# Photonic resource state generation from a minimal number of quantum emitters

Bikun Li<EMAIL_ADDRESS>Department of Physics, Virginia Tech, Blacksburg,
Virginia 24061, USA  Sophia E. Economou<EMAIL_ADDRESS>Department of
Physics, Virginia Tech, Blacksburg, Virginia 24061, USA  Edwin Barnes
<EMAIL_ADDRESS>Department of Physics, Virginia Tech, Blacksburg, Virginia
24061, USA

###### Abstract

Multi-photon entangled graph states are a fundamental resource in quantum
communication networks, distributed quantum computing, and sensing. These
states can in principle be created deterministically from quantum emitters
such as optically active quantum dots or defects, atomic systems, or
superconducting qubits. However, finding efficient schemes to produce such
states has been a long-standing challenge. Here, we present an algorithm that,
given a desired multi-photon graph state, determines the minimum number of
quantum emitters and precise operation sequences that can produce it. The
algorithm itself and the resulting operation sequence both scale polynomially
in the size of the photonic graph state, allowing one to obtain efficient
schemes to generate graph states containing hundreds or thousands of photons.

††preprint: APS/123-QED

##  I Introduction

Entanglement is widely recognized as playing a critical role in quantum
computation, error correction, communication, and sensing. A family of
entangled states that features prominently in these applications are graph (or
cluster) states. They are key resources in one-way quantum computing paradigms
[1, 2] and in quantum error correction [3, 4, 5, 6]. In addition, many quantum
repeater schemes [7, 8, 9, 10, 11] and quantum sensing protocols [12, 13] rely
on graph states. Photonic graph states are especially important because
photons are the predominant platform for measurement- and fusion-based
computing, and, as flying qubits, they are the only viable choice for quantum
networks [14] and quantum imaging [15, 16].

Unfortunately, creating photonic resource states is fundamentally difficult.
Because photons do not interact with each other, most attempts have focused on
probabilistic generation schemes using linear optics and postselection [17],
which are very resource-intensive, severely limiting the size of the resulting
states [18, 19]. This bottleneck can in principle be overcome by instead using
a deterministic approach in which entangled photons are produced directly from
quantum emitters (i.e., matter qubits). One possibility would be to prepare a
graph state on emitters [20, 21] and transduce it to photons, but this
requires a number of emitters equal to the size of the target photonic graph
state. This daunting resource overhead can be avoided by instead using
sequential generation schemes. Refs. [22, 23] put forward such an approach
that works well for one-dimensional (1D) graph states [24] and has led to
experimental demonstrations [25, 26]. However, in the general case where the
entanglement structure is more complicated, this method scales exponentially
in the size of the target state and can lead to long generation circuits,
motivating the search for more efficient approaches. Refs. [27, 28] put
forward protocols for 2D lattice graphs that leverage the principle that
entangled emitters can emit entangled photons. This idea was extended further
to develop protocols that deterministically generate resource states for
quantum repeaters [29, 30, 31, 32]—tailored to color centers in Refs. [33,
34]—and one-way computing [35, 36]. Refs. [35, 32] allowed for the re-
interference of photons with emitters to further enhance flexibility in
entanglement creation.

Despite this progress and the intense interest this approach has generated
among experimentalists, existing graph state generation protocols are limited
to a small subset of graphs or require a number of emitters that scales
linearly with the graph size [37, 36]. This is extremely resource-intensive,
especially in light of the schemes for generating repeater graph states
presented in Refs. [29, 31], which require only two emitters regardless of the
number of photons. The required resources (number of emitters and entangling
gates) is a critical factor that determines the practical feasibility of the
protocol. For a general graph state, finding resource-efficient generation
protocols in polynomial time remains an open problem.

Here, we address this challenge by presenting a general approach to generating
arbitrary photonic graph states from quantum emitters. Given a target graph
state, we show how to determine in polynomial time both the minimal number of
emitters required to create it and an explicit generation protocol. The latter
consists of a sequence of gate operations and measurements performed on the
emitters. Moreover, our protocol naturally takes into account the order in
which photons should be emitted, which can be an important consideration for
applications, as it is generally preferable to emit photons in the order they
are measured to avoid photon storage. Our method provides a recipe for doing
this. The broad applicability of our method, its practical relevance, and its
efficient use of resources make it ideally suited to the generation of any
photonic graph state from various types of quantum emitters.

Figure 1: Illustration of the protocol solver algorithm. (a) An example of a
4-photon graph state. (b) The graph is mapped to a 1D lattice. (c) The height
function is computed and found to have maximum value 2, implying 2 emitters
are needed. These are added to the 1D lattice. (d) Starting from the target
state and decoupled emitters, a time-reversed sequence of emitter gates,
photon absorption events, and time-reversed emitter measurements is
constructed, until all qubits are disentangled. Further details about this
example can be found in the Supplementary Information.

##  II Results and Discussion

###  II.1 Overview of the algorithm

Determining how to efficiently generate an arbitrary photonic graph state from
a set of quantum emitters is highly nontrivial and markedly distinct from the
problem of finding an efficient quantum circuit that creates a target state on
a register of qubits [38]. Several additional challenges arise in the former,
including the fact that qubits are both created and removed, and that
different types of qubits (photons vs. emitters), with different roles and
allowed gates, are involved. Depending on the experimental setup, there may
also be further restrictions, e.g., emitted photons cannot interact with any
other qubits following their emission (although schemes that re-interfere
photons with emitters have been proposed [35, 32]). Our method addresses these
challenges by leveraging three main ingredients: the notion of the height
function (which is related to the entanglement entropy), the stabilizer
formalism, and the concept of time-reversed emission events and measurements,
which we introduce here.

The first insight is to utilize the so-called height function, which is the
entanglement entropy of the system as a function of the partition point when
the system is arranged in a 1D lattice and partitioned into two subsystems
[39, 40]. This function provides information about the entanglement structure
of the target state as well as the number of emitters required to produce it.
The latter is equal to the maximum value of the height function (see below),
which depends on the photon emission order. Optimizing this order is NP-hard
in general, although we show that heuristic approaches exist for more
structured graphs. Moreover, the height function plays a crucial role in
determining the sequence of operations (gates and measurements) needed to
generate the target graph state from the emitters.

A second key ingredient is the use of gates from the Clifford group. Given
that arbitrary graph states can be generated solely with Clifford gates [41,
42], which were also exclusively used in the protocols of Refs. [24, 25, 27,
29, 30, 36, 28, 31], restricting ourselves to this set does not affect the
generality of our approach. Clifford gates enable the use of the stabilizer
formalism, such that we can manipulate Pauli operators instead of keeping
track of the whole state. This makes the problem of finding the emission
operation sequence tractable, reducing it from exponential to polynomial
scaling due to the Gottesman-Knill theorem [43].

A final key element in our algorithm is that we time-reverse the emission
sequence. That is, we start from a target multi-photon graph state and an
appropriate number of decoupled emitters (obtained from the height function
for the target state), and we determine a sequence of emitter gates, “time-
reversed measurements”, and “photon absorption” events such that the target
state is converted to a product state. This is somewhat reminiscent of
disentangling circuits used for quantum state tomography of 1D systems [44].
The final state is a product state because, without loss of generality,
photons that have not yet been emitted can be described by qubits prepared in
the computational basis state $\left|0\right\rangle$. Photon emission is then
modeled as a two-qubit photon-emitter gate that brings the photon from
$\left|0\right\rangle$ into an entangled state with the emitters [24]. Because
the photon absorption steps are time-reversed versions of photon emission,
these too are described by photon-emitter gates.

The run time of the protocol solver algorithm scales as ${\cal O}(n_{p}^{4})$,
where $n_{p}$ is the number of photons in the target graph state. This is a
direct consequence of the fact that the algorithm is based on the stabilizer
formalism (see Methods section). This is in contrast to previous methods [22,
23], which scale exponentially in $n_{p}$ due to the need to perform singular
value decompositions repeatedly. We also show that the number of gates in the
final emission sequence scales at most as ${\cal O}(n_{p}^{2})$ (see Methods).
However, this assumes two-qubit gates can be applied between any pair of
emitters. If this is not the case, then additional SWAP operations are needed,
bringing the gate count up to ${\cal O}(n_{p}^{3})$. Therefore, both the
protocol solver and the resulting gate sequence it obtains scale polynomially
in the size of the target graph state.

Now we provide a more detailed description of the protocol solver algorithm.
We begin with a target graph state $\left|\psi_{p}\right\rangle$ of $n_{p}$
photons and $n_{e}$ decoupled emitters, so that the total state is
$\left|\Psi\right\rangle=\left|\psi_{p}\right\rangle\otimes\left|0\right\rangle^{\otimes
n_{e}}$. An $n_{p}=4$ photon example graph is shown in Fig. 1(a). This is what
the state of the total system should be at the end of the generation sequence.
$n_{p}$ is set by the size of the desired photonic graph state
$\left|\psi_{p}\right\rangle$, while $n_{e}$ remains to be determined. We
assume the graph representing $\left|\psi_{p}\right\rangle$ is connected; if
this is not the case, then the algorithm can be run separately for each
connected subgraph. The state $\left|\Psi\right\rangle$ is fully described by
a set of $n=n_{p}+n_{e}$ stabilizers $g_{m}$, $m=1,\ldots,n$, defined such
that $g_{m}\left|\Psi\right\rangle=\left|\Psi\right\rangle$. The full set of
$n$ qubits can be arranged in a 1D lattice with site index
$x\in\\{0,1,2,\ldots,n\\}$ (see Fig. 1(b)). Sites $x=1,\ldots,n_{p}$
correspond to the photons and are ordered according to the desired photon
emission ordering, while the sites $x=n_{p}+1,\ldots,n$ are the emitters. The
additional $x=0$ site is included as a matter of convention. We can now define
the height function $h(x)=S_{A}$ to be the bipartite entanglement entropy when
the 1D lattice is divided into the subregion $A=\\{1,2,\ldots,x\\}$ and its
complement. Note that
$S_{A}=\frac{1}{1-\alpha}\log_{2}\mathrm{Tr}(\rho^{\alpha}_{A})$ can be any of
the Rényi entropies; for stabilizer states, they are all equal [45]. In Ref.
[22], it was shown that the state of the emitted photons,
$\left|\psi_{p}\right\rangle$, can be represented by a matrix product state
(MPS) with bond dimension $2^{n_{e}}$. Because the entanglement entropy of a
MPS is given by the base-2 logarithm of the bond dimension [46], it follows
that $n_{e}$ is equal to the maximum value of $h(x)$. The height function for
the graph in Fig. 1(a) is shown in Fig. 1(c). In this example, its maximum is
2, implying 2 emitters are needed. In general, the maximum of the height
function is in fact the minimal number of emitters capable of generating the
target graph state, as fewer emitters would be insufficient to match the bond
dimension of any exact MPS representation.

The height function can be computed efficiently from the stabilizers. Because
products of stabilizers are also stabilizers, there are many equivalent
choices for the set $\\{g_{m}\\}$. Here, we focus on a particular choice of
the stabilizers that we refer to as the echelon gauge [47], in which the
stabilizer matrix has a row-reduced echelon form (see Methods). When the
$g_{m}$ are in this gauge, the height function can be expressed as [47]

$h(x)=n-x-\\#\\{g_{m}|\verb|l|(g_{m})>x\\},$ (1)

where $\verb|l|(g_{m})$ is the index of the left-most (smallest index) site on
which $g_{m}$ acts nontrivially. The last term in Eq. (1) counts the number of
stabilizers that act nontrivially only on sites to the right of (i.e., larger
than) $x$. Although Eq. (1) depends on $n_{e}$, this dependence cancels out
for states like $\left|\Psi\right\rangle$ in which the emitters are decoupled.
Therefore we can obtain $n_{e}$ from the maximum of $h(x)$ on the photonic
sites, using only the stabilizers of $\left|\psi_{p}\right\rangle$.

Once we have the number of emitters $n_{e}$, we can run the protocol solver
algorithm to determine the sequence of gates, time-reversed measurements, and
photon absorption events needed to transform the target state
$\left|\Psi\right\rangle$ into the initial state
$\left|0\right\rangle^{\otimes n}$, which corresponds to decoupled emitters
and no photons. We first introduce a photon index $j$ and initialize it to
$j=n_{p}$. The algorithm then consists of four steps:

  1. (i)

Transform the stabilizers $g_{m}$ into echelon gauge if they are not already,
then compute the height function $h(x)$.

  2. (ii)

If $h(j)\geq h(j-1)$, skip to step (iii). Otherwise apply a time-reversed
measurement and update the $g_{m}$ accordingly.

  3. (iii)

Apply a photon absorption operation on the $j$-th photon and update the
$g_{m}$ accordingly. If $j>1$, then set $j\to j-1$ and go to step (i).
Otherwise, go to step (iv).

  4. (iv)

All photons are now in state $\left|0\right\rangle$. Apply a series of gates
on the emitters to disentangle them, bringing the total state to
$\left|0\right\rangle^{\otimes n}$.

This algorithm involves repeated applications of two basic operational
primitives: time-reversed measurement and photon absorption. During the
algorithm, the height function of the current state tells us which of these we
need to perform next to bring the state closer to
$\left|0\right\rangle^{\otimes n}$. Each photon absorption step disentangles
one photon qubit from the rest, starting with the last-emitted photon,
$j=n_{p}$, and working down to the first photon, $j=1$. For our 4-photon
example, the graphs at intermediate steps of the algorithm are shown in Fig.
1(d). A step by step explanation of this example is given in the Supplementary
Information. When the algorithm concludes, we can reverse the entire sequence
to obtain an operation sequence that generates $\left|\psi_{p}\right\rangle$
starting from $n_{e}$ decoupled emitters. We now describe each of the two
operational primitives in more detail, the precise gates they introduce into
the generation sequence, and their connection to the height function.

Figure 2: Results for repeater graph states. (a) 12-photon repeater graph
state in which external photons are emitted first. (b) Same graph state as in
(a), but with “natural” emission ordering. (c) Same graph state as in (b) but
with some unnecessary edges deleted. (d), (e) and (f) show the height
functions of the states in (a), (b) and (c), respectively. (g) Emission
circuit for state shown in (c), where $H$ is the Hadamard gate,
$P=\mathrm{diag}(1,i)$ is the phase gate, and $X\equiv\sigma^{x}$.

Photon absorption of the $j$-th photon refers to a time-reversed version of
photon emission. For concreteness, we focus on the case where emission is
described by a CNOT gate between the photon and its emitter (with the emitter
as the control), as in Ref. [24], although our algorithm can be adapted to any
Clifford gate describing photon emission. Mathematically, the task of
absorbing photon $j$ requires finding a stabilizer $g_{a}$ that can be
transformed to $\sigma_{j}^{z}$ by applying CNOTij, where $i$ is an emitter
site. It is possible to find such a stabilizer when $h(j)\geq h(j-1)$. From
Eq. (1), we see that this condition implies there must be at least one
stabilizer, $g_{a}$, such that $\verb|l|(g_{a})=j$. This stabilizer has the
form

$g_{a}=\sigma^{\alpha}_{j}\sigma_{i_{1}}^{\beta_{1}}\cdots\sigma_{i_{s}}^{\beta_{s}},$
(2)

where $\alpha,\beta_{k}\in\\{x,y,z\\}$ label the nontrivial Pauli operators,
and $1\leq j\leq n_{p}<i_{1}<\cdots<i_{s}\leq n$. Note that we can assume
$g_{a}$ acts trivially on all photons with index larger than $j$ since these
have already been decoupled at this point in the algorithm. We also assume
that $g_{a}$ acts nontrivially on at least one emitter site; if this is not
the case, then photon absorption is unnecessary since the $j$-th photon is
then already disconnected. To transform $g_{a}$ into $\sigma_{j}^{z}$, we can
first apply a local Clifford operation on the $j$-th site and general Clifford
operations on the emitters to transform
$g_{a}\to\sigma_{j}^{z}\sigma_{i}^{z}$, where $i>n_{p}$ is an emitter site.
This can be done for example by applying local Clifford operations to
transform $g_{a}$ to
$\sigma^{z}_{j}\sigma_{i_{1}}^{z}\cdots\sigma_{i_{s}}^{z}$, and then applying
CNOT gates on pairs of emitters to transform this to
$\sigma_{j}^{z}\sigma_{i}^{z}$. Applying CNOTij brings this to
$\sigma_{j}^{z}$, completing the absorption of the $j$-th photon. Note that we
can choose any emitter to absorb the photon; typically, the emitter that
requires the shortest circuit to transform $g_{a}$ into $\sigma_{j}^{z}$ is
preferred. The resulting circuit is included in the time-reversed generation
sequence.

Time-reversed measurements are applied whenever $h(j)<h(j-1)$, in which case
photon absorption is not possible. Indeed, in this case, Eq. (1) implies
$\\#\\{g_{m}|\verb|l|(g_{m})=j\\}=0$, or in other words, a suitable $g_{a}$
does not exist. In order to absorb the next photon, we must therefore first
find a way to increase $h(j)$ relative to $h(j-1)$. This can be accomplished
with a time-reversed measurement on an emitter. To perform this operation, we
first rotate the state to
$\left|\Phi\right\rangle\otimes\left|0\right\rangle_{i}$, where
$\left|\Phi\right\rangle$ is a stabilizer state involving photons $1,\ldots,j$
and emitters other than $i$. This can always be done using
$\mathcal{O}(n_{e})$ Clifford gates on emitters when $h(j)<h(j-1)$ (see
Methods). Now notice that this state is obtained from the pre-measurement
state CNOT${}_{ij}\left|\Phi\right\rangle\otimes\left|+\right\rangle_{i}$ when
emitter $i$ is measured to be in the state $\left|0\right\rangle_{i}$.
Therefore, starting from
$\left|\Phi\right\rangle\otimes\left|0\right\rangle_{i}$, if we perform a
Hadamard gate on emitter $i$ followed by the gate CNOTij, we effectively
reverse the measurement on the emitter. These operations transform the
stabilizers $g_{m}$ in such a way that $h(j)$ now satisfies $h(j)\geq h(j-1)$
(see Methods), and we can proceed with the next photon absorption. The emitter
gates, Hadamard on $i$, and CNOTij are all included in the time-reversed
generation sequence.

###  II.2 Examples

We demonstrate our algorithm with several examples. The first is the important
case of repeater graph states [10], where we use our algorithm to obtain
generation protocols that are more efficient than previously known ones. As a
second example, we consider random graphs containing up to hundreds of photons
and demonstrate the polynomial scaling of the resulting generation circuits.
Additional examples, including modified repeater graph states, error
correcting codes, and a simple example that illustrates the algorithm in
detail can be found in Supplementary Notes 1-4.

Next, we apply our algorithm to find operation sequences that produce repeater
graph states [10]. In addition to its importance in quantum network
applications, this example also illustrates how different photon emission
orderings impact the required number of emitters. Ref. [29] presented a
generation protocol for a particular ordering that was devised essentially
through guesswork. Our algorithm can be used to systematically find protocols
for any ordering. An example of a 12-photon repeater graph state is shown in
Fig. 2(a). The graph contains a fully connected core of 6 photons, each of
which is connected to a single external photon. Bell measurements are
performed on pairs of these external photons, where the two photons in each
pair come from different graph states. If a Bell measurement succeeds, then
the two corresponding core photons are linked by an edge, and entanglement
extends across two nodes of the repeater network. Having multiple external
photons provides built-in redundancy that increases the likelihood that at
least one Bell measurement between two repeater graph states is successful.
Upon success, core photons are then measured in the $z$ or $x$ basis to remove
photons connected to failed measurements or to create entanglement links
between successful measurements, respectively. Because the external photons
are measured first, it may be advantageous to emit these first when generating
the graph state to reduce photon storage requirements. This corresponds to the
photon ordering shown in Fig. 2(a). The height function for this graph and
photon ordering is shown in Fig. 2(d), where it is evident that 6 emitters are
needed to produce the state. However, if efficient photon storage is
available, then the ordering shown in Fig. 2(b) may be preferable, where now
external and core photons are emitted in an alternating sequence. This
ordering reduces the number of emitters down to only 2, as shown in Fig. 2(e).
As we discuss further below, this illustrates our general finding that
“natural” orderings in which neighboring vertices are emitted around the same
time reduce the requisite number of emitters. This reduction in quantum
resources becomes still more dramatic as the size of the graph increases; for
orderings as shown in Fig. 2(a), the number of emitters scales linearly with
photon number, while for the natural ordering of Fig. 2(b), the number of
emitters remains at 2 regardless of the number of photons. This is shown
explicitly in the Supplemental Information.

As discussed in Ref. [36], some of the edges in the repeater graph can be
removed without affecting the functionality of the repeater. Fig. 2(c) shows
an example of this in which 4 of the core edges are deleted. As shown in Fig.
2(f), the number of emitters is still 2. However, removing the redundant edges
reduces the depth of the resulting generation circuit, which is shown in Fig.
2(g). This circuit contains 4 CNOTs between emitters and 1 intermediate
measurement on an emitter, whereas the original protocol presented in Ref.
[29] requires 5 two-qubit gates and 5 intermediate measurements.

To demonstrate how our algorithm scales with the number of photons in the
target state, we run it for random graphs ranging in size from $n_{p}=16$ to
$n_{p}=256$ photons. These graphs are produced randomly using the Erdös–Rényi
model [48]. In this approach, each random graph is constructed by connecting
$n_{p}$ vertices randomly with fixed probability $p$. We discard any graphs
that contain disconnected vertices when sampling these realizations. The
likelihood that such graphs arise becomes very small if $p$ is chosen
sufficiently close to 1. In Fig. 3, we show the maximum value,
$h_{\text{max}}$, of the height function averaged over 1024 realizations for
each value of $n_{p}$. Averaged measurement and gate counts are also shown. It
is evident that $h_{\text{max}}$, and hence the number of emitters, scales
linearly with $n_{p}$ as $n_{p}$ becomes large. The same is also true of the
number of measurements. On the other hand, the number of CNOTs and the total
number of gates in the resulting generation circuits scale quadratically with
the number of photons in the target state. These results confirm both the
polynomial scaling of our algorithm, which allows us to easily find generation
protocols for graph states containing hundreds of photons, and the polynomial
scaling of the resulting protocols, which makes them practical for near-term
experiments.

Figure 3: Scaling of emitter number and generation circuit depth. The maximum
value of the height function $h_{\text{{max}}}$, measurement counts, and gate
counts needed to produce random graphs of size $n_{p}$ are all averaged over
1024 graph realizations for each value of $n_{p}$ drawn from an Erdös–Rényi
ensemble with edge probability $p=0.95$. Dashed curves are included to show
the scaling with $n_{p}$ and $n_{p}^{2}$. The error bars stand for the
standard deviation of these realizations.

###  II.3 Photon emission ordering

A powerful feature of our algorithm is that it readily incorporates a desired
photon emission ordering. This is encoded when we arrange the photons and
emitters in a 1D lattice to define the height function. If no specific
ordering is preferred, then ideally we would want to choose the ordering that
minimizes the number of emitters $n_{e}$. However, the task of finding this
optimal ordering is NP-hard, as we show in Methods. Nevertheless, one can
still look for heuristic solutions to the problem. In fact, the expression for
the height function in Eq. (1) makes it clear that this function is suppressed
for orderings in which the stabilizers, when expressed in the echelon gauge,
are supported predominantly on high-index sites on the right side of the 1D
lattice. This tends to occur for “natural” orderings in which neighboring
photons in the graph are emitted around the same time, because in this case
the stabilizers are localized on the 1D lattice. This was illustrated with our
repeater graph state example in the previous section. The extent to which the
stabilizers can be localized in this way depends on the graph of course. For
an $N\times M$ square lattice, it is inevitable that some neighboring vertices
will be separated by $M$ steps in the emission sequence (assuming $M<N$), and
so the number of emitters is of order $M$. On the other hand, for other graph
structures like those of the repeater graph states, far fewer emitters may be
needed, provided a natural photon ordering is used. Note that in this example,
as for many graphs, edges between remote vertices cannot be avoided (see Fig.
2(b)). Despite this, we showed that optimal orderings for which the height
function remains small can still be found. Thus, emitting neighboring vertices
around the same time is sufficient but not always necessary to keep the number
of emitters small.

In summary, we presented an efficient algorithm to construct polynomial-depth
operation sequences that produce arbitrary multi-photon graph states from a
minimal number of quantum emitters. By reducing both the number of photon
sources and the number of quantum operations that need to be performed on
them, our method brings the wide range of quantum information applications
that rely on entangled photon resource states closer to experimental reality.

##  III Methods

###  III.1 Echelon gauge

The echelon gauge was first defined in Ref. [47], where it was called row
reduced echelon form. In this gauge, the stabilizer tableau has a recursive
row-reduced form based on the following three types of matrices:

$\left(\begin{array}[]{l|cr}\mathbbm{1}&*\cdots*&\\\ \hline\cr\mathbbm{1}&&\\\
\vdots&M&\\\
\mathbbm{1}&&\end{array}\right),\quad\left(\begin{array}[]{l|cr}\sigma&*\cdots*&\\\
\hline\cr\mathbbm{1}&&\\\ \vdots&M&\\\
\mathbbm{1}&&\end{array}\right),\quad\left(\begin{array}[]{l|cr}\sigma_{1}&*\cdots*&\\\
\sigma_{2}&*\cdots*&\\\ \hline\cr\mathbbm{1}&&\\\ \vdots&M&\\\
\mathbbm{1}&&\end{array}\right),$ (3)

where $\sigma$, $\sigma_{1}$, and $\sigma_{2}$ are nontrivial Pauli matrices,
and $\sigma_{1}\neq\sigma_{2}$. In this work, we always choose
$\sigma_{2}=\sigma^{z}$, and $\sigma_{1}$ can be either $\sigma^{x}$ or
$\sigma^{y}$. The full tableau cannot have the first form shown above (with
only identities in the first column), because this case does not apply to pure
states. However, the submatrix $M$ can follow any of the above three patterns,
and the structure iterates recursively. The stabilizers can be transformed
into this gauge starting from any other by performing a series of row
reductions, as described in Ref. [47]. In the echelon gauge, the independent
stabilizers acting on $\bar{A}=\\{x+1,\ldots,n\\}$ appear at the bottom right
of the tableau. Therefore, starting from the formula for the entanglement
entropy for subregion $\bar{A}$ of a stabilizer state [49],
$S_{\bar{A}}=n_{\bar{A}}-|\mathcal{G}_{\bar{A}}|$, where $n_{\bar{A}}$ is the
size of $\bar{A}$ and $|\mathcal{G}_{\bar{A}}|$ is the number of independent
stabilizers acting on $\bar{A}$, and using $h(x)=S_{A}=S_{\bar{A}}$, we obtain
Eq. (1).

###  III.2 Time-reversed measurements

Above, we saw that when the total state of the system has the form
$\left|\Phi\right\rangle\otimes\left|0\right\rangle_{i}$, where $i$ is an
emitter site, we can perform a time-reversed measurement to convert this to
the pre-measurement state
CNOT${}_{ij}\left|\Phi\right\rangle\otimes\left|+\right\rangle_{i}$. Here, we
clarify two important questions regarding this process: (i) When and how can
we bring the system into the state
$\left|\Phi\right\rangle\otimes\left|0\right\rangle_{i}$? (ii) How can we see
that a time-reversed measurement on this state increases $h(j)$, as needed for
a subsequent photon absorption process?

Regarding question (i), when $h(j)<h(j-1)$, we can always find a set of
Clifford gates that act purely on the emitters that will transform the state
of the system into $\left|\Phi\right\rangle\otimes\left|0\right\rangle_{i}$.
To see this, first note that $h(j)=h(n_{p})$, as follows from Eq. (1) when
photons $j+1$ through $n_{p}$ are in state $\left|0\right\rangle$. Using that
the height function is bounded from above by $n_{e}$, we then have
$h(n_{p})=h(j)<h(j-1)\leq n_{e}$. On the other hand, from Eq. (1) we have
$h(n_{p})=n_{e}-\\#\\{g_{m}|\verb|l|(g_{m})>n_{p}\\}$. Together, these results
imply $\\#\\{g_{m}|\verb|l|(g_{m})>n_{p}\\}>0$, or in other words, there is at
least one stabilizer that is supported solely on the emitter sites. We can
therefore transform this stabilizer into $\sigma_{i}^{z}$ using at most
$\mathcal{O}(n_{e})$ Clifford gates on the emitters, bringing the state to
$\left|\Phi\right\rangle\otimes\left|0\right\rangle_{i}$. We can then convert
this stabilizer to $\sigma_{i}^{x}$ by applying a Hadamard gate on site $i$.
This prepares the system for the second part of the time-reversed measurement
process, which is the gate CNOTij.

We answer question (ii) by proving the following theorem:

###### Theorem 1

Theorem 1: If $h(j)<h(j-1)$ and the $i$-th qubit $(i>j)$ is stabilized by
$\sigma^{x}_{i}$, then applying $\mathrm{CNOT}_{ij}$ will boost $h(x)\to
h(x)+1$, $\forall x\in\\{j,j+1,\cdots,i-1\\}$.

#### Proof

We are assuming that $h(j)<h(j-1)$, which from Eq. (1) implies
$\\#\\{g_{m}|\verb|l|(g_{m})=j\\}=0$. Now consider how the stabilizers
transform under $\mathrm{CNOT}_{ij}$. If $\verb|l|(g_{m})<j$ before the gate,
then $\verb|l|(g_{m})$ remains invariant, and the contributions of these
stabilizers to $h(x)$ remain the same after the gate. The only potential
changes to $h(x)$ come from stabilizers $g_{m}$ for which $\verb|l|(g_{m})>j$.
These stabilizers necessarily have $\mathbbm{1}$ on the $j$-th site.
Stabilizers among this set that have $\mathbbm{1}$ or $\sigma_{i}^{z}$ on the
$i$-th site will be unchanged by the $\mathrm{CNOT}_{ij}$ gate. However, if
one or more of these stabilizers have $\sigma_{i}^{x}$ or $\sigma_{i}^{y}$
before the gate, then afterward, these stabilizers will contain
$\sigma_{j}^{x}$. Consequently, $h(j)$ increases, while $h(j-1)$ remains the
same. In the echelon gauge, there can only be one stabilizer with
$\sigma_{j}^{x}$ as the left-most nontrivial Pauli. Therefore, $h(j)\to
h(j)+1$ when $\mathrm{CNOT}_{ij}$ is applied. Moreover, if the $i$-th qubit is
stabilized by $\sigma_{i}^{x}$, then this becomes
$\sigma_{j}^{x}\sigma_{i}^{x}$ after the gate, and so the height function for
all sites between $j-1$ and $i$ increases: $h(x)\to h(x)+1$ $\forall
x\in\\{j,j+1,\cdots,i-1\\}$. $\Box$

###  III.3 Scaling analyses

Here, we determine the complexity of both the protocol solver algorithm itself
and the resulting graph state generation circuit. Regarding the algorithm, the
main factor that determines the complexity is the need to restore the
stabilizers to the echelon gauge after each operation is applied. Transforming
a $n$-qubit stabilizer state into the echelon gauge generally requires
$\mathcal{O}(n^{3})$ steps, which is the complexity of Gaussian elimination.
Another important factor is the process of determining which gates need to be
applied in preparation for photon absorption or time-reversed measurement.
Solving for each set of gates takes no more than $\mathcal{O}(n_{e}n)$ steps,
which is the number of entries in the emitter part of the stabilizer tableau.
Thus, the Gaussian eliminations needed to restore echelon gauge dominate the
scaling. In the worst case where $n_{e}\propto n$, our algorithm will then
take $\mathcal{O}(n^{4})$ steps, where the additional factor of $n$ comes from
the fact that the algorithm requires $\mathcal{O}(n_{p})\sim\mathcal{O}(n)$
iterations.

As for the complexity of the output generation circuit, there are at most
$\mathcal{O}(n_{e})$ operations between any two-photon emissions. For example,
$\mathcal{O}(n_{e})$ gates are needed to transform $g_{a}$ into the
appropriate form for photon absorption. Thus, the depth of the circuit acting
on the emitter qubits is at most $\mathcal{O}(n_{p}n_{e})$. In the worst case
where $n_{e}\sim n_{p}$, the scaling is then $\mathcal{O}(n_{p}^{2})$, which
is consistent with Fig. 3. Nevertheless, due to the fact that some long-range
two-qubit gates may arise, and given that these are usually decomposed as
$\mathcal{O}(n_{e})$ short-ranged two-qubit gates in real devices, the overall
circuit depth may become $\mathcal{O}(n_{p}n_{e}^{2})$.

###  III.4 Complexity of finding optimal photon emission orderings

We can show that the task of finding optimal emission orderings is NP-hard by
mapping this to a known graph theory problem. Define $\Gamma_{ij}$ to be the
adjacency matrix of the graph representing the target state
$\left|\Psi\right\rangle$. Ref. [45] showed that we can obtain the height
function from $\Gamma_{ij}$ using the formula
$h(x)=\mathrm{rank}_{2}(\Gamma_{A\bar{A}})$, where $\Gamma_{A\bar{A}}$ is the
sub-matrix of $\Gamma_{ij}$ with row indices $i\in A=\\{1,2,\cdots,x\\}$ and
column indices $j\in\bar{A}$. Note that this expression does not simplify the
computation of $h(x)$; it can take more steps to find the maximum compared to
using Eq. (1) since the former performs Gaussian eliminations for
$\mathcal{O}(n_{p})$ rounds, while the latter only takes one round. However,
the optimized maximum value of this alternative expression for $h(x)$ (i.e.,
$\mathrm{max}_{x}h(x)$) is precisely equal to a graph theoretic property known
as linear rank-width (LRW) [50]. The task of finding an optimal photon
emission ordering is therefore equivalent to finding LRW through the graph
isomorphism, which has long been studied in coding theory in the context of
optimizing block code trellises [51]. Unfortunately, determining whether a
simple connected graph has an LRW bounded from above by a positive integer $k$
has been shown to be NP-complete [52, 54]. Therefore, it is unlikely this
problem can be solved efficiently for large, arbitrary photonic graph states
unless $\text{P}=\text{NP}$. Nevertheless, if the parameter $k$ is set to $1$,
this problem can be answered in polynomial time [53]. If the parameter $k$ is
set to larger values, a recent work [54] showed that this problem can be
reduced to a fixed parameter tractable problem. Specifically, its answer,
along with the sequence solution (if it exists), can be determined in
$\mathcal{O}(f(k)n^{3}_{p})$ steps, where $f(k)$ is an exponentially large
function of $k$. However, the growth of $f(k)$ is so rapid that this result is
not likely to be of practical use for photonic graph state generation.

## Data Availability

The data that support the findings of this study are available from the
authors upon request.

## Code Availability

A custom MATLAB code to reproduce our results is available on GitHub and
archived in Zenodo (https://doi.org/10.5281/zenodo.5652105).

## Acknowledgments

This work was in part supported by the National Science Foundation (grant no.
1741656). E.B. acknowledges support by National Science Foundation grant no.
2137953. S.E.E. acknowledges support by the Army Research Office (MURI grant
no. W911NF2120214).

## Author Contributions

E.B and S.E.E. conceived and supervised the project. B.L. developed the
general approach and performed all the calculations. E.B. and S.E.E.
contributed to technical components of the project. All authors contributed to
the writing of the manuscript.

## Competing Interests

The authors declare that there are no competing interests.

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Supplementary Information

## Supplementary Note 1

This supplementary material contains several additional examples of generation
protocols produced using our algorithm. We begin with a simple example that
illustrates in detail how our algorithm works. We then provide solutions for
more complicated examples of practical importance, including error correcting
codes and repeater graph states of arbitrary size. These more complicated
examples are solved numerically using MATLAB codes that are available on
GitHub [1]. This code expresses the generation sequence in terms of an MPS [2]
for bookkeeping purposes:

$\left|\Psi\right\rangle=U_{p,\mathrm{tot.}}\langle\psi_{f}|\big{[}\prod_{j=1}^{n_{p}}\left(\hat{M}_{j}U_{e,j}\hat{E}_{\eta_{j}}\right)\big{]}W_{0}|\psi_{0}\rangle,$
(S1)

in which the initial and final states of emitter qubits are simply product
states of $\left|0\right\rangle$:
$\left|\psi_{f}\right\rangle=\left|\psi_{0}\right\rangle=\left|0\right\rangle^{\otimes
n_{e}}$. We denote $\eta_{j}$ as the emitter qubit that emits the $j$-th
photon and $\mu_{j}$ as the emitter qubit that is measured after emitting the
$j$-th photon. $\hat{E}_{\eta_{j}}$ is the emission tensor,
$\hat{E}_{\eta_{j}}=\left|0\right\rangle_{j}\left|0\right\rangle_{\eta_{j}}\\!\\!\left\langle
0\right|_{\eta_{j}}+\left|1\right\rangle_{j}\left|1\right\rangle_{\eta_{j}}\\!\\!\left\langle
1\right|_{\eta_{j}}$, that describes emission of photon $j$ from emitter
$\eta_{j}$, which can be represented as $\mathrm{CNOT}_{\eta_{j},j}$.
$U_{e,j}$ is the unitary operation obtained from the $j$-th photon absorption
step, which transforms $g_{a}$ as explained in the main text (Sec.II.A).
$\hat{M}_{j}$ is identity if no measurement happens ($\mu_{j}$ is not
assigned), otherwise,
$\hat{M}_{j}=W_{j}H_{\mu_{j}}X^{s_{j}}_{\mu_{j}}\hat{\pi}_{\mu_{j}}$, with
projection
$\hat{\pi}_{\mu_{j}}\equiv\frac{1}{2}[\mathbbm{1}+(-1)^{s_{j}}Z_{\mu_{j}}]$
and its random outcome $s_{j}\in\\{0,1\\}$. Here, $W_{j}$ is the unitary
operation that is obtained from time-reversed measurement (Sec.IV.B), and
$W_{0}$ is the unitary operation that disentangles all emitters at the final
stage of the time-reversed sequence. Finally,
$U_{p,\mathrm{tot}}=\prod_{j}\left(X^{s_{j}}_{j}U_{p,j}\right)$ is the local
Clifford operation that acts on photons with conditional $X^{s_{j}}_{j}$
flipping. The profile of the solution is stored as
$\\{U_{e,j},U_{p,j},\mu_{j},\eta_{j},W_{j},W_{0}\\}$. We note that such a
solution is usually not unique due to there being multiple choices for how to
choose the emitter gates and emitter sites in each photon absorption and time-
reversed measurement.

As discussed in the main text, the height function plays a central role in
determining the number of emitters and the operation sequence needed to
generate a target photonic graph state. As shown in Eq. (1) of the main text,
when the stabilizers $g_{m}$ are in the echelon gauge, the height function can
be expressed as

$h(x)=n-x-\\#\\{g_{m}|\verb|l|(g_{m})>x\\},$ (S2)

where $\verb|l|(g_{m})$ is the index of the left-most (smallest index) site on
which $g_{m}$ acts nontrivially. In the main text, we showed that the
difference in the height function across adjacent sites determines whether we
perform a photon absorption or a time-reversed measurement at each step of the
algorithm. Therefore, we define

$\delta h(x)\equiv h(x)-h(x-1)=\\#\\{g_{m}|\verb|l|(g_{m})=x\\}-1,$ (S3)

from which it is apparent that this difference only depends on the number of
stabilizers (in the echelon gauge) that have a left-ending on site $x$.

We begin by demonstrating our protocol solver algorithm in the case of the
simple 4-photon graph state displayed in Supplementary Fig. S1(a). The
stabilizers are given by

$\displaystyle g_{1}$
$\displaystyle=\sigma^{x}_{1}\sigma^{z}_{2}\sigma^{z}_{3},\quad
g_{2}=\sigma^{z}_{1}\sigma^{x}_{2}\sigma^{z}_{3}\sigma^{z}_{4},$ (S4)
$\displaystyle g_{3}$
$\displaystyle=\sigma^{z}_{1}\sigma^{z}_{2}\sigma^{x}_{3}\sigma^{z}_{4},\quad
g_{4}=\sigma^{z}_{2}\sigma^{z}_{3}\sigma^{x}_{4}.$

We can switch to the echelon gauge by redefining $g_{3}\to g_{2}g_{3}$. We
then calculate the height function using Supplementary Eq. (S2), finding that
the maximum is $2$. Therefore, at least $n_{e}=2$ emitter qubits are needed,
and so we assemble a $6$-qubit lattice. We can depict the complete set of 6
stabilizers as a tableau, as shown in Supplementary Fig. S1(b).

In Supplementary Fig. S1(c), we first obtain inset (1) by transforming
Supplementary Fig. S1(b) to the echelon gauge. The upper left sub-block of the
tableau is exactly Supplementary Eq. (S4) with $g_{3}\to g_{2}g_{3}$. Next we
describe in detail how the generator set is updated from inset (1) to inset
(17) step by step. The column label $j$ indicates which photon we are
currently focusing on, and the labels (i),…,(iv) indicate the specific step of
our algorithm. For each photon, we do the following steps:

  * •

$j=4$: (i) Obtain inset (1) by transforming to echelon gauge: $g_{3}\to
g_{2}g_{3}$. (ii) Supplementary Eq. (S3) gives $\delta h(4)=-1$. Perform a
time-reversed measurement on emitter site 5 by applying a Hadamard $H_{5}$
followed by $\mathrm{CNOT}_{54}$, which yields inset (2). (iii) Let
$g_{a}=g_{5}=\sigma^{x}_{4}\sigma^{x}_{5}$ in inset (2). One gets inset (3) by
performing Hadamards on sites 4 and 5. Then the $4$-th photon is absorbed into
the emitter on site 5 by applying $\mathrm{CNOT}_{54}$. Replace $g_{4}\to
g_{4}g_{5}$ to eliminate the redundant $\sigma^{z}_{4}$, yielding inset (4).
($\mu_{4}=5$, $\eta_{4}=5$, $U_{p,4}=H_{4}$, $U_{e,4}=H_{5}$, $W_{4}=H_{5}$.)

  * •

$j=3$: (i) Skip this step since inset (4) is already in echelon gauge. (ii)
Supplementary Eq. (S3) gives $\delta h(3)=-1$. Perform a time-reversed
measurement on emitter site 6 by applying $H_{6}$ followed by
$\mathrm{CNOT}_{63}$. Inset (6) is then obtained by redefining
$g_{5}\leftrightarrow g_{6}$. (iii) Let
$g_{a}=g_{5}=\sigma^{x}_{3}\sigma^{x}_{6}$ in inset (6). One gets inset (7) by
applying Hadamards on sites 3 and 6. Then the $3$rd photon is absorbed by
applying $\mathrm{CNOT}_{63}$. Replace $g_{3}\to g_{3}g_{5}$ to eliminate the
redundant $\sigma^{z}_{3}$. Thus, inset (7) becomes (8). ($\mu_{j}=6$,
$\eta_{3}=6$, $U_{p,3}=H_{3}$, $U_{e,3}=H_{6}$, $W_{3}=H_{6}$.)

  * •

$j=2$: (i) Skip this step since inset (8) is already in echelon gauge. (ii)
Skip this step since Supplementary Eq. (S3) gives $\delta h(2)=1$. (iii)
Choose $g_{a}=g_{4}=\sigma^{z}_{2}\sigma^{z}_{5}\sigma^{x}_{6}$ in inset (10).
One gets inset (11) by applying $H_{6}$ followed by $\mathrm{CNOT}_{65}$ on
the emitters, so that $g_{a}\to\sigma^{z}_{2}\sigma^{z}_{5}$. Then the $2$nd
photon is absorbed into emitter 5 by applying $\mathrm{CNOT}_{52}$. Redefine
$g_{k}\to g_{k}g_{4}$ for $k=1,3$ to eliminate the redundant $\sigma^{z}$’s.
Thus, inset (11) becomes (12). ($\eta_{2}=5$, $U_{p,2}=\mathbbm{1}$,
$U_{e,2}=H_{6}\mathrm{CNOT}_{65}$, $\hat{M}_{2}=\mathbbm{1}$.)

  * •

$j=1$: (i) Obtain inset (13) from (12) by transforming to echelon gauge:
$(g_{3},g_{4},g_{5},g_{6})\rightarrow(g_{4},g_{5},g_{6},g_{3})$. (ii) Skip
this step since Supplementary Eq. (S3) gives $\delta h(1)=1$. (iii) Choose
$g_{a}=g_{2}=\sigma^{z}_{1}\sigma^{x}_{5}\sigma^{z}_{6}$ in inset (14). One
gets inset (15) by applying $H_{5}$ and then $\mathrm{CNOT}_{65}$ to transform
$g_{a}\to\sigma^{z}_{1}\sigma^{z}_{5}$. Then the $1$st photon is absorbed into
emitter site 5 by applying $\mathrm{CNOT}_{51}$. Thus, inset (15) becomes
(16). ($\eta_{1}=5$, $U_{p,1}=\mathbbm{1}$, $U_{e,1}=H_{5}\mathrm{CNOT}_{65}$,
$\hat{M}_{1}=\mathbbm{1}$.)

  * •

(iv) Finally, to recover the state $\left|0\right\rangle^{\otimes n}$, one
needs to disentangle the emitter qubits. This can be done with the following
gate sequence: $H_{5}\mathrm{CNOT}_{56}H_{5}$. In the last step, we permute
the $g_{m}$ to obtain inset (17). ($W_{0}=H_{6}\mathrm{CNOT}_{56}H_{5}$.)

Figure S1: Step-by-step illustration of the protocol solver. (a) A target
graph state with 4 photons. (b) The set of generators
$\mathcal{G}_{f}=\\{g_{m}\\}$ is depicted as a tableau in which each row
corresponds to one generator. Different colors correspond to different Pauli
operators. The first 4 columns correspond to photonic qubits, and the last 2
columns correspond to emitters. (c) Step by step demonstration of how to
obtain the time-reversed generation sequence, where
$\mathcal{G}_{0}=\\{\sigma^{z}_{i}\\}$ is finally obtained. Explanations are
in the main text. (d) Local Clifford equivalent graph state representations of
tableaux in (c). Figure S2:  Graph state generation circuits. In this figure,
$p_{j}\;(j=1,2,3,4)$ labels different photonic qubits, and $e_{1}$ and $e_{2}$
are emitter qubits. At the end of each circuit, the photon qubits are in the
target graph state displayed at the top right, while the emitter qubits are in
state $\left|0\right\rangle$ after the measurements. (a) The emission circuit
obtained from the steps in Supplementary Fig. S1. (b) A different generation
circuit that produces the same target graph state as in (a). This circuit is
obtained by swapping qubits $1\leftrightarrow 3$, resulting in a circuit that
requires only one emitter.

Now that the algorithm is complete, we reverse all the operations to obtain
the final generation sequence. This circuit is shown in Supplementary Fig.
S2(a). It is worth noting that, in this example, the emission sequence can be
further optimized by swapping the 1st and 3rd photons in the emission order,
such that the maximum of $h(x)$ is reduced to $1$. Thus, only one emitter
qubit is needed in this case, and the corresponding generation circuit is
displayed in Supplementary Fig. S2(b).

## Supplementary Note 2

In this subsection, we demonstrate how to generate a useful quantum error
correction code, with some continuous logical rotation. In particular, we
present an emission sequence for the Shor code [3] with 9 photonic qubits,
which is able to protect a qubit from single bit-flip and phase-flip errors.
The stabilizer generators of this code are well known:
$g_{j}=\sigma^{z}_{j}\sigma^{z}_{j+1}$ for $j=1,2,4,5,7,8$, and
$g_{3}=\sigma^{x}_{1}\sigma^{x}_{2}\cdots\sigma^{x}_{6}$,
$g_{6}=\sigma^{x}_{4}\sigma^{x}_{5}\cdots\sigma^{x}_{9}$ [4]. We can also
define the logical operators
$X_{L}\equiv\sigma^{z}_{1}\sigma^{z}_{2}\cdots\sigma^{z}_{9}$,
$Z_{L}\equiv\sigma^{x}_{1}\sigma^{x}_{2}\cdots\sigma^{x}_{9}$ and
$Y_{L}=iX_{L}Z_{L}$. Let the last stabilizer be $g_{9}=\pm X_{L}$, which
determines a pair of logical space basis states $\left|\pm\right\rangle_{L}$.
For both choices, Supplementary Eq. (S2) gives $h_{\max}=2$, and the emission
circuit solutions for $\left|\pm\right\rangle_{L}$ are given in Supplementary
Fig. S3, where $\left|\pm\right\rangle_{L}$ are separately given by
$R=\mathbbm{1}$ and $R=X_{e_{1}}$. Therefore, by replacing $R$ by a more
general $x$-rotation,
$e^{i\frac{\varphi}{2}X_{e_{1}}}\equiv\mathbbm{1}\cos\frac{\varphi}{2}+iX_{e_{1}}\sin\frac{\varphi}{2}$,
we can obtain a rotated logical qubit
$\left|\varphi\right\rangle_{L}=e^{i\frac{\varphi}{2}Z_{L}}\left|+\right\rangle_{L}$,
with an arbitrary angle $\varphi$. That is, the circuit in Supplementary Fig.
S3 allows us to transmit a rotated photonic logical qubit protected by the
Shor code, with merely $2$ emitter qubits, $1$ two-qubit gate and $2$
measurements, which is surprisingly simple.

Figure S3: Shor code example. The emission circuit that generates a logical
state of the Shor code, controlled by a local operation $R$ (the yellow
block). The inset displays the height function of
$\left|\pm\right\rangle_{L}$, which has $h_{\max}=2$.

## Supplementary Note 3

In this section, we generalize the repeater graph state example from the main
text and present explicit generation circuits for repeater graphs with
arbitrarily many photons. As shown in Supplementary Fig. S4, for a repeater
graph state with $2m$ photons, the maximum of the height function indicates
that we need $2$ emitters regardless how large $m$ is $(m\geq 4)$. The unitary
operations $A$, $B$ and $C$ displayed in Supplementary Fig. S5(a) depend on
$m$:

$A=X_{e_{2}}^{\lfloor m/2\rfloor+1},\quad B=X_{e_{1}}^{\lfloor
m/2\rfloor},\quad C=P^{m}_{e_{2}}\;,$ (S5)

where $X_{i}=\sigma_{i}^{x}$, and $P=\mathrm{diag}(1,i)$. Compared to the
approach given in previous work [29], which requires $m-1$ two-qubit gates and
$m$ measurements, the new solution in Supplementary Fig. S4(a) uses $2m-3$
two-qubit gates and $m-1$ measurements, reducing the number of measurements
needed to produce the state. We highlight that our method yields different
solutions with flexible settings, so actually the solution from Ref. [5] can
also be obtained from our algorithm.

Figure S4:  Example of a large RGS. (a) and (b) show the emission circuit and
height function for the repeater graph state of $2m$ photons displayed in (c).
The boxed area of the circuit is repeated multiple times to generate photons
$p_{4},p_{5},p_{6},\cdots,p_{2m-6}$.

## Supplementary Note 4

Finally, we consider a modified repeater graph state that includes some
additional redundancy to further boost the likelihood of successful Bell
measurements [5]. Supplementary Fig. S5(a) shows an example of such a repeater
graph state with $6m$ photons $(m>3)$. Note that compared to the state in
Supplementary Fig. S4(c), this state contains twice as many external photon
arms and is missing those internal edges that are not necessary for the
functionality of this state as a repeater. Supplementary Fig. S5(b) shows that
the height function is at most $2$ for any $m>3$, i.e., only two emitter
qubits are needed to generate the state in (a). We list all operations in the
generation circuit in Supplementary Eq. (S1). Denoting the $e_{1}$-th and the
$e_{2}$-th qubits as the emitter qubits, where $e_{1}=6m+1$ and $e_{2}=6m+2$,
the circuit is given by:

$\begin{aligned} W_{0}&=\mathrm{CNOT}_{e_{1}e_{2}}H_{e_{1}}H_{e_{2}}\;\\\
\eta_{j}&=\left\\{\begin{aligned} &e_{2},\quad 6m-5\leq j\leq 6m-3\\\
&e_{1},\quad\text{otherwise}\end{aligned}\right.\\\
U_{p,j}&=\left\\{\begin{aligned} &\mathbbm{1},\quad j=1\text{ (mod 3)}\\\
&H_{j},\quad\text{otherwise}\\\ \end{aligned}\right.\\\
U_{e,j}&=\left\\{\begin{aligned} &\mathrm{CNOT}_{e_{2}e_{1}},\quad j=3\\\
&H_{e_{1}},\quad j=3k\text{ with }2\leq k\leq 2m\\\ &H_{e_{2}},\quad j=6m-3\\\
&\mathbbm{1},\quad\text{otherwise}\end{aligned}\right.\end{aligned},\quad\begin{aligned}
\mu_{j}&=\left\\{\begin{aligned} &e_{1},\quad j=3k\text{ with }2\leq k\leq
2m,\\\ &\qquad\qquad\qquad\text{and}\;k\neq 2m-1\\\ &e_{2},\quad j=6m-3\\\
&\text{not assigned},\quad\text{otherwise}\end{aligned}\right.\\\
W_{j}&=\left\\{\begin{aligned} &\mathrm{CNOT}_{e_{1}e_{2}},\quad j=3k\text{
with }2\leq k\leq 2m-2,\\\ &\qquad\qquad\qquad\text{and}\;k\neq m-1\\\
&H_{e_{2}}\mathrm{CNOT}_{e_{1}e_{2}},\quad j=3m-3\\\ &H_{e_{2}},\quad
j=6m-3\\\ &H_{e_{1}},\quad j=6m\\\
&\mathbbm{1},\quad\text{otherwise}\end{aligned}\right.\end{aligned}.$ (S6)

In the above circuit, there are $2m-1$ measurements, and $2m-1$
$\mathrm{CNOT}$ gates. Plugging these operations into Supplementary Eq. (S1)
gives the full sequence.

Figure S5: Modified RGS example. (a) The graph for RGS, which has $6m$
vertices. The labels represent the emission sequence. (b) The height function
$h(x)$ for the target graph state in (a).

## References

  * [1] Bikun Li.  Photon-emission-circuit-solver: Emission circuit solver for photonics stabilizer state, (2021).  (https://doi.org/10.5281/zenodo.5652105).
  * [2] C. Schön, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf.  Sequential generation of entangled multiqubit states.  Phys. Rev. Lett., 95:110503, (2005).
  * [3] Peter W. Shor.  Scheme for reducing decoherence in quantum computer memory.  Phys. Rev. A, 52:R2493–R2496, (1995).
  * [4] Michael A. Nielsen and Isaac L. Chuang.  Quantum Computation and Quantum Information: 10th Anniversary Edition.  Cambridge University Press, (2010).
  * [5] Donovan Buterakos, Edwin Barnes, and Sophia E. Economou.  Deterministic generation of all-photonic quantum repeaters from solid-state emitters.  Phys. Rev. X, 7:041023, (2017).


# Discovery of CWISE J052306.42-015355.4, an Extreme T Subdwarf Candidate

Hunter Brooks Department of Astronomy and Planetary Science, Northern Arizona
University, Flagstaff, AZ 86011, USA  J. Davy Kirkpatrick IPAC, Mail Code
100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Dan
Caselden Backyard Worlds: Planet 9, USA  Adam C. Schneider United States Naval
Observatory, Flagstaff Station, 10391 West Naval Observatory Rd., Flagstaff,
AZ 86005, USA  Department of Physics and Astronomy, George Mason University,
MS3F3, 4400 University Drive, Fairfax, VA 22030, USA  Aaron M. Meisner NSF’s
National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave.,
Tucson, AZ 85719, USA  Jacqueline K. Faherty Department of Astrophysics,
American Museum of Natural History, Central Park West at 79th Street, New
York, NY 10024, USA  S.L.Casewell School of Physics and Astronomy, University
of Leicester, University Road, Leicester, LE1 7RH, UK  Marc J. Kuchner NASA
Goddard Space Flight Center, Exoplanets and Stellar Astrophysics Laboratory,
Code 667, Greenbelt, MD 20771, USA  The Backyard Worlds: Planet 9
Collaboration

###### Abstract

We present the discovery of CWISE J052306.42$-$015355.4, which was found as a
faint, significant proper motion object (0.52 $\pm$ 0.08 arcsec yr-1) using
machine learning tools on the unWISE re-processing on time series images from
the Wide-field Infrared Survey Explorer. Using the CatWISE2020 W1 and W2
magnitudes along with a $J-$band detection from the VISTA Hemisphere Survey,
the location of CWISE J052306.42$-$015355.4 on the W1$-$W2 vs. $J-$W2 diagram
best matches that of other known, or suspected, extreme T subdwarfs. As there
is currently very little knowledge concerning extreme T subdwarfs we estimate
a rough distance of $\leq$ 68 pc, which results in a tangential velocity of
$\leq$ 167 km s-1, both of which are tentative. A measured parallax is greatly
needed to test these values. We also estimate a metallicity of $-1.5<$ [M/H]
$<-0.5$ using theoretical predictions.

Brown Dwarfs, Extreme T Subdwarf, Low-Mass Star

††software: WiseView (Caselden et al. 2018).

##  1 Introduction

Data from NASA’s Wide-field Infrared Survey Explorer (hereafter, WISE; Wright
et al. 2010) and Near-Earth Object WISE (hereafter, NEOWISE; Mainzer et al.
2011) missions have made searches for nearby, low-temperature objects
considerably easier. The long time baseline between the earliest WISE data in
early 2010 and the most recent (ongoing) NEOWISE data means that objects with
small proper motion can now be detected using only WISE data. The first WISE
searches for nearby objects were reliant mainly on colors (Kirkpatrick et al.
2011), but the addition of motion information has enabled the discovery of
brown dwarfs with unusual colors, a subset of which are believed to be old and
metal-poor (Schneider et al. 2020; Meisner et al. 2020, 2021, and Kirkpatrick
et al. 2021b).

Several tools have been developed to exploit these motions. First, the
CatWISE2020 Catalog (Marocco et al. 2021) uses a long, 8-yr time baseline
between WISE and NEOWISE data to compute proper motion measurements for all
detected objects. Second, the WiseView user interface (Caselden et al. 2018),
which was designed to aid the Backyard Worlds: Planet 9 citizen science group
(hereafter, BYW; Kuchner et al. 2017), allows for visual confirmation of
motion objects. It is particularly useful for those citizens doing their own
targeted color or motion searches on WISE data products. Third, machine
learning algorithms applied to the stack of epochal unWISE coadds (Lang 2014)
from Meisner et al. (2018) can also be trained to identify detections
exhibiting substantial proper motion.

All three methods have been used in the discovery and subsequent
classification of the object CWISE J052306.42$-$015355.4 (hereafter,
J0523$-$0153), a new esdT candidate. In §3 we discuss the discovery of the
object. Below we further describe or estimate other attributes of
J0523$-$0153, including photometry (§4.1), spectral type (§4.2), distance
(§4.3), velocity (§4.4), and metallicity (§4.5). §5 addresses different esdT
candidates and compares them to J0523$-$0153\. Our conclusions are summarized
in §6.

##  2 History of T Subdwarfs and Extreme T Subdwarfs

T subdwarfs (hereafter, sdT) are a low-metallicity counterparts of regular T
dwarfs (hereafter, dT), while extreme T subdwarfs (hereafter, esdT) are those
with even lower metallicity. The first suspected sdT candidate is 2MASS
J09373487+2931409 (Burgasser et al. 2002). However, the first confirmed sdT is
WISE J200520.38+542433.9 (Mace et al. 2013). Its colors and near-infrared
spectrum were unusual, and its common proper motion with a low metallicity
primary, Wolf 1130AB, provided the explanation. This paved the way to the
discovery of other sdTs.

Thereafter, two more sdTs were discovered by Pinfield et al. (2014), WISE
J001354.39+063448.1 and WISE J083337.82+005214.1, both of which were
identified through their unusual colors and high proper motions. Burningham et
al. (2014) discovered the sdT6.5 ULAS J131610.28+075553.0, which was found to
be a potential halo object. These known sdTs all have suppressed $K-$band flux
compared to dTs. Zhang et al. (2019) selected 41 sdT5-9 from the literature by
this feature in their $K-$band spectra, and discussed how sdTs differ from dTs
and L subdwarfs in their characteristics.

Two other spectroscopically unusual T dwarfs, WISE J041451.67$-$585456.7
(hereafter, J0414$-$5854) and WISE J181006.18$-$101000.5 (hereafter,
J1810$-$1010), were then discovered by Schneider et al. (2020). These two
objects have distinctive $J-K$ and $J-$W2 colors from known dTs, sdTs, or
sdLs. They are in the same metallicity range as L extreme subdwarfs (in Table
8 of Zhang et al. 2017), but cooler than esdLs. This suggesting that they are
esdTs. Later, in Meisner et al. (2021) two new esdT candidates were
discovered, WISEA J041451.67$-$585456.7 and WISEA J181006.18$-$101000.5, which
had similar colors to the first esdTs which identified them as candidates.

##  3 Discovery of J0523–0153

J0523$-$0153 was recovered by our team in multiple ways over the course of our
extensive searches for new members of the solar neighborhood. Co-author Dan
Caselden designed a machine learning algorithm, nicknamed SMDET (Caselden et
al. 2020), to aid in the discovery of motion objects across unWISE epochal
coadds (Meisner et al. 2018). SMDET searches for flux groupings whose pixel
positions move across the time sequence of coadds. SMDET is trained on
synthetic motion objects added to the unWISE coadds. In addition, SMDET also
produces a computer-generated mask showing the general movement of the object
across the image, to aid the eye in finding the object in question. Author
Hunter Brooks discovered J0523$-$0153 using SMDET’s list of candidate motion
objects. Figure 1 shows the movement of J0523$-$0153 from $WISE$ in 2010
through $NEOWISE$ in 2019. This object was also submitted as a motion
discovery to BYW by citizen scientist Sam Goodman, using tools provided and
developed for that Zooniverse project. Additionally, J0523$-$0153 was found to
have significant motion in the CatWISE2020 Catalog. We list the proper motion
measurements along with the W1 and W2 magnitudes in Table 1.

Figure 1: The movement of J0523$-$0153 from $WISE$ in 2010 through $NEOWISE$
in 2019. The green circle shows the position of J0523$-$0153 at 2010, and the
red circle shows it at 2019.

##  4 Characteristics of J0523–0153

###  4.1 Photometry

The CatWISE2020 W1 and W2 magnitudes for J0523$-$0153 are 17.27$\pm$0.06 mag
and 15.91$\pm$0.06 mag, respectively, resulting in a color of W1$-$W2 =
1.36$\pm$0.08 mag. This object was not detected in the 2MASS survey (Skrutskie
et al. 2006). We then searched the VISTA Science Archive (Cross et al. 2012)
and WFCAM Science Archive (Hambly et al. 2008). A deep $J-$band detection was
found on the VISTA Hemisphere Survey as shown in Figure 2, with $J_{MKO}$ =
19.75 $\pm$ 0.186 (mag).

Figure 2: VISTA Hemisphere Survey (DR6) $J-$band image is on the left
alongside $WISE$ W2-band image on the right. Both figures show the same patch
of sky at the same scale, which is indicated on the right-hand panel. The
purple circle is J0523$-$0153 and the blue circle is a near-by object that can
be used as a reference.

###  4.2 Spectral Type

With the W1$-$W2 color and $J-$W2 color, we can plot J0523$-$0153 on a color-
color plot like that of Figure 8 of Kirkpatrick et al. (2016). This is
illustrated in Figure 3, showing that the $J-$W2 color is extremely red for
its W1$-$W2 color, compared to the location of normal brown dwarfs. Such
colors imply an extreme T subdwarf as defined by Meisner et al. (2021). As a
result of there being no spectral type vs. color relations for esdTs yet, we
approximate a mid-T type using the relation for normal T dwarfs, as done in
Meisner et al. (2021) and Schneider et al. (2020). We provide more context
with other known and suspected T subdwarfs in §5.1.

On the other hand, when we look at Figure 8(i) from Zhang et al. (2019) at a
fixed spectral type, sdTs are redder in W1$-$W2 compared to normal T dwarfs.
If we assume a similar behavior for esdTs then we would expect J0523$-$0153 to
be an early esdT. This is also supported by WISEA J041451.67$-$585456.7
(hereafter, J0414$-$5854; Schneider et al. 2020), which has similar colors
(Table 2 and Figure 3) to J0523$-$0513, has a tentative spectroscopic
classification of esdT0$\pm$1\. Thus, a slightly preferred estimation for the
spectral type of J0523$-$0512 is $\sim$esdT0.

Figure 3: Color-color diagram adapted from Figure 1 of Meisner et al. (2021)
and Kirkpatrick et al. (2016). The gray box was discussed in Meisner et al.
(2021) to contain all known esdTs, and candidates; therefore, we call it the
esdT box. The location of J0523$-$0153 (red circle) places it well within the
esdT box. L dwarfs (blue downward triangles), T dwarfs (orange upward
triangles), young L (purple diamonds), young T dwarfs (green squares), L
subdwarfs (light blue circles), and T subdwarfs (pink stars) from the 20-pc L,
T, and Y dwarf sample of Kirkpatrick et al. (2021a) are shown for comparison,
as are the extreme T subdwarf candidates, from BYW papers (black crosses), of
Schneider et al. (2020) and Meisner et al. (2021). WISEA J153429.75$-$104303.3
(Kirkpatrick et al. 2021b) has a large $J-$W2 color, falling at a $J-$W2 value
that is too red for this graph. Both CWISEP J050521.29–591311.7 and CWISEP
J090536.35+740009.1 from Meisner et al. (2020) are shown by maroon crosses.
Objects in the esdT box are: 1: J0523$-$0153, 2: J0905+7400, 3: J0414$-$5854,
4: J2217$-$1454, 5: J0505$-$5913, 6: J1810$-$1010, 7: J0738$-$6643 (all of
these objects can be found in Table 2).  Table 1: Compiled Data for
J0523$-$0153

Parameter | Value | Ref.
---|---|---
| Astrometry |
RA | 80.7767549 deg | 2
DEC |  $-$1.8987341 deg | 2
$\mu_{RA}$ |  $0.52{\pm}0.05$ arcsec yr-1 | 2
$\mu_{Dec}$ |  $-0.04{\pm}0.06$ arcsec yr-1 | 2
$\mu_{total}$ |  $0.52{\pm}0.08$ arcsec yr-1 | 1
| Photometry |
$J_{MKO}$ | 19.75$\pm$0.186 mag | 1,3
W1 | 17.27$\pm$0.06 mag | 2
W2 | 15.91$\pm$0.06 mag | 2
$J-$W2 | 3.84$\pm$0.195 mag | 1
W1$-$W2 | 1.36$\pm$0.08 mag | 2
| Derived Quantities |
distance |  $\leq$ 68 pc | 1
$V_{tan}$ |  $\leq$ 167 km s-1 | 1

Reference Codes: (1) This publication, (2) CatWISE2020 Catalog; Marocco et al.
(2021), (3) VISTA Science Archive; Cross et al. (2012)

###  4.3 Distance

As there is no confirmed esdT that has a measured trigonometric parallax, we
have no clear way to estimate the distance. In Figure 7 of Meisner et al.
(2021) the LOWZ models show that W1$-$W2 is not as temperature sensitive as
$J-$W2 at colors similar to esdTs. A better indicator of temperature, or
spectral type, would be obtained from the $J-$W2 color. Comparing our value of
$J-$W2 3.84$\pm$0.195 mag to Figure 20(c) of Kirkpatrick et al. (2021a), we
obtain MW2 $\approx$ 14.0 using Figure 15 of the same paper, where they equate
$Spitzer$ ch2 photometry with $WISE$ W2 photometry, which results in a
distance of $\approx$ 24 pc.

If we use the standard W1$-$W2 vs. spectral type relation of Kirkpatrick et
al. (2021a), which applies only to solar metallicity brown dwarfs, we estimate
a T5 spectral type. From Figure 16(d) of Kirkpatrick et al. (2021a) we find
that a regular T5 has an MW2 $\approx$ 12.5 which results in a distance of
$\approx$ 48 pc. However, it is much more likely that J0523$-$0153 is an esdT0
therefore, using the same process as before we get an MW2 $\approx$ 11.75
which results in a distance of $\sim$68 pc.

Another way to estimate the distance of J0523$-$0153 is looking at Figure
20(c) of Kirkpatrick et al. (2021a). We see that sdTs tend to be brighter in
$M_{ch2}$. If we plot J0523$-$0153 brighter, at about the distance that sdTs
are removed from the regular sequence, we estimate $M_{ch2}$ $\approx$ 13\.
This results in a distance of $\approx$ 38 pc.

We can also estimate other distance limits using the measured proper motion.
As Kirkpatrick et al. (2021b) shows, the highest velocity of any brown dwarf
in the 20 pc sample is $\sim$200 km s-1. If we instead use this value as a
reasonable upper bound, then we find that J0523$-$0153 most likely has a
distance < 84 pc. Even though this is a very loose constraint, it is in
agreement with our previous estimates.

As a result of the total velocity distance estimate being a rough upper bound,
we revert to our previous estimate for esdT0’s of $\leq$ 68 pc for
J0523$-$0153\. A measured trigonometric parallax is needed to better
understand this, and other, esdT candidates.

###  4.4 Proper Motion and Tangential Velocity

J0523$-$0153 is shown in Figure 1 to have significant proper motion, and this
is measured in the CatWISE2020 Catalog as $\mu_{RA}=0.52{\pm}0.05$ arcsec
yr-1, $\mu_{Dec}=-0.04{\pm}0.06$ arcsec yr-1, and $\mu_{tot}=0.52{\pm}0.08$
arcsec yr-1. Using our poorly constrained distance estimate of $\leq$ 68 pc,
we find a tangential velocity of $\leq$ 167 km s-1.

Table 2: Compiled Data for all confirmed or candidate esdTs

|  |  | Photometry |  |  |  |
---|---|---|---|---|---|---|---
Object Name | W1 (mag) | W2 (mag) | J (mag) | W1$-$W1 (mag) |  $J-$W2 (mag) | Ref. |
J0414$-$5854 | 16.71 $\pm$ 0.03 | 15.29 $\pm$ 0.02 | 19.63 $\pm$ 0.11 | 1.42 $\pm$ 0.04 | 4.35 $\pm$ 0.14 | 2 |
J0505$-$5913 | 17.64 $\pm$ 0.06 | 16.13 $\pm$ 0.05 | 20.93 $\pm$ 0.29 | 1.51 $\pm$ 0.08 | 4.80 $\pm$ 0.27 | 3 |
J0523$-$0153 | 17.27 $\pm$ 0.06 | 15.91 $\pm$ 0.06 | 19.75$\pm$0.186 | 1.36 $\pm$ 0.08 | 3.84$\pm$0.195 | 1 |
J0738$-$6643 | 17.22 $\pm$ 0.04 | 15.82 $\pm$ 0.04 | > 19.92 | 1.40 $\pm$ 0.05 | > 4.09 | 4 |
J0905+7400 | 18.25 $\pm$ 0.16 | 16.46 $\pm$ 0.09 | 20.45 $\pm$ 0.14 | 1.79 $\pm$ 0.14 | 3.99 $\pm$ 0.17 | 3 |
J1534$-$1043 | 18.18 $\pm$ 0.19 | 16.15 $\pm$ 0.08 | > 23.80 | 2.04 $\pm$ 0.21 | > 7.66 | 5 |
J1810$-$1010 | 13.65 $\pm$ 0.02 | 12.48 $\pm$ 0.01 | 17.26 $\pm$ 0.02 | 1.17 $\pm$ 0.02 | 4.78 $\pm$ 0.02 | 2 |
J2217$-$1454 | 17.43 $\pm$ 0.08 | 15.78 $\pm$ 0.06 | 20.66 $\pm$ 0.02 | 1.65 $\pm$ 0.10 | 4.88 $\pm$ 0.06 | 4 |
|  |  | Astrometry |  |  |  |
Object Name |  |  $\mu_{RA}$ (arcsec yr-1) |  $\mu_{Dec}$ (arcsec yr-1) |  $\mu_{total}$ (arcsec yr-1) |  | Ref. |
J0414$-$5854 |  | 0.21 $\pm$ 0.03 | 0.65 $\pm$ 0.03 | 0.68 $\pm$ 0.03 |  | 2 |
J0505$-$5913 |  | 0.46 $\pm$ 0.04 | -1.00 $\pm$ 0.04 | 1.10 $\pm$ 0.04 |  | 3 |
J0523$-$0153 |  | 0.52 $\pm$ 0.05 |  $-$0.04 $\pm$ 0.06 | 0.52 $\pm$ 0.08 |  | 1 |
J0738$-$6643 |  | 0.77 $\pm$ 0.03 |  $-$0.43 $\pm$ 0.03 | 0.88 $\pm$ 0.03 |  | 4 |
J0905+7400 |  | 0.47 $\pm$ 0.07 | -1.49 $\pm$ 0.07 | 1.56 $\pm$ 0.07 |  | 3 |
J1534$-$1043 |  |  $-$1.25 $\pm$ 0.01 |  $-$2.38 $\pm$ 0.01 | 2.69 $\pm$ 0.01 |  | 5 |
J1810$-$1010 |  |  $-$1.13 $\pm$ 0.01 |  $-$0.21 $\pm$ 0.01 | 1.16 $\pm$ 0.01 |  | 2 |
J2217$-$1454 |  | 1.64 $\pm$ 0.07 |  $-$0.92 $\pm$ 0.06 | 1.88 $\pm$ 0.07 |  | 4 |
|  |  | Derived Quantities |  |  |  |
Object Name | Distance (pc) |  $V_{tan}$ (km s-1) | Metallicity (dex) |  $T_{\rm eff}$ (K) | Mass ($M_{\astrosun}$) | Ref. |
J0414$-$5854 | 52$-$94 | 170$-$307 |  $\approx$ $-$1 [Fe/H] | 1300 $\pm$ 100 | 0.075$-$0.080 | 2 |
J0505$-$5913 | 34.9$-$46.3 | 181$-$243 | … | 846 $\pm$ 88 | … | 3 |
J0523$-$0153 |  $\leq$ 68 |  $\leq$ 167 |  $-$1.5 to $-$0.5 [M/H] | … | … | 1 |
J0738$-$6643 | 35.2$-$54 | 145$-$225 | … | … | … | 4 |
J0905+7400 | 41.6$-$55.1 | 305$-$410 | … | 930 $\pm$ 87 | … | 3 |
J1534$-$1043 | 15.1$-$17.7 | 191.5$-$223.4 |  $\approx$ $-$1.5 [M/H] |  $-$500 | … | 5 |
J1810$-$1010 | 14$-$67 | 77$-$360 |  $\leq$ $-$1 [Fe/H] | 1300 $\pm$ 100 | 0.075$-$0.080 | 2 |
J2217$-$1454 | 31.6$-$48.9 | 282$-$436 | … | … | … | 4 |

Reference Codes: (1) This publication, (2) Schneider et al. (2020), (3)
Meisner et al. (2020), (4) Meisner et al. (2021), (5) Kirkpatrick et al.
(2021b). Note that only one esdT has a measured parallax, J1534$-$1043\. For
the others, the distances are estimates. This results in the $V_{tan}$ also
being a rough estimate for every object other than J1534$-$1043.

###  4.5 Metallicity

Based on the location of J0523$-$0153 in the color-color plot of Figure 3, we
can attempt to deduce its metallicity. If we plot it on Figure 7 of Meisner et
al. (2021), which is similar to Figure 3 here, we find that it falls along the
LOWZ model with metallicities of [M/H] = $-$2\. However, WISEA
J153429.75$-$104303.3 (hereafter, J1534$-$1043; Meisner et al. 2020 and
Kirkpatrick et al. 2021b) falls in a far more unusual spot on the color-color
diagram and is believed to have a metallicity [M/H] $\approx$ $-$1.5
(Kirkpatrick et al. 2021b), more metal rich than the LOWZ model predicts.

For additional guidance, we can examine the location of WISE
J200520.38+542433.9 (hereafter, J2005+5424; Mace et al. 2013) on Figure 3,
since this object is a co-moving companion to a star with a known subsolar
metallicity of [Fe/H] = $-$0.64 $\pm$ 0.17 dex. If we use the LOWZ model, it
predicts a subsolar metallicity of [M/H] = $-$1.0. However, the actual
measured metallicity from the primary of the system is higher than the
prediction adopting the LOWZ model. Thus, the models are making predictions
that are systematically too metal poor. The location of J2005+5424 on Figure 3
is closer to the normal T dwarf sequence than that of J0523$-$0153, so this
means that this object would have a higher metallicity. But J1534$-$1043 is
further removed from the normal sequence than both of these objects. Since we
believe that J0523$-$0153 is less metal poor than J1534$-$1043 and more than
J2005+5424, then it most likely has a metallicity in the range of $-$1.5 <
[M/H] < $-$0.5.

##  5 Discussion

There are eight other candidates or confirmed esdTs so far published (Table
2), so it is useful to compare J0523$-$0153 to these esdTs. Most esdTs have
been discovered and discussed in papers by the BYW team (Schneider et al.
2020, Meisner et al. 2021, and Kirkpatrick et al. 2021b). There are two esdT
candidates found and discussed in Meisner et al. (2020) that are not from the
BYW community. Since this family of brown dwarfs is relatively new and
exceedingly faint, the follow-up data, which are hard to come by, are sparse.

###  5.1 Known esdTs and other esdT Candidates

The first extreme T subdwarf candidates were discussed in Schneider et al.
(2020). Both J0414$-$5854 and J1810$-$1010 have spectroscopic signatures
indicative of sub-solar metallicity, which identified them as T subdwarfs. The
authors came to this conclusion using their Figure 2, which compares the
spectra of these objects to the spectra of known dTs and sdTs. Accordingly,
their Figure 4, which compares the spectra to model predictions in both
temperature and metallicity.

J0414$-$5854 and J1810$-$1010 were both shown to have similar unusual colors
compared to known brown dwarfs, as shown in Table 2. As shown in Figure 3 of
Schneider et al. (2020) middle to late esdTs are bluer in W1$-$W2 while redder
in $J-$W2, $J-$K, and $J-$H when compared to regular dTs.

Both CWISEP J050521.29$-$591311.7 and CWISEP J090536.35+740009.1 (hereafter,
J0505$-$5913 and J0905+7400) were discussed in Meisner et al. (2020) as mid-T
dwarf candidates, even possibly T subdwarfs. We would now list both objects as
mid-esdT candidates based on their colors. These magnitudes, found in Table 2,
are very similar to other esdTs that are found in the esdT box in Meisner et
al. (2021), shown in Figure 3. They estimated that J0505$-$5913 is a T6.5 and
J0905+7400 is a T6, assuming both are normal T-type dwarfs. With ch1$-$ch2
data from $Spitzer$, they find distance estimates for J0505$-$5913 and
J0905+7400 of 34.9$-$46.3 pc and 41.6$-$55.1 pc, respectively.

Meisner et al. (2021) discussed two new esdT candidates, CWISE
J073844.52$-$664334.6 (hereafter, J0738$-$6643) and CWISE
J221706.28$-$145437.6 (hereafter, J2217$-$1454). J0738$-$6643 was found to
have characteristics of a T5$\pm$1.2. J2217$-$1454 has a spectral type
estimate of T5.5$\pm$1.2, these estimates came from the W1$-$W2 vs. brown
dwarf spectral type, which has been shown to be problematic. This is similar
to J0738$-$6643 and comparable to J0523$-$0153, a reminder that these values
are poor estimates given that these objects are likely not normal-T type
dwarfs.

The most exotic of the extreme subdwarf candidates is WISEA
J153429.75$-$104303.3 (hereafter, J1534$-$1043; Meisner et al. 2020 and
Kirkpatrick et al. 2021b). J1534$-$1043 was so faint in near-infrared that
those authors needed to use the $Hubble$ $Space$ $Telescope$ to obtain a
detection of it. Even with this hurdle, those authors show that J1534$-$1043
is the reddest of all esdTs in $J-$W2, including candidates, and is possibly
an esdY. Even though it has a red $J-$W2, it still falls within the W1$-$W2
color constraint of the esdT box of Figure 3. In §4.5 we discussed how this
object compares to J0523$-$0153 in metallicity. There is a temperature
estimate of $T_{\rm eff}$ < 500 K using the theoretical LOWZ model, but these
predictions have not been tested yet.

###  5.2 Implications on J0523–0153

When looking at J0523$-$0153 we can see that it has many neighbors in Figure
3. Peering at confirmed/candidate esdTs that have similar colors to
J0523$-$0153, we can conclude that they likely have similar characteristics.
For example, as both J0738$-$6653 and J0523$-$0153 have similar colors, we can
assume they have comparable spectral types. Likewise, the metallicities that
we estimated are comparable to similarly colored esdTs that have spectra, with
which a more accurate metallicity can be estimated.

##  6 Conclusion

We have announced the discovery of the proper motion object J0523$-$0153 found
in WISE data. We estimate that the object is an early esdT dwarf candidate.
With our current models, we provide a crude estimate for both a distance and
tangential velocity. Applying different methods for distance estimates, we
find $\leq$ 68 pc. Additionally, using proper motion values from the
CatWISE2020 Catalog and the distance estimate stated before, we calculate a
tangential velocity of $\leq$ 167 km s-1. Using its placement on color-color
diagrams, we estimate the metallicity to fall in the range $-$1.5 to $-$0.5
dex. Follow-up photometry, spectroscopy, and a trigonometric parallax are
needed to help further reveal the properties of J0523$-$0153.

##  7 Acknowledgements

Backyard Worlds research was supported by NASA grant 2017-ADAP17-0067 and by
the NSF under grants AST- 2007068, AST-2009177, and AST-2009136. SLC is
supported by an STFC Ernest Rutherford Research Fellowship. We want to thank
the Student Astrophysics Society111https://www.studentastrophysicssociety.com
for providing the resources that enabled the pairing of high school and
undergraduate students with practicing astronomers and advanced citizen
scientists. This work makes use of data products from WISE/NEOWISE, which is a
joint project of UCLA and JPL/Caltech, funding by NASA. The CatWISE effort is
led by the Jet Propulsion Laboratory, California Institute of Technology, with
funding from NASA’s Astrophysics Data Analysis Program. We would like to thank
our anonymous referee for the suggestions that improved the paper.

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