# Encoding lattice structures in Quantum Computational Basis States

Kalyan Dasgupta

(IBM Research, Bangalore, India
)

##  1 Introduction

Lattice models or structures are geometrical objects with mathematical forms,
that are used to represent physical systems. They have been used widely in
diverse fields, namely, in condensed matter physics, to study degrees of
freedom of molecules in chemistry and in studying polymer dynamics and protein
structures to name a few [1]. In this article we discuss an encoding
methodology of lattice structures in computational basis states of qubits (as
used in quantum computing algorithms). We demonstrate a specific use case of
lattice models in protein structure prediction. A variety of lattice models
have been used for protein structure prediction. Although, proteins have
highly irregular structures, lattice models have been used to predict
structures at a basic level. Lattice models fall under the category of coarse-
grained models. The geometry considered for coarse-grained models could either
have a continuous representation or have a lattice representation [2]).
Lattice models offer significant computational speedup over other
representations. Fig. 1 gives some of the lattice representations.

Figure 1: Lattice structures: (a) cubic, (b) diamond, (c) cubic with planar
diagonals, (d) hexagonal, (e) triangular and (f) face-centred-cubic. Courtesy:
[1]

The protein structure prediction problem suffers due to the large feasible
conformational space. Choosing the correct conformation from such a vast space
is a big challenge. With quantum hardwares, one can create $2^{n}$
computational basis states form $n$ qubits. One can find several papers using
quantum computing for protein structure prediction using lattice models. Cubic
lattices have been used in references [3] and [4]. In reference [5], the
authors presented a quantum model for a coarse-grained protein sequence mapped
onto a tetrahedral lattice structure. In reference [6] the authors propose a
turn based encoding in a cubic lattice with larger degrees of freedom using
the cubic lattice with planar and steric diagonals.

In this article we do not propose any quantum algorithm to solve the protein
structure prediction problem, instead, we propose a generic encoding
methodology of lattice structures. The protein sequence, essentially, consists
of turns or bonds between adjacent monomers or amino acids (also referred as a
bead in coarse-grained models). The turn that the bonds can take are limited
by the degrees of freedom of the lattice model being used. We show how the
encoding methodology encode the turns based on the lattice model selected. We
take two specific lattice models, the cubic with planar diagonals and the
face-centred cubic (FCC), and demonstrate the encoding methodology. The
article is organized as follows. In section 2, we discuss the lattice models
and the coordinate space that the turns can take. In section 3 we discuss how
the coordinates are mapped to the qubit computational basis states. Finally,
we summarize the methodology in the conclusions.

##  2 Lattice models and coordinate space

Fig. 2 (a) gives a section of the structure of the FCC lattice. Given a
protein bead at the origin, we would like to know the possible set of 3D
coordinates that the next bead could take following a turn. Fig. 2 (b) gives
the possible coordinates in a given plane $y-z$.

Figure 2: (a) Section of FCC lattice, (b) coordinate space that a turn could
take

As can be seen from the figure, the bead coloured red in the figure can have
bonds with beads coloured in green. The green beads are at the centre of the
faces of the cubes. The bond distance between a bead at a corner/vertex and a
bead at the centre of a face is $\frac{d}{\sqrt{2}}$. Beads located at the
centre of the faces of the cube can have bonds only with beads that are
located at the vertices. Similarly, beads located at the vertices can have
bonds with beads located at the centre of the faces. Overall, in a given
plane, there are $4$ possible options to choose from. However there are $3$
planes, the $x-y$, $y-z$ and the $z-x$ planes. The turn could be in any one of
those planes. There are $12$ possible directions or degrees of freedom in
which the bond could turn between adjacent beads.

Now, let us have a look at the cubic lattice with planar diagonals. Fig. 3 (a)
gives the section of a cubic lattice and 3 (b) gives the coordinate mapping.

Figure 3: (a) Section of cubic lattice with planar diagonals, (b) coordinate
space that a turn could take

In the cubic lattice, the turns could be in $8$ directions in a given plane.
With 3 planes, there are $24$ possible turns. However, there for every set of
two planes, there are two directions that are repeated. For example, if we
consider the $x-y$ plane and the $y-z$ plane, the $\pm y$ directions are
repeated. As a result, we will have $6$ such planes that are repeated.
Overall, we will have $24-6=18$ degrees of freedom here.

In order to encode, we need to map the qubit states to bead coordinates. Let
us consider the FCC lattice. We can start from the origin as the coordinates
of the first bead. The second bead will take up a position at any one of the
vertices or faces as shown in Fig. 2 (b). If we denote the axes as $x,y$ and
$z$, the location of the second bead could be in any one of the $xy,yz$ or
$zx$ planes. The red bead is at the origin. If the second bead were to be in
the $y-z$ plane, the possible coordinates that it could have are shown in the
figure. The coordinates are the $y$ and $z$ coordinates given as $(y,z)$. This
is a rotation around the $x$-axis at $90^{\circ}$ interval for $4$ different
points, in a counter-clockwise direction. We would see a similar set of
coordinates in the $z-x$ and $x-y$ plane if the second bead were to be in the
those planes. In Fig. 2 (b) the $x$ coordinates of the red and green beads are
the same. We could carry forward this logic to every bead in the protein
sequence. If we consider any bead $i$ with coordinates $(x,y,z)$, the bead
$i+1$ would then have coordinates $(x+\Delta x,y+\Delta y,z+\Delta z)$. If the
bead $i+1$ is in the plane parallel to the $y-z$ plane, $\Delta x=0$ and
$\Delta y$ and $\Delta z$ would take values as shown in Fig. 2 (b). If it is
in the plane parallel to the $z-x$ plane (counter-clockwise rotation around
$y$-axis), $\Delta z$ and $\Delta x$ would take up the same values as $\Delta
y$ and $\Delta z$ in the previous case, and in the same order. Similarly, when
the the bead $i+1$ is in the plane parallel to the $x-y$ plane (counter-
clockwise rotation around $z$-axis), $\Delta x$ and $\Delta y$ would take up
similar values.

If we look at Fig. 2, the values that $\Delta y$ takes are
$\left[\frac{d}{2},-\frac{d}{2},-\frac{d}{2},\frac{d}{2}\right]$. The values
that $\Delta z$ takes on the other hand are
$\left[\frac{d}{2},\frac{d}{2},-\frac{d}{2},-\frac{d}{2}\right]$. Let us
denote the two vectors as $\Delta a$ and $\Delta b$.

$\Delta a=\left[\begin{array}[]{r}\frac{d}{2}\\\ -\frac{d}{2}\\\
-\frac{d}{2}\\\ \frac{d}{2}\end{array}\right]~{}~{}~{}~{}\Delta
b=\left[\begin{array}[]{r}\frac{d}{2}\\\ \frac{d}{2}\\\ -\frac{d}{2}\\\
-\frac{d}{2}\end{array}\right]$ (1)

The coordinates of bead $i+1$ could be any one of the following.

$\displaystyle\begin{aligned} \textrm{Parallel to the $y-z$
plane:}&~{}~{}x_{i+1}=x_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}y_{i+1}=y_{i}+\Delta
a_{k},~{}~{}z_{i+1}=z_{i}+\Delta b_{k}\\\ \textrm{Parallel to the $z-x$
plane:}&~{}~{}x_{i+1}=x_{i}+\Delta
b_{k},~{}~{}y_{i+1}=y_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z_{i+1}=z_{i}+\Delta
a_{k}\\\ \textrm{Parallel to the $x-y$ plane:}&~{}~{}x_{i+1}=x_{i}+\Delta
a_{k},~{}~{}y_{i+1}=y_{i}+\Delta b_{k},~{}~{}z_{i+1}=z_{i}\end{aligned}$ (2)

In the expression above, $\Delta a_{k}$ and $\Delta b_{k}$ are elements with a
given index $k$ (depending upon the bond direction or turn taken) from the
arrays given in (1).

Similarly, with the cubic lattice with planar diagonals, there are $8$
possible directions. The vectors $\Delta a$ and $\Delta b$ will have the form
shown in (3).

$\Delta a=\left[\begin{array}[]{r}d\\\ d\\\ 0\\\ -d\\\ -d\\\ -d\\\ 0\\\
d\end{array}\right]~{}~{}~{}~{}\Delta b=\left[\begin{array}[]{r}0\\\ d\\\ d\\\
d\\\ 0\\\ -d\\\ -d\\\ -d\end{array}\right]$ (3)

##  3 Encoding to qubit states

We will take the example of the cubic lattice with planar diagonals to
demonstrate the encoding methodology. For every plane we need to encode $8$
bits of information to encode the turn. This will be possible with,
$\log_{2}8=3$ qubits. The elements in $\Delta a$ and $\Delta b$ can be
obtained by having a linear combination of the qubit states. The qubit states
will be in a state of superposition and so will the elements in $\Delta a$ and
$\Delta b$. As we will show in lemma 1, any function $f\in\mathcal{R}^{n}$ can
be expressed as a linear combination of product of qubit states. This mixture
of qubit states forms the basis in $\mathcal{R}^{N}$.

###### Lemma 1

Any function $f\in\mathcal{R}^{N}$ can be expressed as a linear combination of
product of $n=\log_{2}N$ qubit states. This set of product of qubit states
will form a basis set and span the space in $\mathcal{R}^{N}$.

###### Proof 1

If we are given $n$ qubits, we can have $2^{n}$ qubit states. For example,
with $3$ qubits the qubit states denoted by $QS_{3}$ will be as follows.

$\displaystyle QS_{3}=\left[\begin{array}[]{ccc}q_{1}&q_{2}&q_{3}\\\ \hline\cr
0&0&0\\\ 0&0&1\\\ 0&1&0\\\ 0&1&1\\\ 1&0&0\\\ 1&0&1\\\ 1&1&0\\\
1&1&1\end{array}\right]$ (13)

The basis that spans $\mathcal{R}^{N}$, can be formed by taking the product of
qubit states. In an $n$-qubit system, there are ${n\choose 1}$ ways to choose
one qubit, ${n\choose 2}$ ways to choose a product of two qubits, ${n\choose
3}$ ways to choose a product of three qubits, and so on till ${n\choose n}$
ways to choose product of $n$ qubits. We will also have a constant term or an
intercept, which is essentially the ${n\choose 0}$ term. Together we will have

${n\choose 0}+{n\choose 1}+{n\choose 2}+\ldots{n\choose n-1}+{n\choose
n}=2^{n}$

terms. These $N=2^{n}$ terms will form the basis of $\mathcal{R}^{N}$. Any
function $f\in\mathcal{R}^{N}$ can then be expressed as linear combination of
these basis terms.

$f=\sum_{i=1}^{N}c_{i}b_{i}$

In the equation above, $b_{i}\in\mathcal{R}^{n}$ is the $i^{th}$ basis and
$c_{i}\in\mathcal{R}$ is the coefficient. Every element of $b_{i}$ can be
obtained by substituting the values of the qubits from a table like $QS_{3}$
above. Since the combinations of qubits are unique, all the basis vectors will
be independent and the basis matrix $B_{n}=\left[b_{1},b_{2},\ldots
b_{N}\right]$ will have full rank. The coefficients can then be obtained from
the equation given in (14).

$c=B_{n}^{-1}f$ (14)

where $c=\left[c_{1},c_{2},\ldots c_{N}\right]^{T}$.

To illustrate this in a $3$-qubit system, let us consider the ways we can get
the product states. Let the qubits in the $3$-qubit system be $q_{1},q_{2}$
and $q_{3}$. There are ${3\choose 1}=3$ ways of choosing one qubit, i.e.,
$q_{1}$, $q_{2}$ or $q_{3}$. There are ${3\choose 2}=3$ ways of choosing two
qubit product states, $q_{1}q_{2}$, $q_{2}q_{3}$ and $q_{3}q_{1}$. There is
${3\choose 3}=1$ way of choosing 3 qubit product states, i.e.
$q_{1}q_{2}q_{3}$. Finally, we have the constant term or the intercept term
which comes from ${3\choose 0}$. Overall, we have $8$ states each with $8$
elements. The structure of the basis matrix is given in Fig. 4. The first
column having all $1$s correspond to the constant term.

Figure 4: Basis matrix structure for a 3-qubit system

If we substitute the values of $q_{1},q_{2}$ and $q_{3}$ from $QS_{3}$ (refer
(13) above) in the matrix given in Fig. 4, we will get the basis matrix
$B_{3}$ as given below.

$B_{3}=\left[\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\ 1&0&0&1&0&0&0&0\\\
1&0&1&0&0&0&0&0\\\ 1&0&1&1&0&0&1&0\\\ 1&1&0&0&0&0&0&0\\\ 1&1&0&1&0&1&0&0\\\
1&1&1&0&1&0&0&0\\\ 1&1&1&1&1&1&1&1\\\ \end{array}\right]$ (15)

We can then represent any vector in $\mathcal{R}^{8}$ as a linear combination
of the columns in $B_{3}$.

###  3.1 Encoding lattice structure in qubit states

The vectors $\Delta a$ and $\Delta b$ in (3) can be represented using the
basis vectors in $B_{3}$. The coefficients can be estimated from the following
operations.

$\displaystyle c_{\Delta a}$ $\displaystyle=B_{3}^{-1}\Delta a$ (16)
$\displaystyle c_{\Delta b}$ $\displaystyle=B_{3}^{-1}\Delta b$

The coefficients so obtained for the cubic lattice are as follows

$c_{\Delta a}=\left[1.0,-2.0,-1.0,0.0,2.0,0.0,-1.0,2.0\right]$

and

$c_{\Delta b}=\left[0.0,0.0,1.0,1.0,-2.0,-2.0,-1.0,2.0\right]$

. In other words we can express $\Delta a$ and $\Delta b$ as follows.

$\displaystyle\Delta a$
$\displaystyle=1-2q_{1}-q_{2}+2q_{1}q_{2}-q_{3}q_{1}+2q_{1}q_{2}q_{3}$ (17)
$\displaystyle\Delta b$
$\displaystyle=q_{2}+q_{3}-2q_{1}q_{2}-2q_{2}q_{3}-q_{3}q_{1}+2q_{1}q_{2}q_{3}$

With $\Delta a$ and $\Delta b$ we can update the coordinates of bead $i+1$
when the coordinates of bead $i$ are known. The coordinates of course will be
in a state of superposition of all the options given in Fig. 3 (b).

We can do a similar exercise for the FCC lattice, where there are 4 options in
a given plane. We will need $\log_{2}4=2$ qubits. The matrix $QS_{2}$ to form
the matrix $B_{2}$ is given as follows.

$\displaystyle QS_{2}=\left[\begin{array}[]{cc}q_{1}&q_{2}\\\ \hline\cr 0&0\\\
0&1\\\ 1&0\\\ 1&1\\\ \end{array}\right]$ (23)

The basis $B_{2}$ will have the form shown in Fig. 5.

Figure 5: Basis matrix structure for a 2-qubit system

By substituting the values of the $q_{1}$ and $q_{2}$ in $QS_{2}$ in the
matrix given in Fig. 5, we will get the basis matrix $B_{2}$ as given below.

$B_{2}=\left[\begin{array}[]{cccc}1&0&0&0\\\ 1&0&1&0\\\ 1&1&0&0\\\ 1&1&1&1\\\
\end{array}\right]$ (24)

As in the previous case, the vectors $\Delta a$ and $\Delta b$ in (1) can be
represented using the basis vectors in $B_{2}$. The coefficients can be
estimated the following way.

$\displaystyle c_{\Delta a}$ $\displaystyle=B_{2}^{-1}\Delta a$ (25)
$\displaystyle c_{\Delta b}$ $\displaystyle=B_{2}^{-1}\Delta b$

The coefficients obtained for the FCC lattice are as follows.

$c_{\Delta a}=\left[0.5,-1.0,-1.0,2.0\right]$

and

$c_{\Delta b}=\left[0.5,-1.0,0,0\right]$

. We can thus express $\Delta a$ and $\Delta b$ as follows.

$\displaystyle\Delta a$ $\displaystyle=0.5-q_{1}-q_{2}+2q_{1}q_{2}$ (26)
$\displaystyle\Delta b$ $\displaystyle=0.5-q_{1}$

###  3.2 Selection of the plane of turn

Since there are $3$ orthogonal planes ($x-y,y-x$ and $z-x$), we need two more
qubits more to make a selection of one of the planes. Let these two qubits be
denoted as $q_{4}$ and $q_{5}$. The selection could be based on the
computational basis states of the $q_{4}$ and $q_{5}$ as follows.

$\begin{array}[]{ccc}\textrm{Plane}&\textrm{Basis state}&\textrm{Qubit
encoding}\\\ \hline\cr y-z&01&(1-q_{4})q_{5}\\\ z-x&10&q_{4}(1-q_{5})\\\
x-y&11&q_{4}q_{5}\end{array}$ (27)

With the given selection criteria, we have to update equation (2) as follows.

$\displaystyle\begin{aligned} x_{i+1}&=x_{i}+q_{4}q_{5}\Delta
a_{k}+q_{4}(1-q_{5})\Delta b_{k}\\\ y_{i+1}&=y_{i}+(1-q_{4})q_{5}\Delta
a_{k}+q_{4}q_{5}\Delta b_{k}\\\ z_{i+1}&=z_{i}+q_{4}(1-q_{5})\Delta
a_{k}+(1-q_{4})q_{5}\Delta b_{k}\\\ \end{aligned}$ (28)

The coordinates of all the beads will be updated using equation (28) and
equation (26). Overall, we will need $5$ qubits to encode a turn going from
one bead to the next in a cubic lattice and $4$ qubits to encode a turn in the
FCC lattice. $2$ qubits for selecting a plane and
$\left\lceil{\log_{2}N}\right\rceil$ qubits for encoding the bond direction
(or turn) in that plane. If the peptide chain is represented by $m$ beads, we
will need $(\left\lceil{\log_{2}N}\right\rceil+2)\times(m-1)$ qubits to encode
the folded structure.

###  3.3 Extending it to other structures

This concept can be extended to other lattice structures as well. We need not
have turns along orthogonal planes like the cubic or FCC lattices. The planes
could be at an angle to the working plane. As long as the every bead or
lattice point has equal and identical degrees of freedom, we could use the
methodology explained here. In case, the lattice involves directions that are
always at angle, we could do away with the qubits involved in plane selection
and directly go for encoding directions to the computational basis states.
Under such circumstances, the coordinate updates following a turn would be of
the form given in (29).

$x_{i+1}=x_{i}+\Delta a_{k},~{}~{}y_{i+1}=y_{i}+\Delta
b_{k},~{}~{}z_{i+1}=z_{i}+\Delta c_{k}$ (29)

The increments $\Delta a_{k},\Delta b_{k}$ and $\Delta c_{k}$ can be obtained
by using a variant of the equation (30).

$\displaystyle c_{\Delta a}$ $\displaystyle=B_{3}^{-1}\Delta a$ (30)
$\displaystyle c_{\Delta b}$ $\displaystyle=B_{3}^{-1}\Delta b$ $\displaystyle
c_{\Delta c}$ $\displaystyle=B_{3}^{-1}\Delta c$

If there are $N$ unique directions, we could use
$\left\lceil{\log_{2}N}\right\rceil$ qubits to encode each of these
directions.

##  4 Conclusions

In this article, we discussed a methodology to encode lattice turns to qubit
computational basis states, that could find use in protein structure
prediction, polymer structure study and other coarse-grained models. We showed
how a combination of qubit states could be used to span the space of
directions. These directions could be along planes that are orthogonal or non-
orthogonal to each other. We took specific examples of cubic and FCC lattice.
However, they could be extended to other lattice structures as well.

## References

  * [1] W. E. Hart and A. Newman, “Protein structure prediction with lattice models,” http://dimacs.rutgers.edu/~alantha/papers2/alantha-bill-bc.pdf.
  * [2] S. Kmiecik and et. al., “Coarse-grained protein models and their applications,” _ACS Chem. Rev._ , vol. 116, pp. 7898–7936, June 2016.
  * [3] M. Fingerhuth, T. Babej, and C. Ing, “A quantum alternating operator ansatz with hard and soft constraints for lattice protein folding,” _arXiv preprint arXiv:1810.13411_ , 2018.
  * [4] A. Perdomo, C. Truncik, I. Tubert-Brohman, G. Rose, and A. Aspuru-Guzik, “Construction of model hamiltonians for adiabatic quantum computation and its application to finding low-energy conformations of lattice protein models,” _Phys. Rev. A_ , vol. 78, p. 012320, Jul 2008.
  * [5] A. Robert, P. Barkoutsos, S. Woerner, and I. Tavernelli, “Resource-efficient quantum algorithm for protein folding,” _npj Quantum Information_ , vol. 7:38, 2021.
  * [6] J. P. Vasavi and et. al., “An approach to solve the coarse-grained protein folding problem in a quantum computer,” _arXiv preprint arXiv:2311.14141v1_ , 2023.


Further author information: Send correspondence to TKS<EMAIL_ADDRESS>or JB,
<EMAIL_ADDRESS>
# Trajectory control using an information engine

Tushar K. Saha  Simon Fraser University, Department of Physics, Burnaby, BC,
Canada  John Bechhoefer  Simon Fraser University, Department of Physics,
Burnaby, BC, Canada

###### Abstract

We have built an information engine that can transport a bead in a desired
direction by using favorable fluctuations from the thermal bath. However, in
its original formulation, the information engine generates a fluctuating
velocity and cannot control the position of the bead. Here, we introduce a
feedback algorithm that can control the bead’s position, to follow a desired
trajectory. The bead can track the path if the maximum desired velocity is
below the engine’s maximum average velocity. Measuring the range of frequency
that the feedback algorithm can track, we find a bandwidth that is slightly
lower than the corner frequency of the bead in the trap.

###### keywords:

Information engines, optical feedback tweezers, stochastic control

##  1 INTRODUCTION

Information engines are a new class of engines that can convert information
about a system into useful energy. This class of engine was first imagined by
Maxwell [1] more than 150 years ago. The considerable subsequent discussion in
the literature to understand their operation has refined our understanding of
the second law of thermodynamics.[2, 3] Recent technical advances have made it
possible to realize these engines in the laboratory.[4, 5, 6, 7, 8, 9, 10, 11,
12] These technological developments include accurate and fast measurement of
the state of microscopic systems and high computation speed, to apply feedback
quickly. In the last decade, experiments have produced working Maxwell demons
[4, 13] and tested the cost of operating the associated measuring device [14,
15, 16, 17]. The focus of research has accordingly shifted from demonstrating
the existence of the “Maxwell demon” to understanding ways to maximize the
performance [18] and efficiency of these engines [19, 20, 21]. Experiments
have also demonstrated related devices such as flashing ratchets [22, 23, 24]
and information ratchets[25, 26, 27]. In almost all of these studies, the
focus was on the conversion of information into directed motion. In Ref. 18,
we designed a “useful” information engine that can both store energy in a work
reservoir and transport a bead at an average velocity in a desired direction.

Although our engine can control the average velocity, it cannot control the
bead’s position. As we will see below, random fluctuations lead to a diverging
bead position with time. This observation is a consequence of the previously
used ratcheting algorithms, which do not control the bead trajectory in space
and time. In this article, we introduce an improved feedback algorithm to
limit the position divergence and control the bead’s trajectory. Trajectory
control makes possible practical applications such as transporting a bead to a
specific site at a particular time. Although other techniques such as photon
nudging[28], fluid flow[29], and magnetic steering [30] can also control the
bead trajectory, such engines use conventional feedback algorithms that
directly apply external forces to the bead. By contrast, information engines
are powered by rectifying thermal fluctuations. No additional forces are
applied to the bead, and no external work is done: The engine is powered
purely by information. Here, we present a heuristic algorithm that can control
the bead trajectory while respecting the no-work constraint, and we
characterize the performance of the algorithm using techniques from control
theory.[31] We find that the performance of the engine is essentially limited
by the engine’s material properties and not by the algorithm. We also find
that the force constraint in the information engine does not affect the
tracking performance significantly when compared to driving the bead by
directly applying large forces.

##  2 INFORMATION ENGINE

In previous work[18], we developed an information engine that could lift an
optically trapped bead at an average velocity against gravity. In this
article, to simplify the discussion, we move the bead perpendicular to gravity
to remove the effect of gravitational forces from the bead dynamics. Note that
the same algorithm will also work for raising it against gravity.
Conceptually, the engine can be viewed as a microscopic spring-mass system in
contact with a thermal bath, as shown in Fig. 1a. Because of the thermal
noise, the mass fluctuates about an equilibrium position. Let the desired
directed velocity be to the left. Occasionally, the bead fluctuates left,
compressing the spring. The favorable “left” fluctuation is captured by
immediately shifting the trap left, decompressing the spring. The
corresponding energy of the spring-mass system is shown in Fig. 1b. The energy
is visualized as a bead in a harmonic well. One important but subtle point is
that the distance the spring (bottom of the harmonic well) is shifted should
be chosen so that no external work is done by the spring on the bead. This is
ensured by shifting the minimum of the well so that the bead is reset to the
equi-potential state on the opposite side of the well, as shown in Fig. 1b
(iii). In effect, the information engine converts heat from the bath into
directed motion.

Figure 1:  Schematic diagram of the information-engine concept. (a) Ratcheted
spring-mass system moves to the left; (b) equivalent dynamics in a harmonic
well.  Figure 2:  Schematic diagram of the experimental set-up. AOD $=$
Acousto-Optic Defector, FPGA $=$ Field-Programmable Gate Array, and QPD $=$
Quadrant Photodiode. Representative components are not to scale.

Figure 2 shows the experimental set-up, which consists of a 3-µm-diameter bead
trapped by optical tweezers. Optical trapping is produced by a tightly focused
532-nm green laser (HÜBNER Photonics, Cobolt Samba, 1.5 W, 532 nm) using a
microscope objective (60x Olympus water-immersion objective 1.2 NA). A 660-nm
red laser (Thorlabs, LP660-SF20, 20 mW, 660 nm) is used as the detection laser
by loosely focusing the laser beam on the bead using a second objective (40x
Nikon, NA = 0.5). The two objectives are aligned to match their focal planes
inside the sample chamber. Two acousto-optic deflectors (AODs, DTSXY-250-532,
AA Opto Electronic) are used to steer the position of the trapping beam
parallel and perpendicular to gravity. The red (660 nm) light collected from
the trapping plane is focused on a quadrant photodiode (QPD) for position
detection. The position of the bead is measured every 20 µs (50 kHz) by the
A/D converter of a Field Programmable Gate Array (FPGA, National Instruments,
NI PCIe-7857), which calculates the feedback control signal response for the
AOD. Further details on the experimental apparatus can be found in Ref. 32.

##  3 SYSTEM DYNAMICS

###  3.1 Bead dynamics

The equation of motion of the trapped bead is described, in one dimension, by
the overdamped Langevin equation

$\displaystyle\gamma\dot{x}(t)=\underbrace{-\kappa\left(x(t)-\lambda(t)\right)}_{\text{restoring
force}}+\underbrace{\sqrt{2k_{\rm{B}}T\gamma}\ \nu(t)}_{\text{thermal noise}}\
,$ (1)

where $x(t)$ is the position of the bead at time t, $\lambda(t)$ the trap
center, $\gamma$ the friction coefficient, and $\nu(t)$ is Gaussian white
noise, with $\langle\nu(t)\rangle=0$ and
$\langle\nu(t)\nu(t^{\prime})\rangle=\delta(t-t^{\prime})$. The bead
measurement and feedback occur at discrete time intervals $t_{\mathrm{s}}$. By
integrating the dynamics over the interval $t_{\mathrm{s}}$ and accounting for
the relaxation of the bead during the time interval, we can derive a discrete-
time equation of motion[33]

$\displaystyle
x_{n+1}=\operatorname{e}^{-t_{\mathrm{s}}/\tau_{\mathrm{r}}}x_{n}+(1-\operatorname{e}^{-t_{\mathrm{s}}/\tau_{\mathrm{r}}})\lambda_{n}+\sqrt{1-\operatorname{e}^{-2t_{\mathrm{s}}/\tau_{\mathrm{r}}}}\zeta_{n},$
(2)

where $x_{n}\equiv x(nt_{\mathrm{s}})$ is the position at the
$n^{\mathrm{th}}$ time step, $\tau_{\mathrm{r}}=\gamma/\kappa$ the trap
relaxation time, and $\zeta_{n}$ is a Gaussian random variable with zero mean
and unit variance. That is, $\langle\zeta_{n}\rangle=0$, and
$\langle\zeta_{m}\zeta_{n}\rangle=\delta_{mn}$.

###  3.2 Trap dynamics

Figure 3:  Bead trajectories for constrained and unconstrained feedback
algorithms. (a) Mean bead position (solid red line shows average over 184
trajectories) for unconstrained feedback algorithm, with ten representative
trajectories. (b) Mean bead position (solid red line shows average over 132
trajectories) for the tracking feedback algorithm, with five representative
trajectories. Thin solid red lines represent two standard deviations of the
bead trajectories in (a), (b). The data in (a) is from Ref. 18.

As described earlier, the bead is propelled in the medium by the operation of
the information engine, which exploits favorable fluctuations. Since these
fluctuations occur randomly in time, beads in different trials follow
different paths. Figure 3a shows the bead positions (light red lines) when
using the previously described algorithm,
$\lambda_{n+1}=\lambda_{n}+\alpha(x_{n}-\lambda_{n})$ when
$x_{n}>\lambda_{n}$; else, $\lambda_{n+1}=\lambda_{n}$. [18] For a fixed
diameter and trap stiffness, the bead moves at an average, but fluctuating
velocity (red solid line). Note that individual trajectories diverge from the
average; the thin red lines represent two standard deviations of the bead
trajectories. To restrict the divergence of trajectories with time, we
introduce a tracking feedback control, given by

$\displaystyle\lambda_{n+1}=\begin{cases}\lambda_{n}+\alpha(x_{n}-\lambda_{n})\,,&\quad
r_{n}>x_{n}>\lambda_{n}\,,\,\text{or }\;r_{n}<x_{n}<\lambda_{n}\,,\\\
\lambda_{n}\,,&\quad\mathrm{otherwise}\,,\end{cases}$ (3)

where $\alpha$ is the feedback gain. The feedback algorithm depends on the
desired reference position $r_{n}$ and can track it in either direction. In
the previous algorithm, the bead could ratchet only in one direction.

The feedback gain is tuned to $\alpha=1.7$, which ensures that the average
input work is zero [18]. Figure 3b shows the average position (solid red line)
of the bead and five individual trajectories (light red lines). The
trajectories do not diverge with time. The desired velocity in the constrained
algorithm was chosen to be slower than the average velocity obtained in the
unconstrained case – hence the smaller slope in Fig. 3b. The information
engine cannot track velocities faster than the unconstrained feedback
algorithm.

A schematic diagram of the feedback algorithm is shown in Fig. 4. To
understand it, we note that if, after a position measurement, the trap force
is already in the “right” direction (i.e., that it will push the particle
towards the reference), then no action is taken by the controller. But for
forces in the “wrong” direction (away from the reference), the controller
ratchets to correct the bead position. Notice that the algorithm ratchets when
the bead position is measured to be between the reference and the trap. The
algorithm thus captures all fluctuations that push the trap towards the
reference signal. Some fluctuations overshoot the reference or push the bead
in the opposite direction. In such cases, the trap position stays constant
until the next favorable fluctuation. By neglecting the unfavorable
fluctuations, the feedback rule constrains the position variance to a narrow
band (thin red lines), as shown in Fig. 3b. The trap stiffness for all the
experiments is 40 pN/µm, and the corresponding average velocity of the
unconstrained information engine is $v_{\mathrm{max}}=12$ µm/s[18]. We also
tried a feedback algorithm that ratcheted after every measurement, so as to
always push the particle towards the reference. However, we found the $\alpha$
values required for zero input work led to persistent oscillation of the bead
about the desired trajectory.

Figure 4:  Schematic of the feedback algorithm. (a) Trap position ($\lambda$),
bead position ($x$) and reference ($r$) for a trapped bead. (b) Trap position
ratchets when $\lambda<x<r$ or when $r<x<\lambda$. (c) No ratcheting for other
combinations of $r$ and $\lambda$, where the red circles represent the bead
positions.

##  4 Trajectory control

The average velocity of an information engine depends on the trap stiffness
and bead diameter[18]. Therefore, a bead powered by an information engine
cannot track a reference signal that requires a higher velocity than that
allowed by the thermal fluctuations. Figure 5 shows the trajectory of the bead
(gray) for the desired sine wave reference (black). We can change both the
amplitude and the frequency of the sine wave that is followed by the bead. We
also show the error signal (residual), $r-x$, for each of the sine waves. For
comparison, the position of a bead in a static trap is shown in Fig. 5d. This
is equivalent to $r(t)=0$, with $\lambda(t=0)=0$ as the trap does not ratchet
in this case.

Figure 5:  Controlled trajectory of the bead to follow sine waves. The bead
position ($x$, gray) and trap position ($\lambda$, blue) for different
reference ($r$, black) signals, and the corresponding residual. The plot
represents $r(t)=$ (a) $2A\,\text{sin}(\omega t)$, (b) $A\,\text{sin}(\omega
t)$, (c) $A\,\text{sin}(5\omega t)$, and (d) $r(t)=\lambda(t)=0$, where
$A=165$ nm and $\omega=2\pi\times 1$ Hz.

Figures 3b and 5 show example trajectories where the reference waves are well
tracked. In all these cases, the magnitude of the velocity of the reference,
$|\dot{r}(t)|$, is always below the maximum unconstrained velocity,
$v_{\mathrm{max}}$. However, the algorithm cannot respond to a reference that
changes arbitrarily quickly, even when $|\dot{r}(t)|<v_{\mathrm{max}}$. To
quantify the range of frequency the feedback algorithm can track, we measure
the transfer function of the controller.[31] The schematic block diagram of
the feedback loop is shown in Fig. 6. Based on the error $r_{n}-x_{n}$, the
controller $K$ sends an input to the system $G$. The controller is the
feedback algorithm in Eq. 3, and the input is the force due to the trap on the
bead. The corresponding output response $x$, the bead position, is measured.
In addition to the reference input, there is also an input to the system that
arises from the thermal noise $\zeta$. The closed-loop response of this system
is then given by

$\displaystyle x$ $\displaystyle=\frac{KG}{1+KG}\,r+\frac{1}{1+KG}\,\zeta$
(4a) $\displaystyle\equiv T\,r+S\,\zeta$ (4b)

where $S$ and $T$ are the sensitivity and complementary sensitivity functions,
respectively.[31] Assuming that thermal fluctuations are always smaller than
the reference at each reference frequency $f$, we can neglect the contribution
of $S$. The closed-loop transfer function is then given by

$\displaystyle T(f)=\frac{x(f)}{r(f)}\,,$ (5)

where $x(f)$ and $r(f)$ are the Fourier transforms of the bead position $x(t)$
and reference $r(t)$, respectively.

Figure 6:  Block diagram of the feedback loop. K denotes the controller, G the
system, and $\zeta$ the thermal noise.

To experimentally measure the transfer function, we input reference sine
waves. For each frequency, the amplitude of the reference sine wave is chosen
so that it is always below the maximum achievable velocity at the given
frequency. In particular, for an input reference $r(t)=A\sin{\omega t}$, with
$\omega=2\pi f$, the amplitude $A$ is chosen to be below
$v_{\mathrm{max}}/\omega$. We record the reference signal and the
corresponding bead trajectory and then calculate the power spectrum density of
the bead position and reference signal. Figure 7a shows the power spectrum
density of the trapped bead, $|x(f)|^{2}$, when it is driven by thermal
fluctuations from the surrounding water bath.

Figure 7:  Transfer functions. (a) Thermal noise power spectrum density of a
trapped bead (gray) calculated from a 20-s time series. Black solid line shows
fit to an aliased Lorentzian[34] with the corner frequency
$f_{\mathrm{c}}=254\pm 2$ Hz. (b) Experimental data for $|T|^{2}$ for
different reference frequency for information engine (blue solid markers) and
open-loop (red solid markers). Blue and red dashed lines are fit to Eq. 6 with
$f_{0}=186\pm 11$ Hz for the information engine and
$f_{0}^{\mathrm{OL}}=240\pm 13$ Hz for direct driving by sinusoidal reference
inputs. Black solid line in (b) is the amplitude-normalized fit in (a).

##  5 Measurement of the feedback bandwidth

Figure 7b shows the reference-bead transfer function, defined as the ratio of
the power of the bead and the reference as a function of reference frequency.
We find that the bead follows the reference trajectory at low frequencies but
not at high frequencies (blue solid markers). The feedback algorithm thus acts
as a low-pass filter. After 1000 Hz, the power of the bead response to the
reference goes below the thermal noise and cannot be measured. The measured
frequency response is well-fit by a first-order low-pass filter [31]

$\displaystyle|T|^{2}=\left|\frac{x(f)}{r(f)}\right|^{2}=\frac{1}{1+(f/f_{0})^{2}},$
(6)

where $f_{0}$ is the frequency bandwidth of the algorithm. The DC gain
(numerator) is 1, as the reference and bead position are measured in the same
units. The fit to the experimental data is shown as the blue dashed line in
Fig. 7b. For comparison, we measure the transfer function of the trapped bead
(red solid markers) by driving the trap with the desired reference sine wave
without feedback i.e., in open-loop. We fit the data to Eq. 6, shown as red
dashed line, to find the corresponding bandwidth $f_{0}^{\mathrm{OL}}$. We
find that the bandwidth $f_{0}=186\pm 11$ Hz of the feedback algorithm is
slightly lower than the bandwidth $f_{0}^{\mathrm{OL}}=240\pm 13$ Hz of the
open-loop driving. Naively, one would expect that a conventional open-loop
driving, without any force constraints, would track the reference
significantly better than the force-constrained information engine. However,
we find that the open-loop driving has a bandwidth comparable to that of an
information engine.

The amplitude-normalized power spectrum found from the fit is also shown in
Fig. 7b (black solid line). Note that the open-loop bandwidth
$f_{0}^{\mathrm{OL}}=240\pm 13$ Hz and corner frequency $f_{\mathrm{c}}=254\pm
2$ Hz of the trap match well, as they are equivalent methods to measure the
bandwidth. Since the bead response, when driven by an information engine, will
be limited by the thermal fluctuations in the system, a close match between
the controller bandwidth $f_{0}=186\pm 11$ Hz and the corner frequency
$f_{\mathrm{c}}$ suggests that any other feedback algorithm will only
marginally improve the tracking performance. Thus, the information engine’s
response seems limited by the rate of thermal fluctuations in the system, and
not by the feedback algorithm. The amplitude and frequency of the input sine
wave in Fig. 7b are given in Table 1.

Figure 8:  Trajectory control to follow a sine wave. The bead position (gray)
for different reference sine waves (black) of amplitude (a) 8 nm and (c) 1 nm
at a frequency of 10 Hz. The standard deviation of the bead in the trap
$\sigma=10$ nm. (b), (d) show the corresponding power spectra calculated from
time series of duration 20 s.

With the feedback algorithm in Eq. 3, the bead could follow a sine wave of
amplitude 8 nm. The amplitude is comparable to the standard deviation of the
bead in the trap, $\sigma=10$ nm. The bead trajectory (gray) and the reference
(black) are shown in Fig. 8a. The corresponding power spectrum density shown
in Fig. 8b, has a peak at 10 Hz which corresponds to the frequency of the
reference sine wave. We found that the bead could also track a sine wave with
amplitude 1 nm, an order of magnitude smaller than $\sigma$, at 10 Hz (Fig.
8c). As the amplitude is smaller than $\sigma$, one cannot see in Fig. 8c that
the bead does in fact follow the reference signal. However, a peak at 10 Hz in
the corresponding power spectrum density (Fig. 8d), confirms that the bead
follows the reference signal. Of course, tracking motion at such a low
amplitude is possible only because the desired trajectory is periodic, which
allows averaging that reduces the effect of noise. We would expect to be able
to track a non-repeating waveform only if its amplitude exceeds the thermal
noise level, $\sigma$.

##  6 Conclusion

We have developed a feedback algorithm that can constrain a bead to follow a
desired trajectory, with zero input trap work. The feedback algorithm
continuously ratchets when the bead lags the reference wave in either
direction and stops when it reaches the reference. The algorithm is
reminiscent of the bang-bang algorithm from control theory[31] where the
controller applies zero or maximum force in either direction to control the
dynamics. It might be possible to further improve the tracking bandwidth, but
we do not expect a significant increase, as the bandwidth $f_{0}$ of our
algorithm is already close to the corner frequency $f_{\mathrm{c}}$ of the
trap. Although conventional feedback algorithms can reduce the position
variance of the bead below the trap standard deviation $\sigma$, the
constraint that the algorithm not allow any work to be done on the particle is
a severe one, and it is not clear whether reducing the variance of
fluctuations around a reference to values below that of the natural trap is
possible for an information-fueled engine.

Controlled manipulation of microscopic systems has potential application in
drug delivery.[35] A major challenge for controlling the trajectory of
microscopic systems is the thermal fluctuations that push them away from the
target. On the contrary, information engines can convert these fluctuations
into “useful” motion. With our feedback algorithm, an information engine can
control the bead trajectory and limit the position variance to a width
comparable to $\sigma$. This feature could help transport a bead to a specific
location, with a precision of $\sigma$, at a time defined to a precision
$\tau_{\mathrm{R}}$. For our experiment, we recall that $\sigma=10$ nm and
$\tau_{\mathrm{R}}=0.6$ ms. Experiments have also shown that carefully
designed feedback algorithms can reduce the dissipation in stochastic
systems.[36]. Integrating the ability to efficiently use thermal fluctuations
and reducing thermal dissipation could also further improve the performance of
current micro-bots.[35]

###### Acknowledgements.

We thank Jannik Ehrich for useful comments on the manuscript. This research
was supported by grant number FQXi-IAF19-02 from the Foundational Questions
Institute Fund, a donor-advised fund of the Silicon Valley Community
Foundation.

##  Appendix A Experimental parameters

Information engine | Frequency (Hz) | 1 | 3.3 | 10 | 40 | 100 | 150 | 300 | 500 | 750 | 1000
---|---|---|---|---|---|---|---|---|---|---|---
| Amplitude (nm) | 165 | 165 | 165 | 45.4 | 17.3 | 13.2 | 4.9 | 3.3 | 2.0 | 1.7
Open-loop | Frequency (Hz) | 2 | 5 | 20 | 50 | 120 | 180 | 350 | 600 | 800 | 1200
| Amplitude (nm) | 165 | 165 | 165 | 165 | 165 | 165 | 165 | 82.5 | 82.5 | 82.5

Table 1: Amplitude and frequency of the reference wave for the information
ratchet and open-loop driving.

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