# High-fidelity quantum gates for OAM qudits on quantum memory

Vashukevich E.A., Golubeva T.Yu., Golubev Yu.M  Saint Petersburg State
University, Universitetskaya emb. 7/9, St. Petersburg, 199034 Russia

###### Abstract

The application of high-dimensional quantum systems (qudits) in quantum
computing and communications seems to be a promising avenue due to the
possibility of increasing the amount of information encoded in one physical
carrier. In this work, we propose a method for implementing single-qudit gates
for qudits based on light modes with orbital angular momentum. Method for
logical qudits encoding, which ensures the quasi-cyclicity of operations, is
introduced. Based on the protocol for converting the orbital angular momentum
of light in the Raman quantum memory scheme [Vashukevich E.A. et. al. PRA,
101, 033830 (2020)], we show that the considered gates provide an extremely
high level of fidelity of single-qudit transformations. We also compare
quantum gates’ properties for systems of different dimensions and find the
optimal conditions for carrying out transformations in the protocol under
consideration.

###### pacs:

42.50.Dv, 42.50.Gy, 42.50.Ct, 32.80.Qk, 03.67.-a

##  I Introduction

Several reasons explain the currently tremendous interest in high-dimensional
quantum systems (qudits). Researchers are attracted by the possibility of
increasing the information capacity of the channel – the amount of information
that can be encoded in one physical carrier, which turns out to be very useful
in the problems of quantum communication [1]. Significant advantages of qudits
over qubits in quantum cryptography protocols have been demonstrated, where
the security of the protocol appears to be the higher, the larger the
dimension of the system [2]. Many variants of various physical systems that
could encode qudits have been proposed, for example, time-bin [3], the orbital
angular momentum of light (OAM) [4], polarization multi-photon states [5] and
optical frequency comb [6]. Nevertheless, there are still blind spots in the
problem of highly efficient manipulation of multidimensional quantum states.

The orbital angular momentum is an exciting resource for constructing a qudit
since the OAM can take any integer values, which allows us to work in the
Hilbert space of high dimension [7]. Laguerre-Gaussian (LG) modes with OAM
show high stability and a relatively high decoherence time when propagating in
a turbulent atmosphere [8]. Since LG modes are well localized in the spatial
domain, several well-proven experimental techniques for generating [9, 10],
separating and detecting such multimode radiation [11, 12, 13] exist. Many
methods of OAM manipulation based on phase holograms [14], q-plates [15], and
a system of cylindrical lenses [16] have also been proposed. However,
performing efficient mode conversions with different OAMs applying such
optical elements requires a mode-specific change of the system parameters,
which cannot satisfy quantum computing needs. Although successful attempts to
construct quantum logic gates over qudits based on OAM have been made [17],
the construction of universal gates that transform an arbitrary quantum state
of light with OAM in a controlled manner with high fidelity and efficiency
remains an open problem.

In the work [18], we demonstrated conversion of the OAM of light in the scheme
of Raman quantum memory on cold atoms. We have shown that by varying the
spatial profile of the driving field at the writing and read-out stages, the
OAM of the quantum field can be changed by a certain amount in a wide range of
values. Such a conversion scheme has relatively high efficiency (of the same
order as the Raman memory protocols for light without OAM) and allows storing
information simultaneously with the conversion.

In this work, we construct quantum single-qudit gates $\hat{X}_{d}^{m}$ based
on the OAM transformation on a quantum memory cell for different dimensions of
the qudits encoded in the OAM values. We show that with a specific encoding of
the logical values of a qudit, it is possible to perform a cyclically closed
operation $\hat{X}_{d}^{m}$. The proposed method is remarkable since it shows
an extremely high fidelity, provides relatively high probabilities of success.
In the last section, we compare systems of different dimensions in terms of
both the properties of the gate $\hat{X}_{d}^{m}$ and the information capacity
of the channel in an attempt to answer the question: ”systems of what
dimension are optimal for quantum computing?” in the presented protocol.

##  II Single-qudit multidimensional quantum gates

Let us begin with recalling the general rules for constructing logical
operations on high-dimensional systems. Following [19] we introduce
$d$-dimensional logical operations on qudits $\hat{X}_{d}$ and $\hat{Z}_{d}$
using projection operators:

$\displaystyle\hat{X}_{d}=\sum\limits_{l=1}^{d}|l\oplus_{d}1\rangle\langle
l|,\;\;l\oplus_{d}1=l+1\;\;\hbox{mod}(d),$ (1)
$\displaystyle\hat{Z}_{d}=\sum\limits_{l=1}^{d}|l\rangle\omega^{l}\langle
l|,\;\;\omega=\exp{2\pi i/d},$ (2)

that is, the $\hat{X}_{d}$ gate performs modulo $d$ addition of the value of
the qudit with one, and the $\hat{Z}_{d}$ gate adds a relative phase to the
members of the superposition. For $d=2$, the matrices of such operators
coincide with the Pauli matrices
$\hat{\sigma}_{x}=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right),\hat{\sigma}_{z}=\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)$, and their action on an arbitrary qudit state
$|\psi\rangle$ can be described by the operation ”NOT” and the operation of
adding the phase $\exp{i\pi}=-1$ to the second term of the superposition,
respectively. It is important to note that integer powers of mentioned above
$X_{d},Z_{d}$ operators should also be taken into account when considering $d$
-dimensional operations on qudits:

$\displaystyle\hat{X}^{m}_{d}=\sum\limits_{l=1}^{d}|l\oplus_{d}m\rangle\langle
l|,$ (3)
$\displaystyle\hat{Z}^{m}_{d}=\sum\limits_{l=1}^{d}|l\rangle\omega^{ml}\langle
l|.$ (4)

The power $m$ can range from $1$ to $d-1$, since
$\hat{X}_{d}^{d}=\hat{Z}_{d}^{d}=I$. It is not difficult to write down the
commutation relations for this operators, defining the Lie algebra $su(d)$:

$\displaystyle\left[\hat{X}_{d}^{k},\hat{X}_{d}^{m}\right]=\left[\hat{Z}_{d}^{k},\hat{Z}_{d}^{m}\right]=0,$
(5)
$\displaystyle\left[\hat{X}_{d}^{k},\hat{Z}_{d}^{m}\right]=\sum\limits_{j=1}^{d}\left(\omega^{mj}-\omega^{m(j\oplus_{d}k)}\right)|j\oplus_{d}k\rangle\langle
j|.$ (6)

Commutation relations for the Hermitian conjugate operators
$(\hat{X}_{d}^{k})^{\dagger},(\hat{Z}_{d}^{m})^{\dagger}$ can be obtained from
(5), (6) and following expressions:

$\displaystyle(\hat{X}_{d}^{k})^{\dagger}=\sum\limits_{j=1}^{d}|j\rangle\langle
j\oplus_{d}k|=\sum\limits_{j=1}^{d}|j\ominus_{d}k\rangle\langle
j|=\hat{X}_{d}^{-k},$ (7)
$\displaystyle(\hat{Z}_{d}^{m})^{\dagger}=\sum\limits_{j=1}^{d}|j\rangle(\omega^{mj})^{*}\langle
j|=\sum\limits_{j=1}^{d}|j\rangle\omega^{-mj}\langle j|=\hat{Z}_{d}^{-m}.\;\;$
(8)

It is necessary to clarify the reason for considering only the operators
$\hat{X}_{d},\hat{Z}_{d}$ and their integer powers. As shown in the work [17],
any unitary transformation over a single qudit $\hat{U}$ can be represented as
a decomposition:

$\displaystyle\hat{U}=\sum\limits_{k=0}^{d-1}\sum\limits_{j=0}^{d-1}g_{j,k}\hat{X}^{j}_{d}\hat{Z}^{k}_{d}.$
(9)

For the sake of completeness, here we briefly recall the reasoning from [17],
so consider the so-called Heisenberg-Weil operators:

$\displaystyle\mathcal{\hat{D}}(j,k)=\exp{\\{i\frac{\pi
kj}{2}\\}}\hat{Z}_{d}^{j}\hat{X}_{d}^{k}.$ (10)

From these operators, one can compose a complete orthonormal set of operators
in the Hilbert-Schmidt space $\mathcal{L}(\mathcal{H}^{d})$, which is the
linear shell of the Hilbert space of dimension $d$:

$\displaystyle\mathcal{\hat{Q}}_{j,k}=\frac{1+i}{2}\mathcal{\hat{D}}(j,k)+\frac{1-i}{2}\mathcal{\hat{D}}^{\dagger}(j,k).$
(11)

Any Hermitian operator $\hat{A}\in\mathcal{L}(\mathcal{H}^{d})$ can be
represented as a decomposition in a complete orthonormal set:

$\displaystyle\hat{A}=\sum\limits_{k=0}^{d-1}\sum\limits_{j=0}^{d-1}C_{j,k}\mathcal{\hat{Q}}_{j,k}.$
(12)

Here $C_{j,k}$ are real decomposition coefficients. Any unitary transformation
$\hat{U}$ can be written as a matrix exponent of the Hermitian operator:

$\displaystyle\hat{U}=\exp{\\{iA\\}}=\sum\limits_{n=0}^{\infty}\frac{i^{n}}{n!}\hat{A}^{n}.$
(13)

Using the commutation relations (5), (6) and expressions (10)-(13), one can
always write the unitary transformation matrix $\hat{U}$ in the form (9).

##  III Orbital angular momentum conversion

From the previous section, one can see that the necessary universal set of
single-qudit transformations consists of the operators
$\hat{X}_{d},\hat{Z}_{d}$ and their integer powers, while the number of
operations in the universal set increases with the dimension of the space. If
we consider qudits based on orbital angular momentum, it turns out that the
gate $\hat{Z}_{d}$ and its powers can be easily obtained using the Dove prism
[20], so we will focus here on describing the protocol for implementing the
gate $\hat{X}_{d}^{m}$ based on the method developed by us for transforming
the OAM of the quantum field on the quantum memory cell [18].

Let us briefly recall the essence of the transformation of modes with OAM on a
Raman quantum memory cell containing cold three-level atoms with
$\Lambda$-configuration of energy levels. We have shown that it is possible to
select the physical conditions of interaction that are consistent with the
experimental possibilities, such that the modes of the quantum field with
different OAM interact with the atomic ensemble independently of each other.
If the driving field at the writing stage is a LG mode with OAM $m$, and at
the read-out stage it is a plane wave, the following expression describes the
transformation of the quantum field in a complete memory cycle:

$\displaystyle\hat{a}^{out}_{l-m}(\tilde{t})$ $\displaystyle=$
$\displaystyle\chi_{l,m}\int\limits_{0}^{\tilde{T}}d\tilde{t}^{\prime}\hat{a}^{in}_{l}(\tilde{t}^{\prime})K(\tilde{t},\tilde{t}^{\prime})+\hat{F}_{l}.$
(14)

Under the action of the driving field, the orbital angular momentum of the
output radiation changes. Here $\hat{a}^{out}_{l-m},\hat{a}^{in}_{l}$ are the
annihilation operators in LG modes with a certain OAM (${l-m}$ and $l$) at the
output and at the input of the memory cell, respectively, $\hat{F}_{l}$ are
the operators of noise, which should be inevitably added during lossy
transformations, $K(\tilde{t},\tilde{t}^{\prime})$ is the kernel of a complete
memory cycle. We assume that the memory protocol for spatial modes is
optimized by choosing the effective cell length and temporal field profiles (a
detailed analysis of the kernel is presented in [21]), and the integral
transformation with the kernel $K(\tilde{t},\tilde{t}^{\prime})$ can be
replaced with its unit eigenvalue. We want to highlight the coefficients in
front of the integral $\chi_{l,m}$ – the mode overlap integrals normalized to
the cross-sectional area of the LG mode. The coefficients $\chi_{l,m}$ are
determine the efficiency of the entire conversion. That is, we need to ensure
a good overlap of spatial modes. To do this, we shift the waist of the driving
and quantum fields by a certain amount $z_{S}$, assuming other parameters
(waist width $w_{0}$ and Rayleigh range $z_{R}$) of the beams to be the same.
(see Fig. 1).

Figure 1: Schematic draw of the transformation geometry: the waist of the
driving beam and the waist of the signal beam are shifted by $z_{S}$, while
the atomic ensemble is located in the waist of the signal field

Such a parameter is introduced to follow the overlap of fields, and its
variation allows us to control the efficiency of the transformation.

As one can see, in the process under consideration, the conversion of the
orbital angular momentum of the quantum field occurs. Furthermore, if a field
with a certain angular momentum $l$ was written on the memory cell, then the
OAM of the field at the output of the cell is equal to $l-m$. Similarly, if
the writing process is carried out by a plane wave, and the read-out is
carried out by a field with OAM, then the orbital angular momentum of the
field at the output will be equal to $l+m$. The calculation shows that the
conversion of any OAM up or down by 1 can be performed with high efficiency:
about 0.9 for conversion with $m=-1$, and $0.6-0.8$ for $m=+1$ [18]. Since the
kernel of a complete memory cycle $K(\tilde{t},\tilde{t}^{\prime})$ does not
depend on the index $l$ of the signal field, we can transform the
superposition of OAM states, where the OAM of all terms of the superposition
increase or decrease by the same value $m$.

Figure 2: Coefficients $\chi_{l,m}$ depending on the normalized relative shift
of the beam waist $z_{S}/z_{R}$ for different moments of the driving field
$m=-1,-2,1,2$ (from left to right).

Fig. 2 shows the dependence of the coefficients $\chi_{l,m}$ for cases of
writing by the driving field with OAM (from left to right) $m=-1,-2,1,2$ for
different values of $l$ depending on control parameter $z_{s}/z_{R}$. Such a
configuration of the fields makes it possible to change the OAM of the quantum
field by the value $m$ with the efficiency $\chi_{l,m}$. We will denote by the
lowercase letter $\hat{x}_{d}$ the lossy transformations, and by the capital
letter $\hat{X}_{d}$ –the ideal one. Then the proposed configuration of the
fields allows one to perform nonideal
$\hat{x}^{-1}_{d},\hat{x}_{d}^{-2},\hat{x}_{d},\hat{x}_{d}^{2}$, respectively.
It can be noted that for the $l=0$ the value of the coefficient $\chi_{0,m}$
is rather small compared to the coefficients with nonzero $l$. Since we are
interested in performing transformations with high efficiency, we can conclude
that using a state with $l=0$ to build a state of a qudit is impractical. In
addition, as it will be shown in the next section, transformations with
$m=1,2$ are noticeably loses to those with $m=-1,-2$ due to small values of
the coefficients $\chi_{l,m}$ in the region of small values of $l$.

##  IV Orbital angular momentum qudits and quantum gates

###  IV.1 Cyclicity of the gates and the concept of logical qudit

As a starting point for describing the action of the OAM transformation on the
states of the qudit, we consider an arbitrary multi-dimensional quantum state
based on modes with OAM, which can be written in the following form:

$\displaystyle|\Psi\rangle=\sum\limits_{l=1}^{N}B_{l}|l\rangle.$ (15)

Here $B_{l}$ are the decomposition coefficients, which obey the normalization
condition $\sum\limits_{l=1}^{N}B^{2}_{l}=1$, $l$ is the index associated with
the orbital angular momentum.

We consider a state with a certain OAM $|l\rangle$ as a one-photon state
generated as a result of the action of the creation operator
$\hat{a}^{\dagger}_{l}$ on the vacuum state:

$\displaystyle|l\rangle=\hat{a}_{l}^{\dagger}|0\rangle.$ (16)

It should be noted here that we restrict the summation in the expression (15)
within the range from $1$ to $N$ since consideration of the entire mode
continuum with $l\in(-\infty,\infty)$ is inconvenient for build a qudit for
the reasons that we will discuss below.

Considering the $N$ ”physical” modes, let us single out several ”logical” sets
of dimensions $d$ on this set:

$\displaystyle l=\underbrace{1,2,...,d}_{\mbox{first logical
set}},\underbrace{d+1,..2d}_{\mbox{second logical
set}},...,\underbrace{...,N.}_{\mbox{N/d logical set}}$ (17)

The state in which the index $l$ varies within one logical set, we will assign
to one qudit:

$\displaystyle|\psi\rangle_{n}=\sum\limits_{l=(n-1)d+1}^{nd}C^{n}_{l}|l\rangle.$
(18)

The index $n$ numbers the ”logical sets”. The division into logical sets was
made by us in order to cyclically close single-qudit operation $\hat{X}_{d}$:

$\displaystyle
X_{d}|d\rangle_{physical}=|d+1\rangle_{physical}\equiv|1\rangle_{logical}$
(19)

If we consider as a qudit the whole state (15) (the entire continuum of modes
with $l\in(-\infty,\infty)$), then, obviously, no cyclically closed operation
can be obtained. In our case, we can choose the ”region” within the complete
set of modes with OAM arbitrarily, depending on the experimental capabilities
and computational needs.

Without loss of generality, let us suppose that $N/d$ is an integer, then the
state (15) can be rewritten as

$\displaystyle|\Psi\rangle=\sum\limits_{n=1}^{N/d}|\psi\rangle_{n}.$ (20)

We will further assume that different logical sets are indistinguishable. This
assumption means that the state $|l\rangle$ with OAM equal, for example, $1$,
$d+1$ and $2d+1$ encode a logical unit. The states $2$, $d+2$ and $2d+2$
encode logical ”2”, and $d$, $2d$ and $3d$ encode logical ”$d$”. Further we
will consider only the state $|\Psi\rangle_{n}$ as the state of the qudit and
will omit the index $n$ where possible.

Since proposed protocol for converting modes with OAM on a quantum memory cell
[18] use the notation of the creation and annihilation operators in modes with
a specific OAM it seems reasonable to use the operator representation. The
initial qudit state (18) can be generated from a vacuum as follows:

$\displaystyle|\psi\rangle=\sum\limits_{l=1}^{d}C_{l}\hat{a}^{\dagger}_{l}|0\rangle=\hat{A}^{\dagger}|0\rangle.$
(21)

Let the initial state of the qudit to be pure and described by the density
matrix $\hat{\rho}_{in}$:

$\displaystyle\hat{\rho}_{in}=|\psi\rangle\langle\psi|=\hat{A}^{\dagger}|0\rangle\langle
0|\hat{A}.$ (22)

The state of the qudit after applying the ideal gate $X_{d}^{m}$ can be
written as

$\displaystyle\hat{\rho}_{ideal}=\hat{X}_{d}^{m}|\psi\rangle\langle\psi|\hat{X}_{d}^{-m}=\hat{X}_{d}^{m}\hat{A}^{\dagger}|0\rangle\langle
0|\hat{A}\hat{X}_{d}^{-m}=$
$\displaystyle=\sum\limits_{l=1}^{d}C_{l}\hat{a}^{\dagger}_{l\oplus_{d}m}|0\rangle\langle
0|\sum\limits_{l=1}^{d}C^{*}_{l}\hat{a}_{l\oplus_{d}m}.$ (23)

The ideal transformation $\hat{X}_{d}^{m}$ ”shifts” the mode index and does
not change the coefficient $C_{l}$ – the contribution to the superposition of
each state with a certain OAM. Such a transformation has two essential
properties: it occurs in a deterministic way (if we are talking about states
with a few photons like (15)) and ”coherently” affects the state of the qudit,
consistently transforming each member of the superposition. At the same time,
we should ensure a ”uniform” change in all physical states encoding the same
logical qudit. We will subsequently evaluate the imperfect transformation,
which we will denote by $\hat{x}_{d}^{m}$, according to these criteria.

Moving on to the consideration of the nonideal transformation
$\hat{x}_{d}^{m}$, from the expression (14) we can conclude that the gate
$\hat{x}_{d}^{m}$ has a probabilistic nature, since the coefficients
$\chi_{l,m}$, which determine the efficiency of the transformation of the OAM
$l$ by the value $m$ is always less than one (see Fig. 2). In this case,
having at the input a pure state of qudit (22), at the output we will obtain a
statistical mixture described by the density matrix $\hat{\rho}_{out}$:

$\displaystyle\hat{\rho}_{out}=p_{1}\hat{\rho}_{1}+(1-p_{1})\hat{\rho}_{2},$
(24)
$\displaystyle\hat{\rho}_{1}=\frac{\hat{x}_{d}^{m}\hat{\rho}_{in}\hat{x}_{d}^{-m}}{\hbox{Tr}\left[\hat{x}_{d}^{m}\hat{\rho}_{in}\hat{x}_{d}^{-m}\right]}.$
(25)

Here we especially highlighted the first term corresponding to the realization
of the case when the entire state is successfully transformed, $p_{1}$ is the
probability of the transformation, $\hat{\rho}_{2}$ is the density matrix
corresponding to all other outcomes (when not all modes were transformed).
Renormalization to
$\hbox{Tr}\left[\hat{x}_{d}^{m}\hat{\rho}_{in}\hat{x}_{d}^{-m}\right]$ is
necessary to preserve the unit trace of the density matrix.

We will evaluate the conversion quality by the fidelity $F$, calculated as
follows:

$\displaystyle
F=\left(\hbox{Tr}\sqrt{\sqrt{\hat{\rho}_{ideal}}\hat{\rho}_{1}\sqrt{\hat{\rho}_{ideal}}}\right)^{2}.$
(26)

It should be noted that the expression above contains only the density matrix
$\hat{\rho}_{1}$, which corresponds to the outcome when a complete
transformation of all modes occurred, and the fidelity written in this way
shows only the degree of coherence of our transformation. The probability
$p_{1}$ of the outcome we are interested in should be calculated individually.
Based on the physical meaning of the density matrix $\hat{\rho}_{1}$, we can
write:

$\displaystyle p_{1}=\prod\limits_{l=1}^{d}\chi^{2}_{l,m}.$ (27)

Although the probabilities do not exceed 0.6, as shown in the next section,
this problem can be eliminated in the experiment using the methods of
postselection [22, 23].

Summarizing all of the above, we can emphasize several key aspects of the
problem under consideration. The application of high-dimensional systems
increases the channel’s information capacity and the cryptographic security of
the channel. However, as the dimension grows, the number of gates, which is
necessary to construct an arbitrary unitary transformation (see (9)), also
increases, and the probability of successful operation of the gate is reduced.
Next, we will evaluate the influence of various factors and compare the
results with a well-studied case of qubits.

###  IV.2 Computation of qutrit gates’ properties

This section will illustrate calculations of fidelity and probability for the
case of qutrit, which, moreover, can be easily generalized to systems of
higher dimensions. For space dimension $d=3$, the set of operations necessary
for constructing arbitrary unitary transformation includes gates
$\hat{X}_{3},\hat{X}_{3}^{2},\hat{Z}_{3},\hat{Z}_{3}^{2}$. As mentioned
earlier, the gates $\hat{Z}_{d}^{m}$ can be implemented using the Dove prism.

Based on the definition of (3), we can conclude that the transformation
$\hat{X}_{3}^{2}$ is equivalent to the transformation $\hat{X}_{3}^{-1}$, and
$\hat{X}_{3}$ is equivalent to $\hat{X}_{3}^{-2}$ in the way of acting on an
arbitrary state of qutrit, so in the further analysis, we will compare pairs
of transformations with positive and negative powers.

Let us recall that not only the high probability of the gate success or the
absolute values of the coefficients $\chi_{l,m}$, is crucial to building the
most optimal gate, but also the general ”coherence” of the transformation,
which can be formulated as the nearness of the values of the conversion
coefficients for different members of the superposition. For example, if we
work with the qutrit, which is encoded by the physical values of the OAM
$l-1,l,l+1$, then we must require the fulfilment of the following approximate
equality:

$\displaystyle\chi_{l-1,m}\approx\chi_{l,m}\approx\chi_{l+1,m}.$ (28)

Hence, a natural question arises: in what physical values of $l$ should the
qutrit be encoded so that the transformations of $\hat{x}_{3}^{m},m=\pm 1,\pm
2$ occur in the most optimal way? The calculation shows that as $l$ grows, the
conversion coefficients of different members of the superposition become
closer to each other. That is, the condition (28) is fulfilled the better, the
bigger is $l$. This result seems logical since $\chi_{l,m}$ is accounted with
geometric overlapping of the transverse spatial profiles of the modes (see
[18] for more details). However, the generation of qutrit states with large
values of $l$ is problematic from the experimental point of view. Therefore,
we are especially interested in cases when relatively high values of
probability and fidelity are attainable in the region of small $l$.

Figure 3: The probabilities of success of the qutrite gates $x_{3},x^{2}_{3}$
(top), $x^{-1}_{3},x^{-2}_{3}$ (bottom) depending on the values of $l$ and of
the parameter $z_{s}/z_{R}$ that controls the geometry of the modes. As one
can see, transformations with negative degrees have more attractive behaviour
in the region of small OAM.

Fig. 3 shows the probabilities of success of gates $x_{3},x^{2}_{3}$ and
$x^{-1}_{3},x^{-2}_{3}$. Areas of high probabilities, as can be seen from the
figure, show interesting behaviour. The regions expand with an increase of the
control parameter $z_{s}$ and an increase of the OAM $l$. Moreover, with an
increase in $l$, they become more and more bent towards the vertical. This
fact implies two remarkable properties of our transformation: firstly, for
large $l$ the probabilities will be high in a certain wide range of $z_{s}$,
and secondly, having an unknown state at the input of the system, we can fix
some specific value of $z_{s}$, which will provide an equally high probability
of the gate success, practically (in the limit) independent of the OAM values
of the input state when working in the region of large OAM values. However,
since today the generation of light with a high OAM remains an experimentally
laborious task, we will focus on the properties of transformations in the
region of small $l$. It can be noted that for $x^{-1}_{3},x^{-2}_{3}$ the
probabilities are higher than for the corresponding transformations with
positive powers and, more importantly, high probabilities can be achieved in
the region of small $l$, that is a significant advantage over conversions
$x_{3},x^{2}_{3}$.

One can also calculate fidelity values for these gates. The calculation shows
that the proposed conversion method shows extremely high fidelity ($F\geq
0.97$) in a wide range of values of the control parameter $z_{s}/z_{R}$ for
various values of $l$. Below we give specific values of the maximum fidelity
(by varying the control parameter) for qutrit with values of $l=4,5,6$:

$\displaystyle F_{x_{3}}=1-8,9\times 10^{-6},$ $\displaystyle
F_{x^{2}_{3}}=1-26,2\times 10^{-6},$ $\displaystyle F_{x^{-1}_{3}}=1-5,5\times
10^{-6},$ $\displaystyle F_{x^{-2}_{3}}=1-33,2\times 10^{-6}.$

Since the fidelity values turn out to be equally high for all transformations,
we can conclude that it is most reasonable to perform transformations of the
qutrit state with gates $x^{-1}_{3},x^{-2}_{3}$.

Similar patterns can be seen for the transformation of quantum systems with a
dimension greater than 3. Therefore, in the next section, we will compare the
parameters of different gates for high-dimensional systems with the case of a
qubit.

###  IV.3 Comparison of the gates for qudits with different dimension

We have performed calculations of transformation properties for qudits of
different dimensions (from $d=2$ to $5$) for the following states (the state
numbers $|l\rangle$ denote the physical value of the OAM projection):

$\displaystyle|\psi\rangle^{d=2}=\frac{1}{\sqrt{2}}\left(|4\rangle+|5\rangle\right),$
(29)
$\displaystyle|\psi\rangle^{d=3}=\frac{1}{\sqrt{3}}\left(|4\rangle+|5\rangle+|6\rangle\right),$
(30)
$\displaystyle|\psi\rangle^{d=4}=\frac{1}{\sqrt{4}}\left(|4\rangle+|5\rangle+|6\rangle+|7\rangle\right),$
(31)
$\displaystyle|\psi\rangle^{d=5}=\frac{1}{\sqrt{5}}\left(|4\rangle+|5\rangle+|6\rangle+|7\rangle+|8\rangle\right).$
(32)

Such projections of the OAM $l$ were chosen to encode the qudit state because
for small values of $l=0,\pm 1,\pm 2$, we will need to change the OAM of a
state with $l=0$ when performing transformations. These transformations occur
with low efficiency due to weak overlap of the transverse profiles of the
fundamental Gaussian mode with any LG modes with a nonzero OAM projection. The
transition to the region of higher OAM values, as Fig. 3 shows, gives even
higher values of probabilities, so the presented calculation can be considered
as a lower bound.

Table 1: Comparison of gate $x^{-1}_{d}$ parameters for qudits of different dimensions Qudit dim d | Probability P | Fidelity F | $\log_{2}d\times F\times P$
---|---|---|---
$2$ | $0,59$ | $1$ | $0,59$
$3$ | $0,44$ | $1-5,5\times 10^{-6}$ | $0,70$
$4$ | $0,32$ | $1-37,4\times 10^{-6}$ | $0,65$
$5$ | $0,23$ | $1-80,6\times 10^{-6}$ | $0,54$
Table 2: Comparison of gate $x^{-2}_{d}$ parameters for qudits of different dimensions Qudit dim d | Probability P | Fidelity F | $\log_{2}d\times F\times P$
---|---|---|---
$2$ | $0,64$ | $1$ | $0,64$
$3$ | $0,49$ | $1-33,2\times 10^{-6}$ | $0,78$
$4$ | $0,37$ | $1-218,0\times 10^{-6}$ | $0,74$
$5$ | $0,27$ | $1-457,9\times 10^{-6}$ | $0,62$

Table 1 shows a comparison of the properties of the $x_{d}^{-1}$
transformation for systems of various dimensions described by states
(29)-(32). It can be noted that with an increase in the dimension (for $d\geq
3$), both the probability of the gate success and the fidelity decrease, but
the latter decreases very insignificantly. At the same time, in the last
column of the table, the values of the product of fidelity, the probability,
and the information capacity of the channel – the number of bits of
information encoded by one qudit of dimension $d$, is shown. Thus, we wanted
to emphasize that the loss in probabilities turns out to be not as significant
as the gain in information capacity: indeed, it can be noted that for the case
of qutrit and ququart, the value constructed in this way turns out to be
larger than for other dimensions. Of particular interest are the results shown
in Table 2, where the properties of the $x_{d}^{-2}$ gate are calculated. Its
probability turns out to be greater than that of $x_{d}^{-1}$, which can be
quite easily explained by purely geometric reasons of overlap of modes with
close numbers.

Summarizing all of the above, we have shown that using a quantum memory scheme
as a base for implementing single-qudit gates of various dimensions has
significant advantages in extremely high fidelity and the ability to perform
the storing of quantum state simultaneously with its transformation. The
calculated probability of successful operation of the gates also remains high
enough, and the scheme can be easily supplemented using postselection methods.

##  V Conclusion

We have developed multidimensional quantum gates for qudits with OAM of
different dimensions, based on the transformation of light modes with OAM in a
quantum memory scheme. To ensure the cyclicity of the operations, we have
identified several indistinguishable logical sets of dimension $d$, encoding
one qudit on the set of ”physical” modes with OAM. Since the $\hat{Z}_{d}$
transformation over light with OAM can be performed using Dove prisms, we
focused on constructing the lossy gate $\hat{x}_{d}^{m}$ and comparing it with
the ideal one $\hat{X}_{d}^{m}$. We carried out the comparison by calculating
the probability of success and the fidelity of the transformation.

Calculations carried out for the particular case of the qudit dimension $d=3$
showed that the considered gates reveal a relatively high probability of
success (from 0.3 to 0.5). Moreover, for $\hat{x}^{-1}_{3},\hat{x}^{-2}_{3}$
high probability values are achieved in the region of small OAM $l$, which
seems to be a powerful argument in favor of working with transformations with
negative powers of $m$, since the experimental generation of light with high
values of $l$ has been currently facing with some difficulties. However, if
efficient work with large $l$ someday will be possible, then the proposed
gates can provide equal probabilities when transforming an unknown state for a
specific value of the control parameter $z_{s}$ in a wide range of values of
$l$. This effect is achieved since the change in the regions of high
probability with increasing $z_{s}$ approaches the vertical line more and
more. That is, the value of the coordinate $z_{s}$, providing the highest
probability, begins to depend weakly on the OAM values. A significant result
is that all transformations provide an exceptionally high level of fidelity
($F\geq 97$), which is also weakly dependent on the OAM.

We have compared the characteristics of the gates
$\hat{x}^{-1}_{3},\hat{x}^{-2}_{3}$ for different dimensions of qudits with
the trivial case of a qubit. The values of probability and fidelity were taken
into account, along with the potential gain from information capacity
increased with the dimension of a qudit. An estimation based on all these
factors shows that working with qudits of dimensions $d=3$ and $d=4$, despite
the lower probabilities, turns out to be preferable for performing quantum
computations in the proposed protocol. A feature of the proposed
transformations is that one can carry out computational procedures
simultaneously with storing quantum-statistical properties of light in a
memory cell.

This work was supported by the RFBR (grants 19-32-90059, 19-02-00204) and by
the Foundation for the Advancement of Theoretical Physics and Mathematics
”BASIS” (grant 20-1-5-120-1).

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249731291.01 | 15 | 10.2053773 | 6.310 | 3160.91941 | 0.0245 | 12.92 | 0.97 | - | Y
249731291.02 | 15 | 23.3367386 | 5.824 | 3179.76842 | 0.0226 | 12.92 | 0.97 | - | Y
249822095.01 | 15 | 1.69671578 | 5.738 | 3157.01732 | 0.0264 | 12.80 | 1.13 | - | Y
250106132.01 | 15 | 22.1223653 | 3.715 | 3157.32569 | 0.0351 | 11.19 | 0.90 | Y | -
211396431.01 | 16 | 8.51418384 | 38.30 | 3266.73236 | 0.0418 | 11.77 | 1.18 | - | Y
251304634.01 | 16 | 4.39250294 | 1.110 | 3264.36688 | 0.0152 | 12.25 | 1.02 | Y | -
251399501.01 | 16 | 11.6349883 | 2.605 | 3272.15823 | 0.0218 | 12.10 | 1.25 | Y | -
212479997.01 | 17 | 8.9317272 | 9.390 | 3348.03092 | 0.0478 | 10.80 | 1.07 | - | -
211408558.01 | 18 | 9.96768882 | 2.005 | 3427.65262 | 0.0178 | 11.51 | 1.13 | - | Y

##  5 Conclusions

In total, we validate 60 new _K2_ planets in 46 systems, out of 91 total
candidates. This is a validation rate of 66%, having already down-selected the
full catalog in Paper IV to the most likely planetary signals by visual
inspection. The most promising new planets for transmission spectroscopy are
K2-384 f, K2-387 b, K2-390 b, K2-403 b, and K2-398 c, while the most promising
new planets for emission spectroscopy are K2-371 b, K2-370 b, and K2-399 b.
K2-406 band K2-405 bshow interesting TSM and ESM values, making them
potentially valuable atmospheric characterization targets. In addition to
these new planets, in Table 7 we highlight 37 planet candidates orbiting
bright host stars that pass visual inspection and warrant further follow-up.

We note that many additional planets remain in the _K2_ data. Besides the
planet candidates listed above, Paper IV also includes 109 planet candidates
that pass visual inspection orbiting stars $13<Kp<15$, and 90 planet
candidates that pass visual inspection orbiting stars $Kp>15$, that require
additional follow-up observations. There are also hundreds of additional _K2_
planet candidates at the NASA Exoplanet Archive; although Paper IV presented a
large, uniform catalog of planet candidates, comparison with larger catalogs
such as Kruse et al. (2019) show that our catalog was incomplete. Given the
plethora of archival data from multiple missions now available, there remain
strong reasons to pursue _K2_ planet confirmation. There are significant
ongoing and upcoming opportunities to exploit and expand upon the _K2_ data,
including: (i) a re-observation of the ecliptic plane by the TESS mission
(Ricker et al., 2015) during its first extended mission; (ii) opportunities
for measuring transit timing variations or searching for additional transits
with CHEOPS (Broeg et al., 2013); (iii) a new generation of extreme precision
radial velocity instruments to map out the composition of different classes
and sub-classes of planets; and (iv) opportunities for characterizing
exoplanet atmospheres to an unprecedented level of detail with JWST and high-
resolution ground-based spectroscopy for statistical samples of planets.

##  6 Acknowledgements

The analysis described here were performed on the UCLA Hoffman2 shared
computing cluster and using the resources provided by the Bhaumik Institute.
This research has made use of the NASA Exoplanet Archive and the Exoplanet
Follow-up Observation Program website, which are operated by the California
Institute of Technology, under contract with the National Aeronautics and
Space Administration under the Exoplanet Exploration Program. This publication
makes use of data products collected by the _Kepler_ mission and obtained from
the MAST data archive at the Space Telescope Science Institute (STScI); the
K2SFF apertures can be accessed via https://doi.org/10.17909/T9BC75 (catalog
https://doi.org/10.17909/T9BC75), and the K2 target pixel files via
https://doi.org/10.17909/T9K30X (catalog https://doi.org/10.17909/T9K30X).
STScI is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS5–26555. Support to MAST for these
data is provided by the NASA Office of Space Science via grant NAG5–7584 and
by other grants and contracts. Funding for the _Kepler_ mission was provided
by the NASA Science Mission Directorate. This publication makes use of data
products from the Two Micron All Sky Survey, which is a joint project of the
University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation. This dataset is
made available by the Infrared Science Archive (IRSA) at IPAC, which is
operated by the California Institute of Technology under contract with the
National Aeronautics and Space Administration. The specific data products can
be accessed via https://doi.org/10.26131/IRSA2 (catalog
https://doi.org/10.26131/IRSA2).

Funding for this work for CH is provided by grant number 80NSSC20K0874,
through NASA ROSES. M.T. is supported by JSPS KAKENHI grant
Nos.18H05442,15H02063, and 22000005. Support for this work was provided by
NASA through the NASA Hubble Fellowship grant #51497.001 awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555. The material is based upon work supported by NASA under award
number 80GSFC21M0002 and by the GSFC Sellers Exoplanet Environments
Collaboration (SEEC), which is funded by the NASA Planetary Science Division’s
Internal Scientist Funding Mode.

The observations in the paper made use of the NN-EXPLORE Exoplanet and Stellar
Speckle Imager (NESSI). NESSI was funded by the NASA Exoplanet Exploration
Program and the NASA Ames Research Center. NESSI was built at the Ames
Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett
Quigley. The authors are honored to be permitted to conduct observations on
Iolkam Du’ag (Kitt Peak), a mountain within the Tohono O’odham Nation with
particular significance to the Tohono O’odham people.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

This work made use of the gaia-kepler.fun crossmatch database created by Megan
Bedell.

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##  Appendix A Validated planets

###  A.1 EPIC 204750116

K2-365 b is a sub-Neptune (2.906 $R_{\oplus}$) planet orbiting a bright
($V$=11.37 mag, $K_{s}$=9.62 mag) G4V star (1.05 $R_{\odot}$, 0.82
$M_{\odot}$) observed in Campaign 2. It was noted as EPIC 204750116.01 by
Vanderburg et al. (2016), Rizzuto et al. (2018), Mayo et al. (2018), and Kruse
et al. (2019), and it orbits the star at a distance of 0.150 au, with a period
of 23.448172 days and an equilibrium temperature of 710 K. The host star has a
clean, single-lined FLWO/TRES spectrum, and Keck/NIRC2 imaging which shows no
contaminating stellar companions. The vespa FPP value is 0.0016, without
requiring use of the available contrast curves. The centroid p-value is
0.5219, which is consistent with the source of the transiting signal being on
the target star. Using the Chen & Kipping (2017) mass-radius relation predicts
a mass of $\sim$8.7 $M_{\oplus}$. This would result in a measurable RV semi-
amplitude (_K_ $\sim$2.2 m/s), but predicts TSM ($\sim$22) and ESM ($\sim$1.3)
values that indicate atmospheric follow-up would be challenging, compared to
the thresholds recommended by Kempton et al. (2018). We note that our pipeline
was able to marginally detect the shorter period candidate, EPIC 204750116.02,
reported by Kruse et al. (2019), but with insufficient signal-to-noise to
attempt validation, and it remains a candidate.

###  A.2 EPIC 205111664

K2-366 is a moderately bright ($V$=12.49 mag, $K_{s}$=9.88 mag) G4V star (0.94
$R_{\odot}$, 0.83 $M_{\odot}$) that was observed in Campaign 2. It is orbited
by a sub-Neptune-sized planet that was previously noted as EPIC 205111664.01
by Vanderburg et al. (2016) and Kruse et al. (2019). K2-366 b is a 2.359
$R_{\oplus}$ planet, orbiting at a distance of 0.116 au, with a period of
15.937039 days and an equilibrium temperature of $\sim$770 K. The host star
has a clean, single-lined FLWO/TRES spectrum, and Keck/NIRC2 AO imaging that
shows no contaminating stellar companions. The vespa FPP value is $1.40\times
10^{-4}$, without requiring use of the contrast curve. The centroid p-value is
0.4962, which is consistent with the source of the transiting signal being on
the target star. Using the Chen & Kipping (2017) mass-radius relation predicts
a mass of $\sim$6.3 $M_{\oplus}$. This would result in a challenging RV semi-
amplitude (_K_ $\sim$1.8 m/s), and predicts TSM ($\sim$18) and ESM ($\sim$1.1)
values that indicate atmospheric follow-up would be challenging, compared to
the thresholds recommended by Kempton et al. (2018).

###  A.3 EPIC 206055981

K2-367 is a moderately faint ($V$=13.80 mag, $K_{s}$=10.96 mag) K5V star (0.63
$R_{\odot}$, 0.76 $M_{\odot}$) that was observed in Campaign 3. It is orbited
by a sub-Neptune-size planet, K2-367 b which was previously identified as EPIC
206055981.01 by Vanderburg et al. (2016) and Kruse et al. (2019). It is a
2.413 $R_{\oplus}$ planet, orbiting at a distance of 0.134 au, with a period
of 20.645099 days and a relatively cool (for our sample) equilibrium
temperature of $\sim$450 K. The host star has clean, single-lined Keck/HIRES
and IRTF/SpeX spectra, and WIYN/NESSI speckle imaging at 562nm and 832nm that
show no contaminating stellar companions. The vespa FPP value is 0.000721,
using the available contrast curves. The centroid p-value is 0.5842, which is
consistent with the source of the transiting signal being on the target star.
Using the Chen & Kipping (2017) mass-radius relation predicts a mass of
$\sim$6.5$M_{\oplus}$. This would result in a challenging RV semi-amplitude
(_K_ $\sim$1.8m/s), especially given the faint host star, and predicts TSM
($\sim$15) and ESM ($\sim$0.4) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.4 EPIC 206146957

K2-369 is a bright ($V$=11.79 mag, $K_{s}$=9.93 mag) G3V star (0.88
$R_{\odot}$, 0.74 $M_{\odot}$) that was observed in Campaign 3. It is orbited
by a super-Earth-size planet, K2-369 b which was previously identified as EPIC
206146957.01 by Vanderburg et al. (2016), Mayo et al. (2018), and Kruse et al.
(2019). It is a 1.313 $R_{\oplus}$ planet, orbiting at a distance of 0.057 au,
with a period of 5.761612 days and an equilibrium temperature of $\sim$1080 K.
The host star has a clean, single-lined FLWO/TRES spectra, and Palomar/PHARO
AO imaging that shows no contaminating stellar companions. The vespa FPP value
is $3.02\times 10^{-5}$, using the available contrast curve. The centroid
p-value is 0.4584, which is consistent with the source of the transiting
signal being on the target star. Using the Chen & Kipping (2017) mass-radius
relation predicts a mass of $\sim$2.2$M_{\oplus}$. This would result in a
challenging RV semi-amplitude (_K_ $\sim$1.0m/s), and predicts TSM ($\sim$2.3)
and ESM ($\sim$0.8) values that indicate atmospheric follow-up would be
challenging, compared to the thresholds recommended by Kempton et al. (2018).

###  A.5 EPIC 206317286

EPIC 206317286 (K2-303) is a moderately faint ($V$=14.05 mag, $K_{s}$=11.64
mag) K3V star (0.76 $R_{\odot}$, 0.83 $M_{\odot}$) that was observed in
Campaign 3. It was previously found to host a short period (1.58 d) Earth-
sized (0.96 $R_{\oplus}$) planet, EPIC 206317286 b (EPIC 206317286.02) (Heller
et al., 2019). An additional longer period candidate, EPIC 206317286.01, was
noted by Vanderburg et al. (2016) and Kruse et al. (2019), and was recovered
by our pipeline. EPIC 206317286 c is a 2.464 $R_{\oplus}$ sub-Neptune-sized
planet, orbiting at a distance of 0.124 au, with a period of 17.515472 days
and an equilibrium temperature of $\sim$570 K. The host star has a clean,
single-lined IRTF/SpeX spectrum. Heller et al. (2019) validated EPIC 206317286
b in part by using Pan-STARRS imaging (with a resolution down to 200
mas/pixel) to eliminate background flux contaminants; we therefore did not
obtain additional high-resolution imaging. The vespa FPP value of EPIC
206317286 c is $8.16\times 10^{-5}$ using the Campaign 3 multiplicity boost.
The centroid p-value is 0.1811, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of planet c of $\sim$5.4 $M_{\oplus}$.
This would result in a challenging RV semi-amplitude (_K_ $\sim$1.5 m/s), and
predicts TSM ($\sim$8.0) and ESM ($\sim$0.3) values that indicate atmospheric
follow-up would be challenging, compared to the thresholds recommended by
Kempton et al. (2018).

###  A.6 EPIC 211490999

K2-372 is a moderately faint ($V$=13.60 mag, $K_{s}$=11.87 mag) G5V star (0.92
$R_{\odot}$, 0.85 $M_{\odot}$) that was observed in Campaigns 5, 16, and 18.
It is orbited by a sub-Neptune-sized planet, K2-372 b previously noted as EPIC
211490999.01 by Pope et al. (2016), Petigura et al. (2018a), Kruse et al.
(2019), and de Leon et al. (2021). It is a 2.945 $R_{\oplus}$ planet, orbiting
at a distance of 0.085 au, with a period of 9.844077 days and an equilibrium
temperature of $\sim$870 K. The host star has clean, single-lined Keck/HIRES
and McDonald/TS23 spectra, and Gemini/NIRI AO and WIYN/NESSI speckle imaging
at 562nm and 832nm that show no contaminating stellar companions. The vespa
FPP value is $3.39\times 10^{-3}$, using the available contrast curves. The
centroid p-value is 0.4752, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$9.2 $M_{\oplus}$. This would
result in a measurable RV semi-amplitude (_K_ $\sim$3.1 m/s), but predicts TSM
($\sim$13) and ESM ($\sim$1.0) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.7 EPIC 211539054

K2-373 is a bright ($V$=10.47 mag, $K_{s}$=9.16 mag) G1V star (1.49
$R_{\odot}$, 1.16 $M_{\odot}$) that was observed in Campaign 5. It is orbited
by a sub-Neptune-size planet, K2-373 b which was first noted by Kruse et al.
(2019) as EPIC 211539054.01. It is a 2.061 $R_{\oplus}$ planet, orbiting at a
distance of 0.102 au, with a period of 11.020660 days and an equilibrium
temperature of $\sim$1130 K. The host star has a clean, single-lined FLWO/TRES
spectra, and Palomar/PHARO AO imaging that shows no contaminating stellar
companions. The vespa FPP value is 0.00213, using the available contrast
curve. The centroid p-value is 0.3779, which is consistent with the source of
the transiting signal being on the target star. Using the Chen & Kipping
(2017) mass-radius relation predicts a mass of $\sim$5.2$M_{\oplus}$. This
would result in a challenging RV semi-amplitude (_K_ $\sim$1.3m/s), and
predicts TSM ($\sim$14) and ESM ($\sim$1.0) values that indicate atmospheric
follow-up would be challenging, compared to the thresholds recommended by
Kempton et al. (2018).

###  A.8 EPIC 211732116

K2-374 is a moderately bright ($V$=12.80 mag, $K_{s}$=11.29 mag) G3V star
(0.91 $R_{\odot}$, 0.73 $M_{\odot}$) that was observed in Campaign 16. It is
orbited by a super-Earth and a sub-Neptune, which were first noted in Paper
IV. K2-374 b (EPIC 211732116.01) is a 2.597 $R_{\oplus}$ planet, orbiting at a
distance of 0.055 au, with a period of 4.521953 days and a high equilibrium
temperature of $\sim$1210 K. K2-374 c (EPIC 211732116.02) is a 2.213
$R_{\oplus}$ planet, orbiting at a distance of 0.114 au, with a period of
16.434450 days and an equilibrium temperature of $\sim$790 K. The host star
has a clean, single-lined FLWO/TRES spectrum, and Keck/NIRC2 AO imaging that
shows no contaminating stellar companions. The vespa FPP values of planets b
and c are $6.80\times 10^{-5}$ and $2.42\times 10^{-3}$ respectively, using
the available contrast curve and the multiplicity boost. The centroid p-values
are 0.5403 and 0.5749 respectively, consistent with the source of the
transiting signals being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts masses of $\sim$3.6 $M_{\oplus}$ and $\sim$5.7
$M_{\oplus}$ for planets b and c respectively. These mass estimates for the
two planets result in challenging RV semi-amplitudes ($K$ = 1.7 m/s, and 1.8
m/s), and TSM ($\sim$10, $\sim$11) and ESM ($\sim$0.7, $\sim$0.6) values below
the thresholds of interest recommended by Kempton et al. (2018).

###  A.9 EPIC 212006318

K2-375 is a moderately faint ($V$=13.04 mag, $K_{s}$=11.56 mag) G2V star (1.54
$R_{\odot}$, 1.12 $M_{\odot}$) that was observed in Campaign 3. It is orbited
by a sub-Neptune-sized planet, K2-375 b, which was previously noted as EPIC
212006318.01 by Pope et al. (2016), Mayo et al. (2018), Kruse et al. (2019),
and de Leon et al. (2021). K2-375 b is a 2.243 $R_{\oplus}$ planet, orbiting
at a distance of 0.121 au, with a period of 14.453895 days and an equilibrium
temperature of $\sim$1000 K. The host star has clean, single-lined Keck/HIRES
and FLWO/TRES spectra, and Palomar/PHARO AO and WIYN/NESSI speckle imaging at
562nm and 832nm that show no contaminating stellar companions. The vespa FPP
value is $4.11\times 10^{-3}$, without using the available contrast curves.
The centroid p-value is 0.1458, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$5.8$M_{\oplus}$. This would
result in a challenging RV semi-amplitude (_K_ $\sim$1.4 m/s), and predicts
TSM ($\sim$4.3) and ESM ($\sim$0.31) values that indicate atmospheric follow-
up would be challenging, compared to the thresholds recommended by Kempton et
al. (2018).

###  A.10 EPIC 212222383

K2-376 is a bright ($V$=10.42 mag, $K_{s}$=9.01 mag) G1V star (1.15
$R_{\odot}$, 0.91 $M_{\odot}$) that was observed in Campaign 16. It is orbited
by a super-Earth-size planet that was first noted in Paper IV as EPIC
212222383.01. K2-376 b is a 1.736 $R_{\oplus}$ planet, orbiting at a distance
of 0.061 au, with a period of 5.776475 days and an equilibrium temperature of
$\sim$1240 K. The host star has a clean, single-lined FLWO/TRES spectrum, and
Palomar/PHARO AO imaging that shows no contaminating stellar companions. The
vespa FPP value is 0.00170, using the available contrast curve. The centroid
p-value is 0.6183, which is consistent with the source of the transiting
signal being on the target star. Using the Chen & Kipping (2017) mass-radius
relation predicts a mass of $\sim$4.1$M_{\oplus}$. This would result in a
challenging RV semi-amplitude (_K_ $\sim$1.6m/s), and predicts TSM ($\sim$21)
and ESM ($\sim$1.6) values that indicate atmospheric follow-up would be
challenging, compared to the thresholds recommended by Kempton et al. (2018).

###  A.11 EPIC 212530118

K2-377 is a moderately faint ($V$=14.05 mag, $K_{s}$=10.99 mag) K7V star (0.69
$R_{\odot}$, 0.82 $M_{\odot}$) that was observed in Campaign 6. It is orbited
by a super-Earth-sized planet that was previously noted as EPIC 212530118.01
by Barros et al. (2016) and Kruse et al. (2019). K2-377 b is a 1.693
$R_{\oplus}$ planet, orbiting at a distance of 0.100 au, with a period of
12.932235 days and an equilibrium temperature of $\sim$530 K. The host star
has a clean, single-lined IRTF/SpeX spectrum, and WIYN/NESSI speckle imaging
at 692nm and 880nm that show no contaminating stellar companions. The vespa
FPP value is $1.14\times 10^{-3}$, using the available contrast curves. The
centroid p-value is 0.5774, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$3.9$M_{\oplus}$. This would
result in a challenging RV semi-amplitude (_K_ $\sim$1.2 m/s), and predicts
TSM ($\sim$7.7) and ESM ($\sim$0.3) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.12 EPIC 212575828

K2-378 is a faint ($V$=15.79 mag, $K_{s}$=13.39 mag) K3V star (0.73
$R_{\odot}$, 0.92 $M_{\odot}$), observed in Campaign 6, that is orbited by a
sub-Neptune-sized planet. K2-378 b, previously noted by Pope et al. (2016),
Crossfield et al. (2016), and Kruse et al. (2019) as EPIC 212575828.01, is a
2.880 $R_{\oplus}$ planet, orbiting at a distance of 0.031 au, with a period
of 2.060438 days and an equilibrium temperature of $\sim$1135 K. The host star
has a clean, single-lined Hale/TripleSpec spectrum, and Gemini/DSSI speckle
imaging at both 692nm and 880nm that show no contaminating stellar companions.
The vespa FPP value is $3.78\times 10^{-3}$, using the available contrast
curves. The centroid p-value is 0.4968, which is consistent with the source of
the transiting signal being on the target star. Using the Chen & Kipping
(2017) mass-radius relation predicts a mass of $\sim$9.0 $M_{\oplus}$. This
would result in a measurable RV semi-amplitude (_K_ $\sim$4.5 m/s), albeit
around a faint target, but predicts TSM ($\sim$12) and ESM ($\sim$1.6) values
that indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018).

###  A.13 EPIC 214173069

K2-379 is a moderately faint ($V$=13.21 mag, $K_{s}$=10.51 mag) K3V star (0.75
$R_{\odot}$, 0.86 $M_{\odot}$) that was observed in Campaign 7. It is orbited
by a sub-Neptune-sized planet, K2-379 b which was first noted as EPIC
214173069.01 by Kruse et al. (2019). It is a 2.181 $R_{\oplus}$ planet,
orbiting at a distance of 0.079 au, with a period of 8.777247 days and an
equilibrium temperature of $\sim$680 K. The host star has clean, single-lined
IRTF/SpeX and FLWO/TRES spectra, and Gemini/DSSI speckle imaging at 692nm and
880nm that show no contaminating stellar companions. The vespa FPP value is
$5.38\times 10^{-3}$, using the available contrast curves. The centroid
p-value is 0.8487, which is consistent with the source of the transiting
signal being on the target star. Using the Chen & Kipping (2017) mass-radius
relation predicts a mass of $\sim$5.6$M_{\oplus}$. This would result in a
challenging RV semi-amplitude (_K_ $\sim$1.9 m/s), and predicts TSM ($\sim$17)
and ESM ($\sim$0.8) values that indicate atmospheric follow-up would be
challenging, compared to the thresholds recommended by Kempton et al. (2018).
We note that our pipeline did not recover the smaller (1.4$R_{\oplus}$),
longer period (14.1297-day) candidate, EPIC 214173069.02, reported by Kruse et
al. (2019).

###  A.14 EPIC 214419545

K2-380 is a bright ($V$=11.65 mag, $K_{s}$=9.68 mag) F8V star (1.32
$R_{\odot}$, 0.99 $M_{\odot}$) that was observed in Campaign 7. It is orbited
by a sub-Neptune-sized planet, K2-380 b, which was first noted as EPIC
214419545.01 by Kruse et al. (2019). It is a 2.729 $R_{\oplus}$ planet,
orbiting at a distance of 0.087 au, with a period of 9.401312 days and an
equilibrium temperature of $\sim$1110 K. The host star has a clean, single-
lined FLWO/TRES spectrum, and Keck/NIRC2 AO imaging that show no contaminating
stellar companions. The vespa FPP value is $8.82\times 10^{-3}$, without using
the available contrast curves. The centroid p-value is 0.6540, which is
consistent with the source of the transiting signal being on the target star.
Using the Chen & Kipping (2017) mass-radius relation predicts a mass of
$\sim$8.0$M_{\oplus}$. This would result in a measurable RV semi-amplitude
(_K_ $\sim$2.4 m/s), but predicts TSM ($\sim$19) and ESM ($\sim$1.8) values
that indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018).

###  A.15 EPIC 217977895

K2-382 b is a sub-Neptune (2.072 $R_{\oplus}$) planet in the small planet
radius valley, orbiting a moderately faint ($V$=12.98 mag, $K_{s}$=11.05 mag)
G8V star (0.81 $R_{\odot}$, 0.92 $M_{\odot}$). It was observed in Campaign 7,
and was first noted in Mayo et al. (2018) and Kruse et al. (2019) as EPIC
217977895.01. It orbits the star at a distance of 0.148 au, with a period of
21.700156 days and an equilibrium temperature of 615 K. The host star has a
clean, single-lined FLWO/TRES spectrum, and Gemini/DSSI speckle imaging at 692
nm and 880 nm which show no contaminating stellar companions. The vespa FPP
value is $2.62\times 10^{-3}$ using the available contrast curves. The
centroid p-value is 0.3721, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$5.2 $M_{\oplus}$. This would
result in a challenging RV semi-amplitude (_K_ $\sim$1.3 m/s), and predicts
TSM ($\sim$10) and ESM ($\sim$0.4) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.16 EPIC 218668602

K2-383 is a moderately bright ($V$=12.68 mag, $K_{s}$=10.46 mag) K0V star
(0.80 $R_{\odot}$, 1.06 $M_{\odot}$) that was observed in Campaign 7. It is
orbited by a super-Earth-sized planet, K2-383 b which was previously noted by
Mayo et al. (2018) and Kruse et al. (2019) at EPIC 218668602.01. It is a 1.564
$R_{\oplus}$ planet, orbiting at a distance of 0.030 au, with a period of
1.865962 days and an equilibrium temperature of $\sim$1270 K. The host star
has a clean, single-lined FLWO/TRES spectrum, and Keck/NIRC2 AO imaging that
show no contaminating stellar companions. The vespa FPP value is $2.12\times
10^{-3}$, without using the available contrast curves. The centroid p-value is
0.1391, which is consistent with the source of the transiting signal being on
the target star. Using the Chen & Kipping (2017) mass-radius relation predicts
a mass of $\sim$3.3$M_{\oplus}$. This would result in a challenging RV semi-
amplitude (_K_ $\sim$1.6 m/s), but predicts TSM ($\sim$18) and ESM ($\sim$1.7)
values that indicate atmospheric follow-up would be challenging, compared to
the thresholds recommended by Kempton et al. (2018).

###  A.17 EPIC 220459477

K2-385 b is a super-Earth (1.539 $R_{\oplus}$) planet orbiting a moderately
faint ($V$=14.65 mag, $K_{s}$=12.24 mag) K3V star that was observed in
Campaign 8. It was first noted as EPIC 220459477.01 by Kruse et al. (2019). It
orbits the star at a distance of 0.034 au, with a period of 2.380867 days and
an equilibrium temperature of $\sim$1100 K. The host star has a clean, single-
lined TripleSpec spectrum, and WIYN/NESSI speckle imaging at both 562nm and
832nm that show no contaminating stellar companions. The vespa FPP value is
$1.09\times 10^{-5}$ using the available contrast curves. The centroid p-value
is 0.1107, which is consistent with the source of the transiting signal being
on the target star. Using the Chen & Kipping (2017) mass-radius relation
predicts a mass of $\sim$3.2 $M_{\oplus}$. This would result in a challenging
RV semi-amplitude (_K_ $\sim$1.6 m/s), especially given the faintness of the
target, and predicts TSM ($\sim$8) and ESM ($\sim$0.7) values that indicate
atmospheric follow-up would be challenging, compared to the thresholds
recommended by Kempton et al. (2018).

###  A.18 EPIC 220510874

K2-386 is a moderately faint ($V$=13.17 mag, $K_{s}$=11.52 mag) F9V star (0.97
$R_{\odot}$, 0.91 $M_{\odot}$) that was observed in Campaign 8. It is orbited
by a sub-Neptune-sized planet, K2-386 b which was previously noted as EPIC
220510874.01 by Kruse et al. (2019) and Zink et al. (2019). K2-386 b is a
2.320 $R_{\oplus}$ planet, orbiting at a distance of 0.073 au, with a period
of 7.473223 days and an equilibrium temperature of $\sim$1010 K. The host star
has a clean, single-lined McDonald TS23 spectrum, and WIYN/NESSI speckle
imaging at 562nm and 832nm that show no contaminating stellar companions. The
vespa FPP value is $3.34\times 10^{-5}$, using the available contrast curves.
The centroid p-value is 0.6971, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$6.2$M_{\oplus}$. This would
result in a measurable RV semi-amplitude (_K_ $\sim$2.2 m/s), but predicts TSM
($\sim$12) and ESM ($\sim$0.9) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.19 EPIC 245943455

K2-388 b is a sub-Neptune (3.660 $R_{\oplus}$) planet orbiting a moderately
bright ($V$=12.82 mag, $K_{s}$=10.93 mag) G8V star (0.90 $R_{\odot}$, 0.84
$M_{\odot}$). The target was observed in Campaign 12, and the transiting
signal was previously noted in Dattilo et al. (2019). The planet orbits the
star at a distance of 0.063 au, with a period of 6.339069 days and an
equilibrium temperature of $\sim$970 K. The host star has a clean, single-
lined Keck/HIRES spectrum, and Palomar/PHARO, Gemini/NIRI, and WIYN/NESSI
imaging which all show no contaminating stellar companions. The vespa FPP
value is $8.02\times 10^{-8}$, using the available contrast curves. The
centroid p-value is 0.0938, which is moderately inconsistent with the source
of the transiting signal being on the target star at the $\sim$2-$\sigma$
level, but does not fall below our $p=0.05$ threshold for revoking a planet’s
validation status. In Dattilo et al. (2019), they note that transit depth
changes as a function of aperture size for this signal, which may indicate a
contaminating background star. Although this planet passes our statistical
thresholds for validation, it may warrant additional ground-based time series
photometry to confirm the transit is indeed on the target star. Using the Chen
& Kipping (2017) mass-radius relation predicts a mass of $\sim$13.1
$M_{\oplus}$. This would result in a measurable RV semi-amplitude (_K_
$\sim$5.1 m/s), but predicts TSM ($\sim$30a) and ESM ($\sim$3.4) values that
indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018).

###  A.20 EPIC 245991048

K2-389 is a moderately bright ($V$=12.30 mag, $K_{s}$=10.18 mag) G3V star
(1.08 $R_{\odot}$, 0.80 $M_{\odot}$). It was observed in Campaign 12 and is
orbited by two sub-Neptunes, which were first noted Paper IV. K2-389 b is a
2.118 $R_{\oplus}$ planet, orbiting at a distance of 0.076 au, with a period
of 8.583566 days and an equilibrium temperature of $\sim$1050 K. Using the
Chen & Kipping (2017) mass-radius relation predicts a mass of $\sim$5.7
$M_{\oplus}$. This would result in a measurable RV semi-amplitude (_K_
$\sim$2.0 m/s), but predicts TSM ($\sim$16) and ESM ($\sim$1.3) values that
indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018). K2-389 c is a 2.040
$R_{\oplus}$ planet, orbiting at a distance of 0.138 au, with a period of
20.851337 days and an equilibrium temperature of $\sim$780 K. Using the Chen &
Kipping (2017) mass-radius relation predicts a mass of $\sim$5.3 $M_{\oplus}$.
This would result in a challenging RV semi-amplitude (_K_ $\sim$1.4 m/s), and
predicts TSM ($\sim$11) and ESM ($\sim$0.6) values that indicate atmospheric
follow-up would be challenging, compared to the thresholds recommended by
Kempton et al. (2018). The host star has a clean, single-lined FLWO/TRES
spectrum, as well as Gemini/NIRI and Palomar/PHARO AO imaging and WIYN/NESSI
speckle imaging that show no contaminating stellar companions. The vespa FPP
value for planet b is $6.41\times 10^{-5}$ using the available contrast curves
and the application of the Campaign 12 multiplicity boost. The centroid
p-value for planet b of is 0.2457, which is consistent with the source of the
transiting signal being on the target star. The vespa FPP value for planet c
is $5.75\times 10^{-5}$ using the available contrast curves and the
application of the Campaign 12 multiplicity boost. The centroid p-value for
planet c of is 0.2053, which is consistent with the source of the transiting
signal being on the target star.

###  A.21 EPIC 246074314

K2-391 b is a near-Earth-sized (1.371 $R_{\oplus}$) planet orbiting a
moderately faint ($Kep$=12.36 mag, $K_{s}$=10.71 mag) G7V star (0.57
$R_{\odot}$, 0.76 $M_{\odot}$), observed in Campaign 12. It was first noted in
Paper IV; it orbits the star at a distance of 0.049 au, with a period of
4.622654 days and an equilibrium temperature of $\sim$900 K. The host star has
a clean, single-lined FLWO/TRES spectrum, and WIYN/NESSI speckle imaging at
562 nm and 832 nm which show no contaminating stellar companions. The vespa
FPP value is $9.87\times 10^{-5}$ using the available contrast curves. The
centroid p-value is 0.3581, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$2.5 $M_{\oplus}$. This would
result in a challenging RV semi-amplitude (_K_ $\sim$1.2 m/s), and predicts
TSM ($\sim$3.3) and ESM ($\sim$1.1) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.22 EPIC 246084398

K2-392 b is a super-Earth (1.850 $R_{\oplus}$) planet orbiting a moderately
bright ($V$=12.82 mag, $K_{s}$=11.24 mag) G3V star (1.06 $R_{\odot}$, 0.84
$M_{\odot}$), observed in Campaign 12. It was first noted in Paper IV; it
orbits the star at a distance of 0.063 au, with a period of 15.399 days and an
equilibrium temperature of $\sim$840 K. The host star has a clean, single-
lined FLWO/TRES spectrum, and WIYN/NESSI speckle imaging at 562 nm and 832 nm
which show no contaminating stellar companions. The vespa FPP value is
$2.12\times 10^{-3}$, using the available contrast curves. The centroid
p-value is 0.5614, which is consistent with the source of the transiting
signal being on the target star. Using the Chen & Kipping (2017) mass-radius
relation predicts a mass of $\sim$4.5 $M_{\oplus}$. This would result in a
challenging RV semi-amplitude (_K_ $\sim$1.3 m/s), and predicts TSM
($\sim$6.4) and ESM ($\sim$0.4) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.23 EPIC 246429049

K2-393 is a bright ($V$=11.84 mag, $K_{s}$=10.28 mag) F9V star (1.07
$R_{\odot}$, 0.88 $M_{\odot}$) that was observed in Campaign 12. It is orbited
by a sub-Neptune-sized planet, K2-393 b which was previously noted by Zink et
al. (2019) as EPIC 246429049.01. It is a 2.352 $R_{\oplus}$ planet, orbiting
at a distance of 0.089 au, with a period of 10.413181 days and an equilibrium
temperature of $\sim$950 K. The host star has a clean, single-lined FLWO/TRES
spectrum, and WIYN/NESSI speckle imaging at 562nm and 832nm that show no
contaminating stellar companions. The vespa FPP value is $9.00\times 10^{-3}$,
using the available contrast curves. The centroid p-value is 0.3512, which is
consistent with the source of the transiting signal being on the target star.
Using the Chen & Kipping (2017) mass-radius relation predicts a mass of
$\sim$6.3$M_{\oplus}$. This would result in a measurable RV semi-amplitude
(_K_ $\sim$2.0m/s), but predicts TSM ($\sim$16) and ESM ($\sim$1.1) values
that indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018).

###  A.24 EPIC 246891819

K2-395 is a moderately faint ($V$=14.67 mag, $K_{s}$=11.37 mag) K0V star (0.78
$R_{\odot}$, 0.95 $M_{\odot}$) observed in Campaign 13 that is orbited by two
close-in sub-Neptune planets, which were first noted in Paper IV. K2-395 b is
a 2.597 $R_{\oplus}$ planet, orbiting at a distance of 0.055 au, with a period
of 4.803352 days and an equilibrium temperature of $\sim$910 K. K2-395 c is a
3.395 $R_{\oplus}$ planet, orbiting at a distance of 0.080 au, with a period
of 8.491282 days and an equilibrium temperature of $\sim$750 K. The host star
has a clean, single-lined TripleSpec spectrum, and Keck/NIRC2 AO imaging that
shows no contaminating stellar companions. The vespa FPP values of planets b
and c are $6.35\times 10^{-10}$ and $1.74\times 10^{-5}$ respectively, using
the available contrast curve and the Campaign 13 multiplicity boost. The
centroid p-values are 0.4413 and 0.4640 respectively, consistent with the
source of the transiting signals being on the target star. Using the Chen &
Kipping (2017) mass-radius relation predicts masses of $\sim$7.6 $M_{\oplus}$
and $\sim$11.8 $M_{\oplus}$ for planets b and c respectively. These mass
estimates for the two planets result in potentially measurable RV semi-
amplitudes ($K$ = 3.0 m/s, and 3.8 m/s), but TSM ($\sim$16, $\sim$20) and ESM
($\sim$1.7, $\sim$1.9) values below the thresholds of interest recommended by
Kempton et al. (2018).

###  A.25 EPIC 246953392

K2-396 is a moderately faint ($V$=13.24 mag, $K_{s}$=10.81 mag) G9V star (0.92
$R_{\odot}$, 0.95 $M_{\odot}$) observed in Campaign 13. It is orbited by an
ultra-short period super-Earth, K2-396 b, first reported in Paper IV as EPIC
246953392.01, and a longer-period sub-Neptune, K2-396 c(EPIC 246953392.02),
reported here for the first time. K2-396 b orbits the star at a distance of
0.015 au, with an ultra-short period of 0.673862 days. With a high equilibrium
temperature of $\sim$1990 K, it is the second hottest of the validated planets
in this analysis. K2-396 c orbits the star at a distance of 0.168 au, with a
period of 25.760509 days, and an equilibrium temperature of $\sim$590 K. The
host star has a clean, single-lined FLWO/TRES spectrum, and WIYN/NESSI speckle
imaging at both 562nm and 832nm that show no contaminating stellar companions.
The vespa FPP values of planets b and c are $4.16\times 10^{-4}$ and
$1.07\times 10^{-5}$ using the available contrast curves and Campaign 13
multiplicity boost. The centroid p-values are 0.4124 and 0.3421, which is
consistent with the sources of the transiting signals being on the target
star. Using the Chen & Kipping (2017) mass-radius relation predicts masses of
$\sim$3.4 $M_{\oplus}$ and $\sim$10.0 $M_{\oplus}$ for planets b and c
respectively. This would result in measurable RV semi-amplitudes (_K_
$\sim$2.6 m/s and $\sim$2.2 m/s), but predicts TSM ($\sim$18, $\sim$15) and
ESM ($\sim$2.3, $\sim$0.7) values that indicate atmospheric follow-up would be
challenging, compared to the thresholds recommended by Kempton et al. (2018).

###  A.26 EPIC 247383003

K2-397 is a bright ($V$=11.83 mag, $K_{s}$=9.95 mag) G7V star (0.97
$R_{\odot}$, 0.94 $M_{\odot}$) that was observed in Campaign 13. It is orbited
by a sub-Neptune-size planet that was first noted in Paper IV. K2-397 b is a
2.417 $R_{\oplus}$ planet, orbiting at a distance of 0.045 au, with a period
of 3.572326 days and an equilibrium temperature of $\sim$1210 K. The host star
has a clean, single-lined FLWO/TRES spectrum, and Keck/NIRC2 AO imaging and
WIYN/NESSI speckle imaging at 562nm and 832nm that show no contaminating
stellar companions. The vespa FPP value is 0.00799, using the available
contrast curves. The centroid p-value is 0.5226, which is consistent with the
source of the transiting signal being on the target star. Using the Chen &
Kipping (2017) mass-radius relation predicts a mass of $\sim$6.7$M_{\oplus}$.
This would result in a measurable RV semi-amplitude (_K_ $\sim$2.9m/s), but
predicts TSM ($\sim$28) and ESM ($\sim$3.0) values that indicate atmospheric
follow-up would be challenging, compared to the thresholds recommended by
Kempton et al. (2018).

###  A.27 EPIC 248518307

K2-400 is a faint ($V$=15.13 mag, $K_{s}$=10.41 mag) M3V star (0.38
$R_{\odot}$, 0.37 $M_{\odot}$) that was observed in Campaign 14. It is orbited
by an Earth-size planet that was first noted in Paper IV as EPIC 248518307.01.
K2-400 b is a 1.163 $R_{\oplus}$ planet, orbiting at a distance of 0.035 au,
with a period of 3.865053 days and an equilibrium temperature of $\sim$570 K.
The host star has a clean, single-lined Hale/TripleSpec spectrum, and
WIYN/NESSI speckle imaging at 562nm and 832nm that shows no contaminating
stellar companions. The vespa FPP value is $1.35\times 10^{-3}$, using the
available contrast curve. The centroid p-value is 0.5332, which is consistent
with the source of the transiting signal being on the target star. Using the
Chen & Kipping (2017) mass-radius relation predicts a mass of
$\sim$1.7$M_{\oplus}$. This would result in a challenging RV semi-amplitude
(_K_ $\sim$1.3m/s), and predicts TSM ($\sim$4.0) and ESM ($\sim$1.0) values
that indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018).

###  A.28 EPIC 248527514

K2-401 is a moderately faint ($V$=13.72 mag, $K_{s}$=11.11 mag) K5V star (0.68
$R_{\odot}$, 0.87 $M_{\odot}$) that was observed in Campaign 14. It is orbited
by a sub-Neptune that was first noted in Paper IV. K2-401 b is a 2.262
$R_{\oplus}$ planet, orbiting at a distance of 0.064 au, with a period of
6.293099 days and an equilibrium temperature of $\sim$660 K. The host star has
a clean, single-lined TripleSpec spectrum, and WIYN/NESSI speckle imaging at
both 562nm and 832nm that show no contaminating stellar companions. We note
that Dressing et al. (2019) find that K2-401 lies above the main sequence and
may therefore be an unresolved eclipsing binary; however the clean spectrum
and high-resolution imaging, as well as the low _Gaia_ RUWE value of 1.09, all
point to a single host star. The vespa FPP value is $3.78\times 10^{-3}$,
using the available contrast curves. The centroid p-value is 0.3051, which is
consistent with the source of the transiting signal being on the target star.
Using the Chen & Kipping (2017) mass-radius relation predicts a mass of
$\sim$5.8 $M_{\oplus}$. This would result in a measurable RV semi-amplitude
(_K_ $\sim$2.2 m/s), but predicts TSM ($\sim$15) and ESM ($\sim$1.1) values
that indicate atmospheric follow-up would be challenging, compared to the
thresholds recommended by Kempton et al. (2018).

###  A.29 EPIC 248621597

K2-402 b is a sub-Neptune (2.678 $R_{\oplus}$) planet orbiting a moderately
faint ($V$=13.03 mag, $K_{s}$=11.38 mag) G2V star (1.28 $R_{\odot}$, 0.94
$M_{\odot}$), observed in Campaign 14. It was first noted in Paper IV; it
orbits the star at a distance of 0.128 au, with a period of 17.274740 days and
an equilibrium temperature of $\sim$870 K. The host star has a clean, single-
lined FLWO/TRES spectrum, and WIYN/NESSI speckle imaging at 562 nm and 832 nm
which show no contaminating stellar companions. The vespa FPP value is
$2.07\times 10^{-4}$ using the available contrast curves. The centroid p-value
is 0.4934, which is consistent with the source of the transiting signal being
on the target star. Using the Chen & Kipping (2017) mass-radius relation
predicts a mass of $\sim$7.8 $M_{\oplus}$. This would result in a measurable
RV semi-amplitude (_K_ $\sim$2.0 m/s), and predicts TSM ($\sim$7.4) and ESM
($\sim$0.5) values that indicate atmospheric follow-up would be challenging,
compared to the thresholds recommended by Kempton et al. (2018).

###  A.30 EPIC 248861279

K2-404 b is a sub-Neptune (2.530 $R_{\oplus}$) planet orbiting a moderately
faint ($V$=14.56 mag, $K_{s}$=10.75 mag) early M dwarf (M1V, 0.55 $R_{\odot}$,
0.55 $M_{\odot}$), observed in Campaign 14. It was first noted in Paper IV; it
orbits the star at a distance of 0.089 au, with a period of 13.115365 days.
With a relatively low equilibrium temperature of $\sim$450 K, it is one of the
coolest validated planets in this analysis. The host star has a clean, single-
lined TripleSpec spectrum, and both Keck/NIRC2 AO imaging and WIYN/NESSI
speckle imaging which show no contaminating stellar companions. The vespa FPP
value is $8.62\times 10^{-4}$ using the available contrast curves. The
centroid p-value is 0.4292, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$7.3 $M_{\oplus}$. This would
result in a measurable RV semi-amplitude (_K_ $\sim$3.0 m/s), and predicts TSM
($\sim$20) and ESM ($\sim$0.8) values that indicate atmospheric follow-up
would be challenging, compared to the thresholds recommended by Kempton et al.
(2018).

###  A.31 EPIC 249403651

K2-407 is a bright ($V$=11.97 mag, $K_{s}$=10.21 mag) early G4V star (0.93
$R_{\odot}$, 0.88 $M_{\odot}$) that was observed in Campaign 15 and is orbited
by two small, close-in planets. The two candidates were first noted in Paper
IV; K2-407 b is a super-Earth (1.283 $R_{\oplus}$) planet, orbiting at a
distance of 0.054 au, with a period of 4.941907 days and an equilibrium
temperature of $\sim$1105 K, and K2-407 c is a super-Earth (1.360
$R_{\oplus}$) planet, orbiting at a distance of 0.083 au, with a period of
9.224530 days and an equilibrium temperature of $\sim$900 K. The host star has
a clean, single-lined FLWO/TRES spectrum, and Gemini/DSSI speckle imaging at
692nm and 880nm which show no contaminating stellar companions. The vespa FPP
values for planets b and c are $3.23\times 10^{-5}$ and $1.03\times 10^{-4}$
respectively, incorporating the available contrast curves and Campaign 15
multiplicity boost. The centroid p-value for planet b is 0.0826, which is
marginally consistent with the source of the transiting signal being on the
target star. The centroid p-value for planet c is 0.0372, which is marginally
inconsistent with the source of the transiting signal being on the target star
given our threshold of 0.05. However, visual inspection of the centroid plots
shows no systematic offset, with the transit cadences that fall outside the
significance contours spread around a large fraction of the exterior of the
contours, and the presence of multiple signals in the same light curve gives
credence to their planetary nature. We consider planet c to be validated, but
note that additional ground-based time series photometry, while challenging
given the shallow depths of these transits, would provide additional
assurance. From the Chen & Kipping (2017) mass-radius relation, we estimate
masses of $\sim$2.1 $M_{\oplus}$ and $\sim$2.5 $M_{\oplus}$ for planets b and
c respectively, leading to very low radial velocity semi-amplitude estimates
and atmospheric spectroscopy metrics.

###  A.32 EPIC 249924395

K2-409 b is a sub-Neptune (2.482 $R_{\oplus}$) planet orbiting a moderately
faint ($V$=12.76 mag, $K_{s}$=11.04 mag) G4V star (1.23 $R_{\odot}$, 0.89
$M_{\odot}$), observed in Campaign 15. It was first noted in Paper IV; it
orbits the star at a distance of 0.029 au, with a period of 1.908084 days and
a very high equilibrium temperature of $\sim$1800 K. The host star has a
clean, single-lined FLWO/TRES spectrum, and Gemini/DSSI speckle imaging which
shows no contaminating stellar companions. Ground-based time-series photometry
of K2-409 during a predicted transit with the 0.6-m ULMT found no detectable
event on the target star (consistent with the shallow K2 detection) and no
nearby binaries eclipsing at the predicted transit time within $\sim$120” down
to $\Delta$mag = 6.5, eliminating many blended eclipsing binary scenarios. The
vespa FPP value is $3.92\times 10^{-5}$ using the available contrast curves.
The centroid p-value is 0.4360, which is consistent with the source of the
transiting signal being on the target star. Using the Chen & Kipping (2017)
mass-radius relation predicts a mass of $\sim$6.7 $M_{\oplus}$. This would
result in a measurable RV semi-amplitude (_K_ $\sim$3.8 m/s), but given the
relative faintness of the target, predicts TSM ($\sim$18) and ESM ($\sim$2.2)
values that indicate atmospheric follow-up would be challenging, compared to
the thresholds recommended by Kempton et al. (2018).