# Artificial magnetism for a harmonically trapped Fermi gas in a synthetic
magnetic field

Shyamal Biswas  sbsp [at] uohyd.ac.in Avijit Ghosh  Soumyadeep Majumder
School of Physics, University of Hyderabad, C.R. Rao Road, Gachibowli,
Hyderabad-500046, India

###### Abstract

We have analytically explored the artificial magnetism for a 3-D spin-
polarized harmonically trapped ideal Fermi gas of electrically neutral
particles exposed to a uniform synthetic magnetic field. Though polarization
of the spin is necessary for trapping electrically neutral atoms in a magneto-
optical trap, Pauli paramagnetism can not be studied for the spin-polarized
Fermi system. However, it is possible to study Landau diamagnetism and de
Haas-van Alphen effect for such a system. We have unified the artificial
Landau diamagnetism and the artificial de Haas-van Alphen effect in a single
framework for all temperatures as well as for all possible magnitudes of the
synthetic magnetic field in the thermodynamic limit. Our prediction is
testable in the present-day experimental setup for ultracold fermionic atoms
in magneto-optical trap.

###### pacs:

05.30.Fk (Fermion systems and electron gas), 75.20.-g (Diamagnetism,
paramagnetism, and superparamagnetism), 67.85.-d (Ultracold gases, trapped
gases)

##  I Introduction

It is well known that the Coriolis force
($\vec{F}_{C}=2\bar{m}\vec{v}\times\vec{\Omega}$) acting on an electrically
neutral particle in a rotating frame is analogues to the Lorentz force
($\vec{F}_{L}=q\vec{v}\times\vec{B}$) acting on a charged particle due to an
external magnetic field in a rest frame of reference Juzeliunas . This analogy
gives rise to the artificial magnetism for a rotating harmonically trapped gas
of electrically neutral particles in the rotating frame of reference Lin2 , in
particular, when the angular speed of rotation approaches the angular trap
frequency ($\Omega\rightarrow\omega_{\perp}$) Juzeliunas . When the angular
speed of rotation approaches the angular trap frequency, the trapping effect
is cancelled by the centrifugal effect and the problem reduces to the
cyclotron motion of a charged particle in a constant magnetic field Juzeliunas
. This analogy finds application of Landau level physics Landau2 ; Schweikhard
; Stock , such as the quantum Hall effect Wilkin , in the rotating trapped
quantum gases of electrically neutral atoms. However, in reality Matthews ;
Madison ; Abo-Shaeer ; Haljan ; Zwierlein ; Lin2 , the angular speed of
rotation is always less than the angular trap frequency
($\Omega<\omega_{\perp}$). This is a requirement for the stability of the
system in the rotating frame Stock . The significant difference between the
magnitude of the centrifugal force ($\bar{m}\Omega^{2}\vec{r}_{\perp}$) and
that of the trapping force ($-\bar{m}\omega_{\perp}^{2}\vec{r}_{\perp}$) can
drastically change the properties of artificial magnetism because the
centrifugal force is analogous to a term
$\bar{m}(\frac{qB}{2\bar{m}})^{2}\vec{r}_{\perp}$ which is nonlinear in
artificial magnetic field Das . Hence it is impossible to study Landau
diamagnetism Landau  and de Haas-van Alphen effect DHVA  in a rotating frame
for the trapped Fermi gas of neutral atoms even for $\Omega<\omega_{\perp}$.

The artificial magnetism can be alternatively studied by exposing the
harmonically trapped gas of neutral atoms to an artificial (or synthetic)
magnetic field. An external artificial magnetic field has already been created
for a neutral gas of trapped ultracold atoms Lin . Hereafter any effect
related to the artificial magnetic field will be coined as artificial effect
over the existing name. Artificial quantum Hall effect has been observed in
such a system of neutral atoms moving perpendicular to the artificial magnetic
field LeBlanc . Creation of artificial magnetic field for neutral atoms
stimulated a lot of experimental and theoretical progress in the field
Spielman ; Dalibard ; Jaksch ; Huo ; Yu ; Babik . However, the elementary
aspects of the artificial magnetism, such as artificial Landau diamagnetism
Landau  and artificial de Haas-van Alphen effect DHVA , have not been observed
in trapped gas of neutral fermionic atoms Chaturvedi ; Biswas3  exposed to an
artificial magnetic field. A few such elementary aspects, on the other hand,
were theoretically explored for both the 2-D Fermi system Farias  and the 3-D
Fermi system Suzuki ; Li  for low temperatures and weak & strong fields.
However, unification of the artificial Landau diamagnetism and the artificial
de Haas-van Alphen effect have not been even theoretically achieved for all
temperatures and all strengths of the field. Hence we take up the discussion
of artificial magnetism for a 3-D harmonically trapped Fermi gas in an
artificial magnetic field.

Pauli paramagnetism often comes into the discussion of magnetism in terms of
the spin of individual particles (fermions) Pauli . The particles we are
considering though have individual spin, the spin does not couple with the
artificial magnetic field. Hence we are not discussing the artificial Pauli
paramagnetism. Moreover, Pauli paramagnetism can not be studied for a spin-
polarized Fermi system although the polarization of the spin is required
Giorgini  for trapping electrically neutral atoms in a magneto-optical trap.
On the other hand, Pauli paramagnetism, Landau diamagnetism and de Haas-van
Alphen effect have been unified for all temperatures and all magnitudes of the
magnetic field for an untrapped ideal gas of electrons exposed to a uniform
magnetic field Biswas . We adopt and extend the method described in Ref.Biswas
for the unification of the artificial Landau diamagnetism and the artificial
de Haas-van Alphen effect for all temperatures and all possible magnitudes of
the artificial magnetic field.

Calculation in this article begins with the single-particle Hamiltonian of a
3-D harmonically trapped ideal Fermi gas of uncharged particles exposed to a
uniform artificial magnetic field. We consider cylindrical symmetry of
harmonic oscillations. We also consider that the artificial magnetic field is
applied to the axis of the cylindrical symmetry. We calculate the grand free
energy of the Fermi gas. We unify the artificial Landau diamagnetism and the
artificial de Haas-van Alphen effect in a single framework for all
temperatures and all magnitudes of the artificial magnetic field in the
thermodynamic limit. Here-from we evaluate magnetic moment of the Fermi gas.
We plot artificial magnetic field dependence of the magnetic moment of the
Fermi system. We show the cross-over of de Haas-van Alphen oscillation and the
saturation in this regard. Finally, we conclude.

##  II Grand free energy for a harmonically trapped ideal Fermi gas exposed to
an artificial magnetic field

Let us consider a 3-D harmonically trapped ideal Fermi gas of particles in
thermodynamic equilibrium with a heat and particle reservoir of temperature
$T$ and chemical potential $\mu$. Let the average total number of particles in
the gas be $N$. Let the mass and charge of each particle of the gas be
$\bar{m}$ and $0$, respectively. We consider the (angular) trap frequency
($\omega_{\perp}$) to be the same for oscillations along the $x$-axis and that
along the $y$-axis. Let us now apply a uniform artificial magnetic field
$\vec{B}=B\hat{k}$ to the Fermi system along the $z$-axis. Single-particle
Hamiltonian of the system under the application of the artificial magnetic
field becomes Lin

$\displaystyle
H=\frac{(\vec{p}-q\vec{A})^{2}}{2\bar{m}}+\frac{1}{2}\bar{m}\omega_{\perp}^{2}r_{\perp}^{2}+\frac{1}{2}\bar{m}\omega_{z}^{2}z^{2}$
(1)

where $\vec{p}$ is the generalized momentum of a particle, $q$ is the
artificial charge of the particle which either can be set to $1$ or can be
absorbed to $\vec{A}$ Stock , $\vec{A}=-\frac{1}{2}\vec{r}\times B\hat{k}$ is
the artificial gauge field, $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is the
position of the particle, $\vec{r}_{\perp}$ is the position of the particle in
the $x-y$ plane, $\omega_{z}$ is the trap frequency of the particle along the
$z$-axis. While the first term of the single-particle Hamiltonian leads to the
orbital motion, the second and third terms lead to the simple harmonic
oscillations.

Energy eigenvalues for the single-particle Hamiltonian of the system with
respect to the lab-fixed (non-rotating) frame are given by the combination of
the Fock-Darwin energy levels111Fock-Darwin energy levels are linked to
$\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}\pm\Omega_{B}$ for the orbital motion
and harmonic motion together in the $x-y$ plane. and 1-D simple harmonic
oscillator energy levels222Simple harmonic oscillator’s energy levels are
linked to $\omega_{z}$ for the harmonic motion along the $z$-axis., as Halonen

$\displaystyle\epsilon_{n,m,j}$ $\displaystyle=$
$\displaystyle(n+1/2)\hbar\big{(}\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}\big{)}+(m+1/2)$
(2)
$\displaystyle\times\hbar\big{(}\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}\big{)}+(j+1/2)\hbar\omega_{z}$

where $\Omega_{B}=\frac{qB}{2\bar{m}}$ is half of the cyclotron frequency,
$n=0,1,2,...$ represents the Landau level quantum number for
$\omega_{\perp}\rightarrow\Omega_{B}$, $n-m=n,n-1,n-2,...$ represents the
magnetic quantum number for $m=0,1,2,...$, and $j=0,1,2,...$ represents 1-D
simple harmonic oscillator energy level quantum number. The grand free energy
of the Fermi system is given by

$\displaystyle\Phi=-k_{B}T\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{j=0}^{\infty}\ln(1+\text{e}^{-[\epsilon_{n,m,j}-\mu]/k_{B}T}).$
(3)

Taylor expansion of the above logarithm about $\text{e}^{\mu/k_{B}T}=0$ and
successive evaluation of the above summations, results in Dey

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle-
k_{B}T\bigg{[}\sum_{l=1}^{\infty}\frac{(-1)^{l-1}z^{l}}{l}\frac{1}{2\sinh(\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{2k_{B}T})}$
(4)
$\displaystyle\times\frac{1}{2\sinh(\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{2k_{B}T})}\frac{1}{2\sinh(\frac{l\hbar\omega_{z}}{2k_{B}T})}\bigg{]}$

where $z=\text{e}^{\mu/k_{B}T}$ is the fugacity of the system. This result is
exact for all temperatures and all magnitudes of the artificial magnetic
field.

The average total number of particles
($N=-\frac{\partial\Phi}{\partial\mu}|_{T,\omega_{\perp},\omega_{z},\Omega_{B}}$)
in the system can be obtained from Eqn.(4), as Dey

$\displaystyle N$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{\infty}(-1)^{l-1}z^{l}\frac{1}{2\sinh(\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{2k_{B}T})}$
(5)
$\displaystyle\times\frac{1}{2\sinh(\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{2k_{B}T})}\frac{1}{2\sinh(\frac{l\hbar\omega_{z}}{2k_{B}T})}.$

This form of the average total number of particle, however, can be simplified
in the thermodynamic limit and in the limiting case of $\Omega_{B}\rightarrow
0$, as

$\displaystyle
N\simeq\bigg{(}\frac{k_{B}T}{\hbar\bar{\omega}}\bigg{)}^{3}(-)\text{Li}_{3}(-z)$
(6)

where $\bar{\omega}=[\omega_{\perp}^{2}\omega_{z}]^{1/3}$ is the geometric
mean of the trap frequencies and
$\text{Li}_{\nu}(z)=z+\frac{z^{2}}{2^{\nu}}+\frac{z^{3}}{3^{\nu}}+...$ is the
poly-logarithmic function of the order $\nu$ and argument $z$. Here, by the
thermodynamic limit, we mean $N\rightarrow\infty$,
$\frac{\hbar\omega_{\perp}}{k_{B}T}\rightarrow 0$,
$\frac{\hbar\omega_{z}}{k_{B}T}\rightarrow 0$ and
$\frac{\hbar\bar{\omega}}{k_{B}T}N^{1/3}=constant$. The fugacity, as well as
the chemical potential, is usually determined from Eqn.(6) in terms of $T$,
$\bar{\omega}$ and $N$ in the thermodynamic limit for $\Omega_{B}\rightarrow
0$ Biswas2 . The Fermi temperature of the system can also be determined from
Eqn.(6) as $T_{F}=\frac{\hbar\bar{\omega}}{k_{B}}[6N]^{1/3}$ Biswas2 .
However, extraction of the properties of the artificial magnetism needs to go
beyond $\Omega_{B}\rightarrow 0$ limit.

##  III Artificial magnetic moment of the harmonically trapped ideal Fermi gas

The artificial magnetic moment of the system of our interest can be defined by
$\vec{M}=-\frac{\partial\Phi}{\partial\vec{B}}|_{\mu,T}=-\hat{k}\frac{q}{2\bar{m}}\frac{\partial\Phi}{\partial\Omega_{B}}|_{\mu,T}$.
From Eqn.(4), by following this definition, we get value of the artificial
magnetic moment of the Fermi system along the $z$-direction, as

$\displaystyle M$ $\displaystyle=$ $\displaystyle\frac{\hbar
q}{2\bar{m}}\bigg{[}\sum_{l=1}^{\infty}\frac{(-1)^{l-1}z^{l}\text{csch}(\frac{l\hbar\omega_{z}}{2k_{B}T})}{4}\times$
(7)
$\displaystyle\frac{\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}\sinh(\frac{l\hbar\Omega_{B}}{k_{B}T})-\Omega_{B}\sinh(\frac{l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T})}{\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}\big{[}\cosh(\frac{l\hbar\Omega_{B}}{k_{B}T})-\cosh(\frac{l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T})\big{]}^{2}}\bigg{]}.~{}~{}~{}~{}$

This form of artificial magnetic moment is exact in all respects. This form,
of course, takes the finite-size effect into account. It is expected that the
artificial magnetic moment would be proportional to the average total number
of particles, at least, in the thermodynamic limit. Exact form of the
artificial magnetic moment per particle can now be obtained from Eqns.(7) and
(5), as

$\displaystyle\frac{M}{N}$ $\displaystyle=$ $\displaystyle\frac{\hbar
q}{2\bar{m}}\bigg{[}\sum_{l=1}^{\infty}\frac{(-1)^{l-1}z^{l}\text{csch}(\frac{l\hbar\omega_{z}}{2k_{B}T})}{4}\times$
(8)
$\displaystyle\frac{\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}\sinh(\frac{l\hbar\Omega_{B}}{k_{B}T})-\Omega_{B}\sinh(\frac{l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T})}{\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}\big{[}\cosh(\frac{l\hbar\Omega_{B}}{k_{B}T})-\cosh(\frac{l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T})\big{]}^{2}}\bigg{]}/$
$\displaystyle\bigg{[}\sum_{l=1}^{\infty}(-1)^{l-1}z^{l}\frac{1}{2\sinh(\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{2k_{B}T})}$
$\displaystyle\times\frac{1}{2\sinh(\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{2k_{B}T})}\frac{1}{2\sinh(\frac{l\hbar\omega_{z}}{2k_{B}T})}\bigg{]}.$

This equation suggests that the artificial magnetic moment of the system has a
saturation value $-M_{s}$ where $M_{s}=N\frac{\hbar q}{2\bar{m}}$. The
saturation value, of course, is reached at $B\rightarrow\infty$. Let us now
define the artificial magneton as $\mu_{B}^{\prime}=\frac{\hbar q}{2\bar{m}}$
which is the counter of the Bohr magneton for the artificial magnetism. The
artificial magnetic moment can be expressed in terms of this quantity
specially for the case of the weak artificial magnetic field
($\mu_{B}^{\prime}B\ll k_{B}T$). Plotting of Eqn.(8) with respect to the
magnitude of the artificial magnetic field needs the chemical potential to be
evaluated from Eqn.(5) as a function of $N$, $T$, and the trap frequencies. We
plot Eqn.(8) (solid line) in figure 1 for the parameters as shown in the plot-
label and that mentioned in the figure caption. These parameters, in
particular, $\omega_{z}$, $\omega_{\perp}$, and $T_{F}$ are taken from the
experimental values for $N=3\times 10^{5}$ Stock . It is clear from figure 1
that the artificial magnetic moment is negative and reaches the saturation as
$B\rightarrow\infty$. This result is analogous to the phenomenon of orbital
magnetism. For the case of the low temperature ($T\ll T_{F}$) and strong
artificial magnetic field ($k_{B}T\lesssim\mu_{B}^{\prime}B\lnsim
k_{B}T_{F}$), the artificial magnetic moment would be found to be oscillating
with respect to the changes in $B$ due to the existence of alternating series
of the powers of $z$ ($\gg 1$) in Eqn.(7) and the competition of the two terms
$\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}\sinh(\frac{l\hbar\Omega_{B}}{k_{B}T})$
and
$\Omega_{B}\sinh(\frac{l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T})$
in numerator of the same equation. This phenomenon would be coined as the
artificial de Haas-van Alphen effect.

Figure 1: The plot of the artificial magnetic moment versus artificial
magnetic field for the harmonically trapped ideal Fermi gas for Fermi
temperature $T_{F}=27.0976~{}\mu$K Stock . While the solid line follows
Eqn.(8), the dotted line follows Eqn.(9).

###  III.1 Artificial Landau diamagnetism

We recast Eqn.(8) to the lowest leading order in $\Omega_{B}$ after the
Laurent series expanding the hyperbolic functions (e.g.
$\sinh(x)=x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+...$ and
$\text{cosh}(x)=\frac{1}{x}-\frac{x}{6}+\frac{7x^{3}}{360}-...$), as

$\displaystyle M\simeq-
M_{s}\frac{-\text{Li}_{2}(-z)}{-\text{Li}_{3}(-z)}\frac{\hbar\Omega_{B}}{3k_{B}T}=-M_{s}\frac{-\text{Li}_{2}(-z)}{-\text{Li}_{3}(-z)}\frac{\mu_{B}^{\prime}B}{3k_{B}T}.$
(9)

This result corresponds to Landau diamagnetism. Temperature dependence of the
fugacity of Eqn.(9) can be determined from Eqn.(6). There-from it can be shown
that $\frac{-\text{Li}_{2}(-z)}{-\text{Li}_{3}(-z)}\rightarrow 1$ for
$T/T_{F}\rightarrow\infty$. Hence Eqn.(9) follows Curie’s law at the high
temperatures. For $T/T_{F}\rightarrow 0$, on the hand, we have
$\frac{-\text{Li}_{2}(-z)}{-\text{Li}_{3}(-z)}\rightarrow\frac{3T}{T_{F}}$
according to the Sommerfeld’s asymptotic expansions of both the numerator
(Fermi-Dirac integral of order $2$) and the denominator (Fermi-Dirac integral
of order $3$) Biswas2 . Thus for $T/T_{F}\rightarrow 0$, the artificial
magnetic moment takes the form $M\simeq-
M_{s}\frac{\mu_{B}^{\prime}B}{k_{B}T_{F}}$ to the lowest leading order in $B$.
We plot Eqn.(9) (dotted line) in figure 1 for the parameters same as that for
the solid line in the same figure. Linearity of the plot with $B$ and negative
values of $M$ confirm the existence of the artificial Landau diamagnetism for
the harmonically trapped ideal Fermi gas of uncharged particles exposed to an
artificial magnetic field. This result would, however, be okay for the case of
the weak field ($\frac{\mu_{B}^{\prime}B}{k_{B}T}\ll 1$). The solid line in
the same figure, on the other hand, captures the finite-size effect and the
higher order effect of $B$, and consequently, exhibits the saturation of the
artificial magnetic moment of the system.

###  III.2 Artificial de Haas-van Alphen effect

Artificial magnetic field dependence of the artificial magnetic moment would
be significantly different at low temperature and the strong artificial
magnetic field regime ($k_{B}T\lesssim\mu_{B}^{\prime}B\lnsim k_{B}T_{F}$).
Artificial de Haas-van Alphen oscillation would be observed in this regime. We
will, however, investigate the same in the thermodynamic limit. Properties of
the artificial de Haas-van Alphen effect is supposed to be extracted from
Eqn.(7) because it represents an exact result even for low temperatures and
strong fields. Although Eqn.(7) is plausible for high temperature and weak
field regime ($\mu_{B}^{\prime}B\ll k_{B}T$), it is not plausible for low
temperatures due to the existence of a pole type singularity in $M/N$ at
either $T\rightarrow 0$ or at $\Omega_{B}\rightarrow\infty$. We can avoid such
a problem by recasting the grand free energy in Eqn.(4) for low temperatures
and strong artificial magnetic fields.

Since we have $\frac{1}{2\sinh(x/2)}=\frac{\text{e}^{-x/2}}{1-\text{e}^{-x}}$
$\forall~{}x\in\mathbb{R}$, we recast Eqn.(4), as

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle-
k_{B}T\bigg{[}\sum_{l=1}^{\infty}\frac{(-1)^{l-1}z^{l}}{l}\frac{\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{2k_{B}T}}}{1-\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{k_{B}T}}}$
(10)
$\displaystyle\times\frac{\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{2k_{B}T}}}{1-\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{k_{B}T}}}\frac{\text{e}^{-\frac{l\hbar\omega_{z}}{2k_{B}T}}}{1-\text{e}^{-\frac{l\hbar\omega_{z}}{k_{B}T}}}\bigg{]}.$

The above equation can be further recast by truncating the Taylor expansions
of the exponentials in the last two fractions and taking the binomial
expansion of
$\frac{1}{1-\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{k_{B}T}}}$,
as

$\displaystyle\Phi$ $\displaystyle\simeq$ $\displaystyle-
k_{B}T\bigg{[}\sum_{l=1}^{\infty}\frac{(-1)^{l-1}z^{l}}{l}\frac{1}{\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{k_{B}T}}\frac{1}{\frac{l\hbar\omega_{z}}{k_{B}T}}$
(11)
$\displaystyle\times\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{2k_{B}T}}\sum_{k=1}^{\infty}\text{e}^{-\frac{l(k-1)\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{2k_{B}T}}\bigg{]}$
$\displaystyle\simeq$
$\displaystyle-\frac{(k_{B}T)^{3}[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{\hbar^{2}\omega_{\perp}^{2}\omega_{z}}$
$\displaystyle\times\sum_{k=1}^{\infty}(-)\text{Li}_{3}\big{(}-\text{e}^{\frac{\mu-(k-1)\hbar\Omega_{B}-k\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T}}\big{)}$

in the thermodynamic limit. Here-from we get the average total number of
particles
($N=-\frac{\partial\Phi}{\partial\mu}|_{T,\omega_{\perp},\omega_{z},\Omega_{B}}$),
as

$\displaystyle N$ $\displaystyle\simeq$
$\displaystyle\frac{(k_{B}T)^{2}[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{\hbar^{2}\omega_{\perp}^{2}\omega_{z}}$
(12)
$\displaystyle\times\sum_{k=1}^{\infty}(-)\text{Li}_{2}\big{(}-\text{e}^{\frac{\mu-k\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-(k-1)\hbar\Omega_{B}}{k_{B}T}}\big{)}.$

Exact expression of magnitude of the artificial magnetic moment
($\vec{M}=-\frac{\partial\Phi}{\partial\vec{B}}|_{\mu,T}=-\hat{k}\frac{q}{2\bar{m}}\frac{\partial\Phi}{\partial\Omega_{B}}|_{\mu,T}$)
can be obtained from Eqn.(10), as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{\infty}\frac{\mu_{B}^{\prime}(-1)^{l}\text{e}^{\frac{\mu
l-l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T}}}{\big{[}1-\text{e}^{\frac{-l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{k_{B}T}}\big{]}^{2}\big{[}1-\text{e}^{\frac{-l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{k_{B}T}}\big{]}^{2}}$
(13)
$\displaystyle\times\frac{\text{e}^{-\frac{l\hbar\omega_{z}}{2k_{B}T}}}{1-\text{e}^{-\frac{l\hbar\omega_{z}}{k_{B}T}}}\bigg{(}\frac{\Omega_{B}}{\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}\bigg{[}1-\text{e}^{-\frac{2l\hbar\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}{k_{B}T}}\bigg{]}$
$\displaystyle-\bigg{[}\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{k_{B}T}}-\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{k_{B}T}}\bigg{]}\bigg{)}.$

It is clear from the above equation that the artificial magnetic moment of the
system is always negative. It is also clear from the terms in the parenthesis
of the above equation that the artificial magnetic moment would be linear in
$\Omega_{B}$ for small $\Omega_{B}$s as expected from the artificial Landau
diamagnetism.

Figure 2: The plot of the artificial magnetic moment versus artificial
magnetic field for the harmonically trapped ideal Fermi gas for Fermi
temperature $T_{F}=27.0976~{}\mu$K Stock . The solid line follows Eqn.(14).
The dotted line follows Eqn.(9). The solid line in the inset also follows
Eqn.(14).

Now approximating
$\frac{\text{e}^{-\frac{l\hbar\omega_{z}}{2k_{B}T}}}{1-\text{e}^{-\frac{l\hbar\omega_{z}}{k_{B}T}}}$
as $\frac{1}{\frac{l\hbar\omega_{z}}{k_{B}T}}$,
$\frac{1}{\big{[}1-\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{k_{B}T}}\big{]}^{2}}$
as
$\frac{1}{\big{[}\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}{k_{B}T}\big{]}^{2}}$,
binomial expanding
$\frac{1}{\big{[}1-\text{e}^{-\frac{l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{k_{B}T}}\big{]}^{2}}$
as
$\sum_{k=1}^{\infty}k\text{e}^{-\frac{(k-1)l\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+\Omega_{B}]}{k_{B}T}}$,
and evaluating the summation over $l$, we recast Eqn.(13) with the further use
of Eqn.(12), as

$\displaystyle\frac{M}{N}$ $\displaystyle\simeq$
$\displaystyle-\mu_{B}^{\prime}\frac{k_{B}T}{\hbar[\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}-\Omega_{B}]}\bigg{\\{}\sum_{k=1}^{\infty}k\bigg{(}\frac{\Omega_{B}}{\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}}$
(14)
$\displaystyle\times\bigg{[}(-)\text{Li}_{3}\big{(}-\text{e}^{\frac{\mu-\hbar[k\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+(k-1)\Omega_{B}]}{k_{B}T}}\big{)}$
$\displaystyle-(-)\text{Li}_{3}\big{(}-\text{e}^{\frac{\mu-\hbar[(k+2)\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+(k-1)\Omega_{B}]}{k_{B}T}}\big{)}\bigg{]}$
$\displaystyle+\bigg{[}(-)\text{Li}_{3}\big{(}-\text{e}^{\frac{\mu-\hbar[(k+1)\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+k\Omega_{B}]}{k_{B}T}}\big{)}$
$\displaystyle-(-)\text{Li}_{3}\big{(}-\text{e}^{\frac{\mu-\hbar[(k+1)\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+(k-2)\Omega_{B}]}{k_{B}T}}\big{)}\bigg{]}\bigg{)}\bigg{\\}}/$
$\displaystyle\bigg{[}\sum_{n=1}^{\infty}(-)\text{Li}_{2}\big{(}-\text{e}^{\frac{\mu-\hbar[n\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+(n-1)\hbar\Omega_{B}]}{k_{B}T}}\big{)}\bigg{]}$

in the thermodynamic limit. Eqn.(14) represents the artificial magnetic moment
per particle of the Fermi system. This equation is useful for describing the
artificial de Haas-van Alphen effect for strong artificial magnetic fields.
This equation is also useful for describing the artificial magnetism for weak
artificial magnetic fields in the thermodynamic limit. Eqn.(12) becomes same
as Eqn.(6) once the summation over $k$ is replaced by integration over $k$ in
the thermodynamic limit for $\Omega_{B}\rightarrow 0$. Now we evaluate the
chemical potential from Eqn.(6) according to the method described in
Ref.Biswas2 . We plot the artificial magnetic moment of Eqn.(14) in figure 2
(solid line) with respect to the artificial magnetic field for the parameters
same as that used in figure 1 except the temperature $T=0.01~{}T_{F}$ and
chemical potential $\mu=0.999671~{}k_{B}T_{F}$. We also plot the artificial
magnetic moment of Eqn.(9) (dotted) line in the same figure for the same
parameters as mentioned in the plot-label and the figure caption. We also plot
the artificial magnetic moment with respect to inverse of the artificial
magnetic field in the inset of figure 2 for same set of parameters.
Oscillations in the artificial magnetic moment about its weak field results,
which are indicated by the dotted line, represent the artificial de Haas-van
Alphen effect. While de Haas-van Alphen oscillations are observed to be
periodic with the inverse of the external magnetic field in a gas of free
electrons in a metal DHVA , the artificial de Haas-van Alphen oscillations are
not appearing periodic with the inverse of the artificial magnetic field,
rather quasi-periodic due to the inhomogeneity of the Fermi system, as clear
from the inset of figure 2. Both the artificial Landau diamagnetism and the
artificial de Haas-van Alphen effect are described by the solid line in figure
2. Hence Eqn.(14) unifies the artificial Landau diamagnetism and the
artificial de Haas-van Alphen effect in the thermodynamic limit.

Figure 3: The plot of the $k$th term ($M_{k}$) of the artificial magnetic
moment (AMM) versus the artificial magnetic field for $k=1$ (solid line), $2$
(dotted line), and $3$ (dashed line). The $k$th term $M_{k}$ comes from
Eqn.(14) for the parameters as mentioned in figure 3. The inset represents the
sum of all the $M_{k}$ terms ($M=\sum_{k=1}^{\infty}M_{k}$) and the summation
follows Eqn.(14) for the same parameters.

What makes the artificial de Haas-van Alphen oscillation for the trapped
system of our interest is not very clear from Eqn.(14). We can analyse the
oscillation by focusing on the individual terms
($\\{M_{k}f_{2}(\mu,T,B)/M_{s}\\}$, $k=1,2,3,...$) of the artificial magnetic
moment ($M$) in units of $M_{s}/f_{2}(\mu,T,B)$ where
$f_{2}(\mu,T,B)=\sum_{n=1}^{\infty}(-)\text{Li}_{2}\big{(}-\text{e}^{\frac{\mu-\hbar[n\sqrt{\omega_{\perp}^{2}+\Omega_{B}^{2}}+(n-1)\hbar\Omega_{B}]}{k_{B}T}}\big{)}$
is the non-oscillating denominator of $M$ as expressed in Eqn.(14). We plot
$M_{k}$ in figure 3 for $k=1$ (solid line), $2$ (dotted line) and, $3$ (dashed
line) with respect to the artificial magnetic field. The polylogarithmic
functions (Fermi integrals $\\{-\text{Li}_{3}\\}$) in $M_{k}$ though
individually diverge very slowly (cubic power of logarithm Biswas2 ) for
$T\rightarrow 0$ and $B\rightarrow 0$, the operations on the polylogarithmic
functions result in $M_{k}$ to start decreasing at $B=0$ and increasing at
around $\frac{\mu_{B}^{\prime}B}{k_{B}T_{F}}/k\lnsim 1$ (strong field regime)
before slowly vanishing beyond $\frac{\mu_{B}^{\prime}B}{k_{B}T_{F}}/k\simeq
1$ at a low temperature ($T/T_{F}\ll 1$, $\mu\simeq k_{B}T_{F}$) for
$\hbar\omega_{\perp}\ll k_{B}T\lesssim\mu_{B}^{\prime}B$. This leads to an
oscillation of the artificial magnetic moment ($M=\sum_{k=1}^{\infty}M_{k}$)
with a quasi-period inversely proportional to the artificial magnetic field in
the interval $(0,k_{B}T_{F}/\mu_{B}^{\prime})$, as clear from the inset of
figure 3.

##  IV Conclusion

To conclude, we have analytically explored the artificial magnetism for a 3-D
spin-polarized harmonically trapped ideal Fermi gas of electrically neutral
particles exposed to a uniform synthetic magnetic field. We have unified
artificial Landau diamagnetism and artificial de Haas-van Alphen effect for
such a system. Our unification is applicable for all temperatures as well as
for all strengths of the synthetic magnetic field in the thermodynamic limit.
Our prediction is testable within the existing experimental setup for
ultracold fermionic atoms in magneto-optical trap Stock ; Lin ; Lin2 .

We could study artificial magnetism rotating harmonically trapped Bose gas of
uncharged particles. All the results would be unaltered for this case except
the replacement of $(-1)^{l}z^{l}$ by $z^{l}$ and the replacement of
$-\text{Li}_{j}(-x)$ by $\text{Li}_{j}(x)$ $\forall~{}x\in~{}\mathbb{R}$.
However, necessary condition for observing the artificial de Haas-van Alphen
effect would be $\mu>\epsilon_{0,0,0}$. Such a condition is not reached by a
Bose system. Hence one can not observe artificial de Haas-van Alphen effect in
a Bose system.

Properties of rotating trapped Bose and Fermi systems are often analysed with
respect to the body-fixed rotating frame of reference Stock ; Das . We have,
however, analysed the artificial magnetic properties of the Fermi system from
the lab-fixed (non-rotating) frame. The artificial magnetism would have been
studied with respect to the body-fixed rotating frame without applying the
artificial magnetic field Das . However, the centrifugal force additionally
appears in the rotating frame and consequently, it changes the entire physics
because of appearance of a pole type singularity at
$\Omega_{B}\rightarrow\omega_{\perp}$ in the grand free energy of the system.
It also brings the question of stability of the system in the rotating frame
for $\Omega_{B}\geq\omega_{\perp}$. We would not get the artificial Landau
diamagnetism, as well as Curie’s law, in the rotating frame.

We have considered the grand canonical ensemble for obtaining our results.
Consequently, we have fixed the chemical potential ($\mu$) for our analyses.
However, the average number of particles ($N$) becomes fixed once we fix the
chemical potential and the temperature ($T$) of the system.

We did not directly consider any accidental degeneracy because the single-
particle energy eigenvalues as mentioned in Eqn.(2) are non-degenerate due to
the trapping effect for a finite strength of the artificial magnetic field.
The accidental degeneracy significantly modifies the density of states and
consequently plays a key role in the de Haas-van Alphen effect on an untrapped
ideal gas of electrons Landau ; Biswas . However, a similar accidental
degeneracy is also there with quantum number $m$ (Eqn.(2)) of the trapped
system of our interest in the limiting case of
$\frac{\omega_{\perp}}{\Omega_{B}}\rightarrow 0$ for a strong artificial
magnetic field.

We have not considered the inter-particle interaction in our analysis. While
the weak attractive inter-particle interactions lead to the formation of the
Cooper pairs in the harmonically trapped Fermi gas Giorgini ; Parish , the
strong attractive interactions result in the BEC-BCS crossover in the same
system Parish ; Randeria . The study of the artificial magnetism for the
harmonically trapped interacting Fermi gas of uncharged particles exposed to
an artificial magnetic field is kept as an open problem.

Pauli paramagnetism, Landau diamagnetism, and de Haas-van Alphen effect have
already been unified for an untrapped ideal gas of electrons Biswas . Our
consideration of only one component of spin of the uncharged fermionic atoms
stopped us from studying Pauli paramagnetism. One component of the spin is
required for trapping the atoms in a magneto-optical trap Pitaevskii ;
Giorgini . However, how to unify Pauli paramagnetism, Landau diamagnetism, and
de Haas-van Alphen effect for the artificial magnetism of a multi-spin
component trapped Fermi gas, is kept as an open problem.

###### Acknowledgements.

S. Biswas acknowledges partial financial support of the SERB, DST, Govt. of
India under the EMEQ Scheme [No. EEQ/2019/000017]. Useful discussions with
Prof. J. K. Bhattacharjee (IACS, Kolkata) and Mr. Samir Das (UoH, Hyderabad)
are gratefully acknowledged. We thank the anonymous reviewers for their
thorough review and highly appreciate their comments and suggestions which
significantly contributed to improving the quality of the presentation.

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# Diffusion-based Quantum Error Mitigation using Stochastic Differential
Equation

Joo Yong Shim and Joongheon Kim  Department of Electrical and Computer
Engineering, Korea University, Seoul, Republic of Korea
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Unlike closed systems, where the total energy and information are conserved
within the system, open systems interact with the external environment which
often leads to complex behaviors not seen in closed systems. The random
fluctuations that arise due to the interaction with the external environment
cause noise affecting the states of the quantum system, resulting in system
errors. To effectively concern quantum error in open quantum systems, this
paper introduces a novel approach to mitigate errors using diffusion models.
This approach can be realized by noise occurrence formulation during the state
evolution as forward-backward stochastic differential equations (FBSDE) and
adapting the score-based generative model (SGM) to denoise errors in quantum
states.

##  I Introduction

Quantum computing is anticipated to complement classical algorithms in various
industry sectors that require large-scale computations or demand high-speed
operations [1, 2, 3, 4, 5, 6]. Discussions of quantum computing in an ideal
scenario often overlook the inherent challenges associated with the practical
implementation of quantum technologies. Quantum computers are inherently prone
to errors caused by interactions between their qubits and the environment [7].
The interaction of quantum systems with their environment causes significant
noises, dramatically affecting their operational efficiency and accuracy. This
noise arises from various sources, including quantum decoherence and quantum
interference, posing a substantial barrier to the reliability of quantum
computations.

For a long period, several types of research on quantum error mitigation (QEM)
[8], ranging from fundamental limits and theoretical frameworks to practical
simulation tools for QEM, have been proposed such as zero-noise extrapolation
which uses data at different noise levels to extrapolate the zero-noise limit
[9]. Probabilistic error cancellation, which models the actual quantum
operations as combinations of ideal and erroneous operations, compensating for
errors by sampling different operations based on their likelihood of
occurrence [10]. The measurement error mitigation addresses errors in the
measurement phase, which can distort the final results of quantum computations
[11]. This paper excludes state preparation and measurement (SPAM) errors,
which occur at the beginning and end of actual operations, however focuses on
mitigating gate noise occurring in open quantum systems.

For the practical applications of quantum computing in real-world scenarios,
several factors must be considered, in particular, i) computational overhead
[12], ii) accuracy, and iii) generality (generally applicable) [13]. Each of
these factors plays a crucial role in transitioning quantum computing from
theory to practical, scalable applications. Quantum error correction and
mitigation techniques typically require additional qubits and computational
steps, increasing complexity and overhead. For example, implementing quantum
error correction methods often requires a significant number of auxiliary
qubits for encoding and measurement. In particular, this overhead can be a
challenge, especially in the current noisy intermediate scale quantum (NISQ)
era [14], where the limited number of qubits available in modern quantum
computers [IBM2021Quantum]. The other important consideration is accuracy in
quantum computing, which is directly impacted by the effectiveness of the
error mitigation strategy. QEM aims to preserve the fidelity of quantum states
by reducing or correcting errors as they occur, thereby increasing the
accuracy of the computation. However, the accuracy of these techniques depends
on their ability to correctly identify and eliminate errors without additional
error occurrence during the process. This is challenging in practice due to
imperfect gate operations, finite coherence times, and the probabilistic
nature of quantum measurements. Finally, QEM techniques should be broadly
applicable across different quantum computing platforms and algorithms in an
ideal manner. However, many approaches can be applied only to the specific
types of errors or system architectures. The generality of a technique also
depends on how easily it can be integrated with various quantum computing
tasks and algorithms. The corresponding techniques that require the extensive
modification of the quantum circuit or are only compatible with specific types
of quantum gates may be less useful. Developing universally applicable QEM
strategies that can adapt to different noise environments and error types is a
key area of research, aiming to make quantum computing more robust and
flexible.

Accordingly, this paper presents a novel QEM approach in open systems
utilizing diffusion-based models. Diffusion models, which are predominantly
used in generative modeling for high-fidelity image generation, audio
synthesis, text generation, and molecular design, leverage the concept of
simulating a Markov chain. This simulation involves modeling the gradual
transition of data from a complex distribution to a tractable one (typically
Gaussian) and subsequently learning to reverse this process. Seminal works
like the introduction to denoising diffusion probabilistic models (DDPMs) by
[15] and improvements by [16] have reduced sampling times and enhanced sample
quality. Additionally, the continuous-time framework proposed in score-based
generative modeling through stochastic differential equations [17] provides a
robust theoretical foundation, further enhancing efficiency in diffusion
models.

In the context of QEM, this work adapts these principles to address noise in
quantum systems by formulating the noisy gate dynamics into linear stochastic
differential equations (SDE), thus framing the problem as a forward-backward
SDE (FBSDE) challenge. This method deviates from traditional approaches that
rely on complex equations or additional quantum gates for noise mitigation.
Instead, it simplifies the master equation into an SDE format, which enables
simpler computational models using state vectors, thereby reducing system
complexity [18]. A score-based generative diffusion model is employed to
analyze and mitigate noise patterns predicted by the linear SDEs.

##  II Modeling Open Quantum Systems

The fundamental understanding of quantum systems and their dynamics often
begins with the Shrödinger equation, which describes the time evolution of a
pure state of closed quantum systems. It can be fundamentally expressed as
follows,

$\frac{d}{dt}\ket{\psi_{t}}=-\frac{i}{\hbar}\ket{\psi_{t}}$ (1)

where $\ket{\psi_{t}}$ represents the state of the system at time $t$, $\hbar$
is the Planck’s constant, and $i$ is the imaginary unit, respectively.

To extend the Shrödinger equation to explain quantum systems that include
mixed states (but still assume closed systems without environmental
interaction), the concept of density matrices is essentially required. A
density matrix, denoted by
$\rho=\sum_{i}p_{i}\outerproduct{\psi_{i}}{\psi_{i}}$, represents the mixed
state of a quantum system in a set of states ${\ket{\psi_{i}}}$ with
probabilities $p_{i}$. The von Neumann equation gives the evolution of a mixed
state, as follows,

$\frac{d}{dt}\rho_{t}=-\frac{i}{\hbar}[H_{t},\rho_{t}]$ (2)

where $H$ is the Hamiltonian operator. In addition, $[A,B]$ denotes the
commutator, defined as $AB-BA$. In open quantum systems, interaction with the
environment leads to non-unitary evolution. This cannot be described solely by
the Schrödinger equation or von Neumann equation. Thus, the additional term
$\mathcal{L}(\rho_{t})$ is used, as follows,

$\frac{d}{dt}\rho_{t}=-\frac{i}{\hbar}[H_{t},\rho_{t}]+\mathcal{L}(\rho_{t})$
(3)

and this can be also described as follows for more explicitly,

$\frac{d}{dt}\rho_{t}=-\frac{i}{\hbar}[H_{t},\rho_{t}]+\gamma\sum_{n}\left[L_{n}\rho
L_{n}^{\dagger}-\frac{1}{2}\\{L_{n}^{\dagger}L_{n},\rho_{t}\\}\right]$ (4)

where the first term $-\frac{i}{\hbar}[H_{t},\rho_{t}]$ is the Hamiltonian
part, which describes the unitary evolution. In addition, the second term
$\mathcal{L}(\rho_{t})$ expresses the dissipating effects on the open system.
Lastly, the $L_{n}$ usually represents jump operators and the
$\sum_{n}L_{n}\rho
L_{n}^{\dagger}-\frac{1}{2}\\{L_{n}^{\dagger}L_{n},\rho_{t}\\}$ represents the
effect caused by the interaction to the external environment, adding
probability to certain states. The Lindblad equation [7] thus derived is the
most general form for the generator of quantum dynamics for Markovian systems.
This formulation allows to explain a broad range of physical phenomena,
including relaxation and decoherence, to be modeled accurately.

The dissipating term $\mathcal{L}(\rho_{t})$ can be defined by the summation
of the effects of the device depolarization and the relaxation process. The
depolarization, which arises due to the imperfections of the device,
represents a process where a qubit gradually loses its purity and moves
towards a maximally mixed state, irrespective of its initial state [19]. The
Lindblad operators for depolarization in a single-qubit system are defined by
the Pauli matrices $\sigma_{x},\sigma_{y},\sigma_{z}$ as follows,

$\mathcal{L}_{\text{depolarization}}(\rho_{t})=\\\
\sum_{k\in\\{x,y,z\\}}\gamma_{d,k}\left(\sigma_{k}\rho\sigma_{k}-\frac{1}{2}\\{\sigma_{k}^{2},\rho\\}\right)$
(5)

where $\gamma_{d,k}$ is the rate of depolarization over axes $x,y,z$. In
addition, $\rho$ is the density matrix of the system and $\sigma_{k}$ are the
Pauli matrices. Because $\sigma_{k}^{2}=I$ holds for all Pauli matrices, it
can be simplified as follows,

$\mathcal{L}_{\text{depolarization}}(\rho_{t})=\sum_{k\in\\{x,y,z\\}}\gamma_{d,k}\left(\sigma_{k}\rho_{t}\sigma_{k}-\frac{1}{2}\rho_{t}\right).$
(6)

Relaxation error results from interactions between quantum bits (qubits) and
their environment. This interaction leads to an energy exchange that typically
pushes the qubit towards a lower energy state, known as the ground state
$|0\rangle$. The primary effects of relaxation noise include amplitude and
phase damping [19]. Amplitude damping [20] is one significant effect of
relaxation where a qubit loses energy to its surroundings and gradually
transitions from a higher energy state (like the excited state) $|1\rangle$ to
the ground state $|0\rangle$. The amplitude damping is characterized by the
Lindblad operator $\sigma^{-}$ (the lowering operator) modeled as:

$\mathcal{L}_{\text{amp}}(\rho_{t})=\gamma_{a}\left(\sigma^{-}\rho_{t}\sigma^{+}-\frac{1}{2}\left\\{\sigma^{+}\sigma^{-},\rho_{t}\right\\}\right),$
(7)

where $\gamma_{a}$ is the rate of amplitude damping and $\rho$ is the density
matrix of the system. Moreover, $\sigma^{-}=(X-iY)/2$ and
$\sigma^{+}=(X+iY)/2$ are the lowering and raising operators, respectively.
Lastly, $\\{A,B\\}$ denotes the anti-commutator, defined as $AB+BA$.

Besides energy loss, relaxation can also cause dephasing, where the coherence
between quantum states (the phase relationships) is lost. This affects the
off-diagonal elements of the qubit’s density matrix, leading to a loss of
quantum information that is not necessarily related to energy exchange.
Dephasing is characterized by the operator $\sigma_{z}$ (the Pauli Z
operator), affecting the coherence between the computational basis states
without energy exchange, modeled as follows,

$\mathcal{L}_{\text{phase}}(\rho_{t})=\gamma_{p}\left(\sigma_{z}\rho_{t}\sigma_{z}-\rho_{t}\right),$
(8)

where $\gamma_{p}$ is the dephasing rate.

Combining these two effects, the total relaxation term that describes both
amplitude damping and dephasing in a single qubit system can be expressed as
follows,

$\mathcal{L}_{\text{relaxation}}(\rho_{t})=\mathcal{L}_{\text{amp}}(\rho_{t})+\mathcal{L}_{\text{phase}}(\rho_{t}),$
(9)

and this can be as follows for more explicitly,

$\mathcal{L}_{\text{relaxation}}(\rho_{t})=\gamma_{a}\left(\sigma^{-}\rho_{t}\sigma^{+}-\frac{1}{2}\left\\{\sigma^{+}\sigma^{-},\rho_{t}\right\\}\right)+\\\
\gamma_{p}\left(\sigma_{z}\rho_{t}\sigma_{z}-\rho_{t}\right).$ (10)

In practical terms, relaxation noise is a critical factor to consider in
quantum computing and other quantum technologies because it can degrade the
performance of quantum algorithms, leading to errors in quantum computations.

The combined Lindblad term for both depolarization and relaxation can be as
follows,

$\mathcal{L}(\rho_{t})=\mathcal{L}_{\text{depolarization}}(\rho)+\mathcal{L}_{\text{relaxation}}(\rho),$
(11)

and finally, the dissipator term can be obtained by reformulating [21] using
Lindblad operators $L_{n}$, i.e.,

$\mathcal{L}(\rho_{t})=\gamma\sum_{n}\left[L_{n}\rho
L_{n}^{\dagger}-\frac{1}{2}\\{L_{n}^{\dagger}L_{n},\rho_{t}\\}\right].$ (12)

##  III Forward-Backward Stochastic Differential Equation (FBSDE) Approach

This paper proposes an FBSDE-based noise mitigation method, where the FBSDE is
a class of stochastic processes consisting of two linked differential
equations, i.e., one evolving forward in time and the other backward.
Combining with score-based generative modeling (SGM) [17], this work
theoretically shows the noise mitigation problem can be framed within the
mathematical structure of FBSDE and solved by SGM. During this computation
setup, the forward equation might model the progressive addition of noise to
the data (thus simulating the forward evolution of the noise process), and the
backward equation would represent the learned denoising path (the reverse-time
generative model). This could provide a robust mathematical framework for
understanding and analyzing the dynamics of diffusion models, potentially
enhancing effectiveness by exploiting the properties of FBSDE.

###  III-A Problem Formulation in SDE

Let’s consider (4), i.e., the Lindblad Master Equation, which was previously
defined as follows,

$\frac{d}{dt}\rho_{t}=-\frac{i}{\hbar}[H_{t},\rho_{t}]+\\\
\gamma\sum_{n}\left[L_{n}\rho
L_{n}^{\dagger}-\frac{1}{2}\\{L_{n}^{\dagger}L_{n},\rho_{t}\\}\right].$ (13)

Unraveling the defined density matrix into the equation for the state vector,
it can be represented in terms of linear SDE [18], as follows,

${d|\psi_{t}\rangle}=\left[-\frac{i}{\hbar}H\,{dt}+\right.\\\
\left.\sum_{n}\left[i\sqrt{\gamma}L\,{dW_{t}}-\frac{\gamma}{2}L^{\dagger}L\,{dt}\right]\right]|\psi_{t}\rangle.$
(14)

###  III-B Score-based Generative Model (SGM)

The proposed SGM is an advanced method in the field of machine learning for
generating new data samples that are statistically similar to a given dataset.
SGM is based on the concept of gradually adding noise to data until a simple,
known distribution is reached (often a Gaussian distribution). The ‘score’ in
SGM modeling refers to the gradient of the log probability density of the data
concerning the data itself. The key insight of SGM is that this score can
guide the process of generating new data samples by indicating how to denoise
or reverse the noise added to the data. Its process can be described in three
steps, as follows.

  1. 1.

Adding Noise: A series of transformations is performed, gradually adding noise
to the data, moving from a complex data distribution to a simpler, noise-
dominated distribution. This process is called the diffusion process,
typically defined through an SDE.

  2. 2.

Training the Model: The neural network is trained to estimate the score, which
quantifies the direction and rate at which the probability density of the data
increases. This training involves learning to predict the gradient of the log
density at different noise levels.

  3. 3.

Generating Samples: To generate new samples, the model starts with samples
from the simple noise distribution (like Gaussian) and then applies a reverse
process, guided by the learned scores, to progressively remove the noise. This
reverse process can also be modeled by a reverse SDE.

It can be noted that SGM can be described in the FBSDE problem. Then, the
diffusion process can be formulated in the form of forward SDE, as follows,

${dX_{t}}=f(X_{t},t)\,{dt}+g(t)\,dW_{t},$ (15)

where $W$ represents a standard Wiener process based on Brownian motion,
$f(X_{t},t)$ is the drift coefficient, and $g(t)$ is the diffusion
coefficient. If $f(X_{t},t)$ and $g(t)$ satisfy Lipschitz conditions (i.e. if
their derivatives are bounded),

$\displaystyle|\nabla_{x}f(x,t)|$ $\displaystyle\leq\epsilon_{1},$ (16)
$\displaystyle|\nabla_{t}g(t)|$ $\displaystyle\leq\epsilon_{2},$ (17)

then, a unique strong solution can be derived [22].

If the prior is known (i.e., predefined distribution), one can sample $X_{T}$
from the prior sample. If the prior is predefined as Gaussian, $X_{T}$ will be
the sample from the Gaussian distribution. The reverse process of diffusion
can also be given by the backward SDE [23], as follows,

${dX_{t}}=\left[f(X_{t},t)-g^{2}(t)\nabla_{x}\log{p_{t}(X_{t})}\right]\,dt+\\\
g(t)\,{d\bar{W_{t}}},$ (18)

where $\bar{W_{t}}$ is the time reversal of the Wiener process in (15). While
the forward process gradually adds noise in very small increments as $t$
increases, the reverse formulation is traced as $t$ decreases in very small
increments. If the score $\nabla_{x}\log{p_{t}(X_{t})}$ from the marginal
distribution $p_{t}(X_{t})$ can be determined, then $X_{0}$ can be obtained
from a random noise sample $X_{T}$.

###  III-C SGM for QEM

In QEM, adapting SGM would involve conceptualizing quantum states and
operations in a probabilistic framework similar to how data distributions are
treated in classical SGM. The primary steps include the following items.

  1. 1.

Modeling the Noise in Quantum States: Similar to the Adding Noise process in
SGM, one could model the evolution of quantum decoherence and operational
errors as a stochastic process that transforms an ideal quantum state into a
noisy state. This could also be described using forward SDE.

  2. 2.

Training the Model for Score Estimation: The score in SGM corresponds to the
gradient of the log probability density. For quantum states, this could
translate to gradients of quantum state probabilities or amplitudes, providing
a direction in which to adjust quantum states to mitigate errors. Learning
these gradients would involve training neural networks capable of estimating
these derivatives from noisy quantum states.

  3. 3.

Denoising Quantum States: Using the learned scores, one could then attempt to
reverse the noise process. This involves applying a series of quantum
operations that gradually ‘denoise’ the quantum error, moving it closer to its
intended noise-free state.

The QEM problem can be defined as an FBSDE problem. First, it can reformulate
the (14) into the form of (15), as follows,

${d|\psi_{t}\rangle}=f(\psi_{t},t)\,{dt}+g(\psi_{t},t)\,{dW_{t}}.$ (19)

where the drift and diffusion terms can be expressed as,

$\displaystyle f(\psi_{t},t)$
$\displaystyle=-\frac{i}{\hbar}H|\psi_{t}\rangle-\sum_{n}\frac{\gamma}{2}L_{n}^{\dagger}L_{n}|\psi_{t}\rangle,$
(20) $\displaystyle g(\psi_{t},t)$
$\displaystyle=\sum_{n}i\sqrt{\gamma}L_{n}|\psi_{t},\rangle,$ (21)

then, by applying (18), the denoising process of the state of noisy gates can
be derived as follows,

${d|\psi_{t}\rangle}=\left[f(\psi_{t},t)\,-g^{2}(\psi_{t},t)\nabla_{\psi}\log{p_{t}(|\psi_{T}\rangle)}\right]\,dt+\\\
g(\psi_{t},t)\,{d\bar{W_{t}}}.$ (22)

###  III-D Solving the FBSDE using SGM

This section introduces the process to solve the FBSDE problem using SGM. The
first step, which is Adding Noise in classical SGM and Modeling the Noise in
Quantum States in the proposed method is conducted in previous subsections,
where (15) and (19) are obtained as the solutions. After this step, in order
to solve the FBSDE, prior distribution $p_{T}(\cdot)$ and the score function
$\nabla\log p_{t}(\cdot)$ should be given. Then, the $p_{T}(x)$ is usually
assumed to be Gaussian, which is fully tractable using the diffusion process.
In the QEM problem, $p_{T}(|\psi\rangle)$ is the noisy state. To estimate the
score $\nabla\log p_{t}(\cdot)$, time-dependent neural network
$s_{\theta}(\cdot,t)$ is implemented. This is trained to approximate the score
function $\nabla\log p_{t}(\cdot)$ such that,

$s_{\theta}(\cdot,t)\approx\nabla\log p_{t}(\cdot).$ (23)

The objective function for $s_{\theta}(\cdot,t)$ is the continuous weighted
combination of Fisher divergences, given by,

$\mathbb{E}_{U(0,T)}\mathbb{E}_{p_{\mathbf{x}}(t)}[\lambda(t)\|\nabla\log
p_{t}(\mathbf{x})-s_{\theta}(\mathbf{x},t)\|^{2}],$ (24)

where $U(0,T)$ denotes a uniform distribution over time interval $[0,T]$ and
$\lambda$ is a positive weighting function to balance the magnitude of
different score-matching losses.

Once the score-based model $s_{\theta}(\cdot,t)$ is trained, numerically
solving the FBSDE is the same as the prediction of the form of a function
(trajectory) that becomes the solution. In the literature, there are various
methods to solve FBSDE, such as the Euler-Maruyama method and stochastic
Runge-Kutta method. The key point here is that once a score predictor is
obtained, then any form of SDE solver can be used to solve the reverse SDE
numerically.

This paper particularly employs the Euler-Maruyama method to perform the
backward SDE. This algorithm provides a method for denoising data by
simulating the backward trajectory of the stochastic process. The Euler-
Maruyama method is effective for integrating the backward SDE, provided the
score function can be accurately estimated, which can be described as follows,

$dX_{t}=[f(X_{t},t)-g(t)^{2}\nabla_{x}\log p_{t}(X_{t})]dt+g(t)d\bar{W}_{t},$
(25)

where $\nabla_{x}\log p_{t}(X_{t})$ is the score, obtained by a neural network
defined as $s_{\theta}(X_{t},t)\approx\nabla\log p_{t}(X_{t})$.

For noise mitigation problem, the backward SDE process can be re-defined as
follows,

${d|\psi_{t}\rangle}=\left[f(\psi_{t},t)\,-g^{2}(\psi_{t},t)\nabla_{\psi}\log{p_{t}(|\psi_{t}\rangle)}\right]\,dt+\\\
g(\psi_{t},t)\,{d\bar{W_{t}}}.$ (26)

The backward process starts with $X_{T}$ sampled from a predefined noise
distribution, typically Gaussian at the final noise level $T$. The final noisy
state $|\psi_{T}\rangle$ at time $T$ corresponds to $X_{T}$ in the noise
mitigation problem, which is to be denoised eventually.

The underlying dynamic in score-based diffusion models [17] can be described
by the Ornstein-Uhlenbeck diffusion process, i.e.,

$\displaystyle dX_{t}$ $\displaystyle=-\alpha
X_{t}\,dt+\sqrt{2}\beta\,dW_{t},$ (27) $\displaystyle X_{0}$
$\displaystyle\sim p_{0},$ (28)

and $\alpha>0$.

The noising process $(X_{t})_{t\in[0,T]}$ begins with data from $p_{0}$ and is
perturbed into the invariant distribution $p_{T}(p_{\text{prior}})$.
Generally, assumes $p_{\text{prior}}$ to be Gaussian, i.e.,

$p_{\text{prior}}=N(0,1/\alpha).$ (29)

When $t$ approaches infinity, the distribution $X_{t}$ tends towards
$p_{\text{prior}}$, which is Gaussian (Ornstein-Uhlenbeck process). This can
be seen as a situation where noise is added as time evolution progresses,
which is defined in the dissipative term in (14). The evolution
$(|\psi_{t}\rangle)_{t\in[0,T]}$ starting from the noiseless state
$|\psi_{0}\rangle$ in open quantum system problem to the noisy state
$|\psi_{T}\rangle$.

The $(X_{t})_{t\in[0,T]}$ dynamics reverses $p_{t}$ through time-reversal
operations, using an explicit drift and diffusion matrix to precisely reverse
the noise process, as follows,

$\displaystyle dX_{t}$ $\displaystyle=[\alpha X_{t}\,dt+2\nabla\log
p_{T-t}(X_{t})]\,dt+\sqrt{2}\beta\,dW_{t},$ (30) $\displaystyle X_{0}$
$\displaystyle\sim p_{T},$ (31)

which can be formulated in our problem over the quantum computing domain, as
follows,

$\displaystyle{d|\psi_{t}\rangle}=$
$\displaystyle[\alpha|\psi_{t}\rangle\,dt+2\nabla\log
p_{T-t}(|\psi_{t}\rangle)]\,dt$ $\displaystyle+\sqrt{2}\beta\,dW_{t},$ (32)
$\displaystyle|\psi_{0}\rangle\sim$ $\displaystyle p_{T}.$ (33)

The reconstructed data, $X_{0}$, represents an estimate of the original sample
before noise was added. Moreover, $|\psi_{0}\rangle$ will be the noise-
eliminated state in the noise mitigation problem.

##  IV Concluding Remarks

The primary aim of the paper is to mitigate errors in open quantum systems, a
critical challenge as quantum computing transitions from theoretical
exploration to practical application. This paper presents a new approach using
diffusion models, specifically through the formulation of noise as forward-
backward stochastic differential equations (FBSDE) and using the score-based
generative model (SGM) for error mitigation. The proposed method leverages SGM
to effectively denoise quantum states. This is achieved by dynamically
adjusting the system’s state towards an ideal, noise-free trajectory, thereby
improving the precision and reliability of quantum computations.

In conclusion, this paper not only advances our understanding of quantum error
mitigation in open systems but also presents a practical and scalable approach
that could significantly enhance the performance and reliability of quantum
computers. This makes it a valuable contribution to the field of quantum
computing, particularly in the era of NISQ.

## Acknowledgment

Joongheon Kim is a corresponding author.

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