# Quantum annealing with symmetric subspaces

Takashi Imoto  Research Center for Emerging Computing Technologies, National
Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono,
Tsukuba, Ibaraki 305-8568, Japan.  Yuya Seki  Graduate School of Science and
Technology, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522,
Japan  Yuichiro Matsuzaki<EMAIL_ADDRESS>Research Center for
Emerging Computing Technologies, National Institute of Advanced Industrial
Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568,
Japan.  NEC-AIST Quantum Technology Cooperative Research Laboratory, National
Institute of Advanced Industrial Science and Technology (AIST), Tsukuba,
Ibaraki 305-8568, Japan

###### Abstract

Quantum annealing (QA) is a promising approach for not only solving
combinatorial optimization problems but also simulating quantum many-body
systems such as those in condensed matter physics. However, non-adiabatic
transitions constitute a key challenge in QA. The choice of the drive
Hamiltonian is known to affect the performance of QA because of the possible
suppression of non-adiabatic transitions. Here, we propose the use of a drive
Hamiltonian that preserves the symmetry of the problem Hamiltonian for more
efficient QA. Owing to our choice of the drive Hamiltonian, the solution is
searched in an appropriate symmetric subspace during QA. As non-adiabatic
transitions occur only inside the specific subspace, our approach can
potentially suppress unwanted non-adiabatic transitions. To evaluate the
performance of our scheme, we employ the XY model as the drive Hamiltonian in
order to find the ground state of problem Hamiltonians that commute with the
total magnetization along the $z$ axis. We find that our scheme outperforms
the conventional scheme in terms of the fidelity between the target ground
state and the states after QA.

††preprint: APS/123-QED

##  I Introduction

Quantum annealing (QA) is a technique for solving combinational optimization
problems Kadowaki and Nishimori (1998); Farhi  _et al._ (2000, 2001). The
solution of the problem is mapped to the ground state of the Ising
Hamiltonian, and it is searched in a large Hilbert space during QA. The
adiabatic theorem guarantees that the solution can be found as long as the
dynamics is adiabatic. Many other applications have been proposed for finding
the ground state of the Ising Hamiltonian using QA. One such application is
prime factorization Jiang  _et al._ (2018); Dridi and Alghassi (2017). The
shortest-vector problem is a promising candidate for post quantum
cryptography, and an approach for solving this problem using QA has been
proposed Joseph  _et al._ (2020, 2021). Another application of QA is machine
learning Adachi and Henderson (2015); Wilson  _et al._ (2021); Li  _et al._
(2019); Sasdelli and Chin (2021); Date and Potok (2020); Neven  _et al._
(2008, 2012); Willsch  _et al._ (2020); Winci  _et al._ (2020). In addition,
an application to clustering has been reported Kumar  _et al._ (2018);
Kurihara  _et al._ (2014). Further, a method of using QA to perform
calculations for topological data analysis (TDA) has been investigated Berwald
_et al._ (2018). Moreover, a method for mapping the graph coloring problem to
the Ising model for the QA format has also been proposed Kudo (2018, 2020).

Several applications of QA to the simulation of condensed matter have also
been reported. For example, simulation of the $\mathbb{Z}_{2}$ spin liquid has
been investigated using QA Zhou  _et al._ (2021). Further, the
Kosterlitz–Thouless (KT) phase transition has been experimentally observed
King  _et al._ (2018). In addition, the Shastry–Sutherland Ising Model has
been simulated Kairys  _et al._ (2020). Moreover, Harris et al. investigated
the spin-glass phase transition using QA Harris  _et al._ (2018).

However, it is difficult to prepare the exact ground state using QA with the
existing technology. The main causes of deterioration are non-adiabatic
transitions and decoherence. In the case of QA for Ising models to solve
combinatorial optimization problems, the probability of successfully obtaining
the ground state increases with the number of trials. For a given density
matrix $\rho=\sum_{n}p_{n}|E_{n}\rangle\langle E_{n}|$, where $|E_{n}\rangle$
denotes the eigenvector of the problem Hamiltonian and $p_{n}$ denotes the
population, single-shot measurements in the computational basis randomly
select $E_{n}$ and let us know this energy $E_{n}$. Hence, as long as the
ground state has a finite population, we can obtain the ground state energy if
we perform a large number of measurements. Meanwhile, if we consider a
Hamiltonian having non-diagonal matrix elements with the computational basis,
the measurements in the computational basis do not let us know the energy of
the Hamiltonian. Instead, we need to measure an observable of $H$ with Pauli
measurements. In this case, we obtain $\langle H\rangle=\sum_{n}p_{n}E_{n}$
for a given density matrix $\rho=\sum_{n}p_{n}|E_{n}\rangle\langle E_{n}|$ in
the limit of a large number of measurements, and this is certainly different
from the true ground state if the density matrix has a finite population of
any excited state. Therefore, the imperfection of the ground-state preparation
induces a crucial error in the estimation of the energy of the ground state of
the Hamiltonian having non-diagonal elements. The objective of this study is
to address this problem. Furthermore, there have been reported cases where
quantum annealing fails due to the existence of symmetryFrancis  _et al._
(2022); Imoto  _et al._ (2021a).

In this paper, we propose the use of a drive Hamiltonian that preserves the
symmetry of the problem Hamiltonian for more efficient QA, which can
potentially suppress non-adiabatic transitions. The relevant Hamiltonians in
condensed matter physics occasionally commute with the total magnetization
$S_{z}=\sum_{j}\sigma_{j}^{(z)}$, and we aim to obtain the ground state of
such a problem Hamiltonian. We employ the XY model (flip-flop interaction) as
the drive Hamiltonian. As both the problem and the drive Hamiltonian commute
with the total magnetization, the unitary evolution is restricted to the
symmetric subspace with a specific magnetization. As the total Hamiltonian
commutes with the total magnetization, the non-adiabatic transitions are
restricted to the same sector with respect to the total magnetization. Thus,
the non-adiabatic transitions can potentially be suppressed. Through numerical
simulations, we demonstrate that our method can suppress non-adiabatic
transitions for some problem Hamiltonians.

The remainder of this paper is organized as follows. Section II reviews QA.
Section III describes our method of using the flip-flop interaction for the
drive Hamiltonian. Section IV describes numerical simulations conducted to
evaluate the performance of our scheme, where the deformed spin star and a
random XXZ spin chain are selected as the problem Hamiltonians. Finally,
Section V concludes the paper.

##  II Quantum annealing

In this section, we review QA Kadowaki and Nishimori (1998); Farhi  _et al._
(2000, 2001). QA was proposed by Kadowaki and Nishimori as a method for
solving combinatorial optimization problems Kadowaki and Nishimori (1998). We
choose the magnetic transverse field

$\displaystyle H_{D}=-B\sum_{j=1}^{L}\sigma_{j}^{(x)}$ (1)

as the drive Hamiltonian, where $B$ denotes the amplitude of the magnetic
field. In the original idea of QA Kadowaki and Nishimori (1998), the problem
Hamiltonian was chosen as an Ising model mapped from a combinatorial
optimization problem to be solved. We set the ground state of this Hamiltonian
as

$\displaystyle\ket{\Phi(0)}=\ket{+\cdots+},$ (2)

where the state $\ket{+}$ denotes the eigenstate of $\sigma^{(x)}$ with an
eigenvalue of $+1$. The total Hamiltonian for QA is written as follows:

$\displaystyle H=\frac{t}{T}H_{P}+\biggl{(}1-\frac{t}{T}\biggr{)}H_{D},$ (3)

where $T$ is the annealing time, $H_{D}$ is the drive Hamiltonian, and $H_{P}$
is the problem Hamiltonian. First, we prepare the ground state of the drive
Hamiltonian. Second, the drive Hamiltonian is adiabatically changed into the
problem Hamiltonian according to Eq. (3). As long as this dynamics is
adiabatic, we can obtain the ground state of the problem Hamiltonian, which is
guaranteed by the adiabatic theorem Morita and Nishimori (2008).

Non-adiabatic transitions and decoherence are the main causes of performance
degradation of QA. Considerable effort has been devoted toward suppressing
such non-adiabatic transitions and decoherence. It is reported that a non-
stoquastic Hamiltonian with off-diagonal matrix elements improves the
performance of QA for some problem Hamiltonians Seki and Nishimori (2012,
2015); Susa  _et al._ (2022). Susa et al. suggested that the accuracy of QA
can be improved using an inhomogeneous drive Hamiltonian for specific cases
Susa  _et al._ (2018a, b). Further, it has been shown that Ramsey interference
can be used to suppress non-adiabatic transitions Matsuzaki  _et al._ (2021).
Other methods have been reported for suppressing non-adiabatic transitions and
decoherence using variational techniques Susa and Nishimori (2021); Matsuura
_et al._ (2021); Imoto  _et al._ (2021b). Error correction of QA has been
investigated for suppressing decoherence Pudenz  _et al._ (2014). In addition,
the use of decoherence-free subspaces for QA has been proposed Albash and
Lidar (2015); Suzuki and Nakazato (2020). Moreover, spin-lock techniques have
been theoretically proposed to use long-lived qubits for QA Chen  _et al._
(2011); Nakahara (2013); Matsuzaki  _et al._ (2020).

Experimental demonstrations of QA have also been reported. D-Wave Systems Inc.
realized QA machines composed of thousands of qubits by using superconducting
flux qubits Johnson  _et al._ (2011). Many experimental demonstrations of QA
using such devices have been reported, such as machine learning, graph
coloring problems, and magnetic material simulation Kudo (2018); Adachi and
Henderson (2015); Hu  _et al._ (2019); Kudo (2020); Harris  _et al._ (2018).

##  III Method

In this section, we describe our proposal of QA with symmetric subspaces. In
general, when a Hamiltonian has symmetry, it commutes with an observable that
is a conserved quantity. In this case, the Hamiltonian is block-diagonalized,
and the eigenstates of the Hamiltonian are divided into subspaces
characterized by the conserved quantity. When the drive Hamiltonian and
problem Hamiltonian have the same conserved quantities, any non-adiabatic
transitions during QA will occur inside the same subspace, which is in stark
contrast to the conventional scheme where non-adiabatic transitions can occur
in any subspace. Owing to such a restriction on non-adiabatic transitions, our
scheme can potentially suppress non-adiabatic transitions.

We employ the one-dimensional XY model as the drive Hamiltonian. The one-
dimensional XY model is defined by

$\displaystyle
H=g\sum_{j=1}^{L}\bigl{(}\sigma_{j}^{(+)}\sigma_{j+1}^{(-)}+\sigma_{j}^{(-)}\sigma_{j+1}^{(+)}\bigr{)},\
\ \sigma_{L+1}^{(k)}=\sigma_{1}^{(k)},$ (4)

where $g$ denotes the coupling strength, $L$ is the number of sites,
$\sigma_{j}^{(\alpha)}(\alpha=x,y,z)$ are the Pauli matrices defined on the
$j$-th site, and $\sigma_{j}^{(\pm)}=\sigma_{j}^{(x)}\pm i\sigma_{j}^{(y)}$.
Further, $S_{z}$ is a conserved quantity of the XY model. This model can be
mapped to the free fermion system by the Jordan–Wigner transformation; hence,
the ground-state energy of this Hamiltonian can be easily calculated using a
classical computer. Therefore, when we try to prepare the ground state of this
Hamiltonian (by using thermalization, for example), we can experimentally
verify how close the state is to the true ground state by measuring the
expectation value and variance of Imoto  _et al._ (2021c). It is worth
mentioning that, although the XY model was employed as the drive Hamiltonian
in previous studies Kudo (2018, 2020), the purpose was to add a proper
constraint to solve graph coloring problems. To the best of our knowledge, our
study is the first to use the XY model as the drive Hamiltonian for the
suppression of non-adiabatic transitions.

We choose a problem Hamiltonian that commutes with $S_{z}$. This guarantees
that the non-adiabatic transitions can occur only inside the symmetric
subspace. Some Hamiltonians used in condensed matter physics also commute with
$S_{z}$.

From the Lieb–Schultz–Mattis theorem Lieb  _et al._ (1961), it is known that
the energy gap between the ground state and the first excited state of the XY
model is $O(1/L)$ if we fix the value of $g$. Thus, as the number of qubits
increases, this energy gap becomes smaller, which can make it difficult to
realize adiabatic dynamics. However, as we are going to optimize the amplitude
of the drive Hamiltonian (as explained below), the problem of the small energy
gap can be solved by setting an appropriate value of $g$.

##  IV Numerical results

This section describes numerical simulations performed to compare our scheme
with the conventional scheme. We employ the
Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation for the numerical
simulations to account for the decoherence. For the drive Hamiltonian, the
transverse magnetic field Hamiltonian described in Eq. (1) is used in the
conventional scheme, while we employ the one-dimensional XY model in Eq. (4).
Moreover, we choose a deformed spin star model and a random XXZ spin chain as
the problem Hamiltonians. We define the estimation error as
$|E^{(\rm{true})}_{g}-E^{(\rm{QA})}_{g}|$, where $E^{(\rm{true})}_{g}$ denotes
the true ground state energy and $E^{(\rm{QA})}_{g}$ denotes the energy
obtained by QA. For fair comparison, we optimize the values of $B$ and g to
minimize the expectation value of the problem Hamiltonian after QA (or
equivalently, to minimize the estimation error).

We introduce the GKSL master equation to consider the decoherence Gorini  _et
al._ (1976); Lindblad (1976). The GKSL master equation is given by

$\displaystyle\frac{d\rho(t)}{dt}=-i[H(t),\rho(t)]+\sum_{n}\gamma[\sigma^{(k)}_{n}\rho(t)\sigma^{(k)}_{n}-\rho(t)],$
(5)

where $\sigma^{(k)}_{j}(k=x,y,z)$ denotes the Lindblad operators acting on
site $j$, $\gamma$ denotes the decoherence rate, and $\rho(t)$ is the density
matrix of the quantum state at time $t$. We solve the GKSL master equation
using Qutip Johansson ; Johansson  _et al._ (2012). Throughout this paper, we
choose $\sigma^{(z)}$ as the Lindblad operator.

###  IV.1 Deformed spin star model

(a) Energy spectrum of QA for the deformed spin star model with the drive
Hamiltonian of the transverse field.

(b) Energy spectrum of QA for the deformed spin star model with the drive
Hamiltonian of the XY model.

Figure 1: Energy spectrum during QA plotted against time $t$ to prepare the
ground state of the deformed spin star model. (a) The transverse field is
chose. (b) The XY model is chosen.

In this subsection, we consider the deformed spin star model as the problem
Hamiltonian. The Hamiltonian of the deformed spin star model is given by

$\displaystyle H$
$\displaystyle=\omega\hat{\sigma}_{0}^{z}+\omega_{1}\hat{J}^{z}+J(\hat{\sigma}_{0}^{+}\hat{J}^{-}+\hat{\sigma}_{0}^{-}\hat{J}^{+}),$
(6)

where $\hat{J}^{+}\equiv\sum_{j=1}^{L}e^{2\pi\frac{j}{L}}\sigma_{j}^{+}$ and
$\hat{J}^{-}\equiv\sum_{j=1}^{L}e^{-2\pi\frac{j}{L}}\sigma_{j}^{-}$. This kind
of model has been investigated, because this describes a dynamics of a hybrid
system composed of a superconducting flux qubit and nitrogen-vacancy centers
in a diamond Marcos  _et al._ (2010); Twamley and Barrett (2010); Zhu  _et
al._ (2011, 2014); Matsuzaki  _et al._ (2015); Cai  _et al._ (2015).

The energy spectra of the annealing Hamiltonians for the conventional scheme
and our scheme are plotted in Fig. 1. It is worth mentioning that the
performance of QA depends on the amplitude of the drive Hamiltonian, i.e., $B$
and $g$. Hence, we sweep these parameters and compare the performance of our
scheme with that of the conventional one when we optimize the amplitude of the
drive Hamiltonian. We performed the numerical simulations with decoherence
rate of $\gamma=2.5\times 10^{-5}$ GHz. In the case, we choose the optimal
amplitude of the drive Hamiltonian at each time. We can see that the
estimation error of our scheme is just three times smaller than that of the
conventional scheme, as shown in Fig 2. We find that the optimal annealing
time of our scheme, which provides the smallest estimation error, is much
shorter than that of the other scheme. This implies that, as the effect of the
non-adiabatic transitions is significantly reduced, we can shorten the
annealing time and thus suppress the effect of the decoherence.

Figure 2:  Estimation error $|E^{(\rm{true})}_{g}-E^{(\rm{QA})}_{g}|$ for the
deformed spin star model plotted against the annealing time, where
$E^{(\rm{true})}_{g}$ denotes the true ground state energy and
$E^{(\rm{QA})}_{g}$ denotes the energy obtained by QA. The blue (orange) line
denotes the case when we choose the XY model (transverse field) with an
amplitude of $g$ ($B$) as the drive Hamiltonian. Here, for a optimal amplitude
with a decoherence rate of $\gamma=2.5\times 10^{-5}$ GHz. In addition, we
choose the model parameters $\omega=\omega_{1}=0.5$ GHz, $J=5$ GHz, and
$L=3$(i.e. the total spin number is $4$).

###  IV.2 Random XXZ spin chain

In this subsection, we consider a random XXZ spin chain as the problem
Hamiltonian. The Hamiltonian of the random XXZ spin chain is given by

$\displaystyle
H=\sum_{j=1}^{L-1}J_{j}\biggl{(}\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{y}_{j}\sigma^{y}_{j+1}+\Delta\sigma^{z}_{j}\sigma^{z}_{j+1}\biggr{)},$
(7)

where $\Delta$ is an anisotropic parameter and $\\{J_{j}\\}_{j=1}^{L-1}$ is a
random interaction. We employ an open boundary condition, and we assume that
$\\{J_{j}\\}_{j=1}^{L-1}$ is chosen from independent uniform distributions on
the interval $[0,2]$ (i.e.,
$\\{J_{j}\\}_{j=1}^{L-1}\overset{\text{\small}\textrm{iid}}{\sim}U(0,2)$). It
is worth mentioning that this Hamiltonian commutes with the total
magnetization $S_{z}$; hence, we can use our scheme to obtain the ground state
of this Hamiltonian.

$J_{1}$ | $0.8441683664299817$
---|---
$J_{2}$ | $0.47574391516586223$
$J_{3}$ | $0.06980280523824778$
$J_{4}$ | $0.6197240483819366$
Table 1: Coupling constant of the random XXZ spin chain
$\\{J_{j}\\}_{j=1}^{4}$ used in Fig. 3. The unit of these values is GHz, as
described in the main text.

(a) Energy spectrum of QA for the random XXZ spin chain with the drive
Hamiltonian of tshe random transverse field.

(b) Energy spectrum of QA for the random XXZ spin chain with the drive
Hamiltonian of the random XY model.

Figure 3: Energy spectrum during QA plotted against time to prepare the ground
state of the random XXZ spin chain. The anisotropic parameter is $\Delta=0.7$
GHz and the number of qubits is $L=4$. The coupling strength of the XXZ spin
chain is described in Table 1. (a)The uniform transverse field is chosen as
the drive Hamiltonian with an amplitude of $B=1.0$ GHz. (b)The XY model is
chosen as the drive Hamiltonian with an amplitude of $g=1.0$ GHz.  Figure 4:
Estimation error plotted against the annealing time. We choose the drive
Hamiltonian as the uniform transverse field (blue) and the XY model (orange)
with a decoherence rate of $\gamma=10^{-4}$ GHz. We use the same parameters as
Fig. 3.

The energy spectrum of the annealing Hamiltonian for each scheme is plotted in
Fig. 3. Furthermore, the estimation error is plotted against the annealing
time in Fig. 4. We performed numerical simulations with a decoherence rate of
$\gamma=10^{-4}$ GHz. In this case, we choose the optimal amplitude of the
drive Hamiltonian at each time. The estimation error of our scheme is around
10 times smaller than that of the other schemes. Similar to the case of the
deformed spin star model, we find that the optimal annealing time of our
scheme becomes shorter than that of the conventional schemes, which confirms
the suppression of the non-adiabatic transitions in our scheme.

##  V Conclusion

We proposed the use of a drive Hamiltonian that preserves the symmetry of the
problem Hamiltonian. Owing to the symmetry, we can search the solution in an
appropriate symmetric subspace during QA. As the non-adiabatic transitions
occur only inside the specific subspace, our approach can potentially suppress
unwanted non-adiabatic transitions. To evaluate the performance of our scheme,
we employed the XY model as the drive Hamiltonian in order to find the ground
state of problem Hamiltonians that commute with the total magnetization along
the z-axis. We found that our scheme outperforms the conventional scheme in
terms of the estimation error of QA.

###### Acknowledgements.

We thank a useful comment from Shiro Kawabata. This work was supported by
MEXT’s Leading Initiative for Excellent Young Researchers and JST PRESTO
(Grant No. JPMJPR1919), Japan. This paper is partly based on the results
obtained from a project, JPNP16007, commissioned by the New Energy and
Industrial Technology Development Organization (NEDO), Japan.

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# Time-Spatial Mode Selective Quantum Frequency Converter

Santosh Kumar<EMAIL_ADDRESS>He Zhang  Prajnesh Kumar  Malvika
Garikapati  Yong Meng Sua  Yu-Ping Huang<EMAIL_ADDRESS>Department of
Physics, Stevens Institute of Technology, Hoboken, NJ, 07030, USA  Center for
Quantum Science and Engineering, Stevens Institute of Technology, Hoboken, NJ,
07030, USA

(August 27, 2024)

###### Abstract

We experimentally demonstrate a mode-selective quantum frequency converter
over a compound spatio-temporal Hilbert space. We show that our method can
achieve high-extinction for high-dimensional quantum state tomography by
selectively upconverting the signal modes with a modulated and delayed pump.
By preparing the pump in optimized modes through adaptive feedback control,
selective frequency conversion is demonstrated with up to 30 dB extinction.
The simultaneous operations over high-dimensional degrees of freedom in both
spatial and temporal domains can serve as a viable resource for photon-
efficient quantum communications and computation.

Suggested keywords

##  I Introduction

High-dimensional (HD) quantum information carriers have been studied with
superconducting circuits [1], atomic ensembles [2, 3, 4], photons [5, 6], and
so on. They can increase the information capacity, improve the noise
resistance, and make the quantum cryptographic schemes more difficult to hack
despite errors [7, 8]. In photonic systems, single photons can carry quantum
information in a HD Hilbert space subtending polarization, spatial, temporal,
frequency, and path degrees of freedom (DOF), with broad applications in long-
distance quantum communication, quantum key distribution, quantum gate
operations, and quantum teleportation [9, 10, 11, 12, 13, 14, 13].

Meanwhile, in modern quantum information architectures, quantum frequency
conversion (QFC) has become an essential element with numerous applications in
quantum communications and quantum information processing [15, 16]. It allows
us to convert the frequency of the photons without disturbing their quantum
properties [17]. Through QFC, quantum information can be transmitted through
optical fiber links using telecom wavelength at low loss while interfaced with
quantum atomic memories [18, 19, 20]. Many previously studied nonlinear optics
based QFCs have been demonstrated in two-dimensional Hilbert space for qubit
teleportation and communications [21, 22, 23, 24, 25]. Recently, this study
has been extended to a HD Hilbert space using spatial, frequency, or temporal
DOF due to its potential in long distance quantum communications [26, 27, 28].
For example, HD QFC was demonstrated for orbital angular momentum (OAM) qudits
via sum frequency generation (SFG) with a flat-top Gaussian pump beam [29].
The HD quantum information of the single photons were shown to be transferred
from the polarization DOF to the orbital angular momentum DOF and orbital
angular momentum to temporal DOF or vice versa [30, 31, 32].

In view of the two vivid directions of pursuit, in this paper, we propose and
experimentally demonstrate a HD mode-selective QFC in a compound spatio-
temporal Hilbert space. We show that the mutually unbiased basis (MUB) sets in
a HD Hilbert space can be selectively up-converted according to their spatial
and temporal modes. In contrast to previous demonstrations [29], we use an
optimized pump beam which is the superposition of many higher-order orthogonal
modes. The optimization is performed by using an adaptive feedback technique
[33]. Our HD mode-selective QFC provides an ingenious and effective solution
for quantum information processing using single photons in HD Hilbert spaces,
with potential applications in quantum communication and quantum networks [34,
35, 36]. In future, it can be extended to hyper-entangled photon generation
for promising quantum applications [37, 38, 39, 40].

##  II Model

We consider a signal and a pump beam in the spatio-temporal domain as

$\displaystyle\Psi_{i\equiv
p,s}(x,y,z,t)=A_{i}E^{r}_{i}(x,y,z)E^{t}_{i}(t)\exp[-i(k_{i}z-\omega_{i}t)],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (1)

where $A_{i\equiv p,s}$ is the input electric-field amplitude, and
$E^{r}_{i}(x,y,z)$ and $E^{t}_{i}(t)$ are the electric fields in the spatial
and temporal domain, respectively. Here, we consider that the full electric-
field is the product of the spatial and temporal DOF, i.e. we ignore the
inter-coupling of the space and time DOF [41, 42]. Under the slowly-varying-
envelope approximation with negligible group velocity dispersion, the coupled
wave differential equations can be written as:

$\displaystyle
2ik_{s}(\partial_{z}+\beta^{\prime}_{s}\partial_{t})\Psi_{s}+(\partial_{x}^{2}+\partial_{y}^{2})\Psi_{s}=-2\frac{\omega_{s}^{2}}{c^{2}}\chi\Psi_{p}^{*}\Psi_{f}e^{i\triangle
kz},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (2) $\displaystyle
2ik_{p}(\partial_{z}+\beta^{\prime}_{p}\partial_{t})\Psi_{p}+(\partial_{x}^{2}+\partial_{y}^{2})\Psi_{p}=-2\frac{\omega_{p}^{2}}{c^{2}}\chi\Psi_{s}^{*}\Psi_{f}e^{i\triangle
kz},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (3) $\displaystyle
2ik_{f}(\partial_{z}+\beta^{\prime}_{f}\partial_{t})\Psi_{f}+(\partial_{x}^{2}+\partial_{y}^{2})\Psi_{f}=-2\frac{\omega_{f}^{2}}{c^{2}}\chi\Psi_{p}\Psi_{s}e^{-i\triangle
kz},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (4)

where $\beta^{\prime}_{i=s,p,f}$ and $\omega_{i}$ are the inverse group
velocities, and frequencies of signal, pump and SF light, respectively.
$k_{s}={n_{s}\omega_{s}}/{c}$ , $k_{p}={n_{p}\omega_{p}}/{c}$ and
$k_{f}={n_{f}\omega_{f}}/{c}$ are their wave numbers. $\chi$ is the second-
order nonlinear susceptibility. $\Delta k=k_{s}+k_{p}-k_{f}-2\pi/\Lambda$ is
the momentum mismatching condition, where $\Lambda$ is the poling period of
the nonlinear crystal. Equations (2)-(4) can be solved numerically using the
split-step Fourier method with adaptive step-size [43, 33].

Figure 1 shows an example of the simulated SF output for two different signal
modes in the combined spatial and temporal domain. Both signals are converted
by the same spatio-temporal pump field. Figure 1(a) plots the electric field
amplitudes in the spatial and temporal domain. These amplitudes are used to
create the spatio-temporal signal and pump fields. Figure 1(b) and (c) show
the SF output for the signal field $\Psi_{s1}=E_{1}(x,y)E_{3}(t)$ and
$\Psi_{s2}=E_{2}(x,y)E_{4}(t)$ with pump field $\Psi_{p}=E_{1}(x,y)E_{4}(t)$,
respectively. $E_{1}(x,y)=e^{-\frac{x^{2}+y^{2}}{W_{0}^{2}}}$ and
$E_{3}(t)=e^{-\frac{(t-t_{0})^{2}}{(\tau_{0})^{2}}}$ are the Gaussian electric
field amplitude in the spatial and temporal domain with widths $W_{0}$ and
$\tau_{0}$, respectively. $E_{2}(x,y)$ is a Laguerre-Gaussian mode with
angular mode number $l=1$ and radial mode number $p=0$ (see Appendix A). Each
plot shows the distribution of the modes in the spatial and temporal domain.

Figure 1: Generated spatio-temporal SF modes. (a) shows the electric field
amplitude in the spatial and temporal domain for composing different signal
and pump modes. (b) SF output for signal mode $\Psi_{s}=E_{1}E_{3}$ with pump
mode $\Psi_{p}=E_{1}E_{4}$. (c) SF output for signal mode
$\Psi_{s}=E_{2}E_{4}$ with pump mode $\Psi_{p}=E_{1}E_{4}$.

Next, we show that the optimized pump field can selectively upconvert a
desired signal mode over other overlapping modes. In general, the optimized
pump is considered as a superposition of the higher-order modes. It can be
written as

$\displaystyle
P_{opt}=[C_{11}LG^{0}_{0}+C_{01}LG^{0}_{1}+...+C_{pl}LG^{p}_{l}(x,y)]\times$
$\displaystyle(\tau_{1}\Phi_{1}(t)+\tau_{2}\Phi_{2}(t)+...+\tau_{j}\Phi_{j}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (5)

where $LG^{p}_{l}$ are the Laguerre-Gaussian modes, $\Phi_{j}$ are the Schmidt
decomposed temporal modes, $C_{pl}$ and $\tau_{j}$ are the normalized
coefficients of the spatial and temporal modes, respectively.

For a given optimized pump, to select a $i^{th}$ signal mode among other
overlapping modes. We first define the normalized SF photon, $\bar{N_{i}}$, of
the $i^{th}$ signal as:

$\displaystyle\bar{N_{i}}=10log_{10}\left(\frac{N_{i}}{\sum^{n}_{i=1}N_{i}}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (6)

where $N_{i}$ is the measured SF photons of the $i^{th}$ signal mode and $n$
is the total number of signal modes. Note that, the input incident photons for
each signal mode are equal. The selectivity, $\eta$, between desired signal
mode and other overlapping modes can be defined as

$\eta=\bar{N}_{i}-\bar{N}_{j\neq i},$ (7)

where $\bar{N}_{i}$ and $\bar{N}_{j\neq i}$ are the normalized SF photons of
the desired signal and the other overlapping signal modes, respectively.

Figure 2: An example of selection among different spatio-temporal modes
$S_{k}(k=1,2,3,4)$. (a) four different spatio-temporal modes $S_{1}$, $S_{2}$,
$S_{3}$, and $S_{4}$, where the spatial and temporal components are shown in
Fig. 1(a). (b) The optimized pump mode in the spatial and temporal domain. (c)
Normalized SF photon for different signal modes with optimized pump mode.
Figure 3:  Experimental setup for high-dimensional mode selective frequency
up-conversion process. ODL: programmable optical delay line, SLM: Spatial
Light Modulator, BS: Beamsplitter, WDM: Wavelength-division multiplexing,
PPLN: Magnesium-doped Periodic Poled Lithium Niobate crystal, BF: Shortpass
bandpass filters and PM: power meter.

Figure 2 shows an example in the simulation to select a signal mode ($S_{4}$)
with respect to other overlapping signal modes ($S_{1},S_{2}$, and $S_{3}$).
For simplicity, we used a random-walk optimization method to optimize the pump
field in spatio-temporal domain [33, 43]. We consider four signal modes
$S_{k}(k=1,2,3,4)$, which are composed of two spatial ($E_{1}$, $E_{2}$) and
two temporal ($E_{3}$, $E_{4}$) electric field amplitudes, as shown in Fig.
1(a). One optimized pump, shown in Fig. 2(b), is designed to select $S_{4}$
and suppress the others simultaneously. With this optimized pump, the
normalized up-converted SF output for different signals are shown in Fig.
2(c). The selectivity of the signal $S_{4}$ against $S_{1}$, $S_{2}$, and
$S_{3}$ are 20.2 dB, 8.4 dB, and 11.8 dB, respectively. To further improve the
selectivity, we can use a particle-swarm optimization algorithm in the spatial
and temporal domain [44].

##  III Experimental Setup

Figure 3 outlines our present experimental setup. A mode-lock laser is used to
generate the pulse train with pulse width $\sim$0.3 ps at 50 MHz repetition
rate. We use two inline narrow-band wavelength division multiplexers (WDMs)
with bandwidth 100 GHz to select two wavelengths, one at 1545 nm as a pump and
another at 1558 nm as a signal. A programmable optical delay line (ODL) in the
signal arm is used to scan the temporal shape of the optical pulses. The pump
optical pulses are amplified using an Erbium-doped fiber amplifier (Thorlabs,
EDFA100S) to obtain maximum average power $\sim$ 10 mW incident on the
nonlinear crystal. On the other hand, the signal optical pulses are attenuated
with neutral density filters to produce ultra-low optical signal. In free
space, both arms select the horizontal polarization for the pump and signal.
The transverse FWHM of the pump and signal are 2.8$\pm 0.05$ mm and 2.6$\pm
0.05$ mm, respectively. Then, these two beams are incident on separate
spatial-light modulators (Santec SLM-100, 1440 $\times$ 1050 pixels, pixel
pitch 10.4 $\times$ 10.4 $\mu$m) [45]. The pump and signal SLMs are used to
upload desired (for example: HG or LG) phase masks. A beam splitter is used to
combines the two pulse trains which are focused (focus length $F=$200 mm) into
a temperature-stabilized second-order nonlinear crystal with a poling period
of 19.36 $\mu$m (5 mol.% MgO doped PPLN, 10 mm length, 3 mm width, and 1 mm
height) for SF generation process. The normalized conversion efficiency of the
nonlinear crystal is 1%. Half waveplates on both arm, before the nonlinear
crystal, are used to ensure the vertically polarized optical field parallel to
the crystal’s optical axis. The output pulses are then filtered with a short-
pass filter to remove any residual fundamental light and bandpass filters to
provide a total $>$180 dB extinction in rejecting the second harmonic of the
pump [43, 46]. The visible SF photons pass through a half waveplate and
polarizing beam splitter. The output SF photons are then coupled into a
single-mode fiber (SMF-28) using a fiber collimator consisting of aspheric
lens (Thorlabs A375TM-B). A silicon avalanche photodiode (Si-APD, ID100) is
used to detect the photons which has 15% quantum efficiency. The SF photons
are then collected by the field programmable gate array (FPGA, Zedboard
ZYNQ-7000) and sent to a computer for post processing. A MATLAB program is
used to optimize the pump mode using adaptive feedback technique based on a
random-walk optimization [33, 44].

##  IV Results

Figure 4: Measured SF output matrix with higher-order LG modes for the signal
and the pump. The values in the matrix are representing the normalized SF
output $\bar{N}$ dB (using Eq.6).

Before presenting the results for mode-selective QFC with optimized pump, we
first show the preparation of the quantum states in the spatial HD Hilbert
space. Figure 4 shows the experimentally measured matrix for higher-order
signal and pump modes. In this case, both signal and pump modes have the
radial index $p=0,1$ and azimuthal index $l=-3$ to 3. The values in the color
bar and matrix represent the normalized SF photon($\bar{N}$), which is
normalized with the sum of the elements in each row (i.e. with same pump)
according to Eq.6. It is clear from this figure that the angular momentum with
opposite sign for the signal and pump modes have higher extinction compare to
other modes. We can also observe that the nearest neighbour spatial modes are
difficult to distinguish [33, 7].

Figure 5: Qubit tomography for MUB of signal modes: (a)
$\\{|LG^{0}_{+1}\rangle,|LG^{0}_{-1}\rangle\\}$ and (b)
$\\{(|LG^{0}_{+1}\rangle+|LG^{0}_{-1}\rangle)/\sqrt{2},(|LG^{0}_{+1}\rangle-|LG^{0}_{-1}\rangle)/\sqrt{2}\\}$.
Normalization is done by summation of each row. $P_{1}$ and $P_{2}$ are the
optimized pump modes to selectively upconvert signal $S_{1}$ and $S_{2}$,
respectively.

Figure 6: Qudit tomography for MUB of signal modes: (a)
$\\{(|LG^{0}_{+1}\rangle+|LG^{0}_{-1}\rangle+|LG^{1}_{+1}\rangle+|LG^{1}_{-1}\rangle)/2,(|LG^{0}_{+1}\rangle+|LG^{0}_{-1}\rangle-|LG^{1}_{+1}\rangle-|LG^{1}_{-1}\rangle)/2,(|LG^{0}_{+1}\rangle-|LG^{0}_{-1}\rangle-|LG^{1}_{+1}\rangle+|LG^{1}_{-1}\rangle)/2,(|LG^{0}_{+1}\rangle-|LG^{0}_{-1}\rangle+|LG^{1}_{+1}\rangle-|LG^{1}_{-1}\rangle)/2\\}$
and (b)
$\\{|LG^{0}_{+1}\rangle,|LG^{0}_{-1}\rangle,|LG^{1}_{+1}\rangle,|LG^{1}_{-1}\rangle\\}$.
$P_{i}$ are the optimized pump modes to selectively upconvert signal $S_{i}$,
where $i=1$ to 4.

After that, optimized pump modes are prepared to selectively upconvert the MUB
of qubit and qudit states. The optimized pump modes are in the superposition
of the 18 higher-order LG modes ($l\in[-5:5],p\in[0,1]$). To show the
selectivity for each optimized pump, the up-converted photons are collected
and normalized. As an example, in Fig. 5, we use two set of MUB for signal
qubit states (a) {$|LG^{0}_{+1}\rangle,|LG^{0}_{-1}\rangle$} and (b)
{$(|LG^{0}_{+1}\rangle+|LG^{0}_{-1}\rangle)/\sqrt{2},(|LG^{0}_{+1}\rangle-|LG^{0}_{-1}\rangle)/\sqrt{2}$}.
The qubit tomography shows that the optimized pump modes can selectively
pickup the desired modes. It achieves the selectively of $\sim$ 28 dB on
average. Similarly, Fig. 6 shows the superposition of 4 LG modes, i.e. qudit,
to prepare 2 MUB consisting of 4 signal modes. Fig. 6 (a) shows the first MUB,
we use signals
{$(|LG^{0}_{+1}\rangle+|LG^{0}_{-1}\rangle+|LG^{1}_{+1}\rangle+|LG^{1}_{-1}\rangle)/2,(|LG^{0}_{+1}\rangle+|LG^{0}_{-1}\rangle-|LG^{1}_{+1}\rangle-|LG^{1}_{-1}\rangle)/2,(|LG^{0}_{+1}\rangle-|LG^{0}_{-1}\rangle-|LG^{1}_{+1}\rangle+|LG^{1}_{-1}\rangle)/2,(|LG^{0}_{+1}\rangle-|LG^{0}_{-1}\rangle+|LG^{1}_{+1}\rangle-|LG^{1}_{-1}\rangle)/2$}
to selectively upconverts each with different optimized pump modes. The
selectivity $\eta$ gives $\sim$17 dB on average extinction. In Fig. 6 (b), we
show the second MUB, which consists 4 signals
$|LG^{0}_{+1}\rangle,|LG^{0}_{-1}\rangle,|LG^{1}_{+1}\rangle$, and
$|LG^{1}_{-1}\rangle$. We selectively upconvert these four signal with
different optimized pump mode. The value of selectivity $\eta$ gives $\sim$16
dB extinction.

Figure 7 demonstrates the selectivity among five higher-order LG modes (a)
$LG_{0}^{0}$, $LG_{1}^{0}$, $LG_{2}^{0}$, $LG_{3}^{0}$, $LG_{4}^{0}$ and (b)
$LG_{0}^{1}$, $LG_{1}^{1}$, $LG_{2}^{1}$, $LG_{3}^{1}$, $LG_{4}^{1}$ with two
different optimized pumps, respectively. (a) and (b) show the selectivity of
signal modes {$LG_{1}^{0}$ & $LG_{2}^{0}$ }, and {$LG_{1}^{1}$ & $LG_{2}^{1}$}
with simultaneously suppressing other overlapping modes. Here also the
selectivity for each optimized pump is determined by the collected and
normalized SF photons, as defined in Eq. 7. We can see that the selectivity
for 5 signal modes in Fig. 7(a) and (b) are in between 20 dB to 30 dB and 15
dB to 25 dB, respectively.

Figure 7: Spatial mode-selective upconversion of the signal modes with
optimized superposition of the pump LG modes. The values in the table are the
normalized SF output $\bar{N}$ dB, which are evaluated from Eq.6. Five signal
modes from left to right are (a) {$LG_{0}^{0}$, $LG_{1}^{0}$, $LG_{2}^{0}$,
$LG_{3}^{0}$, $LG_{4}^{0}$}, and (b) {$LG_{0}^{1}$, $LG_{1}^{1}$,
$LG_{2}^{1}$, $LG_{3}^{1}$, $LG_{4}^{1}$}, respectively.

To show the performance of mode selective QFC in the spatial and temporal
domain, we introduce the time-bin manipulation by changing the delay of the
signal pulses with respect to the pump pulses. The upconverted photons are
detected as a function of the temporal delay scanned by the programmable ODL
between the synchronous signal and pump pulses. In Fig. 8, we define signal
modes in three dimensions, $x,y$, and $t$ and consider six signal modes. The
spatial profile of these six signals are $S_{1}$:
$(LG^{0}_{-1}+LG^{0}_{1})/\sqrt{2}$, $S_{2}$:
$(LG^{0}_{-1}-LG^{0}_{1})/\sqrt{2}$, $S_{3}$:
$(LG^{0}_{-1}-LG^{0}_{1})/\sqrt{2}$, $S_{4}$:
$(LG^{0}_{-1}-iLG^{0}_{1})/\sqrt{2}$, $S_{5}$: $LG^{0}_{-1}$, and $S_{6}$:
$LG^{0}_{1}$, and corresponding temporal delays are $t_{0}=0$ ps, $5$ ps, $20$
ps, $15$ ps, $10$ ps, and $25$ ps, respectively. The optimized pump $P_{1}$
($P_{5}$) is designed to selectively up-convert $S_{1}$ ($S_{5}$) and suppress
the others. The selectivities for $P_{1}$ and $P_{5}$ are 15 dB and 14 dB on
average, respectively. These selectivities can further improve with other
efficient adaptive optimization methods [43, 47, 48].

Figure 8: Spatio-temporal selectivity of the signal modes with optimized pump
modes. The values in the table are the normalized SF output $\bar{N}$ dB. The
spatial modes of these six signals are $S_{1}$:
$(LG^{0}_{-1}+LG^{0}_{1})/\sqrt{2}$, $S_{2}$:
$(LG^{0}_{-1}-LG^{0}_{1})/\sqrt{2}$, $S_{3}$:
$(LG^{0}_{-1}-LG^{0}_{1})/\sqrt{2}$, $S_{4}$:
$(LG^{0}_{-1}-iLG^{0}_{1})/\sqrt{2}$, $S_{5}$: $LG^{0}_{-1}$, and $S_{6}$:
$LG^{0}_{1}$. The corresponding temporal delay, $t_{0}$, of the signals are
$0$ ps, $5$ ps, $20$ ps, $15$ ps, $10$ ps, and $25$ ps, respectively.

##  V Discussion

In the current setup, the spatial mode shaping of the pump is provided by the
SLM and the temporal shaping is realized by the time delay. In future, optical
arbitrary waveform generator based on spectral line-by-line pulse shaping can
be utilized to arbitrarily manipulate the amplitude and phase profiles of the
time-frequency modes [43]. This can improve the selectivity of the signal
modes by preparing the pump pulses in the optimized amplitude and phase
profiles in time. Merging the selectivity in spatial and time-frequency modes
will realize the full advantage of spatio-temporal mode selective QFC. We
envision that spatio-temporal mode selectivity will be effective against
channel noise in both measurable spatial and temporal dimensionality thus
enables robust and realistic entanglement distribution with high-dimensional
encoding [49] with minimum trade-off. Note that, the selectivity and
conversion efficiency of the QFC can be improved while suppressing it’s
intrinsic noise level by using longer nonlinear crystal with well defined
phase matching profile [16]. The signal-to-noise ratio in our current
nonlinear crystal is characterized in Appendix B.

Thanks to excellent selectivity among traverse spatial modes, our technique
can be used for measurement of the traverse spatial mode joint probability
distribution of a bi-photon state produced in spontaneous parametric down-
conversion [50, 51] by performing mode selective measurement on the photon
pair separately. Subsequently, this will allow one to directly quantify and
certifying high-dimensional entanglement in traverse spatial mode.

We can also incorporate polarization degree of freedom to extend the mode
selective QFC [18, 29]. The preliminary measurements in the polarization
degree of freedom, as shown in Appendix C, would extend the dimensionality of
our mode-selective QFC.

##  VI Conclusion

We have demonstrated a high-dimensional mode-selective QFC in spatio-temporal
DOF. Our experimental results showed that a high-dimensional QFC can be
realized with high-extinction (up to 30 dB) by selectively upconverting the
signal modes using optimized pump. This technique is potentially useful for
multi-fold quantum applications [52, 53] such as long-distance quantum
communication, quantum key distribution (QKD), quantum state tomography and so
on. In future, it could be extended to other DOF to increase the information
capacity of the quantum signals.

## Acknowledgement

This research was supported in part by National Science Foundation (Grant No.
1842680) and in part by the Earth Science Technology Office, NASA, through the
Instrument Incubator Program.

## Appendix

###  VI.1 Spatial modes

The Laguerre-Gaussian (LG) modes can be written in the cylindrical coordinates
as

$LG_{l}^{p}(r,\phi,z)=\frac{C_{lp}}{w}\left(\frac{r\sqrt{2}}{w}\right)^{|l|}L^{p}_{|l|}\left(\frac{2r^{2}}{w^{2}}\right)\exp(-il\phi)\\\
\times\exp\left(i\zeta-ikz-\frac{ikr^{2}}{2R}-\frac{r^{2}}{w^{2}}\right),$
(A.1)

where $\phi=\arctan(y/x)$ is the azimuthal coordinate, ${\displaystyle
C_{lp}}=\sqrt{\frac{2p!}{\pi(p+|l|)!}}$ is a normalization constant,
$\\{\displaystyle L^{p}_{|l|}\\}$ are the generalized Laguerre polynomials
with the azimuthal mode index $l$ and the radial index $p$. The LG modes can
be expressed as the superposition of Hermite-Gaussian (HG) modes and vice-
versa [54]. The $LG_{l}^{p}$ modes with the angular mode number $l=n-m$ and
radial mode number $p=min(m,n)$. These can be written as

$\displaystyle LG_{\pm 1}^{0}=\frac{1}{\sqrt{2}}\left(HG_{01}\pm
iHG_{10}\right),$ (A.2) $\displaystyle
HG_{01}(\theta)=\frac{1}{\sqrt{2}}\left(LG_{+1}^{0}e^{-i\theta}+LG_{-1}^{0}e^{i\theta}\right),$
(A.3)

where $\theta$ is the angle of rotation of the coordinate frame from $(x,y)$
to $(x^{\prime},y^{\prime})$.

Figure A1:  SF photon counts are plotted by rotating the Hermite-Gaussian
($HG_{10}$) phase mask of the pump and the signal in (a) simulation and (b)
experiment. The simulated result is normalized using the maximum value in the
plot. The color bar, shown in (b), represents the detected SF counts.

The rotation of the signal phase mask with respect to pump can manipulate the
generated SF photons. As an example, in Fig. A1, we plots the simulated and
experimentally measured SF photons by rotating the $HG_{10}$ pump and signal
masks. It shows that the SF photons are minimum, when the pump and signal
phase masks are orthogonal to each other. This flexible control of phase mask
rotation can be utilize for the HD quantum applications [55, 56].

Figure A2: Experimentally measured SF photon counts (blue astric) and
background noise photon counts (red cross) vs average pump power. Solid red
and blue curves are the corresponding fits.

###  VI.2 Signal-to-Noise ratio

We characterize the quantum noise generated in our nonlinear frequency
conversion process in our experiment. We measured the upconverted SF photon
per pulse and background noise photon per pulse vs average pump power as shown
in Fig. A2. The noise photon observed in Fig. A2 indicates that noise photon
counts per pulse is $\sim$ $10^{-6}$, less than the noise level attained in LN
waveguide [57]. The background noise photons could be generated by the pump
field due to unwanted second harmonic and spontaneous parametric down
conversion processes [58, 57]. In this work, the background noise $\sim$ 46 dB
lower than the upconverted SF photons.

###  VI.3 Polarization degree of freedom

We show the polarization dependence of the nonlinear crystal (black curve) and
a visible PBS (blue curve) in Fig. A3. First measurement is performed by
rotating the signal half-wave plate ($\theta_{s}$) in front of the nonlinear
crystal and the second measurement (linear) is performed by rotating half-wave
plate ($\theta_{SF}$) in front of visible PBS. In both cases, we record the
photon counts using same visible APD detector. Visibility can be evaluated as

$V=\frac{C_{max}-C_{min}}{C_{max}+C_{min}},$ (A.4)

where $C_{max}$ and $C_{min}$ are the maximum and minimum number of SF photon
counts.

Our result in Fig. A3 shows that the visibility of the photon counts in linear
and nonlinear cases are 0.9832 and 0.9804, respectively. It shows that our
system could be utilized for HD quantum states in polarization DOF [18, 59,
29].

Figure A3: SF photon counts vs polarization rotation of the pump half-wave
plate ($\theta_{p}$) with respect to the crystal. $\theta_{SF}$ is the
polarization rotation of the half-wave plate with respect to the visible PBS.

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