# Fast differentiable evolution of quantum states under Gaussian
transformations

Yuan Yao  Institut Polytechnique de Paris  Télécom Paris, LTCI, 19 Place
Marguerite Perey 91120 Palaiseau, France  Filippo M. Miatto  Xanadu Quantum
Technologies, 777 Bay St. Toronto, Canada  Institut Polytechnique de Paris
Télécom Paris, LTCI, 19 Place Marguerite Perey 91120 Palaiseau, France

###### Abstract

In a recent work we presented a recursive algorithm to compute the matrix
elements of a generic Gaussian transformation in the photon-number basis. Its
purpose was to evolve a quantum state by building the transformation matrix
and subsequently computing the matrix-vector product. Here we present a faster
algorithm that computes the final state without having to generate the full
transformation matrix first. With this algorithm we bring the time complexity
of computing the Gaussian evolution of an $N$-dimensional $M$-mode state from
$O(MN^{2M})$ to $O(M(N^{2}/2)^{M})$, which is an exponential improvement in
the number of modes. In the special case of high squeezing, the evolved state
can be approximated with complexity $O(MN^{M})$. Our new algorithm is
differentiable, which means we can use it in conjunction with gradient-based
optimizers for circuit optimization tasks. We benchmark our algorithm by
optimizing circuits to produce single photons, Gottesman-Kitaev-Preskill
states and NOON states, showing that it is up to one order of magnitude faster
than the state of the art.

##  I Introduction

Parametrized optical quantum circuits process quantum information by encoding
it in quantum states of light and by propagating them through a sequence of
parametrized optical gates [1, 2, 3]. Our main goal is to design a fast,
automated procedure to design quantum optical circuits. Our focus is on the
speed of the optimization: by using faster algorithms we can optimize larger
devices, or similarly-sized devices to a higher precision.

Circuits in the optical domain are interesting because they enjoy of a few
advantages over other technologies, such as no strict requirements for low
temperatures or vacuum, and the potential of building compact and robust
devices with integrated optics [4, 5, 6]. Furthermore, optical quantum devices
may be easily plugged into existing fiber networks for quantum communication
purposes [7, 8].

When using our algorithm, we model a quantum state propagating through an
optical quantum circuit and we craft a differentiable loss function that
evaluates how much the output of the circuit deviates from the target output.
We then compute the gradient of the loss function with respect the parameters
of the circuit and apply gradient descent to train the parameters until the
output matches the target. After convergence we can read out the value of the
parameters, effectively obtaining a design for a circuit that behaves as
desired.

The circuit architecture that we consider is layered and reminiscent of dense
neural networks, in that we alternate between a Gaussian (linear)
transformation and a Kerr (non-linear) transformation. All Gaussian
transformations are linear in the sense that they are always equivalent to a
linear (affine) transformation of the optical phase space coordinates [3].
Note that there are ways in which Gaussian transformations can be considered
non-linear, for example by the fact that their Hamiltonian can be non-linear
in the quadrature operators. Our choice of architecture combines simplicity
(each Gaussian+Kerr layer is functionally the same), and generality (we can
implement any quantum transformation by stacking enough layers).

Parameterized optical circuits can already be trained via machine learning
techniques for state preparation, gate synthesis, function simulation, image
generation, hybrid classical-quantum classifiers, Hamiltonian simulation and
quantum repeaters [3, 9, 10, 11]. However, current methods can quickly become
very demanding in terms of computational resources, and this makes it
challenging to apply them to the design of devices that span several optical
modes. Moreover, as the size of the Hilbert space grows, there can be
vanishing-gradient effects that can impact the optimization of a parametrized
circuit [12].

Our solution in [13] was up to 100 times faster than the previous state of the
art, and in this work we increase the speedup exponentially in the number of
modes $M$ by a factor $2^{M}$. Here is a summary of the contributions
presented in this work:

  1. 1.

We present an algorithm to directly compute the output state of a quantum
optical circuit by computing only a fraction $1/2^{M}$ of tensor elements with
respect to the whole transformation tensor. This exponential improvement
benefits both the forward and the backward pass.

  2. 2.

Our algorithm is differentiable, as it relies on a linear recurrence relation.

  3. 3.

Backpropagated gradient tensors have the shape of the state, rather than the
shape of the transformation. Such quadratic improvement benefits the backward
pass.

This paper is organized as follows. In section 2 we recall the recurrence
relation that was at the core of our previous algorithm, and that we will
adapt for our new algorithm, which we present in section 3. In section 4 we
first illustrate the backpropagation mechanism with complex parameters and
explain how we obtain the gradient of a transformed state with respect to the
parameters of the transformation. In section 5 we compare the runtime of the
forward pass of our algorithm with respect to the current state of the art,
and we benchmark the backward pass on a circuit optimization task. Finally, we
discuss the results and future research directions in section 6.

##  II Preliminaries

###  II.1 Definition of an $M$-mode general Gaussian transformation

Here we present the circuits that implement general Gaussian transformations,
which are based on the Bloch-Messiah decomposition [14, 15].

In the single-mode case we use a sequence of squeezer, phase rotation and
displacement gates:

$\displaystyle\mathcal{G}^{(1)}(\gamma,\phi,\zeta)=D(\gamma)R(\phi)S(\zeta).$
(1)

$\Qcircuit@C=1em@R=.7em{&\gate{S(\zeta)}\gate{R(\phi)}\gate{D(\gamma)}\qw}$
Figure 1: Single-mode Gaussian transformation architecture (states move
through the circuit left to right).

In the two-mode case we have:

$\displaystyle\mathcal{G}^{(2)}(\bm{\gamma},\bm{\phi},\theta^{\prime},\varphi^{\prime},\bm{\zeta},\theta,\varphi)=D(\bm{\gamma})R(\bm{\phi})B(\theta^{\prime},\varphi^{\prime})S(\bm{\zeta})B(\theta,\varphi).$
(2)

$\Qcircuit@C=1em@R=.7em{&\multigate{1}{B(\theta,\varphi)}\gate{S(\zeta_{1})}\multigate{1}{B(\theta^{\prime},\varphi^{\prime})}\gate{R(\phi_{1})}\gate{D(\gamma_{1})}\qw\\\
\ghost{B(\theta,\varphi)}\gate{S(\zeta_{2})}\ghost{B(\theta^{\prime},\varphi^{\prime})}\gate{R(\phi_{2})}\gate{D(\gamma_{2})}\qw}$
Figure 2: Two-mode Gaussian transformation architecture.

For $M$ modes, we have an $M$-mode interferometer, $M$ single-mode squeezers,
a second $M$-mode interferometer and $M$ single-mode displacements. In this
case, the interferometers need to be decomposed as a suitable arrangement of
beamsplitters, for example following Clements decomposition [16, 17, 18].

###  II.2 Notation and definitions

We adopt a multi-index notation, i.e.,
$\mathcal{G}_{\bm{k}}=\mathcal{G}_{k_{1},k_{2},\dots}$, but we will show all
indices when necessary. Transformations on $M$ modes are tensors of rank $2M$,
where the first half of their indices correspond to the output and the second
half correspond to the input (but we maintain the terminology “matrix
elements”). We also indicate as $\bm{0}$ the vector of all zeros and by
$1_{i}$ a vector with all zeros and a single 1 in position $i$.

Eq. (29) from our previous work [13] presents a recurrence relation to express
the matrix elements of a general Gaussian transformation up to a global phase:

$\displaystyle\mathcal{G}_{\bm{0}}=C,$
$\displaystyle\mathcal{G}_{\bm{k}+1_{i}}=\frac{1}{\sqrt{k_{i}+1}}\left(\mathcal{G}_{\bm{k}}\mu_{i}-\sum_{l=1}^{2M}\sqrt{k_{l}}\mathcal{G}_{\bm{k}-1_{l}}\Sigma_{il}\right).$
(3)

Here $\mathcal{G}_{\bm{k}}$ denotes the matrix elements of a Gaussian
transformation and $C,\mu$ and $\Sigma$ are defined as follows:

$\displaystyle
C=\frac{\exp\left(-\tfrac{1}{2}\left[||\bm{\gamma}||^{2}+\bm{\gamma}^{\dagger}\bm{W}\operatorname{\mathrm{diag}}(e^{i\bm{\delta}}\tanh\bm{r})\bm{W}^{T}\bm{\gamma}^{*}\right]\right)}{\sqrt{\prod_{i=1}^{M}\cosh
r_{i}}},$ (4) $\displaystyle\bm{\mu}^{T}=$
$\displaystyle\left[\bm{\gamma}^{\dagger}\bm{W}\operatorname{\mathrm{diag}}(e^{i\bm{\delta}}\tanh\bm{r})\bm{W}^{T}+\bm{\gamma}^{T},-\bm{\gamma}^{\dagger}\bm{W}\operatorname{\mathrm{diag}}(\operatorname{\mathrm{sech}}\bm{r})\bm{V}\right],$
(5) $\displaystyle\bm{\Sigma}=$
$\displaystyle\left[\begin{array}[]{c|c}\bm{W}\operatorname{\mathrm{diag}}(e^{i\bm{\delta}}\tanh\bm{r})\bm{W}^{T}&-\bm{W}\operatorname{\mathrm{diag}}(\operatorname{\mathrm{sech}}\bm{r})\bm{V}\\\
\hline\cr-\bm{V}^{T}\operatorname{\mathrm{diag}}(\operatorname{\mathrm{sech}}\bm{r})\bm{W^{T}}&-\bm{V}^{T}\operatorname{\mathrm{diag}}(e^{-i\bm{\delta}}\tanh\bm{r})\bm{V}\end{array}\right],$
(8)

where $\bm{\gamma}$ is the vector of displacement parameters, $\bm{\delta}$
and $\bm{r}$ are the polar coordinates of the complex vector $\bm{\zeta}$ of
squeezing parameters, and $\bm{W}$ and $\bm{V}$ are unitary covariance
matrices describing the two interferometers.

##  III Generating the transformed state directly

###  III.1 Single-mode

The matrix of a single-mode Gaussian transformation has rank 2. According to
Eq. (3), the first row is given by:

$\displaystyle\mathcal{G}_{0,n}=\frac{\mu_{2}}{\sqrt{n}}\mathcal{G}_{0,n-1}-\Sigma_{22}\sqrt{\frac{n-1}{n}}\mathcal{G}_{0,n-2},$
(9)

where the first element is $\mathcal{G}_{0,0}=C$.

We indicate the first row in Dirac notation as $\langle\mathcal{G}_{0}|$. By
using (3) again, each successive row of the Gaussian transformation can be
computed recursively:

$\displaystyle\langle\mathcal{G}_{m}|=\frac{\mu_{1}}{\sqrt{m}}\langle\mathcal{G}_{m-1}|-\Sigma_{11}\sqrt{\frac{m-1}{m}}\langle\mathcal{G}_{m-2}|-\frac{\Sigma_{12}}{\sqrt{m}}\langle\mathcal{G}_{m-1}|a,$
(10)

where $a$ is the annihilation operator.

Given an input state $|\psi\rangle$, the $m$-th amplitude of the output state
is given by $\langle\mathcal{G}_{m}|\psi\rangle$. Therefore, using the
recurrence relation for the rows $\langle\mathcal{G}_{m}|$ in Eq. (10), we can
write:

$\displaystyle\underbrace{\langle\mathcal{G}_{m}|\psi\rangle}_{R_{m}^{(0)}}=$
$\displaystyle\frac{\mu_{1}}{\sqrt{m}}\underbrace{\langle\mathcal{G}_{m-1}|\psi\rangle}_{R_{m-1}^{(0)}}-\Sigma_{11}\sqrt{\frac{m-1}{m}}\underbrace{\langle\mathcal{G}_{m-2}|\psi\rangle}_{R_{m-2}^{(0)}}$
$\displaystyle-\frac{\Sigma_{12}}{\sqrt{m}}\underbrace{\langle\mathcal{G}_{m-1}|a|\psi\rangle}_{R_{m-1}^{(1)}},$
(11)

where we have defined $R_{m}^{(k)}=\langle\mathcal{G}_{m}|a^{k}|\psi\rangle$.

This relation must hold for any $k\geq 0$:

$\displaystyle
R_{m}^{(k)}=\frac{\mu_{1}}{\sqrt{m}}R_{m-1}^{(k)}-\Sigma_{11}\sqrt{\frac{m-1}{m}}R_{m-2}^{(k)}-\frac{\Sigma_{12}}{\sqrt{m}}R_{m-1}^{(k+1)},$
(12)

which means that we can write all the final amplitudes $R_{m}^{(0)}$ as linear
combinations of elements in $R_{0}^{(k)}$.

Imagine now $R_{m}^{(k)}$ as a matrix whose rows are indexed by $m$ and whose
columns are indexed by $k$. From Eq. (12) we know that each element in $R$
only depends on its three neighbouring elements (see Fig. 3).

Figure 3: Any element in the $R$ matrix (e.g. the blue one at the bottom) can
be calculated as a linear combination of the three neighbouring matrix
elements shown in orange.

As we are interested in the first column of $R$, we start by filling the first
row $R_{0}^{(k)}$ and then apply the rule iteratively, stopping at one fewer
element per row, effectively filling half of the matrix. This discount in
computation carries over to each mode, i.e., for $M$ modes we need to compute
only a fraction $2^{-M}$ of elements. As we need to combine $2M+1$ numbers to
compute each new element of $R$, the time complexity of our algorithm is
$O(MN^{2M}/2^{M})$, which is exponentially faster in the number of modes with
respect to computing the whole tensor $\mathcal{G}$, which has complexity
$O(MN^{2M})$.

In our Python code, we proceed as follows:

  1. 1.

We build a vector of square roots of integers up to $N$ to avoid having to
recompute them often.

  2. 2.

We build the first row of the transformation matrix,
$\langle\mathcal{G}_{0}|$.

  3. 3.

We build the first row $R_{0}^{(k)}$ by computing
$\langle\mathcal{G}_{0}|a^{k}|\psi\rangle$ for $k\leq N$. To do so we take
inner products only between the relevant part of the vector
$\langle\mathcal{G}_{0}|$ and we update the state as $a^{k}|\psi\rangle$ for
the next value of $k$.

  4. 4.

Starting from $m=1$, we compute each row $R_{m}^{(k)}$ using Eq. (12), but we
only need $N-m$ elements.

  5. 5.

After we have built the last row $R_{N}^{(0)}$ (consisting of a single
element), we read out the final result from the first column $R_{m}^{(0)}$.

The code can then be sped-up via the Numba [19] decorator `numba.njit`, and it
will be compiled at the first call:

⬇

@njit

def new_state(C, mu, Sigma, state):

# 1. Setup

N = len(state)

dtype = state.dtype

sqrt = np.sqrt(np.arange(N, dtype=dtype))

\par# 2. First row of the transformation matrix

G = np.zeros(N, dtype=dtype)

G[0] = C

for n in range(1, N):

G[n] = (mu[1]/sqrt[n]*G[n-1] \-

Sigma[1,1]*sqrt[n-1]/sqrt[n]*G[n-2])

\par# 3. First row of the R matrix

R = np.zeros((N, N), dtype=dtype)

for k in range(N):

R[0,k] = np.dot(G[:N-k], state)

state = state[1:]*sqrt[1:N-k]

\par# 4. Other rows of the R matrix

for m in range(1, N):

for k in range(N-m):

R[m, k] = (mu[0]*R[m-1, k]/sqrt[m]

-Sigma[0,0]*R[m-2, k]*sqrt[m-1]/sqrt[m]

-Sigma[0,1]*R[m-1, k+1]/sqrt[m])

\par# 5. Readout

return R[:, 0]

####  III.1.1 Special case: large squeezing

If the squeezing parameter is large we don’t need to generate the whole $R$
matrix to find the transformed state. To see that this is the case, notice
that in (8) we can approximate $\operatorname{\mathrm{sech}}(r)\sim 0$ and
$\tanh(r)\sim 1$ as $\mathbf{r}$ grows. We do maintain
$\sqrt{\operatorname{\mathrm{sech}}(r)}$ in $C$, as it is quadratically larger
than $\operatorname{\mathrm{sech}}{r}$.

For a single-mode Gaussian transformation, we have:

$\displaystyle C$
$\displaystyle\sim\sqrt{\operatorname{\mathrm{sech}}r}\exp\left(-\frac{1}{2}|\gamma|^{2}-\frac{1}{2}{\gamma^{*}}^{2}e^{i(\delta+2\phi)}\right),$
(13) $\displaystyle\bm{\mu}^{T}$
$\displaystyle\sim[\gamma^{*}e^{i(\delta+2\phi)}+\gamma,0],$ (14)
$\displaystyle\bm{\Sigma}$
$\displaystyle\sim\left[\begin{array}[]{c|c}e^{i(\delta+2\phi)}&0\\\ \hline\cr
0&-e^{-i\delta}\end{array}\right].$ (17)

We can hence rewrite the recurrence relations of Eq. (9) and (12) for a
single-mode Gaussian transformation with large squeezing:

$\displaystyle\mathcal{G}_{0,n}$ $\displaystyle\sim
e^{-i\delta}\sqrt{\frac{n-1}{n}}\mathcal{G}_{0,n-2}$ (18) $\displaystyle
R_{m}^{(0)}$
$\displaystyle\sim\frac{\gamma^{*}e^{i(\delta+2\phi)}+\gamma}{\sqrt{m}}R_{m-1}^{(0)}-e^{i(\delta+2\phi)}\sqrt{\frac{m-1}{m}}R_{m-2}^{(0)}$
(19)

Notice that in the second relation, $R^{(0)}_{m}$ does not depend on
$R^{(k)}_{m}$ for $k>0$. We then use the first relation to compute $R_{0}^{0}$
and the second to compute the output state directly. We remark that under
these conditions one can (differentiably) approximate the transformed state up
to a cutoff in the order of thousands in milliseconds.

To assess the quality of the approximation within the cutoff, in Fig. 4 we
show the deviation from the normalized overlap with respect to the exact state
as a function of the parameter $r$ of the squeezing for a few random states.
Note that we need the renormalization because we compute the exact output
state only up a fixed cutoff dimension, but large squeezing populates also
Fock states with very large photon numbers.

Figure 4: The error in the the normalized overlap between the approximated
state and the exact state (here a set of 10 random states) goes to zero as the
magnitude of the squeezing parameter $r=|z|$ increases. Note that the overlap
only includes photon number states up to $n=50$.

###  III.2 Two-mode Gaussian transformed state

The transformation operator of a two-mode Gaussian transformation has rank 4,
so we will build its elements by using our recursive equations four times.

The first two indices are filled by:

$\displaystyle\mathcal{G}_{0,0,0,q}$
$\displaystyle=\frac{1}{\sqrt{q}}\left(\mathcal{G}_{0,0,0,q-1}\mu_{4}-\sqrt{q-1}\mathcal{G}_{0,0,0,q-2}\Sigma_{44}\right),$
(20) $\displaystyle\mathcal{G}_{0,0,p,q}$
$\displaystyle=\frac{1}{\sqrt{p}}(\mathcal{G}_{0,0,p-1,q}\mu_{3}-\sqrt{p-1}\mathcal{G}_{0,0,p-2,q}\Sigma_{33}$
$\displaystyle-\sqrt{q}\mathcal{G}_{0,0,p-1,q-1}\Sigma_{34}).$ (21)

Using Dirac notation $\langle\mathcal{G}_{m,n}|$ we can express the following
identities:

$\displaystyle\mathcal{G}_{m,n,p-1,q}\sqrt{p}$
$\displaystyle=\langle\mathcal{G}_{mn}|a,$ (22)
$\displaystyle\mathcal{G}_{m,n,p,q-1}\sqrt{q}$
$\displaystyle=\langle\mathcal{G}_{mn}|b,$ (23)

where $a$ and $b$ are annihilation operators in the first and second mode,
respectively.

We then apply the recurrence relation again to fill the third and fourth
indices:

$\displaystyle\langle\mathcal{G}_{0,n}|$
$\displaystyle=\frac{1}{\sqrt{n}}(\mu_{2}\langle\mathcal{G}_{0,n-1}|-\Sigma_{22}\sqrt{n-1}\langle\mathcal{G}_{0,n-2}|$
$\displaystyle\
-\Sigma_{23}\langle\mathcal{G}_{0,n-1}|a-\Sigma_{24}\langle\mathcal{G}_{0,n-1}|b),$
(24) $\displaystyle\langle\mathcal{G}_{m,n}|$
$\displaystyle=\frac{1}{\sqrt{m}}(\mu_{1}\langle\mathcal{G}_{m-1,n}|-\Sigma_{11}\sqrt{m-1}\langle\mathcal{G}_{m-2,n}|$
$\displaystyle\
-\Sigma_{12}\sqrt{n}\langle\mathcal{G}_{m-1,n-1}|-\Sigma_{13}\langle\mathcal{G}_{m-1,n}|a$
$\displaystyle\ -\Sigma_{14}\langle\mathcal{G}_{m-1,n}|b).$ (25)

We now assume a pure two-mode input state $|\bm{\psi}\rangle$ and we compute
the $(m,n)$-th amplitude of the output state:
$\langle\mathcal{G}_{mn}|\bm{\psi}\rangle$. Using the recurrence relation in
Eq. (24) and Eq. (25), and defining
$R_{m,n}^{(j,k)}=\langle\mathcal{G}_{m,n}|a^{j}b^{k}|\bm{\psi}\rangle$, we
get:

$\displaystyle\underbrace{\langle\mathcal{G}_{0,n}|\bm{\psi}\rangle}_{R_{0,n}^{(0,0)}}$
$\displaystyle=\frac{1}{\sqrt{n}}(\mu_{2}\underbrace{\langle\mathcal{G}_{0,n-1}|\bm{\psi}\rangle}_{R_{0,n-1}^{(0,0)}}-\Sigma_{22}\sqrt{n-1}\underbrace{\langle\mathcal{G}_{0,n-2}|\bm{\psi}\rangle}_{R_{0,n-1}^{(0,0)}}$
$\displaystyle-\Sigma_{23}\underbrace{\langle\mathcal{G}_{0,n-1}|a|\bm{\psi}\rangle}_{R_{0,n-1}^{1,0}}-\Sigma_{24}\underbrace{\langle\mathcal{G}_{0,n-1}|b|\bm{\psi}\rangle}_{R_{0,n-1}^{0,1}}),$
(26)
$\displaystyle\underbrace{\langle\mathcal{G}_{m,n}|\bm{\psi}\rangle}_{R_{m,n}^{0,0}}$
$\displaystyle=\frac{1}{\sqrt{m}}(\mu_{1}\underbrace{\langle\mathcal{G}_{m-1,n}|\bm{\psi}\rangle}_{R_{m-1,n}^{0,0}}-\Sigma_{11}\sqrt{m-1}\underbrace{\langle\mathcal{G}_{m-2,n}|\bm{\psi}\rangle}_{R_{m-2,n}^{0,0}}$
$\displaystyle-\Sigma_{12}\sqrt{n}\underbrace{\langle\mathcal{G}_{m-1,n-1}|\bm{\psi}\rangle}_{R_{m-1,n-1}^{0,0}}-\Sigma_{13}\underbrace{\langle\mathcal{G}_{m-1,n}|a|\bm{\psi}\rangle}_{R_{m-1,n}^{1,0}}$
$\displaystyle-\Sigma_{14}\underbrace{\langle\mathcal{G}_{m-1,n}|b|\bm{\psi}\rangle}_{R_{m-1,n}^{0,1}}).$
(27)

The relation holds for any $j$ and $k$, so we finally obtain:

$\displaystyle R_{0,n}^{(j,k)}$
$\displaystyle=\frac{1}{\sqrt{n}}(\mu_{2}R_{0,n-1}^{(j,k)}-\Sigma_{22}\sqrt{n-1}R_{0,n-2}^{(j,k)}$
$\displaystyle-\Sigma_{23}R_{0,n-1}^{(j+1,k)}-\Sigma_{24}R_{0,n-1}^{(j,k+1)}),$
(28) $\displaystyle R_{m,n}^{(j,k)}$
$\displaystyle=\frac{1}{\sqrt{m}}(\mu_{1}R_{m-1,n}^{(j,k)}-\Sigma_{11}\sqrt{m-1}R_{m-2,n}^{(j,k)}$
$\displaystyle-\Sigma_{12}\sqrt{n}R_{m-1,n-1}^{(j,k)}-\Sigma_{13}R_{m-1,n}^{(j+1,k)}-\Sigma_{14}R_{m-1,n}^{(j,k+1)}).$
(29)

Similarly to the single-mode case, because of the dependencies among
neighboring elements, we compute the output state without needing to generate
the whole rank-4 tensor.

In our python code for the two-mode case we proceed as follows :

  1. 1.

We build a vector of square roots of integers up to $N$ to avoid having to
recompute them often.

  2. 2.

We compute the matrix $\langle\mathcal{G}_{00}|$ using Eq. (20) and (21).

  3. 3.

We compute $R_{00}^{(j,k)}$ by multiplying
$\langle\mathcal{G}_{00}|a^{j}b^{k}$ and $|\bm{\psi}\rangle$. This step is
analogous to the single-mode case except we loop over two indices instead of
one.

  4. 4.

We compute $R_{0n}^{(j,k)}$ using Eq. (28) and $R_{mn}^{(j,k)}$ using Eq.
(29). We do this under the conditions: $m<N,\ n<N,\ j<N-m,\ k<N-m-j$, which
allow us to compute only one quarter of the possible $N^{4}$ elements of
$R_{mn}^{(j,k)}$.

  5. 5.

We read out the final result from $R_{mn}^{(0,0)}$.

One can extend this method to any number of modes by following a similar
procedure with a larger number of indices. In the $M$-mode case, the
conditions will allow one to obtain the final state by computing only a
fraction of $2^{-M}$ elements of the rank $2M$ tensor $R$.

##  IV Gradients of the transformed state

The recurrence relations that we have presented are all linear, and therefore
they can be easily differentiated. This gives us a direct way of computing the
gradient of the transformed state with respect to the parameters of the
transformation.

In our Gaussian transformations we use a mix of real and complex parameters.
We will therefore follow Wirtinger calculus to compute the gradients with
respect to complex parameters: Wirtinger derivatives are computed by
considering a complex variable $\xi$ and its complex conjugate $\xi^{*}$ as
_independent_ variables. This allows us to compute derivatives of non-
holomorphic functions, which can be thought of as functions that explicitly
depend on $\xi^{*}$ such as the absolute value $|\xi|=\sqrt{\xi\xi^{*}}$. If a
function is holomorphic, its derivative with respect to $\xi^{*}$ is zero and
in this case the derivative with respect to $\xi$ is sufficient. This is a
consequence of the fact that holomorphic functions satisfy the Cauchy-Riemann
equations, establishing a dependence between the derivatives of the real and
imaginary parts of the function and thereby removing a degree of freedom.

Assuming we have defined a real loss function
$L:\mathbb{C}^{n}\rightarrow\mathbb{R}$ that maps the output state of the
Gaussian transformation to a real value, we can choose to use gradient descent
to optimize it. Note that for a complex parameter $\xi$, the correct gradient
to utilize is the one with respect to the _conjugate_ of the parameter we want
to optimize, i.e., the gradient update rule reads:

$\displaystyle\xi\leftarrow\xi-\epsilon\frac{\partial L}{\partial\xi^{*}}$
(30)

for a learning rate $\epsilon$. This gradient update rule falls back to the
usual one in case $\xi$ is a real variable.

We consider quantum transformations that depend on a complex parameter $\xi$
as functions $f:\mathbb{C}^{n}\times\mathbb{C}\rightarrow\mathbb{C}^{n}$, such
that $\bm{\psi}^{(out)}=f(\bm{\psi}^{(in)},\xi)$. Therefore, we may want to
backpropagate the upstream gradient $\partial L/\partial\bm{\psi}^{(out)}$ to
the gradient with respect to the parameter $\partial L/\partial\xi^{*}$ or to
the gradient with respect to the input $\partial L/\partial\bm{\psi}^{(in)}$.
As we need to treat any complex variable and its conjugate as independent, we
need to consider both when applying the chain rule.

For our code, we use the automatic differentiation framework of TensorFlow,
which computes the gradient $\frac{\partial L}{\partial\psi_{\bm{k}}^{*}}$ for
us, where $\psi_{\bm{k}}^{*}$ is the conjugate of the input to the loss
function (note that TensorFlow computes the appropriate gradient as per Eq.
(30)). From this gradient we obtain $\frac{\partial
L}{\partial\psi_{\bm{k}}}=\left(\frac{\partial
L}{\partial\psi_{\bm{k}}^{*}}\right)^{*}$ by conjugation.

Then, if we want to backpropagate the upstream gradient to the input of the
previous transformation, we have to provide the conjugate of the
transformation itself:
$\frac{\partial{\psi^{(out)}}^{*}_{\bm{k}}}{\partial{\psi^{(in)}}^{*}_{\bm{k}^{\prime}}}=\mathcal{G}_{{\bm{k}},{\bm{k}}^{\prime}}^{*}$:

$\displaystyle\frac{\partial L}{\partial{\psi^{(in)}}_{\bm{k}^{\prime}}^{*}}$
$\displaystyle=\sum_{\bm{k}}\frac{\partial
L}{\partial{\psi^{(out)}_{\bm{k}}}^{*}}\frac{\partial{\psi^{(out)}}_{\bm{k}}^{*}}{\partial{\psi^{(in)}}_{\bm{k}^{\prime}}^{*}}+\frac{\partial
L}{\partial{\psi^{(out)}_{\bm{k}}}}\frac{\partial{\psi^{(out)}}_{\bm{k}}}{\partial{\psi^{(in)}}_{\bm{k}^{\prime}}^{*}}$
$\displaystyle=\sum_{\bm{k}}\frac{\partial
L}{\partial{\psi^{(out)}_{\bm{k}}}^{*}}\mathcal{G}_{{\bm{k}},{\bm{k}}^{\prime}}^{*},$
(31)

the second term being always zero.

If instead we want to backpropagate the upstream gradient to the parameter of
the transformation, we have to provide
$\frac{\partial\psi_{k}}{\partial\xi^{*}}$ and
$\frac{\partial\psi_{k}^{*}}{\partial\xi^{*}}=\left(\frac{\partial\psi_{k}}{\partial\xi}\right)^{*}$:

$\displaystyle\frac{\partial L}{\partial\xi^{*}}=\sum_{\bm{k}}\frac{\partial
L}{\partial\psi_{\bm{k}}^{*}}\frac{\partial\psi_{\bm{k}}^{*}}{\partial\xi^{*}}+\frac{\partial
L}{\partial\psi_{\bm{k}}}\frac{\partial\psi_{\bm{k}}}{\partial\xi^{*}}.$ (32)

To compute the desired gradients (here for a single-mode), we differentiate
Eq. (10) and Eq. (12) with respect to a parameter $\xi$ (or its conjugate
$\xi^{*}$):

$\displaystyle\partial_{\xi}\mathcal{G}_{0,n}=$
$\displaystyle\frac{1}{\sqrt{n}}\left[\mu_{2}(\partial_{\xi}\mathcal{G}_{0,n-1})+(\partial_{\xi}\mu_{2})\mathcal{G}_{0,n-1}\right]$
$\displaystyle-\sqrt{\frac{n-1}{n}}\left[(\partial_{\xi}\Sigma_{22})\mathcal{G}_{0,n-2}+\Sigma_{22}(\partial_{\xi}\mathcal{G}_{0,n-2})\right],$
(33) $\displaystyle\partial_{\xi}R_{m}^{(k)}=$
$\displaystyle\frac{1}{\sqrt{m}}\left[(\partial_{\xi}\mu_{1})R_{m-1}^{(k)}+\mu_{1}(\partial_{\xi}R_{m-1}^{(k)})\right]$
$\displaystyle-\sqrt{\frac{m-1}{m}}\left[(\partial_{\xi}\Sigma_{11})R_{m-2}^{(k)}+\Sigma_{11}(\partial_{\xi}R_{m-2}^{(k)})\right]$
$\displaystyle-\frac{1}{\sqrt{m}}\left[(\partial_{\xi}\Sigma_{12})R_{m-1}^{(k+1)}+\Sigma_{12}(\partial_{\xi}R_{m-1}^{(k+1)})\right].$
(34)

And we obtain the gradient from the first column:

$\displaystyle\frac{\partial\psi_{m}}{\partial\xi}=\partial_{\xi}R_{m}^{(0)}.$
(35)

These recurrence relations can be computed once the $R$ matrix is known.
Therefore, after generating the matrix $R$, we keep it in memory for the
computation of the gradient. To speed things up even further, we pre-compute
and compile the derivative functions of $C$, $\bm{\mu}$ and $\bm{\Sigma}$ with
respect to the parameters.

From a practical point of view, the backpropagated gradients have the same
shape of the state (i.e., $N^{M}$ for an $N$-dimensional Fock space on
$M$-modes) rather than the shape of the transformation (i.e., $N^{2M}$), which
is a quadratic advantage over previous approaches. We still have to compute
the transformation itself in order to backpropagate the gradients, but instead
of the gradient of the transformation we can just compute the gradient of $R$,
which is only a fraction $2^{-M}$ in size.

##  V Numerical experiments

In this section we present benchmarks of various experiments based on our new
method.

First, we benchmark the runtime to compute the transformed state (i.e., the
forward pass) against the current state of the art. We obtain the transformed
state by three methods: constructing the Gaussian transformation matrix by
using a sequence of gates in StrawberryFields 0.17 [10] and taking the matrix-
vector product, by getting the Gaussian transformation matrix from our
previous work in the Ggate branch of The Walrus [20] and taking the matrix-
vector product, and with the method in this work to get the transformed state
directly. The results are summarized in Fig. 5.

Figure 5: Our implementation (lowest line, green) is up to an order of
magnitude faster at generating the output of a Gaussian transformation than
the other methods. For the two-mode case, Strawberry Fields has the advantage
of exploiting the photon-number conservation of the beamsplitter to spare one
nested loop. In the other two implementations we build the tensors all at
once, and therefore we don’t have access to the same savings. Despite this,
our current implementation is still faster and even more so when the backward
pass is computed (see Table I), as the backpropagated gradients have the shape
of the state rather than the shape of the transformation.

Second, we benchmark a circuit optimization task for various one- and two-mode
states (i.e., we benchmark both the forward and the backward pass). Our
architecture is made of a sequence of $L$ layers, each formed by a Gaussian
gate followed by a single-mode Kerr gate in each mode:

$\displaystyle
U(\bm{\beta})=\prod_{\ell=1}^{L}\mathcal{G^{\ell}(\bm{\alpha}_{\ell})}K(\bm{\kappa}_{\ell}),$
(36)

where $\bm{\alpha}_{\ell}$ denotes the parameters of the $\ell$-th Gaussian
layer and $\bm{\beta}$ denotes the parameters of the whole circuit. Each
single-mode Kerr transformation is given by

$\displaystyle K(\kappa)=e^{i\kappa(a^{\dagger}a)^{2}},$ (37)

where $\kappa$ is a positive parameter.

Suppose we have a set of $S$ required input and target pairs
$\\{(\Psi^{\mathrm{in}}_{s},\Psi^{\mathrm{target}}_{s})\\}$, our goal is to
find a circuit that can satisfy the requirements. The optimization is
performed by minimizing the loss function:

$\displaystyle
L(\bm{\beta})=1-\frac{1}{S}\sum_{s=1}^{S}|\langle\Psi^{\mathrm{target}}_{s}|U(\bm{\beta})|\Psi^{\mathrm{in}}_{s}\rangle|^{2}.$
(38)

Note that if the number of required input-output pairs is large, the optimizer
might content itself with satisfying only a few of them. In that case, we can
include in the loss function a term that favours a uniform probability over
all the required input-output pairs, for example the KL divergence with
respect to the uniform distribution:
$-\sum_{s}\log|\langle\Psi^{\mathrm{target}}_{s}|U(\bm{\beta})|\Psi^{\mathrm{in}}_{s}\rangle|^{2}$,
which diverges rapidly as soon as either of the probabilities tries to go to
zero.

We ran benchmarks on generating a single-photon
($|\Psi^{\mathrm{in}}\rangle=|0\rangle,|\Psi^{\mathrm{target}}\rangle=|1\rangle$),
NOON state with $N=5$
($|\Psi^{\mathrm{in}}\rangle=|0,0\rangle,|\Psi^{\mathrm{target}}\rangle=(|5,0\rangle+|0,5\rangle)/\sqrt{2}$)
and Hex GKP state as in [9] with $d=2$, $\mu=1$, $\delta=0.3$. The results are
shown in the table 1. One can compare them with our previous implementation in
table I of [13] and with the original implementation in table I of [9], while
taking into account differences in computational hardware.

Hyperparams | Single photon | NOON state | Hex GKP
---|---|---|---
Modes M | 1 | 2 | 1
Cutoff N | 100 | 10 | 50
Layers L | 8 | 20 | 25
Steps | 1500 | 3000 | 10000
Results |  |  |
Fidelity | 99.992% | 99.905% | 99.537%
Runtime(s) | 24s | 341s | 410s
Table 1: Runtime for three circuit optimization tasks (single-core on an Apple
M1 chip). We achieve about twice the performance as compared to our previous
implementation, which was more than one order of magnitude faster than
previous methods.

##  VI Conclusions

In this work we introduced a new algorithm to compute how a quantum state
evolves under a Gaussian transformation. In comparison to previous approaches,
our algorithm has an advantage in runtime and memory that is exponential in
the number of optical modes. Moreover it is differentiable, which means we can
calculate gradients of the resulting state with respect to the transformation
parameters and use them to train or optimize quantum optical circuits. We made
available our algorithm through our library _poenta_ [21].

The main outstanding challenge of our work is that we assume the input state
is pure. If the input state is mixed, we would have to run the algorithm on
each significant eigenstate of the density matrix, thus getting a performance
hit proportional to the rank of the state. However, in preliminary numerical
benchmarks we have noticed no impact, probably thanks to automatic code
vectorization. We will investigate this avenue in future work.

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##  Appendix A COSMOS galaxy overdensity maps

Figs. 16a-16z show galaxy overdensity maps (colored contours) using the WAK
(left panels) and WVT estimators (right panels). Starting from the top row the
overdensity maps shown are for the following redshift bins: a-b) $z=6.05\pm
0.10$ (595 galaxies), c-d) $z=6.25\pm 0.10$ (226 galaxies), e-f) $z=6.47\pm
0.11$ (95 galaxies), g-h) $z=6.69\pm 0.11$ (123 galaxies), i-j) $z=6.92\pm
0.12$ (187 galaxies), k-l) $z=7.17\pm 0.12$ (193 galaxies), m-n) $z=7.42\pm
0.13$ (102 galaxies), o-p) $z=7.69\pm 0.14$ (26 galaxies), q-r) $z=7.97\pm
0.14$ (22 galaxies), s-t) $8.26\pm 0.15$ (20 galaxies), u-v) $8.57\pm 0.16$
(21 galaxies), w-x) $8.89\pm 0.17$ (23 galaxies), y-z) $9.23\pm 0.18$ (33
galaxies). Contours indicate regions of increasing overdensity values in steps
of $1\sigma$. Red contours indicate regions where the galaxy overdensity is
significant by $4\sigma$ or higher and meet our selection criteria. These are
indicated by numbers that refer to their name in Table 2. The location of the
individual galaxies are marked by cyan dots, and the number of galaxies within
each redshift bin is indicated in the parentheses above.

Figure 16: Galaxy overdensity maps (colored contours in steps of $2\sigma$)
using the WAK (left panels) and WVT estimators (right panels) as described in
Section 3. The overdensity maps are for the indicated redshift bins. Red
contours indicate regions where the galaxy overdensity is significant by
$4\sigma$ or higher and meet our selection criteria. These $\geq 4\sigma$
overdensities are labeled by numbers that refer to their IDs in Table 2. The
location of the individual galaxies are marked by black dots, and the number
of galaxies within each redshift bin is indicated in parentheses.

##  Appendix B Weight distribution for galaxy sample of all bins

Figure 17: Weight distribution of every $z$-bin using LePhare

##  Appendix C Properties of galaxies associated with $z=6-10$ overdensities
in COSMOS

In this appendix we list the properties of the galaxies associated with each
of the 15 $\geq 4\sigma$ overdensities given in Table 2. In Tables 4 to 18
given below, * indicates galaxies that fullfil the LBG dropout criteria
expected for their photometric redshift.

COSMOS2020-PCz6.05-01: We identify 19 sources inside the $4\sigma$ contour of
this overdensity, of which 14 are $i$-band dropouts (Table 4). The galaxies
span a volume of $10.7\times 14.3\times 66.3\,{\rm cMpc}$ (Fig. 18c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 2-13\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 19 galaxies
is shown in Fig. 18b along with the CDF of the field population. A two-sample
K-S test of the two CDFs yields a statistic of $0.28$ and a $p$-value of
$0.10$.

Table 4: Same as table 1 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare}/\mbox{$\rm M_{\odot}\,$})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1* | 127337 | 09:59:22.49 | 02:12:53.54 | 6.02${}^{+0.09}_{-0.08}$ | 6.06${}^{+0.05}_{-0.05}$ | 9.4${}^{+0.1}_{-0.2}$ | 48${}^{+16}_{-15}$ | 26.57
2* | 142959 | 09:59:20.56 | 02:13:10.62 | 6.12${}^{+0.08}_{-0.10}$ | 6.12${}^{+0.06}_{-0.05}$ | 9.1${}^{+0.1}_{-0.2}$ | 9${}^{+1}_{-1}$ | 26.17
3 | 156780 | 09:59:29.14 | 02:08:08.38 | 6.08${}^{+0.10}_{-0.10}$ | 6.08${}^{+0.05}_{-0.04}$ | 9.3${}^{+0.1}_{-0.2}$ | 13${}^{+1}_{-1}$ | 26.68
4 | 187778 | 09:59:23.08 | 02:13:20.10 | 6.06${}^{+0.09}_{-0.09}$ | 6.09${}^{+0.04}_{-0.04}$ | 8.7${}^{+0.2}_{-0.2}$ | 5${}^{+1}_{-1}$ | 26.28
5 | 220530 | 09:59:27.36 | 02:13:27.56 | 6.01${}^{+0.12}_{-0.11}$ | 6.03${}^{+0.05}_{-0.06}$ | 8.2${}^{+0.3}_{-0.2}$ | 2${}^{+1}_{-1}$ | 28.08
6* | 225263 | 09:59:33.21 | 02:08:23.24 | 6.07${}^{+0.14}_{-0.18}$ | 5.91${}^{+0.05}_{-0.06}$ | 9.4${}^{+0.2}_{-0.2}$ | 32${}^{+6}_{-7}$ | 25.47
7* | 361608 | 09:59:30.58 | 02:08:57.78 | 6.00${}^{+0.09}_{-0.08}$ | 5.99${}^{+0.05}_{-0.04}$ | 10.4${}^{+0.1}_{-0.1}$ | 15${}^{+1}_{-1}$ | 25.46
8 | 369661 | 09:59:23.33 | 02:14:04.42 | 6.01${}^{+0.13}_{-0.11}$ | 6.05${}^{+0.06}_{-0.06}$ | 9.4${}^{+0.2}_{-0.2}$ | 41${}^{+4}_{-6}$ | 27.05
9* | 413243 | 09:59:23.09 | 02:09:04.73 | 6.01${}^{+0.08}_{-0.08}$ | 5.99${}^{+0.05}_{-0.05}$ | 9.8${}^{+0.1}_{-0.2}$ | 44${}^{+44}_{-24}$ | 25.80
10* | 441761 | 09:59:35.16 | 02:09:53.31 | 5.98${}^{+0.12}_{-0.12}$ | 5.90${}^{+0.05}_{-0.06}$ | 10.0${}^{+0.1}_{-0.2}$ | 125${}^{+41}_{-26}$ | 25.43
11* | 444487 | 09:59:30.56 | 02:09:10.48 | 6.13${}^{+0.11}_{-0.12}$ | 6.08${}^{+0.05}_{-0.05}$ | 10.0${}^{+0.1}_{-0.1}$ | 16${}^{+1}_{-1}$ | 25.89
12* | 482804 | 09:59:33.50 | 02:09:17.17 | 6.00${}^{+0.09}_{-0.09}$ | 6.00${}^{+0.06}_{-0.06}$ | 9.3${}^{+0.2}_{-0.1}$ | 10${}^{+1}_{-1}$ | 25.32
13* | 573604 | 09:59:36.65 | 02:10:37.28 | 6.06${}^{+0.15}_{-0.20}$ | 5.94${}^{+0.20}_{-0.20}$ | 10.1${}^{+0.1}_{-0.1}$ | 9${}^{+1}_{-1}$ | 26.57
14* | 582186 | 09:59:33.25 | 02:09:45.81 | 6.05${}^{+0.11}_{-0.12}$ | 6.10${}^{+0.07}_{-0.07}$ | 10.2${}^{+0.1}_{-0.1}$ | 117${}^{+27}_{-21}$ | 25.88
15* | 694706 | 09:59:38.29 | 02:11:12.90 | 6.00${}^{+0.11}_{-0.09}$ | 5.98${}^{+0.05}_{-0.04}$ | 9.8${}^{+0.2}_{-0.2}$ | 73${}^{+61}_{-37}$ | 25.83
16* | 742465 | 09:59:35.88 | 02:11:29.06 | 6.08${}^{+0.10}_{-0.13}$ | 6.05${}^{+0.05}_{-0.05}$ | 9.6${}^{+0.1}_{-0.2}$ | 24${}^{+1}_{-1}$ | 25.08
17* | 759747 | 09:59:31.11 | 02:10:31.32 | 5.99${}^{+0.15}_{-0.22}$ | 6.04${}^{+0.06}_{-0.07}$ | 9.2${}^{+0.2}_{-0.2}$ | 11${}^{+1}_{-1}$ | 25.97
18* | 783817 | 09:59:27.96 | 02:10:39.35 | 6.06${}^{+0.07}_{-0.09}$ | 6.03${}^{+0.07}_{-0.06}$ | 9.8${}^{+0.1}_{-0.1}$ | 59${}^{+25}_{-15}$ | 25.48
19 | 958367 | 09:59:24.54 | 02:12:27.15 | 5.97${}^{+0.23}_{-0.19}$ | 5.96${}^{+0.16}_{-0.09}$ | 9.6${}^{+0.2}_{-0.2}$ | 39${}^{+4}_{-4}$ | 27.17
Figure 18: Same figures as a) 5, b) 7, c) 4, d) 2b, but for PCz6.05-01.

COSMOS2020-PCz6.05-02: We identify 8 sources inside the $4\sigma$ contour of
this overdensity, of which 7 are $i$-band dropouts and 1 has no $i$-band data.
(Table 5). The galaxies span a volume of $6.9\times 4.8\times 54.4\,{\rm
cMpc}$. (Fig. 19c). We infer a dark matter halo mass of $M_{\rm DM}\approx
8-26\times 10^{11}\,M_{\odot}$ using the methods described in 5.1. The stellar
mass CDF for the 8 galaxies is shown in Fig. 19b along with the CDF of the
field population. A two-sample K-S test of the two CDFs yields a statistic of
$0.45$ and a $p$-value of $0.05$.

Table 5: Same as table 1, but for PCz6.05-02 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare}/\mbox{$\rm M_{\odot}\,$})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 28317 | 09:59:29.55 | 01:57:39.75 | 5.99${}^{+0.07}_{-0.06}$ | 6.03${}^{+0.03}_{-0.04}$ | 9.4${}^{+0.1}_{-0.2}$ | 6${}^{+1}_{-1}$ | 27.56
2* | 644571 | 09:59:30.12 | 01:55:40.81 | 6.05${}^{+0.08}_{-0.09}$ | 6.05${}^{+0.05}_{-0.04}$ | 9.4${}^{+0.1}_{-0.1}$ | 8${}^{+1}_{-1}$ | 25.90
3* | 683055 | 09:59:23.16 | 01:55:56.08 | 6.09${}^{+0.06}_{-0.07}$ | 5.99${}^{+0.07}_{-0.04}$ | 9.9${}^{+0.1}_{-0.1}$ | 71${}^{+19}_{-14}$ | 25.18
4* | 743713 | 09:59:30.80 | 01:56:23.88 | 6.03${}^{+0.10}_{-0.10}$ | 6.03${}^{+0.09}_{-0.06}$ | 9.3${}^{+0.2}_{-0.2}$ | 28${}^{+12}_{-10}$ | 26.00
5* | 763833 | 09:59:21.88 | 01:56:31.77 | 6.10${}^{+0.05}_{-0.10}$ | 5.89${}^{+0.04}_{-0.04}$ | 10.3${}^{+0.1}_{-0.1}$ | 79${}^{+17}_{-14}$ | 24.87
6* | 828608 | 09:59:27.63 | 01:57:07.11 | 6.01${}^{+0.15}_{-0.27}$ | 6.06${}^{+0.05}_{-0.06}$ | 9.3${}^{+0.1}_{-0.2}$ | 11${}^{+1}_{-1}$ | 25.80
7* | 862479 | 09:59:33.29 | 01:57:25.31 | 6.00${}^{+0.10}_{-0.12}$ | 6.02${}^{+0.06}_{-0.09}$ | 9.5${}^{+0.1}_{-0.1}$ | 14${}^{+10}_{-5}$ | 26.02
8* | 871368 | 09:59:28.44 | 01:57:29.36 | 5.97${}^{+0.11}_{-0.10}$ | 5.93${}^{+0.07}_{-0.06}$ | 9.5${}^{+0.1}_{-0.1}$ | 23${}^{+18}_{-11}$ | 26.11

Figure 19:

COSMOS2020-PCz6.05-03: We identify 6 sources inside the $4\sigma$ contour of
this overdensity, of which 3 are $i$-band dropouts (Table 6). The galaxies
span a volume of $2.3\times 2.3\times 71.4\,{\rm cMpc}$. (Fig. 20c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 1\times 10^{12}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 8 galaxies is
shown in Fig. 20b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.43$ and a $p$-value of $0.17$.

Table 6: Same as table 1, but for PCz6.05-03 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 460349 | 09:59:25.61 | 02:24:51.99 | 5.99${}^{+0.13}_{-0.13}$ | 6.03${}^{+0.08}_{-0.09}$ | 9.8${}^{+0.1}_{-0.2}$ | 48${}^{+45}_{-35}$ | 26.68
2* | 465581 | 09:59:25.80 | 02:24:53.99 | 5.96${}^{+0.10}_{-0.08}$ | 5.96${}^{+0.04}_{-0.04}$ | 10.0${}^{+0.1}_{-0.1}$ | 145${}^{+61}_{-31}$ | 25.76
3 | 465988 | 09:59:25.65 | 02:24:54.26 | 6.13${}^{+0.12}_{-0.15}$ | 6.10${}^{+0.07}_{-0.09}$ | 10.6${}^{+0.1}_{-0.0}$ | 30${}^{+5}_{-5}$ | 25.50
4* | 545707 | 09:59:29.51 | 02:25:15.82 | 5.96${}^{+0.11}_{-0.10}$ | 5.90${}^{+0.06}_{-0.05}$ | 9.2${}^{+0.2}_{-0.2}$ | 7${}^{+1}_{-1}$ | 26.50
5 | 591296 | 09:59:28.36 | 02:25:32.79 | 6.06${}^{+0.11}_{-0.11}$ | 6.08${}^{+0.06}_{-0.05}$ | 9.8${}^{+0.1}_{-0.2}$ | 57${}^{+13}_{-12}$ | 26.18
6* | 650067 | 09:59:26.52 | 02:25:49.17 | 6.05${}^{+0.07}_{-0.11}$ | 5.99${}^{+0.06}_{-0.05}$ | 9.4${}^{+0.1}_{-0.2}$ | 12${}^{+1}_{-1}$ | 25.31

Figure 20:

COSMOS2020-PCz6.05-05: We identify 6 sources inside the $4\sigma$ contour of
this overdensity, of which 2 are $i$-band dropouts (Table 7). The galaxies
span a volume of $2.3\times 3.2\times 62.7\,{\rm cMpc}$. (Fig. 21c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 8-58\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 8 galaxies is
shown in Fig. 21b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.45$ and a $p$-value of $0.20$.

Table 7: Same as table 1, but for PCz6.05-05 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1* | 84673 | 10:00:38.07 | 02:12:54.33 | 5.98${}^{+0.19}_{-0.16}$ | 5.79${}^{+0.04}_{-0.04}$ | 9.6${}^{+0.1}_{-0.1}$ | 16${}^{+4}_{-3}$ | 26.76
2 | 90989 | 10:00:37.90 | 02:12:53.40 | 6.13${}^{+0.13}_{-0.14}$ | 6.18${}^{+0.07}_{-0.09}$ | 9.6${}^{+0.1}_{-0.1}$ | 58${}^{+5}_{-5}$ | 25.98
3 | 102863 | 10:00:38.90 | 02:12:59.67 | 6.10${}^{+0.14}_{-0.18}$ | 5.77${}^{+0.18}_{-0.49}$ | 9.7${}^{+0.1}_{-0.2}$ | 65${}^{+6}_{-5}$ | 25.51
4 | 862755 | 10:00:35.09 | 02:11:39.56 | 5.99${}^{+0.17}_{-0.14}$ | 5.88${}^{+0.08}_{-0.08}$ | 10.4${}^{+0.1}_{-0.2}$ | 154${}^{+5}_{-5}$ | 25.96
5* | 936376 | 10:00:35.66 | 02:12:13.21 | 5.98${}^{+0.09}_{-0.08}$ | 5.97${}^{+0.05}_{-0.05}$ | 9.2${}^{+0.1}_{-0.2}$ | 8${}^{+4}_{-4}$ | 27.24

Figure 21:

COSMOS2020-PCz6.05-06: We identify 8 sources inside the $4\sigma$ contour of
this overdensity, of which 5 are $i$-band dropouts (Table 8). The galaxies
span a volume of $4.9\times 6.1\times 43.0\,{\rm cMpc}$. (Fig. 22c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 5\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 8 galaxies is
shown in Fig. 22b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.24$ and a $p$-value of $0.66$.

Table 8: Same as table 1, but for PCz6.05-06 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 117712 | 10:00:58.43 | 01:47:56.73 | 6.00${}^{+0.09}_{-0.09}$ | 6.05${}^{+0.04}_{-0.04}$ | 9.6${}^{+0.1}_{-0.1}$ | 5${}^{+1}_{-1}$ | 31.44
2 | 199719 | 10:00:51.17 | 01:48:32.19 | 6.03${}^{+0.09}_{-0.08}$ | 6.05${}^{+0.04}_{-0.05}$ | 9.5${}^{+0.1}_{-0.1}$ | 8${}^{+3}_{-2}$ | 26.14
3* | 305408 | 10:00:56.50 | 01:48:37.47 | 6.03${}^{+0.06}_{-0.05}$ | 6.02${}^{+0.08}_{-0.04}$ | 9.3${}^{+0.2}_{-0.2}$ | 29${}^{+16}_{-10}$ | 25.36
4* | 839753 | 10:00:59.25 | 01:46:05.58 | 6.01${}^{+0.07}_{-0.06}$ | 6.05${}^{+0.03}_{-0.03}$ | 9.7${}^{+0.1}_{-0.1}$ | 8${}^{+1}_{-1}$ | 26.76
5* | 912236 | 10:00:53.13 | 01:46:23.26 | 6.07${}^{+0.09}_{-0.10}$ | 6.06${}^{+0.04}_{-0.04}$ | 8.4${}^{+0.2}_{-0.1}$ | 5${}^{+1}_{-1}$ | 26.32
6* | 944694 | 10:00:57.64 | 01:46:51.49 | 6.02${}^{+0.10}_{-0.10}$ | 6.06${}^{+0.05}_{-0.05}$ | 9.6${}^{+0.1}_{-0.1}$ | 6${}^{+1}_{-1}$ | nan
7* | 946072 | 10:00:55.03 | 01:46:56.03 | 6.05${}^{+0.05}_{-0.06}$ | 6.04${}^{+0.04}_{-0.04}$ | 9.7${}^{+0.1}_{-0.1}$ | 16${}^{+1}_{-1}$ | 25.56
8 | 947720 | 10:00:55.38 | 01:46:58.40 | 5.97${}^{+0.12}_{-0.11}$ | 5.96${}^{+0.08}_{-0.06}$ | 9.0${}^{+0.2}_{-0.2}$ | 6${}^{+2}_{-1}$ | 26.21

Figure 22:

COSMOS2020-PCz6.05-08: We identify 5 sources inside the $4\sigma$ contour of
this overdensity, of which 3 are $i$-band dropouts (Table 9). The galaxies
span a volume of $2.2\times 4.5\times 39.2\,{\rm cMpc}$. (Fig. 23c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 8\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 5 galaxies is
shown in Fig. 23b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.32$ and a $p$-value of $0.61$.

Table 9: Same as table 1, but for PCz6.05-08 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1* | 85122 | 10:01:02.38 | 01:57:50.42 | 6.07${}^{+0.05}_{-0.07}$ | 6.05${}^{+0.06}_{-0.04}$ | 10.1${}^{+0.1}_{-0.2}$ | 137${}^{+39}_{-29}$ | 24.93
2 | 89385 | 10:01:02.39 | 01:57:52.21 | 6.09${}^{+0.11}_{-0.11}$ | 6.15${}^{+0.06}_{-0.06}$ | 9.6${}^{+0.1}_{-0.1}$ | 33${}^{+25}_{-16}$ | nan
3* | 333962 | 10:01:06.00 | 01:59:02.17 | 5.99${}^{+0.11}_{-0.10}$ | 5.94${}^{+0.06}_{-0.06}$ | 9.3${}^{+0.2}_{-0.2}$ | 18${}^{+1}_{-2}$ | 26.03
4* | 368321 | 10:01:05.94 | 01:59:13.33 | 6.03${}^{+0.07}_{-0.06}$ | 6.03${}^{+0.05}_{-0.06}$ | 10.4${}^{+0.1}_{-0.1}$ | 16${}^{+1}_{-1}$ | 25.02
5 | 916028 | 10:01:02.94 | 01:57:22.12 | 6.05${}^{+0.13}_{-0.13}$ | 6.09${}^{+0.07}_{-0.06}$ | 8.8${}^{+0.2}_{-0.2}$ | 5${}^{+1}_{-1}$ | 26.59

Figure 23:

COSMOS2020-PCz6.69-02: We identify 5 sources inside the $4\sigma$ contour of
this overdensity, of which 2 are $z$-band dropouts (Table 10). The galaxies
span a volume of $2.3\times 6.4\times 15.0\,{\rm cMpc}$. (Fig. 24c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 1\times 10^{12}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 5 galaxies is
shown in Fig. 24b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.41$ and a $p$-value of $0.30$.

Table 10: Same as table 1, but for PCz6.69-02 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 188158 | 10:00:42.73 | 02:03:15.34 | 6.75${}^{+0.08}_{-0.05}$ | 6.91${}^{+0.01}_{-0.02}$ | 8.5${}^{+0.1}_{-0.1}$ | 8${}^{+1}_{-1}$ | 25.27
2* | 231335 | 10:00:43.87 | 02:03:19.66 | 6.74${}^{+0.12}_{-0.10}$ | 6.74${}^{+0.04}_{-0.04}$ | 10.4${}^{+0.1}_{-0.1}$ | 286${}^{+76}_{-80}$ | 25.48
3 | 318344 | 10:00:45.43 | 02:03:45.64 | 6.75${}^{+0.10}_{-0.07}$ | 6.74${}^{+0.04}_{-0.03}$ | 8.9${}^{+0.2}_{-0.2}$ | 8${}^{+2}_{-1}$ | 26.16
4 | 750274 | 10:00:45.86 | 02:01:10.13 | 6.78${}^{+0.29}_{-1.88}$ | 6.75${}^{+0.02}_{-0.02}$ | 10.0${}^{+0.2}_{-0.1}$ | 78${}^{+6}_{-8}$ | nan
5* | 894738 | 10:00:42.13 | 02:01:56.92 | 6.78${}^{+0.09}_{-0.10}$ | 6.70${}^{+0.07}_{-0.03}$ | 10.1${}^{+0.1}_{-0.1}$ | 190${}^{+43}_{-24}$ | 25.10

Figure 24:

COSMOS2020-PCz6.92-01: We identify 6 sources inside the $4\sigma$ contour of
this overdensity, of which 0 are $z$-band dropouts (Table 11). The galaxies
span a volume of $2.5\times 5.7\times 56.8\,{\rm cMpc}$. (Fig. 25c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 9-39\times 10^{10}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 6 galaxies is
shown in Fig. 25b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.51$ and a $p$-value of $0.06$.

Table 11: Same as table 1, but for PCz6.92-01 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 184953 | 09:57:46.04 | 01:48:46.31 | 6.93${}^{+0.24}_{-0.28}$ | 7.10${}^{+0.06}_{-0.08}$ | 9.2${}^{+0.2}_{-0.2}$ | 62${}^{+14}_{-12}$ | 26.03
2 | 185324 | 09:57:46.05 | 01:48:43.93 | 6.87${}^{+0.38}_{-0.27}$ | 6.98${}^{+4.73}_{-0.13}$ | 8.2${}^{+0.3}_{-0.3}$ | 2${}^{+1}_{-1}$ | nan
3 | 383012 | 09:57:47.89 | 01:49:48.22 | 6.84${}^{+0.19}_{-0.22}$ | 7.04${}^{+0.05}_{-0.06}$ | 8.5${}^{+0.3}_{-0.2}$ | 4${}^{+1}_{-1}$ | nan
4 | 426584 | 09:57:48.87 | 01:49:56.33 | 7.01${}^{+0.13}_{-6.44}$ | 7.11${}^{+0.05}_{-0.05}$ | 8.6${}^{+0.2}_{-0.2}$ | 7${}^{+1}_{-1}$ | nan
5 | 450133 | 09:57:46.82 | 01:50:04.89 | 6.93${}^{+0.20}_{-0.17}$ | 6.76${}^{+0.03}_{-0.02}$ | 9.3${}^{+0.1}_{-0.2}$ | 42${}^{+35}_{-23}$ | 27.92
6 | 651664 | 09:57:44.85 | 01:51:00.99 | 6.99${}^{+0.14}_{-0.15}$ | 6.91${}^{+0.03}_{-0.05}$ | 8.2${}^{+0.2}_{-0.1}$ | 3${}^{+1}_{-1}$ | 26.01

Figure 25:

COSMOS2020-PCz6.92-04 We identify 5 sources inside the $4\sigma$ contour of
this overdensity, of which 2 are $z$-band dropouts (Table 12). The galaxies
span a volume of $3.5\times 3.5\times 37.4\,{\rm cMpc}$. (Fig. 26c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 2-5\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 5 galaxies is
shown in Fig. 26b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.34$ and a $p$-value of $0.51$.

Table 12: Same as table 1, but for PCz6.92-04 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 445086 | 10:00:34.08 | 02:10:13.61 | 6.93${}^{+0.13}_{-0.13}$ | 6.94${}^{+0.03}_{-0.03}$ | 8.2${}^{+0.2}_{-0.1}$ | 5${}^{+1}_{-1}$ | 27.62
2* | 493215 | 10:00:32.63 | 02:10:25.58 | 6.85${}^{+0.14}_{-0.19}$ | 6.91${}^{+0.03}_{-0.03}$ | 8.9${}^{+0.1}_{-0.2}$ | 11${}^{+1}_{-1}$ | 25.66
3* | 567863 | 10:00:30.81 | 02:10:42.46 | 6.83${}^{+0.09}_{-0.08}$ | 6.76${}^{+0.03}_{-0.02}$ | 8.8${}^{+0.2}_{-0.1}$ | 12${}^{+1}_{-1}$ | 25.67
4 | 610764 | 10:00:36.42 | 02:10:24.91 | 6.89${}^{+0.64}_{-0.41}$ | 6.75${}^{+0.05}_{-0.05}$ | 9.4${}^{+0.3}_{-0.3}$ | 26${}^{+8}_{-7}$ | nan
5 | 750581 | 10:00:34.03 | 02:11:37.84 | 6.94${}^{+0.26}_{-0.17}$ | 6.76${}^{+0.02}_{-0.02}$ | 9.7${}^{+0.1}_{-0.2}$ | 40${}^{+3}_{-3}$ | 26.63
Figure 26:

COSMOS2020-PCz6.92-05: We identify 7 sources inside the $4\sigma$ contour of
this overdensity, of which 3 are $z$-band dropouts (Table 13). The galaxies
span a volume of $6.6\times 4.6\times 55.9\,{\rm cMpc}$. (Fig. 27c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 1\times 10^{12}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 7 galaxies is
shown in Fig. 27b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.50$ and a $p$-value of $0.05$.

Table 13: Same as table 1, but for PCz6.92-05 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1* | 134434 | 10:02:00.48 | 02:33:16.52 | 6.87${}^{+0.25}_{-0.14}$ | 6.83${}^{+0.03}_{-0.07}$ | 9.7${}^{+0.1}_{-0.2}$ | 92${}^{+45}_{-32}$ | 25.63
2 | 462038 | 10:02:05.87 | 02:34:37.81 | 6.93${}^{+0.13}_{-0.20}$ | 6.93${}^{+0.09}_{-0.11}$ | 9.7${}^{+0.1}_{-0.2}$ | 12${}^{+8}_{-2}$ | 26.60
3* | 507497 | 10:02:05.88 | 02:34:39.53 | 6.96${}^{+0.24}_{-0.26}$ | 6.89${}^{+0.04}_{-0.05}$ | 9.1${}^{+0.2}_{-0.2}$ | 21${}^{+2}_{-2}$ | 25.49
4 | 509093 | 10:02:06.34 | 02:34:43.20 | 6.96${}^{+0.13}_{-0.19}$ | 6.51${}^{+0.06}_{-0.05}$ | 10.0${}^{+0.1}_{-0.1}$ | 49${}^{+4}_{-4}$ | 25.44
5* | 554918 | 10:02:06.74 | 02:34:51.33 | 6.97${}^{+0.25}_{-0.21}$ | 6.94${}^{+0.03}_{-0.02}$ | 9.9${}^{+0.2}_{-0.2}$ | 86${}^{+6}_{-6}$ | 25.07
6 | 555718 | 10:02:06.68 | 02:34:44.61 | 7.03${}^{+0.22}_{-0.22}$ | 6.76${}^{+0.04}_{-0.03}$ | 10.3${}^{+0.1}_{-0.1}$ | 120${}^{+114}_{-65}$ | 26.02
7 | 592899 | 10:02:10.95 | 02:35:06.62 | 7.00${}^{+0.10}_{-0.17}$ | 7.04${}^{+0.05}_{-0.05}$ | 9.6${}^{+0.1}_{-0.1}$ | 25${}^{+8}_{-12}$ | 26.89
Figure 27:

COSMOS2020-PCz7.17-01: We identify 5 sources inside the $4\sigma$ contour of
this overdensity, of which 0 are $z$-band dropouts (Table 14). The galaxies
span a volume of $3.4\times 4.9\times 52.6\,{\rm cMpc}$. (Fig. 28c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 8\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 5 galaxies is
shown in Fig. 28b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.31$ and a $p$-value of $0.64$.

Table 14: Same as table 1, but for PCz7.17-01 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 43737 | 09:57:46.98 | 02:02:44.79 | 7.23${}^{+0.11}_{-0.11}$ | 7.24${}^{+0.05}_{-0.04}$ | 9.0${}^{+0.1}_{-0.2}$ | 10${}^{+1}_{-1}$ | 25.46
2 | 204754 | 09:57:43.51 | 02:03:40.26 | 7.20${}^{+0.11}_{-0.12}$ | 7.20${}^{+0.04}_{-0.04}$ | 8.7${}^{+0.2}_{-0.2}$ | 8${}^{+1}_{-1}$ | 26.23
3 | 258823 | 09:57:41.55 | 02:03:56.50 | 7.22${}^{+0.14}_{-0.18}$ | 6.86${}^{+0.05}_{-0.06}$ | 8.8${}^{+0.2}_{-0.2}$ | 11${}^{+2}_{-2}$ | 26.28
4 | 368191 | 09:57:42.57 | 02:04:27.35 | 7.12${}^{+0.13}_{-0.17}$ | 7.15${}^{+0.05}_{-0.06}$ | 10.6${}^{+0.2}_{-0.2}$ | 270${}^{+27}_{-26}$ | 25.46
5 | 871193 | 09:57:41.62 | 02:02:30.18 | 7.28${}^{+0.12}_{-0.11}$ | 7.17${}^{+0.05}_{-0.22}$ | 8.5${}^{+0.1}_{-0.1}$ | 7${}^{+1}_{-1}$ | 26.22
Figure 28:

COSMOS2020-PCz7.17-03: We identify 10 sources inside the $4\sigma$ contour of
this overdensity, of which 0 are $z$-band dropouts (Table 15). The galaxies
span a volume of $8.7\times 17.3\times 76.0\,{\rm cMpc}$. (Fig. 29c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 2-7\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 10 galaxies
is shown in Fig. 29b along with the CDF of the field population. A two-sample
K-S test of the two CDFs yields a statistic of $0.36$ and a $p$-value of
$0.13$.

Table 15: Same as table 1, but for PCz7.17-03 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 12090 | 09:58:54.28 | 02:47:38.65 | 7.16${}^{+0.09}_{-0.09}$ | 7.17${}^{+0.04}_{-0.04}$ | 8.7${}^{+0.2}_{-0.2}$ | 11${}^{+1}_{-1}$ | 25.77
2 | 15707 | 09:59:00.08 | 02:42:42.19 | 7.09${}^{+0.09}_{-0.11}$ | 6.96${}^{+0.05}_{-0.03}$ | 8.3${}^{+0.1}_{-0.1}$ | 4${}^{+1}_{-1}$ | 27.11
3 | 65232 | 09:58:52.51 | 02:42:50.36 | 7.27${}^{+0.05}_{-0.05}$ | 7.21${}^{+0.03}_{-0.03}$ | 8.9${}^{+0.2}_{-0.1}$ | 23${}^{+1}_{-1}$ | 24.89
4 | 425359 | 09:58:54.30 | 02:40:47.76 | 7.06${}^{+0.10}_{-0.12}$ | 7.08${}^{+0.05}_{-0.08}$ | 8.3${}^{+0.2}_{-0.1}$ | 5${}^{+1}_{-1}$ | 26.05
5 | 561655 | 09:58:54.44 | 02:45:03.72 | 7.17${}^{+0.10}_{-0.10}$ | 6.94${}^{+0.03}_{-0.03}$ | 8.4${}^{+0.1}_{-0.1}$ | 4${}^{+1}_{-1}$ | 25.45
6 | 572331 | 09:58:46.26 | 02:41:25.08 | 7.21${}^{+0.03}_{-0.06}$ | 7.14${}^{+0.01}_{-0.01}$ | 9.5${}^{+0.1}_{-0.1}$ | 203${}^{+32}_{-25}$ | 24.62
7 | 573770 | 09:58:52.58 | 02:41:25.24 | 7.05${}^{+0.10}_{-0.10}$ | 6.94${}^{+0.03}_{-0.02}$ | 8.4${}^{+0.1}_{-0.1}$ | 6${}^{+1}_{-1}$ | 26.39
8 | 720685 | 09:58:52.60 | 02:42:06.86 | 7.21${}^{+0.04}_{-0.04}$ | 7.20${}^{+0.02}_{-0.02}$ | 9.4${}^{+0.1}_{-0.2}$ | 53${}^{+1}_{-1}$ | 24.01
9 | 748702 | 09:58:52.29 | 02:46:19.69 | 7.20${}^{+0.16}_{-0.16}$ | 7.16${}^{+0.06}_{-0.08}$ | 8.4${}^{+0.2}_{-0.2}$ | 5${}^{+1}_{-1}$ | 25.9
10 | 931635 | 09:58:52.26 | 02:47:36.93 | 7.24${}^{+0.13}_{-0.17}$ | 7.23${}^{+0.06}_{-0.05}$ | 8.9${}^{+0.2}_{-0.1}$ | 13${}^{+1}_{-1}$ | 26.57
Figure 29:

COSMOS2020-PCz7.17-04: We identify 5 sources inside the $4\sigma$ contour of
this overdensity, of which 0 are $z$-band dropouts (Table 16). The galaxies
span a volume of $1.6\times 1.9\times 31.7\,{\rm cMpc}$. (Fig. 30c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 9-35\times 10^{10}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 5 galaxies is
shown in Fig. 30b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.34$ and a $p$-value of $0.52$.

Table 16: Same as table 1, but for PCz7.17-04 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 300458 | 10:00:53.42 | 02:24:01.42 | 7.25${}^{+0.68}_{-0.35}$ | 7.26${}^{+0.09}_{-0.08}$ | 8.6${}^{+0.3}_{-0.3}$ | 3${}^{+1}_{-1}$ | 27.51
2 | 308815 | 10:00:53.60 | 02:24:03.06 | 7.24${}^{+0.15}_{-0.20}$ | 7.28${}^{+0.06}_{-0.06}$ | 8.8${}^{+0.2}_{-0.3}$ | 6${}^{+1}_{-1}$ | 26.98
3 | 354781 | 10:00:54.00 | 02:24:28.83 | 7.29${}^{+1.68}_{-0.19}$ | 7.29${}^{+0.07}_{-0.07}$ | 9.4${}^{+0.2}_{-0.2}$ | 34${}^{+21}_{-15}$ | 25.95
4 | 368364 | 10:00:51.55 | 02:24:34.59 | 7.29${}^{+0.22}_{-0.18}$ | 7.26${}^{+0.09}_{-0.07}$ | 8.4${}^{+0.2}_{-0.2}$ | 4${}^{+1}_{-1}$ | 26.39
5 | 409511 | 10:00:52.95 | 02:24:46.54 | 7.19${}^{+0.08}_{-0.09}$ | 7.17${}^{+0.04}_{-0.04}$ | 8.7${}^{+0.2}_{-0.2}$ | 9${}^{+1}_{-1}$ | 25.89
Figure 30:

COSMOS2020-PCz7.42-01: We identify 8 sources inside the $4\sigma$ contour of
this overdensity, of which 0 are $z$-band dropouts (Table 17). The galaxies
span a volume of $6.5\times 4.3\times 54.1\,{\rm cMpc}$. (Fig. 31c). We infer
a dark matter halo mass of $M_{\rm DM}\approx 1-4\times 10^{11}\,M_{\odot}$
using the methods described in 5.1. The stellar mass CDF for the 8 galaxies is
shown in Fig. 31b along with the CDF of the field population. A two-sample K-S
test of the two CDFs yields a statistic of $0.53$ and a $p$-value of $0.02$.

Table 17: Same as table 1, but for PCz7.42-01 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 39068 | 09:57:55.62 | 02:37:49.20 | 7.40${}^{+1.20}_{-0.56}$ | 7.45${}^{+0.19}_{-0.17}$ | 8.4${}^{+0.3}_{-0.2}$ | 4${}^{+1}_{-1}$ | 26.99
2 | 40168 | 09:57:46.25 | 02:37:52.44 | 7.39${}^{+1.01}_{-0.57}$ | 7.27${}^{+1.16}_{-0.46}$ | 8.6${}^{+0.3}_{-0.4}$ | 1${}^{+1}_{-1}$ | nan
3 | 41007 | 09:57:56.39 | 02:37:49.26 | 7.39${}^{+0.41}_{-0.21}$ | 7.30${}^{+0.13}_{-7.18}$ | 8.5${}^{+0.2}_{-0.2}$ | 5${}^{+1}_{-1}$ | 28.48
4 | 42562 | 09:57:53.76 | 02:37:47.65 | 7.34${}^{+1.13}_{-0.50}$ | 6.98${}^{+0.23}_{-0.07}$ | 8.3${}^{+0.2}_{-0.2}$ | 2${}^{+1}_{-1}$ | 27.94
5 | 43841 | 09:57:53.86 | 02:37:47.76 | 7.29${}^{+0.48}_{-0.25}$ | 7.23${}^{+0.11}_{-0.11}$ | 8.6${}^{+0.3}_{-0.3}$ | 4${}^{+1}_{-1}$ | 27.6
6 | 44452 | 09:57:53.62 | 02:37:49.56 | 7.41${}^{+0.44}_{-0.27}$ | 7.39${}^{+0.11}_{-0.11}$ | 8.8${}^{+0.3}_{-0.3}$ | 5${}^{+1}_{-1}$ | 26.17
7 | 85572 | 09:57:49.69 | 02:38:05.73 | 7.46${}^{+0.71}_{-0.47}$ | 7.47${}^{+0.11}_{-0.14}$ | 8.9${}^{+0.3}_{-0.3}$ | 5${}^{+1}_{-1}$ | 26.42
8 | 384723 | 09:57:55.33 | 02:39:29.21 | 7.41${}^{+0.23}_{-0.22}$ | 7.46${}^{+0.07}_{-0.07}$ | 9.4${}^{+0.3}_{-0.4}$ | 12${}^{+2}_{-2}$ | nan
Figure 31:

COSMOS2020-PCz7.69-01: We identify 5 sources inside the $4\sigma$ contour of
this overdensity, of which 0 are $z$-band dropouts (Table 18). The galaxies
span a volume of $12.0\times 13.6\times 49.4\,{\rm cMpc}$. (Fig. 32c). We
infer a dark matter halo mass of $M_{\rm DM}\approx 9\times
10^{11}\,M_{\odot}$ using the methods described in 5.1. The stellar mass CDF
for the 5 galaxies is shown in Fig. 32b along with the CDF of the field
population. A two-sample K-S test of the two CDFs yields a statistic of $0.24$
and a $p$-value of $0.92$.

Table 18: Same as table 1, but for PCz7.69-01 # | ID | R.A. (J2000) | Decl. (J2000) | $z_{\rm Lephare}$ | $z_{\rm EAZY}$ | $\log(M_{\rm\star,LePhare})$ | $SFR_{\rm UV}$ | $K_{\rm S}$
---|---|---|---|---|---|---|---|---
|  | hh:mm:ss.ss | dd:mm:ss.ss |  |  | ${\rm dex}$ | ${\rm[M_{\odot}\,yr^{-1}]}$ | ${\rm[AB\,mag]}$
1 | 20030 | 09:57:44.11 | 02:27:38.97 | 7.64${}^{+0.51}_{-0.24}$ | 7.39${}^{+0.12}_{-0.12}$ | 10.5${}^{+0.1}_{-0.2}$ | 17${}^{+1}_{-1}$ | 25.92
2 | 71035 | 09:57:54.70 | 02:27:54.93 | 7.80${}^{+0.33}_{-0.16}$ | 7.67${}^{+0.08}_{-0.07}$ | 10.0${}^{+0.1}_{-0.1}$ | 16${}^{+5}_{-1}$ | 25.84
3 | 550618 | 09:57:55.46 | 02:30:12.19 | 7.66${}^{+1.01}_{-6.56}$ | 7.65${}^{+1.22}_{-0.08}$ | 8.9${}^{+0.2}_{-0.3}$ | 7${}^{+1}_{-1}$ | 26.42
4 | 605774 | 09:57:36.81 | 02:24:54.68 | 7.64${}^{+0.65}_{-0.17}$ | 7.59${}^{+0.06}_{-0.06}$ | 8.9${}^{+0.2}_{-0.3}$ | 9${}^{+1}_{-1}$ | 27.48
5 | 801177 | 09:57:45.23 | 02:25:57.54 | 7.69${}^{+1.15}_{-0.30}$ | 7.51${}^{+0.09}_{-0.10}$ | 9.6${}^{+0.4}_{-0.4}$ | 21${}^{+17}_{-3}$ | 26.21
Figure 32: