# Kinetic Exchange Models of Income and Wealth Distribution : Self
Organization and Poverty Level

Sanjukta Paul  Department of Physics, Bankura University, Purandarpur,
Bankura, West Bengal 722155, India  Bikas K. Chakrabarti  Saha Institute of
Nuclear Physics, Kolkata 700064, India  Economic Research Unit, Indian
Statistical Institute, Kolkata 700108, India

##  1 Introduction

Our thought process is often wired to rely heavily on a reductionist approach
while trying to explain any observed complex physical phenomena. This helps
especially when we fail to understand the basic principles governing them.
What we essentially do is break up a physical event into the smallest possible
snapshots of incidents, dismantle a machine to its constituent parts and tear
apart an atom to find its elementary particles, so that we know how the basic
units actually behave. While reaching the roots of a physical problem can at
times give us piecewise information, that can add up to generate the bigger
picture, it may not always give a one-to-one correspondence between the two.
Thus, conjecturing the characteristics of the collective behavior by always
studying the properties of the elements can be quite misleading! Here come the
features of any many-body theory, which differs considerably from the single-
body behavior. A simplistic guess could be that the additive laws our mental
imagery uses to yield a many-body picture from a collection of single bodies,
can often give way to laws of multiplication which drastically change the
results. In the context of physics, this is termed “emergence”, which means
giving rise to combined or cooperative properties that are starkly different
from the individual characteristics. In the literary language, it simply
translates to the child being very different from the parents!

The field of econophysics is one such emergent interdisciplinary area, which
briefly stated, tries to employ established and tested laws of physics (in
particular the many-body physics or statistical physics of condensed matter
systems) to model and comprehend social behavior or, in particular, market
behavior. Both these social or economic systems are many-body systems, though
the number of basic constituents, $N$, (“Social atoms”$\sim 10^{2}$ in case of
rural markets and $\sim 10^{9}$ in case of global markets) is much less than
the Avogadro number $\sim 10^{23}$ (as in physical condensed matter systems).
The reason for the expected success of econophysics comes from the observation
that no established law in one branch of natural science (say physics) becomes
invalid in another (say in biology or chemistry), though the importance may
not be uniform in all the branches of natural sciences. Also, physics has been
the oldest of all the natural sciences to get firmly established and so has
already acquired a lot of armory over more than three centuries. Consequently,
the branches like physical chemistry, chemical physics, biophysics, geophysics
etc. have developed and contributed significantly to the respective mother
disciplines. In a similar way, the recent attempts to develop
interdisciplinary fields like econophysics or sociophysics have followed quite
inevitably. Indeed, the early approaches to develop econophysics (see e.g.,
Saha and Srivastava [1] in 1931, Majorana [2] in 1942, Mandelbrot [3] in 1960,
Mantegna and Stanley [4] in 1995, and Chakrabarti and Marjit[5] in 1995)
proposed precisely this natural path. Continuing this, the recent attempts to
develop econophysics (see e.g., [6, 7, 8] for reviews) and sociophysics (see
e.g., [9, 10] for reviews) have made considerable progress.

The kinetic theory has been the oldest (more than a century old) many-body
theory, starting with the classical ideal gas model (see e.g., [1]). The
theory has undergone several tests over ages and has made important analytical
progress over the years. It has therefore been a instinctive choice of
econophysicists for developing the first-order models and science of markets,
and society in general. In particular, the model studies on income or wealth
distribution in societies have received major attention and obtained
reasonable success (see e.g., [11, 12, 13, 14, 15, 16]; see also [16] arguing
why econophysics, economics for that matter, has to become an integral part of
natural science).

We will talk about Kinetic Exchange Models (KEMs) in detail later (see Sec. 3)
after touching upon the concepts of Inequality indices (in Sec. 2), since
their existence has predated these models in other fields of study and which
play pivotal roles in understanding realistic aspects of the models.
Discussions on the concept of Self-Organization and its relevance in KEMs
follows in Sec. 4, with the summary in Sec. 5 winding up the current chapter .

##  2 Inequality Indices

Societies, since time immemorial, have been rife with inequalities and it is
important to study them carefully since they have far-flung consequences like
fluctuations in economy, political instability, civil wars and global
conflicts. Inequalities can be broadly categorized into inequalities of
conditions that imply inequality in the wealth, asset and income distributions
and inequalities of opportunities referring to unequal access to “life
chances” like education, health facilities and so on. Inequalities arise
because human interactions, in the realm of trade, are quite complex and the
resulting dynamics often lead to them quite stochastically. However, the
inequalities can be easily computed (see for e.g. [17, 18]) with relative
measures of income using certain indices, so that we can easily regulate them,
rather than studying broader distributions of wealth or income, involving log-
normals and power laws.

Figure 1:  $L(x)$, the Lorenz function, shown as a red curve, represents the
fraction of overall income or wealth assumed by the fraction $x$ of the
people, when arranged from lowest to highest income. The Gini index $g$ is the
ratio of the area between the line of equality and Lorenz curve, normalized by
the total area under the line of equality ($g=2S$). The complementary Lorenz
function $L^{c}(x)\equiv 1-L(x)$, given by the green curve, is essential for
defining the $k$ index. The $k$ index is given by the ordinate value of the
intersection of $L^{c}(k)$ with the diagonal perpendicular to the equality
line. It is a fixed point $L^{c}(k)=k$ of $L^{c}(x)$.

###  2.1 Gini Index ($g$)

The most traditional and commonly used inequality index is the Gini index $g$
([17, 18]). To define $g$, we need to remind ourselves of the Lorenz curve
which represents the cumulative fraction $L(x)$
($L=\int_{0}^{m}mP(m)dm/[\int_{0}^{\infty}mP(m)dm]$) of income or wealth
accumulated (or citations of articles, cumulative proportion of species in a
biodiversity, etc.) to the cumulative proportion $x$ ($\int_{0}^{m}P(m)dm/$
$[\int_{0}^{\infty}P(m)dm]$) of individuals (or entries) in a population,
ordered from poorest to richest. We can thus readily infer that if every
individual or trader possessed equal wealth or income, then we would get the
equality line $L(x)=x$ to be passing through the origin (refer Fig. 1). An
inequality measure would immediately suggest that we consider the deviation
from this perfect equality line. Since we are considering a dispersion of the
income or wealth on a 2D plane, it is common to use the area enclosed by the
straight line and the curve to represent the degree of inequality. However, it
is imperative to use ratios of areas to reduce the Gini index ($g$) to just a
number so that $g=0$ represents perfect equality, where every individual has
the same money. On the other extreme, $g=1$ represents perfect inequality,
where the total money gets acquired by a single individual in the market. $g$
is thus defined as the ratio of the area between the equality curve and the
Lorenz curve $S$ to the area below the equality line $(S+S^{\prime})$ (see
Fig. 1). A simple observation suggests that $g$ is actually equal to $2S$.

###  2.2 Kolkata Index ($k$)

A relatively newer inequality index than $g$, called the Kolkata index $k$,
first proposed in [17] to account for acute social inequalities in Kolkata, is
given by the y-coordinate $k$ of the intersection point of the Lorenz curve
and the diagonal perpendicular to the equality line (see Fig. 1 and [19] for
some details). $k$ is the non-trivial fixed point of the complementary Lorenz
function $L^{c}(x)=1-L(x)$, similar to $x=0$ ,$1$ being the non-trivial fixed
points of $L(x)$. Kolkata index states that fraction $(1-k)$ of the population
possesses $k$ fraction of the total wealth. This index essentially vouches for
the validity of the 80 - 20 fractions of Pareto’s law by indicating that
fraction $(1-k)$ of the population earns more money or possesses more wealth
than fraction $k$ of the people (for some early report on observations and
data analysis, see [20]). The two limits of the $k$ index are $k=1/2$ for
$g=0$ and $k=1$ for $g=1$.

##  3 Kinetic Exchange Models for Economy

The kinetic theory of gases deals with the behavior of a large number of
particles, by studying their interactions in the microscopic (small scale)
detail to calculate macroscopic (large scale) parameters [1]. Thermodynamics,
however, has been the oldest discipline that establishes relationships between
the macroscopic parameters in equilibrium, using few postulates but without
delving into the microscopic details. Statistical mechanics, being the most
advanced in this lineage, stems from these two fields and using the laws of
mechanics, makes some valid statements about the microscopic properties of the
system in equilibrium and connects them to the macroscopic properties.

###  3.1 Maxwell-Boltzmann and Gibbs distribution in an ideal gas

In the conventional kinetic theory for gas models, the gas molecules or atoms,
at equilibrium and at a particular temperature $T$ collide with one another
and exchange kinetic energy (they are treated as free particles neglecting the
potential energy), keeping the total energy conserved in the process. The
energy distribution function in this case can be seen to be an exponential of
the form $f(\epsilon)=\exp({-\epsilon/{\Delta}})$ (from the condition
$f(\epsilon_{1})f(\epsilon_{2})=f(\epsilon_{1})+f(\epsilon_{2})$,
$\epsilon_{1}$ and $\epsilon_{2}$ being energies of any two particles
respectively) [1], where $\epsilon$ denotes the energy of a gas particle and
$\Delta$ can be estimated from the equation of state of the gas. It must be
noted that the gas particles have a chaotic motion and that the change in
velocity (or energy) of the particles after a collision is fully random. This
implies that at an instant of time, even though particles may have the same
velocity or energy, after a collision process some have these quantities
increased and the others reduced. From the distribution function, we can
easily calculate the number density of gas particles [1] having energy
$\epsilon$ to be $n(\epsilon)=\sqrt{\epsilon}\exp({-\epsilon/{KT}})$, where
$\Delta=KT$. The factor $\sqrt{\epsilon}$ ensues from the density of states
(in 3 dimensions) [1], which when multiplied to the distribution function
$f(\epsilon)$, gives rise to the number density $n(\epsilon)$. This gives the
Gamma Distribution or the Maxwell-Boltzmann Distribution, where a peak or most
probable energy corresponding to a non-zero energy possessed by the largest
number of gas particles, exists. Particles with very low and very high energy
are considerably less in number, following energy the distribution function
(see [18] for an elaborate discussion).

###  3.2 Gibbs Distribution in Kinetic Exchange Models of market

Translating the kinetic theory for gases to the domain of market models in an
economy (see for e.g. [21, 22, 23, 24]), we can find equivalences and
differences between the two. In the KEMs for wealth distribution, we have
random traders or agents that act as socially interacting atoms taking part in
a trade, where random exchanges of wealth is conducted, much like the
collisions between the gas particles resulting in exchange of energies.
Wealth, in general, is usually considered to be the solid assets, one
individual or an organization possesses, having an economic value. However, it
can broadly represent the fortune or well being of a particular individual. In
these models, wealth denotes the exchange of a more fluid entity called money,
that traders use for bartering intangible services or tangible objects of
interest. It thus is an analogy for economic energy and is associated with an
economic temperature $\Delta^{\prime}$, a constant for a trading process.
Since there is no counterpart of the particle momenta in the trade market, the
density of states is a constant and hence as before (see Sec. 3.1), the number
density $n(m)=cf(m)$, where $f(m)\sim\exp(-m/{\Delta^{\prime}})$ (with the
condition that $f(m_{1})f(m_{2})=f(m_{1})+f(m_{2})$, $m_{1}$ and $m_{2}$ being
wealth of any two agents respectively) is the money distribution function. We
note that the total number of traders $\int_{0}^{M}n(m)dm=N$, is a constant in
a market, although the buyer of one trading transaction may be the seller of
the other and vice versa. The total money is $\int_{0}^{M}mn(m)dm=M$ and
without considering debt in the system, it is a constant for a market.
Therefore, $\Delta^{\prime}$, identified as $M/N$ (the average money per
person) is a constant. Thus we conclude that if the traded amount is a
fraction of the wealth possessed by an agent, subject to the conservation of
$M$, the resulting distribution at equilibrium is undoubtedly the Gibbs or
exponential distribution. An exponential distribution immediately points at
the largest number of paupers in the system to be the condition for the maxima
in the distribution, indicating an unequal distribution of wealth, where most
of the money gets concentrated in a few rich hands. This inequality strongly
underpins the absence of an equivalent equipartition theorem, applicable for
gas models, that renders equal kinetic energies of $\frac{1}{2}KT$ for each
degree of freedom in a polyatomic gas molecule. The equipartition law, stems
from the ability to cast the kinetic energy of a gas molecule in terms of
squares of momenta, which has no analogy in the market model[18].

Now, we present from [11], a purely observational income distribution data in
Fig. 2 to illuminate the nature of the different kinds of distribution
elaborated in Secs. 3.1 and 3.2. This plot beautifully demarcates the lowest
income group exponential distribution from the highest income section of the
society, following a power law decay (Pareto tail).

###  3.3 Estimates of $g$ and $k$ for the Gibbs distribution

To understand $g$ and $k$ quantitatively, we can evaluate them for the
simplest case where P(m) is the normalized Gibb’s distribution,
$P(m)=\exp(-m)$. Now as,
$x=\int_{0}^{m}\exp(-m^{\prime})dm^{\prime}=1-\exp(-m)$, it yields
$m=-\ln(1-x)$, and hence
$L=\int_{0}^{m}m^{\prime}\exp(-m^{\prime})dm^{\prime}$ = $1-(m+1)\exp(-m)$.
Thus, $L(x)=$ $1-(1-x)[1-\ln(1-x)]$. Since, the area under the equality line
is the area of one of the two triangles with equal areas, so it is 1/2. The
corresponding $g=$ $1-2\int_{0}^{1}L(x)dx=1/2$, $L(x)$ being the Lorenz curve.
The Kolkata index $k$ for this distribution is therefore given by the self-
consistent equation $1-k=L(k)$ or $1-2k=(1-k)[\ln(1-k)]$, that produces
$k\simeq 0.68$.

Figure 2: The US income distribution data (obtained from income tax returns)
collapse for the period 1983–2001. The cumulative percentage of returns has
been plotted against normalized income in log-log scale. The fitting values of
the effective temperature (obtained by fitting the data for the exponential
part of the distribution; see Sec. 3.5) have been given against each year.
This figure has been reproduced from [11] (permission from American Physical
Society).

###  3.4 Basic computational framework for Kinetic Exchange Models

Two agent interactions in a closed economy, with total number of agents $N$
and total money $M$ can be considered analogous to a closed thermodynamic
system with total energy and total number of gas particles as constants. Any
two randomly selected agents can carry out trade among themselves following a
Markovian dynamics, where exchange of money at any time $t+1$ does not depend
on any other exchanges other than $t$. The exchange mechanism $\mathcal{M}$
can be defined as

$\displaystyle\begin{bmatrix}m_{i}(t+1)\\\
m_{j}(t+1)\end{bmatrix}=\mathcal{M}\begin{bmatrix}m_{i}(t)\\\
m_{j}(t)\end{bmatrix}$ (1)

Here, each agent or trader possesses money or wealth $m_{i}(t)$ at time $t$
and any two random traders $i$ and $j$ exchange money by participating in the
trading process, with no provision for debt ($m_{i}(t)\geq 0$). Since
conservation of total money is maintained in the whole trading process, we
have

$\displaystyle m_{i}(t+1)=m_{i}(t)+\Delta m$ $\displaystyle
m_{j}(t+1)=m_{j}(t)-\Delta m$ (2)

and

$\displaystyle m_{i}(t)+m_{j}(t)=m_{i}(t+1)+m_{j}(t+1)$ (3)

$\Delta m$ represents the fraction of money that is used for trade. Each trade
corresponds to an increase by an unit time interval (see [28] for more
realistic saving behaviors).

###  3.5 Without-saving income exchange models

It is more realistic to use a fraction of the money a trader possesses, rather
than invest the whole of it in the interaction. If $0<\epsilon_{ij}<1$
represents a uniform stochastic fraction, with an average precisely equal to
0.5,

$\displaystyle\Delta m=\epsilon_{ij}[m_{i}(t)+m_{j}(t)]-m_{i}(t)$ (4)

The steady state always gives the Gibbs Distribution, $P(m)=1/T\exp(-m/T),$
$T=M/N$, where it is immediately obvious that most of the population has very
less money corresponding to the exponentially decaying curve, irrespective of
whether the initial distribution was uniform (see [24] for a detailed
description).

###  3.6 Uniform-saving income exchange models with inequality indices

Although the conservation of total money is a necessary condition for the
above models to work as reliable econophysical models, the number of paupers
is too large for them to point towards a sustainable economy. By this we mean
that a trader is prone to lose all his money to a fellow trader in a single
trade since the exchanges are completely random, keeping the total money
conserved. This is in synchronism to the kinetic theory of gases, where one
gas particle can lose the entire energy in a single collision. This brings us
to a more realistic structure of these exchange models, where savings appears
quite organically for individuals unlike gas molecules. If a saving propensity
is introduced, traders will save a fraction of their money before involving
the rest of the money in trading and hence will avoid being paupers. A trader
can only become a pauper if he loses in every trade and gradually loses the
saved fraction of the money, which is not possible because exchanges are
random. Thus, for any non-vanishing saving propensity that remains uniform
over time, the steady state distribution transitions from an exponential or
Gibbs to a Gamma distribution, where we have zero number density of paupers.
Let $\lambda$ be the uniform or fixed saving propensity of the traders. The
exchange of money between two randomly chosen pairs $i$ and $j$ can be
expressed as

$\displaystyle m_{i}(t+1)$ $\displaystyle=\lambda
m_{i}(t)+\epsilon_{ij}((1-\lambda)(m_{i}(t)+m_{j}(t)))$ $\displaystyle
m_{j}(t+1)$ $\displaystyle=\lambda
m_{j}(t)+(1-\epsilon_{ij})((1-\lambda)(m_{i}(t)+m_{j}(t)))$ (5)

where $0\leq\epsilon_{ij}\leq 1$ is a random fraction varying in every
interaction.

The important aspect to note in Fig. 3 here is that introduction of a saving
propensity immediately introduces a most probable income (MPI) in the
distribution and we have the MPI shifting from $m=0$ for $\lambda=0$ to
$m=M/N$ for $\lambda\rightarrow 1$. Thus, the Markovian dynamics slowly gives
way to a self-organizing feature of the whole market, even with the saving
fractions on individual levels. The market thus essentially turns interacting
for non-zero $\lambda$, with a global order and a critical behavior emerging,
very different from the non-cooperative nature of the $\lambda=0$ limit. A
natural extension of this would be the inclusion of a “distributed-saving
propensity” , which has been studied in [24]. For non-uniform saving
propensity, the Gamma distribution collapses to a power law decay called the
Pareto tail (see Fig. 2). The saving propensities readily generate the cross-
over point for the Pareto distribution.

Figure 3: Kinetic exchange dynamics with uniform saving propensity : (a)
$P(m)$, the steady state income distribution, for fixed saving propensity
$\lambda$ has been plotted. The exponential nature of the tail end of the
distributions is presented in semi-log scale in the inset. (b) The variations
of the Kolkata index ($k$), the Gini index ($g$) and location of the Self-
Organized Poverty Level (SOPL) are plotted against fixed saving propensity
$\lambda$ (maximum value of $\rm\lambda=1_{-}$).

###  3.7 Indices for pure kinetic exchange model with two annealed choices of
$\lambda$

Figure 4: The pure KEM for two choices of $\lambda$. (a) $P(m)$, the steady
state income distribution, for fixed saving propensity $\lambda$ has been
plotted. The exponential nature of the tail end of the distributions is
presented in semi-log scale in the inset. (b) The variations of the Kolkata
index ($k$), the Gini index ($g$) and location of the Self-Organized Poverty
Level (SOPL) are plotted against the probability for taking $\lambda=1$ i.e.
$\rho$ (maximum value of $\rho=1_{-}$).

In order to incorporate the inherent volatility of human saving behavior or
take into account impalpable circumstances that might affect the constancy of
the saving behavior, it would be wise to include a time dependence of
$\lambda$. Hence, mathematically, treating the $\lambda$ probabilistically
should apparently solve the problem. The multiplicative probability factor
$\rm\rho$ thus determines the “luck ”for the saving behavior since we consider
only the two extreme conditions of $\lambda_{\rho}=1$ with probability $\rho$
and $\lambda_{\rho}=0$ with probability $1-\rho$; $\rho<1$. Therefore, the
slightly modified (from Eq. 5) exchange dynamics goes as follows:

$\displaystyle
m_{i}(t+1)=\lambda_{\rho}m_{i}(t)+\epsilon_{ij}((1-\lambda_{\rho})(m_{i}(t)+m_{j}(t)))$
$\displaystyle
m_{j}(t+1)=\lambda_{\rho}m_{j}(t)+(1-\epsilon_{ij})((1-\lambda_{\rho})(m_{i}(t)+m_{j}(t)))$
(6)

It is to be noted that $\lambda_{\rho}$ will be treated as $\lambda$ from now
on, provided Eqn. 6 is satisfied.

Eqn. 6 suggests that each trader has two choices of $\lambda$ over time. We
observe in Fig. 4 that the money distribution regains its exponential nature
pushing the MPI to zero again and the inequality indices $g$ and $k$ increase
minimally with $\rho$. It can be readily inferred that although a
probabilistic factor has been introduced into the picture, $\lambda$ can only
oscillate between two distinct and extreme values of 1 (with probability
$\rho$) and 0 (with probability $1-\rho$) . It may be noted that when
$\lambda=0$, we have the well known Gamma Distribution (see Fig. 3) where the
MPI is close to zero and for $\lambda=1$, the exchange process stops since
money is only saved and not traded. Realistically, for $\lambda\sim 1$ (that
has been studied for simulations in [28]), very few people save the total
money in the system and most of the population go penniless. Although, the
time-averaged distribution may look similar for both the limits, the number of
people being paupers is far higher for $\lambda\sim 1$. We thus see in Fig. 4
that for $\rho=0$ (which implies $\lambda=0$ with probability 1) we regain the
$\lambda=0$ exponential of Fig. 3. Also, for higher $\rho\sim 1$, we have a
very high probability of having $\lambda\sim 1$, leading to a steeper
exponential. Therefore, there can be no intermediate $\rho$ value possible for
which we get a non-zero MPI and the inequality indices ($g$ and $k$) only
increase with $\rho$ (Fig. 4).

##  4 Self-Organization

The concept of self-organization is not unique to the field of market models
but to various other fields where “avalanches” occur due to dynamical
interactions of individual particles, without the help of any external
disturbance. In the market models, it can be readily inferred from Sec. 3.7
and Fig. 4 that there should be a bias in the selection of agents, so that the
non-zero MPI condition can be revived (for important observations, see [25,
26, 27]). In [28], this has been explored by taking one of the agents to be
the poorest of all in the trade and keeping the saving propensity time-
independent and time-dependent respectively (see Figs. 5 and 6 respectively).
There appears a self-organized threshold value of money, with no trader having
money less than that, arising purely due to the spontaneous interaction
between the agents called the Self-Organized Poverty Level (SOPL).

Figure 5: Self-Organized Poverty Level model: (a) $P(m)$, the steady state
income distribution, for fixed saving propensity $\lambda$ has been plotted.
The exponential nature of the tail end of the distributions is presented in
semi-log scale in the inset. (b) The variations of the Kolkata index ($k$),
the Gini index ($g$) and location of the Self-Organized Poverty Level (SOPL)
are plotted against fixed saving propensity $\lambda$ (maximum value of
$\rm\lambda=1_{-}$).

###  4.1 Self-organized poverty level in Kinetic Exchange models and
inequality indices

For a time-independent $\lambda$, one of the agents is favorably chosen,
provided the agent has the lowest money while the other agent is randomly
chosen from the rest of the population. The money exchange mechanism is the
same as in Eqn. 6. It has been observed in [27, 28] that not only does the MPI
shift to the right end of the money distribution (close to $m=1$ for larger
values of $\lambda$) but also an SOPL appears (as in Figs. 5 and 6), even for
$\lambda=0$, that was previously absent. The SOPL has been found to increase
with the increase in $\lambda$ (see Fig. 5). It is to be emphasized that the
inequality indices $g$ and $k$ are also seen to decrease with $\lambda=0$
(Fig. 5), establishing the fact that the mere advantage of being afloat in the
trading market on a local scale, without losing the ability to trade because
of recurrent losses in the exchange, brings about a collective change in the
global economy.

###  4.2 An estimate of the SOPL for $\lambda=0$

We denote the SOPL (for saving propensity $\lambda=0$) by $\delta$ and assume
the money or wealth distribution $P(m)$ remains exponential ($\rm\exp(-m)$),
with average money = $M/N=1$ (see Fig. 3 of [28]). Assuming a random (50 - 50
sharing) exchange between the agent at money $\delta$ and another with the
average money $(\delta+1)$ with probability $\exp(-1)$, one gets
$\delta=(1/2)[\delta+(1+\delta)\exp(-1)$, giving
$\delta=\exp(-1)/(1-\exp(-1))$, or $\delta\simeq 0.58$, compared to
numerically estimated value ([27, 28]) of $\delta\simeq 0.60$.

###  4.3 Self-organized minimum poverty level model: Indices for two choices
of $\lambda$

Figure 6: Emergence of a self-organized minimum poverty level model is shown
for two choices of saving propensity: (a) $P(m)$, the steady state income
distribution, for fixed saving propensity $\lambda$ has been plotted. The
exponential nature of the tail end of the distributions is presented in semi-
log scale in the inset. (b) The variations of the Kolkata index ($k$), the
Gini index ($g$) and the Self-Organized Poverty-Line or SOPL are plotted
against fixed saving propensity $\lambda$. The inequality indices gradually
increase along with the SOPL as a function of $\rho$ (maximum value of
$\rho=1_{-}$).

In this section we revisit the final case investigated in [28]. $\lambda$, as
before in Sec. 3.7 can take two values of 1 or 0, depending on the probability
$\rho$ or ($1-\rho$). However, the biasness of always selecting one of the
agents to be the poorest of all in the population remains fixed. The steady
state and averaged distribution $P(m)$ varying with $m$ is provided in Fig. 6.
The remarkable change noted here is that the time dependence does in no way
kill the finite MPI and the SOPL formation (unlike in Fig. 4) ! Both are
rather retained, making them robust with change in $\lambda$. However, the
inequality indices $g$ and $k$ increase very slowly with $\rho$ and the SOPL
reduces gradually with $\rho$.

##  5 Summary and Discussions

In summary, we have invited the reader to engage in a broader discussion about
the captivating field of econophysics (see Sec. 1) and how, sitting on the
fertile confluence of physics and economics, it seeks quantitative answers
about the questions of inequalities in economies, spontaneous re-organization
of money distributions and triggers research on methods to improve the overall
quality of economies in general. We have tried to provide a nuance of the KEMs
(refer Sec. 3) and the relevant inequality indices (see Sec. 2) needed to
model any many-body economical market, using basic laws in physics. We have
also extended our ideas on how they can be tuned to generate self-organizing
features in the wealth distribution of a population, leading to a minimum
poverty line and reduced inequality measures. Since we have visualized
economies through the lens of physics, governed by precise laws that have been
tested, we have more freedom to introduce certain constraints or biases in the
system that can mimic real world policies framed for the reduction of
inequalities in income or wealth distributions among people. We have thus
enumerated various cases (mostly numerically; see for e.g. [28, 29]) where the
basic KEMs have been manipulated (see Secs. 3, 4) to include a saving behavior
locally, with an impact that would grow manifold and show up in the global
MPI; favor the poorest in the trading market, which in the real world could
mean giving them incentives; or render the saving behavior time-dependent,
which in real economies too could put a cap on the reduction of the inequality
in possession of wealth.

While the world economy, in recent times, is abuzz with reports on
condensation of almost half the population’s total wealth in the hands of a
countable few rich, it is essential to investigate whether the KEMs can
capture such observations. In the acclaimed Chakraborti model [30] or the so-
called Yard Sale model, where for each trade the richer one offers the same
amount as that possessed by the other, the market dynamics stops when the
entire money ends up in the hands of one trader (and no further trade is
possible with the paupers). The KEM of Goswami and Sen [31], where the trading
probability between any pair of traders decreases with their wealth difference
(following an inverse power of the wealth difference between the two traders),
indicated considerable wealth concentration towards the richer. Following this
indication, the Banerjee model was proposed [32], where the trade between just
nearest neighbors in wealth at any stage showed [33] that about $99.8\%$ of
the total wealth gets accumulated in the hands of exactly 10 traders (fortunes
of these traders are not everlasting) in the steady state for the macroscopic
limit of the market size $N$. The novelty of such KEMs ([30, 31, 32]) for
these quasi-oligarchic societies is provided by the observation that the
presence of a non-vanishing probability of random exchanges can destroy the
condensation phenomena and can lead to exponential wealth distributions in the
large $N$ limit (see [33]).

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based input model with the high-level sparse matrix simulation algorithms, we
can perform prototype structure and dynamics calculations for these model
problems on near-term noisy intermediate scale quantum devices. Generalization
to complex and realistic many-nucleon problems is straightforward.

Going forward, we plan to explore the applications of our framework to perform
dynamics simulations and structure calculations of simple systems such as the
simplified pairing models described in this work. These calculations serve as
benchmark tests for more complicated applications, such as the four-nucleon
calculations in a restricted model space. On the other hand, we also plan to
improve the performance of our input model. While better options of basis set
may reduce the counts of the terms in the second-quantized Hamiltonian
Babbush:2018BasisChoice , the incorporation of simple arithmetic operations
Vedral:1996 ; Abhari:2014  would also be promising.

##  IX Acknowledgments

We acknowledge fruitful discussions with William M. Kirby, Peter J. Love,
Michael Kreshchuk, Pieter Maris, and Chao Yang. WD thanks Morten Hjorth-Jensen
and Dean Lee for valuable suggestions. This work was supported by the U.S.
Department of Energy under Grants No. DE-SC0018223 (SciDAC4/NUCLEI), DE-
SC0023495 (SciDAC5/NUCLEI), and No. DE-FG02-87ER40371.

##  Appendix A Hamiltonian input model

###  A.1 Isometry construction

For completeness and self-consistency, we present the construction of the
isometry $\mathcal{T}$ that performs the mapping defined in Eq. (49).

######  Lemma 1.

For an input state $|\mathcal{F}\rangle|b\rangle$, where $|\mathcal{F}\rangle$
denotes the many-nucleon state and $b\in\\{0,1\\}$, the isometry $\mathcal{T}$
defined in Eq. (45) can be implemented with $\mathcal{O}(1)$ queries to
$O_{\rm F}$ and $O_{\rm H}$ defined in Eqs. (43) and (44). (This Lemma follows
$Lemma$ 1 in Ref. Kirby:2021ajp  and $Lemma$ 4 in Ref. DWBerry:2012 .)

###### Proof.

The isometry $\mathcal{T}$ maps the state $|\mathcal{F}\rangle|b\rangle$ to
$|\mathcal{F}\rangle|b\rangle|\phi_{\mathcal{F},b}\rangle$ as shown in Eq.
(49). We give the proof by showing the construction of $\mathcal{T}$ as
follows.

Provided the many-nucleon state $|\mathcal{F}\rangle$, we initialize the input
state to be

$\displaystyle|\Psi_{\rm
in}\rangle=|\mathcal{F}\rangle|b\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle,$
(92)

where we write down the state of each quantum register explicitly.

##### Case 1:

If $|b\rangle=|1\rangle$, we flip the last qubit register from $|0\rangle$ to
$|1\rangle$ controlled on the ancilla qubit $|b\rangle=|1\rangle$. In this
way, we prepare the output state as

$\displaystyle|\Psi_{\rm
out}\rangle=|\mathcal{F}\rangle|1\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|1\rangle.$
(93)

##### Case 2:

If $|b\rangle=|0\rangle$, we start with the input $|\Psi_{\rm
in}\rangle=|\mathcal{F}\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle$
and operate the registers as follows.

  1. 1.

Prepare a uniform superposition of the indices $i=0,1,2,\cdots,\mathcal{D}-1$
on the third register. The overall state of the quantum register $|\Psi_{\rm
in}\rangle$ becomes

$\displaystyle|\Psi_{1}\rangle=\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}|\mathcal{F}\rangle|0\rangle|i\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle.$
(94)

  2. 2.

For a specific index $i$ and $|\mathcal{F}\rangle$, we apply the $O_{\rm F}$
oracle [Eq. (43)] on the first (leftmost), third, fourth, fifth, sixth, and
seventh registers to obtain

$\displaystyle|\Psi_{2}\rangle=\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle.$
(95)

  3. 3.

Controlled on $|y_{i}\rangle=|0\rangle$, we apply the $O_{\rm H}$ oracle [Eq.
(44)] to the first, fourth, fifth and sixth registers. The kernel
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|H^{\prime}|\mathcal{F},P_{i}\rangle$ is
computed in an ancilla register that is initiated as $|0\rangle$. Controlled
on the value
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|H^{\prime}|\mathcal{F},P_{i}\rangle$,
we rotate the last (rightmost) single qubit (initialized as $|0\rangle$) in
$|\Psi_{2}\rangle$ as

$\displaystyle|0\rangle\mapsto\sqrt{\frac{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}{\Lambda_{m}}}|0\rangle+\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|1\rangle,$
(96)

where the operation of the square root is defined below [Eq. (112)]. We choose
$\Lambda_{m}$ to be larger than the largest absolute value of the two-body
kernel:
$\Lambda_{m}\geq|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|=|\langle
p_{i}q_{i}|H|r_{i}s_{i}\rangle|$ for any $i\in[0,\mathcal{D}-1]$, such that
$0\leq\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}\leq
1$ is satisfied. Then, uncompute the ancilla register that computes
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle$
by another controlled query to $O_{\rm H}$ after the above single-qubit
rotation [Eq. (96)] is completed.

In the case of $|y_{i}\rangle=|1\rangle$, we anticipate that
$|\mathcal{F}_{i}^{\prime}\rangle$ does not connect to $|\mathcal{F}\rangle$
via the many-nucleon Hamiltonian, and
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle=0$
(due to the violation of, e.g., the symmetry, the Pauli principle, etc.). In
this case, we flip the last qubit from $|0\rangle$ to $|1\rangle$ controlled
on $|y_{i}\rangle=|1\rangle$.

After this step, without loss of generality, the full state of the quantum
registers becomes

$|\Psi_{\rm
out}\rangle=\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle\Bigg{]}\\\
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|1\rangle\Bigg{]}.$
(97)

It is understood that the necessary condition for the first term of the above
equation to exist is $|y_{i}\rangle=|0\rangle$. When
$|y_{i}\rangle=|1\rangle$, we should take
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle=0$
and flip the last qubit register from $|0\rangle$ to $|1\rangle$. In our
construction, we make use an additional, redundant qubit, the second last
qubit which is initialized as $|0\rangle$, to guarantee the orthogonalization
between the terms with $|y_{i}\rangle=|0\rangle$ and those with
$|y_{i}\rangle=|1\rangle$ [see discussion of the swap operator [Eq. (107)]
below].

In viewing the fact that only nonvanishing kernels
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle$
contribute to the summation in the first term of $|\Psi_{\rm out}\rangle$.
Therefore, we can simplify Eq. (97) as

$|\Psi_{\rm
out}\rangle=\frac{1}{\sqrt{\mathcal{D}}}\sum_{i\in\mathcal{I}(\mathcal{F})}\Bigg{[}\sqrt{\frac{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle\Bigg{]}\\\
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|1\rangle\Bigg{]},$
(98)

where $\mathcal{I}(\mathcal{F})$ denotes the set of indices $i$ with
$\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle\neq
0$. Eq. (98) can also be rewritten as

$\displaystyle|\Psi_{\rm
out}\rangle=\sqrt{\frac{1}{\mathcal{D}\Lambda_{m}}}\sum_{i\in\mathcal{I}(\mathcal{F})}\Bigg{[}\sqrt{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle\Bigg{]}+\sqrt{1-\frac{\sigma_{\mathcal{F}}}{\mathcal{D}\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|\zeta_{\mathcal{F}}\rangle|1\rangle,$
(99)

where we define

$\displaystyle|\zeta_{\mathcal{F}}\rangle=\sqrt{\frac{1}{1-\frac{\sigma_{\mathcal{F}}}{\mathcal{D}\Lambda_{m}}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle\Bigg{]},$
(100)

with
$\sigma_{\mathcal{F}}=\sum_{i=0}^{\mathcal{D}-1}|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|$.
Note that $\sigma_{\mathcal{F}}$ serves as a parameter to normalize
$|\Psi_{\rm out}\rangle$ in Eq. (99). This can be checked based on the
orthogonality relation between any term in the summand of the first term and
the second term in Eq. (98).

To summarize, we have present the construction of $\mathcal{T}$ that operates
as Eq. (49) based on one query of $O_{\rm F}$ [Eq. (43)] and two queries of
$O_{\rm H}$ [Eq. (44)].

∎

###  A.2 Block-encoding scheme

With the isometry $\mathcal{T}$ [Eq. (45)], we can access the Hamiltonian via
the block-encoding scheme following the approach in Refs. DWBerry:2012 ;
AMChilds:2009 . Indeed, it is straightforward to show that

$\displaystyle\mathcal{T}^{{\dagger}}S\mathcal{T}=\frac{1}{\mathcal{D}\Lambda_{m}}H\otimes|0\rangle\langle
0|+|0\rangle\langle 0|\otimes|1\rangle\langle 1|,$ (101)

where $|0\rangle\langle 0|$ and $|1\rangle\langle 1|$ (following the symbols
“$\otimes$”) project the subspace that corresponds to the state $|b\rangle$
(in either $|0\rangle$ or $|1\rangle$ state) in Eq. (45). $H$ is the many-
nucleon Hamiltonian defined in Eq. (4) that acts on the many-nucleon state, as
does the operator $|0\rangle\otimes\langle 0|$ in the second term above. $S$
is the swap operator defined in Eq. (107) below.

To prove the identity [Eq. (101)], we compute $\langle\mathcal{F}|\langle
b|\big{(}T^{{\dagger}}ST\big{)}|\mathcal{G}\rangle|b^{\prime}\rangle$ with
$b,b^{\prime}=$ 0 or 1. Based on the construction of the isometry
$\mathcal{T}$ [Eq. (97)], we have

$\displaystyle\mathcal{T}|\mathcal{F}\rangle|b=0\rangle=$
$\displaystyle\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle\Bigg{]}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|1\rangle\Bigg{]},$
(102) $\displaystyle\mathcal{T}|\mathcal{F}\rangle|b=1\rangle=$
$\displaystyle|\mathcal{F}\rangle|1\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|1\rangle.$
(103)

Similarly, we also have

$\displaystyle\mathcal{T}|\mathcal{G}\rangle|b^{\prime}=0\rangle=$
$\displaystyle\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle}{\Lambda_{m}}}|\mathcal{G}\rangle|0\rangle|{0}\rangle|P_{j}\rangle|\mathcal{G}^{\prime}_{j}\rangle|Q_{j}\rangle|y_{j}\rangle|0\rangle|0\rangle\Bigg{]}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle|}{\Lambda_{m}}}|\mathcal{G}\rangle|0\rangle|{0}\rangle|P_{j}\rangle|\mathcal{G}^{\prime}_{j}\rangle|Q_{j}\rangle|y_{j}\rangle|0\rangle|1\rangle\Bigg{]},$
(104) $\displaystyle\mathcal{T}|\mathcal{G}\rangle|b^{\prime}=1\rangle=$
$\displaystyle|\mathcal{G}\rangle|1\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|0\rangle|1\rangle.$
(105)

##### Case 1.

We first compute the case with $b=b^{\prime}=0$. We have

$\displaystyle\langle\mathcal{F}|\langle
0|(\mathcal{T}^{{\dagger}}S\mathcal{T})|\mathcal{G}\rangle|0\rangle=$
$\displaystyle\Bigg{\\{}\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle\Bigg{]}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|1\rangle\Bigg{]}\Bigg{\\}}^{{\dagger}}$
$\displaystyle
S\Bigg{\\{}\frac{1}{\sqrt{D}}\sum_{j=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle}{\Lambda_{m}}}|\mathcal{G}\rangle|0\rangle|{0}\rangle|P_{j}\rangle|\mathcal{G}^{\prime}_{j}\rangle|Q_{j}\rangle|y_{j}\rangle|0\rangle|0\rangle\Bigg{]}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle|}{\Lambda_{m}}}|\mathcal{G}\rangle|0\rangle|{0}\rangle|P_{j}\rangle|\mathcal{G}^{\prime}_{j}\rangle|Q_{j}\rangle|y_{j}\rangle|0\rangle|1\rangle\Bigg{]}\Bigg{\\}}.$
(106)

We define the swap operator $S$ as

$\displaystyle
S|r_{0}\rangle|r_{1}\rangle|r_{2}\rangle|r_{3}\rangle|r_{4}\rangle|r_{5}\rangle|r_{6}\rangle|r_{7}\rangle|r_{8}\rangle=|r_{4}\rangle|r_{8}\rangle|r_{2}\rangle|r_{5}\rangle|r_{0}\rangle|r_{3}\rangle|r_{7}\rangle|r_{6}\rangle|r_{1}\rangle,$
(107)

where we swap states stored in the registers as
$|r_{0}\rangle\leftrightarrow|r_{4}\rangle$,
$|r_{1}\rangle\leftrightarrow|r_{8}\rangle$,
$|r_{3}\rangle\leftrightarrow|r_{5}\rangle$, and
$|r_{6}\rangle\leftrightarrow|r_{7}\rangle$. With the swap operations, we
obtain

$\displaystyle\langle\mathcal{F}|\langle
0|(\mathcal{T}^{{\dagger}}S\mathcal{T})|\mathcal{G}\rangle|0\rangle=$
$\displaystyle\Bigg{\\{}\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|0\rangle\Bigg{]}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{i=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle|}{\Lambda_{m}}}|\mathcal{F}\rangle|0\rangle|{0}\rangle|P_{i}\rangle|\mathcal{F}^{\prime}_{i}\rangle|Q_{i}\rangle|y_{i}\rangle|0\rangle|1\rangle\Bigg{]}\Bigg{\\}}^{{\dagger}}$
$\displaystyle\Bigg{\\{}\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{\frac{\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle}{\Lambda_{m}}}|\mathcal{G}^{\prime}_{j}\rangle|0\rangle|{0}\rangle|Q_{j}\rangle|\mathcal{G}\rangle|P_{j}\rangle|0\rangle|y_{j}\rangle|0\rangle\Bigg{]}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\Bigg{[}\sqrt{1-\frac{|\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle|}{\Lambda_{m}}}|\mathcal{G}^{\prime}_{j}\rangle|1\rangle|{0}\rangle|Q_{j}\rangle|\mathcal{G}\rangle|P_{j}\rangle|0\rangle|y_{j}\rangle|0\rangle\Bigg{]}\Bigg{\\}}.$
(108)

Implementing the orthonormality condition, the above equation can be
simplified as

$\displaystyle\langle\mathcal{F}|\langle
0|(\mathcal{T}^{{\dagger}}S\mathcal{T})|\mathcal{G}\rangle|0\rangle=$
$\displaystyle\frac{1}{\mathcal{D}\Lambda_{m}}\sum_{i=0}^{\mathcal{D}-1}\sum_{j=0}^{\mathcal{D}-1}\Big{(}\sqrt{\langle\mathcal{F}_{i}^{\prime},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}\Big{)}^{\ast}\sqrt{\langle\mathcal{G}_{j}^{\prime},Q_{j}|\mathcal{H}|\mathcal{G},P_{j}\rangle}\delta_{\mathcal{F},\mathcal{G}^{\prime}_{j}}\delta_{P_{i},Q_{j}}\delta_{\mathcal{F}_{i}^{\prime},\mathcal{G}}\delta_{Q_{i},P_{j}}\delta_{y_{i},0}\delta_{y_{j},0}.$
(109)

We note that the swap $|r_{6}\rangle\leftrightarrow|r_{7}\rangle$ guarantees
that only those terms with $|y_{i}\rangle=|y_{j}\rangle=|0\rangle$ contribute;
this swap operation eliminates the contributions from terms with other
possible combinations of $|y_{i}\rangle$ and $|y_{j}\rangle$. The summation
indices $i$ and $j$ enumerates the monomials in the second-quantized
Hamiltonian [Eq. (14)]. By observation, the above equation enumerates the
contributions from two types of monomials (see notations in Eq. (15)):

  1. 1.

$b_{Q_{i}}^{{\dagger}}b_{P_{i}}$, which corresponds to the kernel
$\langle\mathcal{F}^{\prime}_{i}|b_{Q_{i}}^{{\dagger}}b_{P_{i}}|\mathcal{F}\rangle$
with $P_{i}\mapsto\\{r_{i},s_{i}\\}$ ($r_{i}<s_{i}$) and
$Q_{i}\mapsto\\{p_{i},q_{i}\\}$ ($p_{i}<q_{i}$);

  2. 2.

$b_{Q_{j}}^{{\dagger}}b_{P_{j}}$, which corresponds to the kernel
$\langle\mathcal{G}^{\prime}_{j}|b_{Q_{j}}^{{\dagger}}b_{P_{j}}|\mathcal{G}\rangle$
with $P_{j}\mapsto\\{r_{j},s_{j}\\}$ ($r_{j}<s_{j}$) and
$Q_{j}\mapsto\\{p_{j},q_{j}\\}$ ($p_{j}<q_{j}$).

The delta functions in Eq. (109) (resulted from the orthonormality relations
of the register states) enforce the conditions that
$|\mathcal{F}\rangle=|\mathcal{G}^{\prime}_{j}\rangle$,
$|\mathcal{F}^{\prime}_{i}\rangle=|\mathcal{G}\rangle$, and retain monomials
that are conjugate transpose to each other, i.e.,
$(b_{Q_{i}}^{{\dagger}}b_{P_{i}})^{{\dagger}}=b_{Q_{j}}^{{\dagger}}b_{P_{j}}$
with $P_{i}=Q_{j}$ and $Q_{i}=P_{j}$. Note that we define the mapping
$P_{i}\mapsto\\{r_{i},s_{i}\\}$ ($r_{i}<s_{i}$) and
$Q_{j}\mapsto\\{p_{j},q_{j}\\}$ ($p_{j}<q_{j}$) such that $P_{i}=Q_{j}$ if and
only if $r_{i}=p_{j}$ and $s_{i}=q_{j}$.

For the trivial case with $\langle\mathcal{F}|H|\mathcal{G}\rangle=0$, then
Eq. (109) produces $0$ in the right-hand side, which is as expected. As for
the non-trivial case with $\langle\mathcal{F}|H|\mathcal{G}\rangle\neq 0$,
there exists at least one choice of $i\in\mathcal{I}_{\mathcal{F}}$ such that
the right-hand side of Eq. (109) is non-vanishing. In this case, we can
rewrite Eq. (109) as

$\displaystyle\langle\mathcal{F}|\langle
0|(\mathcal{T}^{{\dagger}}S\mathcal{T})|\mathcal{G}\rangle|0\rangle=\frac{1}{\mathcal{D}\Lambda_{m}}\sum_{i=0}^{\mathcal{D}-1}\Big{(}\sqrt{\langle\mathcal{G},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}\Big{)}^{\ast}\sqrt{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}.$
(110)

It is also noteworthy that we need to choose an appropriate convention for the
square-root operation, especially for the cases where
${\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}$ has negative
or complex values. In this case, we define

$\displaystyle{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}=|{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}|\exp[i\theta_{\mathcal{F},P_{i};\mathcal{G},Q_{i}}],$
(111)

with
$\theta_{\mathcal{F},P_{i};\mathcal{G},Q_{i}}={\arg}[\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle]\in(-\pi,\pi]$.
Then the quantity
$\sqrt{{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}}$ can
be uniquely defined as:

$\displaystyle\sqrt{{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}}=$
$\displaystyle\sqrt{|{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}|}\exp[i\theta_{\mathcal{F},P_{i};\mathcal{G},Q_{i}}/2],$
(112)

With the standard operation of Hermitian matrix elements

$\displaystyle\Big{(}\sqrt{\langle\mathcal{G},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}\Big{)}^{\ast}=\sqrt{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle},$
(113)

we have

$\displaystyle\Big{(}\sqrt{\langle\mathcal{G},Q_{i}|\mathcal{H}|\mathcal{F},P_{i}\rangle}\Big{)}^{\ast}\sqrt{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}=|{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}|\exp[i\theta_{\mathcal{F},P_{i};\mathcal{G},Q_{i}}].$
(114)

With the definition of the square-root operation [Eq. (112)], Eq. (110)
becomes

$\displaystyle\langle\mathcal{F}|\langle
0|(\mathcal{T}^{{\dagger}}S\mathcal{T})|\mathcal{G}\rangle|0\rangle=\frac{1}{\mathcal{D}\Lambda_{m}}\sum_{i=0}^{\mathcal{D}-1}{\langle\mathcal{F},P_{i}|\mathcal{H}|\mathcal{G},Q_{i}\rangle}=\frac{1}{\mathcal{D}\Lambda_{m}}\langle\mathcal{F}|H|\mathcal{G}\rangle,$
(115)

where we have used the relation in Eq. (16). The summation in the above
equation enumeratess all the nonvanishing matrix elements
${\langle\mathcal{F},P_{i}|H^{\prime}|\mathcal{G},Q_{i}\rangle}$ which
contribute to the Hamiltonian matrix element
$\langle\mathcal{F}|H|\mathcal{G}\rangle$.

##### Case 2.

For the cases with $(b,b^{\prime})=(0,1)$ and $(1,0)$, analogous calculation
yields

$\displaystyle\langle\mathcal{F}|\langle
b|\big{(}\mathcal{T}^{{\dagger}}S\mathcal{T}\big{)}|\mathcal{G}\rangle|b^{\prime}\rangle=0.$
(116)

##### Case 3.

For the case with $(b,b^{\prime})=(1,1)$, we can also calculate

$\displaystyle\langle\mathcal{F}|\langle
1|(\mathcal{T}^{{\dagger}}S\mathcal{T})|\mathcal{G}\rangle|1\rangle=\delta_{\mathcal{F},0}\delta_{\mathcal{G},0},$
(117)

with the application of Eq. (103) and Eq. (107).

Combining Case 1, Case 2, and Case 3, we verify Eq. (101).

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