# Symmetric teleparallel cosmology with boundary corrections

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Departamento de
Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280
Antofagasta, Chile

###### Abstract

We investigate the geometrodynamical effects of introducing the boundary term
in symmetric teleparallel gravity. Specifically, we consider a homogeneous and
isotropic universe in $f\left(Q,B\right)$, where $Q$ is the non-metricity
scalar, and $B$ is the boundary term that relates the non-metricity and Ricci
scalars. For the connection in the coincidence gauge, we find that the field
equations are of fourth-order, and the fluid components introduced by the
boundary are attributed to a scalar field. In the coincidence gauge, the
cosmological field equations are equivalent to those of teleparallelism with a
boundary term. Nevertheless, for the connection defined in the non-coincidence
gauge, the geometrodynamical fluid consists of three scalar fields. We focus
on the special case of $f\left(Q,B\right)=Q+F\left(B\right)$ theory, and we
determine a new analytic cosmological solution that can explain the late-time
acceleration of the universe and provide a geometric mechanism for the
unification of dark energy with dark matter.

Symmetric teleparallel; non-metricity gravity; non-coincidence gauge; scalar
field description.

##  I Introduction

General Relativity is a well-tested theory gw1 ; however, it faces challenges
on cosmological scales ten1 ; ten1a ; ten2 . Recent cosmological observations
rr1 ; Teg ; Kowal ; Komatsu ; suzuki11  suggest that the universe is currently
experiencing an acceleration phase driven by an exotic matter source known as
dark energy jo1 ; jo . On the other hand, to address the homogeneity and
flatness problems, it has been proposed that the universe underwent an
inflationary epoch in its early stages driven by the inflaton field inf1 .

The nature and physical properties of the inflaton and dark energy remain
subjects of debate among cosmologists. Cosmologists have proposed different
models to explain cosmic acceleration, which can be broadly categorized into
two groups. In the first category, a matter source with negative pressure is
introduced into the field equations of General Relativity ratra ; de1 ; de2 ;
de3 ; de4 ; de5 . In the second category of models, cosmologists focus on
determining a new gravitational theory by modifying the gravitational Action
Integral de01 ; de02 ; de03 .

The consideration of quantum-gravitational effects in the one-loop
approximation in gravity df1 ; df2  leads to the introduction of the $R^{2}$
term in the gravitational Lagrangian. This quadratic theory of gravity qua1 ;
qua2 ; qua3  has been used as a mechanism to explain inflation star ; barc ;
stt1 . Indeed, when the $R^{2}$ term dominates, it leads to a de Sitter
expansion bar1 . The success of the $R^{2}$ term in describing acceleration
has given rise to a family of theories known as $f(R)$-gravity Buda . For a
comprehensive review, we recommend referring to Sotiriou .

The symmetric teleparallel $f(Q)$-theory of gravity has garnered the attention
of cosmologists lav2 ; lav3 . $f(Q)$-theory is defined within the framework of
symmetric teleparallel theory nester ; nester1 , where the physical space is
described by the metric tensor. However, the fundamental connection in this
theory is not the Levi-Civita connection but a symmetric and flat connection,
leading to the definition of a nonzero non-metricity scalar $Q$. When $f(Q)$
is linear, the gravitational theory becomes equivalent to General Relativity,
known as Symmetric Teleparallel General Relativity (STGR) nester . In STGR,
the Ricci scalar $R$ and the non-metricity scalar $Q$ differ only by a
boundary term $B$, which means that the variation of the boundary term is
neglected, resulting in equivalent field equations. There is a plethora of
studies in the literature on $f(Q)$-theory; for example, see fq1 ; fq2 ; fq3 ;
fq4 ; fq5 ; fq6 ; fq7 ; fq8 ; fq9 ; fq10 ; fq11 , and references therein.

In the process of constructing a gravitational theory, the boundary term $B$
has been utilized to modify the gravitational Action Integral. Inspired by
teleparallelism bah , the symmetric teleparallel theory with a boundary term,
expressed as $f(Q,B)$-gravity, was introduced in ftc1 ; ftc2 . It was
discovered that the introduction of the boundary term further modifies the
field equations of the gravitational model. In this study, we aim to
investigate the effects of the boundary term on cosmological dynamics.

Because the definition of the symmetric and flat connection is not unique,
there can be multiple different connections that describe the same
gravitational model. The corresponding non-metricity constants differ by a
boundary term, which means that they lead to the same equations in STGR.
However, this is not true in the case of a nonlinear Lagrangian. As a result,
the definition of the connection affects the boundary term, implying that
$f(Q,B)$-gravity depends on the definition of the connection.

Recently, in min , it was proven that $f(Q)$-theory of gravity admits a
minisuperspace description, and the theory is of fourth- or sixth-order,
depending on the connection used. The higher-order derivatives can be
attributed to a scalar field, and the gravitational Lagrangian can be written
as that of a higher-dimensional second-order dynamical system. This approach
is applied in this study in the context of $f(Q,B)$-gravity for a homogeneous
and isotropic geometry. The structure of the paper is as follows.

In Section II, we present the definition of symmetric teleparallel gravity and
its generalization, the $f(Q)$-theory, where $Q$ represents the non-metricity
scalar. The extension of symmetric teleparallel theory with a boundary term
$B$ is discussed in Section III. For the $f(Q,B)$-theory, we discuss when it
is equivalent to teleparallel $f(T,B_{T})$ theory and when it approaches the
limit of $f(R)$-gravity. In Section IV, we present the gravitational field
equations for a homogeneous and isotropic universe. Specifically, we employ
Lagrange multipliers to demonstrate that the field equations in
$f(Q,B)$-theory can be described within a minisuperspace framework. It’s
important to note that the definition of the symmetric and flat connection in
a FLRW universe is not unique. Therefore, we provide the field equations for
all four different families of connections. One novel aspect of the Lagrange
multiplier approach is the introduction of scalar fields to account for the
higher-order derivatives in the gravitational model. Consequently,
$f(Q,B)$-cosmology is a fourth-order gravitational theory, equivalent to
teleparallel $f(T,B_{T})$-cosmology, when the symmetric and flat connection is
defined in the coincidence gauge. However, when the connection is defined in
the non-coincidence gauge, the field equations become eighth-order and are
described by three scalar fields.

Furthermore, in Section V, we focus on the particular case of $f(Q,B)=Q+F(B)$
cosmology, where the gravitational Action Integral is modified by nonlinear
terms of the boundary. For this specific gravitational model, the order of the
gravitational field equations is reduced by two in the non-coincidence gauge.
Consequently, the geometric fluid is described by two scalar fields. In
Section VI, we present a new analytic solution for symmetric teleparallel
cosmology with a boundary term in the non-coincidence gauge. This new solution
is capable of describing the $\Lambda$CDM universe at the present time and
possesses a de Sitter attractor. Finally, in Section VII, we summarize our
findings and draw our conclusions.

##  II Symmetric teleparallel gravity

We examine a gravitational model described by the four-dimensional metric
tensor $g_{\mu\nu}$ and the covariant derivative $\nabla_{\lambda}~{}$ which
is defined using the generic connection $\Gamma_{\mu\nu}^{\kappa}\,$, such
that the autoparallels are defined as auto

$\frac{d^{2}x^{\mu}}{ds^{2}}+\Gamma_{\kappa\nu}^{\mu}\frac{dx^{\kappa}}{ds}\frac{dx^{\nu}}{ds}=0.$

Connection $\Gamma_{\mu\nu}^{\kappa}$ determines the nature of the geometry.

For the general connection we can define the the Riemann tensor

$R_{\;\lambda\mu\nu}^{\kappa}=\frac{\partial\Gamma_{\;\lambda\nu}^{\kappa}}{\partial
x^{\mu}}-\frac{\partial\Gamma_{\;\lambda\mu}^{\kappa}}{\partial
x^{\nu}}+\Gamma_{\;\lambda\nu}^{\sigma}\Gamma_{\;\mu\sigma}^{\kappa}-\Gamma_{\;\lambda\mu}^{\sigma}\Gamma_{\;\mu\sigma}^{\kappa},$
(1)

the torsion tensor

$\mathrm{T}_{\mu\nu}^{\lambda}=\Gamma_{\;\mu\nu}^{\lambda}-\Gamma_{\;\nu\mu}^{\lambda},$
(2)

and the non-metricity tensor Eisenhart

$Q_{\lambda\mu\nu}=\nabla_{\lambda}g_{\mu\nu}=\frac{\partial
g_{\mu\nu}}{\partial
x^{\lambda}}-\Gamma_{\;\lambda\mu}^{\sigma}g_{\sigma\nu}-\Gamma_{\;\lambda\nu}^{\sigma}g_{\mu\sigma}.$

In General Relativity, $\Gamma_{\mu\nu}^{\kappa}$ is recognized as the Levi-
Civita connection, denoted as $\tilde{\Gamma}{\mu\nu}^{\kappa}$. Consequently,
in this framework, $\mathrm{T}{\mu\nu}^{\lambda}=0$ and $Q_{\lambda\mu\nu}=0$.
Therefore, the primary scalar in General Relativity is the Ricci scalar $R$.

On the other hand, in the Teleparallel Equivalent of General Relativity (TEGR)
de03 , the connection $\Gamma_{\mu\nu}^{\kappa}$ is replaced by the
antisymmetric Weitzenb”ock connection, resulting in
$R_{;\lambda\mu\nu}^{\kappa}=0$ and $Q_{\lambda\mu\nu}=0$. In this context,
the torsion scalar $T$ takes on the role of the fundamental geometric object
in teleparallel gravity.

In the theory under consideration, which is STGR, $\Gamma_{\mu\nu}^{\kappa}$
possesses the property of being both flat and torsionless. This implies that
$R_{;\lambda\mu\nu}^{\kappa}=0$ and $\mathrm{T}{\mu\nu}^{\lambda}=0$.
Additionally, it inherits the symmetries of the metric tensor $g{\mu\nu}$.
Thus, the nonmetricity scalar $Q$ defined as nester

$Q=Q_{\lambda\mu\nu}P^{\lambda\mu\nu}$ (3)

is the fundamental geometric quantity of gravity.

Tensor $P^{\lambda\mu\nu}$ is defined as nester

$P_{\;\mu\nu}^{\lambda}=-\frac{1}{4}Q_{\;\mu\nu}^{\lambda}+\frac{1}{2}Q_{(\mu\phantom{\lambda}\nu)}^{\phantom{(\mu}\lambda\phantom{\nu)}}+\frac{1}{4}\left(Q^{\lambda}-\bar{Q}^{\lambda}\right)g_{\mu\nu}-\frac{1}{4}\delta_{\;(\mu}^{\lambda}Q_{\nu)},$
(4)

which is written with the help of the traces111Parenthesis in the indices
denote symmetrization, that is,
$A_{(\mu\nu)}=\frac{1}{2}\left(A_{\mu\nu}+A_{\nu\mu}\right)$; and
$\delta_{\;\nu}^{\mu}$ is the Kroncker delta.
$Q_{\mu}=Q_{\mu\nu}^{\phantom{\mu\nu}\nu}$ and
$\bar{Q}_{\mu}=Q_{\phantom{\nu}\mu\nu}^{\nu\phantom{\mu}\phantom{\mu}}$.

The Ricci scalar $R$ correspond to the Levi-Civita connection
$\tilde{\Gamma}_{\mu\nu}^{\kappa}$ of the metric tensor $g_{\mu\nu}$, and the
nonmetricity scalar $Q$ for a symmetric and flat connection
$\Gamma_{\mu\nu}^{\kappa}$ differ by a boundary term $B,$defined as
$B=R-Q~{}$.

The gravitational Action Integral of STGR reads nester

$\int d^{4}x\sqrt{-g}Q\simeq\int d^{4}x\sqrt{-g}R+\text{boundary terms.}$ (5)

from where it follows that STGR is dynamically equivalent to GR.

However, when nonlinear components of the non-metricity scalar $Q$ are
introduced in the gravitational Action, as in $f(Q)$-gravity, this equivalence
is lost. Nevertheless, the corresponding gravitational theory does not possess
any dynamical equivalence with General Relativity or its generalization,
$f(R)$-gravity.

The Action Integral in symmetric teleparallel $f(Q)$-gravity is defined lav2 ;
lav3

$S_{f\left(Q\right)}=\int d^{4}x\sqrt{-g}f(Q).$ (6)

The resulting field equations are

$\frac{2}{\sqrt{-g}}\nabla_{\lambda}\left(\sqrt{-g}f_{,Q}P_{\;\mu\nu}^{\lambda}\right)-\frac{1}{2}f(Q)g_{\mu\nu}+f_{,Q}\left(P_{\mu\rho\sigma}Q_{\nu}^{\;\rho\sigma}-2Q_{\rho\sigma\mu}P_{\phantom{\rho\sigma}\nu}^{\rho\sigma}\right)=0,$
(7)
$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}f_{,Q}P_{\phantom{\mu\nu}\sigma}^{\mu\nu}\right)=0.$
(8)

Equations (7) represent the modified Einstein field equations in
$f(Q)$-gravity, while equation (8) corresponds to the equation of motion for
the connection. . When equation (8) holds true for a specific connection at
all times, that connection is referred to as the “coincidence gauge”. However,
if the equation (8) is not always satisfied for a particular connection, then
that connection is defined within the so-called “non-coincidence gauge”as
discussed in lav3 .

##  III Symmetric teleparallel boundary gravity

Recently, a generalization of the $f(Q)$ theory was introduced ftc1 ; ftc2  by
incorporating a boundary term into the gravitational Action Integral. More
precisely, this extended framework includes the gravitational Action Integral
as follows

$S_{f\left(Q,B\right)}=\int d^{4}x\sqrt{-g}f(Q,B)~{}$ (9)

where $B=R-Q$.

In the notation of ftc2 , the boundary term is noted as $C$. This
gravitational theory is equivalent to General Relativity, when
$f\left(Q,B\right)$ is a linear function, that is,
$f\left(Q,B\right)=f_{1}Q+f_{2}B-2\Lambda$; the limit of $f\left(Q\right)$ is
recovered when $f\left(Q,B\right)=f\left(Q\right)+f_{2}B$; while the fourth-
order $f\left(R\right)$ theory of gravity is recovered when
$f\left(Q,B\right)=f\left(Q+B\right)$.

The gravitational field equations which correspond to the Action Integral (9)
are

$\displaystyle 0$
$\displaystyle=\frac{2}{\sqrt{-g}}\nabla_{\lambda}\left(\sqrt{-g}f_{,Q}P_{\;\mu\nu}^{\lambda}\right)-\frac{1}{2}f(Q,B)g_{\mu\nu}+f_{,Q}\left(P_{\mu\rho\sigma}Q_{\nu}^{\;\rho\sigma}-2Q_{\rho\sigma\mu}P_{\phantom{\rho\sigma}\nu}^{\rho\sigma}\right)$
$\displaystyle+\left(\frac{B}{2}g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu}g^{\kappa\lambda}\nabla_{\kappa}\nabla_{\lambda}-2P_{~{}~{}\mu\nu}^{\lambda}\nabla_{\lambda}\right)f_{,B}$
(10)

$f\left(Q,B\right)$ has been inspired by the teleparallel boundary gravity
$f\left(T,B_{T}\right)~{}$bah , where now $B_{T}$ is the boundary term relates
the Ricciscalar $R$ and the torsion scalar $T$ for the Weitzenböck connection
Weitzenb23 . In a similar way, $f\left(T,B_{T}\right)$ recovers
$f\left(R\right)$-gravity when
$f\left(T,B_{T}\right)=f\left(T+B_{T}\right)~{}$bah , while the limit of GR is
recovered when $f\left(T,B_{T}\right)=f_{1}T+f_{2}B_{T}-2\Lambda;$ and
$f\left(T\right)$-theory follows when
$f\left(T,B_{T}\right)=f\left(T\right)+f_{2}B_{T}$. There are many
astrophysical and cosmological applications of $f\left(T,B_{T}\right)$-theory
in the literature ftb1 ; ftb2 ; ftb3 ; ftb4 ; ftb5 ; ftb6 . From these results
it is clear that the boundary $B_{T}$ plays an important role in geometric
description of dark energy. Thus, the generalization if symmetric teleparallel
theory seems natural.

$f\left(T,B_{T}\right)$ theory of gravity is a fourth-order theory, similar to
$f\left(R\right)$-theory, while when
$f\left(T,B_{T}\right)=f\left(T\right)+f_{2}B_{T}$ the resulting field
equations are of second-order ftb7 . The order of symmetric teleparallel
$f\left(Q\right)$-theory depends on the connection which is used for the
definition of the nonmetricity scalar $Q$. For the connection defined in the
coincidence gauge $f\left(Q\right)$ is a second-order theory, while for a
connection in the non-coincidence gauge $f\left(Q\right)$-theory is a sixth-
order theory of gravity.

To comprehend the degrees of freedom within $f(Q,B)$-theory, we turn our
attention to a cosmological model representing an isotropic and homogeneous
universe. Employing the Lagrange multipliers method, we introduce scalar
fields that account for the dynamical degrees of freedom within the
$f(Q,B)$-theory. Consequently, our analysis reveals that the theory introduces
either one or three scalar fields.

##  IV Isotropic and Homogeneous Universe

The isotropic and homogeneous universe is described by the FLRW line element

$ds^{2}=-N(t)^{2}dt^{2}+a(t)^{2}\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right],$ (11)

in which $N\left(t\right)$ is the lapse function and $a\left(t\right)$ is the
scale factor denotes the radius of the universe. Hence,
$H=\frac{1}{N}\frac{\dot{a}}{a}$, where $\dot{a}=\frac{da}{dt}$ is the Hubble
function. Parameter $k$ is the spatial curvature, for $k=0$, the universe is
spatially flat, $k=+1$ corresponds to a closed FLRW geometry and $k=-1$
describes an open universe.

FLRW spacetime admits a sixth-dimensional Killing algebra consisted by the
vector fields

$\zeta_{1}=\sin\varphi\partial_{\theta}+\frac{\cos\varphi}{\tan\theta}\partial_{\varphi},\quad\zeta_{2}=-\cos\varphi\partial_{\theta}+\frac{\sin\varphi}{\tan\theta}\partial_{\varphi},\quad\zeta_{3}=-\partial_{\varphi}$
(12)
$\begin{split}\xi_{1}&=\sqrt{1-kr^{2}}\sin\theta\cos\varphi\partial_{r}+\frac{\sqrt{1-kr^{2}}}{r}\cos\theta\cos\varphi\partial_{\theta}-\frac{\sqrt{1-kr^{2}}}{r}\frac{\sin\varphi}{\sin\theta}\partial_{\varphi}\\\
\xi_{2}&=\sqrt{1-kr^{2}}\sin\theta\sin\varphi\partial_{r}+\frac{\sqrt{1-kr^{2}}}{r}\cos\theta\sin\varphi\partial_{\theta}+\frac{\sqrt{1-kr^{2}}}{r}\frac{\cos\varphi}{\sin\theta}\partial_{\varphi}\\\
\xi_{3}&=\sqrt{1-kr^{2}}\cos\theta\partial_{r}-\frac{\sqrt{1-kr^{2}}}{r}\sin\theta\partial_{\varphi}.\end{split}$
(13)

###  IV.1 Non-zero spatial curvature $k\neq 0$

For the FLRW with $k\neq 0$, there exist a unique connection defined in the
non-coincidence gauge with non-zero components Heis2 ; Zhao

$\begin{split}&\Gamma_{\;tr}^{r}=\Gamma_{\;rt}^{r}=\Gamma_{\;t\theta}^{\theta}=\Gamma_{\;\theta
t}^{\theta}=\Gamma_{\;t\varphi}^{\varphi}=\Gamma_{\;\varphi
t}^{\varphi}=-\frac{k}{\gamma(t)},\quad\Gamma_{\;rr}^{r}=\frac{kr}{1-kr^{2}},\\\
&\Gamma_{\;\theta\theta}^{r}=-r\left(1-kr^{2}\right),\quad\Gamma_{\;\varphi\varphi}^{r}=-r\sin^{2}(\theta)\left(1-kr^{2}\right)\quad\Gamma_{\;r\theta}^{\theta}=\Gamma_{\;\theta
r}^{\theta}=\Gamma_{\;r\varphi}^{\varphi}=\Gamma_{\;\varphi
r}^{\varphi}=\frac{1}{r},\\\
&\Gamma_{\;\varphi\varphi}^{\theta}=-\sin\theta\cos\theta,\quad\Gamma_{\;\theta\varphi}^{\varphi}=\Gamma_{\;\varphi\theta}^{\varphi}=\cot\theta,\end{split}$
(14)

and

$\Gamma_{\;tt}^{t}=-\frac{k+\dot{\gamma}(t)}{\gamma(t)},\quad\Gamma_{\;rr}^{t}=\frac{\gamma(t)}{1-kr^{2}}\quad\Gamma_{\;\theta\theta}^{t}=\gamma(t)r^{2},\quad\Gamma_{\;\varphi\varphi}^{t}=\gamma(t)r^{2}\sin^{2}(\theta).$

For this connection, the calculation of the non-metricity scalar is produced

$Q_{k}=-6\left(H^{2}-\frac{k}{a^{2}}\right)+\frac{3}{a^{3}N}\left(aN\gamma-k\frac{a^{3}}{\gamma
N}\right)^{\cdot}.$ (15)

From the Levi-Civita connection of spacetime (11) we calculate the Ricciscalar

$R_{k}=6\left(2H^{2}+\frac{k}{a^{2}}\right)+\frac{6}{N}\dot{H}.$ (16)

Consequently, the boundary term$~{}B=R-Q~{}$ftc2  is

$B_{k}=R_{k}-Q_{k}=3\left(6H^{2}+\frac{2}{N}\dot{H}-\frac{1}{a^{3}N}\left(aN\gamma-k\frac{a^{3}}{\gamma
N}\right)^{\cdot}\right).$ (17)

In order to derive the gravitational field equations we apply the mathematical
manipulation introduced in min  and we introduce the scalar field $\Psi$ such
that $\gamma=\frac{1}{\dot{\Psi}}$.

We introduce in (9) the Lagrange multipliers $\lambda_{1}$ and $\lambda_{2}$
such that

$S_{f\left(Q,B\right)}=\int
d^{4}x\sqrt{-g}\left(f(Q,B)~{}-\lambda_{1}\left(Q-Q_{k}\right)-\lambda_{2}\left(B-B_{k}\right)\right).$
(18)

Variation with respect to the non-metricity scalar $Q$ and the boundary term
$B$, gives $\lambda_{1}=f_{,Q}$ and $\lambda_{2}=f_{,B}$.

Thus, by replacing in (18) it follows

$S_{f\left(Q,B\right)}=\int
dt\left(Na^{3}\left(f-f_{,Q}-f_{,B}\right)+Na^{3}f_{,Q}Q_{k}+Na^{3}f_{,B}B_{k}\right).$
(19)

Hence, integration by parts gives

$\int dt\left(Na^{3}f_{,Q}Q_{k}\right)=\int
dt\left(-6Na^{3}f_{,Q}\left(H^{2}-\frac{k}{a^{2}}\right)+3f_{,Q}\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)^{\cdot}\right),$

$\int dt\left(Na^{3}f_{,B}B_{k}\right)=\int
dt\left(18Na^{3}f_{,B}H^{2}+6a^{3}f_{,B}\dot{H}-3f_{,B}\left(aN\gamma-k\frac{a^{3}}{\gamma
N}\right)^{\cdot}\right).$

Then

$S_{f\left(Q,B\right)}=\int
dt\left(\begin{array}[c]{c}Na^{3}\left(f-f_{,Q}-f_{,B}\right)-6\left(f_{,Q}-3f_{,B}\right)\left(Na^{3}H^{2}\right)\\\
+6Na^{3}f_{,Q}\frac{k}{a^{2}}+6f_{,B}a^{3}\dot{H}+3\left(f_{,Q}-f_{,B}\right)\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)^{\cdot}\end{array}\right).$

It follows

$\int
6f_{,B}a^{3}\dot{H}dt=\int\left(-18Nf_{,B}a^{3}H^{2}-6\dot{f}_{,B}a^{3}H\right)dt,$
$\int
3\left(f_{,Q}-f_{,B}\right)\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)^{\cdot}dt=\int-3\left(\dot{f}_{,Q}-\dot{f}_{,B}\right)\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)dt.$

The minisuperspace Lagrangian is

$\displaystyle L\left(N,a,\dot{a},Q,\dot{Q},B,\dot{B},\Psi,\dot{\Psi}\right)$
$\displaystyle=-\frac{6}{N}f_{,Q}a\dot{a}^{2}+6Naf_{,Q}k-\frac{6}{N}a^{2}\dot{f}_{,B}\dot{a}$
$\displaystyle-3\left(\dot{f}_{,Q}-\dot{f}_{,B}\right)\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)+Na^{3}\left(f-f_{,Q}-f_{,B}\right)$
(20)

or equivalently

$L\left(N,a,\dot{a},\phi,\dot{\phi},\zeta,\dot{\zeta},\Psi,\dot{\Psi}\right)=-\frac{6}{N}\phi
a\dot{a}^{2}+6Na\phi
k-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}-3\left(\dot{\phi}-\dot{\zeta}\right)\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)+Na^{3}V\left(\phi,\zeta\right).$
(21)

in which $\phi=f_{,Q}$, $\zeta=f_{,B}$ and
$V\left(\phi,\zeta\right)=\left(f-f_{,Q}-f_{,B}\right)$.

In the four-dimensional space ${a,\phi,\zeta,\Psi}$, we calculate
$\left|\frac{\partial^{2}L}{\partial q\partial
q}\right|=\frac{324a^{6}}{N^{4}\dot{\Psi}^{4}}(N^{2}+ka^{2}\dot{\Psi}^{2})^{2}\neq
0$. This implies that the field equations are of eighth-order and are
described by the three scalar fields $\phi,\zeta,\Psi$. The Lagrangian
function (21) represents a singular dynamical system in which variation with
respect to the lapse function $N$ yields the modified Friedmann equation.

Moreover, variation with respect to the dynamical variables
$\left\\{a,\phi,\zeta,\Psi\right\\}$ gives four second-order differential
equations.

For a constant lapse function, i.e. $N=1$, the cosmological field equation are

$0=3\phi
H^{2}+3\phi\frac{k}{a^{2}}+3H\dot{\zeta}-\frac{3}{2}\left(\dot{\phi}-\dot{\zeta}\right)\left(\frac{1}{a^{2}\dot{\Psi}}+k\dot{\Psi}\right)+\frac{1}{2}V\left(\phi,\zeta\right),$
(22)
$0=\dot{H}+\frac{3}{2}H^{2}+\frac{k}{2a^{2}}+\frac{V}{4\phi}+H\frac{\dot{\phi}}{\phi}+\frac{\dot{\zeta}-\dot{\phi}}{\phi}\left(\frac{1}{4a^{2}\dot{\Psi}}-\frac{3k}{4}\dot{\Psi}\right)+\frac{1}{2}\ddot{\zeta},$
(23)
$0=3\left(1+ka^{2}\dot{\Psi}^{2}\right)\frac{\ddot{\Psi}}{\Psi}-\left(3H+6k\dot{\Psi}-a^{2}\dot{\Psi}\left(6H^{2}+9kH\dot{\Psi}-V_{,\phi}\right)\right),$
(24)
$0=6\dot{H}+9H\left(2H+k\dot{\Psi}\right)+3k\ddot{\Psi}+V_{,\zeta}-\frac{3}{a^{2}\dot{\Psi}^{2}}\left(H\dot{\Psi}-\ddot{\Psi}\right),$
(25)
$0=3a\dot{\Psi}\left(1+a^{2}k\dot{\Psi}^{2}\right)\left(\ddot{\zeta}-\ddot{\phi}\right)+a\left(\dot{\zeta}-\dot{\phi}\right)\left(\dot{\Psi}H\left(1+3a^{2}k\dot{\Psi}^{2}\right)-2\ddot{\Psi}\right).$
(26)

The modified Friedmann’s equations (22), (23) can be written in the equivalent
form

$\displaystyle 3\left(H^{2}+\frac{k}{a^{2}}\right)$
$\displaystyle=G_{eff}\rho_{f\left(Q,B\right)}^{\Gamma_{k}},$ (27)
$\displaystyle-2\dot{H}-3H-\frac{k}{a^{2}}$
$\displaystyle=G_{eff}p_{f\left(Q,B\right)}^{\Gamma_{k}},$ (28)

in which $\rho_{f\left(Q,B\right)}$, $p_{f\left(Q,B\right)}$ are the
components for the geometric fluid which follows by the nonlinear
$f\left(Q,B\right)$-theory defined as

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{k}}$
$\displaystyle=-\left(3H\dot{\zeta}-\frac{3}{2}\left(\dot{\phi}-\dot{\zeta}\right)\left(\frac{1}{a^{2}\dot{\Psi}}+k\dot{\Psi}\right)+\frac{1}{2}V\left(\phi,\zeta\right)\right),$
(29) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{k}}$
$\displaystyle=\frac{V}{2}+2H\dot{\phi}+\frac{\dot{\zeta}-\dot{\phi}}{2}\left(\frac{1}{4a^{2}\dot{\Psi}}-\frac{3k}{4}\dot{\Psi}\right).$
(30)

and

$G_{eff}=\frac{1}{\phi}\text{.}$ (31)

We remark that scalar field $\phi$ is defined in the Jordan frame.

###  IV.2 Spatially flat case $k=0$

For the spatially flat case, $k=0$, it has been found that there exist three
families of connections. The common non-zero coefficients of the three
connections are

$\Gamma_{\theta\theta}^{r}=-r~{},~{}\Gamma_{\varphi\varphi}^{r}=-r\sin^{2}\theta$
$\Gamma_{\varphi\varphi}^{\theta}=-\sin\theta\cos\theta~{},~{}\Gamma_{\theta\varphi}^{\varphi}=\Gamma_{\varphi\theta}^{\varphi}=\cot\theta$
$\Gamma_{\;r\theta}^{\theta}=\Gamma_{\;\theta
r}^{\theta}=\Gamma_{\;r\varphi}^{\varphi}=\Gamma_{\;\varphi
r}^{\varphi}=\frac{1}{r}$

while the additional components for connections $\Gamma_{1},~{}\Gamma_{2}$ and
$\Gamma_{3}$ are Heis2 ; Zhao

$\Gamma_{1}:\Gamma_{\;tt}^{t}=\gamma(t),$
$\Gamma_{2}:\Gamma_{\;tt}^{t}=\frac{\dot{\gamma}(t)}{\gamma(t)}+\gamma(t),\quad\Gamma_{\;tr}^{r}=\Gamma_{\;rt}^{r}=\Gamma_{\;t\theta}^{\theta}=\Gamma_{\;\theta
t}^{\theta}=\Gamma_{\;t\varphi}^{\varphi}=\Gamma_{\;\varphi
t}^{\varphi}=\gamma(t),$

and

$\Gamma_{\;tt}^{t}=-\frac{\dot{\gamma}(t)}{\gamma(t)},\quad\Gamma_{\;rr}^{t}=\gamma(t),\quad\Gamma_{\;\theta\theta}^{t}=\gamma(t)r^{2},\quad\Gamma_{\;\varphi\varphi}^{t}=\gamma(t)r^{2}\sin^{2}\theta.$

The non-metricity scalars for each connection are calculated

$Q_{1}\left(\Gamma_{1}\right)=-6H^{2}$ (32)
$Q_{2}\left(\Gamma_{2}\right)=-6H^{2}+\frac{3}{a^{3}N}\left(\frac{a^{3}\gamma}{N}\right)^{\cdot}$
(33)

and

$Q_{3}\left(\Gamma_{3}\right)=-6H^{2}+\frac{3}{a^{3}N}\left(aN\gamma\right)^{\cdot}.$
(34)

Connection $\Gamma_{1}$ is the one defined in the coincidence gauge, while
connections $\Gamma_{2}$ and $\Gamma_{3}$ are defined in the non-coincidence
gauge. We observe that connection $\Gamma_{k}$ in the limit $k=0,$ reduces to
that of $\Gamma_{3}$, that is, $\Gamma_{k}\left(k\rightarrow
0\right)=\Gamma_{3}$ and $Q_{k}\left(k\rightarrow
0\right)=Q_{3}\left(\Gamma_{3}\right)$.

We proceed with the derivation of the minisuperspace Lagrangian and the field
equations for each family of connections in $f\left(Q,B\right)$-theory of
gravity.

####  IV.2.1 Connection $\Gamma_{1}$

For connection $\Gamma_{1}$, and the Ricciscalar (16) for the spatially flat
FLRW geometry we derive the boundary term

$B_{1}=B\left(\Gamma_{1}\right)=3\left(6H^{2}+\frac{2}{N}\dot{H}\right).$ (35)

For the coincidence gauge, scalars $Q_{1}$ and $B_{1}$ have the same
functional form with the torsion scalar $T$ and the boundary $B_{T}$. As a
result $f\left(Q,B\right)-$gravity for the connection $\Gamma_{1}$ is
equivalent with the teleparallel $f\left(T,B_{T}\right)$-gravity.

The Lagrangian of the field equations is

$L\left(N,a,\dot{a},Q,B,\dot{B}\right)=-\frac{6}{N}f_{,Q}a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{f}_{,B}\dot{a}+Na^{3}\left(f-f_{,Q}-f_{,B}\right)$
(36)

or equivalently in scalar field description

$L\left(N,a,\dot{a},\phi,\zeta,\dot{\zeta}\right)=-\frac{6}{N}\phi
a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}+Na^{3}V\left(\phi,\zeta\right).$
(37)

where similarly as before $\phi=f_{,Q}$, $\zeta=f_{,B}$ and
$V\left(\phi,\zeta\right)=\left(f-f_{,Q}-f_{,B}\right)$.

For the Lagrangian (37) in the three-dimensional space
$\left\\{a,\phi,\zeta\right\\}$ we derive $\left|\frac{\partial^{2}L}{\partial
q\partial q}\right|=0$. Hence, the field equations are of fourth-order.

We select the constant lapse function $N=1$, and we derive the field equations

$0=6H\left(\phi H+\dot{\zeta}\right)+V\left(\phi,\zeta\right),$ (38)
$0=2\phi\left(2\dot{H}+3H^{2}\right)+4H\dot{\phi}+2\ddot{\zeta}+V\left(\phi,\zeta\right),$
(39) $0=6H^{2}-V_{,\phi},$ (40) $0=\dot{H}+3H^{2}+\frac{1}{6}V_{,\zeta}.$ (41)

Modified Friedmann’s equations are written in the equivalent form

$\displaystyle 3H^{2}$
$\displaystyle=G_{eff}\rho_{f\left(Q,B\right)}^{\Gamma_{1}},$
$\displaystyle-2\dot{H}-3H^{2}$
$\displaystyle=G_{eff}p_{f\left(Q,B\right)}^{\Gamma_{1}},$

with energy density$~{}\rho_{f\left(Q,B\right)}^{\Gamma_{1}}$ and pressure
$p_{f\left(Q,B\right)}^{\Gamma_{1}}$ for the effective fluid

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{1}}$
$\displaystyle=-\left(3\dot{\zeta}H+\frac{1}{2}V\left(\phi,\zeta\right)\right),$
(42) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{1}}$
$\displaystyle=\left(2H\dot{\phi}+\ddot{\zeta}+\frac{1}{2}V\left(\phi,\zeta\right)\right).$
(43)

and $G_{eff}=\frac{1}{\phi}$.

####  IV.2.2 Connection $\Gamma_{2}$

For the non-coincidence connection $\Gamma_{2}$ it follows the boundary term

$B_{2}=B\left(\Gamma_{2}\right)=3\left(6H^{2}+\frac{2}{N}\dot{H}-\frac{3}{a^{3}N}\left(\frac{a^{3}\gamma}{N}\right)^{\cdot}\right).$
(44)

Hence, by introducing Lagrange multipliers as we did for the generic
connection $\Gamma_{k}$, we determine the point-like Lagrangian

$L\left(N,a,\dot{a},Q,\dot{Q},B,\dot{B},\psi,\dot{\psi}\right)=-\frac{6}{N}f_{,Q}a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{f}_{,B}\dot{a}+3\left(\dot{f}_{,Q}-\dot{f}_{,B}\right)\frac{a^{3}\dot{\psi}}{N}+Na^{3}\left(f\left(Q,B\right)-f_{,Q}-f_{,B}\right),$
(45)

or equivalently

$L\left(N,a,\dot{a},\phi,\dot{\phi},\zeta,\dot{\zeta},\psi,\dot{\psi}\right)=-\frac{6}{N}\phi
a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}+3\left(\dot{\phi}-\dot{\zeta}\right)\frac{a^{3}\dot{\psi}}{N}+Na^{3}V\left(\phi,\zeta\right),$
(46)

in which $\gamma=\dot{\psi}$, $\phi=f_{,Q}$, $\zeta=f_{,B}$ and
$V\left(\phi,\zeta\right)=\left(f-f_{,Q}-f_{,B}\right)$.

The field equations are of eight-order described by three scalar fields. For
$N=1$, the equations of motions for the scale factor and the three scalar
fields are

$0=6H\left(\phi
H+\dot{\zeta}\right)-3\left(\dot{\phi}-\dot{\zeta}\right)\dot{\psi}+NV\left(\phi,\zeta\right),$
(47)
$0=2\phi\left(2\dot{H}+3H^{2}\right)+4H\dot{\phi}-3\left(\dot{\phi}-\dot{\zeta}\right)\dot{\psi}+2\ddot{\zeta}+V\left(\phi,\zeta\right),$
(48) $0=3\ddot{\psi}+9H\dot{\psi}-6H^{2}+V_{,\phi}$ (49)
$0=6\dot{H}+18H^{2}-9H\dot{\psi}-3\ddot{\psi}+V_{,\zeta}$
$0=\ddot{\phi}-\ddot{\zeta}+3H\left(\dot{\phi}-\dot{\zeta}\right).$ (50)

Hence. the effective geometric fluid has the following energy density and
pressure components

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{2}}$
$\displaystyle=-\left(3\dot{\zeta}H+\frac{3}{2}\left(\dot{\phi}-\dot{\zeta}\right)\dot{\psi}+\frac{1}{2}V\left(\phi,\zeta\right)\right),$
(51) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{2}}$
$\displaystyle=\left(2H\dot{\phi}-\frac{3}{2}\left(\dot{\phi}-\dot{\zeta}\right)\dot{\psi}+\ddot{\zeta}+\frac{1}{2}V\left(\phi,\zeta\right)\right),$
(52)

such that

$\displaystyle 3H^{2}$
$\displaystyle=G_{eff}\rho_{f\left(Q,B\right)}^{\Gamma_{2}}$
$\displaystyle-2\dot{H}-3H^{2}$
$\displaystyle=G_{eff}p_{f\left(Q,B\right)}^{\Gamma_{2}},$

and $G_{eff}=\frac{1}{\phi}$.

####  IV.2.3 Connection $\Gamma_{3}$

We set $k=0$ in (21), thus, the minisuperspace Lagrangian function is

$L\left(N,a,\dot{a},\phi,\dot{\phi},\zeta,\dot{\zeta},\Psi,\dot{\Psi}\right)=-\frac{6}{N}\phi
a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}-3Na\frac{\left(\dot{\phi}-\dot{\zeta}\right)}{\dot{\Psi}}+Na^{3}V\left(\phi,\zeta\right).$
(53)

in which $\phi=f_{,Q}$, $\zeta=f_{,B}$ and
$V\left(\phi,\zeta\right)=\left(f-f_{,Q}-f_{,B}\right)$, and the field
equations are

$0=3\phi
H^{2}+3H\dot{\zeta}-\frac{3}{2}\left(\dot{\phi}-\dot{\zeta}\right)\left(\frac{1}{a^{2}\dot{\Psi}}\right)+\frac{1}{2}V\left(\phi,\zeta\right),$
(54)
$0=\dot{H}+\frac{3}{2}H^{2}+\frac{V}{4\phi}+H\frac{\dot{\phi}}{\phi}+\frac{\dot{\zeta}-\dot{\phi}}{\phi}\left(\frac{1}{4a^{2}\dot{\Psi}}\right)+\frac{1}{2}\ddot{\zeta},$
(55)
$0=3\frac{\ddot{\Psi}}{\Psi}-\left(3H-a^{2}\dot{\Psi}\left(6H^{2}+9kH\dot{\Psi}-V_{,\phi}\right)\right),$
(56)
$0=6\dot{H}+18H^{2}+V_{,\zeta}-\frac{3}{a^{2}\dot{\Psi}^{2}}\left(H\dot{\Psi}-\ddot{\Psi}\right),$
(57)
$0=3a\dot{\Psi}\left(\ddot{\zeta}-\ddot{\phi}\right)+a\left(\dot{\zeta}-\dot{\phi}\right)\left(\dot{\Psi}H-2\ddot{\Psi}\right).$
(58)

We conclude that the field equations are of eighth-order. Furthermore, the
geometric fluid has the energy density and pressure given by expressions (29)
and (30).

##  V $f\left(Q,B\right)=Q+F\left(B\right)$-Cosmology

Based on the above analysis, we observe that $f\left(Q,B\right)$-theory
introduces a varying parameter $G_{eff}=\frac{1}{\phi}$, $\phi=f_{,Q}$. Of
special interest are the $f\left(Q,B\right)=Q+F\left(B\right)$ models, in
which $G_{eff}=const$. and $\phi$ is always a constant. In this theory the
nonlinear terms of the Action Integral follow correspond to boundary
corrections. This approach has been previously explored in teleparallel
$f(T,B_{T})$-gravity, yielding numerous interesting results. Specifically,
$f(T,B_{T})=T+F(B_{T})$ has the potential to explain both the late and early-
time acceleration phases of the universe ftb5 .

In the $f(Q,B)=Q+F(B)$ theory, the number of field equations is reduced by
one, as $\phi$ is not a dynamic parameter but a constant, i.e., $\phi=1$.
Below, we provide the sets of field equations in $f(Q,B)=Q+F(B)$-theory for
the four different families of connections.

###  V.1 Connection $\Gamma_{1}$

For the connection $\Gamma_{1}$ defined in the coincidence gauge, the
minisuperspace Lagrangian is

$L\left(N,a,\dot{a},\phi,\zeta,\dot{\zeta}\right)=-\frac{6}{N}a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}+Na^{3}V\left(\zeta\right).$
(59)

where $\zeta=f_{,B}$ and $V\left(\zeta\right)=\left(f-f_{,B}\right)$.

Thus, for $N=1$, the field equations are

$\displaystyle 3H^{2}$ $\displaystyle=\rho_{f\left(Q,B\right)}^{\Gamma_{1}},$
(60) $\displaystyle-2\dot{H}-3H^{2}$
$\displaystyle=p_{f\left(Q,B\right)}^{\Gamma_{1}},$ (61)

in which

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{1}}$
$\displaystyle=-\left(3\dot{\zeta}H+\frac{1}{2}V\left(\zeta\right)\right),$
(62) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{1}}$
$\displaystyle=\left(\ddot{\zeta}+\frac{1}{2}V\left(\zeta\right)\right),$ (63)

and the scalar field $\zeta$ satisfies the equation of motion (41). Because
the theory is equivalent to the teleparallel
$f\left(T,B_{T}\right)=T+F\left(B_{T}\right)$ model, the results of the latter
theory are valid for the the symmetric teleparallel theory with boundary term.

###  V.2 Connection $\Gamma_{2}$

For connection $\Gamma_{2}$ defined in the non-coincidence gauge, the
minisuperspace Lagrangian reads

$L\left(N,a,\dot{a},\phi,\dot{\phi},\zeta,\dot{\zeta},\psi,\dot{\psi}\right)=-\frac{6}{N}a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}-3a^{3}\frac{\dot{\zeta}\dot{\psi}}{N}+Na^{3}V\left(\zeta\right),$
(64)

in which $\gamma=\dot{\psi}$, $\zeta=f_{,B}$ and
$V\left(\phi,\zeta\right)=\left(f-f_{,B}\right)$.

For $N=1$, modified Friedmann’s equations are

$\displaystyle 3H^{2}$ $\displaystyle=\rho_{f\left(Q,B\right)}^{\Gamma_{2}}$
(65) $\displaystyle-2\dot{H}-3H^{2}$
$\displaystyle=p_{f\left(Q,B\right)}^{\Gamma_{2}},$ (66)

with

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{2}}$
$\displaystyle=-\left(3\dot{\zeta}H-\frac{3}{2}\dot{\zeta}\dot{\psi}+\frac{1}{2}V\left(\zeta\right)\right),$
(67) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{2}}$
$\displaystyle=\left(\frac{3}{2}\dot{\zeta}\dot{\psi}+\ddot{\zeta}+\frac{1}{2}V\left(\zeta\right)\right),$
(68)

where the scalar fields $\zeta$ and $\psi$ satisfy the field equations

$\displaystyle 0$
$\displaystyle=6\dot{H}+18H^{2}-9H\dot{\psi}-3\ddot{\psi}+V_{,\zeta}$ (69)
$\displaystyle 0$ $\displaystyle=\ddot{\zeta}+3H\dot{\zeta}.$ (70)

###  V.3 Connection $\Gamma_{3}$

For the connection $\Gamma_{3}$ the minisuperspace Lagrangian becomes

$L\left(N,a,\dot{a},\phi,\dot{\phi},\zeta,\dot{\zeta},\Psi,\dot{\Psi}\right)=-\frac{6}{N}a\dot{a}^{2}-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}+3Na\frac{\dot{\zeta}}{\dot{\Psi}}+Na^{3}V\left(\zeta\right).$
(71)

with $\gamma=\frac{1}{\dot{\Psi}}$,, $\zeta=f_{,B}$ and
$V\left(\zeta\right)=\left(f-f_{,B}\right)$.

Furthermore, for $N=1$, the gravitational field equations are

$\displaystyle 3H^{2}$ $\displaystyle=\rho_{f\left(Q,B\right)}^{\Gamma_{3}}$
(72) $\displaystyle-2\dot{H}-3H^{2}$
$\displaystyle=p_{f\left(Q,B\right)}^{\Gamma_{3}},$ (73)

where

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{k}}$
$\displaystyle=-\left(3H\dot{\zeta}+\frac{3}{2}\frac{\dot{\zeta}}{a^{2}\dot{\Psi}}+\frac{1}{2}V\left(\zeta\right)\right),$
(74) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{k}}$
$\displaystyle=\frac{V}{2}+\frac{\dot{\zeta}}{8a^{2}\dot{\Psi}}.$ (75)
$0=3\phi
H^{2}+3H\dot{\zeta}-\frac{3}{2}\left(\dot{\phi}-\dot{\zeta}\right)\left(\frac{1}{a^{2}\dot{\Psi}}\right)+\frac{1}{2}V\left(\phi,\zeta\right),$
(76)

where the scalar fields $\zeta$ and $\Psi$ satisfy the equations of motion

$0=6\dot{H}+18H^{2}+V_{,\zeta}-\frac{3}{a^{2}\dot{\Psi}^{2}}\left(H\dot{\Psi}-\ddot{\Psi}\right),$
(77)
$0=3\dot{\Psi}\ddot{\zeta}+\dot{\zeta}\left(\dot{\Psi}H-2\ddot{\Psi}\right).$
(78)

###  V.4 Connection $\Gamma_{k}$

Finally, for the fourth connection $\Gamma_{k}$ where curvature is nonzero,
the minisuperspace Lagrangian is written

$L\left(N,a,\dot{a},\phi,\dot{\phi},\zeta,\dot{\zeta},\Psi,\dot{\Psi}\right)=-\frac{6}{N}a\dot{a}^{2}+6Nak-\frac{6}{N}a^{2}\dot{a}\dot{\zeta}+3\dot{\zeta}\left(a\frac{N}{\dot{\Psi}}-k\frac{a^{3}}{N}\dot{\Psi}\right)+Na^{3}V\left(\zeta\right).$
(79)

in which $\gamma=\frac{1}{\dot{\Psi}}$,, $\zeta=f_{,B}$ and
$V\left(\zeta\right)=\left(f-f_{,B}\right)$.

The gravitational field equations in the presence of curvature are

$\displaystyle 3\left(H^{2}+\frac{k}{a^{2}}\right)$
$\displaystyle=G_{eff}\rho_{f\left(Q,B\right)}^{\Gamma_{k}},$ (80)
$\displaystyle-2\dot{H}-3H-\frac{k}{a^{2}}$
$\displaystyle=G_{eff}p_{f\left(Q,B\right)}^{\Gamma_{k}}.$ (81)

The energy density and pressure components of the geometric fluid are defined
as

$\displaystyle\rho_{f\left(Q,B\right)}^{\Gamma_{k}}$
$\displaystyle=-\left(3H\dot{\zeta}+\frac{3}{2}\dot{\zeta}\left(\frac{1}{a^{2}\dot{\Psi}}+k\dot{\Psi}\right)+\frac{1}{2}V\left(\zeta\right)\right),$
(82) $\displaystyle p_{f\left(Q,B\right)}^{\Gamma_{k}}$
$\displaystyle=\frac{V}{2}+\frac{\dot{\zeta}}{2}\left(\frac{1}{4a^{2}\dot{\Psi}}-\frac{3k}{4}\dot{\Psi}\right).$
(83)

For the scalar fields we derive the equations of motion

$0=6\dot{H}+9H\left(2H+k\dot{\Psi}\right)+3k\ddot{\Psi}+V_{,\zeta}-\frac{3}{a^{2}\dot{\Psi}^{2}}\left(H\dot{\Psi}-\ddot{\Psi}\right),$
(84)
$0=3a\dot{\Psi}\left(1+a^{2}k\dot{\Psi}^{2}\right)\ddot{\zeta}+\dot{\zeta}\left(\dot{\Psi}H\left(1+3a^{2}k\dot{\Psi}^{2}\right)-2\ddot{\Psi}\right).$
(85)

From the above results we remark that in $f\left(Q,B\right)=Q+F\left(B\right)$
gravity the field equations are of fourth-order in the coincidence gauge and
of sixth-order in the non-coincidence gauge.

##  VI New solution in the non-coincidence gauge

We focus on the field equations (65)-(70) for the connection $\Gamma_{2}$ in
the case of $f\left(Q,B\right)=Q+F\left(B\right)$ theory. From equation (70)
we construct the conservation law

$I_{0}=a^{3}\dot{\zeta}.$ (86)

For the exponential potential $V\left(\zeta\right)=V_{0}e^{\lambda\zeta}$,
i.e. $F\left(B\right)=-\frac{B}{\lambda}\ln\left(-\frac{B}{\lambda
V_{0}}\right)-\frac{B}{\lambda}$; we are able to write the second conservation
law

$I_{1}=a^{2}\left(2\left(\lambda-3\right)\dot{a}+a\left(\lambda\dot{\zeta}-3\dot{\psi}\right)\right).$
(87)

In order to determine the latter conservation law we applied the method of
variational symmetries which has been widely used in modified theories of
gravity, for more details we refer the reader to nos1 ; nos2  and references
therein.

With the use of the two conservation laws and of the constraint equation (65),
the second Friedmann equation reads

$2a^{5}\ddot{a}-2a^{2}\dot{a}\left(I_{0}\lambda+a^{2}\dot{a}\right)-I_{0}\left(I_{0}\lambda-
I_{1}\right)=0.$ (88)

For $I_{0}\lambda-I_{1}=0$, we are able to determine the closed form solution

$a\left(t\right)=\left(a_{1}e^{a_{0}t}+I_{0}\lambda\right)^{\frac{1}{3}}.$
(89)

Consequently, the Hubble function is derived

$H^{2}\left(a\right)=\sqrt{\left(\frac{a_{0}}{3}\right)^{2}-\frac{2a_{0}I_{0}\lambda}{9}a^{-3}+\frac{\left(I_{0}\lambda\right)^{2}}{9}a^{-6}}.$
(90)

This analytic solution describes a universe with a cosmological constant, dark
matter and a stiff fluid. Indeed when
$\frac{\left(I_{0}\lambda\right)^{2}}{9}a^{-6}$ is neglected, i.e.
$\frac{\left(I_{0}\lambda\right)^{2}}{9}a^{-6}\rightarrow 0$ , the limit of
$\Lambda$CDM universe is recovered. Furthermore, we calculate the deceleration
parameter

$q\left(a\right)=2-\frac{3a^{3}}{a^{3}-I_{0}\lambda}\text{,}$

from where it follows that for large values of $a$,
$q\left(a\right)\rightarrow-1$, that is, the de Sitter universe is recovered.
Acceleration point is occurred when $2<\frac{3a^{3}}{a^{3}-I_{0}\lambda}$.

In order to solve equation (88) we apply the Lie symmetry analysis nos1 . We
find that equation (88) is invariant under the action of the elements of a
two-dimensional Lie algebra consisted by the vector fields
$X_{1}=\partial_{t}$ and $X_{2}=3t\partial_{t}+a\partial_{a}$. The application
of the Lie invariants of $X_{2}$ indicate the existence of the exact solution
$a\left(t\right)=\bar{a}_{0}t^{\frac{1}{3}}$ with constraint equation
$2a_{0}^{3}\left(a_{0}^{3}+\lambda I_{0}\right)=3I_{0}\left(I_{0}\lambda-
I_{1}\right)$.

On the other hand, the application of the Lie symmetry vector $X_{1}$ provides
the reduced equation

$\frac{dA}{da}=\frac{I_{0}\left(I_{1}-I_{0}\lambda\right)}{2a^{5}}A^{3}-\frac{I_{0}\lambda}{a^{3}}A^{2}-\frac{1}{a}A~{},~{}A\left(t\right)=\frac{1}{\dot{a}}~{}.$
(91)

This is an Abel type equation.

In Figs. 1 and 2 we present the qualitative evolution of the deceleration
parameter $q=-1-\frac{\dot{H}}{H^{2}}$ and of the function
$\gamma=\dot{\psi}$, as it is given after the numerical simulation of equation
(88). We observe that the de Sitter universe is a future attractor for the
cosmological model, and in the limit of the de Sitter universe connection
$\Gamma_{2}$ takes the form of $\Gamma_{1}$.

Figure 1: Qualitative evolution for the effective deceleration parameter
$q=-1-\frac{\dot{H}}{H^{2}}~{}$as it is given by the numerical solution of
equation (88). The plots are for different values of the free parameters.
Figure 2: Qualitative evolution for the function $\gamma=\dot{\psi}~{}$as it
is given by the numerical solution of equation (88). The plots are for
different values of the free parameters. We observe that as the solution
reaches the de Sitter universe connection $\Gamma_{2}$ reach to the limit of
$\Gamma_{1}$.

##  VII Conclusions

We conducted a study on FLRW cosmology within the framework of symmetric
teleparallel theory, considering boundary corrections in the gravitational
Lagrangian. In this context, and for the four different families of
connections, we determined the minisuperspace description of the field
equations. To achieve this, we introduced Lagrange multipliers, enabling us to
express the higher-order derivatives of the field equations in terms of scalar
fields. As a result, we were able to recast the cosmological field equations
into the equivalent form of multi-scalar field cosmology. In the case of
connections defined in the non-coincidence gauge, $f(Q,B)$-gravity is
characterized by three scalar fields. However, in the limiting case of the
$f(Q,B)=Q+F(B)$ model, the field equations are described by two scalar fields.
Conversely, for the connection defined in the coincidence gauge, there exists
only one scalar field, and the field equations are of fourth-order. It’s worth
noting that $f(Q)$-gravity introduces two scalar fields when the connection is
defined in the non-coincidence gauge.

This scalar field description and the derivation of the minisuperspace
representation are crucial for further investigations into the dynamic
evolution of physical variables within the theory. Moreover, the
minisuperspace Lagrangian can be employed to establish the Hamiltonian
formalism of the model and derive the Wheeler-DeWitt equation of quantum
cosmology.

To illustrate the practical application of the minisuperspace description, we
employed the method of variational symmetries and successfully determined an
integrable cosmological model. We were able to express the analytic solution
in terms of the Abel equation. This particular cosmological model not only
accounts for cosmic acceleration but also includes a dark matter component in
the Hubble function.

These results suggest that $f(Q,B)$-theory holds promise as a viable
cosmological framework. However, one notable implication is the significant
increase in degrees of freedom introduced by this theory. Therefore, the new
scalar fields must be capable of describing a wide range of cosmological
phenomena. In future work, we plan to investigate whether boundary correction
terms in symmetric teleparallel theory can resolve cosmological tensions and
whether the theory can provide explanations for eras in the cosmological
history beyond late-time acceleration.

Data Availability Statements: Data sharing is not applicable to this article
as no datasets were generated or analyzed during the current study.

###### Acknowledgements.

The author thanks the support of Vicerrectoría de Investigación y Desarrollo
Tecnológico (Vridt) at Universidad Católica del Norte through Núcleo de
Investigación Geometría Diferencial y Aplicaciones, Resolución Vridt No -
098/2022.

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# Quantum state tomography of qudits via Hong-Ou-Mandel interference

Yoshiaki Tsujimoto<EMAIL_ADDRESS>Advanced ICT Research Institute,
National Institute of Information and Communications Technology (NICT),
Koganei, Tokyo 184-8795, Japan  Rikizo Ikuta  Graduate School of Engineering
Science, Osaka University, Toyonaka, Osaka 560-8531, Japan  Center for Quantum
Information and Quantum Biology, Osaka University, Osaka 560-8531, Japan
Kentaro Wakui  Advanced ICT Research Institute, National Institute of
Information and Communications Technology (NICT), Koganei, Tokyo 184-8795,
Japan  Toshiki Kobayashi  Graduate School of Engineering Science, Osaka
University, Toyonaka, Osaka 560-8531, Japan  Center for Quantum Information
and Quantum Biology, Osaka University, Osaka 560-8531, Japan  Mikio Fujiwara
Advanced ICT Research Institute, National Institute of Information and
Communications Technology (NICT), Koganei, Tokyo 184-8795, Japan

###### Abstract

We propose a method to perform the quantum state tomography (QST) of an
$\mathit{n}$-partite qudit state embedded in single photons using the Hong-Ou-
Mandel (HOM) interference between the target state and probe state. This
method requires only passive beam splitters for the HOM interference and
removes all active optical devices in the target modes to control the
measurement bases needed in conventional QST. Hence, it is applicable to
various degree of freedom of the target state without altering the measurement
setup. Moreover, a faithful estimation is realized even with classical probe
light such as laser and thermal light. As a proof-of-principle, we performed
the experimental demonstration using a polarization qubit. Regardless of the
photon statistics of the probe light, the estimated results of state
reconstruction are as accurate as those verified by conventional QST.

Introduction.- The Hong-Ou-Mandel (HOM) interference is a bunching phenomenon
based on the bosonic nature of photons Hong1987 . When two photons that are
perfectly indistinguishable from each other are mixed using a half beam
splitter (HBS), they always bunch and exhibit the absence of coincidence
PhysRevLett.100.133601 . This feature has been used in a variety of quantum
information processing applications, including quantum computation Knill2001 ;
Kaltenbaek2010 ; Vigliar2021  and quantum communication Halder2007 ; Wang2015
; Chen2017 ; RevModPhys.83.33 ; Yu2020 . The decrease in coincidence reflects
the degree of indistinguishability, i.e., it provides information regarding
the probability of overlap of the input photons. Thus, HOM interference
enables the estimation of the unknown quantum state of a target photon under
test using a probe photon with a known quantum state. Although this concept
has been applied for estimating photon spectra PhysRevLett.99.123601 ;
Thiel:20 , spatial phase-amplitude structures Chrapkiewicz2016 , and photon
number distributions PhysRevA.79.020101 ; PhysRevLett.113.070403 ; Tiedau_2018
, there are no reported studies on the reconstruction of the complete density
operators of photonic qubits.

In this Letter, we propose and demonstrate the quantum state tomography (QST)
of photonic qubits based on HOM interference, which we refer to as HOM-QST.
This concept is illustrated in Fig. 1(a). While the conventional QST James2001
is performed by applying a set of projective measurements to the target
photon, HOM-QST is performed by interference of the target photon with a set
of probe photons. HOM-QST offers the following outstanding advantages over
conventional QST: (i) Only passive optical HBSs are added to the photonic
circuits, and there is no need for active control of the target photons under
test. This feature enables a low-loss and compact circuit design and is hence
ideal for integrated photonic quantum circuits Najafi2015 ; Schuck2016 ;
Gaggero:19 ; Gyger2021 , as shown in Fig. 1(b). (ii) This method can be used
to estimate the states of various degrees of freedom, such as polarization,
time, frequency, and orbital angular momentum, without changing the
measurement setup for the target photon. This is because the HBS allows for
wide operation over these degrees of freedom Harnchaiwat:20 ; Lingaraju:19 ;
Nagali2009 . To compensate for the above advantages, HOM-QST needs preparation
of the probe light; nonetheless, HOM-QST has the following useful feature:
(iii) Except for the degree of freedom of interest for QST, there is no need
to calibrate the probe photon information, such as the average photon number,
photon statistics, and mode mismatch with the target photon. This feature
mitigates the measurement cost of the proposed HOM-QST significantly.

Figure 1:  (a) Concept of HOM-QST. (b) Possible application of HOM-QST.

Theory.- We explain the HOM-QST of a $d$-dimensional qudit in the Hilbert
space spanned by $\\{|m\rangle\\}_{m=1,\ldots,d}$ embedded in a single photon.
The polarization qubit corresponds to the case of $d=2$. As shown in Fig.
1(a), we consider the case wherein the target light under test is injected
into the HBS from input port 1, and the probe light is injected from input
port 2. The typical HOM interferogram (HOM dip) based on the two-fold
coincidence probabilities between detectors ${\rm D}_{1}$ and ${\rm D}_{2}$ is
shown in the inset graph. The depth of the HOM dip reflects the degree of the
bosonic bunching effect due to the indistinguishability of the input photons.
Here, we consider the HOM experiment with the probe light in state $k$ which
is a superposition of the basis states. The depth of the dip is defined by
$\Delta_{k}:=P_{\infty}-P_{k}$, where $P_{\infty}$ and $P_{k}$ are the two-
fold coincidence probabilities with infinite and zero time delays of the input
photons, respectively.

We assume that the detection efficiencies of the detectors are not dependent
on $m$ and are significantly lower than 1, such that the detection
probabilities are proportional to the photon numbers in the detected modes.
When we measure the photons without resolving the state $m$, the coincidence
probability, $P$, between ${\rm D}_{1}$ and ${\rm D}_{2}$ in Fig. 1(a) is
given by

$P=\eta_{1}\eta_{2}\left\langle:\sum_{m}N_{1,m}^{\rm
out}\sum_{m^{\prime}}N_{2,m^{\prime}}^{\rm out}:\right\rangle,$ (1)

using the normal order representation, where $\hat{N}_{i,m}^{\rm out}$ is the
number operator of the photon in state $m$ at the output port $i$ of the HBS
and $\eta_{i}$ is a detection efficiency of ${\rm D}_{i}$.

When there is no mode mismatch between the target and probe lights, the
transformation of the HBS is given by $N_{1,m}^{\rm
out}\rightarrow(a_{1,m}^{\dagger}+a_{2,m}^{\dagger})(a_{1,m}+a_{2,m})/2$ and
$N_{2,m^{\prime}}^{\rm
out}\rightarrow(a_{1,m^{\prime}}^{\dagger}-a_{2,m^{\prime}}^{\dagger})(a_{1,m^{\prime}}-a_{2,m^{\prime}})/2$,
where $a_{i,m}$ is the annihilation operator of the photon in state $m$ at
input port $i$ of the HBS. Assuming that the target and probe lights have no
phase correlations and are statistically independent Tsujimoto2017 ;
Tsujimoto2018 , $P_{k}$ is calculated as follows:

$\displaystyle P_{k}$ $\displaystyle=$
$\displaystyle\frac{\eta_{12}}{4}\left(\left\langle:(N_{1}+N_{2,k})^{2}:\right\rangle-2\left\langle
N_{1,k}\right\rangle\left\langle N_{2,k}\right\rangle\right)$ (2)
$\displaystyle=$
$\displaystyle\frac{\eta_{12}}{4}\left(n_{s}^{2}g^{(2)}_{s}+n_{p}^{2}g^{(2)}_{p}+2n_{s}n_{p}(1-\rho_{k})\right),$
(3)

where $\eta_{12}:=\eta_{1}\eta_{2}$, $N_{i,m}:=a_{i,m}^{\dagger}a_{i,m}$,
$N_{i}:=\sum_{m}N_{i,m}$, and $\rho_{k}:=\left\langle
N_{1,k}\right\rangle/\left\langle N_{1}\right\rangle$. We also used
$\left\langle:N_{1}N_{2}:\right\rangle=\left\langle
N_{1}\right\rangle\left\langle N_{2}\right\rangle$, $\left\langle
N_{1}\right\rangle=n_{s}$, $\left\langle N_{2}\right\rangle=\left\langle
N_{2,k}\right\rangle=n_{p}$,
$\left\langle:N_{1}^{2}:\right\rangle=n_{s}^{2}g^{(2)}_{s}$, and
$\left\langle:N_{2}^{2}:\right\rangle=n_{p}^{2}g^{(2)}_{p}$, where $n_{s(p)}$
and $g^{(2)}_{s(p)}$ are an average photon number and intensity correlation
function of the target (probe) light. Here, $\rho_{k}$ is the probability of
the $k$-portion of the target photon that is to be estimated. Since
$P_{\infty}$ is equal to the first term in Eq. (2), $\Delta_{k}$ is given by

$\Delta_{k}=\frac{\eta_{12}}{2}\left\langle N_{1,k}\right\rangle\left\langle
N_{2,k}\right\rangle=\frac{\eta_{12}n_{s}n_{p}}{2}\rho_{k}.$ (4)

As the coefficient of $\rho_{k}$ is independent of $k$, the target state
$\hat{\rho}$ can be reconstructed from the HOM experiments with various
settings of $k$. Furthermore, we note that Eq. (4) holds even when dark counts
of the detectors are considered, assuming that the contribution of the dark
counts is the same between $P_{0}$ and $P_{\infty}$. When the target light is
a polarization qubit, $\rho_{k}$ corresponds to the probability obtained from
projective measurements, as follows:

$\rho_{k}=\langle k|\hat{\rho}|k\rangle,$ (5)

where $|k\rangle$ is the $k$-polarized single-photon state. By measuring
$\Delta_{k}$ for four independent polarization configurations, namely
horizontal ($H$), vertical ($V$), diagonal ($D$), and right-circular ($R$)
polarizations, we can reconstruct the target state. The feature of HOM-QST is
that no prior knowledge of the probe light is needed, such as $n_{p}$ and
$g^{(2)}_{p}$ in $P_{\infty}$ and $P_{k}$; moreover, there is no theoretical
requirement on these values.

Next, we consider a more practical situation wherein there exists a non-zero
mode mismatch between the target and probe photons even when both are in state
$k$. We describe the operators for the target and probe photons that are mode-
matched to the target by the same notations as those used above. Moreover, we
introduce operators for the probe light that express perfectly distinguishable
mode mismatches with the target photon by adding overlines to the related
operators. Eq. (1) is expanded to include the mode mismatch as

$P=\eta_{12}\left\langle:\sum_{m}(N_{1,m}^{\rm out}+\overline{N}_{1,m}^{\rm
out})\sum_{m^{\prime}}(N_{2,m^{\prime}}^{\rm
out}+\overline{N}_{2,m^{\prime}}^{\rm out}):\right\rangle.$

In input modes, $\left\langle N_{1}\right\rangle=n_{s}$ and
$\left\langle\overline{N}_{1}\right\rangle=0$ hold for the target photon, and
$\left\langle N_{2,k}+\overline{N}_{2,k}\right\rangle=n_{p}$ and
$\left\langle:(N_{2,k}+\overline{N}_{2,k})^{2}:\right\rangle=n_{p}^{2}g^{(2)}_{p}$
hold for the probe light. By defining the mode overlap probability between the
target and probe light photons as $M:=\left\langle
N_{2,k}\right\rangle/n_{p}$, the coincidence probability $P_{k}$ is described
as

$\displaystyle P_{k}$ $\displaystyle=$
$\displaystyle\frac{\eta_{12}}{4}\left(\left\langle:(N_{1}+N_{2,k}+\overline{N}_{2,k})^{2}:\right\rangle-2\left\langle
N_{1,k}\right\rangle\left\langle N_{2,k}\right\rangle\right)$ (7)
$\displaystyle=$
$\displaystyle\frac{\eta_{12}}{4}\left(n_{s}^{2}g^{(2)}_{s}+n_{p}^{2}g^{(2)}_{p}+2n_{s}n_{p}(1-M\rho_{k})\right).$

Note that the case of $M=1$ in Eq. (7) corresponds with $P_{k}$ in Eq. (3),
and the case of $M=0$ corresponds with

$\displaystyle P_{\infty}$ $\displaystyle=$
$\displaystyle\frac{\eta_{12}}{4}\left\langle:(N_{1}+N_{2,k}+\overline{N}_{2,k})^{2}:\right\rangle$
(8) $\displaystyle=$
$\displaystyle\frac{\eta_{12}}{4}\left(n_{s}^{2}g^{(2)}_{s}+n_{p}^{2}g^{(2)}_{p}+2n_{s}n_{p}\right).$
(9)

The depth $\Delta_{k}$ of the HOM dip is described as

$\displaystyle\Delta_{k}=\frac{\eta_{12}n_{s}n_{p}M}{2}\rho_{k}.$ (10)

Since $M$ is independent of $k$, we can reconstruct $\hat{\rho}$ without
estimating $M$.

Figure 2: The experimental setup. The setup consists of two stages, namely
preparation of the target photon and probe light as well as preparation of the
HOM interferometer for HOM-QST. In the first stage, the HSPS is generated as
the target photon by SPDC. The probe light is prepared in a HSPS or a thermal
state via SPDC with or without using heralding signals from D4, or in a
coherent state via difference frequency generation. SPF: short-pass filter;
DM: dichroic mirror; SHG: second harmonic generation; PPLN/W; periodically
poled lithium niobate waveguide; VHG: volume holographic grating.

Finally, we consider the BS imbalance and state dependency of the detection
efficiency. The transformations of the BS with transmittance $T$ and
reflectance $R$ are as follows: $N_{1,m}^{\rm
out}\rightarrow(\sqrt{T}a_{1,m}^{\dagger}+\sqrt{R}a_{2,m}^{\dagger})(\sqrt{T}a_{1,m}+\sqrt{R}a_{2,m})$
and $N_{2,m^{\prime}}^{\rm
out}\rightarrow(\sqrt{R}a_{1,m^{\prime}}^{\dagger}-\sqrt{T}a_{2,m^{\prime}}^{\dagger})(\sqrt{R}a_{1,m^{\prime}}-\sqrt{T}a_{2,m^{\prime}})$.
In addition, to treat the state-dependent detection efficiencies, $\eta_{1}$,
$\eta_{2}$, and $\eta_{12}=\eta_{1}\eta_{2}$ in Eq. (1) are replaced with
$\eta_{1m}$, $\eta_{2m^{\prime}}$, and
$\eta_{mm^{\prime}}=\eta_{1m}\eta_{2m^{\prime}}$, respectively. A
straightforward calculation is thus derived as

$\displaystyle P_{k}$
$\displaystyle=\sum_{m,m^{\prime}}\eta_{mm^{\prime}}\left\langle:(TN_{1,m}+RN_{2,m}+R\overline{N}_{2,m})\right.$
$\displaystyle\times\left.(RN_{1,m^{\prime}}+TN_{2,m^{\prime}}+T\overline{N}_{2,m^{\prime}}):\right\rangle-2\eta_{kk}TRn_{s}n_{p}M\rho_{k}.$

Then, the depth of the HOM dip is described as

$\displaystyle\Delta_{k}=2\eta_{kk}TRn_{s}n_{p}M\rho_{k},$ (11)

which shows that we can reconstruct $\hat{\rho}$ regardless of $T$ and $R$
using only the additional estimation of the relative values of the relative
detection efficiencies.

In conventional QST, an $n$-partite qudit state can be estimated by preparing
$n$ sets of the measurement configuration for the 1-qudit state. Similarly,
$n$ sets of HOM interferometers, each with the probe lights, enable performing
HOM-QST of the $n$-partite qudit state, including the entangled states.
Accordingly, the extension of Eq. (11) to the $n$-partite qudit states is
described in Supplemental material. We observe that, compared with the
conventional QST settings, the number of measurements in HOM-QST is the same
as scaling by $d^{2n}$, while the coincidence rate changes from
$\mathcal{O}(\eta^{n})$ to $\mathcal{O}(\eta^{2n}n_{p}^{n})$. To shorten the
measurement time, larger values of $n_{p}$ and $\eta$ are effective as long as
$n_{p}\eta\ll 1$ holds as a prerequisite for HOM-QST. In addition, the use of
an $n$-partitioned thermal state as the probe is effective. For the thermal
state, the $n$-th order cross-correlation function satisfies the condition
$g^{(n)}_{p}=n!$ barnett2002methods , which implies that the $n$-fold
coincidence probability is $n!$ times larger than that with the coherent-state
probe ($g^{(n)}_{p}=1$).

Target state | $H$ | $V$ | $D$ | $A$ | $R$ | $L$
---|---|---|---|---|---|---
Conventional method | 0.993(00) | 0.992(00) | 0.983(00) | 0.999(00) | 0.998(00) | 0.995(00)
HOM-based QST (a) | 0.950(20) | 0.970(16) | 0.999(06) | 0.994(08) | 0.942(19) | 0.957(17)
HOM-based QST (b) | 0.990(03) | 0.959(14) | 0.995(03) | 0.999(01) | 0.967(06) | 0.976(05)
HOM-based QST (c) | 0.996(01) | 0.969(02) | 0.996(02) | 0.994(02) | 0.975(03) | 0.980(02)
Table 1: The fidelities estimated by the conventional 1-qubit tomography
James2001  and the HOM-based QST using (a) the HSPSs, (b) thermal states and
(c) coherent states. The digits in parentheses represent the standard
deviation.

Figure 3: (a)–(d) Observed HOM dips between the $D$-polarized target photons
and (a) $D$-polarized HSPS, (b) $H$-polarized HSPS, (c) $D$-polarized thermal-
state, and (d) $D$-polarized coherent-state probes. In the HOM-QST experiment,
we measure only three data points rather than observing the HOM dip for each
polarization setting to estimate the coincidence count rates $C_{k}$ (purple
point) at the zero delay point and $C_{k\infty}$ (two red points) at large
delay points. The depth of the HOM dip is calculated as
$\tilde{\Delta}_{k}=C_{k\infty}-C_{k}$. The measurement times for the data
points in HOM-QST are $30\text{\,}\mathrm{s}$ each for (a), (b) HSPS and (c)
thermal-state probes and $10\text{\,}\mathrm{s}$ for the (d) coherent-state
probe. (e)–(h) Real parts of the reconstructed density matrices of the
$D$-polarized target states using (e) conventional 1-qubit QST and HOM-QST
with (f) HSPS, (g) thermal-state, and (h) coherent-state probes. Each value on
the vertical axis corresponds to the real part of the matrix element, e.g.,
the value for label HV corresponds to $\mathrm{Re}\langle
H|\hat{\rho}|V\rangle$. The error bars indicate the standard deviations
estimated using the Monte Carlo method under the assumption of Poisson
statistics for the photon counts.

Experiment.- We experimentally demonstrate HOM-QST for polarization qubits. We
prepared three types of probe lights, namely the heralded-single-photon-state
(HSPS) ($g^{(2)}_{p}\ll 1$), thermal-state ($g^{(2)}_{p}\sim 2$), and
coherent-state ($g^{(2)}_{p}\sim 1$) probes. The sketch of the experimental
setup is shown in Fig.2. A fiber-based mode-locked laser emitting
$1550\text{\,}\mathrm{n}\mathrm{m}$ optical pulses at a repetition rate of
$250\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ is used as a pump laser. This
setup consists of an electro-optical comb generator Wakui:20 ; Tsujimoto2021
followed by a high-speed pulse picker and a fiber amplifier. After the output
is passed to the free-space system, it is frequency-doubled via second
harmonic generation (SHG) using a type-0
$10\text{\,}\mathrm{m}\mathrm{m}$-long periodically poled lithium niobate
waveguide (PPLN/W). Then, the SHG light at $775\text{\,}\mathrm{n}\mathrm{m}$
is divided into two spatial modes using a half waveplate (HWP) and
polarization beam splitter (PBS) before being used to pump the subsequent
PPLN/Ws for spontaneous parametric down-conversion (SPDC). One pump beam is
used to prepare the target photon; this beam is coupled with a
$34\text{\,}\mathrm{m}\mathrm{m}$ long PPLN/W (PPLN/W 1) and generates a
$1535\text{\,}\mathrm{n}\mathrm{m}$ and $1565\text{\,}\mathrm{n}\mathrm{m}$
photon pair via SPDC. The $1535\text{\,}\mathrm{n}\mathrm{m}$ photon is
detected by $\mathrm{D}_{3}$ after frequency-filtering using a volume
holographic grating (VHG) of bandwidth
$58\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ to herald the
$1565\text{\,}\mathrm{n}\mathrm{m}$ photon. The HSP at
$1565\text{\,}\mathrm{n}\mathrm{m}$ is frequency-filtered using another VHG of
bandwidth $32\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, which is narrower than
the pump bandwidth of $75\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, and is used
as the target photon. The other pump at $775\text{\,}\mathrm{n}\mathrm{m}$ is
coupled with a second $34\text{\,}\mathrm{m}\mathrm{m}$-long PPLN/W (PPLN/W 2)
and used to prepare the 1565 nm probe light. The HSPS probe is prepared in a
similar manner to that of the target photon; the thermal-state probe is
prepared by discarding the heralding signal in the configuration, and the
coherent-state probe is prepared by difference frequency generation of the
$775\text{\,}\mathrm{n}\mathrm{m}$ light with the input of another
$1535\text{\,}\mathrm{n}\mathrm{m}$ continuous-wave (cw) laser beam as the
pump light. In the second stage of the HOM-QST, the target photon and probe
light are obtained from modes 1 and 2, respectively. Each photon is then
initialized to the $H,V,D,A,R$, and $L$ polarization states using a HWP and
quarter waveplate (QWP) after a PBS, where $A$ and $L$ are the anti-diagonal
and left-circular polarizations, respectively. HOM-QST is then performed by
mixing the target photon and probe light at the HBS, followed by the photon
detections at $\mathrm{D}_{1}$ and $\mathrm{D}_{2}$. All the detectors are
superconducting nanowire single-photon detectors (SSPDs) Miki:17 , whose
relative joint detection efficiencies, dependent on polarization, are
$\eta_{kk}/\eta_{HH}=$ 1, 1.39, 1.19, 0.77, 1.19, and 1.19 for $k=H,V,D,A,R$,
and $L$, respectively. The electrical signals from the SSPDs are fed to a
time-to-digital converter (not shown). The signal from $\mathrm{D}_{3}$ is
used as the start signal, and the signals from $\mathrm{D}_{4}$,
$\mathrm{D}_{1}$, and $\mathrm{D}_{2}$ are used as the stop signals. For the
HSPS probe, the four-fold coincidence events are used as successful events.
For the thermal-state and coherent-state probes, the three-fold coincidence
events among $\mathrm{D}_{3}$, $\mathrm{D}_{1}$, and $\mathrm{D}_{2}$ are
used.

As a preliminary experiment, we characterized the quantum state of the target
photon by the conventional 1-qubit QST James2001 . The QST was performed by
inserting a QWP, a HWP, and a PBS between the HBS and $\mathrm{D}_{1}$ in Fig.
2, while blocking the probe light. Using the diluted maximum-likelihood
algorithm Rehacek2007 , we reconstructed the density operator $\hat{\rho}$ of
the target photon. We show the real part of the reconstructed density matrix
of the $D$-polarized target photon in Fig. 3(e). We next evaluated the
fidelity $F_{k}:=\langle k|\hat{\rho}|k\rangle$ of the experimentally prepared
$k$-polarized target state $\hat{\rho}$ to $|k\rangle$ for
$k\in\\{H,V,D,A,R,L\\}$. These results are summarized in Table. 1. The
fidelity in each case is close to unity, thereby indicating that the target
states are almost ideal states.

We show the experimental results of HOM-QST using the HSPS probe. For example,
we show the HOM dip between the $D$-polarized target photon and
$D$($H$)-polarized probe photon in Fig. 3(a)((b)). The depth of each dip was
proportional to the probability $\Delta_{D(H)}$ in Eq. (11). We observed the
largest count rate as $\tilde{\Delta}_{D}=$6.1\text{\,}\mathrm{H}\mathrm{z}$$
and the smallest count rate as
$\tilde{\Delta}_{A}=$-0.7\text{\,}\mathrm{H}\mathrm{z}$$, indicating that the
target state is close to $|D\rangle$. The count rates for the other probe
states are provided in the Supplemental material. The negative value of the
depth, such as $\tilde{\Delta}_{A}$, occurs when the HOM dip is almost zero
because of statistical fluctuations of the photon counts and optical
misalignment. The latter effect could be removed by calibrating the dependency
of the photon count rate on the optical delay. In the demonstration, we
reconstructed the density operator based on two strategies to treat the
negative value as either zero or using only the results of the positive values
to avoid the negative values. For each case, the fidelity was estimated as
$F_{D}=0.999(06)$, and the difference between the two strategies was
negligibly small. Below, we show the results based on the latter strategy. The
real part of the reconstructed density matrix of the $D$-polarized target
photons is shown in Fig. 3(f). The fidelities are summarized in Table 1. Each
fidelity value is close to that estimated by 1-qubit QST, thus clearly showing
that HOM-QST provides reliable estimations of the input quantum states. We
show the reconstructed density matrices for the other target states in the
Supplemental material.

Finally, we show the experimental results of HOM-QST using the thermal-state
and coherent-state probes. The HOM dip with the $D$-polarized target photon
and $D$-polarized thermal (coherent)-state probe is shown in Fig. 3(c)((d)).
These HOM visibilities are smaller than that in the case of the HSPS probe in
Fig. 3(a) owing to the difference between $n_{p}$ and $g^{(2)}_{p}$ (see
Supplemental material). Nonetheless, the reconstructed density operators and
their fidelities are similar for all cases, as shown in Table 1 and Figs. 3(g)
and (f). These results elucidate our theoretical prediction that precise
estimations can be achieved by HOM interference regardless of the values of
$n_{p}$ and $g^{(2)}_{p}$.

Conclusion.- In conclusion, we proposed the QST of photonic qudits via HOM
interference and demonstrated it successfully using polarization qubits as the
proof-of-principle. The proposed method shifts the complexity of the
measurement system to the probe light preparation system, which then enables
us to construct the optical circuit with only passive elements. We
theoretically showed that the HOM-QST does not need information regarding the
probe light, such as intensity and photon statistics, and allows for
experimental imperfections, such as mode mismatch between the photons and use
of an unbalanced BS. We demonstrated the HOM-QST using the HSPS, thermal-
state, and coherent-state probes. The estimated fidelities were of a similar
order as those obtained via conventional QST, which demonstrates the precise
estimation of the density operator regardless of the intensity and photon
statistics of the probe light. In the demonstration, we applied HOM-QST to the
polarization qubits. However, our method exhibits its power for high-
dimensional systems in which the measurement setup tends to be complex. For
example, in the case of frequency qudits, a line-by-line waveform shaping
technique for frequency combs Cundiff2010  can be used to prepare the probe
light without requiring any nonlinear optical interactions in a measurement
setup, e.g. as shown in Refs. Kobayashi2016 ; Gil-Lopez:21 ; Kues2019 . Our
method serves as a new alternative to perform QST with a passive measurement
setup and is expected to be applicable to integrated quantum photonics.

Y.T. is supported by JSPS KAKENHI Grant No. JP20K14393. Y.T. and K.W. are
supported by CREST, JST JPMJCR1772. R.I. and T.K. are supported by CREST, JST
JPMJCR1671, and JSPS KAKENHI Grant Nos. JP21H04445 and JP20H01839.

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##  I Supplemental material

##  II HOM-QST of a composite system

We consider the HOM-QST of an $n$-qudit system. For convenience, we treat the
detection efficiencies of the photon detectors as constants in the derivation
and consider their state dependencies in the end. The setup for $n$-qudit
tomography consists of $n$ sets of HOM interferometers. We label the input
ports of the target and probe light for the $i$-th interferometer as $2i-1$
and $2i$, respectively. We define the set of serial numbers of the HOM
interferometers as $X_{n}:=\\{1,\ldots,n\\}$. We also define the power set of
$X_{n}$ as ${\mathfrak{P}}(X_{n})$ to separate the interferometers into two
groups: interferometers in $S\in{\mathfrak{P}}(X_{n})$ satisfy the condition
of zero time delay between the target and probe light, and those in
$X_{n}\backslash S$ satisfy the condition of large time delays. We denote the
list of probe states affecting the coincidence probabilities as
$\kappa(S):=k_{1},\ldots,k_{n}$, where $k_{i}$ is the probe state of the
$i$-th interferometer for $i\in S$, and $k_{i}=\emptyset$ for $i\in
X_{n}\backslash S$. Based on these definitions, the 2$n$-fold coincidence
probability $P_{\kappa(S)}^{(n)}$ is given by extending Eq. (LABEL:eq:Ck_MM)
in the main text as

$\displaystyle P_{\kappa(S)}^{(n)}$
$\displaystyle=\frac{\eta}{4^{n}}\left\langle:\left(\prod_{i\in
X_{n}\backslash
S}(N_{2i-1}+N_{2i,k_{i}}+\overline{N}_{2i,k_{i}})^{2}\right)\left(\prod_{j\in
S}(N_{2j-1}+N_{2j,k_{j}}+\overline{N}_{2j,k_{j}})^{2}-2N_{2j-1,k_{j}}N_{2j,k_{j}}\right):\right\rangle$
$\displaystyle=\sum_{S^{\prime}\in{\mathfrak{P}}(S)}(-1)^{|S^{\prime}|}\Delta_{\kappa(S^{\prime})}^{(n)},$
(S1)

where $\eta$ is the total detection efficiency from $2n$ photon detectors, and

$\displaystyle\Delta_{\kappa(S^{\prime})}^{(n)}$
$\displaystyle:=\frac{\eta}{4^{n}}\left\langle:\left(\prod_{i\in
X_{n}\backslash
S^{\prime}}(N_{2i-1}+N_{2i,k_{i}}+\overline{N}_{2i,k_{i}})^{2}\right)\left(2^{|S^{\prime}|}\prod_{j\in
S^{\prime}}N_{2j-1,k_{j}}N_{2j,k_{j}}\right):\right\rangle,$ (S2)

with $\prod_{i(j)\in\varnothing}\cdots:=1$.
$P_{\kappa(\varnothing)}^{(n)}=\Delta_{\kappa(\varnothing)}^{(n)}$ is the
probability obtained when the photons in all the interferometers are
distinguishable corresponding to $P_{\infty}$ in Eq. (8) in the main text for
$n=1$. We note that $\Delta_{\kappa(X_{n})}^{(n)}$, included in the observable
probability $P_{\kappa(X_{n})}^{(n)}$, contains information pertaining to

$\displaystyle\rho_{\kappa(X_{n})}$ $\displaystyle:=$
$\displaystyle\left\langle\prod_{j\in
X_{n}}N_{2j-1,k_{j}}\right\rangle/n_{s,n}$ (S3) $\displaystyle\sim$
$\displaystyle\langle k_{1}\cdots k_{n}|\hat{\rho}|k_{1}\cdots k_{n}\rangle,$
(S4)

which is to be determined, where $n_{s,n}:=\left\langle\prod_{j\in
X_{n}}N_{2j-1}\right\rangle$ is the $n$-th order cross-correlation of the
target state. $\Delta_{\kappa(X_{n})}^{(n)}$ is given by

$\displaystyle\Delta_{\kappa(X_{n})}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{\eta}{4^{n}}\left\langle:\left(2^{|X_{n}|}\prod_{j\in
X_{n}}N_{2j-1,k_{j}}N_{2j,k_{j}}\right):\right\rangle$ (S5) $\displaystyle=$
$\displaystyle\frac{\eta}{2^{n}}\left\langle\prod_{j\in
X_{n}}N_{2j-1,k_{j}}\right\rangle\left\langle\prod_{j\in
X_{n}}N_{2j,k_{j}}\right\rangle.$ (S6)

Similar to the expansion of Eq. (4) to Eq. (11) in the main text for $n=1$, by
considering the mode mismatch effects, state dependency of the detection
efficiency, and BS imbalance, Eq. (S6) can be expressed as

$\displaystyle\Delta_{\kappa(X_{n})}^{(n)}=2^{n}T^{n}R^{n}\eta_{X_{n}}n_{s,n}n_{p,n}M_{X_{n}}\rho_{\kappa(X_{n})},$
(S7)

where $n_{p,n}:=\left\langle\prod_{j\in
X_{n}}(N_{2j,k_{j}}+\overline{N}_{2j,k_{j}})\right\rangle=g_{p,X_{n}}^{(n)}\prod_{j\in
X_{n}}\left\langle N_{2j,k_{i}}+\overline{N}_{2j,k_{j}}\right\rangle$ is the
$n$-th order cross-correlation of the probe state, $\eta_{X_{n}}:=\prod_{j\in
X_{n}}\eta_{k_{j}}\eta_{k_{j}}$ is the total state-dependent detection
efficiency of the $2n$ photon detectors, $M_{X_{n}}:=\prod_{j\in X_{n}}M_{j}$
is the product of mode-matching between the target and probe light in the
$j$-th interferometer with $M_{j}:=\left\langle
N_{2j,k_{j}}\right\rangle/\left\langle
N_{2j,k_{j}}+\overline{N}_{2j,k_{j}}\right\rangle$, and
$g_{p,X_{n}}^{(n)}=n_{p,n}/\prod_{j\in X_{n}}\left\langle
N_{2j,k_{j}}+\overline{N}_{2j,k_{j}}\right\rangle=\left\langle\prod_{j\in
X_{n}}N_{2j,k_{j}}\right\rangle/\prod_{j\in X_{n}}\left\langle
N_{2j,k_{j}}\right\rangle$ is the $n$-th order cross-correlation function of
the probe light. The coefficient of $\rho_{\kappa(X_{n})}$ in the equation
does not depend on $\kappa(X_{n})$. Thus, we can reconstruct $\hat{\rho}$ from
$(1+d^{2})^{n}$ different experimental settings.

We present examples for the $n=1$ and $n=2$ cases, as follows. From Eqs. (S1)
and (S2), for $n=1$, we see that
$P_{\kappa(X_{1})}^{(1)}=P_{k_{1}}^{(1)}=\Delta_{\kappa(\varnothing)}^{(1)}-\Delta_{k_{1}}^{(1)}$,
where $\Delta_{\kappa(\varnothing)}^{(1)}=\Delta_{\emptyset}^{(1)}=P_{\infty}$
in Eq. (8) and $\Delta_{k_{1}}^{(1)}=\eta n_{s,1}n_{p,1}M_{1}\rho_{k_{1}}/2$
is equivalent to Eq. (10). For $n=2$, the four-fold coincidence probability
$P_{\kappa(X_{2})}^{(2)}=P_{k_{1},k_{2}}^{(2)}$ can be expressed as

$\displaystyle P_{k_{1},k_{2}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{\kappa(\varnothing)}^{(2)}-\Delta_{\kappa(\\{1\\})}^{(2)}-\Delta_{\kappa(\\{2\\})}^{(2)}+\Delta_{\kappa(\\{1,2\\})}^{(2)}$
(S8) $\displaystyle=$
$\displaystyle\Delta_{\emptyset,\emptyset}^{(2)}+(P_{k_{1},\emptyset}^{(2)}-\Delta_{\emptyset,\emptyset}^{(2)})+(P_{\emptyset,k_{2}}^{(2)}-\Delta_{\emptyset,\emptyset}^{(2)})+\Delta_{k_{1},k_{2}}^{(2)}$
(S9) $\displaystyle=$ $\displaystyle
P_{k_{1},\emptyset}^{(2)}+P_{\emptyset,k_{2}}^{(2)}-\Delta_{\emptyset,\emptyset}^{(2)}+4n_{s,2}n_{p,2}M_{1}M_{2}\rho_{k_{1},k_{2}}.$
(S10)

Thus, $\rho_{k_{1},k_{2}}$ is estimated by measuring $P_{k_{1},k_{2}}^{(2)}$,
$\Delta_{\emptyset,\emptyset}^{(2)}$, $P_{k_{1},\emptyset}^{(2)}$ and
$P_{\emptyset,k_{2}}^{(2)}$.

##  III HOM visibility

We consider the HOM visibility between the $k$-polarized target photon and
probe light. From Eqs. (9) and (10) in the main text, the HOM visibility is
expressed as

$V_{\rm th}:=\frac{\Delta_{k}}{P_{\infty}}=M\frac{1}{1+\frac{\zeta
g^{(2)}_{p}+1/\zeta g^{(2)}_{s}}{2}},$ (S11)

where $\zeta:=n_{p}/n_{s}$. In Figs. 3(a), (c) and (d), we show the HOM dips
between the $D$-polarized target photons and $D$-polarized probes. The
experimental HOM visibilities defined by $V_{\rm
ex}:=(C_{k\infty}-C_{k})/C_{k\infty}$ are given by $V_{\rm ex}=0.707(57)$,
$0.311(29)$, and $0.575(16)$ for the HSPS, thermal-state, and coherent-state
probes, respectively. In the HOM experiment with the HSPS probe, we
additionally measured the experimental parameters as $\zeta=0.694(01)$,
$g^{(2)}_{s}=0.211(02)$, and $g^{(2)}_{p}=0.355(03)$. By substituting these
values into Eq. (S11), and assuming $V_{\rm th}=V_{\rm ex}=0.707(57)$,
$M=0.901(73)$ was estimated. For the thermal-state probe, $\zeta=0.0646(06)$
was observed. Using the estimated value of $M$ and assuming that
$g^{(2)}_{p}=2$, we obtained $V_{\rm th}=0.334(27)$. In the coherent-state
probe experiment, $\zeta=0.382(01)$ and $g^{(2)}_{s}=0.191(03)$ were observed.
Assuming $g^{(2)}_{p}=1$, we obtained $V_{\rm th}=0.626(51)$. These values are
in good agreement with $V_{\rm ex}$ within the margin of errors, which
indicates that our model successfully explains the influence of $\zeta$ and
$g^{(2)}_{s(p)}$ on HOM visibility.

##  IV Count rates

We list all the count rate data for the conventional QST and HOM-QST in Table
S1.

|  | Measurement polarization $k$
---|---|---
QST method | Probe | $H$ | $V$ | $D$ | $A$ | $R$ | $L$
Conventional QST | - | 10885 | 8417 | 18969 | - | 9910 | -
| HSP | 2.3 | 2.5 | 6.1 | -0.7 | 2.7 | 1.9
HOM-QST | Thermal state | 11.5 | 14.6 | 32.2 | -1.9 | 13.1 | 13.8
| Coherent state | 145.3 | 166.3 | 391.7 | -84.8 | 197.7 | 129.1
Table S1:  Observed count rates per $1\text{\,}\mathrm{s}$ for the
$D$-polarized target photon. For conventional QST, single count rates are
listed. For HOM-QST, $\tilde{\Delta}_{k}/\eta_{kk}$ is listed. The measurement
polarization $k$ is chosen using the WPs and PBS for conventional QST and
probe light polarization for HOM-QST.

##  V Results of HOM-QST

We present all the density matrices reconstructed using 1-qubit tomography and
HOM-QST in Fig. S1.

Figure S1: Density matrices reconstructed using (a) 1-qubit tomography as well
as HOM-QST with (b) HSPS, (c) thermal-state, and (d) coherent-state probes.
The left and right parts of each figure correspond to the real and imaginary
parts of the reconstructed density matrix, respectively.