# Neutrino oscillation with minimal length uncertainty relation via wave
packet approach

M.M. Ettefaghi<EMAIL_ADDRESS>Department of Physics, University of Qom,
Ghadir Blvd., Qom 371614-611, I.R. Iran

###### Abstract

Theories of Quantum Gravity as well as string theory suggest the existence of
a minimal measurable length and the related Generalized Uncertainty Principle
(GUP). The universality of Quantum Gravity implies that the GUP influences
every quantum mechanical process. Neutrino oscillation as a quantum phenomenon
exhibits quantumness at macroscopic distances and could provide potentially a
suitable room for quantum foundation explorations. In this paper, we
perturbatively derive the neutrino oscillation probability based on the GUP
and by treating neutrinos as wave packets. We see that the GUP modifications
are dependent on the effective position width of the transition amplitude
$\sigma_{x}$ such that with the smaller $\sigma_{x}$ we can obtain a stronger
bound on the minimal length scale in comparison to what is expected from
standard model interactions. More explicitly, one can obtain an upper bound
about $5\times 10^{25}$ for the deformation parameter, $\beta_{0}$, with
accelerator neutrino experiments such as MINOS, provided that $\sigma_{x}\sim
10^{-15}\text{m}$ which is reasonable since the energy of these neutrinos is
of the order of a few GeV.

##  I Introduction

At least at energy scales as large as the Planck scale, gravity as a
fundamental interaction must be compatible with quantum mechanics. However,
the quantization of gravity using the quantum field prescriptions leads to a
non-renormalizable theory. Hence, it has been believed that in the theories of
quantum gravity there may exist an effective cutoff in the ultraviolet or
equivalently a minimal measurable distance at the Planck scale gravity1 ;
gravity2 ; gravity3 . Indeed, the belief that the minimal length exists is the
result of a conjecture that gravity causes the structure of space-time to be
disturbed at the Plank scale. Although, at the electroweak scale the effects
of gravity interactions are so weak that they cannot be measured directly by
recent and upcoming experiments, they are universal and anything which has
energy creates gravity and is effected by it. The existence of a minimal
measurable length, predicted by theories of Quantum Gravity, influences all
quantum Hamiltonians das . Therefore, it is suitable to consider these quantum
gravity corrections to various quantum phenomena and obtain experimental bound
on the scale at which these corrections appear.

When there exists a minimal measurable length, the increase in energy cannot
reduce the amount of uncertainty of the position measurement to arbitrary
size. So the Heisenberg uncertainty principle, $\Delta x\Delta p\geq 1/2$,
breaks down for energies close the minimal length scale and it must be
generalized GUP1 ; GUP2 ; GUP3 . In one dimension, the Generalized Uncertainty
Principle (GUP) is given by

$\Delta x\Delta p\geq\frac{1}{2}(1+\beta(\Delta p)^{2}+\beta\langle
p\rangle^{2}),$ (1)

where $\beta$ is a positive parameter defined as
$\beta=\beta_{0}/M^{2}_{\text{Pl}}$ with $M_{\text{Pl}}\sim 10^{19}GeV$. If
the dimensionless parameter $\beta_{0}$ is of the order of unity, the new
physics coming from minimal length correction cannot be measured even in
future experiments. However, the hypothesis of extra dimensions paved the way
for the investigation of quantum gravity at the achievable energy at the LHC
ed1 ; ed2 ; ed3 ; ed4 . For instance, according to Arkani-Hamed, Dimopolous
and Dvali (ADD) model ed1 , the gauge interactions are confined to a 3-brane
while gravity propagates in all the dimensions. In this case, the effective
scale of the appearance of quantum gravitational interactions $M_{S}$ is
related to the usual Planck scale by $M_{Pl}^{2}=R^{d}M_{S}^{d+2}$ where $R$
is the radius of compactification and $d$ is the number of extra dimensions.
Thus it is possible to choose $R$ for a given $d$ such that the quantum
gravity scale comes down to $M_{S}\sim 1$ TeV. Hence, it was proposed that
high energy particle colliders could turn out to be huge BH factories bh1 ;
bh2 ; bh3 . Therefore, the production of a microscopic black hole has been
explored by the LHC data; the CMS collaboration excluded the production of a
microscopic black hole with masses below 4.3 to 6.2 TeV (depending on the
models) with 95 % confidence level cms  and the ATLAS collaboration indicated
that the threshold mass must be larger than 5.3 TeV atlas . Hereupon, so far
we have gained the ability to survey experimentally lengths with a scale up to
two orders of magnitude smaller than the electroweak length scale, and no new
physics, compared to the standard model, has been discovered with certainty.
In Refs. das3 ; ali , it has been shown that the GUP brings important
qualitative and quantitative changes to black hole thermodynamics. This causes
the minimum energy for formation to be increased so that it could be pushed
beyond the reach of LHC even if $M_{S}\sim 1$TeV. Therefore, the length scale
of new physics and gravity modifications shall be smaller than this scale.
This implies that the best upper bound on $\beta_{0}$ which one can obtain
potentially from LHC data should be of the order of $10^{30}$.

Otherwise, considering the universality of the GUP, one can obtain an upper
bound for $\beta_{0}$ from high-resolution measurements such as spectroscopy
on the hydrogen atom, considering the ground state Lamb shift lamb , the 1S-2S
level difference 1s2s  and Landau levels using a scanning tunnel microscope
(STM) das ; ett . The values obtained by these methods in real conditions are
greater than $10^{32}$. In the case of the upper bound obtained from the
AURIGA detector AURIGA1 ; AURIGA2 , the situation is the same. However, the
upper bound obtained from the lack of violation of the equivalence principle
is in the order of $10^{21}$ ep .

Neutrino oscillation is a fantastic quantum phenomenon occurring at
macroscopic distances. In fact, this process can be used to study quantum
aspects in large distances even up to several hundred kilometers away blasone1
; blasone2 ; ette2020 ; alok ; Banerjee ; naikoo ; ette2022 ; ette2022D ; li ;
jha ; yadav ; blasone3 ; ettef5 . The idea of neutrino oscillation was first
put forward more than half a century ago and since then, using various
neutrino sources, many experimental indications showing the transition between
different neutrino flavors have been discovered historical . Meanwhile, there
are some debates on the basic issues of the theory of neutrino oscillations
paradoxes . At first glance and not so accurately, treating neutrinos as a
plane wave or stationary state and assuming some additional and non-physical
constraint such as equal either energy or momentum for mass eigenstates, one
can obtain the probability of oscillation. However, a wave function with
definite momentum spread throughout the space and a stationary state does not
depend on time. While neutrino oscillation is the result of dependence on
spatial position and time evolution. In order to provide a more consistent
theoretical explanation for neutrino oscillation, we must take into account
the localization of microscopic processes by which a neutrino is produced and
detected based on the uncertainty principle and treat neutrinos as wave
packets paradoxes ; Nussinov ; Kayser ; giunti1 ; giunti2 ; Kiers ; giunti3 .

An upper bound on the scale of a minimal length induced by a quantum gravity
theory has already been obtained through neutrino oscillation study based on
the stationary description which is inconsistent Sprenger . Meanwhile, in this
paper, we will restudy this problem with taking into account the GUP to obtain
the momentum wave functions of produced and detected neutrinos. The produced
state evolves with time in position space to reach the detector.
Transformation of the momentum wave functions to the position wave functions
is problematic; since there exists an measurable minimal length, $\Delta
x_{0}=\sqrt{\beta}$, one cannot define a position eigenstate with a zero
position uncertainty. Therefore, one needs to formulate appropriate Hilbert
space representation of the modified Heisenberg algebra in quantum mechanics,
which we call the GUP position eigenstate kempf . Given that neutrino
oscillation experiments are performed in energies very less than the Planck
scale, the GUP induced effects are very small in comparison to the usual one
so that we will keep the linear terms of $\beta$ and we obtain perturbatively
the transition amplitude and consequently the transition probability.

In principle, it is expected that the effects of the measurable minimal length
occur in the Planck scale. However, the GUP is universal and similar to das ;
lamb ; 1s2s , one can infer an upper bound on $\beta_{0}$ by neutrino
oscillation experiments. For instance, we will use the set up requirements of
the far MINOS detectorminos  to estimate an lower bound on $\beta_{0}$.

The paper is organized as follows: in the following section, we give a brief
review of GUP. The oscillation probability is obtained in section III. We give
a numerical analyzing in section IV. Finally, we summarize our results in
section V.

##  II GUP

The Heisenberg algebra corresponding to the GUP given in Eq. (1) can be
written as follows111Since detected neutrinos propagate along a macroscopic
distance, we consider only one spatial dimension along the neutrino path.:

$[{\bf x},{\bf p}]=i(1+\beta{\bf p}^{2}).$ (2)

In this paper, we consider the existence of a minimal position length and
ignoring any restriction on the momentum space. Hence, in the momentum space,
the operators ${\bf p}$ and ${\bf x}$ have the following representations:

${\bf p}\phi(p)=p\phi(p),$ (3)

and

${\bf x}\phi(p)=i(1+\beta p^{2})\partial_{p}\phi(p).$ (4)

This representation satisfies Eq. (2). A nonzero minimal uncertainty in the
position implies that we cannot have a physical state which is a position
eigenstate kempf . Nevertheless, let us use Eq. (4) to obtain the position
eigenstate in the momentum space

$i(1+\beta p^{2})\partial_{p}\phi_{x}(p)=x\phi_{x}(p),$ (5)

where $x$ is the position eigenvalue. Because $\beta p^{2}\ll 1$ and we are
interested in the first-order corrections, we can solve Eq. (5) by the
perturbation method. Therefore, up to the first order of $\beta$, the position
eigenstate in the momentum space is given by:

$\phi_{x}(p)=\frac{1}{\sqrt{2\pi}}(1+\frac{i\beta}{3}xp^{3})e^{-ipx}.$ (6)

Furthermore, in order to preserve the symmetry property of ${\bf x}$ and ${\bf
p}$, the identity operator in momentum space must be written as kempf

${\bf 1}=\int\frac{dp}{1+\beta p^{2}}|p\rangle\langle p|.$ (7)

From Eq. (2), one can show that the momentum operator in position space is
given by

${\bf p}\psi(x)=-i(1-\frac{\beta}{3}\partial_{x}^{2})\partial_{x}\psi(x).$ (8)

Accordingly, the modified Dirac equation in one dimension can be written as
das2 ; pedram

$i\partial_{0}\psi=-\gamma_{0}\Big{(}i(1+\frac{\beta}{3}\partial_{x}^{2})\gamma^{1}\partial_{x}+m\Big{)}\psi$
(9)

where $\gamma_{0}$ and $\gamma_{1}$ are two of Diac matrices. Therefore, the
dispersion relation that results from this relation up to the first order of
$\beta$ is written as follows:

$E^{2}=p^{2}(1-\frac{2}{3}\beta p^{2})+m^{2}.$ (10)

In order to explain neutrino oscillation in quantum mechanics framework, we
must take into account the localization of microscopic processes by which a
neutrino is produced and detected. The localization in quantum mechanics is
based on the uncertainty principle. For each state in the representation of
the Heisenberg algebra, the uncertainty principle can be deduced from the
positivity of the norm

$||\Big{(}{\bf x}-\langle{\bf x}\rangle+\frac{\langle[{\bf x},{\bf
p}]\rangle}{2(\Delta p)^{2}}({\bf p}-\langle{\bf p}\rangle)\Big{)}||\geq 0.$
(11)

In fact, given that $\langle[{\bf x},{\bf p}]\rangle$ is imaginary, this
relation is written as

$\langle\psi|({\bf x}-\langle{\bf x}\rangle)^{2}+\Big{(}\frac{\langle[{\bf
x},{\bf p}]\rangle}{2(\Delta p)^{2}}\Big{)}^{2}({\bf p}-\langle{\bf
p}\rangle)^{2}|\psi\rangle\geq 0,$ (12)

which immediately implies that

$\Delta x\Delta p\geq\frac{|\langle[{\bf x},{\bf p}]\rangle|}{2}.$ (13)

In the case of boundary of the physically allowed region, a state
$|\psi\rangle$ must fulfill the following condition:

$\Big{(}{\bf x}-\langle{\bf x}\rangle+\frac{\langle[{\bf x},{\bf
p}]\rangle}{2(\Delta p)^{2}}({\bf p}-\langle{\bf
p}\rangle)\Big{)}|\psi\rangle=0,$ (14)

Using Eq. (2) and Eq. (4), one casts the recent equation in the momentum
representation as follows

$\Big{(}i(1+\beta p^{2})\partial_{p}-\langle{\bf
X}\rangle+i\frac{1+\beta(\Delta p)^{2}+\beta\langle{\bf
p}\rangle^{2}}{2(\Delta p)^{2}}(p-\langle{\bf p}\rangle)\Big{)}\phi(p)=0.$
(15)

##  III Neutrino oscilation with GUP

There are two approaches for explanation of neutrino oscillation; field theory
and quantum mechanics approaches beuthe . In the field theory approach,
neutrinos are treated as intermediate particles between the source and the
detector, and the neutrino oscillation is the result of considering the
details of the localization of particles participating in the production and
detection processes. This leads to the uncertainty in determining the momentum
of the mass eigenstates, such that a coherent superposition of the various
mass eigenstate propagators contributes in the transition amplitude.
Therefore, the oscillation pattern comes form the interference terms in the
transition probability. In this approach, we should note that neutrinos are
described by a Feynman propagator which is modified due the induced GUP
corrections feynmanp . Moreover, the interactions during the production and
detection processes may be affected by the GUP due to the production of
(microscopic) black hole productions which are predicted by ADD theoryed1 . A
detailed study of this theory is beyond the scope of the present work and is
left for future studies.

In the case of quantum mechanical approach which is followed in the this
paper, neutrinos are treated as localized particles traveling between source
and detector. Since the GUP affects a host of quantum mechanical problems, we
must modify our considerations about the localization properties in this
approach of the neutrino oscillation explanation as well. But the question may
arise, why do we expect the uncertainty principle for neutrinos to be
reformulated under quantum gravity as GUP? In the absence of a consistent
theory to merge quantum mechanics with general relativity, we can use a semi-
classical approximation through which the canonical commutation relations
between the momentum operator $p^{\mu}$ and position operator $x^{\mu}$, which
are $[x^{\mu},p^{\nu}]=i\hbar\eta^{\mu\nu}$ in Minkowski space-time, in a
curved space-time with metric $g_{\mu\nu}$ have been generalized as

$[x^{\mu},p^{\nu}]=i\hbar g^{\mu\nu}.$ (16)

In Ref. metric , by representing the position and momentum operators as
covariant derivatives with an appropriate connection in the eight dimensional
manifold, quantization has been geometrically interpreted as the curvature of
phase space. Through this approach, implying the existence of an upper limit
on the acceleration of particles moving along their worldlines, one can show
that the corresponding metric tensor can be written as follows:

$g^{\mu\nu}\sim\left(1-c^{4}\frac{\ddot{x}^{\sigma}\ddot{x}_{\sigma}}{\cal
A}\right)\eta^{\mu\nu},$ (17)

where $\ddot{x}^{\sigma}$ and $\cal A$ denote the four-acceleration which can
be given in terms of momentum dispersion and the maximal acceleration,
respectively. Substituting $g^{\mu\nu}$ from Eq. (17) in Eq. (16) and using
the uncertainty relation $\Delta A\Delta B\leq 1/2|\langle[A,B]\rangle|$, one
can obtain GUP

$\Delta x\Delta p\leq\frac{\hbar}{2}+\frac{\alpha}{c^{3}}G\Delta p^{2},$ (18)

where $\alpha$ is a free parameter. Moreover, as another viewpoint, the GUP
can arise provided that relaxing the postulates of quantum mechanics, we
ignore the position operator that satisfies the usual Heisenberg commutation
relation kazemi . Nevertheless, we base it on the universality of quantum
gravity and exert the modifications due to GUP on the quantum mechanics
approach of neutrino oscillation explanation.

The existence of minimal length has been explored previously using neutrino
oscillation phenomenon with treating neutrinos as stationarySprenger .
However, strictly speaking, the probability of finding a particle described by
stationary states does not depend on time. In quantum theory, propagating
particles must be described generally by moving wave packets which fulfill the
uncertainty principle deduced from Heisenberg algebra. As we saw, the quantum
gravity induced minimal length leads to a modification of Heisenberg’s algebra
(Eq. 2) and the corresponding uncertainty principle called GUP. Here, we will
assume the neutrino wave packets satisfy the boundary allowed region of the
GUP (Eq. 15). Accordingly, the scenario that we follow to obtain the
probability of neutrino oscillation is as follows: using Eq. 15 we must obtain
the neutrino wave function in momentum representation at the source and
detector. Then after temporal evolution of the source wave function, we must
project the source and detector wave functions into the position
representation. The amplitude of oscillation is equal to the spatial
correlation of these wave functions. The squire of absolute value of the
amplitude is the oscillation probability. Of course, we must keep in mind that
the produced and detected neutrinos do not have a specific mass and they are
said to be in the flavor eigenstate. Therefore, their states must be written
as a superposition of all mass eigenstates. So we begin by calculating the
corresponding wave functions.

###  III.1 Propagating neutrino wave function

Our basis for calculating the related wave functions is Eq. (15). In this
equation, $\langle{\bf x}\rangle$ and $\langle{\bf p}\rangle$ are the mean
value of the position and momentum operators, respectively. We chose
$\langle{\bf x}\rangle=0$ and $\langle{\bf p}\rangle=p_{k}$ in which $p_{k}$
is the average amount of momentum for the $k$’th mass eigenstate of neutrinos.
Moreover, $\Delta p$ is taken the momentum uncertainty of neutrino during the
creation process and we denote it by ${\sigma_{p}}_{P}$. Since we are
interested in the GUP corrections up to the first order of $\beta$, we solve
Eq. (15) by the perturbation method. In this manner, the created momentum wave
function of neutrinos in $k$’th mass eigenstate is obtained as follows:

$\phi^{\text{P}}_{k}(p;p_{k},{\sigma_{p}}_{\text{P}})=\frac{e^{-\frac{(p-p_{k})^{2}}{4{\sigma_{p}^{2}}_{\text{P}}}}}{\sqrt[4]{2\pi\sigma_{pP}^{2}}}\left(1-\frac{1}{4}\beta(p-p_{k})^{2}+\frac{\beta}{24{\sigma_{p}^{2}}_{\text{P}}}(p-p_{k})^{3}(3p+5p_{k})-\frac{1}{8}\beta{\sigma_{p}^{2}}_{\text{P}}\right),$
(19)

which is normalized up to the first order of $\beta$. What participates in
weak interactions are the flavor eigenstates of neutrinos, which are written
in terms of mass eigenstates as follows:

$|\nu_{\alpha}\rangle=\sum_{k=1}^{3}U^{*}_{\alpha k}|\nu_{k}\rangle,$ (20)

where $U_{\alpha k}$ denotes the mixing unitary matrix elements. Therefore,
using Eq. (6), Eq. (7) and Eq. (10), one can give the time evaluated of the
neutrino wave function with definite flavor in the Schrodinger picture by

$\psi^{\text{P}}_{\alpha}(x,t)=\frac{1}{\sqrt{2\pi}}\sum_{k=1}^{3}U^{*}_{\alpha
k}\int\frac{dp}{1+\beta
p^{2}}\phi^{\text{P}}_{k}(p;p_{k},{\sigma_{p}}_{\text{P}})(1+\frac{i}{3}\beta
xp^{3})\text{e}^{-i\left(E_{k}(p)t-px\right)},$ (21)

where $E_{k}(p)=\sqrt{p^{2}(1-\frac{2}{3}\beta p^{2})+m_{k}^{2}}\approx
E_{k}^{0}(p)-\frac{1}{3}\frac{\beta p^{4}}{E_{k}^{0}(p)}$ in which
$E_{k}^{0}(p)=\sqrt{p^{2}+m_{k}^{2}}$. We assume the momentum wave functions,
Eq. (19), are sharply peaked around the corresponding average momentum, i.e.
${\sigma_{p}}_{\text{P}}\ll p_{k}$. Under this condition, we can keep the
terms which are liner with respect to $\beta$ for energies below the minimal
length scale. Moreover, $E_{k}^{0}(p)$ can be approximated by
$E_{k}^{0}(p)\approx E_{k}-v_{k}(p-p_{k})$ where
$E_{k}=\sqrt{p_{k}^{2}+m_{k}^{2}}$ and $v_{k}=p_{k}/E_{k}$ is the group
velocity of each wave packet with definite mass. The other point which should
be noted is that due to the large energy scale for the emergence of the
effects of the minimal measurable length and due to the limited neutrino
energy at different sources, we can use the usual uncertainty relation,
${\sigma_{x}}_{P}{\sigma_{p}}_{P}=1/2$, to replace momentum uncertainty with
position uncertainty. After taking into account all these issues, the wave
function of the propagated neutrinos up to the first order of $\beta$ is
obtained as follows:

$\displaystyle\psi^{\text{P}}_{\alpha}(x,t)=\frac{1}{\sqrt[4]{2\pi\sigma_{xP}^{2}}}\sum_{k=1}^{3}U^{*}_{\alpha
k}\text{e}^{-i(E_{k}t+p_{k}x)-\frac{(x-v_{k}t)^{2}}{4\sigma_{xP}^{2}}}\bigg{[}1+\beta\bigg{(}$
$\displaystyle\frac{i(x-v_{k}t)}{\sigma_{xP}^{2}}-\frac{5(x-v_{k})^{2}}{16\sigma_{xP}}-\frac{ip_{k}(x-v_{k}t)^{3}}{6\sigma_{xP}^{4}}$
(22)
$\displaystyle+\frac{(x-v_{k}t)^{4}}{32\sigma_{xP}^{6}}+\frac{7}{32\sigma_{xP}^{2}}\bigg{)}\bigg{]}.$

###  III.2 Detected neutrino wave function

The wave function of the detected neutrino must be localized around the
detector position, a distance $L$ from the source. In addition, we assume that
the mean momentum of the detected neutrino is equal to the mean momentum of
the propagated neutrino due to the momentum conservation. Therefore, in Eq.
(15) for detected neutrino, we take $\langle{\bf x}\rangle=L$ and $\langle{\bf
p}\rangle=p_{k}$. Similar to the previous case, the wave function of detected
neutrinos which is the solution of Eq. (15) is given by

$\displaystyle\phi^{\text{D}}_{k}(p;p_{k},{\sigma_{p}}_{\text{D}})=\frac{1}{{\sqrt[4]{2\pi{\sigma_{p}}_{\text{D}}^{2}}}}\text{e}^{-\frac{(p-p_{k})^{2}}{4{\sigma_{p}}_{\text{D}}^{2}}-ipL}\left[1+\frac{\beta}{24}\left(-3{\sigma_{p}}_{\text{D}}^{2}+2\left(-3(p-p_{k})^{2}+4ip^{3}L\right)+\frac{(p-p_{k})^{3}(3p+5p_{k})}{{\sigma_{p}}_{\text{D}}^{2}}\right)\right].$

This wave function does not represent a propagating neutrino, therefore, it
does not evolve in time. Consequently, using Eq. (6) and Eq. (7), one can
write the position wave function of the detected neutrinos as follows:

$\psi^{\text{D}}_{\alpha}(x)=\frac{1}{\sqrt{2\pi}}\sum_{k=1}^{3}U^{*}_{\alpha
k}\int\frac{dp}{1+\beta
p^{2}}\phi^{\text{D}}_{k}(p;p_{k},{\sigma_{p}}_{\text{D}})(1+\frac{i}{3}\beta
xp^{3})\text{e}^{ipx}.$ (24)

With assumption similar to what considered for Eq. (22), up to the first order
of $\beta$, one can simplify the position wave function of the detected
neutrinos as follows:

$\displaystyle\psi^{\text{D}}_{\alpha}(x,t)=\frac{1}{24\sqrt[4]{2\pi{\sigma_{x}}_{\text{D}}^{2}}}\sum_{k=1}^{3}U^{*}_{\alpha
k}e^{i(x-L)p_{k}-\frac{(x-L)^{2}}{4{\sigma_{x}}_{\text{D}}^{2}}}\Big{(}$
$\displaystyle 24+8i\beta Lp_{k}^{3}+\frac{12\beta
Lp_{k}^{2}(L-x)}{{\sigma_{x}}_{\text{D}}^{2}}$ (25)
$\displaystyle-\frac{2i\beta
p_{k}\left(L^{3}+6L{\sigma_{x}}_{\text{D}}^{2}-3Lx^{2}+2x^{3}-12{\sigma_{x}}_{\text{D}}^{2}x\right)}{{\sigma_{x}}_{\text{D}}^{4}}$
$\displaystyle-\frac{\beta\left(6{\sigma_{x}}_{\text{D}}^{2}\left(L^{2}-6Lx+5x^{2}\right)+(L-x)^{3}(L+3x)-21{\sigma_{x}}_{\text{D}}^{4}\right)}{4{\sigma_{x}}_{\text{D}}^{6}}\Big{)}.$

###  III.3 Oscillation probability

The oscillation amplitude is obtained through calculating the correlation of
the wave functions given by Eqs. (22) and (25)

$A_{\alpha\beta}=\int{\psi^{D}_{\beta}}^{*}(x,t)\psi^{P}_{\alpha}(x,t)dx,$
(26)

and consequently the transition probability is given by

$P_{\alpha\beta}=|A_{\alpha\beta}|^{2}.$ (27)

Because the neutrino emission and arrival time are not measured, we should
integrate over the time during which a neutrino travels from the source to the
detector. Hence, by direct calculation, one can obtain the transition
probability up to the first order of $\beta$ as follows:

$P_{\alpha\beta}\simeq\sum_{i}\sum_{j}U_{\alpha i}U^{*}_{\beta i}U^{*}_{\alpha
j}U_{\beta j}\text{e}^{-2\pi
i\frac{L}{L^{\text{osc}}_{ij}}-\frac{L^{2}}{{L^{\text{coh}}_{ij}}^{2}}-2\pi^{2}\rho^{2}\frac{\sigma_{x}^{2}}{{L^{\text{osc}}_{ij}}^{2}}}f_{ij}(\beta),$
(28)

where

$f_{ij}(\beta)=1-\beta\left(\frac{i\pi\left(4E^{2}\sigma_{x}^{2}+1\right)}{2\sigma_{x}^{2}}\frac{L}{L^{\text{osc}}_{ij}}+\frac{\left(8E^{2}\sigma_{x}^{2}-5\right)}{4\sigma_{x}^{2}}\frac{L^{2}}{{L^{\text{coh}}_{ij}}^{2}}+\frac{i\pi^{3}}{6E^{2}\sigma_{x}^{4}}\frac{L^{3}}{{L^{\text{osc}}_{ij}}^{3}}+\frac{3}{4\sigma_{x}^{2}}\frac{L^{4}}{{L^{\text{coh}}_{ij}}^{4}}\right),$
(29)

with $L^{\text{osc}}_{ij}=(4\pi E)/\Delta m_{ij}^{2}$ and
$L^{\text{coh}}_{ij}=(4\sqrt{2}\sigma_{x}E^{2})/|\Delta m^{2}_{ij}|$.
$\sigma_{x}$ is an effective width of the shape factor of the transition
amplitudes. Therefore, it depends on the widths of both the production and
detection wave packets ${\sigma_{x}}_{P}$ and ${\sigma_{x}}_{D}$. For the
conditions assumed here, establishing the Eq. (15) in both production and
detection processes, $\sigma_{x}$ is given by the following relation paradoxes
:

$\sigma_{x}=\sqrt{{\sigma_{x}}_{P}^{2}+{\sigma_{x}}_{D}^{2}}.$ (30)

Hence, it is of the order of the largest between ${\sigma_{x}}_{P}$ and
${\sigma_{x}}_{D}$.

In this calculation, we use the following statements:

  * •

We assume that neutrinos are extremely relativistic. Therefore, their mass
eigenstate energy and momentum are approximated by the following relations:

$E_{i}=E+\rho\frac{m_{i}^{2}}{2E},$ (31)

and

$p_{i}=E-(1-\rho)\frac{m_{i}^{2}}{2E},$ (32)

where $E$ is the neutrino energy in the limit of zero mass and $\rho$ is a
dimensionless quantity that can be estimated from energy-momentum conservation
in the production processes rho .

  * •

In the limit of extremely relativistic, the group velocity can be estimated as
follows:

$v_{j}\simeq 1-\frac{m_{j}^{2}}{2E}.$ (33)

  * •

Up to first order of $\beta$, the transition probability is dependent on the
propagation length $L$ maximum up to the forth power of it. For every power of
$L$, we keep those terms which are dependent on the neutrino mass upmost in
the order to the twice of the $L$ power.

##  IV Numerical analysis

Although quantum gravity has been of interest to scientists for several
decades, no significant success has been achieved in establishing a precise
and definitive theory. The reason is that the effects of quantum gravity
usually appear at the Planck scale, an energy scale beyond the reach of any
kind of experiment. However, as it was said, in order to construct a
consistent quantum gravity theory, it is necessary to add new fundamental
principles such as the existence of minimal measurable length, space-time
noncommutativity, etc. to our fundamental principles. As a rule, these new
principles are universal and they can affect theories that can be investigated
with the available experiments. For instance, the neutrino oscillation theory
is affected by the GUP as it was studied in the previous section. In this
section, we want to give a numerical estimation for the GUP modifications to
the neutrino oscillation probability obtained in the previous section.

As one can see from Eqs. (28) and (29), the GUP modifications affects on both
the oscillation wave length and the decoherence factors. We take the following
values for the usual neutrino oscillation parameters oscillation parameters :

$\displaystyle\Delta m_{21}^{2}\approx 7.39\times 10^{-5}eV^{2};\hskip
28.45274pt\Delta m_{31}^{2}\approx 2.45\times 10^{-3}eV^{2};$
$\displaystyle\theta_{12}\approx 33.82;\hskip 28.45274pt\theta_{13}\approx
8.61;\hskip 28.45274pt\theta_{23}\approx 49.7.$ (34)

The other effective parameter that plays an important role in the magnitude of
the GUP modifications is $\sigma_{x}$. As it can be seen from Eqs. (28) and
(29), the smaller $\sigma_{x}$ leads to the more visible effects of the GUP
modifications. However, it should be noted that in order to see the phenomenon
of neutrino oscillation, $\sigma_{x}$ should not be so small that the
coherence length becomes smaller than the oscillation length. For a fixed
teravel distance, the GUP modifications increase with the amount of neutrino
energy. If the energy of neutrinos is in order of $1\text{GeV}$ like MINOS
experiment in which both production and detection processes are performed
through nuclear reactions, it is natural to take $\sigma_{x}\sim
10^{-15}\text{m}$. Comparing with the MINOS results, we estimate an upper
bound on the GUP parameter $\beta$. In this experiment, the energy of the
neutrinos varies from several hundreds MeV to several tens GeV. In this energy
interval, the coherence length is certainly larger than the oscillation one.

Figure 1: Survival probability in terms of neutrino energy for initial muon
neutrino with fixed travel distance, $L=735\text{km}$. Blue (dashed) and red
(dashed) lines illustrate the theoretical predictions based on usual and
generalized uncertainty principle (GUP), respectively. We also show the data
of MINOS data (black, circle) taken from Ref. minos .

In the MINOS experiment, the distance between the neutrino source and the far
detector is 735 km. Taking $\beta_{0}=5\times 10^{25}$, we compare the long
range survival probability for an initial muon neutrino in theories both with
and without GUP modifications through red (dashed) and blue (solid) line,
respectively, in Fig. 1. The data of MINOS experiment (black, circle) is also
shown in this figure. Therefore, the accelerator neutrino oscillation
experiments such as MINOS with more precision can provide a more suitable
opportunity to study the GUP.

##  V Summary and conclusions

Although experimental research of quantum gravity is far from available, its
effects are universal and can be investigated in existing experiments. The
existence of the measurable minimal length is one of the effects of quantum
gravity that can affect low energy physics and provide us with the possibility
to study experimentally its effects das . In fact, due to quantum gravity it
is no longer possible to spatially localize a wave function to arbitrary
precision and it leads to GUP given in Eq. (1). On the other hand, to give an
appropriate and accurate explanation of the neutrino oscillation phenomena,
one needs to consider the localization properties of neutrinos during
production, propagation and detection processes paradoxes . The wave packet
approach in quantum mechanics provides us with a convenient framework for
considering this issues Kayser . In this paper, we have obtained the GUP
effects on the neutrino oscillation via wave packet approach. Given that the
GUP induce an measurable minimal length, $\Delta x_{0}=\sqrt{\beta}$, one
cannot define a position eigenstate with a zero position uncertainty.
Therefore, using the method developed in Ref. kempf , we have formulated
appropriate Hilbert space representation to construct the produced and
detected neutrino state in the position representation. Meanwhile, since the
GUP effects are expected to be very small at the energy scale at which
neutrino oscillation experiments are performed, we have treated these effects
perturbatively and obtained the corrections on the oscillation probabilities
at the leading order.

Next, we performed a numerical analysis on the GUP corrections, which shows
that they depend on the effective width of the shape factor of the transition
amplitudes, $\sigma_{x}$; smaller $\sigma_{x}$ makes these corrections more
prominent. Therefore, through neutrino oscillation experiments with energies
of about a few GeV such as the MINOS experiment in which one can take
$\sigma_{x}\sim 10^{-15}\text{m}$, we can obtain a stronger upper bound on the
GUP deformation parameter $\beta_{0}$ in comparison to other experiments. In
particular, through Fig. 1, we have illustrated a deviation due to GUP
modifications from the standard theoretical predictions for the survival
probability of muon neutrinos by taking $\beta_{0}=5\times 10^{25}$. The
comparison with MINOS experimental data shows that if the length scale of the
GUP is of the order of $l_{GUP}\sim 10^{13}l_{Pl}$ where $l_{Pl}\equiv
1.6\times 10^{-35}\text{m}$, by measuring and analyzing the data from these
types of experiments more precisely, the effects of the measurable minimal
length can be explored. Hence, in this way, one can obtain a very stronger
upper bound on $\beta_{0}$ compared to the bounds obtained in Refs. lamb ;
1s2s ; das ; ett ; AURIGA1 ; AURIGA2 . But the bound obtained from the effect
of GUP on the equivalence principle ep  is stronger than this value.

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# Training a T5 Using Lab-sized Resources

Manuel R. Ciosici
USC Information Sciences Institute
USA
<EMAIL_ADDRESS>
Leon Derczynski
ITU Copenhagen
Denmark
<EMAIL_ADDRESS>

###### Abstract

Training large neural language models on large datasets is resource- and time-
intensive. These requirements create a barrier to entry, where those with
fewer resources cannot build competitive models. This paper presents various
techniques for making it possible to (a) train a large language model using
resources that a modest research lab might have, and (b) train it in a
reasonable amount of time. We provide concrete recommendations for
practitioners, which we illustrate with a case study: a T5 model for Danish,
the first for this language.

Training a T5 Using Lab-sized Resources

Manuel R. Ciosici USC Information Sciences Institute USA<EMAIL_ADDRESS>Leon
Derczynski ITU Copenhagen Denmark<EMAIL_ADDRESS>

##  1 Introduction

Pre-training large language models (LLMs) is costly. There is an environmental
cost Strubell et al. (2019); Patterson et al. (2021); money, in running costs
Sharir et al. (2020); and potentially money in hardware, where the equipment
to provide the needed computational power is often outside of a modest
research lab’s budget.

This paper presents two contributions. The first is a set of recommendations
to practitioners for training a modern LLM on modest hardware, limited data,
and a reasonable compute time. The second is a demonstration of the
recommendations resulting in DanT5, a T5 model for Danish, the first for this
language.111DanT5-small and DanT5-large are available at
https://huggingface.co/strombergnlp/dant5-small and
https://huggingface.co/strombergnlp/dant5-large.

##  2 Computational challenges

GPU memory is the most significant hurdle for training LLMs. State-of-the-art
optimizers like Adam Kingma and Ba (2015); Loshchilov and Hutter (2018)
converge much faster than traditional Stochastic Gradient Descent due to
tracking the first and second order momentum, but the performance comes at a
cost. However, to track the momentum, Adam must keep in memory two additional
values for each parameter in the model, thus adding a 2X memory overhead.

Recent research has investigated smaller-than-default number representations,
taking a floating point value (typically 32 bits) down to 16, 8, or fewer bits
Lin et al. (2017); Wang et al. (2018); Kalamkar et al. (2019). While these
advances have the potential to ease and so democratize large model training,
they require hardware and software support, which is still “just over the
horizon".

Using float16 representations typically reduces a model’s GPU memory
consumption by half Micikevicius et al. (2018) and require a relatively modern
GPU and software stack.222NVIDIA Pascal architecture GPU or newer, CUDA 8+.
Modern PyTorch or TensorFlow. However, these reduced-range representations can
overflow if training a model that expects a higher dynamic range. This can
happen, for example, when fine-tuning a model initially trained using TPUs,
which default to bfloat16 for float representation.

The bfloat16 representation should reduce compute needs Kalamkar et al.
(2019), though using it requires hardware333An NVIDIA Ampere GPU or newer. and
software support, and implementations across common software frameworks are
still “not quite finished just yet". Using even coarser representations, such
as bitsandbytes Dettmers et al. (2021) does, should also help, but these are
just beginning to be implemented in popular NLP frameworks.444For example, the
Hugging Face transformers library only added support for 8-bit optimizers
starting with version $4.19$, released on May 12, 2022.

One approach to training LLMs with modern GPUs is model parallelism, where a
language model is split over multiple GPUs, with each GPU storing one or more
layers. However, model parallelism brings the challenge of data communication.
GPU-to-GPU communication typically goes through the slow PCI Express bus and
sometimes the even slower CPU-to-CPU NUMA interconnect. Communication effects
can be ameliorated via hardware and software. In hardware, one can use NVLink
bridges to support high-speed GPU-to-GPU communication. NVLink bridges are a
cost-effective solution for linking GPU pairs in a machine. NVLink bridges
have a bandwidth of 50-100GB/s (depending on the GPU generation) compared to
31.5 GB/s for PCIe 4.0. On the software side, DeepSpeed Ren et al. (2021);
Rajbhandari et al. (2020, 2021) allows complex GPU offloading – supporting,
for example, moving the expensive Adam optimizer from GPU memory and into
(much larger) CPU memory. DeepSpeed CPU optimizers are implemented in
efficient C++ code that uses the CPU’s native SIMD support. While offloading
the optimizer to system memory increases the system memory requirements, RAM
capacity is usually extendable at a reasonable cost. GPU memory is not
extendable.

Given the challenges discussed above, we recommend the following when choosing
compute hardware for training Large Language Models:

  * •

Prioritize GPU memory over computation speed. LLM training is usually limited
by: the effects of low memory; communication bandwidth between GPUs; and
gradient accumulation over small batches.

  * •

Install NVLink bridges to speed up GPU-to-GPU communication. Modern ML
frameworks automatically use NVLink for GPU-to-GPU data transfer.

  * •

Prioritize GPUs with support for float16 and bfloat16 in order to take
advantage of optimizations whose widespread support is very close/nascent.

  * •

Plan for system memory of 512GB or more. If the budget is tight, buy larger
capacity RAM modules and leave empty RAM slots for future upgrades.

  * •

Leaving PCIe slots for future GPU acquisitions can extend the life of a lab’s
computing machines. For example, buying computers with support for eight GPUs
even though budget only permits four, allows for staggered GPU acquisition.

##  3 Working with sub-gargantuan corpora

There is a large disparity in the size of corpora available for English and
for other languages. While English has, for example, “The Pile” Gao et al.
(2020), an automatically-gathered corpus of over 800GiB, most other languages
have to get by with datasets orders of magnitude smaller Dunn and Adams
(2020).

Apart from size, corpus quality matters. Large corpora, such as the Common
Crawl, include significant amounts of undesirable/abusive/illegal material
Luccioni and Viviano (2021); Birhane et al. (2021), which leads to harmful
model behavior Bender et al. (2021); web-spider-derived corpora also tend to
duplicate data, requiring de-duplication, a deceptively complex and often
challenging task Lee et al. (2021). While these issues may have partly reduced
impact with massive corpora, the effect of each unwanted piece of content is
likely to be larger the smaller the corpus. When dealing with corpora of a few
gigabytes or less, quality becomes much more important, and the advantages of
automatic data collection can wane.

As a case study, we look at building a model for Danish using the T5 Raffel et
al. (2020) architecture, DanT5. Danish is a modestly-resourced language
Kirkedal et al. (2019). The original T5 used 34B sentence piece tokens from
C4. The Norwegian T5,555https://huggingface.co/NbAiLab/nb-t5-base-v2 for a
language similar to Danish, used the _Norwegian Colossal Corpus_ (NCC,
Kummervold et al. (2021), 7B words666https://github.com/NBAiLab/notram).
Unfortunately, there is no high-quality Danish language text corpus the size
of C4 or even the NCC.

The largest freely available high-quality Danish corpus is the _Danish
Gigaword Corpus_ (DAGW, Strømberg-Derczynski et al. (2021)), a curated 1B word
corpus. Like the NCC, DAGW assembles text spanning several domains, dialects,
time periods, and modalities.

A single training pass over DAGW is too little training data for a modern LLM,
even when warm-starting the training (see Section 4). To compensate for this,
when training DanT5, we perform 10 epochs over the DAGW with _dynamic
masking_. Before training, we split the corpus into sequences of 512 word
pieces, but do not mask any token. During training, when each batch is
assembled, we randomly mask 15% of each sequence’s tokens. The random, just-
in-time masking results in different masks for each epoch. Exposing DanT5 to
constantly changing masks attempts to compensate for DAGW’s size by generating
multiple masked configurations for any given sequence in the source corpus. We
mask 15% of the tokens in each sequence, following T5’s original training
procedure.

##  4 Warm starting from a different language

Warm-starting is a shortcut to training a large language model Rothe et al.
(2020). Warm-starting does not train a model from scratch but continues
training a pre-existing LLM. Therefore, warm-starting reduces computation
costs and environmental impact by taking advantage of some of the computation
that went into the source LLM. We used the original T5 checkpoints Raffel et
al. (2020) as the basis for DanT5. Warm-starting DanT5 from an English T5 also
supports transfer learning as the two languages are somewhat
related.777English is a West Germanic language, while Danish is North
Germanic.

Language models trained from pre-existing models are constrained to using the
source model’s tokenizer and, therefore, its vocabulary of tokens. The English
T5’s vocabulary contains sub-word tokens representing mostly English, with a
minority of tokens specialized for Romanian, German, and French.888These
tokens are present in the original T5 to allow the model to perform machine
translation for some language pairs. Reusing the sub-word token vocabulary for
Danish works but introduces excessive word splitting and leaves many tokens
rarely used. Splitting words excessively results in long input sequences,
reducing the amount of content that can fit T5’s maximum input length of 512.
Rarely-used tokens occupy space in the embedding layer, displacing tokens
relevant to the target language.

Several approaches have been proposed to modify a pre-trained language model’s
vocabulary. But, most methods cannot convert a mostly-monolingual vocabulary
into another mostly-monolingual vocabulary, as needed for DanT5. For multi-
language machine translation models, Garcia et al. (2021) extend a highly-
multilingual vocabulary with tokens for a new language, but assume that the
vocabulary grows by only a tiny fraction. Chronopoulou et al. (2020) first
train a language model using a tokenizer trained only on the source language.
Then they train a tokenizer on the concatenation of the source and target
language corpora. The target language model then reuses word embeddings for
the tokens appearing in both tokenizers. This approach cannot support adapting
a pretrained language model for warm-start training in a new language. More
recently, Xue et al. (2022) proposed doing away with tokenizers altogether and
training T5 models directly on byte encodings. This method promises token-free
models, but the requirement for a deeper encoder and the 30% larger training
cost push this approach even further away from what can be achieved with the
resources of a modest research lab.

Below, we present a new but simple way to adapt the vocabulary and parameters
of English T5 to create a good (warm-)starting point for training DanT5. Our
approach can generate warm-starting models for any language and transformer
architecture (encoder-only, decoder-only, or encoder-decoder, like T5).

Warm-start tokenization translation. We first follow Raffel et al. (2020) and
train a mostly Danish sentence-piece tokenizer on a mixture of 90% Danish and
10% English. We include English for the same reason as Raffel et al. (2020):
so that we can later fine-tune the model to support machine translation, in
this case Danish-English. DanT5’s tokenizer has exactly the same size as T5’s
to make the warm starting process easier, but one could extend or reduce the
vocabulary as needed. For warm-starting, we adapt the original T5 to the
Danish tokenizer using a simple approach. First, we attempt to translate into
English every token in the Danish tokenizer using Google Translate. We record
the successful translations, and where the translation fails, we set the
translation to equal the Danish word. We then create the starter DanT5 model
by cloning T5 and adjusting the input embeddings.999T5’s input and output
embeddings are identical, so we effectively adjust both the input and output
embeddings. We set each embedding in DanT5 to the mean of the English
embeddings of the English translation. Table 1 shows examples of Danish words
translating to single or multiple English words and the contingency in case of
translation failure (last row).

Danish | English |  en tokens
---|---|---
doktor | doctor | doctor
dokumentet | the document | the; document
værsgo | here you go | here; you; go
Yndling | Favorite | Favor; it; e
Aarhus | ERROR! | A; ar; hus
Table 1: Example Danish to English token mappings. The last row shows our
approach for cases where translation fails.

##  5 DanT5 Training

We follow T5’s original training procedure Raffel et al. (2020) with a few
exceptions. When we trained DanT5, there was no bfloat16 support in DeepSpeed
or transformers, so we trained our models with float32 representations.

_Optimizer._ We use DeepSpeed’s AdamW implementation instead of Adafactor
Raffel et al. (2020); Shazeer and Stern (2018). We use a learning rate of
$4e-3$, with warmup over the first $5\ 000$ steps and a linear learning rate
decay for the remaining. We chose AdamW over Adafactor due to the availability
of a CPU-offloadable implementation of AdamW through DeepSpeed, an essential
requirement when constrained by GPU memory. Even so, to achieve a batch size
of $128$, we accumulate gradients over $8$ forward passes.

_Batch Packing._ Unlike T5’s original training, we do not pack batches with
multiple sequences to obtain batches of roughly the same number of tokens.
Instead, we dynamically pad each batch so that all inputs in a batch are
padded to the maximal length within the batch. Therefore, length varies across
batches but minimizes padding within each batch. As opposed to static padding,
this technique speeds up computation on GPUs as it reduces the time wasted
computing activations for pad tokens. While batch packing can be adapted to
GPU computation, it is best suited for computations of TPUs which do not
support dynamic padding.

_Dynamic Masking._ The original T5’s training consumed 1 trillion tokens from
the _Colossal Clean Crawled Corpus (C4)_ , a procesed subset of _Common
Crawl_. Since no equivalently large corpus exists for Danish, we used the
_Danish Gigaword Corpus (DAGW, Strømberg-Derczynski et al. (2021))_ together
with _dynamic masking_ (see Section 3).

_Hardware._ We train DanT5 on a single machine with two AMD Epyc 7252 8-Core
CPUs, 128 GB of RAM, and four NVIDIA A100 GPUs with 40 GB of memory each. We
did not have NVLink bridges for this project.

##  6 Case Study Results

We trained two models: dant5-small, a 60M parameter model based on T5-small,
and dant5-large, a 770M parameter model based on T5-large. Both models trained
for 10 epochs, as described in Section 3. Training dant5-small took 91 hours,
while dant5-large trained in 508 hours.

Figure 1a indicates that dant5-large fit the training data better. The
training loss decreases up to the end, suggesting that the models are still
learning. The evaluation loss curve (shown in Figure 1b) closely tracks the
training loss, indicating that the models are not overfitting the data.

(a) Training loss

(b) Evaluation loss

Figure 1: Training and Evaluation loss for dant5-small (orange, 60M
parameters) and dant5-large (red, 770M parameters).

##  7 Conclusion

This paper presents a set of concrete recommendations to enable training LLMs
using modest research lab resources, in a reasonable amount of time. We
provide parameter values, software and hardware configuration strategies, and
techniques for enhancing the use of available data. These recommendations are
then demonstrated in the creation and release of “DanT5 ", the first T5 model
for Danish.

## Acknowledgements

This work was conducted with support from the Independent Danish Research
under the project VerifAI, 9131-00131B, and the Novo Nordisk Foundation
project ClinRead, NNF19OC0059138.

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# Pattern-recalling processes in quantum Hopfield networks far from saturation

Jun-ichi Inoue  Graduate School of Information Science and Technology,
Hokkaido University, N14-W9, Kita-ku, Sapporo 060-0814, Japan
<EMAIL_ADDRESS>
###### Abstract

As a mathematical model of associative memories, the Hopfield model was now
well-established and a lot of studies to reveal the pattern-recalling process
have been done from various different approaches. As well-known, a single
neuron is itself an uncertain, noisy unit with a finite unnegligible error in
the input-output relation. To model the situation artificially, a kind of
‘heat bath’ that surrounds neurons is introduced. The heat bath, which is a
source of noise, is specified by the ‘temperature’. Several studies concerning
the pattern-recalling processes of the Hopfield model governed by the Glauber-
dynamics at finite temperature were already reported. However, we might extend
the ‘thermal noise’ to the quantum-mechanical variant. In this paper, in terms
of the stochastic process of quantum-mechanical Markov chain Monte Carlo
method (the quantum MCMC), we analytically derive macroscopically
deterministic equations of order parameters such as ‘overlap’ in a quantum-
mechanical variant of the Hopfield neural networks (let us call quantum
Hopfield model or quantum Hopfield networks). For the case in which non-
extensive number $p$ of patterns are embedded via asymmetric Hebbian
connections, namely, $p/N\to 0$ for the number of neuron $N\to\infty$ (‘far
from saturation’), we evaluate the recalling processes for one of the built-in
patterns under the influence of quantum-mechanical noise.

##  1 Introduction

Basic concept of associative memories in artificial neural networks was
already proposed in early 70’ s by a Japanese engineer Kaoru Nakano [1].
Unfortunately, in that time, nobody interested in his model, however in 80’s,
J.J. Hopfield [2, 3] pointed out that there exists an energy function
(Lyapunov function) in the so-called associatron (the Nakano model) and the
system can be treated as a kind of spin glasses. After his study, a lot of
researchers who were working in the research field of condensed matter physics
picked the so-called Hopfield model up for their brand-new ‘target materials’.

Among these studies, a remarkable progress has been done by three theoretical
physicists Amit, Gutfreund and Sompolinsky [4] who clearly (mathematically)
defined the concept of storage capacity in the Hopfield model as a critical
point at which system undergoes phase transitions from ferromagnetic retrieval
to spin glass phases by utilizing the replica method. They also introduced a
noise to prevent the network from retrieving one of the built-in patterns as a
‘heat bath’ which surrounds the neurons. They draw the phase diagram which is
composed of three distinct phases, namely, ferromagnetic-retrieval,
paramagnetic and spin glass phases. These phase boundaries are specified by
two control parameters, namely, storage capacity and temperature of the heat
bath.

To evaluate the storage capacity of the Hopfield model without energy function
(for instance, Hopfield model having a non-monotonic input-output function
[5]), Shiino and Fukai [6] proposed the so-called Self-Consistent Signal-to-
Noise Analysis (SCSNA) which enables us to derive a couple of self-consistent
macroscopic equations of state by making use of the concept of the TAP
equations [3].

As we mentioned, these theoretical arguments are constructed for the case in
which each neuron is surrounded by a heat bath at finite temperature. In this
sense, the above studies revealed the robustness of the associative memories
against thermal noises in the artificial brain.

However, we might consider a different kind of such noises, that is, quantum-
mechanical noise. As such successful attempts, Ma and Gong [7], Nishimori and
Nonomura [8] independently introduced the quantum-mechanical noise to the
conventional Hopfield model by adding the transverse field to the classical
Hamiltonian (from now on, we shall refer the model as quantum Hopfield model).
Especially, Nishimori and Nonomura [8] investigated the structure of retrieval
phase diagrams by using of the standard replica method with the assistance of
the static approximation. Therefore, we might say that the equilibrium
properties of the Hopfield model were now well-understood and the methodology
to investigate the model was already established.

On the other hand, theory of the dynamics to evaluate the pattern-recalling
processes is not yet well-established. However, up to now, several powerful
approaches were proposed. For instance, Amari and Maginu [9] pointed out that
the relevant macroscopic quantities in the synchronous neuro-dynamics are the
overlap (the direction cosine) and the noise variance. They derived the update
equations with respect to these quantities. After their study, the so-called
Amari-Maginu theory was improved by taking into account the correlations in
the noise variances by Okada [10].

Whereas for the asynchronous dynamics, Coolen and his co-authors established a
general approach, so-called dynamical replica theory [11, 12]. They utilized
two assumptions, namely, equipartitioning in the sub-shells and self-averaging
of the intrinsic noise distribution to derive the deterministic flow equations
for relevant macroscopic quantities. However, there is no such theoretical
framework so far to investigate the pattern-recalling process of the quantum
Hopfield model systematically.

In this paper, we propose such a candidate of dynamical theory to deal with
the pattern-recalling processes in the quantum Hopfield model. We shall
consider the stochastic process of quantum Monte Carlo method which is applied
for the quantum Hopfield model and investigate the quantum neuro-dynamics
through the differential equations with respect to the macroscopic quantities
such as the overlap.

This paper is organized as follows. In the next section 2, we explain the
basics of the conventional Hopfield model and its generic properties. Then, we
categorize the model into two distinct classes according to the origin of
noises in artificial brain. The quantum Hopfield model is clearly defined. In
section 3, we explain the quantum Monte Carlo method based on the Suzuki-
Trotter decomposition [13] and consider the stochastic process in order to
investigate the pattern-recalling dynamics of the quantum Hopfield model. In
section 4, we derive the macroscopic deterministic flow of the overlap between
the neuronal state and one of the built-in patterns from the microscopic
master equation which describes the stochastic processes in the quantum Monte
Calro method for the quantum Hopfield model [14]. The general solution of the
dynamics is obtained under the so-called static approximation. In the next
section 5, we apply our general solution to a special case, namely, sequential
recalling the built-in two patterns via asymmetric Hebb connections. The
effect of quantum-mechanical noise is compared with that of the conventional
thermal noise. The last section is summary.

##  2 The Hopfield model

In this section, we briefly explain the basics of the conventional Hopfield
model. Then, we shall divide the model into two classes, namely, the Hopfield
model put in thermal noises (the model is referred to as classical systems)
and the same model in the quantum-mechanical noise (the model is referred to
as quantum systems). In following, we define each of the models explicitly.

###  2.1 The classical system

Let us consider the network having $N$-neurons. Each neuron $S_{i}$ takes two
states, namely, $S_{i}=+1$ (fire) and $S_{i}=-1$ (stationary). Neuronal states
are given by the set of variables $S_{i}$, that is,
$\bm{S}=(S_{1},\cdots,S_{N}),\,S_{i}\in\\{+1,-1\\}$. Each neuron is located on
a complete graph, namely, graph topology of the network is ‘fully-connected’.
The synaptic connection between arbitrary two neurons, say, $S_{i}$ and
$S_{j}$ is defined by the following Hebb rule:

$J_{ij}=\frac{1}{N}\sum_{\mu,\nu}\xi_{i}^{\mu}A_{\mu\nu}\xi_{j}^{\nu}$ (1)

where $\bm{\xi}^{\mu}=(\xi_{1},\cdots,\xi_{N}),\xi_{i}^{\mu}\in\\{+1,-1\\}$
denote the embedded patterns and each of them is specified by a label
$\mu=1,\cdots,P$. $A_{\mu\nu}$ denotes $(P\times P)$-size matrix and $P$
stands for the number of built-in patterns. We should keep in mind that there
exists an energy function (a Lyapunov function) in the system if the matrix
$A_{\mu\nu}$ is symmetric.

Then, the output of the neuron $i$, that is, $S_{i}$ is determined by the sign
of the local field $h_{i}$ as

$h_{i}=\sum_{\mu,\nu=1}^{p}\xi_{i}^{\mu}A_{\mu\nu}m^{\nu}+\frac{1}{N}\sum_{\mu^{{}^{\prime}},\nu^{{}^{\prime}}=p+1}^{P}\xi_{i}^{\mu^{{}^{\prime}}}A_{\mu^{{}^{\prime}}\nu^{{}^{\prime}}}\sum_{j}\xi_{j}^{\nu^{{}^{\prime}}}S_{j}$
(2)

where $A_{\mu\nu}$ and $A_{\mu^{{}^{\prime}}\nu^{{}^{\prime}}}$ are elements
of $p\times p,(P-p)\times(P-p)$-size matrices, respectively. We also defined
the overlap (the direction cosine) between the state of neurons $\bm{S}$ and
one of the built-in patterns $\bm{\xi}^{\nu}$ by

$\displaystyle m^{\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N}\,(\bm{S}\cdot\bm{\xi}^{\nu})=\frac{1}{N}\sum_{i}\xi_{i}^{\nu}S_{i}.$
(3)

Here we should notice that the Hamiltonian of the system is given by
$-\sum_{i}h_{i}S_{i}$. The first term appearing in the left hand side of
equation (2) is a contribution from $p\sim{\cal O}(1)$ what we call ‘condensed
patterns’, whereas the second term stands for the so-called ‘cross-talk
noise’. In this paper, we shall concentrate ourselves to the case in which the
second term is negligibly small in comparison with the first term, namely, the
case of $P=p\sim\mathcal{O}(1)$. In this sense, we can say that the network is
‘far from its saturation’.

###  2.2 The quantum system

To extend the classical system to the quantum-mechanical variant, we rewrite
the local field $h_{i}$ as follows.

$\bm{\phi}_{i}=\sum_{\mu,\nu=1}^{p}\xi_{i}^{\mu}A_{\mu\nu}\left(\frac{1}{N}\sum_{i}\xi_{i}^{\nu}\bm{\sigma}_{i}^{z}\right)$
(4)

where $\bm{\sigma}^{z}_{i}$ ($i=1,\cdots,N$) stands for the $z$-component of
the Pauli matrix. Thus, the Hamiltonian
$\bm{H}_{0}\equiv-\sum_{i}\bm{\phi}_{i}\bm{\sigma}_{i}^{z}$ is a diagonalized
$(2^{N}\times 2^{N})$-size matrix and the lowest eigenvalue is identical to
the ground state of the classical Hamiltonian $-\sum_{i}\phi_{i}S_{i}$
($S_{i}$ is an eigenvalue of the matrix $\bm{\sigma}_{i}^{z}$).

Then, we introduce quantum-mechanical noise into the Hopfield neural network
by adding the transverse field to the Hamiltonian as follows.

$\bm{H}=\bm{H}_{0}-\Gamma\sum_{i=1}^{N}\bm{\sigma}_{i}^{x}$ (5)

where $\bm{\sigma}_{i}^{x}$ is the $x$-component of the Pauli matrix and
transitions between eigenvectors of the classical Hamiltonian $\bm{H}_{0}$ are
induced due to the off-diagonal elements of the matrix $\bm{H}$ for
$\Gamma\neq 0$. In this paper, we mainly consider the system described by (5).

##  3 Quantum Monte Carlo method

The dynamics of the quantum model (5) follows Schr$\ddot{\rm o}$dinger
equation. Thus, we should solve it or investigate the time dependence of the
state $|\psi(t)\rangle$ by using the time-evolutionary operator $\mbox{\rm
e}^{-i\bm{H}\Delta t/\hbar}$ defined for infinitesimal time $\Delta t$ as

$\displaystyle|\psi(t+\Delta t)\rangle$ $\displaystyle=$
$\displaystyle\mbox{\rm e}^{-i\bm{H}\Delta t/\hbar}|\psi(t)\rangle.$ (6)

However, even if we carry it out numerically, it is very hard for us to do it
with reliable precision because $(2^{N}\times 2^{N})$-size Hamilton matrix
becomes huge for the number of neurons $N\gg 1$ as in a realistic brain.
Hence, here we use the quantum Monte Carlo method to simulate the quantum
system in our personal computer and consider the stochastic processes of
Glauber-type to discuss the pattern-recalling dynamics of the quantum Hopfield
model.

###  3.1 The Suzuki-Trotter decomposition

The difficulty to carry out algebraic calculations in the model system is due
to the non-commutation operators appearing in the Hamiltonian (5), namely,
$\bm{H}_{0},\bm{H}_{1}\equiv-\Gamma\sum_{i}\bm{\sigma}_{i}^{x}$. Thus, we use
the following Suzuki-Trotter decomposition [13] in order to deal with the
system as a classical spin system.

$\mbox{\rm tr}\,\mbox{\rm
e}^{\beta(\bm{H}_{0}+\bm{H}_{1})}=\lim_{M\to\infty}\mbox{\rm
tr}\left({\exp}\left(\frac{\beta\bm{H}_{0}}{M}\right){\exp}\left(\frac{\beta\bm{H}_{1}}{M}\right)\right)^{M}$
(7)

where $\beta$ denotes the ‘inverse temperature’ and $M$ is the number of the
Trotter slices, for which the limit $M\to\infty$ should be taken. Thus, one
can deal with $d$-dimensional quantum system as the corresponding
($d+1$)-dimensional classical system.

##  4 Derivation of the deterministic flows

In the previous section, we mentioned that we should simulate the quantum
Hopfield model by means of the quantum Monte Carlo method to reveal the
quantum neuro-dynamics through the time-dependence of the macroscopic
quantities such as the overlap. However, in general, it is also very difficult
to simulate the quantum-mechanical properties at the ground state by a
personal computer even for finite size systems ($N,M<\infty$).

With this fact in mind, in this section, we attempt to derive the macroscopic
flow equations from the microscopic master equation for the classical system
regarded as the quantum system in terms of the Suzuki-Trotter decomposition.
This approach is efficiently possible because the Hopfield model is a fully-
connected mean-field model such as the Sherrington-Kirkpatrick model [15] for
spin glasses and its equilibrium properties are completely determined by
several order parameters.

###  4.1 The master equation

After the Suzuki-Trotter decomposition (7), we obtain the local field for the
neuron $i$ located on the $k$-th Trotter slice as follows.

$\displaystyle\beta\phi_{i}(\bm{\sigma}_{k}:\sigma_{i}(k\pm 1))$
$\displaystyle=$
$\displaystyle\frac{\beta}{M}\sum_{\mu,\nu}\xi_{i}^{\nu}A_{\mu\nu}\left\\{\frac{1}{N}\sum_{j}\xi_{j}^{\nu}\sigma_{j}(k)\right\\}+\frac{B}{2}\left\\{\sigma_{i}(k-1)+\sigma_{i}(k+1)\right\\}$
(8)

where parameter $B$ is related to the amplitude of the transverse field (the
strength of the quantum-mechanical noise) $\Gamma$ by

$B=\frac{1}{2}\log\coth\left(\frac{\beta\Gamma}{M}\right).$ (9)

In the classical limit $\Gamma\to 0$, the parameter $B$ goes to infinity. For
the symmetric matrix $A_{\mu\nu}$, the Hamiltonian (scaled by $\beta$) of the
system is given by
$-\sum_{i}\beta\phi_{i}(\mbox{\boldmath$\sigma$}_{k}:\sigma(k\pm
1))\sigma_{i}(k)$.

Then, the transition probability which specifies the Glauber dynamics of the
system is given by
$w_{i}(\mbox{\boldmath$\sigma$}_{k})=(1/2)[1-\sigma_{i}(k)\tanh(\beta\phi_{i}(\mbox{\boldmath$\sigma$}_{k}:\sigma(k\pm
1)))]$. More explicitly, $w_{i}(\mbox{\boldmath$\sigma$}_{k})$ denotes the
probability that an arbitrary neuron $\sigma_{i}(k)$ changes its state as
$\sigma_{i}(k)\to-\sigma_{i}(k)$ within the time unit. Therefore, the
probability that the neuron $\sigma_{i}(k)$ takes $+1$ is obtained by setting
$\sigma_{i}(k)=-1$ in the above $w_{i}(\mbox{\boldmath$\sigma$}_{k})$ and we
immediately find $\sigma_{i}(k)=\sigma_{i}(k-1)=\sigma_{i}(k+1)$ with
probability $1$ in the limit of $B\to\infty$ which implies the classical limit
$\Gamma\to 0$.

Hence, the probability that a microscopic state including the $M$-Trotter
slices
$\\{\bm{\sigma}_{k}\\}\equiv(\bm{\sigma}_{1},\cdots,\bm{\sigma}_{M}),\bm{\sigma}_{k}\equiv(\sigma_{1}(k),\cdots,\sigma_{N}(k))$
obeys the following master equation:

$\displaystyle\frac{dp_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})}{dt}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{M}\sum_{i=1}^{N}[p_{t}(F_{i}^{(k)}(\mbox{\boldmath$\sigma$}_{k}))w_{i}(F_{i}^{(k)}(\mbox{\boldmath$\sigma$}_{k}))-p_{t}(\mbox{\boldmath$\sigma$}_{k})w_{i}(\mbox{\boldmath$\sigma$}_{k})]$
(10)

where $F_{i}^{(k)}(\cdot)$ denotes a single spin flip operator for neuron $i$
on the Trotter slice $k$ as $\sigma_{i}(k)\to-\sigma_{i}(k)$. When we pick up
the overlap between neuronal state $\bm{\sigma}_{k}$ and one of the built-in
patterns $\bm{\xi}^{\nu}$, namely,

$\displaystyle m_{k}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N}\,(\bm{\sigma}_{k}\cdot\bm{\xi}^{\nu})=\frac{1}{N}\sum_{i}\xi_{i}^{\nu}\sigma_{i}(k)$
(11)

as a relevant macroscopic quantity, the joint distribution of the set of the
overlaps $\\{m_{1},\cdots,m_{M}\\}$ at time $t$ is written in terms of the
probability for realizations of microscopic states
$p_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})$ at the same time $t$ as

$P_{t}(m_{1}^{\nu},\cdots,m_{M}^{\nu})=\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})\prod_{k=1}^{M}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))$
(12)

where we defined the sums by

$\displaystyle\sum_{\\{\bm{\sigma}_{k}\\}}(\cdots)$ $\displaystyle\equiv$
$\displaystyle\sum_{\bm{\sigma}_{1}}\cdots\sum_{\bm{\sigma}_{M}}(\cdots),\,\,\,\,\,\sum_{\bm{\sigma}_{k}}(\cdots)\equiv\sum_{\sigma_{1}(k)=\pm
1}\cdots\sum_{\sigma_{N}(k)=\pm 1}(\cdots).$ (13)

Taking the derivative of equation (12) with respect to $t$ and substituting
(10) into the result, we have the following differential equations for the
joint distribution

$\displaystyle\frac{dP_{t}(m_{1}^{\nu},\cdots,m_{M}^{\nu})}{dt}$
$\displaystyle=$ $\displaystyle\sum_{k}\frac{\partial}{\partial
m_{k}^{\nu}}\\{m_{k}^{\nu}P_{t}(m_{1}^{\nu},\cdots,m_{k}^{\nu},\cdots,m_{M}^{\nu})\\}$
(14) $\displaystyle-$ $\displaystyle\sum_{k}\frac{\partial}{\partial
m_{k}^{\nu}}{\Biggr{\\{}}P_{t}(m_{1}^{\nu},\cdots,m_{k}^{\nu},\cdots,m_{M}^{\nu})\int_{-\infty}^{\infty}D[\xi^{\nu}]d\xi^{\nu}$
$\displaystyle\times$
$\displaystyle\frac{\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})\xi^{\nu}\tanh[\beta\phi(k)]\prod_{k,i}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))}{\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})\prod_{k}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))}{\Biggr{\\}}}$
$\displaystyle\times$
$\displaystyle\delta(\sigma(k+1)-\sigma_{i}(k+1))\delta(\sigma(k-1)-\sigma_{i}(k-1))$

where we introduced several notations

$D[\xi^{\nu}]\equiv\frac{1}{N}\sum_{i}\delta(\xi^{\nu}-\xi_{i}^{\nu})$ (15)
$\beta\phi(k)\equiv\frac{\beta\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}}{M}m_{k}^{\nu}+\frac{B}{2}\sigma(k-1)+\frac{B}{2}\sigma(k+1)$
(16)

for simplicity.

Here we should notice that if the local field $\beta\phi(k)$ is independent of
the microscopic variable $\\{\bm{\sigma}_{k}\\}$, one can get around the
complicated expectation of the quantity $\tanh[\beta\phi(k)]$ over the time-
dependent Gibbs measurement which is defined in the sub-shell:
$\prod_{k}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))$. As
the result, only procedure we should carry out to get the deterministic flow
is to calculate the data average (the average over the built-in patterns).
However, unfortunately, we clearly find from equation (16) that the local
field depends on the $\\{\bm{\sigma}_{k}\\}$. To overcome the difficulty and
to carry out the calculation, we assume that the probability
$p_{t}(\\{\bm{\sigma}_{k}\\})$ of realizations for microscopic states during
the dynamics is independent of $t$, namely,

$\displaystyle p_{t}(\\{\bm{\sigma}_{k}\\})$ $\displaystyle=$ $\displaystyle
p(\\{\bm{\sigma}_{k}\\}).$ (17)

Then, our average over the time-dependent Gibbs measurement in the sub-shell
is rewritten as

$\displaystyle\frac{\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})\xi^{\nu}\tanh[\beta\phi(k)]\prod_{k,i}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))}{\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p_{t}(\\{\mbox{\boldmath$\sigma$}_{k}\\})\prod_{k}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))}$
(18) $\displaystyle\times$
$\displaystyle\delta(\sigma(k+1)-\sigma_{i}(k+1))\delta(\sigma(k-1)-\sigma_{i}(k-1))$
$\displaystyle\equiv$
$\displaystyle\langle\xi^{\nu}\tanh[\beta\phi(k)]\prod_{i}\delta(\sigma(k+1)-\sigma_{i}(k+1))\delta(\sigma(k-1)-\sigma_{i}(k-1))\rangle_{*}$

where $\langle\cdots\rangle_{*}$ stands for the average in the sub-shell
defined by $m_{k}^{\nu}=m_{k}^{\nu}(\bm{\sigma}_{k})\,(\forall_{k})$:

$\langle\cdots\rangle_{*}\equiv\frac{\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p(\\{\mbox{\boldmath$\sigma$}_{k}\\})(\cdots)\prod_{k}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))}{\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p(\\{\mbox{\boldmath$\sigma$}_{k}\\})\prod_{k}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))}$
(19)

If we notice that the Gibbs measurement in the sub-shell is rewritten as

$\displaystyle\sum_{\\{\mbox{\boldmath$\sigma$}_{k}\\}}p(\\{\mbox{\boldmath$\sigma$}_{k}\\})\prod_{k}\delta(m_{k}^{\nu}-m_{k}^{\nu}(\mbox{\boldmath$\sigma$}_{k}))$
$\displaystyle=$ $\displaystyle\mbox{\rm
tr}_{\\{\sigma\\}}{\exp}\left[\beta\sum_{l=1}^{M}\phi(l)\sigma(l)\right]$ (20)

($\mbox{\rm
tr}_{\\{\sigma\\}}(\cdots)\equiv\prod_{k}\sum_{\mbox{\boldmath$\sigma$}_{k}}(\cdots)$),
and the quantity

$\tanh\left[\beta\phi(k)\right]=\frac{\sum_{\sigma(k)=\pm
1}\sigma(k)\,{\exp}[\beta\phi(k)\sigma(k)]}{\sum_{\sigma(k)=\pm
1}{\exp}[\beta\phi(k)\sigma(k)]}$ (21)

is independent of $\sigma(k)$, the average appearing in (18) leads to

$\displaystyle\langle\xi^{\nu}\tanh[\beta\phi(k)]\prod_{i}\delta(\sigma(k\pm
1)-\sigma_{i}(k\pm 1))\rangle_{*}$ $\displaystyle=$
$\displaystyle\frac{\mbox{\rm
tr}_{\\{\sigma\\}}\xi^{\nu}\\{\frac{1}{M}\sum_{l=1}^{M}\sigma(l)\\}\exp[\beta\phi(k)\sigma(k)]}{{\rm
tr}_{\\{\sigma\\}}\exp[\beta\phi(k)\sigma(k)]}$ (22) $\displaystyle\equiv$
$\displaystyle\xi^{\nu}\langle\sigma\rangle_{path}^{(\xi^{\nu})}$

in the limit of $M\to\infty$. This is nothing but a path integral for the
effective single neuron problem in which the neuron updates its state along
the imaginary time axis: $\mbox{\rm
tr}_{\\{\sigma\\}}(\cdots)\equiv\sum_{\sigma(1)=\pm
1}\cdots\sum_{\sigma(M)=\pm 1}(\cdots)$ with weights
${\exp}[\beta\phi(k)\sigma(k)],\,(k=1,\cdots,M)$.

Then, the differential equation (14) leads to

$\displaystyle\frac{dP_{t}(m_{1}^{\nu},\cdots,m_{M}^{\nu})}{dt}$
$\displaystyle=$ $\displaystyle\sum_{k}\frac{\partial}{\partial
m_{k}^{\nu}}\\{m_{k}^{\nu}P_{t}(m_{1}^{\nu},\cdots,m_{k}^{\nu},\cdots,m_{M}^{\nu})\\}$
(23) $\displaystyle-$ $\displaystyle\sum_{k}\frac{\partial}{\partial
m_{k}^{\nu}}{\Biggr{\\{}}P_{t}(m_{1}^{\nu},\cdots,m_{k}^{\nu},\cdots,m_{M}^{\nu})\int_{-\infty}^{\infty}D[\xi^{\nu}]d\xi^{\nu}\xi^{\nu}\langle\sigma\rangle_{path}^{(\xi^{\nu})}{\Biggr{\\}}}.$

In order to derive the compact form of the differential equations with respect
to the overlaps, we substitute
$P_{t}(m_{1}^{\nu},\cdots,m_{M}^{\nu})=\prod_{k=1}^{M}\delta(m_{k}^{\nu}-m_{k}^{\nu}(t))$
into the above (23) and multiplying $m_{l}^{\nu}$ by both sides of the
equation and carrying out the integral with respect to $dm_{1}^{\nu}\cdots
dm_{M}^{\nu}$ by part, we have for $l=1,\cdots,M$ as

$\displaystyle\frac{dm_{l}^{\nu}}{dt}$ $\displaystyle=$ $\displaystyle-
m_{l}^{\nu}+\int_{-\infty}^{\infty}D[\xi^{\nu}]d\xi^{\nu}\xi^{\nu}\langle\sigma\rangle_{path}^{(\xi^{\nu})}.$
(24)

Here we should notice that the path integral
$\xi^{\nu}\langle\sigma\rangle_{path}^{(\xi^{\nu})}$ depends on the embedded
patterns $\bm{\xi}^{\nu}$. In the next subsection, we carry out the quenched
average explicitly under the so-called static approximation.

###  4.2 Static approximation

In order to obtain the final form of the deterministic flow, we assume that
macroscopic quantities such as the overlap are independent of the Trotter
slices $k$ during the dynamics. Namely, we must use the so-called static
approximation:

$\displaystyle m_{k}^{\nu}$ $\displaystyle=$ $\displaystyle
m^{\nu}\,(\forall_{k}).$ (25)

Under the static approximation, let us use the following inverse process of
the Suzuki-Trotter decomposition (7):

$\lim_{M\to\infty}Z_{M}=\mbox{\rm
tr}\,{\exp}\left[\beta\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}\sigma_{z}+\beta\Gamma\sigma_{x}\right]$
(26) $\displaystyle Z_{M}$ $\displaystyle\equiv$ $\displaystyle\mbox{\rm
tr}_{\\{\sigma\\}}{\exp}{\Biggr{[}}\frac{\beta\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}}{M}\sum_{k}\sigma(k)+B\sum_{k}\sigma(k)\sigma(k+1){\Biggr{]}}$
(27)

In our previous study [14], we numerically checked the validity of static
approximation by computer simulations and found that the approximation is
successfully valid for the pure-ferromagnetic system but it is deviated from
the good approximation for the disordered systems. The validity of the static
approximation was recently argued by Takahashi and Matsuda [16] from the
different perspective.

Then, one can calculate the path integral immediately as

$\displaystyle\langle\sigma\rangle_{path}^{(\xi^{\nu})}$ $\displaystyle=$
$\displaystyle\frac{\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}}{\sqrt{(\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu})^{2}+\Gamma^{2}}}\tanh\beta\sqrt{\left(\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}\right)^{2}+\Gamma^{2}}.$
(28)

Inserting this result into (24), we obtain

$\displaystyle\frac{dm^{\nu}}{dt}$ $\displaystyle=$
$\displaystyle-m^{\nu}+\mathbb{E}_{\bm{\xi}}{\Biggr{[}}\frac{\xi^{\nu}\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}}{\sqrt{(\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu})^{2}+\Gamma^{2}}}\tanh\beta\sqrt{\left(\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}\right)^{2}+\Gamma^{2}}{\Biggr{]}}$
(29)

where we should bear in mind that the empirical distribution $D[\xi^{\nu}]$ in
(24) was replaced by the built-in pattern distribution
$\mathcal{P}(\xi^{\nu})$ as

$\lim_{N\to\infty}\frac{1}{N}\sum_{i}\delta(\xi_{i}^{\nu}-\xi^{\nu})=\mathcal{P}(\xi^{\nu})$
(30)

in the limit of $N\to\infty$ and the average is now carried out explicitly as

$\displaystyle\int
D[\xi^{\nu}]d\xi^{\nu}(\cdots)=\int\mathcal{P}(\xi^{\nu})d\xi^{\nu}(\cdots)\equiv\mathbb{E}_{\bm{\xi}}[\cdots].$
(31)

Equation (29) is a general solution for the problem in this paper.

###  4.3 The classical and zero-temperature limits

It is easy for us to take the classical limit $\Gamma\to 0$ in the result
(29). Actually, we have immediately

$\frac{dm^{\nu}}{dt}=-m^{\nu}+\mathbb{E}_{\bm{\xi}}\left[\xi^{\nu}\tanh\left(\beta\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}\right)\right].$
(32)

The above equation is identical to the result by Coolen and Ruijgrok [17] who
considered the retrieval process of the conventional Hopfield model under
thermal noise.

We can also take the zero-temperature limit $\beta\to\infty$ in (29) as

$\frac{dm^{\nu}}{dt}=-m^{\nu}+\mathbb{E}_{\bm{\xi}}\left[\frac{\xi^{\nu}\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu}}{\sqrt{(\sum_{\mu\nu}\xi^{\mu}A_{\mu\nu}m^{\nu})^{2}+\Gamma^{2}}}\right].$
(33)

Thus, the equation (29) including the above two limiting cases is our general
solution for the neuro-dynamics of the quantum Hopfield model in which
$\mathcal{O}(1)$ patterns are embedded. Thus, we can discuss any kind of
situations for such pattern-recalling processes and the solution is always
derived from (29) explicitly.

##  5 Limit cycle solution for asymmetric connections

In this section, we discuss a special case of the general solution (29).
Namely, we investigate the pattern-recalling processes of the quantum Hopfield
model with asymmetric connections $\bm{A}\equiv\\{A_{\mu\nu}\\}$.

###  5.1 Result for two-patterns

Let us consider the case in which just only two patterns are embedded via the
following matrix:

$A=\left(\begin{array}[]{cc}1&-1\\\ 1&1\end{array}\right)$ (34)

Then, from the general solution (29), the differential equations with respect
to the two overlaps $m_{1}$ and $m_{2}$ are written as

$\displaystyle\frac{dm_{1}}{dt}$ $\displaystyle=$ $\displaystyle-
m_{1}+\frac{m_{1}}{\sqrt{(2m_{1})^{2}+\Gamma^{2}}}-\frac{m_{2}}{\sqrt{(2m_{2})^{2}+\Gamma^{2}}}$
$\displaystyle\frac{dm_{2}}{dt}$ $\displaystyle=$ $\displaystyle-
m_{2}+\frac{m_{1}}{\sqrt{(2m_{1})^{2}+\Gamma^{2}}}+\frac{m_{2}}{\sqrt{(2m_{2})^{2}+\Gamma^{2}}}.$

In Figure 1, we show the time evolutions of the overlaps $m_{1}$ and $m_{2}$
for the case of the amplitude $\Gamma=0.01$.

Figure 1: Time evolutions of $m_{1}$ and $m_{2}$ for the case of
$\Gamma=0.01$.

From this figure, we clearly find that the neuronal state evolves as $A\to
B\to\overline{A}\to\overline{B}\to A\to B\to\cdots$
($\overline{A},\overline{B}$ denote the ‘mirror images’ of $A$ and $B$,
respectively), namely, the network behaves as a limit cycle.

To compare the effects of thermal and quantum noises on the pattern-recalling
processes, we plot the trajectories $m_{1}$-$m_{2}$ for
$(T\equiv\beta^{-1},\Gamma)=(0,0.01),(0.01,0)$ (left panel),
$(T,\Gamma)=(0,0.8),(0.8,0)$ (right panel) in Figure 2.

Figure 2: Trajectories $m_{1}$-$m_{2}$ for $(T,\Gamma)=(0,0.01),(0.01,0)$
(left panel), $(T,\Gamma)=(0,0.8),(0.8,0)$ (right panel).

From these panels, we find that the limit cycles are getting collapsed as the
strength of the noise level is increasing for both thermal and quantum-
mechanical noises, and eventually the trajectories shrink to the origin
$(m_{1},m_{2})=(0,0)$ in the limit of $T,\Gamma\to\infty$.

##  6 Summary

In this paper, we considered the stochastic process of quantum Monte Carlo
method applied for the quantum Hopfield model and investigated the quantum
neuro-dynamics through the differential equations with respect to the
macroscopic quantities such as the overlap. By using the present approach, one
can evaluate the ‘inhomogeneous’ Markovian stochastic process of quantum Monte
Carlo method (in which amplitude $\Gamma$ is time-dependent [18, 19]) such as
quantum annealing [20, 21, 22, 23, 24, 25]. In the next step of the present
study, we are planning to extend this formulation to the probabilistic
information processing described by spin glasses including a peculiar type of
antiferromagnet [26]. We thank B.K. Chakrabarti, A.K. Chandra, P. Sen and S.
Dasgupta for fruitful discussion. We also thank local organizers of Statphys-
Kolkata VII for their warm hospitality. This work was financially supported by
Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of
Science, No. 22500195.

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