distribution will result in minimum entropy production. The analysis shows
that the expectation of heat that is being generated by this physical process
during the execution of computation for a given input distribution is
infinite.

To get a clear view of the two physical processes we will give a short
comparison between the physical processes.

(1) For both the physical process, the minimum amount of heat that is required
for the computation to generate output $y$ is bounded by a constant. The
constant defined for these two processes have no relation among them, but the
thermodynamic cost for the dominating physical process is larger in principle
than that of the coin-flipping.

(2) For the coin flipping process, one has to know the shortest route for the
output $y$ to get the bounded form of the heat production, whereas, in the
case of dominating process the condition is quite simple and advantageous. The
condition says that we can get the bounded form if the computation is fed by
input which demands the print of the output $y$.

(3) The heat function happens to be an upper semicomputable function for the
dominating physical process, whereas it is a lower semicomputable function for
the coin-flipping process. So $Q_{dorl}(z)-Q_{cofp}(z)>c_{\gamma}$ where
$c_{\gamma}$ is a real number. The excess amount of heat that is generated in
the dominating physical process is bounded. It is expressed as
$Q_{dorl}(z)-Q_{cofp}(z)\leq lnK(UTM(z))$.

The above methods have provided an insight into the physical realization of
the TM from a thermodynamic viewpoint. Turing machine happens to be the center
of attraction to both physics and computer science. So the physical
realization of TM with a more feasible and realistic model needs further
investigation.

##  X  Error Correction: Thermodynamic Interpretation

After the advent of quantum error correction theory a132 ; a169 ; a170 , it
has experienced a rapid development. Some of the quantum codes were jut mapped
from the classical error correction codes mainly the CSS codes a170 . It was
then further generalized to the stabilizer codes a173 ; a174 ; a175 ; a176 .
Firstly, we will discuss an unsophisticated model for the analysis of error
correction as described in the work a176a . Here in this work, the authors
have considered a reversible cycle to explain the error correction process.
This cycle is a modified version of Bennett’s ‘Maxwell’s demon’. We will
explain the classical codes from a thermodynamics point of view, then we will
explore the quantum error correction theory using this reversible cycle. From
a thermodynamics viewpoint, quantum error correction is equivalent to a
refrigerator process. In this process, one tries to sustain a steady entropy
for the system even when it is subjected to the environmental noises. Due to
the decrease in the entropy, one might intuitively think that the protocol
defined for the quantum error correction violates the second law of
thermodynamics. A careful analysis of the process will show that it does not
violate the second law.

The author in their work a176a  has considered a singlet atom confined in a
box to link the information with thermodynamics. The classical information can
be modeled using this system, where the atom when happens to be in the ‘left-
hand side’ (LHS) is described as 0, and when in ‘right-hand side’ it will be
described as 1. Now we allow the system to expand isothermally considering
that the atom exists in one of the sides. Due to this expansion, one
encounters an increase in the entropy by an amount of $\Delta S=k_{B}\,ln2$.
The atom has an equally likely probability to jump in the either side of the
box from its initial state. This comprises the error that can occur in the
system. The protocol of the error correction is described below:

(1) We will consider that the atoms are in LHS and RHS of the respective
boxes.

(2) Let us consider that some error occurs to the particle in the box
$\mathcal{A}_{\alpha}$. The error is defined as the probability of the atom in
the box $\mathcal{A}_{\alpha}$ to be in the LHS or RHS of the box.

(3) Another system $\mathcal{B}_{\alpha}$ is kept to keep a track on the
system $\mathcal{A}_{\alpha}$. The system $\mathcal{B}_{\alpha}$ correlates
itself with the system $\mathcal{A}_{\alpha}$.

(4) Based on the state of the system $\mathcal{B}_{\alpha}$ one will move the
system $\mathcal{A}_{\alpha}$ to its respective side. This leaves the system
$\mathcal{B}_{\alpha}$ in a randomized state. The execution of this process
needs no work.

(5) This is the last step of the protocol where the system
$\mathcal{B}_{\alpha}$ is brought back to its initial state by isothermal
compression.

The pictorial representation of the protocol is described in Fig. 12. The
energy of the atom in the system $\mathcal{A}_{\alpha}$ is $k_{B}T\,ln2$. Due
to the error, the system undergoes a decrease in the energy by an amount
$\Delta F=-k_{B}T\,ln2$, and the entropy of the system increase by an amount
$S=k_{B}ln2$. In the third stage of the cycle, the system
$\mathcal{B}_{\alpha}$ has the information of the $\mathcal{A}_{\alpha}$ as
they happen to be correlated. And in the final stage of the cycle, the
$k_{B}T\,ln2$ amount of work is done in the system $\mathcal{B}_{\alpha}$ to
reset the system. The amount of entropy change during this stage of the
process is $S=-k_{B}\,ln2$. So the error that has been described in this model
is nothing but the inability to do work. So to make the system free of this
inability the system is correlated to another system. This makes the second
system to possess the same inability. In the reset, step entropy is wasted so
the system regains its initial state. Now we will analyze the error correction
in the quantum realm using this protocol.

Figure 12: A pictorial representation of the reversible cycle which implement
the classical error correction protocol is depicted. The detail description of
each stage of the cycle are given in the text.

The main motivation of the quantum error correction theory is to preserve the
quantum state. The protocol is described below for the pure states. One can
similarly describe the protocol for the mixed states. The protocol is as
follows:

(1) The state of the system after the introduction of the error is described
as $\sum_{j}Err_{j}|\psi_{cs}\rangle|m\rangle|env_{j}\rangle$, where
$|\psi_{co}\rangle$ describes the encoded state, $|m\rangle$ represents the
state of the measurement device, and the environmental state is described
$|env\rangle_{j}$. The environmental states are orthogonal in nature. Here, it
has been assumed that $\langle env_{j}|env_{k}\rangle=\delta(j,k)$.

(2) Now in the next phase of the protocol, the environmental effect is traced
out. The state of the system after tracing out is
$\sum_{j}Err_{j}|\psi_{co}\rangle\langle\psi_{co}|Err_{j}^{\dagger}\otimes|m\rangle\langle
m|$.

(3) In this phase of the protocol, one observes the system, due to which
correlation is generated between the measurement apparatus and the system. The
state of the system is described as
$\sum_{j}Err_{j}|\psi_{co}\rangle\langle\psi_{co}|Err_{j}^{\dagger}\otimes|m_{j}\rangle\langle
m_{j}|$, where $\langle m_{j}|m_{k}\rangle=\delta(j,k)$. We have to keep in
mind that if the observation is not perfect, then the protocol will not work
properly, as we will not be able to get the orthogonal states of the system.

(4) During this phase of the cycle, the system gets uncorrelated. In this
phase, the correction of the error is done. The state of the system after the
execution of this step is
$|\psi_{co}\rangle\langle\psi_{co}|\otimes\sum_{j}|m_{j}\rangle\langle
m_{j}|$. This is not the exact state that we have started with. We have to
reset the measurement state to its initial state.

(5) This is the last stage of the protocol, where we reset the measurement
system to its initial state. To do this, we include a garbage system to the
system. Now, the state of the system is
$|\psi_{co}\rangle\langle\psi_{co}|\otimes\sum_{j}|m_{j}\rangle\langle
m_{j}|\otimes|m\rangle\langle m|$. The state of the system by swapping the
garbage system and the measurement system can be expressed as
$|\psi_{co}\rangle\langle\psi_{co}|\otimes|m\rangle\langle
m|\otimes\sum_{j}|m_{j}\rangle\langle m_{j}|$. Now the system is reset for
another cycle of quantum error correction.

Similar to the analysis done for the classical code, we will analyze the
entropy change that takes place throughout the protocol. The entropy of the
system after the first stage of the cycle is $\Delta S=S(\rho_{ab})$, where
$\rho_{ab}=\sum_{j}Err_{j}|\psi_{co}\rangle\langle\psi_{co}|Err_{j}^{\dagger}$
is the density matrix. We don’t encounter a change in the entropy for the
other steps except for the reset step. The entropy of the system during this
reset step is $\Delta S=-S(\rho_{ab})$. The garbage system encounters a gain
in the entropy of the system. The information gain by the system during the
correlation step is $S(\rho_{ab})$. So from the observation, we can infer that
the gain in the entropy by the garbage system is more than the information
gain. This confirms that the error correction protocol is successful from a
thermodynamic viewpoint.

In the work a176b , the author has further extended the model described in the
work a176a  to include the non-maximally mixed states for the analysis of
quantum error correction. In their work, they have conveyed that their system
exhibits a constant entropy even when subjected to the impact of the
environmental noises. In their model, they have introduced some ancilla qubits
which keeps a track of the error. In the refrigeration process, this is
equivalent to the transfer of information of the error from the data qubit to
the ancillary qubit. This causes an entropy change and cools downs the data
qubits. They have also explored the changes that occur in the quantum error
correction cycle when the measurement system is not perfect, and also the
information gain is not optimal. Another important aspect, i.e., how the non-
orthogonal states will have an impact on the thermodynamic analysis of the
error cycle is also addressed in this work.

Now we will explore a modern formalism to describe quantum error-correcting
condition from a thermodynamic viewpoint as studied in the work a78 .
Different interaction in a system has been modeled through various
Hamiltonian, like spin chain model. One have encounter applicability of spin
model in quantum information theory like in space free from decohorence a177 ,
entanglement a178 , gates a179 , and even in topological quantum computing
a180 ; a181 . Here in this work, the authors have used spin chain formalism to
develop the condition of the quantum codes.

In the seminal works a133 ; a136 ; a171 ; a172 , they have proposed the
necessary and the sufficient condition for the quantum codes to correct the
errors. The condition to correct the errors $err$ that appears during
transmission, or during storage in quantum devices can be defined as
$\langle\psi|Err_{a}^{\dagger}Err_{b}|\psi\rangle$, where $\psi$ is a vector
$\in\mathbb{C}$, and $Err_{a}$, $Err_{b}\in err$. The errors are defined as a
set of linear operators $err=\\{Err_{a}:\mathbb{B}\rightarrow\mathbb{B}\\}$,
where $\mathbb{B}$ represents the encoding space. This encoding space has a
set of orthonormal basics defined as $\\{|\psi_{i}\rangle\\}$. So, one can
alternatively define the condition for quantum error correction as
$\langle\psi_{j}|Err_{a}^{\dagger}Err_{b}|\psi_{i}\rangle=\varpi_{ab}\delta(i,j)$,
where $\varpi_{ab}$ is a constant.

For the analysis, the authors have first considered the XXO model. The Hilbert
space for this model is $\mathbb{B}\simeq(\mathbb{C}^{2})^{\otimes n_{c}}$,
where $n_{c}$ represents the size of the model or one can convey it as the
length of the lattice. The Hamiltonian of the system is described as

$\mathcal{H}=-\sum_{i=1}^{n_{c}}\left(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}+\iota\,\sigma_{j}^{z}\right),$
(43)

where $\sigma^{a}$ for $a=x,y,z$ describes the Pauli matrices and $\iota$
describes the magnetic field, and it is real in its form. This spin chain
model is also called as “isotropic XY model”, and was proposed in the seminal
work a182 . For any thermodynamical analysis, we need the partition function
$Z$ of the system, then we can extract the information of other thermodynamic
variables. The partition function for this system for $T>0$ can be evaluated
as

$Z=tr(2^{-\mathcal{H}/T})=\sum_{En}2^{n_{c}S}2^{-En/T},$ (44)

where $2^{nS}$ describes the degeneracy that exists within the energy levels,
and S depicts the entropy of the system. For this system, we observe that the
energy and the entropy are directly proportional to $n_{c}$.

Now for this model the partition function of the system can be evaluated as

$\displaystyle Z$ $\displaystyle=$ $\displaystyle
tr_{\mathcal{C}1}(2^{-\mathcal{H/T}})$ $\displaystyle\sim$
$\displaystyle\frac{n}{\pi}\int_{-\pi}^{\pi}ln(1+2^{-e(p)/T})\,dp,$

where $e(p)=-4\cos\,p+2\iota$. Here, the trace is taken over the subspace
$\mathcal{C}1$. So, the free energy for the system can be described as
$F_{en}=\frac{1}{2\pi}\int_{-\pi}^{\pi}ln(1+2^{-e(p)/T})$. So the entropy of
the system can be described as

$\displaystyle
S=\int_{-\pi}^{\pi}\Big{(}\frac{1}{2\pi}ln\left(\frac{1}{2\pi}\right)-\mathcal{J}(p)\,ln[\mathcal{J}(p)]$
$\displaystyle-\\{\left(\frac{1}{2\pi}\right)-\mathcal{J}(p)\\}\,ln\Big{[}\left(\frac{1}{2\pi}\right)-\mathcal{J}(p)\Big{]}\Big{)},$
(45)

where $\mathcal{J}(p_{i})=\frac{1}{n_{c}(p_{i+1}-p_{i})}$. Here $[p]$ denotes
the collection of the momentum of the particles. Now we define a correlation
function $\langle\mathbb{O}\rangle_{T}$, where $\mathbb{O}$ represents the
product of Pauli operators in linear combination. The mathematical definition
of this function is defined as

$\langle\mathbb{O}\rangle_{T}=tr\frac{2^{-\mathcal{H}/T}\mathbb{O}}{Z}.$

This is also called as the thermodynamic correlation function. For this model,
the correlation is evaluated as

$\langle\mathbb{O}\rangle_{T}=\lim_{n_{c}\rightarrow\infty}\frac{\langle[p]|\mathbb{O}|[p]\rangle}{\langle[p]|[p]\rangle}.$

The eigenvector for this system happen to be in thermo-equilibrium space with
same $\mathcal{J}$. The thermodynamic correlation function is independent of
the wave vectors $|\psi\rangle$.

In the field of statistical mechanics the probability that the system
collapses in an eigenstate $|\phi_{b}\rangle\in\mathbb{B}$ with eigenvalue
$E(|\phi_{b}\rangle\in\mathbb{B})$ is $\frac{2^{-\mathcal{H}/T}}{Z}$. So the
subspace of this system is at the thermo-equilibrium state and one can compare
this with Schumacher’s subspace. This subspace asymptotically converges to the
quantum error correction criteria for all types of errors. One can
equivalently define the quantum codes to be an approximation of the ‘thermo-
equilibrium space’. To get a clear idea about this spin chain model in quantum
error correction, practical implementation of this process using gates and
measurement tools are required.

##  XI Brownian computer: Thermodynamic interpretation

Here we will explore the thermodynamic properties of the Brownian computer
a143 . To analyze, we will first study the expansion of a single molecule gas.
Using this expansion model we will explore Brownian computers with different
constraints. Finally, from all this analysis, we will be able to infer that
Bennett’s claim about the necessary condition of thermodynamic reversibility
for the operation of Brownian computer is unsustainable.

###  XI.1 Single molecule gas expansion

Let us consider an ideal singlet gas molecule at a particular temperature $T$.
We will execute a process, where the molecule will have $m$-fold expansion.
The thermodynamics of this process will be analyzed for this particular
temperature $T$. A large chamber is divided into $m$ parts by partitions. At
the initial stage, the gas molecule is at the first cell of volume $V$ of the
$mV$ volume chamber. The partitions are removed and the single gas molecule
expands into the large volume chamber.

The system Hamiltonian is

$H=\mathbb{L}(p),$ (46)

where $p$ is the momentum of the molecule. The Hamiltonian of a system is
generally a function of spatial and momentum coordinate. Interestingly for
this system, the Hamiltonian is independent of the spatial coordinate of the
molecule. It has been assumed in this model that, the axis of the chamber and
the spatial coordinate are aligned.

The molecule at thermal equilibrium exhibits Boltzmann distribution over the
considered phase space. Mathematically,

$\Pi_{a}(x,p)=\frac{exp(-H/k_{B}T)}{Z},$ (47)

where the chamber is considered to extend from $x=0$ to $x=L$. $Z$ represents
the partition function of the system and $k$ is the Boltzmann constant. The
partition function for this system is

$Z=VL\,\int_{p}e^{\Big{(}-\mathbb{L}(p)/k_{B}T\Big{)}}dp.$ (48)

The information of the partition function provides sufficient information for
the analysis of the thermodynamic variables. With the help of Eq. (48), we can
describe the thermodynamic entropy for the system as

$S=\frac{\partial}{\partial T}(k_{B}T\,\,lnZ)=k_{B}\,ln(VL)+C_{p}(T),$ (49)

where the contribution from the momentum perspective is included in the
constant $C_{p}(T)$. The entropy exhibits logarithmic dependence over volume
as the Hamiltonian of the system shown in Eq. (46) has no dependence on
position coordinate.

So, the entropy change in this n-folded expansion of the gas molecule is
$S=k_{B}\,ln(mV)-k_{B}\,lnV=k_{B}\,ln\,m$. There is no change in the mean
energy while this process is executed. So, the free energy $F=E-TS$ (where $E$
is the mean energy) of the system is

$F=-k_{B}T\,ln\,m.$ (50)

This is equivalent to $F=-k_{B}T\,lnZ$. So, we can infer that the system is
not driven by any external force but by the entropic force.

###  XI.2 Undriven Brownian computer

From the point of view of thermodynamics, one can consider a Brownian computer
to be equivalent to the singlet molecule expansion. First, we will describe
the undriven Brownian computer, i.e., the system is not driven by any external
energy force. We have subdivided the analysis into two parts. One is without
the trap and the other is with a trap. A trap is the energy gradient from the
existing energy of the particle where the molecule gets trapped.

Without Trap

Here we discuss about the Brownian computer which is undriven by any energy
force, and also without any trap. Alike singlet molecule expansion, the
Brownian particle has the freedom to expand irreversibly into its confined
configuration space. Though it is an irreversible process, it will terminate
the computational process of the Brownian computer. This model describes the
minimum amount of entropy that is being created in all forms of Brownian
computers.

In this model, we consider that the spatial configuration of the Brownian
computer has the same energy throughout the channel of the phase space. No
external energy force to drive the system is applied throughout the process.
So the Hamiltonian of this model is equivalent to the Hamiltonian of the
singlet molecule shown in Eq. (46). So, this model is independent of the
spatial coordinate, and have only one degree of freedom, i.e., the momentum of
the Brownian particle. The analysis for this model is thus equivalent to that
of the singlet molecule expansion. The pictorial representation of this model
is shown in Fig. 13.

Figure 13: A schematic diagram of the Brownian computer without any energy
trap is depicted.

For our analysis, similar to that of singlet molecule expansion, we distribute
the configuration space of the computer into $m$ stages. Each of these states
is equivalent to the square blocks of the Turing machine. We will consider
that each state of the configuration space of the computer occupies volume
$V$. We will set a counter $\lambda_{a}$ which will keep an update of the
progress, and the states that have been accessed.

To execute the process, the initial state of the computer is defined by
confining the Brownian particle to the first stage, i.e., $\lambda_{a}=0$ to
$\lambda_{a}=1$. The other stages of the computer are unlocked in the next
phase of the process. This is equivalent to the case of singlet molecule gas
expansion, where the partitions are removed so that the molecule gets access
to the whole volume of the system. Similarly, the Brownian particle undergoes
a random walk and explore the full volume $mV$ of the configuration space. The
entropy of the system is equivalent to Eq. (49). So, the entropy change of the
system is

$S=k_{B}\,ln(mV)-k_{B}\,lnV=k_{B}\,ln\,m.$ (51)

This is the minimum amount of thermodynamic entropy required for the operation
of the Brownian computer. The free energy for this model of Brownian computer
is equivalent to Eq. (50).

With Trap

The model studied in the previous section the system at equilibrium is evenly
distributed over the whole channel. This model is not computationally useful.
The simple remedy to this model is to add an extra stage to the channel of the
configuration space of the system, i.e., $\lambda_{a}=m$ to $\lambda_{a}=m+1$.
This stage is what we call the trap stage for the system. The energy of the
system during this stage will be $E_{trap}$. The energy $E_{trap}$ will be
less than the position energy than that of the other stage of the system. So,
unlike the case of the previous section, the energy of the system is now
dependent on the spatial configuration. The schematic representation of this
model is shown in Fig. 14. So this stage represents the halting state of the
computation in this model, i.e., when the computer reaches this state it
defines the completion of the process of computation.

Figure 14: Schematic diagram of the Brownian computer with an energy trap.

The hamiltonian of the system thus changes from the previous model, and its
form is described as

$H=\mathbb{L}(p)+\Psi(\lambda_{a}),$ (52)

where

$\displaystyle\Psi(\lambda_{a})=\left\\{\begin{array}[]{lc}0,&\
0<\lambda_{a}<m,\\\ -E_{trap},&\ m<\lambda_{a}<m+1.\end{array}\right.$ (55)

The Boltzmann distribution for this model similar to Eq. (48) is expressed as

$\Pi_{a}(\lambda_{a},p)=\frac{exp(-H/k_{B}T)}{Z},$ (56)

where the partition function for this model is expressed as

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int exp(-H/k_{B}T)dxdp,$
(57) $\displaystyle=$
$\displaystyle\int_{p}e^{(-\mathbb{L}(p)/k_{B}T)}dp\times\int_{\lambda_{a}=0}^{m+1}e^{(-\Psi(\lambda_{a})/k_{B}T)}Vd\lambda_{a},$
$\displaystyle=$ $\displaystyle N_{b}V(m+e^{E_{trap}}/k_{B}T).$

Here the momentum contribution of the system is represented by
$N_{b}=\int_{p}e^{(-\mathbb{L}(p)/k_{B}T)}dp$. The probability density that
the state is in a specific stage specified by the counter $\lambda_{a}$ can be
described as

$\displaystyle
p(\lambda_{a})=\left\\{\begin{array}[]{lc}\frac{1}{m+exp({E_{trap}}/k_{B}T)},&\
0<\lambda_{a}<m,\\\ \\\
\frac{exp({E_{trap}}/k_{B}T)}{m+exp({E_{trap}}/k_{B}T)},&\
m<\lambda_{a}<m+1.\end{array}\right.$ (61)

The probability that the computer is at the trap stage is
$P_{a}=\frac{1}{1+m\,\,exp(-{E_{trap}}/k_{B}T)}$.

At equilibrium, state the entropy for this model can be calculated as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
T}(k_{B}T\,lnZ),$ (62) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
T}(k_{B}T\,lnN_{a})+\frac{\partial}{\partial
T}(k_{B}T\,ln[m+exp({E_{trap}}/k_{B}T)])$ $\displaystyle+$
$\displaystyle\frac{\partial}{\partial T}(k_{B}T\,lnV),$ $\displaystyle=$
$\displaystyle C_{p}(T)+k_{B}\,lnV+k_{B}\,ln[m+e^{({E_{trap}}/k_{B}T)}]$
$\displaystyle-$ $\displaystyle\frac{P_{a}\,E_{trap}}{T}.$

The momentum contribution gets absorbed in the first term of the Eq. (62). So
one can calculate the entropy gain due to the inclusion of the trap to the
system, and it is expressed as

$S_{add}=k_{B}ln[m+e^{({E_{trap}}/k_{B}T)}]-\frac{P_{a}\,E_{trap}}{T}.$

In the case of singlet molecule and the Brownian computer without a trap, we
have encountered no change of entropy for the environment. But in this case,
when the system reaches the halting state it releases energy to the
environment. So the thermodynamic entropy of the environment gets increased by
an amount of $S_{envr}=\frac{P_{a}\,E_{trap}}{T}$. So the total change in the
entropy that occurs due to the process is

$S_{total}=k_{B}ln[m+e^{({E_{trap}}/k_{B}T)}].$

The free energy for this model of Brownian computer can be calculated as

$F=-k_{B}T\,ln\,m-k_{B}T\,{({E_{trap}}/k_{B}T)}.$

The results are equivalent when derived using the definition
$F=-k_{B}T\,ln\,Z$. So the trap increases the amount of entropic force
required to drive this model of the Brownian computer.

###  XI.3 Driven Brownian computer

So far we have discussed about two models of Brownian computer. The model with
a trap is sufficient enough to operate as a Brownian computer. But to speed up
the process of computation we can add some external energy force to the
system. Though it complicates the system, we get an advantage due to this
driving energy. First, we will analyze the simpler version of this energy-
driven model, i.e., we will consider the driven Brownian computer without an
energy trap and then we will analyze the computer with an energy trap.

Without Trap

In this section, we will explore the driven Brownian computer without an
energy trap. Similar to the previously studied models we will consider that
the parameter $\lambda_{a}$ will keep a track of the progress the system makes
during the computation within the accessible channel of the confined phase
space. For this model, we will consider the energy gradient to be a function
of this spatial parameter $\lambda_{a}$. Let us assume that for each stage,
the energy ramp is $\epsilon_{r}$. This can be pictorially represented similar
to the Fig. 13 with driven energy acting on the system. So the Hamiltonian of
the models results to

$H=\mathbb{L}(p)-\epsilon_{r}\,\lambda_{a}.$ (63)

The work of the energy ramp is to boost the computation and help to reach the
halting state much earlier than previously studied models. The Boltzmann
distribution for this model similar to Eq. (48) is expressed as

$\Pi_{a}(\lambda_{a},p)=\frac{exp(-H/k_{B}T)}{Z},$ (64)

where the partition function takes the form

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int exp(-H/k_{B}T)dxdp,$
(65) $\displaystyle=$
$\displaystyle\int_{p}e^{(-\mathbb{L}(p)/k_{B}T)}dp\times\int_{\lambda_{a}=0}^{m}e^{(-\epsilon_{r}\,\lambda_{a})/k_{B}T}Vd\lambda_{a},$
$\displaystyle=$ $\displaystyle
N_{b}\,V\,(k_{B}T/\epsilon_{r})\,\,(exp(\epsilon_{r}m/k_{B}T)-1).$

Similar to the previous models, we are not concerned about the momentum
contribution which is depicted by $N_{b}$ in Eq. (65). To calculate the
probability density of the configuration space accessible to the system we
integrate over the momentum contribution. It is expressed as

$p(\lambda_{a})=(\epsilon_{r}/k_{B}T)\,\,\times\frac{exp(\epsilon_{r}\lambda_{a}/k_{B}T)}{exp(\epsilon_{r}m/k_{B}T)-1}.$
(66)

With the help of the probability density defined in Eq. (66), we can calculate
the probability of the computer being in the halting state of the system

$\displaystyle P$ $\displaystyle=$
$\displaystyle\int_{m-1}^{m}p(\lambda_{a})d\lambda_{a},$ (67) $\displaystyle=$
$\displaystyle\frac{exp(\epsilon_{r}m/k_{B}T)-exp(\epsilon_{r}(m-1)/k_{B}T)}{exp(\epsilon_{r}m/k_{B}T)-1},$
$\displaystyle\approx$ $\displaystyle 1-exp(-\epsilon_{r}/k_{B}T).$

Here we have considered that $\epsilon_{r}m/k_{B}T>>1$. The energy ramp
considered for this model can be expressed in terms of the success probability
as $\epsilon_{r}=k_{B}T\,ln\,(1/(1-P))$. So if we want a high success
probability $P$, we observe that the energy ramp results to a high value. The
convention of the equilibrium condition of the Brownian computer conveys that
the driving force $\epsilon_{r}$ should be less than $k_{B}T$. If the driving
force is able to comply with this condition, then the Brownian computer spends
most of its time in the final state (halting state). To satisfy this condition
for this model we have to consider the energy ramp $\epsilon_{r}<<k_{B}T$. So
the probability $P$ takes the form
$P=1-exp(-\epsilon_{r}/k_{B}T)\approx\epsilon_{r}/k_{B}T$.

The mean energy of the model which is associated with spatial configuration
can be evaluated as

$\displaystyle E(m)$ $\displaystyle=$ $\displaystyle
k_{B}T^{2}\,\frac{\partial}{\partial T}lnZ,$ $\displaystyle=$ $\displaystyle
E_{mom}(T)+k_{B}T-\frac{\epsilon_{r}m}{1-exp(-\epsilon_{r}m/k_{B}T)},$

where $E_{mom}(T)$ is the contribution that comes from the momentum. The
thermodynamic entropy for this system for the halting state at equilibrium is
expressed as

$\displaystyle S(m)$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
T}(k_{B}T\,lnZ(m)),$ $\displaystyle=$ $\displaystyle
C_{p}(T)-\frac{E_{mom}}{T}+k_{B}\,ln\,V+k_{B}\,ln\Big{(}\frac{k_{B}T}{\epsilon_{r}}\Big{)}$
$\displaystyle+$ $\displaystyle
k_{B}\,ln\,(exp(\epsilon_{r}m/k_{B}T)-1)+E(m)/T,$ $\displaystyle=$
$\displaystyle k_{B}ln\,V+\frac{E(m)}{T}$ $\displaystyle+$ $\displaystyle
k_{B}\,ln\Big{(}\frac{k_{B}T}{\epsilon_{r}}\Big{)}+k_{B}\,ln\,(exp(\epsilon_{r}m/k_{B}T)-1).$

where we considered the momentum contribution $C_{p}(T)$ to be equivalent to
$\frac{E_{mom}}{T}$. Entropy for the initial state can be calculated by
substituting $m=1$ in Eq. (XI.3). It results to

$\displaystyle S(1)$ $\displaystyle=$ $\displaystyle
k_{B}ln\,V+k_{B}\,ln\Big{(}\frac{k_{B}T}{\epsilon_{r}}\Big{)}+k_{B}\,ln\,(exp(\epsilon_{r}/k_{B}T)-1)$
$\displaystyle+$ $\displaystyle\frac{E(1)}{T}.$

So, the increase in the entropy of the computer when it transits from the
initial stage of the process to the final stage of the computational process
can be calculated as

$\displaystyle S_{add}$ $\displaystyle=$ $\displaystyle
k_{B}\,ln\,(exp(\epsilon_{r}m/k_{B}T)-1)$ $\displaystyle+$
$\displaystyle\frac{E(m)-E(1)}{T}-k_{B}\,ln\,(exp(\epsilon_{r}/k_{B}T)-1).$

Unlike the case of the Brownian computer without a trap, this model will
transfer heat to the environment while the computation is executed. The
entropy change that occurs in the environment can be described as

$S_{envr}=-\frac{E(m)-E(1)}{T}.$

So, the total change in the thermodynamic entropy of the system can be
expressed as

$\displaystyle S_{total}$ $\displaystyle=$ $\displaystyle S_{add}+S_{envr},$
(70) $\displaystyle=$ $\displaystyle
k_{B}\,ln\left(\frac{exp(\epsilon_{r}m/k_{B}T)-1}{exp(\epsilon_{r}/k_{B}T)-1}\right).$

Now we describe the free energy of the model under analysis. The free energy
can be evaluated as

$\displaystyle
F=-k_{B}T\,ln\left(\frac{exp(\epsilon_{r}m/k_{B}T)-1}{exp(\epsilon_{r}/k_{B}T)-1}\right).$
(71)

The thermodynamic cost for the execution of the process is high as one
requires the driven energy to be on the higher side for the higher probability
of the computation to halt at its final state.

With Trap

Now we will analyze the model where the Brownian computer will be driven by an
external energy force along with an energy trap. So this model is the most
complicated version of the models studied so far. The pictorial representation
of this model is equivalent to Fig. 14 with the external force which drives
the system for fast computation. The Hamiltonian of this model can be
expressed as

$H=\mathbb{L}(p)+\Psi(\lambda_{a}),$ (72)

where

$\displaystyle\Psi(\lambda_{a})=\left\\{\begin{array}[]{lc}\epsilon_{r}\lambda_{a},&\
0<\lambda_{a}<m,\\\ -\epsilon_{r}m-E_{trap},&\
m<\lambda_{a}<m+1.\end{array}\right.$ (75)

Following the procedures used to derive the partition function for the model
under analysis, we can describe the partition function for the system is

$\displaystyle Z(m)$ $\displaystyle=$ $\displaystyle
N_{b}\,V\,(k_{B}T/\epsilon_{r})\,\,(exp(\epsilon_{r}m/k_{B}T)-1),$

and

$Z(m+1)=N_{b}V(m+e^{E_{trap}}/k_{B}T).$

So the partition function for the system is equivalent to the partition
functions of the driven Brownian computer, and with the model undriven
Brownian computer with an energy trap. So the analysis of the thermodynamic
entropy for the system can be equivalently executed from the previous two
models. So we skip the detailed analysis for this model. The thermodynamic
entropy to drive this model of the Brownian computer is the most out of all
the models analyzed so far.

If we summarize the results of the thermodynamical analysis of the Brownian
computer we can infer that the $m$ stage Brownian computer is actually a
thermodynamically irreversible process with a minimum entropy amount
$k\,ln\,m$. With the addition of different parameters (like energy trap,
external energy force) to the systems, we observe additional thermodynamic
entropy change in the system. According to Bennett, the Brownian computer is a
reversible process in a sense equivalent to the Carnot engine. It is pre-
assumed that the system remains stable for all effects. So, according to the
convention proposed by Bennett, every Brownian computer should operate in a
thermodynamically reversible way. But the thermodynamics analysis of the
Brownian computer shows that it is equivalent to the process of singlet
molecule expansion which represents an irreversible process. So, the Brownian
computer executes a thermodynamically irreversible process when it explores
the accessible space. During this process, the motion of the system is
mechanically reversible, i.e., at any instantaneous moment, we will be unable
to distinguish whether the system is moving in the forward or reverse
direction. To get an understanding of the evolution of the system (i.e.,
whether the system is exploring the confined space or it’s reverting back to
the initial state), we have to observe the system for a long time.

The misidentification of the Brownian computer as a reversible thermodynamic
process by Bennett is due to the tracking of the internal energy. Tracking of
internal energy to analyze thermodynamic reversibility instead of
thermodynamic entropy is not a sufficient criterion. The track of the total
entropy of the system provides the necessary condition for the verification of
a process being reversible or not. If the total entropy
($S_{system}+S_{environment}$) remains constant throughout the execution of
the process, then it represents a thermodynamically reversible process.
Bennett and Landauer didn’t track the entropy of the Brownian computer which
led to the misidentification of the Brownian computer as a thermodynamically
reversible process.

##  XII Miscellaneous

In this section, some disparate aspects of thermodynamic computation are
collated here.

###  XII.1 Engine as computer

In a simple sense, one can convey that the Carnot heat engine extracts an
amount of heat $Q_{hot}$ from the hot reservoir at temperature $T_{hot}$ and
transfers an amount of heat $Q_{cold}$ to the sink at temperature $T_{cold}$.
The work done to execute this process is $W_{hot\rightarrow
cold}=Q_{hot}-Q_{cold}$. One will observe optimal efficiency if

$\mathcal{L}_{hot}-\mathcal{L}_{cold}=0,$

where $\mathcal{L}_{hot}=-Q_{hot}/T_{hot}$ is the negentropy that is imprinted
in the engine. So the loss that occurs in the process is just throwing away of
the negentropy of amount $Q_{cold}/T_{cold}$. Now we move on to analyze the
computer equivalently as a Carnot cycle as shown in the work a184 . We know
from the work a5 ; a183 , that one encounters zero work balance for an ideal
computer and the amount of information that is delivered during a process is
$ln2\,dl=\mathcal{L}_{hot}-\mathcal{L}_{cold}$ where $l$ denotes the amount of
information.

So in the case of an ideal computer, we also encounter a loss by throwing out
an equivalent amount of negentropy as in the case of the Carnot cycle.

###  XII.2 Nonergodic systems and its thermodynamics

A memory device can be designed using a multistable system applied to the
condition that, the transition between the states is not allowed.
Alternatively, we can say that information can be stored only in a nonergodic
system where the ‘time average’ of the system differs from the ‘phase space
average’ of the system. In the work a185 , the authors have analyzed the
thermodynamics of computation with the notion that information is recorded in
a multistable system. These multistable states are considered as the
nonergodic state.

We will interpret the memory register in terms of thermodynamics. According to
the Landauer principle, the system will confront a decrease in entropy due to
restore to one (RTO) operation. Whereas in the work a185 , the authors have
proved that there is no change in the thermodynamic entropy even after the RTO
operation. The Clausius definition of thermodynamic entropy for a system in
two states $ST_{a}$ and $ST_{b}$ is expressed as

$\Delta S=\int_{ST_{a}}^{ST_{b}}dQ/T,$

where $dQ$ represents the heat change in the system and $T$ the temperature of
the system. So to prove the statement proposed in the work a185  the author
considered a thought experiment. The thought experiment is that we will
consider a system where the particle is in a bistable-monostable potential
well. This system will interact with a heat reservoir. Goto and his co-workers
in their work a186 ; a187  coined this model as “quantum flux parametron”. For
the analysis, the state of the system will be described as one when the
particle is found on the right side of the potential well and zero when one
finds the particle to be on the left-hand side of the well. The thought
experiment is as follows:

  * •

The system is considered to be initiated in the state zero. This will be the
initial state of the system. The entropy of the system is $S_{zero}$.

  * •

The left well of the system is lowered by applying a bias potential.

  * •

Now the ‘double-well potential’ is converted to a single well. This is
executed by removing the partition between this well.

  * •

In the next stage of the process, we will remove the bias potential that was
applied to the left well of the system. The system is now in a neutral state
and we denote it as $ST_{ne}$. The state of the system of being in one or zero
is denoted by $ST_{m}$.

  * •

The bias potential is again applied, but now on the right side of the single
potential well. This lowers the right side of the well.

  * •

In the next stage of the protocol, we then revert back the single potential
well into the double potential well just by recreating the partition between
the left and right side of the well.

  * •

In the final stage of the process, we remove the bias potential. The system is
now in state one, and the entropy of the system is $S_{one}$.

The schematic representation of the thought experiment is shown in Fig. 15.

Figure 15: A schematic representation of the thought experiment step by step
is depicted. The red solid sphere is the single atom of the system

The first four steps of the thought experiment are called as the erasing
process, and the final three steps represent the writing process. If we apply
RTO to the system, and the system is in the state zero, we will observe the
same configuration even after the RTO operation ($S_{zero}=S_{one}$),
subjected to a condition that the state is known before the execution of this
experiment.

From an information point of view, entropy is defined as a measure of lack of
information, i.e., the lesser the information we gather about the system the
higher is the entropy of the system. So, the entropy of the unknown state
$ST_{ukn}$ should have large entropy than that of the known state. Based on
this notion, Landauer argued that the system will observe a decrease in
entropy after RTO operation.

It is a well-known fact that when a computer runs repeated recordings and
erasure occurs in a memory device. The physical equivalence of this process
can be thought of as a sequence of evolution of the system from an ergodic
state to a nonergodic state, and vis versa. Now we will analyze the
thermodynamics of this evolution using two different models. The first model
that we are going to analyze is the Szilard engine model, and the second one
is the bistable-monostable potential well.

Szilard engine with one atom is of the widely studied model for a memory
device. This model helps us to understand the physics behind the computation.
The Szilard engine model is associated with a heat reservoir at temperature
$T$. When the system has the partition wall the state of the system is in
nonergodic state $ST_{nerg}$ and it will be in the ergodic state $ST_{erg}$
when one removes the partition. Now we will describe the writing process for
this model. The initial state of the system is considered as $ST_{erg}$. Now
the partition is inserted in the middle of the chamber, so work in done on the
atom in the system. The work done can be expressed as $\langle
W_{writing}\rangle=k_{B}T\,ln2$ and the heat that is generated to execute the
process of writing is

$\langle Q_{writing}\rangle=T(S_{ST_{erg}}-S_{ST_{nerg}})=k_{B}T\,ln2.$ (76)

The process is executed for a large number of times to evaluate the
expectation of the work done and the heat generation during the execution of
the process.

Along with the writing process, the system goes through the erasure process.
For the writing process, we have two types of erasure process. One is
reversible in nature and the other is irreversible in its form. For the
reversible process the work done and the creation of heat in the surrounding
can be expressed as $\langle W_{erasure}\rangle=-\langle W_{writing}\rangle$
and $\langle Q_{erasure}\rangle=-\langle Q_{writing}\rangle$. So for the
process, where one executes the writing process as well as the erasure process
using the reversible form, one can infer that the total work is done and the
heat generation is zero. One will be able to execute the reversible erasure
process if the state of the system is known before the process is executed.

The second model of the erasure process is the irreversible process. The
initial state of the system is now not known to us. To execute this process
one removes the partition from the chamber without any prior information about
the state of the memory. The removal of the partition evolves the state of the
system from the nonergodic state to the ergodic state. The work done and the
heat generation for this erasure process is zero but we encounter an increase
in the entropy of the system. The increase of this entropy can be evaluated as
$S_{erg}=k_{B}ln2$. So for the total process of writing and as well as the
erasure process, the total work done and the heat production can be expressed
as

$\langle W_{tirr}\rangle=\langle Q_{tirr}\rangle=k_{B}\,ln2,$ (77)

where $\langle W_{tirr}\rangle$ and $\langle Q_{tirr}\rangle$ represents the
work and heat for the total process.

Now we will explore the second model, i.e., the bistable-monostable potential
well model. In this process, the partition of the Szilard engine gets replaced
by a potential wall. The writing process and the erasure process for this
model are described in the thought experiment. For the writing process, the
amount of work done and the heat production can be evaluated as

$\displaystyle\langle W_{writing}\rangle$ $\displaystyle=$ $\displaystyle
k_{B}Tln\left(\frac{Z_{ST_{ne}}}{Z_{ST_{m}}}\right),$ $\displaystyle\langle
Q_{writing}\rangle$ $\displaystyle=$ $\displaystyle
E_{ST_{ne}}-E_{ST_{m}}+k_{B}Tln\left(\frac{Z_{ST_{ne}}}{Z_{ST_{m}}}\right),$

where $E$ and $Z$ represents the energy and partition function of the system.
The partition function for the system can be evaluated as

$\displaystyle Z_{ST_{ne}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}e^{-V_{ne}/kT}dx,$ $\displaystyle
Z_{ST_{m}}$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}e^{-V_{m}/kT}dx,$

where $V_{ne}$, $V_{m}$ describes the potential energy of the double and the
single potential well. The partition function of the system is defined based
on the assumption that we have considered the nonergodicity of the system. The
potential wall should be chosen in such a way that it is greater than
$k_{B}T$. The potential wall is chosen in this manner to reduce the error
during the operation.

The erasure process considered for this model is the irreversible one. The
irreversible erasure process is executed by diminishing the potential wall
height (i.e. reducing the height of the hill), and along with that, we will
not use the bias potential in this process. Due to this lowering of the height
of the wall, we encounter entropy production $S_{ern}$ in the system by an
amount of $k_{B}ln2$. The work done and the heat production for the execution
of the erasure process can be expressed as

$\displaystyle\langle W_{erasure}\rangle$ $\displaystyle=$
$\displaystyle-\langle W_{writing}\rangle+TS_{ern},$ $\displaystyle\langle
Q_{erasure}\rangle$ $\displaystyle=$ $\displaystyle-\langle
Q_{writing}\rangle+TS_{ern}.$

So for the total process of writing and as well as the erasure process, the
total work done and the heat production is equivalent to the results (Eq.
(77)) obtained in the irreversible process of the Szilard model.

###  XII.3 Thermodynamics of algorithm

It is generally believed that the advent of quantum computers will help to
solve some age-old problems in number theory, physical as well as
combinatorial search faster than the existing classical computers. To get a
better understanding of quantum speedups, one has to consider a realistic
model of computation. The realistic model should consider the time-complexity
and time-space tradeoffs during analysis. Some works a189a ; a189b ; a189c ;
a189d  have analyzed these type of models. A recent work a189e  in this
direction has analyzed the quantum speedups from a thermodynamic viewpoint. To
analyze the cost of the algorithm (in classical and quantum regime) from the
thermodynamic perspective Brownian model of computation is considered.

For the thermodynamic interpretation of algorithms, the authors in their work
a189e  have considered collision finding algorithm and preimage search for
their analysis. Before we move on to the thermodynamic analysis of the
algorithm, we would describe the algorithms briefly. First, we will learn
about the collision finding algorithm and then about the preimage search
algorithm.

The parallel collision search algorithm proposed by Van Oorshot and Wiener
a189f  happens to be the best classical collision finding algorithm. This
algorithm can detect a collision in an expected serial depth of
$\mathcal{O}(\sqrt{G}/G_{\alpha})$, where $G$ denotes the range of the
function and $G_{\alpha}$ depicts the parallel processes with memory
$\mathcal{O}(1)$. It is assumed that the communication cost among the thread
is negligible than the overall computation costs. Brassard, Hoyer and Tapp
(BHT) extended this algorithm in the quantum realm in their work a189g . The
operations for this algorithm is $\mathcal{O}(G^{\frac{1}{3}})$ with memory
size $G^{\frac{1}{3}}$. One can further generalize this algorithm to a
parallel algorithm with $n_{\beta}$ parallel processors having memory size
$G_{\beta}$. This generalized algorithm satisfies this condition
$n_{\beta}<G_{\beta}<\mathcal{O}\Big{(}(Gn_{\beta})^{\frac{1}{3}}\Big{)}$ with
a serial complexity
$\mathcal{O}\Big{(}\sqrt{\frac{G}{G_{\beta}n_{\beta}}}\Big{)}$.

Bernstein has shown in their work a189c  that the BHT algorithm, when measured
in terms of memory and serial depth does not encounter any improvement over
the classical algorithm, but one encounters an improvement in terms of query
complexity. Giovanetti in their work a189h , counters about the memory cost by
proposing a quantum RAM model, where they convey that memory access operation
can be executed at logarithmic energy cost despite of large gate complexity.
So the question remains whether one can propose a realistic model where one
encounters improvement of the complexity of the quantum algorithm over the
classical one. Now we will see whether the Brownian model of computation
provides a solution to this proposed problem. To do so, we will first analyze
the quantum version of the algorithm. For a given time $t$ and memory size
$G_{\beta}$, we are going to calculate the total energy that is required for
the collision search. We will consider that in the BHT algorithm the energy
complexity is subjugated by oracle queries over the memory access. So the
energy for per operation is
$\mathcal{O}\Big{(}\frac{\Big{(}\sqrt{\frac{G}{G_{\beta}n_{\beta}}}\Big{)}}{t}\Big{)}$.
So the total energy for the process can be expressed as

$E_{quantum}=\mathcal{O}\Big{(}n_{\beta}\times\sqrt{\frac{G}{G_{\beta}n_{\beta}}}\times\frac{\Big{(}\sqrt{\frac{G}{G_{\beta}n_{\beta}}}\Big{)}}{t}\Big{)}=\mathcal{O}\Big{(}\frac{GG_{\beta}}{t}\Big{)}.$

Now if we consider the classical case the energy for per operation is
$\mathcal{O}(\frac{\sqrt{G}}{G_{\beta}t})$, and the total energy for the
process is

$E_{classical}=\mathcal{O}\Big{(}G_{\beta}\times\frac{\sqrt{G}}{G_{\beta}t}\times\frac{\sqrt{G}}{G_{\beta}}\Big{)}=\mathcal{O}\Big{(}\frac{G}{G_{\beta}t}\Big{)}.$

So we can infer from the results that this realistic model does not provide
any advantage of the quantum algorithm over the classical one. So, it requires
further investigation to verify whether any realistic model will provide an
advantage of the quantum algorithm over the classical algorithm. One can think
of applying similar analysis for the Claw Finding problem. Its objective is to
find the collision between two functions with different domain sizes, e.g.,
$Do_{1}$ and $Do_{2}$. The quantum version of this algorithm was explored in
the work a189j . The energy cost for finding the collision in quantum regime
is
$\mathcal{O}\Big{(}max\,\Big{(}\frac{Do_{1}\,Do_{2}}{G_{b}etat},\frac{Do_{1}}{t},\frac{Do_{2}}{t}\Big{)}\Big{)}$,
wherein the case of classical it is
$\mathcal{O}\Big{(}\frac{(Do_{1}+Do_{2})^{3}}{G_{\beta}^{2}t}\Big{)}$.

Now we will analyze the second algorithm, i.e., the pre-image algorithm. One
can find preimages of a function having a domain size $N_{\gamma}$ using
Grover’s algorithm. The serial complexity for this algorithm is
$\mathcal{O}(N_{\gamma})$. An optimal serial complexity can be obtained if one
considers $M_{\gamma}$ parallel processes with memory $\mathcal{O}(1)$. The
serial complexity of this optimal model is
$\mathcal{O}\Big{(}\sqrt{\frac{N_{\gamma}}{M_{\gamma}}}\Big{)}$. This was
verified in the work a189i . If one implement Grover’s algorithm with Brownian
computation model the energy for per operation is
$\mathcal{O}\left(\frac{\sqrt{\frac{N_{\gamma}}{M_{\gamma}}}}{t}\right)$, and
the total energy of the process for this quantum version is

$E_{quantum}=\mathcal{O}\left(\frac{\sqrt{\frac{N_{\gamma}}{M_{\gamma}}}}{t}\times
M_{\gamma}\times\sqrt{\frac{N_{\gamma}}{M_{\gamma}}}\right)=\mathcal{O}\Big{(}\frac{N_{\gamma}}{t}\Big{)}.$

If we naively apply the Brownian computation for the classical search the
total energy for the process boils down to
$E_{classical}=\mathcal{O}\Big{(}\frac{N_{\gamma}^{2}}{M_{\gamma}t}\Big{)}$.
To improve the complexity of the process, one allows the Brownian motion to
steer the system on a random work. If one implements the computation with
$M_{\gamma}$ parallel processes, then the complexity of the algorithm becomes
equivalent to that of the quantum version. So we encounter the same asymptotic
performance for the algorithm in the classical and quantum regime.

If one compares Grover’s algorithm with the classical search algorithms
instead of quantum versus classical collision search, then one will encounter
that Grover’s algorithm is more efficient from the classical version (where
powered and unpowered Brownian computation model is considered for the
analysis). If the assumption of scale invariance for memory size and the
energy consumption is removed for unpowered primage search, then one
encounters that the preimage algorithm is memory intensive. If one considers
oracle queries then also the unpowered preimage algorithm happens to be less
significant than Grover’s algorithm.

The results that have been discussed so far suggest that the cost of hardware
happens to the prime factor that determines the advantage of the algorithms.
If the costing of the hardware can be reduced, then one can even use the
classical search algorithms over Grover’s algorithm. One can even encounter
that unpowered classical preimage search surpasses Grover’s algorithm provided
that the memory costs are very close to the fundamental thermodynamic limit.
These results also guide us in the field of post-quantum cryptography,
especially in choosing the key size of the block ciphers. It can be inferred
from this analysis that doubling the key size is unnecessary. Instead of that
one can make a small increase from 128 to 192 bits for better protection.

##  XIII Conclusion and Future prospects

The analysis of the computation process from the thermodynamic viewpoint has
been one of the central points of attraction for the researchers of physics as
well as computer science. It got initiated from the physical Church-Turing
thesis, where they conveyed that every computational process is physical.
Various approaches have been considered for the analysis of the different
computational processes. With the advent of the modern statistical theory, the
research in this area got boosted. Not only in the field of computer science
we also encounter its application in different fields from chemical networks
a103 ; a105 ; a188 ; a189 , molecular biology a99 ; a101  and even in
neurology a190 ; a191 .

Further investigation in this direction is required to get a better
understanding of the bond between thermodynamics and the computational
process. For example, in the case of finite automata, one can investigate
about the maximum thermodynamic cost that is required to accept a language for
an automata. Also one can calculate the minimal cost for any deterministic
automata. One can also work on developing a theory to analyze the non-
deterministic finite-state automata in terms of thermodynamics. Models to
describe the complex Turing machine, and also network theory from the
thermodynamic viewpoint is an open area of research.

Error correction is an important part of any computational process. The
analysis of these protocols from a thermodynamic viewpoint is at their baby
stage. Modeling of the error correction models by the physical system to
explain it thermodynamically needs further investigation. In a recent work
a192 , the authors have shown that there a similarity between the quantum heat
engine and quantum error correction codes. They have strengthened their
intuition by making a complete analysis of the thermodynamic properties of the
quantum engine based error correction codes. So the thermodynamic approach to
explain the error correction is an open book to read.

Different systems have different heat signatures. One can utilize this
property of the system for various purposes such as for security in
cryptographic protocols. So one can explore the communications protocols,
crypto-systems from a thermodynamic viewpoint. Algorithms in the form of a
search algorithm from a thermodynamic viewpoint have already been analyzed.
Further exploration in this direction is an open area of research.
Thermodynamic analysis of quantum computations need a rigorous investigation
for a better understanding of quantum computers and to develop hardware with
lower cost functions.

## References

  * (1) R. Clausius. On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat. Poggendorff’s Annalen der Physik, LXXIX (Dover Reprint) (1850).
  * (2) R. Landauer. Information is physical. Physics Today, 44(5), 23-29 (1991).
  * (3) C. P. Kempes, D. Wolpert, Z. Cohen, & J. Pérez-Mercader. The thermodynamic efficiency of computations made in cells across the range of life. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 375(2109), 20160343 (2017).
  * (4) J. von Neumann. Fourth University of Illinois lecture, in Theory of Self-Reproducing Automata, A. W. Burks, ed., p. 66. Univ. of Illinois Press, Urbana, (1966).
  * (5) L. Brillouin. Science and Information Theory (2nd edition, 1962), Academic Press, London (1956).
  * (6) E. Fredkin, and T. Toffoli. “Conservative Logic,” MIT Report MIT/LCS/TM- 197; International Journal of Theoretical Physics, 21, 219 (1982).
  * (7) C. H. Bennett Logical Reversibility of Computation, IBM Journal of Research and Development, 17, 525-532 (1973).
  * (8) R. W. Keyes, and R. Landuer. Minimal energy dissipation in logic, IBM Journal of Research and Development, 14, 152, (1970).
  * (9) K. K. Likharev. Classical and Quantum Limitations on Energy Consumption in Computation, International Journal of Theoretical Physics, 21, 311,(1982).
  * (10) C.H. Bennett. The thermodynamics of computation: a review. Int. J. Theor. Phys. 21, 905–940, 1982.
  * (11) C. H. Bennett. Time/space trade-offs for reversible computation. SIAM J. Comput. 18, 766–776, 1989.
  * (12) U. Seifert. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001, 2012.
  * (13) J. M. Parrondo, J. M. Horowitz, T. Sagawa. Thermodynamics of information. Nat. Phys. 11, 131–139, 2015.
  * (14) G. Diana, G. B. Bagci, M. Esposito. Finite-time erasing of information stored in fermionic bits. Phys. Rev. E 87, 012111, 2013.
  * (15) G. Lan, P. Sartori, S. Neumann, V. Sourjik, Y. Tu. The energy-speed-accuracy trade-off in sensory adaptation. Nat. Phys. 8, 422–428, 2012.
  * (16) T. E. Ouldridge, C. C. Govern, P. R. ten Wolde. Thermodynamics of computational copying in biochemical systems. Phys. Rev. X 7, 021004, 2017.
  * (17) A. Barato, U. Seifert. Thermodynamic uncertainty relation for biomolecular processes. Phys. Rev. Lett. 114, 158101, 2015.
  * (18) D. H. Wolpert. Extending Landauer’s bound from bit erasure to arbitrary computation arXiv:1508.05319.
  * (19) P. Strasberg, J. Cerrillo, G. Schaller, T. Brandes. Thermodynamics of stochastic Turing machines. Phys. Rev. E 92, 042104, 2015.
  * (20) O. J. Maroney. Does a Computer have an Arrow of Time?. Foundations of Physics, 40(2), 205-238, (2010).
  * (21) L. S. Schulman. A computer’s arrow of time. Entropy, 7(4):221–233, 2005.
  * (22) R. Landauer. Irreversibility and heat generation in the computing process. IBM J Res Dev, 5:183–191, 1961.
  * (23) S. W. Hawking. Arrow of time in cosmology. Physical Review D, 32(10):2489–2495, 1985.
  * (24) S. W. Hawking. The no-boundary proposal and the arrow of time. In Hawking On The Big Bang And Black Holes, pp. 266-275. 1993.
  * (25) S. W. Hawking. The no boundary condition and the arrow of time. In Physical Origins of Time Asymmetry, 1994.
  * (26) T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf. Undecidability of the spectral gap. Nature 528, 207 (2015).
  * (27) T. S. Cubitt, J. Eisert, and M. M. Wolf. Extracting dynamical equations from experimental data is NP hard. Physical review letters 108, 120503 (2012).
  * (28) D. Deutsch. Quantum theory, the Church–Turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 400, 97 (1985).
  * (29) P. Benioff. Quantum mechanical Hamiltonian models of Turing machines. Journal of Statistical Physics 29, 515 (1982).
  * (30) M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).
  * (31) M. Baaz, C. H. Papadimitriou, H. W. Putnam, D. S. Scott, and C. L. Harper Jr, eds. Kurt Gödel and the foundations of mathematics: Horizons of truth. Cambridge University Press, 2011.
  * (32) S. Aaronson. Guest column: NP-complete problems and physical reality. ACM Sigact News, 36(1), 30-52, 2005.
  * (33) P. Arrighi and G. Dowek. The physical Church-Turing thesis and the principles of quantum theory. International Journal of Foundations of Computer Science 23, 1131 (2012).
  * (34) G. Piccinini. The physical Church–Turing thesis: Modest or bold?. The British Journal for the Philosophy of Science 62, 733 (2011).
  * (35) I. Pitowsky. The physical Church thesis and physical computational complexity. Iyyun: The Jerusalem Philosophical Quarterly , 81 (1990).
  * (36) N. Rescher. Hypothetical Reasoning: Studies in Logic and the Foundations of Mathematics, (1968).
  * (37) A. Kolchinsky, and D. H. Wolpert. Thermodynamic costs of Turing Machines. Physical Review Research, 2(3), 033312 (2020).
  * (38) S. A. Cook. The complexity of theorem proving procedures, in: Proceedings of the Third Annual ACM Symposium on Theory of Computing, 1971, pp. 151–158.
  * (39) R. J. Lipton, K. W. Regan, People, Problems, and Proofs: Essays from Gödel’s Lost Letter: 2010, Springer, New York, 2013.
  * (40) A.A. Razborov, S. Rudich, Natural proofs, J. Comput. System Sci. 55 (1997) 24.
  * (41) O. Goldreich, The Foundations of Cryptography. Basic Tools, Vol. 1, Cambridge University Press, 2006.
  * (42) S. Aaronson, Is P versus NP formally independent? Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 81 (2003) 109-136.
  * (43) W. H. Zurek, Thermodynamic cost of computation, algorithmic complexity and the information metric, Nature 341 (1989) 119.
  * (44) S. Aaronson, Why philosophers should care about computational complexity, in: B.J. Copeland, C. Posy, O. Shagrir (Eds.), Computability: Gödel, Turing, Church, and Beyond, MIT Press, Cambridge, MA, 2012.
  * (45) R. Raussendorf, H.J. Briegel. A one-way quantum computer. Phys. Rev. Lett. 86 (22) (2001) 5188.
  * (46) C. G. Knott, Life and Scientific Work of Peter Guthrie Tait (London: Cambridge University Press, 1911).
  * (47) M. von Smoluchowski, Experimentell Nachweisbare der Ublichen Thermodynamik Widersprechende Molekularphänemone, Physikalische Zeitung 13, 1069–1080 (1912).
  * (48) M. von Smoluchowski, Gültigkeitsgrenzen des Zweiten Hauptsatzes der Wärmetheorie, in Vorträge über die Kinetische Theorie der Materie undder Elektrizität (Leipzig: Teubner, 1914).
  * (49) L. Szilard. Uber die Entropieverminderung in einem Thermodynamischen System be Eingriffen Intelligenter Wesen, Zeitschrift fur Physik 53, 840-856 (1929). English translation in Leff and Rex, pp. 124–133.
  * (50) T. Toffoli, Reversible Computing, MIT Report MIT/LCS/TM-151.
  * (51) J. Earman and J.D. Norton, Exorcist XIV: The Wrath of Maxwell’s Demon. Part I. From Maxwell to Szilard, Studies in the History and Philosophy of Modern Physics 29, 435-471 (1998).
  * (52) J. Earman and J.D. Norton, Exorcist XIV: The Wrath of Maxwell’s Demon. Part II. From Szilard to Landauer and Beyond, Studies in the History and Philosophy of Modern Physics 30, 1-40 (1999).
  * (53) J. Bub. Maxwell’s Demon and the Thermodynamics of Computation. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 32(4), 569-579 (2001).
  * (54) J. Hani and H. Feldmann. tRNA genes and retroelements in the yeast genome, Nucleic Acids Res. 26, 689 (1998).
  * (55) A. B. Boyd, D. Mandal, P. M. Riechers, and J. P. Crutchfield, Transient Dissipation and Structural Costs of Physical Information Transduction, Phys. Rev. Lett. 118, 220602 (2017).
  * (56) D. Chu. Thermodynamics of quasideterministic digital computers. Physical Review E, 97(2), 022121 (2018).
  * (57) J. Von Neumann. Probabilistic logics and the synthesis of reliable organisms from unreliable components, Automata studies 34, 43–98 (1956).
  * (58) K. A. Johnson. Conformational coupling in dna polymerase fidelity. Annual review of biochemistry 62, 685-713 (1993).
  * (59) J. J. Hopfield. Kinetic proofreading: a new mechanism for reducing errors in biosynthetic processes requiring high specificity. Proceedings of the National Academy of Sciences 71, 4135–4139 (1974).
  * (60) J. Ninio. Kinetic amplification of enzyme discrimination. Biochimie 57, 587–595 (1975).
  * (61) A. Murugan, D. A. Huse, and S. Leibler. Speed, dissipation, and error in kinetic proofreading. Proceedings of the National Academy of Sciences 109, 12034–12039 (2012).
  * (62) A. Murugan, D. A. Huse, and S. Leibler. Discriminatory proofreading regimes in nonequilibrium systems. Physical Review X 4, 021016 (2014).
  * (63) C. H. Bennett. Dissipation-error tradeoff in proof-reading. BioSystems 11, 85–91 (1979).
  * (64) D. Andrieux and P. Gaspard. Nonequilibrium generation of information in copolymerization processes. Proceedings of the National Academy of Sciences 105, 9516–9521 (2008).
  * (65) F. Cady and H. Qian. Open-system thermodynamic analysis of dna polymerase fidelity. Physical biology 6, 036011 (2009).
  * (66) D. Andrieux and P. Gaspard. Molecular information processing in nonequilibrium copolymerizations. The Journal of chemical physics 130, 014901 (2009).
  * (67) M. Esposito, K. Lindenberg, and C. Van den Broeck. Extracting chemical energy by growing disorder: efficiency at maximum power. Journal of Statistical Mechanics: Theory and Experiment 2010, P01008 (2010)
  * (68) P. Sartori and S. Pigolotti. Kinetic versus energetic discrimination in biological copying. Physical review letters 110, 188101 (2013).
  * (69) D. Andrieux and P. Gaspard. Information erasure in copolymers. EPL (Europhysics Letters) 103, 30004 (2013).
  * (70) P. Gaspard and D. Andrieux. Kinetics and thermodynamics of first-order Markov chain copolymerization. The Journal of chemical physics 141, 044908 (2014).
  * (71) C. E. Shannon. A mathematical theory of communications Bell Syst. Tech. J. 27, 379 (1948).
  * (72) C. E. Shannon. Certain results in coding theory for noisy channels. Information and control, 1(1), 6-25, (1957).
  * (73) B. Schumacher. Quantum coding Phys. Rev. A 51, 2738–2747 (1995).
  * (74) P. Sartori, & S. Pigolotti. Thermodynamics of error correction. Physical Review X, 5(4), 041039 (2015).
  * (75) V. Korepin, & J. Terilla. Thermodynamic interpretation of the quantum error correcting criterion. Quantum Information Processing, 1(4), 225-242 (2002).
  * (76) R. Horodecki, M. Horodecki, and P. Horodeck. Balance of information in bipartite quantum-communication systems: Entanglement-energy analogy. Phys. Rev. A 63, 022310 (2001).
  * (77) S. Popescu and D. Rohrlich. Thermodynamics and the measure of entanglement. Phys. Rev. A 56, R3319 (1997).
  * (78) D. Rohrlich. Thermodynamical analogues in quantum information theory. Optics and Spectroscopy, 91(3), 363-367,(2001).
  * (79) J. E. Hopcroft, R. Motwani and J. Ullman. Introduction to Automata Theory, Languages and Computability (Boston, MA: Addison-Wesley) 2000.
  * (80) H. R. Lewis, & C. H. Papadimitriou. Elements of the Theory of Computation. ACM SIGACT News, 29(3), 62-78, (1998).
  * (81) J. Hromkovič. Theoretical computer science: introduction to Automata, computability, complexity, algorithmics, randomization, communication, and cryptography. Springer Science & Business Media, (2003).
  * (82) T. M. Cover, & J. A. Thomas. Elements of information theory. John Wiley & Sons (2012).
  * (83) S. Kullback. Information theory and statistics. Courier Corporation (1997).
  * (84) R. G. Gallager. Information theory and reliable communication. Vol. 2. New York: Wiley, 1968.
  * (85) M. C. Diamantini, L. Gammaitoni, and C. A. Trugenberger. Landauer bound for analog computing systems. Phys. Rev. E 94 012139, 2016.
  * (86) J. Baez, and M. Stay. Algorithmic thermodynamics. Math. Struct. Comput. Sci. 22 771 2012.
  * (87) H. K. Lo, T. Spiller, & S. Popescu. Introduction to quantum computation and information. World Scientific, (1998).
  * (88) S. Deffner, and S. Campbell. Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control. J. Phys. A: Math. Theor. 50 453001 2017.
  * (89) M. B. Pour-El, and I. Richards. Noncomputability in models of physical phenomena. Int. J. Theor. Phys. 21 553 1982.
  * (90) C. Moore. Unpredictability and undecidability in dynamical systems. Phys. Rev. Lett. 64 2354 1990.
  * (91) S. Lloyd. Ultimate physical limits to computation. Nature 406 1047 2000.
  * (92) S. Lloyd. Uncomputability and physical law. In The Incomputable, pp. 95-104. Springer, Cham, 2017.
  * (93) T. E. Ouldridge, and P. R. ten Wolde. Fundamental costs in the production and destruction of persistent polymer copies. Phys. Rev. Lett. 118 158103 2017.
  * (94) T. E. Ouldridge. The importance of thermodynamics for molecular systems, and the importance of molecular systems for thermodynamics. Natural computing, 17(1), 3-29. 2018.
  * (95) R. A. Brittain, N. S. Jones, & T. E. Ouldridge. Biochemical Szilard engines for memory-limited inference. New Journal of Physics, 21(6), 063022, (2019).
  * (96) P. Sartori, L. Granger, C. F. Lee, and J. M. Horowitz. Thermodynamic costs of information processing in sensory adaptation. PLoS Comput. Biol. 10 e1003974 2014.
  * (97) Y. Hasegawa. Multidimensional biochemical information processing of dynamical patterns. Physical Review E, 97(2), 022401, (2018).
  * (98) P. Mehta, and D. J. Schwab. Energetic costs of cellular computation. Proc. Natl Acad. Sci. 109 17978 2012.
  * (99) P. Mehta, A. H. Lang, and D. J. Schwab. Landauer in the age of synthetic biology: energy consumption and information processing in biochemical networks. J. Stat. Phys. 162 1153–66 2016.
  * (100) S. J. Prohaska, P. F. Stadler, and D. C. Krakauer. Innovation in gene regulation: the case of chromatin computation. J. Theor. Biol. 265 27 2010.
  * (101) B. Bryant. Chromatin computation. PloS One 7 e35703 2012.
  * (102) Y. Benenson. Biomolecular computing systems: principles, progress and potential. Nat. Rev. Genet. 13 455 2012.
  * (103) D. Angluin, J. Aspnes, and D. Eisenstat. Proc. of the 25th Annual ACM Symp. on Principles of Distributed Computing (ACM) pp 292-9 2006.
  * (104) H. L. Chen, D. Doty, and D. Soloveichik. Deterministic function computation with chemical reaction networks. Nat. Comput. 13 517 2014.
  * (105) Q. Dong. A bisimulation approach to verification of molecular implementations of formal chemical reaction networks. Master’s Thesis Stony Brook University 2012.
  * (106) D. Soloveichik, M. Cook, E. Winfree, and J. Bruck. Computation with finite stochastic chemical reaction networks. Nat. Comput. 7 615 2008.
  * (107) C. Thachuk, and A. Condon. International Workshop on DNA-Based Computers (Springer) pp 135–49 2012.
  * (108) H. Touchette, and S. Lloyd. Information-theoretic approach to the study of control systems. Physica 331 140 2004.
  * (109) H. Touchette and S. Lloyd. Information-theoretic limits of control. Phys. Rev. Lett. 84 1156 2000.
  * (110) A. C. Barato, and U. Seifert. Thermodynamic cost of external control. New J. Phys. 19 073021 2017.
  * (111) T. Sagawa and M. Ueda. Second law of thermodynamics with discrete quantum feedback control. Phys. Rev. Lett. 100 080403 2008.
  * (112) T. Sagawa and M. Ueda. Nonequilibrium thermodynamics of feedback control. Phys. Rev. E 85 021104 2012.
  * (113) H. Wilming, R. Gallego, and J. Eisert. Second law of thermodynamics under control restrictions. Phys. Rev. E 93 042126 2016.
  * (114) S. J. Large, R. Chetrite, and D. A. Sivak. Stochastic control in microscopic nonequilibrium systems. EPL (Europhysics Letters) 124, no. 2 (2018): 20001.
  * (115) T. R. Gingrich, G. M. Rotskoff, G. E. Crooks, and P. L. Geissler. Near-optimal protocols in complex nonequilibrium transformations. Proc. Natl Acad. Sci. 113 10263 2016.
  * (116) J. M. Horowitz, and J. L. England. Information-theoretic bound on the entropy production to maintain a classical nonequilibrium distribution using ancillary control. Entropy 19 333, 2017.
  * (117) J. E. Savage. Models of Computation vol 136 (Reading, MA: Addison-Wesley) 1998.
  * (118) A. M. TURING. On computable numbers, with an application to the Entscheidungsproblcm. Proceedings of the London Mathematical Society, 2 s. vol. 42 (1936-7), pp. 230-265.
  * (119) D. H. Wolpert. The stochastic thermodynamics of computation. Journal of Physics A: Mathematical and Theoretical, 52(19), 193001, (2019).
  * (120) P. Arrighi. An overview of quantum cellular automata. Natural Computing, 18(4), 885-899, (2019).
  * (121) C. Wüthrich. A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer. Synthese, 192(7), 1989-2008, (2015).
  * (122) C. Moore, and S. Mertens. The Nature of Computation (Oxford: Oxford University Press) 2011.
  * (123) B. J. Copeland, J. P. Carl, and O. Shagrir, eds. Computability: Turing, gödel, church, and beyond. MIT Press, 2013.
  * (124) M. Sipser. Introduction to the Theory of Computation. ACM Sigact News, 27(1), 27-29, (1996).
  * (125) S. Arora, and B. Barak, Computational Complexity: a Modern Approach (Cambridge: Cambridge University Press) 2009.
  * (126) M. Li, and P. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications (New York: Springer) 2008.
  * (127) R. M. Gray, Entropy and information theory. Springer Science & Business Media, 2011.
  * (128) D. J. MacKay. Information theory, inference and learning algorithms. Cambridge university press (2003).
  * (129) M. M. Wilde. Quantum information theory. Cambridge University Press, 2013.
  * (130) M. Hayashi. Quantum Information Theory. Graduate Texts in Physics, Springer (2017).
  * (131) R. W. Hamming. Error Detecting and Error Correcting Codes. Bell System Technical Journal. USA: AT & T. 29 (2): 147–160 1950.
  * (132) F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. North-Holland Publishing Company, 1978.
  * (133) D. G. Hoffman, D. A. Leonard, C. C. Lindner, K. T. Phelps, C. A. Rodger and J. R. Wall. Coding Theory, The Essentials. Marcel Dekker, Inc., 1990.
  * (134) P. W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys Rev A. 1995;52 : R2493.
  * (135) J. Preskill. Reliable quantum computers. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 1998;454(1969):385–410.
  * (136) A. Y. Kitaev. Quantum computations: algorithms and error correction. Russian Mathematical Surveys. 1997;52(6):1191–1249.
  * (137) E. Knill. Resilient quantum computation. Science. 1998; 279(5349):342–345.
  * (138) D. Gottesman. Theory of fault-tolerant quantum computation. Physical Review A. 1998;57(1):127.
  * (139) D. Gottesman. An introduction to quantum error correction and fault-tolerant quantum computation. In Quantum information science and its contributions to mathematics, Proceedings of Symposia in Applied Mathematics, vol. 68, pp. 13-58. 2010.
  * (140) S. J. Devitt, W. J. Munro, K. Nemoto. Quantum error correction for beginners. Reports on Progress in Physics. 2013; 76(7):076001.
  * (141) D. Lidar, T. Brun. Quantum error correction. Cambridge University Press; 2013.
  * (142) B. M. Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics. 2015; 87(2):307–346.
  * (143) J. H. Reif, Complexity of the Mover’s Problem and Generalizations. Proc. 20’th IEEE Syrup. Found. Comp. Sci., San Juan, Puerto Rico, pp. 421-427 (1979).
  * (144) J. D. Norton. Brownian Computation Is Thermodynamically Irreversible. Found Phys 43, 1384–1410 (2013).
  * (145) R.P. Feynman. Simulating physics with computers. Int. J. Theoret. Physics , 21:467-488, 1982.
  * (146) R.P. Feynman. Quantum mechanical computers. Foundations of Physics , 16(1986), 507-531.
  * (147) M. Li, and P. M. B. Vitányi. Mathematical theory of thermodynamics of computation. Amsterdam: Centre for Mathematics and Computer Science, 1992.
  * (148) O. J. E. Maroney. The (absence of a) relationship between thermodynamic and logical reversibility. Stud Hist Philos Mod Phys, 36:355–374, June 2005. arxiv.org://physics/0406137.
  * (149) C. M. Caves. Quantitative limits on the ability of a Maxwell demon to extract work from heat. Phys Rev Lett, 64(18):2111–2114, 1990.
  * (150) C. M. Caves. Information and entropy. Phys Rev E, 47(6):4010–4017, 1993.
  * (151) B. Piechocinska. Information erasure. Phys Rev A, 61:062314, 1–9, 2000.
  * (152) O. J. E. Maroney. Generalising Landauer’s Principle. Phys Rev E, 79:031105-1, 2009. arxiv.org://quant-ph/0702094.
  * (153) J. W. Gibbs. Elementary Principles in Statistical Mechanics. Ox Bow Press, 1902.
  * (154) R. C. Tolman. The Principles of Statistical Mechanics. Dover, 1938. (Dover publication 1979).
  * (155) M. H. Partovi. Irreversibility, reduction and entropy increase in quantum measurements. Phys Lett A, 137(9):445–450, 1989.
  * (156) D. Chu, and R. E. Spinney. A thermodynamically consistent model of finite-state machines. Interface focus 8, no. 6 (2018): 20180037.
  * (157) G. E. Crooks. Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90, 1481–1487 1998.
  * (158) D. H. Wolpert, A. Kolchinsky, & A. J. Owen. A space–time tradeoff for implementing a function with master equation dynamics. Nature communications, 10(1), 1-9,(2019).
  * (159) A. de Castro. One-way-ness in the input-saving (Turing) machine. Physica A: Statistical Mechanics and its Applications, 415, 473-478 (2014).
  * (160) C.H. Bennett, R. Landauer, The fundamental physics limits of computation, Sci. Am. 253 (1985) 48.
  * (161) D. Mandal and C. Jarzynski, Work and information processing in a solvable model of Maxwell’s demon, Proc. Natl. Acad. Sci. 109, 11641 (2012).
  * (162) A. C. Barato and U. Seifert, An autonomous and reversible Maxwell’s demon, Europhys. Lett. 101, 60001 (2013).
  * (163) D. Mandal, H. T. Quan, and C. Jarzynski, Maxwell’s refrigerator: An exactly solvable model, Phys. Rev. Lett. 111, 030602 (2013).
  * (164) S. Deffner and C. Jarzynski, Information Processing and the Second Law of Thermodynamics: An Inclusive, Hamiltonian Approach, Phys. Rev. X 3, 041003 (2013).
  * (165) A. C. Barato and U. Seifert, Stochastic thermodynamics with information reservoirs, Phys. Rev. E 90, 042150 (2014).
  * (166) P. Strasberg, G. Schaller, T. Brandes, and C. Jarzynski, The Second Laws for an Information driven Current through a Spin Valve, Phys. Rev. E 90, 062107 (2014).
  * (167) J. D. Norton, On Brownian computation, Int. J. Mod. Phys. Conf. Ser. 33, 1460366 (2014).
  * (168) C. Van den Broeck. Stochastic thermodynamics: A brief introduction. Physics of Complex Colloids, 184, 155-193 (2013).
  * (169) M. Esposito, & C. Van den Broeck. Three faces of the second law. I. Master equation formulation. Physical Review E, 82(1), 011143 (2010).
  * (170) A. R. Calderbank, & P. W. Shor. Good quantum error-correcting codes exist. Physical Review A, 54(2), 1098, (1996).
  * (171) A. M. Steane. Error correcting codes in quantum theory. Physical Review Letters, 77(5), 793 (1996).
  * (172) C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, & W. K. Wootters, Mixed-state entanglement and quantum error correction. Physical Review A, 54(5), 3824, (1996).
  * (173) E. Knill, & R. Laflamme. Theory of quantum error-correcting codes. Physical Review A, 55(2), 900, (1997).
  * (174) A. R. Calderbank, E. M. Rains, P. W. Shor, & N. J. Sloane. Quantum error correction and orthogonal geometry. Physical Review Letters, 78(3), 405 (1997).
  * (175) A. R. Calderbank, E. M. Rains, P. W. Shor, & N. J. Sloane. Quantum error correction via codes over GF (4). IEEE Transactions on Information Theory, 44(4), 1369-1387 (1998).
  * (176) D. Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Physical Review A, 54(3), 1862 (1996).
  * (177) D. Gottesman, Stabilizer Codes and Quantum Error Correction. PhD thesis, Caltech (1997), quant-ph/9705052.
  * (178) V. Vedral. Landauer’s erasure, error correction and entanglement. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 456(1996), 969-984 (2000).
  * (179) C. Cafaro, & P. van Loock. An entropic analysis of approximate quantum error correction. Physica A: Statistical Mechanics and its Applications, 404, 34-46 (2014).
  * (180) J. Kempe, D. Bacon, D. A. Lidar, & K. B. Whaley. Theory of decoherence-free fault-tolerant universal quantum computation. Physical Review A, 63(4), 042307 (2001).
  * (181) A. Osterloh, L. Amico, G. Falci, & R. Fazio. Scaling of entanglement close to a quantum phase transition. Nature, 416(6881), 608-610 (2002).
  * (182) D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, & K. B. Whaley. Universal quantum computation with the exchange interaction. nature, 408(6810), 339-342 (2000).
  * (183) A. Y. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2-30 (2003).
  * (184) M. Freedman, A. Kitaev, M. Larsen, & Z. Wang. Topological quantum computation. Bulletin of the American Mathematical Society, 40(1), 31-38 (2003).
  * (185) E. Lieb, T. Schultz, & D. Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3), 407-466 (1961).
  * (186) I. Prigogine, and G. Nicolis. Self-organisation in nonequilibrium systems: towards a dynamics of complexity. Bifurcation analysis. Springer, Dordrecht, 1985. 3-12.
  * (187) O. C. de Beauregard. The computer and the heat engine. Foundations of physics, 19(6), 725-727 (1989).
  * (188) S. Ishioka, and N. Fuchikami. Thermodynamics of computing: Entropy of nonergodic systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(3), 734-746 (2001).
  * (189) N. Shimizu, Y. Harada, N. Miyamoto, and E. Goto. A new A/D converter with quantum flux parametron. IEEE Transactions on Magnetics, 25(2), 865-868 (1989).
  * (190) E. Goto, N. Yoshida, K. F. Loe, W. Hioe. A Study on Irreversible Loss of Information without Heat Generation. In Foundations Of Quantum Mechanics In The Light Of New Technology: Selected Papers from the Proceedings of the First through Fourth International Symposia on Foundations of Quantum Mechanics (pp. 389-395) (1996).
  * (191) N. Murphy, R. Petersen, A. Phillips, B. Yordanov, and N. Dalchau. Synthesizing and tuning stochastic chemical reaction networks with specified behaviours. Journal of The Royal Society Interface, 15(145), 20180283 (2018).
  * (192) L. Qian, and E. Winfree. Scaling up digital circuit computation with DNA strand displacement cascades. Science, 332(6034), 1196-1201 (2011).
  * (193) G. Banegas, and D. J. Bernstein. Low-communication parallel quantum multi-target preimage search. International Conference on Selected Areas in Cryptography. Springer, Cham, 2017.
  * (194) R. Beals, et al. Efficient distributed quantum computing. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469.2153 (2013): 20120686.
  * (195) D. J. Bernstein. Cost analysis of hash collisions: Will quantum computers make SHARCS obsolete. SHARCS 9 (2009): 105.
  * (196) S. R. Fluhrer. Reassessing Grover’s Algorithm. IACR Cryptol. ePrint Arch. 2017 (2017): 811.
  * (197) R. Perlner, and Y. K. Liu. Thermodynamic Analysis of Classical and Quantum Search Algorithms. arXiv preprint arXiv:1709.10510 (2017).
  * (198) P. C. Van Oorschot, and M. J. Wiener. Parallel collision search with cryptanalytic applications. Journal of cryptology 12.1 (1999): 1-28.
  * (199) G. Brassard, P. Høyer, and A. Tapp. Quantum cryptanalysis of hash and claw-free functions. Latin American Symposium on Theoretical Informatics. Springer, Berlin, Heidelberg, 1998.
  * (200) V. Giovannetti, S. Lloyd, and L. Maccone. Quantum random access memory. Physical Review Letters, 100(16):160501, 2008.
  * (201) C. Zalka. Grover’s quantum searching algorithm is optimal. Phys. Rev. A, 60:2746–2751, 1999.
  * (202) S. Tani. An Improved Claw Finding Algorithm Using Quantum Walk, pages 536–547. Springer Berlin Heidelberg, Berlin, Heidelberg, 2007.
  * (203) S. B. Laughlin. Energy as a constraint on the coding and processing of sensory information. Current opinion in neurobiology, 11(4), 475-480 (2001).
  * (204) V. Balasubramanian, D. Kimber, and M. J. B. Ii. Metabolically efficient information processing. Neural computation, 13(4), 799-815 (2001).
  * (205) G. T. Landi, A. L. F. de Oliveira, and E. Buksman. Thermodynamic analysis of quantum error-correcting engines. Physical Review A, 101(4), 042106 (2020).