function of $u$ and $\varphi$ (Methods and Supplementary Information). The insets of Fig.~\ref{fig:Energy_analysis}a-d show the surface maps of $U$ for the four models, where dark colored region indicates the valley of the minimum energy level.
 This highlighted region forms a near-curve trajectory in this configuration space, indicating that the TCO-based structure exhibits a near mechanism of single DOF. Thus, simultaneously compressive and rotational motions of the TCO will follow this trajectory to satisfy the minimum potential energy principle (Methods).
 We can also identify the change of the normalized energy ($U/kh_0^2$ where $k$ is the elastic constant of the linear truss element) under non-dimensionalized axial compression ($u/h_0$) by imposing $\partial U / \partial \varphi =0$ (see Supplementary Information and Supplementary Movie 2 for this uni-axial test).
 Figure~\ref{fig:Energy_analysis}a-d shows the energy plots, where solid curves denote analytical results predicted by the minimum potential energy trajectory in the inset surface maps. The experimental measurements (dashed curves) corroborate these analytical results.
 
 Comparing the four plots in Fig.~\ref{fig:Energy_analysis}a-d, we observe remarkably different trends: monostable, bistable, zero-stiffness, and bifurcation behaviors, respectively. If $(h_0,\theta_0)$ = (90 mm, 46$^\circ$), the structure possesses only one minimum energy state at $u=0$ (Fig.~\ref{fig:Energy_analysis}a). 
 Therefore, the total energy increases monotonically as the TCO cell is compressed, implying a monostable property.
 If $(h_0,\theta_0)$ = (150 mm, 40$^\circ$), there exist two local minimum states along the energy valley as shown in Fig.~\ref{fig:Energy_analysis}b, indicating bistability.
 The TCO-based structure can also exhibit zero tangential stiffness, so called zero-stiffness mode, in which the application of axial compression does not create significant axial force or torque at the initial stage.
 Therefore, the total energy increases at an extremely low rate around $u$ = 0 (Fig.~\ref{fig:Energy_analysis}c). The discrepancy between the analytical and experimental results attributes to the friction of the mechanical joints in the truss elements. Nonetheless, we observe much smaller stiffness in this model compared to the previous two cases (0.26\% and 0.18\% in terms of the linearized initial stiffness compared to those of the monostable and bistable cases, respectively; see the force-displacement curves in Supplementary Information). %For example, the ratio of the slope of the zero-stiffness model to that of monostable model is $2.6\times10^{-3}$, which is a significant decrease in stiffness.
  We analytically find that this zero-stiffness mode can be obtained when $\theta_0=\pi/2$. Interestingly, this mode is independent of $k$ and $h_0$ (mathematical proof in Supplementary Information). This zero-stiffness mode can be potentially useful for impact absorption applications of origami, while maintaining its reusable and tailorable feature. 
 
 Lastly, we observe that the TCO-based truss can experience bifurcation if $\theta_0 = 0$ (Fig.~\ref{fig:Energy_analysis}d).
That is, in the initial stage of the folding, the TCO-based structure is axially compressed without rotation. However, if it reaches a bifurcation point, there are three branches: one unstable branch (continuing pure compression without rotation as indicated by arrow 1 in Fig.~\ref{fig:Energy_analysis}d) and two stable branches (starting to develop twisting motions in one or the other direction as pointed by arrow 2 in Fig.~\ref{fig:Energy_analysis}d). The two different trends in the uni-axial testing verify this bifurcation behavior (Fig.~\ref{fig:Energy_analysis}d; see Supplementary Information and Supplementary Movie 3 for the specially devised uni-axial compression setup). Overall, the results from these four prototypes manifest versatile dynamics of the TCO, which can be controlled simply by altering its initial geometry (i.e., $h_0$ and $\theta_0$, more details in Supplementary Information). 
 
 
% \hl{(If Version 2 of FIG.3 is used)} In addition to those four configurations, we numerically analylze the tunable behaviors of the TCO-based unit cell with $\theta_{cal}=9.7^\circ$ by changing its initial height ($h_0$) and angle ($\theta_0$) (Fig.~\ref{fig:Energy_analysis}e, see Supplementary Information).
 %The analysis results show indicates highly tunable mechanical properties of foldable devices.
 
%======================================================================
% Mechanical computing device
%======================================================================
%\section{Mechanical memory device}
Using this TCO-based truss structure as a unit cell, we further investigate the folding mechanism of multi-cell structures composed of serially stacked TCO cells.
 We start with a two-cell structure that consists of identical monostable TCO units with $(h_0, \theta_0) = (90$ mm, $46^\circ)$.
 They are linked together by sharing the interfacial polygon (Fig.~\ref{fig:Singlebit}a). Note that the chirality of the TCO cells is important in the multi-cell architectures.
 In this two-cell level, we arrange the cells in the opposite chirality, i.e., $(h_0, \theta_0) = (90$ mm, $\pm46^\circ)$, such that they collectively show an interesting coupling motion.
 To test the dynamics of the combined structure, we fix the right end of the stacked prototype to the wall and impose precompression $u_C=45$ mm to the left end of the system (Fig~\ref{fig:Singlebit}a).
 Then, the total elastic energy of the system will differ depending on the rotational angles of the two unit cells, characterized by $\varphi_1$ and $\varphi_2$. Note that these angles are measured with respect to the initial positions of the left and central polygons, respectively. The inset of Fig.~\ref{fig:Singlebit}b shows the analytical values of $U$ as a function of $\varphi_1$ and $\varphi_2$, where the highlighted zone represents the valley of the minimum potential energy.
 We find that the pair of TCO cells collectively possess two local minimum states: one with the right cell folded and the other with the left cell folded (see the graphical illustrations in the inset of Fig.~\ref{fig:Singlebit}b). The normalized elastic energy can be re-plotted as a function of $\varphi_1$ by imposing $\partial U / \partial \varphi_2 =0$.
  Figure~\ref{fig:Singlebit}b evidently shows a symmetric double-well potential. This demonstrates that a pair of monostable TCO cells can successfully form a bistable system, requiring energy to overcome the potential barrier for the transition between the two stable states.
 Please note that this potential barrier can be manipulated by controlling precompression, i.e., tunable potential barrier.
 Let `1' be the state where the first unit cell is folded, and `0' be the state where the second unit cell is folded.
 Then we can use this system as a TCO-based mechanical memory device, which can store bit information (`1' or `0') by exploiting the double-well potential.
 One advantage of this mechanical memory is its non-volatility, implying that it can store bit information stably without the necessity of external residual force. To change the states from `0' to `1' or vice versa, we control only $\varphi_1$ so that the two-unit cell system can switch its state (see Supplementary Information and Supplementary Movie 4 for experimental verification).


 
% Two-bit memory ================================================================
%\section{Two-bit operation: Controlled-NOT gate}
%(Previous version)
%Now we demonstrate a logic gate for the CNOT operation.
% The CNOT gate is composed of two bits: one called ``control bit" and the other ``target bit". The CNOT operation flips the target bit if the control bit is $\left| 1 \right\rangle$, in an analogous way to the CNOT operation in a quantum logic gate~\cite{Rieffel}. Here we use two pairs of the TCO cells, i.e., four identical units of the TCO-based truss elements ($h_0$ = 90 mm) in the sequence of $\theta_0$ = [46$^\circ$, -- 46$^\circ$, 46$^\circ$, -- 46$^\circ$]. 
% Similar to the previous setup for the single bit operation, we fix the rightmost polygon to the wall in both translational and rotational directions, while we let the other polygons rotate freely. We apply precompression of $u_C=50$ mm and 47.5 mm to the first and second pairs respectively, such that the two pairs maintain the specified compressed states throughout the operation. Here, we intentionally introduce distinctive $u_C$ values to break symmetry and let one pair act as the control bit while the other as the target bit (further details to be explained later). We first analyze the total elastic energy of the system in relation to the deformed status of the two single-bit memory units. We represent the deformation of these two pairs by measuring the rotational angles $\varphi_1$ and $\varphi_3$, which represent the twisted angles of the first and third polygons respectively with respect to their uncompressed positions (Fig.~\ref{fig:CNOTgate}a).  The analytical results are shown in Fig.~\ref{fig:CNOTgate}b, where we identify four minimum energy states representing $\left| 00 \right\rangle$, $\left| 01 \right\rangle$, $\left| 10 \right\rangle$, and $\left| 11 \right\rangle$. Here the first and second numbers indicate the control (left pair denoted in blue color in the inset of Fig.~\ref{fig:CNOTgate}b) and target (right pair in red color) bit, respectively. For example, $\left| 10 \right\rangle$ is the state where the control bit shows $\left| 1 \right\rangle$, and the target bit shows $\left| 0 \right\rangle$, following the definition of on and off status from the memory operation as illustrated in Fig.~\ref{fig:Singlebit}b.
%
%The next step is to test the CNOT gate operation by applying a pulse input to the control bit and measuring the response from the target bit. For this, we impose a trapezoid-shaped waveform on $\varphi_1$ to alternates the condition of the control bit between $\left| 1 \right\rangle$ and $\left| 0 \right\rangle$ (Supplementary Information). Mathematically, this process determines the state of the control bit in the form of $\alpha \left| 0 \right\rangle +\beta \left| 1 \right\rangle $, where $\alpha$ and $\beta$ are the coefficients satisfying $\alpha+\beta=1$. The key point here is to verify that the onset of the control pulse (i.e., $\left| 1 \right\rangle$) can flip the information stored in the target bit. This process can be represented by the following two cases: $\alpha \left| 00 \right\rangle +\beta \left| 11 \right\rangle $ and $\alpha \left| 01 \right\rangle +\beta \left| 10 \right\rangle $. The former corresponds to the conversion of the target bit from off to on (i.e, $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$), while the latter implies the opposite case that the target bit changes from on to off (i.e, $\left| 1 \right\rangle$ to $\left| 0 \right\rangle$) as the control bit is turned on.  
%
%
%We start with demonstrating the first case ($\alpha \left| 00 \right\rangle +\beta \left| 11 \right\rangle $). Figure~\ref{fig:CNOTgate}c illustrates the conceptual chart of the sequential operation of $\varphi_1$ and the consequential change of $\varphi_3$. The corresponding evolution of the TCO pairs' states is plotted in Figure~\ref{fig:CNOTgate}d, where the red curve denotes the experimental result of $\varphi_1$ and $\varphi_3$ measured by non-contact laser Doppler vibrometry (Supplementary Information). The system is initially positioned at $\left| 00 \right\rangle$, as denoted by the state at (i) in Figs.~\ref{fig:CNOTgate}(c-d). As we apply the pulse input to the control bit (i.e., $\varphi_1 = -62^\circ$ to $62^\circ$), the control bit first changes its state from $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$. See the transition of red curve from point (i) to point (ii) in Fig.~\ref{fig:CNOTgate}d. This onset of the control bit eventually flips the target bit from $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$ (i.e., $\varphi_3 = -31^\circ$ to $28^\circ$, see the state (iii) in Fig.~\ref{fig:CNOTgate}d and Supplementary Movie 5). Thus, we verify the transition from the initial state $\left| 00 \right\rangle$ to the final state $\left| 11 \right\rangle$ without resorting to any direct excitations applied to the target bit.
%
%Next, we move on to demonstrate the second case of the CNOT operation ($\alpha \left| 01 \right\rangle +\beta \left| 10 \right\rangle $). Since the previous operation started from $\left| 00 \right\rangle$, we need to first perform input preparation to change the initial state from $\left| 00 \right\rangle$ to $\left| 01 \right\rangle$. For this, we apply the pulse input directly to the target bit to convert it from $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$. This pulse input to the target bit does not affect the control bit, because $\varphi_1$ is not constrained. Therefore, $\varphi_1$ and $\varphi_3$ rotate in the same direction at the same rate without flipping the status of the control bit (see the input preparation process in Fig.~\ref{fig:CNOTgate}e). Now we apply the pulse input to the control bit, which manipulates the control bit from $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$ as shown by the trajectory of the red curve from the state (i) to the top right corner (ii).
% Sequentially, the energy state moves from the state (ii) to the state (iii), flipping the target bit from $\left| 1 \right\rangle$ to $\left| 0 \right\rangle$. 
% Since distinctive $u_C$ values are applied to the control and target bits, the energy slope from $\left| 11 \right\rangle$ to $\left| 10 \right\rangle$ is less than that from $\left| 11 \right\rangle$ to $\left| 01 \right\rangle$. Therefore, we observe that the state changes from $\left| 11 \right\rangle$ to $\left| 10 \right\rangle$ instead of $\left| 11 \right\rangle$ to $\left| 01 \right\rangle$.
% This serial process eventually changes the combined states of the control and target bits from $\left| 01 \right\rangle$ to $\left| 10 \right\rangle$, successfully verifying the second case of the CNOT operation (Supplementary Information and Supplementary Movie 6).

%(New version)
 Now we demonstrate a two-bit memory operation.
 We use two pairs of the TCO cells, i.e., four identical units of the TCO-based truss elements ($h_0$ = 90 mm) in the sequence of $\theta_0$ = [46$^\circ$, -- 46$^\circ$, 46$^\circ$, -- 46$^\circ$]. 
 Similar to the previous setup for the single bit operation, we fix the rightmost polygon to the wall in both translational and rotational directions, while we let the other polygons rotate freely.
 We apply precompression of $u_C=50$ mm and 47.5 mm to the first and second pairs respectively, such that the two pairs maintain the specified compressed states throughout the operation.
 Here, we intentionally introduce distinctive $u_C$ values to break symmetry, thereby inducing controlled coupling behavior between the two bits. (further details to be explained later).
 We first analyze the total elastic energy of the system in relation to the deformed status of the two single-bit memory units.
 We represent the deformation of these two pairs by measuring the rotational angles $\varphi_1$ and $\varphi_3$, which represent the twisted angles of the first and third polygons respectively with respect to their uncompressed positions (Fig.~\ref{fig:Twobit}a).
 The analytical results are shown in Fig.~\ref{fig:Twobit}b, where we identify four minimum energy states representing `00', `01', `10', and `11'.
 Here the first and second numbers indicate the first bit (left pair denoted in blue color in the inset of Fig.~\ref{fig:Twobit}b) and second bit (right pair in red color) bit, respectively. For example, `10' is the state where the first bit shows `1', and the second bit shows `0', following the definition of on and off status from the memory operation as illustrated in Fig.~\ref{fig:Singlebit}b.

%\COMMENT{This part seems to carry a little bit from the idea of CNOT gate idea. 
%Can we simplify our statement as follows?
%My understanding is that the first bit has a lower barrier between two stable states, which allows for the hysteresis that programs the sequence of `00'  $\to$`10'  $\to$ `11'  $\to$ `01'  $\to$ `00' from the full cycle of trapezoidal input.
%By the way, this indicate that we cannot directly apply this idea to 3 bit memory as the expected sequence would be `000'  `100' `110' `111' `011' `001' `000' (we cannot do `010' !).
%}
%\hl{(HY: I replaced the words "control bit" and "target bit" by "first bit" and "second bit" respectively to avoid carrying the idea of CNOT gate.)}\\
%\hl{(HY: I think the current way of demonstrating the bit-flipping is easier to understand, because important thing is that single operation applied to the first bit can induce bit change in the second bit, and if we choose circular operation, the uniqueness of our system is not clear (e.g., some people may think the cyclic operation is for initializing the memory information). Also, we can explain one-directional coupling, (i.e., the input signal applied to the second bit does not affect the first bit), in the same framework. If we introduce the circular operation, we need to explain this separately, and I think the main feature of our finding may be unclear in this case...)}\\
%\hl{(HY: As for 3-bit memory operations, I think it is interesting in mathematics but I don't think it is important in terms of mechanical computing because to build computing devices, the minimum element is an operation on pairs of input bits (e.g., AND, XOR), so I think that two-bit operation is fundamentally important.)} 

 The next step is to test the operation of the two-bit memory.
 Unlike operations of conventional memories (i.e., manipulation of each bit one by one), we demonstrate a unique operation of the two-bit memory by utilizing controlled coupling behavior between the two bits.
 In particular, this operation flips the second bit if the first bit is `1', which indicates that the first bit can control the state of the second bit. In this process, however, the second bit does not affect the state of the first bit, thus exhibiting a one-directional coupling mechanism.
% Here, we denote the first bit by ``control bit'' and the second bit by ``target bit'',  and we demonstrate this operation by applying a pulse input to the control bit and measuring the response from the target bit.
 We demonstrate this operation by applying a pulse input to the first bit and measuring the response from the second bit.
 For this, we impose a trapezoid-shaped waveform on $\varphi_1$ to alternates the condition of the first bit between `1' and `0' (Supplementary Information).
 The key point here is to verify that the onset of the control pulse (i.e., `1') can flip the information stored in the second bit.
 This process can be represented by the following two cases: `00' $\to$ `11' and `01' $\to$ `10'.
 The former corresponds to the conversion of the second bit from off to on (i.e, `0' to `1'), while the latter implies the opposite case that the second bit changes from on to off (i.e, `1' to `0') as the first bit is turned on.  


We start with demonstrating the first case (`00' $\to$ `11'). Figure~\ref{fig:Twobit}c illustrates the conceptual chart of the sequential operation of $\varphi_1$ and the consequential change of $\varphi_3$. The corresponding evolution of the TCO pairs' states is plotted in Fig.~\ref{fig:Twobit}d, where the red curve denotes the experimental result of $\varphi_1$ and $\varphi_3$ measured by non-contact laser Doppler vibrometry (Supplementary Information). The system is initially positioned at `00', as denoted by the state at (i) in Figs.~\ref{fig:Twobit}c-d. As we apply the pulse input to the first bit (i.e., $\varphi_1 = -62^\circ$ to $62^\circ$), the first bit changes its state from `0' to `1' in the beginning of the operation. See the transition of red curve from point (i) to point (ii) in Fig.~\ref{fig:Twobit}d. This onset of the first bit eventually flips the second bit from `0' to `1' (i.e., $\varphi_3 = -31^\circ$ to $28^\circ$, see the state (iii) in Fig.~\ref{fig:Twobit}d and Supplementary Movie 5). Thus, we verify the transition from the initial state `00' to the final state `11' without resorting to any direct excitations applied to the second bit.

Next, we move on to demonstrate the second case of the two-bit memory operation (`01' $\to$ `10'). Since the previous operation started from `00', we need to first perform input preparation to change the initial state from `00' to `01'. For this, we apply the pulse input directly to the second bit to convert it from `0' to `1'. This pulse input to the second bit does not affect the first bit, because $\varphi_1$ is not constrained. Therefore, $\varphi_1$ and $\varphi_3$ rotate in the same direction at the same rate without flipping the status of the first bit (see the input preparation process in Fig.~\ref{fig:Twobit}e). Now we apply the pulse input to the first bit, which manipulates the first bit from `0' to `1' as shown by the trajectory of the red curve from the state (i) to the top right corner (ii).
 Sequentially, the energy state moves from the state (ii) to the state (iii), flipping the second bit from `1' to `0'. 
 Since distinctive $u_C$ values are applied to the first and second bits, the energy slope from `11' to `10' is less than that from `11' to `01'.
  Therefore, we observe that the state changes from `11' to `10' instead of `11' to `01'.
 This serial process eventually changes the combined states of the first and second bits from `01' to `10', successfully verifying the second case of the two-bit memory operation (Supplementary Information and Supplementary Movie 6).

% The mechanical memory device and logic gate operation have great potential to be a computing device which is extremely robust against external noise unlike conventional electronic devices. 
 
%======================================================================
% Conclusion
%======================================================================
%\section{Conclusion}
Although we showed the versatile nature and potential of the TCO elements hierarchically from a single-cell, double-cell, up to four-cell levels in one dimension, we envision that it can be further extended to multi-dimensions, e.g., honey-comb like 3D clusters. This will function as a layer of mechanical memory storage and computing structures, which can also provide multi-functional features, such as rugged protective layers. Likewise, while this study explored only serial connections of origami cells, they can be also connected in parallel or coaxially, to achieve various functionalities. Moreover, origami can be scaled to miniaturized dimensions. The versatile nature of the TCO together with nanoelectromechanical systems (NEMS) has great potential to develop robust NEMS actuators and sensing devices.
 Also, the intrinsic nature of the TCO cells that interweave axial and torsional motions can be further exploited for dynamic purposes. 
 %The presented study briefly touched the effect of opposite chirality for realizing the bistable two-cell unit for bit storage. From a topological standpoint, the architectures with different chirality will offer highly distinctive wave dispersion relationship, which can be characterized by topological numbers, such as Berry phase [ref]. 
 Conclusively, the volumetric origami cells can pave a new way for designing novel engineering systems for mechanical computing and other purposes relying on their rich constitutive mechanics in a single-cell level and strong cohesion in a multi-cell level. 


% In this study, we employed the triangulated cylindrical origami as a building block for mechanical memory storage and computing units. This volumetric origami offers a rich degree of freedom in motions, interweaving axial and rotational motions. While the paper-version of this origami can suffer from surface deformation, we here demonstrate that the origami-inspired truss structures can operate in a purely elastic manner while inheriting rich mechanics of this volumetric origami. Based on this volumetric origami cell, we designed, fabricated, and tested origami-based truss structures in a hierarchical fashion. In a single-cell level, we found that this structure can exhibit highly tunable behavior depending on its initial geometry, showing monostable, bistable, zero-stiffness, and bifurcation behaviors. In a two-cell level, we found that the combination of two monostable cells allows the bistable, double-well potential by leveraging opposing chirality in twisted origami. This can be potentially related to the topological nature of this cylindrical origami architecture represented by Berry phase. In a four-cell level,   
% 
% Also, although we use monostable TCO-based unit cells, we can designed two-unit cell structures which shows the double-well potential.
% By using the analogy to quantum physics, we proposed the TCO-based mechanical memory device and logic gate.
% Our experimental demonstration shows that our logic gate flips the target bit depending on the state of the control bit.
% This work reveals the unique and rich mechanical response of the TCO-based structure to external inputs such as axial force and torque.
% By leveraging their tunable mechanical properties, the TCO-based mechanical metamaterials can be highly useful for various fields such as architectures, space structures, and computing devices. 
 
 
\section*{Methods}
\textbf{Principle of minimum total potential energy approach.}
 By using the geometry of the TCO-based unit cell, we calculate the length of crease lines $a$ and $b$ (see Fig~\ref{fig:TCO_geometry}(c)) as follows:
\begin{equation} \label{eq:Length}
\begin{split}
	a &= \sqrt{{{\left( {{h}_{0}}-u \right)}^{2}}+4{{{{R}'}}^{2}}{{\sin }^{2}}\left( \frac{\varphi }{2}+\frac{{{\theta }_{0}}}{2}-\frac{\pi }{2n}+{{\theta }_{\text{cal}}} \right)} \\
	b &= \sqrt{{{\left( {{h}_{0}}-u \right)}^{2}}+4{{{{R}'}}^{2}}{{\sin }^{2}}\left( \frac{\varphi }{2}+\frac{{{\theta }_{0}}}{2}+\frac{\pi }{2n}-{{\theta }_{\text{cal}}} \right)}
\end{split}
\end{equation}
where $R'$ and $\theta_{\text{cal}}$ are a modified radius of the circle circumscribing the cross-section and a calibrated angle to compensate for the difference between original and physical prototype models (Please see Supplementary Information for details).
 Then, the total elastic energy is calculated as $U=\frac{1}{2}nk{{\left( a-{{a}_{0}} \right)}^{2}}+\frac{1}{2}nk{{\left( b-{{b}_{0}} \right)}^{2}}$
where $a_0$ and $b_0$ are initial length of $a$ and $b$, and $k$ is a spring constant of the truss members. Also, the work is obtained by $W={{F}}u+{{T}}\varphi$ where $F$ and $T$ are the external force and torque applied to the TCO cell, respectively. Based on these expressions, the total potential energy ($\Pi$) is
\begin{equation} \label{eq:PotentioalEnergy}
	\Pi (u,\varphi )=U-W=\frac{1}{2}nK{{\left( a-{{a}_{0}} \right)}^{2}}+\frac{1}{2}nK{{\left( b-{{b}_{0}} \right)}^{2}}-{{F}}u-{{T}}\varphi 
\end{equation}
 By applying the principle of minimum total potential energy (i.e., $\partial \Pi / \partial u =0$ and $\partial \Pi / \partial \varphi =0$), we obtain the analytical expressions of the two-DOF folding/unfolding motion of the TCO-based structure (Supplementary Information).

\textbf{Prototype fabrication and compression test.}
 We use acrylic plates tailored by a laser cutter for the top and bottom polygons, and 3D printed parts made of polylactic acid for universal joints to attach truss elements to the polygons. 
 %support rotational motion of linear spring elements $a$ and $b$.
Stainless steel shafts (diameter is 3.18 mm) and linear springs ($k$ = 3.32 kN/m for monostable, bistable, zero-stiffness models; $k$ = 1.08 kN/m for bifurcation model) are used to form truss elements that support tension/compression. To obtain the mechanical properties of the prototypes, we build a customized testing setup where the prototype is placed horizontally and its bottom surface is mounted on a fixed wall. The top surface is supported by a ball bearing and stainless steel shaft, so that it can translate and rotate with minimal friction (Supplementary Information and Supplemental Movies 2--6).
 
%\textbf{TCO-based mechanical computing unit.}
% We used two monostable unit cells ($h_0=92$ mm, $\theta_0=\pm 45^\circ$) connected horizontally to form a mechanical memory unit, and four of these unit cells were used for a controlled NOT gate.
% The detail information of the experiment set up is described in Supplementary Information.
 
 
\begin{acknowledgments}
We thank Mike Clark at CoMotion at the University of Washington for technical support. We also thank Hansuek Lee at KAIST and Hyochul Kim at Samsung Advanced Institute of Technology in Korea for helpful discussions. We are grateful for the support from the ONR (N000141410388) and NSF (CAREER-1553202).
\end{acknowledgments}

\section*{Author contributions}
H.Y. and M.L. conducted the research and interpreted the results, and J.Y. and T.T. provided guidance throughout the research. H.Y., T.T., and J.Y. prepared the manuscript.

\section*{Additional information}
The authors declare that they have no competing financial interests. Correspondence and requests for materials should be addressed to Jinkyu Yang~(email: jkyang@aa.washington.edu).



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%==========================================================================
% Figures
%==========================================================================
\begin{figure}[htbp]
\centerline{ \includegraphics[width=0.9\textwidth]{TCO_geometry3.png}}
\caption{ \textbf{Geometry of triangulated cylindrical origami.} \textbf{a}, Folding motion of the TCO. \textbf{b}, The flat sheet with crease patterns consisting of mountain crease lines ($a$ shown as blue solid lines) and valley crease lines ($b$ shown as red dashed lines). \textbf{c}, Truss version of the TCO, where all facets are removed and crease lines are replaced by linear springs with a spring constant of $k$. \textbf{d}, Modified truss structure for the fabrication of physical prototypes. \textbf{e-h}, Graphical illustrations of four different TCO configurations (upper row) and digital images of their physical prototypes (lower row). Their initial configurations are $(h_0,\theta_0)$ = (90 mm, 46$^\circ$), (150 mm, 40$^\circ$), (140 mm, 92$^\circ$), and (119 mm, 0$^\circ$) from left to right.}
\label{fig:TCO_geometry}
\end{figure}

\begin{figure}[htbp]
\centerline{ \includegraphics[width=0.9\textwidth]{TCO_CompressionTest4.png}}
\caption{\textbf{Folding motions of the TCO cells.} \textbf{(a-d)}, The energy analysis for the TCO-based truss structures shows remarkably different behaviors: \textbf{a}, Monostability at $(h_0,\theta_0)$ = (90 mm, 46$^\circ$); \textbf{b}, bistability at (150 mm, 40$^\circ$); \textbf{c}, zero-stiffness mode at (140 mm, 92$^\circ$); and \textbf{d}, bifurcation at (119 mm, 0$^\circ$). The displacement is normalized by $h_0$, and energy is normalized by $kh_0^2$. Experimental results (mean value is shown as dashed curves, and standard deviation is represented by colored areas) show qualitative agreements with the analytical predictions (solid curves). The inset plots show the equi-potential plots of $U/kh_0^2$ as a function of $u/h_0$ and $\varphi$, in which highlighted trajectories indicate the valley of minimum potential energy. In the experimental curves, the range of $u/h_0$ is restricted by the folding motions of the TCO-based truss prototypes. For example, the highly twisted shape of the zero-stiffness TCO prototype (Fig.~\ref{fig:TCO_geometry}g) causes the truss elements overlap in the early stage of folding, allowing only $\sim$15\% of $u/h_0$ as shown in the panel \textbf{c}. The moderately twisted geometry of the monostable and bistable cases (Figs.~\ref{fig:TCO_geometry}e and f) permit more compression, allowing approximately 50\% folding of the truss structure in terms of $u/h_0$ as shown in the panels \textbf{a} and \textbf{b}.}  %by $\partial U / \partial \varphi =0$.} %Black solid lines represent contours of $U/kh_0^2$ whose values are displayed in the plot area. Arrows show paths along which the TCO-based unit cell deforms. Experimental results are compared with $U/kh_0^2$ along these paths.}
\label{fig:Energy_analysis}
\end{figure}

\begin{figure}[htbp]
\centerline{ \includegraphics[width=0.5\textwidth]{TCO_Singlebit6.png}}
\caption{ \textbf{A pair of TCO cells' capability to demonstrate mechanical memory storage.} \textbf{a}, Two monostable TCO-based unit cells are connected horizontally. We fix the distance between leftmost and rightmost polygons by imposing a constant distance between them. Photograph of the corresponding configuration is shown in the right panel. \textbf{b}, The normalized elastic energy as a function of $\varphi_1$ shows the double-well potential numerically. The inset shows the surface map of the elastic energy as a function of both $\varphi_1$ and $\varphi_2$, where the highlighted region denotes the valley of the map corresponding to the minimum potential energy trajectory. There exist two minimum states, and the 
illustrations show the schematic shapes of the pair of TCO cells at these points. Here, the configuration of the folded right cell represents `0', while the one with the folded right cell denotes `1'.}
\label{fig:Singlebit}
\end{figure}


\begin{figure}[htbp]
\centerline{ \includegraphics[width=0.9\textwidth]{TCO_2bit_operation.png}}
\linespread{1.3}\selectfont{}
\caption{ \textbf{Two pairs of TCO cells' to demonstrate two-bit memory operation.} \textbf{a}, Two-bit memory is composed of four TCO-based unit cells as shown in a schematic illustration (left) and photograph (right). The first two units from the left (i.e., left pair in blue color) form a first bit, and the other two unit cells (i.e., right pair in red color) form a second bit. The states of these two bits are determined by the rotational angles of the first and third unit cells, $\varphi_1$ and $\varphi_3$. \textbf{b}, The potential energy of the system as a function of $\varphi_1$ and $\varphi_3$ is analyzed, where a darker region represents a lower level of the normalized potential energy (color map). We divide the $\varphi_1$ and $\varphi_3$ space into four regions: `00', `01', `10', and `11'. Inset illustrations indicate a representative shape of the TCO cells for each region. \textbf{c}, The sequence of the operation for `00' $\to$ `11' is shown. A pulse input is applied only to the first bit. \textbf{d} Experimental result in a red curve indicates that the operation flips the second bit from `0' to `1'. \textbf{e}, The sequence of the operation for `01' $\to$ `10' is shown. A pulse input is applied to the second bit first for the input preparation, and then to the first bit for the operation. \textbf{f} Experimental result indicates that the operation flips the second bit from `1' to `0'.}
\label{fig:Twobit}
\end{figure}

\end{document}




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