#  Bound on the closed quantum dynamics under stochastic noise

Kohei Kobayashi  Quantum Computing Center, Keio University, Hiyoshi 3-14-1,
Kohoku-Ku, Yokohama, Kanagawa 223-8522, Japan

###### Abstract

Quantum information technologies require careful control for preparing a
desired target state used for an information resource. The one of the
obstacles is the stochastic noise on the control Hamiltonian, under which the
realistic control performance is severely limited. Therefore, the reachability
analysis, which in our scenario quantifying the distance between the obtained
state under the noise and the target state, is of great importance. This paper
gives a lower bound of the fidelity for a closed quantum system under the
stochastic noise. Note that this bound is computable without considering the
stochastic process and needing the full time-dependent dynamics of the states.
We demonstrate the actual tightness of this bound via numerical simulation.

##  I  Introduction

In recent years, there has been considerable interests in quantum information
technologies including quantum computing. In order to realize these
technologies, it is necessary to prepare a desired quantum state for an
information resource. Thus, an accurate technique for manipulating qubits and
preparing the desired target state plays an important role [1]. In particular,
it is of great importance to design and implement an ideal dynamics in the
closed system. For example, quantum annealing or quantum adiabatic computation
are modeled by closed quantum dynamics such as the Schr${\rm\ddot{o}}$dinger
equation. Therefore, the control method for designing the control setup
configured by the control Hamiltonian and the control input so as to steer the
state into the target, e.g., open-loop control [2, 3, 4, 5], offers powerful
means.

However, in a realistic situation, there are two types of control errors that
effects on the time evolution of the closed system; deterministic noise and
stochastic noise. Deterministic noise acts as a bias or detuning [6, 7, 8],
which will not be considered in this paper. Our interest is the stochastic
noise, which occurs on the control Hamiltonian at each time. It is one of the
crucial obstacles to overcome toward achieving quantum advantage [9, 10, 11,
12]. Actually, it is difficult to completely eliminate the stochastic noise on
control fields in an experimental setup.

In fact, it is important problem to clarify and quantify the actual
performance of quantum control under some imperfections; for instance,
decoherence, which is induced by the coupling with the environments, is a
typical obstacle to be beaten. There are various studies investigating the
quantum control for open quantum systems under decohrence [13, 14, 15, 16].
This issue on the stochastic noise has been investigated in recent years,
e.g., the adiabatic protocol by the noisy Hamiltonian in quantum annealing
[10], sufficient condition for obtaining a target state under the local noise
[11], and estimation for the performance of the variational quantum
eigensolver (VQE) under the noise [12]. However, there has been no rigorous
approaches for quantitatively evaluating the effect of the stochastic noise.

Motivated by this background, in this paper we theoretically investigate the
influence of the stochastic nosie on the quantum control for closed systems.
More specifically, we derive a lower bound on the fidelity between two quantum
systems obeying the ideal dynamics and noisy one. Note that this bound is
computable without considering the stochastic process and needing the full
time-dependent dynamics of the states. Also it gives a quantitative limit on
the Hamiltonian control for preparing the quantum state under stochastic noise
as a function of the noise strength, as demonstrated via numerical simulation
in two simple settings.

##  II Main result

###  II.1 Setup

We begin with the explanation of the control setup. Let $|{\psi(t)}\rangle$ be
the controlled state in the closed system in the absence of the stochastic
noise. Thus, it obeys the ideal dynamics described by the
Schr${\rm\ddot{o}}$dinger equation:

$\displaystyle\frac{d|{\psi(t)}\rangle}{dt}=-iH(t)|{\psi(t)}\rangle,$ (1)

where $|{\psi(t)}\rangle\in\mathcal{H}$ is the $d$-dimensional quantum state
in the Hilbert space. $H(t)$ is a self-adjoint operator representing the time-
dependent Hamiltonian driving the quantum system (we set $\hbar=1$ in this
paper). We assume that $|{\psi(t)}\rangle$ reaches the target state at final
time $t=T$.

On the other hand, let $|{\phi(t)}\rangle\in\mathcal{H}$ be the another state
we focus on. $|{\phi(t)}\rangle$ is driven by the stochastic noise $B(t)$, in
addition to $H(t)$. $B(t)$ is a self-adjoint operator with the time-dependent
noise strength. In this case, the time evolution of $|{\phi(t)}\rangle$ is
described by the (Itō-type) stochastic Schr${\rm\ddot{o}}$dinger equation:

$\displaystyle d|{\phi(t)}\rangle$
$\displaystyle=-\left(iH(t)+\frac{1}{2}B^{2}(t)dt-
iB(t)dW(t)\right)|{\phi(t)}\rangle.$ (2)

$W(t)$ is the classical stochastic process (i.e., the Winner process)
satisfying the Itō rule $dW^{2}(t)=dt$ and $\mathbb{E}[W(t)]=0$.

Moreover, for simplicity, we assume that $B(t)$ satisfies the following
condition:

$\displaystyle B^{2}(t)=\gamma^{2}(t)I,$ (3)

where $I$ denotes the identity operator. This condition leads to the natural
assumption that the noise acts on the system locally [11]. For instance, when
$B(t)$ is given by the product of the Pauli matrices such as the angular
momentum operators, this condition is satisfied. Then, Eq. (2) becomes

$\displaystyle
d|{\phi(t)}\rangle=-\left(iH(t)dt+\frac{\gamma^{2}(t)}{2}dt+iB(t)dW(t)\right)|{\phi(t)}\rangle.$
(4)

Note that Eq. (4) is the Itō form not the Stratinovich form. Thus, we can take
the expectation of the both sides of Eq. (4) with respect to $W(t)$ as
follows:

$\displaystyle\frac{d\mathbb{E}[|{\phi(t)}\rangle]}{dt}=-\left(iH(t)+\frac{\gamma^{2}(t)}{2}\right)\mathbb{E}[|{\phi(t)}\rangle].$
(5)

In this paper, we assume that the initial states of $|{\psi(t)}\rangle$ and
$|{\phi(t)}\rangle$ are identical $|{\psi_{0}}\rangle=|{\phi_{0}}\rangle$.

Then, we obtain the following relation:

$\displaystyle\mathbb{E}[|{\phi(T)}\rangle]=e^{-\frac{1}{2}\int^{T}_{0}\gamma^{2}(t)dt}|{\psi(T)}\rangle.$
(6)

This result means that the noise $\gamma(t)$ degrades the control performance
exponentially. Hence, even if $\gamma(t)$ is small, the actual control
performance is severely limited as the control time becomes longer. In the
next subsection, we derive the control limit as an explicit form depending on
$\gamma(t)$.

###  II.2 Derivation of the lower bound

We here introduce the fidelity for $|{\psi(t)}\rangle$ and
$|{\phi(t)}\rangle$:

$\displaystyle F(t):=|\langle\psi(t)|\phi(t)\rangle|^{2}\ \ \ (0\leq F(t)\leq
1),$ (7)

where $\||{a}\rangle\|:=\sqrt{\langle a|a\rangle}$ is the Euclidean norm.
$F(t)$ is often employed as a cost function in various quantum science
scenario, because it can be easily computed both analytically and numerically
[1]. If and only if $|{\psi(t)}\rangle=e^{i\alpha}|{\phi(t)}\rangle$
($\alpha\in\mathbb{R}$ is a global phase factor), $F(t)=1$ is achieved.
$F(0)=1$ and $F(t)$ decreases from $1$ under the noise. Therefore, the
objective control limit is a fundamental lower bound of $F(t)$.

Let us consider the time evolution of the overlap
$\langle\psi(t)|\phi(t)\rangle$:

$\displaystyle
d\left(\langle\psi(t)|\phi(t)\rangle\right)=\langle\psi(t)|iH(t)|\phi(t)\rangle
dt$
$\displaystyle-\langle\psi(t)|\left(iH(t)dt+\frac{\gamma^{2}(t)}{2}dt+iB(t)dW(t)\right)|\phi(t)\rangle$
$\displaystyle=-\frac{\gamma^{2}(t)}{2}\langle\psi(t)|\phi(t)\rangle
dt-\langle\psi(t)|iB(t)|\phi(t)\rangle dW(t).$ (8)

Taking the real part and the ensemble average of this equation is thus

$\displaystyle\frac{d\mathbb{E}[\Re\left\\{\langle\psi(t)|\phi(t))\rangle\right\\}]}{dt}=-\frac{\gamma^{2}(t)}{2}\mathbb{E}[\Re\left\\{\langle\psi(t)|\phi(t)\rangle\right\\}].$
(9)

Solving this equation yields

$\displaystyle\mathbb{E}[\Re\left\\{\langle\psi(t)|\phi(t)\rangle\right\\}]=e^{-\frac{1}{2}\int^{t}_{0}\gamma^{2}(t^{\prime})dt^{\prime}}.$
(10)

Further, by using $\mathbb{E}[x^{2}]\geq\mathbb{E}[x]^{2}$,

$\displaystyle\mathbb{E}[\Re\left\\{\langle\psi(t)|\phi(t)\rangle\right\\}]$
$\displaystyle\leq\mathbb{E}[|\langle\psi(t)|\phi(t)\rangle|]$
$\displaystyle\leq\sqrt{\mathbb{E}[|\langle\psi(t)|\phi(t)\rangle|^{2}]}$
$\displaystyle=\sqrt{\mathbb{E}[F(t)]}.$ (11)

From (10) and (II.2), we end up with the following lower bound:

$\displaystyle\mathbb{E}[F(t)]\geq
F_{*}:=e^{-\int^{t}_{0}\gamma^{2}(t^{\prime})dt^{\prime}},$ (12)

for $t\in[0,\infty)$. This inequality is the main result of this paper. The
equality of (12) holds when $t=0$ or $\gamma(t)=0$.

$F_{*}$ imposes a fundamental lower bound on the fidelity between the two
states governed by the ideal dynamics and noisy one. We give some notable
features of $F_{*}$ below:

(i) $F_{*}$ can be calculated without needing the full time-dependent
dynamics, which does not necessitate to solve any equations. This allows for
applying this result to wider classes of quantum control case.

(ii) If the system is subjected to the multi-stochastic noise Hamiltonians
$\sum_{j}B(t)_{j}$, $F_{*}$ are straightforwardly generalized to

$\displaystyle
F_{*}:=e^{-\sum_{j}\int^{t}_{0}\gamma^{2}_{j}(t^{\prime})dt^{\prime}}.$ (13)

(iii) If the integral $\int^{t}_{0}\gamma^{2}(t^{\prime})dt^{\prime}$ cannot
be solved explicitly, by assuming $\gamma_{\rm max}={\rm
max}\\{|\gamma(t)|\\}$, the lower bound becomes the explicit form
$F_{*}=e^{-\gamma^{2}_{\rm max}t}$

(vi) In quantum computational problem, the computing time $T$ is given by a
polynomial of the system size $T=\mathcal{O}(N^{a})$; for instance, $a=1/2$ is
optimal in the quantum annealing. In this case, if the structure of
$\gamma^{2}_{\rm max}$ is given by $\gamma^{2}_{\rm
max}=\mathcal{O}(N^{-1/2})$, the lower bound $F_{*}$ converges a constant
value as the system size increases. This intuition consists with the result in
[11].

(v) Bounding the righthand side of (12) by using $\gamma_{\rm max}={\rm
max}\\{|\gamma(t)|\\}$, we obtain the lower bound on the control time $T$:

$\displaystyle T\geq-\frac{1}{\gamma^{2}_{\rm
max}}\ln\left[\mathbb{E}[F(t)]\right].$ (14)

This inequality gives a fundamental bound on the evolution time of
$|{\psi(t)}\rangle$ and $|{\phi(t)}\rangle$ so as to satisfy a certain value
$\mathbb{E}[F(T)]$. By using it, we have a rough estimate on the control time
under the noise. If $|{\psi(t)}\rangle$ is fixed in the initial state
$|{\psi_{0}}\rangle$, the lower bound on the evolution time of
$|{\phi(t)}\rangle$ is ideantical to the quantum speed limit (QSL). We give
the QSL in our setup in Appendix A.

##  III Actual performance in the simple setting

We investigate the actual performance of $F_{*}$ in two simple settings.

###  III.1 Qubit

We first study the simple example of a qubit system such as a two level atom,
consisting of the excited state $|{0}\rangle=[1,0]^{\top}$ and the ground
state $|{1}\rangle=[0,1]^{\top}$. We consider the bit-flip control where the
state is driven from $|{1}\rangle=[0,1]^{\top}$ to $|{0}\rangle=[1,0]^{\top}$
by the control Hamiltonian

$\displaystyle H=uS_{y},$ (15)

where $S_{x}=|{0}\rangle\langle{1}|+|{1}\rangle\langle{0}|$,
$S_{y}=i(|{1}\rangle\langle{0}|-|{0}\rangle\langle{1}|)$ and
$S_{z}=|{0}\rangle\langle{0}|-|{1}\rangle\langle{1}|$ are the Pauli matrices.
The control input assumed to be time-independent $u(t)=u$. Solving the
equation $d|{\psi(t)}\rangle/dt=-iuS_{y}|{\psi(t)}\rangle$ yields the exact
control time $T=\pi/(2u)$. Further we choose the noise Hamiltonian as
$B=\gamma S_{x}$ with the time-independent noise strength $\gamma$. $B$
fluctuates the state vector around the $x$ axis of the Bloch sphere with
frequency $\gamma$. In this case, $F_{*}$ is calculated as
$F_{*}=e^{-\gamma^{2}\pi/u}$.

Figure 1(a) shows the difference between the lower bound $F_{*}$ and the
simulated values of $\mathbb{E}[F(T)]$ with several $\gamma$. In particular,
$F_{*}$ gives a tighter bound to $\mathbb{E}[F(T)]$ when $\gamma$ is small,
but the distance between $\mathbb{E}[F(T)]$ and $F_{*}$ becomes bigger as
$\gamma$ increases.

Next, we consider the two-noise case where the system is affected by
$B_{1}=\gamma S_{x}$ and $B_{2}=\gamma S_{z}$. As depicted in Fig.1 (b),
$F_{*}=e^{-2\gamma^{2}\pi/u}$ works as a tighter bound for all $\gamma$
compared with Fig.1 (a), which is because $\mathbb{E}[F(T)]$ is far away from
1 under the two noises. From this result, we can say that $F_{*}$ is
relatively tight for multi-noise or small noise.

###  III.2 Two qubits

Here we study a two-qubit system and focus on the SWAP operation permuting the
states of two qubits, which plays an important role in network designs for
quantum computaion [1]. This operation is described by the SWAP gate

$\displaystyle U_{\rm SWAP}=\begin{bmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{bmatrix},$ (16)

which is implemented by the time-independent Hamiltonian

$\displaystyle H=-\frac{u}{2}\left(S_{x}\otimes S_{x}+S_{y}\otimes
S_{y}+S_{z}\otimes S_{z}\right).$ (17)

The ideal state transformation considered here is $|{\psi_{0}}\rangle\to
U_{\rm SWAP}|{\psi_{0}}\rangle=|{\psi(T)}\rangle$ where

$\displaystyle|{\psi_{0}}\rangle=|{+}\rangle\otimes|{0}\rangle,\ \
|{\psi(T)}\rangle=|{0}\rangle\otimes|{+}\rangle,$

where $|{+}\rangle:=(|{0}\rangle+|{1}\rangle)/\sqrt{2}$. In this setting, the
control time is calculated as $T=\pi/(2u)$.

First, let us consider the local noise $B^{L}$ and the global noise $B^{G}$:

$\displaystyle B^{L}=\gamma S_{x}\otimes I,\ \ B^{G}=\gamma S_{x}\otimes
S_{x}.$ (18)

$B^{L}$ acts one qubit only, while $B^{G}=\gamma S_{x}\otimes S_{x}$ acts two
qubits spontaneously. For both cases, the lower bound has same expression
$F_{*}=e^{-\gamma^{2}\pi/u}$. Let us denote $\mathbb{E}[F(T)]$ for $B^{L}$ and
$B^{G}$ as $\mathbb{E}[F(T)]^{L}$ and $\mathbb{E}[F(T)]^{G}$, respectively.
Figure 2 (a) plots $\mathbb{E}[F(T)]^{L}$ and $\mathbb{E}[F(T)]^{G}$ in unit
of $u=1$. It is found that when $\gamma$ is small, the global decay acts more
strongly than the local decay, but from the point $\gamma=1.1$, this behavior
reverses. Note that, however, the lower bound $F_{*}$ cannot reflect this
result in its expression. This is a drawback of our bound.

Figure 2 (b) shows the case where the system experiences the two local noises
$B_{1}=\gamma S_{x}\otimes I$ and $B_{2}=\gamma I\otimes S_{x}$. Likewise the
qubit, $F_{*}$ gives a tighter bound than the plots of $\mathbb{E}[F(T)]^{L}$
in Fig.2 (a).

Figure 1:  Plots of $F_{*}$ (solid red line) and the simulated values of
$\mathbb{E}[F(T)]$ (blue dot) for (a) single noise and (b) two noises, as a
function of $\gamma$ when $u=1$ and $T=\pi/2$.  Figure 2:  Plots of $F_{*}$
(solid red line) and the simulated values of (a) $\mathbb{E}[F(T)]^{L}$ (blue
dot) and $\mathbb{E}[F(T)]^{G}$ (green triangle), and (b) $\mathbb{E}[F(T)]$
for the two noises, as a function of $\gamma$ when $u=1$ and $T=\pi/2$.

##  IV Conclusion

In this paper, we have presented the lower bound $F_{*}$ of the distance
between the controlled quantum state under the stochastic noise and the
another state obeying its ideal dynamics. $F_{*}$ can be straightforwardly
calculated and relatively tight when the noise is small or multiple. Also,
this $F_{*}$ gives an explicit and absolute limit on the set that the quantum
state reaches, hence it is stronger compared than the previous result derived
in [11].
An important remaining work is the generalization of $F_{*}$ to any type of
noise. We have imposed the constraint $B^{2}(t)=\gamma^{2}(t)I$ in order to
derive $F_{*}$. However, due to it, the generality of our bound is limited,
e.g., the collective noise $\gamma(t)(S_{x}\otimes I+I\otimes S_{x})$ cannot
be dealt with, because it does not satisfy this constraint. Also, when $F_{*}$
has same expressions, we cannot distinct the type of noise. Therefore, it is
necessary to exploit another approach for deriving the bound without imposing
any assumption on the noise Hamiltonian.
Nevertheless, we hope that the result shown in this paper will provide a
meaningful insight and scalability in quantum control.
This work was supported by MEXT Quantum Leap Flagship Program Grant Number
JPMXS0118067285 and JPMXS0120319794. Also it is a pleasure to thank Manaka
Okuyama for insightful discussion.

##  Appendix A Quatum speed limit for the stochastic noise

The Quatum speed limit (QSL) is defined as a lower bound of the time for the
quantum state to evolve from the initial state to the final state [17, 18, 19,
20, 21, 22]. Here we derive the QSL for the dynamics under the stochastic
noise given by (4). First, let us focus on the Bures angle

$\displaystyle\mathcal{L}(t)$ $\displaystyle={\rm
arccos}\left\\{|\langle\psi_{0}|\phi(t)\rangle|\right\\}$ $\displaystyle={\rm
arccos}\left\\{\sqrt{{\rm Tr}[\rho_{0}\rho(t)]}\right\\},$ (19)

where we have written $\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$ and
$\rho(t)=|\phi(t)\rangle\langle\phi(t)|$. $\mathcal{L}(t)$ is known as the
shortest path between two states in the Hilbert space [22]. Second, by using
(2), the dynamics of $\rho(t)$ is introduced as follows:

$\displaystyle d\rho(t)=d|\phi(t)\rangle\langle\phi(t)|+|\phi(t)\rangle
d\langle\phi(t)|+d|\phi(t)\rangle d\langle\phi(t)|$
$\displaystyle=\left\\{\left(-iH(t)-\frac{1}{2}B^{2}(t)\right)dt-
iB(t)dW(t)\right\\}|{\phi(t)}\rangle\langle{\phi(t)}|+|{\phi(t)}\rangle\langle{\phi(t)}|\left\\{\left(iH(t)-\frac{1}{2}B^{2}(t)\right)dt+iB(t)dW(t)\right\\}$
$\displaystyle\ \ \ -iB(t)dW(t)|{\phi(t)}\rangle\langle{\phi(t)}|iB(t)dW(t)$
$\displaystyle=-i[H(t),\rho(t)]dt+\mathcal{D}[B(t)]\rho(t)dt-i[B(t),\rho(t)]dW(t),$
(20)

where $\mathcal{D}[B]\rho=B\rho B-B^{2}\rho/2-\rho B^{2}/2$. Therefore, the
infinitesimal change of $\mathcal{L}(t)$ is given by

$\displaystyle
d\mathcal{L}(t)=\frac{-1}{\sqrt{1-F(t)}}\frac{1}{2\sqrt{F(t)}}\left({\rm
Tr}[\rho_{0}d\rho(t)]\right)$
$\displaystyle=\frac{-1}{2\sin\mathcal{L}(t)\cos\mathcal{L}(t)}\left({\rm
Tr}\left[\rho_{0}(-i[H(t),\rho(t)]dt+\mathcal{D}[B(t)]\rho(t)dt-i[B(t),\rho(t)]dW(t))\right]\right)$
$\displaystyle\leq\frac{1}{\sin 2\mathcal{L}(t)}\left(u(t){\rm
Tr}\left\\{i[H(t),\rho_{0}]\rho(t)\right\\}dt+{\rm
Tr}\left\\{(\mathcal{D}[B(t)]\rho_{0})\rho(t)\right\\}dt+i{\rm
Tr}\left\\{[\rho_{0},B(t)]\rho(t)\right\\}dW(t)\right)$
$\displaystyle\leq\frac{1}{\sin
2\mathcal{L}(t)}\left(\|i[\rho_{0},H(t)]\|_{\rm F}\|\rho(t)\|_{\rm
F}dt+\|\mathcal{D}[B(t)]\rho_{0}\|_{\rm F}\|\rho(t)\|_{\rm
F}dt+\|[\rho_{0},B(t)]\|_{\rm F}\|\rho(t)\|_{\rm F}dW(t)\right)$
$\displaystyle=\frac{\sqrt{2}}{\sin
2\mathcal{L}(t)}\left(\sqrt{\langle\psi_{0}|H^{2}(t)|\psi_{0}\rangle-\langle\psi_{0}|H(t)|\psi_{0}\rangle^{2}}dt+\sqrt{\langle\psi_{0}|B^{2}(t)|\psi_{0}\rangle-\langle\psi_{0}|B(t)|\psi_{0}\rangle^{2}}dt+\|[\rho_{0},B(t)]\|_{\rm
F}dW(t)\right),$ $\displaystyle\leq\frac{\sqrt{2}}{\sin
2\mathcal{L}(t)}\left({\rm max}\langle\Delta H(t)\rangle_{0}dt+{\rm
max}\langle\Delta B(t)\rangle_{0}dt+{\rm max}\|[\rho_{0},B(t)]\|_{\rm
F}dW(t)\right),$ (21)

where we have used the Schwarz inequality ${\rm
Tr}(X^{\dagger}Y)\leq\|X\|_{\rm F}\|Y\|_{\rm F}$ and $\|\rho(t)\|_{\rm F}\leq
1$. $\langle\Delta H(t)\rangle_{0}$ and $\langle\Delta B(t)\rangle_{0}$ are
the variances of $H(t)$ and $B(t)$ in the initial state, respectively.

Taking the ensemble average and integrating the both sides of this inequality,
we obtain

$\displaystyle\mathbb{E}\left[\int^{\mathcal{L}_{T}}_{0}\sin
2\mathcal{L}(t)d\mathcal{L}(t)\right]$ $\displaystyle\leq\sqrt{2}T\left({\rm
max}\langle\Delta H(t)\rangle_{0}+{\rm max}\langle\Delta
B(t)\rangle_{0}\right).$ (22)

Further, the lefthand side of (A.4) is evaluated as

$\displaystyle\mathbb{E}\left[\int^{\mathcal{L}_{T}}_{0}\sin
2\mathcal{L}(t)d\mathcal{L}(t)\right]$
$\displaystyle=\mathbb{E}[\sin^{2}\mathcal{L}(t)]$
$\displaystyle\geq\mathbb{E}[\sin\mathcal{L}(T)]^{2}$
$\displaystyle\geq\sin^{2}\mathbb{E}[\mathcal{L}(T)]$ (23)

As a result, we end up with the following QSL:

$\displaystyle T\geq T_{\rm
QSL}:=\frac{\sin^{2}\mathbb{E}[\mathcal{L}(T)]}{\sqrt{2}\left({\rm
max}\langle\Delta H\rangle_{0}+{\rm max}\langle\Delta
B(t)\rangle_{0}\right)}.$ (24)

$T_{\rm QSL}$ gives a fundamental lower bound on the evolution time $T$ for
the state $|{\phi(t)}\rangle$ to any state $|{\phi(T)}\rangle$ satisfying
$\sin^{2}\mathbb{E}[\mathcal{L}(T)]$.

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# The Million Quasars (Milliquas) Catalogue, v8

Eric Wim Flesch  PO Box 15, Dannevirke 4942, New Zealand<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Announcing the final release, v8, of the Milliquas (Million Quasars) quasar
catalogue which presents all published quasars to 30 June 2023, including
quasars from the first releases of the Dark Energy Spectroscopic Instrument
(DESI) and the SDSS-DR18 Black Hole Mapper. Its totals are 907 144 type-I
QSOs/AGN and 66 026 high-confidence ($\approx$99% likelihood) radio/X-ray
associated quasar candidates. Type-II and Bl Lac type objects are also
included, bringing the total count to 1 021 800. Gaia-EDR3 astrometry is given
for most objects. The catalogue is available on multiple sites.

###### keywords:

catalogs — quasars: general

##  1 Introduction

The Million Quasars (Milliquas) catalogue has been available on-line since its
inception in 2009, and is the only quasar catalogue to have kept abreast of
all the latest quasar discoveries from published papers large & small since
the final edition of the Véron-Cetty & Véron quasar catalogue (VCV: Véron-
Cetty & Véron , 2010, 13th edition). With this version v8 of quasars to 30
June 2023, Milliquas now also presents its final edition.

The criteria for including published quasars, pipeline quasars, and candidates
are as given in the Half Million Quasars catalog (HMQ: Flesch, 2015) and
references therein, but it has evolved somewhat over time and an overview is
given below. The counts of this final edition are 907 144 type-I quasars and
AGN, 45 816 type-II objects, 2814 blazars, and 66 026 high-likelihood
(pQSO>=99%) radio/X-ray associated quasar candidates, which totals to 1 021
800 objects presented.

Milliquas can be downloaded from CDS111https://cdsarc.cds.unistra.fr/viz-
bin/cat/VII/294 or from its home page222https://quasars.org/milliquas.htm
which also provides a FITS file. Also CDS provides a query page, as does NASA
HEASARC333https://heasarc.gsfc.nasa.gov/W3Browse/all/milliquas.html. The
ReadMe provided at those sites gives essential information about the data.
Figure 1 shows the Milliquas sky coverage.

The sections below describe salient aspects of this catalogue, rules of
inclusion and exclusion, data fixes and newly discovered quasars, and how
radio/X-ray associations enable the selection of high-confidence photometric
quasar candidates.

Figure 1: Sky coverage of Milliquas, darker is denser. The SDSS footprints
dominate the North with extensions radiating out from them, and the 2QZ stripe
is seen at $\delta=-30^{\circ}$. DESI-EDR coverage is shown by the high-
density beads of coverage sprinkled throughout, including a pair north of
+80∘. The Magellanic Clouds show up at lower right because of multiple surveys
to detect quasars behind them. The large bead at lower left shows the XXL-
South AAOmega field. (chart produced with TOPCAT (Taylor, 2005))

##  2 Milliquas rationale and selection

My original project (circa 1999) was to align and overlay radio & X-ray
surveys on to the optical sky. Quasars of course are prominent in such a task.
I quickly became aware that quasars were often not quite where their authors
had placed them, and could be many arcminutes or even degrees offset in
extreme cases; my later paper Flesch (2013) gave 380 corrections of 8+ arcsec
to the final VCV quasar positions, those fixes subsequently included in the
HMQ. Since the HMQ there have been more such astrometric corrections found,
and Table 1 gives 27 new such corrections of 8+ arcsec to legacy quasar
positions. There are 290 more new corrections of 2+ arcsec in the Milliquas
data since the HMQ, all individually vetted. Total presented astrometry is
60.71% from Gaia-EDR3 (Gaia Collaboration, 2021), 8.34% from Pan-STARRS
(Chambers et al., 2016), 27.79% from SDSS-Sweeps444at
https://data.sdss.org/sas/dr9/boss/sweeps/dr9/, 1.69% from DESI (Abbott et
al., 2021), 0.37% from individual authors, and the remaining 1.10% from the
digitized 20th century photographic surveys collected in the ASP optical
catalogue (Flesch, 2017). All this shows the strong attention given to
astrometric accuracy in the Million Quasars catalogue, both in correctly
siting each quasar including all legacy quasars, and presenting those
locations with the greatest precision available.

Table 1: Legacy quasars: 27 more positional corrections of 8+ arcsec since the HMQ, in order of # arcsec moved # | Name | z |  OA ref∗ | J2000 | moved(") | previous J2000 | comment on move
---|---|---|---|---|---|---|---
1 | NGC 4395 U3 | 1.687 | 0081 | 12 19 17.80 +33 29 58.2 | 4897.2 | 12 25 49.33 +33 30 45.0 |  OA† described position via offsets without giving co-ordinates, SDSS-DR16Q confirms redshift there, has radio
|  |  |  |  |  |  | ILT J121917.80+332958.3. VCV used a finding chart from another author which gave a different object with that same name.
2 | CXOPS J04219+3300 | 1.125 | 1556 | 04 21 55.03 +33 00 36.1 | 629.4 | 04 21 05.00 +33 00 36.0 | VCV transcription error, OA gave correct position, has X-ray CXO J042155.0+330035.
3 | VFQ A4/22 | 1.045 | 1634 | 11 02 05.85 +29 59 14.6 | 212.1 | 11 02 07.20 +30 02 46.0 | apparent OA transcription error, photometry & redshift match to NBCKv3 candidate 210" south, radio ILTJ110206.01+295914.5
4 | Q 0856+406 | 2.276 | 1860 | 08 59 56.33 +40 24 37.2 | 128.8 | 08 59 45.31 +40 25 04.1 | OA gave approximate position and the spectrum, SDSS-DR16Q spectrum here shows exactly-matching spectral features.
5 | NGC 520.D1 | 1.493 | 0088 | 01 25 29.76 +04 11 26.2 | 118.9 | 01 25 25.00 +04 09 51.0 | OA described position via offsets without giving co-ordinates, SDSS-DR16Q confirms redshift there.
6 | NGC 615 UB1 | 1.640 | 0081 | 01 34 50.78 -07 21 43.8 | 74.2 | 01 34 52.15 -07 22 55.1 | OA described position via offsets without giving co-ordinates, DESI to get new spectrum there as targetid 39627610338100732.
7 | NGC 2916 UB2 | 0.796 | 0081 | 09 35 13.08 +21 47 32.7 | 70.3 | 09 35 13.93 +21 48 42.0 | OA described position via offsets, and SDSS-DR16Q confirms redshift there, but OA finding chart pointed to wrong object.
8 | Q 1409+732 | 3.560 | 0051 | 14 09 48.01 +72 59 29.1 | 67.3 | 14 10 03.20 +72 59 39.0 | OA finding chart (which introduced the article) pointed only to a bad pixel, but OA-given co-ordinates were correct.
9 | PKS 1635+159 | 2.145 | 1723 | 16 37 49.04 +15 49 57.4 | 43.7 | 16 37 50.00 +15 49 16.0 | positional correction averred in 1977-ApJ-215-427 without giving co-ordinates, now confirmed by SDSS-DR16Q redshift.
10 | PKS 0159+034 | 0.976 | 0670 | 02 01 51.50 +03 43 09.2 | 43.0 | 02 01 50.13 +03 43 47.0 | OA reported quasar as separate from galaxy but VCV kept them conflated for some reason, has radio FIRST J020151.4+034309
11 | RXS J10478-0113 | 0.435 | 0063 | 10 47 49.99 -01 12 43.5 | 41.0 | 10 47 51.66 -01 13 16.0 | OA-given trail from X-ray detection to quasar was not followed by cataloguers, now fixed.
12 | TOL 1038.2-27.1 | 1.937 | 0187 | 10 40 34.28 -27 22 30.0 | 40.6 | 10 40 33.20 -27 23 08.0 | OA-given position pointed to nothing, but OA finding chart identified the quasar.
13 | KP 1127.9+07.4 | 1.700 | 1756 | 11 30 31.37 +07 12 13.3 | 35.6 | 11 30 32.29 +07 12 46.2 |  OA-given position coarse, nearest eligible (prev) object too bright and Gaia PM, next (this) one has right magnitude and no PM.
14 | ISO J1324-2016 | 1.500 | 1462 | 13 24 47.25 -20 16 12.0 | 21.3 | 13 24 45.73 -20 16 12.0 | OA-given position pointed to nothing, but OA finding chart identified the quasar, has radio RACS J132447.2-201610.
15 | Q 0016-357 | 3.199 | 0087 | 00 18 41.60 -35 29 05.4 | 19.7 | 00 18 40.00 -35 29 04.0 | OA-given position & finding chart were coarse, now radio VLASS J001841.58-352904.9 clarifies position of quasar.
16 | Q 13034+2942 | 1.724 | 0222 | 13 05 47.37 +29 26 44.0 | 17.5 | 13 05 46.33 +29 26 33.0 | OA-given position pointed to nothing, but OA finding chart identified the quasar.
17 | KP 1229.0+07.8 | 1.930 | 1756 | 12 31 34.53 +07 34 25.8 | 17.1 | 12 31 34.00 +07 34 41.0 | OA-given position pointed to nothing, OA finding chart was coarse but adequate to identify the quasar.
18 | LMA 15 | 2.700 | 1080 | 01 05 14.20 +02 11 29.2 | 14.0 | 01 05 13.26 +02 11 29.0 | OA co-ordinates were only to whole time-seconds (i.e., 15 arcsec range), but Pan-STARRS image shows the obvious object.
19 | PC 0027+0513 | 3.005 | 1664 | 00 30 26.48 +05 29 53.7 | 12.7 | 00 30 27.06 +05 30 03.0 | OA-given position was near random faint object, but OA finding chart identified the quasar.
20 | CT 187 | 1.141 | 0315 | 00 54 13.46 -29 55 44.8 | 11.5 | 00 54 13.66 -29 55 56.0 | OA identified quasar correctly but VCV took co-ordinates of nearby galaxy for some reason.
21 | Q 03022-0023 | 2.140 | 0910 | 03 04 46.11 -00 11 27.5 | 10.8 | 03 04 45.93 -00 11 38.0 | Serendipitous but unwanted quasar, OA hurried the J2000, good candidate nearby, X-ray 4XMM J030446.0-001127 confirms.
22 | PC 0027+0525 | 4.099 | 1662 | 00 29 49.99 +05 42 14.5 | 10.6 | 00 29 50.00 +05 42 04.0 | OA-given position pointed to nothing, but OA finding chart identified the quasar.
23 | CTS A11.17 | 2.800 | 1230 | 22 42 30.72 -36 44 14.4 | 10.5 | 22 42 30.80 -36 44 04.0 | OA warned true positions are often 10" south of given, has radio VLASS J224230.71-364414.0
24 | CTS R05.17 | 2.120 | 1230 | 10 45 36.03 -15 28 41.5 | 9.7 | 10 45 35.60 -15 28 49.0 |  OA-given position was near blue doublet of which farther end was picked but has Gaia-EDR3 proper motion, near end does not.
25 | RX J23360+3023 | 2.094 | 2156 | 23 36 04.08 +30 24 00.2 | 9.2 | 23 36 04.66 +30 23 55.0 |  OA confused objects, their pick has proper motion (Gaia-EDR3), other does not and has radio VLASS J233604.06+302400.3
26 | S4 1030+39 | 1.095 | 1023 | 10 33 22.03 +39 35 51.0 | 8.8 | 10 33 22.66 +39 35 56.0 | OA conflated bright object with faint true source of X-ray 2SXPS J103322.0+393547 & radio VLASS J103322.07+393551.0
27 | Q J0641-5050 | 0.618 | 1862 | 06 41 46.00 -50 50 21.7 | 8.2 | 06 41 46.00 -50 50 30.0 | OA identified quasar correctly but VCV somehow took co-ordinates of galaxy to south. Has X-ray 4XMM J064146.0-505022.
∗ references are indexed in the accompanying file "milliquas-references.txt".
† OA = original author(s) of discovery paper.

My motivation in all these fixes of legacy positions was not only to present
the correct astrometry, but also to preserve the work of the legacy
discoverers whose quasars, it seemed, were being submerged by subsequent large
surveys which re-surveyed and re-published those quasars without reference to
the original discoveries. The last large survey to carefully identify quasars
earlier discovered, was SDSS-DR7Q (Schneider et al., 2010), and none since.
Milliquas gives the original discovery citation for all quasars, along with
their original names, although, it must be said, the legacy names are mostly
as inherited from the VCV catalogue which often changed names according to the
quasar co-ordinates. So within that constraint, I have done my best to
preserve the work of the original legacy authors.

An essential issue is to decide which quasar discoveries should be included in
Milliquas, and which not. The predecessor catalogue VCV (Véron-Cetty & Véron ,
2010) included questionable objects (so designated by their discoverers) and
provided a table of "rejected quasars" which had been included as quasars in
earlier editions but had since been found to be not quasars. The Milliquas
approach is different, I include only quasars confidently presented by their
discoverers, and objects subsequently found to be not quasars were simply
dropped from Milliquas. There is however a residue of questionable legacy
objects held over from VCV data not reviewed. Workload prohibited a databasing
of possible-quasars and non-quasars in the accruing data. The general rule for
this final edition is that all Milliquas v8 objects are 99% likely to be true
quasars (or type-II, or blazars as classified), including the 66 026
candidates which are calculated in bulk to be 99% likely to be true quasars –
the candidates selection method is as detailed in HMQ (Flesch, 2015) Section
8.

Some discovery authors present new quasars with a quality flag: one common
such flag is 4=confident (2+ strong lines), 3=good (1 strong line & 1 weak
line), 2=weak (1 strong line only or weak lines only), and 1=poor (1 weak line
or continuum only). For this example, Milliquas accepts only those objects
with flags 3 or 4 as classified quasars. The general rule is that one strong
line plus something extra is required, that something extra being either a
redshift-agreeing weak line or agreeing photometric redshift. Many papers
publish "quasars" with a mix of spectroscopic and photometric redshifts; in
such cases, only the spectroscopic quasars are accepted into Milliquas as
classified quasars because a confident redshift is required as well as a
confident classification. Thus those quasars that are published without
redshift do not appear in Milliquas, unless qualifying as candidates.

##  3 Milliquas since the publication of the Half Million Quasars catalogue in
2015

Milliquas has been published as a "live" catalogue updated at irregular
intervals. Two versions were published on arXiv due to significant changes:
Version 6.4 (Flesch, 2019) presented an upgrade of its optical background data
and other changes since the 2015 HMQ, and version 7.2 (Flesch, 2021b) provided
a detailed accounting of the inclusion of the Sloan Digital Sky Survey (SDSS)
Quasar Catalogue 16th Data Release (DR16Q: Lyke et al., 2020) into Milliquas,
in particular identifying DR16Q objects rejected from Milliquas and also those
SDSS quasars left out of DR16Q but included into Milliquas. Both papers also
described the latest quasar discoveries to the time of publication.

The NASA HEASARC Milliquas page gives a running summary of all the announced
changes since version 0.1 in 2009. All throughout, there was an unpublicized
ongoing campaign of fixits and astrometric/photometric tightening which have
honed Milliquas to its standard of accuracy today. An early such exercise was
to inspect legacy quasars which had non-quasar colours – this yielded many
cases of mistaken identity and the true quasar found and substituted. The most
recent large-scale exercise was to identify classified quasars which were
flagged by Gaia-EDR3 as having likely proper motion or parallax, and which had
poor spectra (especially early grism spectra) and no radio/X-ray association,
and so could easily be dropped. About 110 net legacy quasars and 326 SDSS-DR16
(Ahumada et al., 2020) pipeline-only quasars were dropped in the exercise, but
693 LAMOST555LAMOST QSO Catalog page: https://nadc.china-
vo.org/data/article/20190107155838 objects, previously discounted as low-
confidence, were restored as quasars due to Gaia-EDR3 measuring them but
finding no significant proper motion or parallax. Thus the unexpected net
outcome of the exercise was an $\approx$250 increase in the Milliquas quasar
count. Notable among the legacy deletions were 12 (out of 91) Cyril Hazard
unpublished quasars (cite=0800), having neither radio/X-ray association nor
redshift confirmation from SDSS, dropped due to proper motion detected by
Gaia-EDR3.

Milliquas has always presented quasar candidates in addition to classified
quasars. The HMQ (Flesch, 2015) candidates were only those of 99% pQSO
(confidence of being true quasars) calculated using photometry and required
radio/X-ray association, and in this final edition Milliquas has returned to
that selection. In-between, Milliquas featured a much expanded repertoire of
candidates. Those radio/X-ray associated candidates now excluded from
Milliquas are still available in the MORX v2 (MORX: Flesch, 2023) catalogue.
Candidates without radio/X-ray association were dropped from Milliquas as of
version 7.4 to avoid simply repeating candidates from other catalogues.

##  4 Uptake of the DESI-EDR

The Dark Energy Spectroscopic Instrument Early Data Release (DESI-EDR: DESI
Collaboration, 2023) is of data from their commissioning and Survey Validation
prior to starting their DESI Main Survey. Unfortunately their Quasar Redshift
Value-Added Catalog (VAC) was not released in time for Milliquas v8, but I
have processed the large (2 847 435 rows) DESI-EDR pipeline data along with
three available VACs (Alexander et al. (2023) of 5496 rows, Ramírez-Pérez et
al. (2023) of 20 281 rows, and Filbert et al. (2023) of 83 781 cleaned rows)
to yield 91 011 eligible DESI-classified quasars, of which, however, 1982 were
rejected via my inspections of DESI spectra as described below, leaving 89 029
accepted quasars of which 61 342 are new quasars in Milliquas.

The large DESI-EDR pipeline file presents a cleaned version of its total
available pipeline data, the dropped objects being those bearing certain ZWARN
(observation warning flag) values, including some I would have liked retained.
The 3 VACS use different criteria for data selection, so do have some data
bearing those "forbidden" ZWARN values. The VACs overlap somewhat, and shared
objects can be classified differently, e.g., 467 objects are classified as
galaxies by Filbert et al. (2023) but classified as quasars by Ramírez-Pérez
et al. (2023). Similarly in the large pipeline file, 403 DESI-classified
quasars were also DESI-classified as galaxies or stars, i.e., there were
duplicate rows in the pipeline file for these targetIDs, with different
classifications. Inspection of some of these spectra revealed mostly stars or
inscrutable profiles. Rather than inspect all these dual-classified spectra, I
dropped them all because my priority was to exclude false positives even if
some true quasars were lost as well. The presence of these dual
classifications also brings to light that DESI’s focus is not the absolute
accuracy of each datum, but rather that bulk data (with an acceptable small
error rate) will be used to test cosmological hypotheses. However, Milliquas
does aspire to the absolute accuracy of each datum, so visually inspecting
DESI spectra and consequently editing redshifts & classifications as
appropriate, is necessary. However, visually inspecting 91K DESI spectra is
impractical, so a course of action is outlined below whereby those spectra are
identified which are most likely to have pipeline redshifts discrepant from
the true.

DESI fits their spectral models onto 7 "morphtype" (visual morphology)
classes, and each observed object is allocated to one of those morphtypes for
spectral comparison against that model. The 91 011 eligible quasars defined
above yield these counts per morphtype: 79 559 objects with morphtype="PSF"
(stellar point-spread function), 264 "GPS" (Gaia-positioned PSF), 4939 "REX"
(round exponential galaxy), 2292 "SER" (Sersic galaxy profile), 2287 "DEV"
(DeVaucouleurs: elliptical galaxy), 524 "EXP" (exponential: spiral galaxy),
and 4 "GGA" (Gaia-positioned galaxy). Also there are 1142 with blank
morphtype, of which 549 are listed with DESI grz photometry, and 593 show no
photometry. I treat the blank morphtype with & without photometry as two
notional morphtypes in addition to the 7 DESI morphtypes, thus 9 morphtypes in
all used in my method below for finding key spectra to visually inspect.

Notes on the processing of the DESI-EDR QSOs and the 1982 rejections:

  * •

For each morphtype (see above) I ordered its objects by redshift to find
redshift ranges which were not well-performed, i.e., showed unclear spectra.
When such ranges were identified, I did extensive visual checks on all spectra
at the upper/lower bounds to keep or drop individual objects found there; in
total, I did about 1000 visual DESI spectrum inspections.

  * •

About 28K of the objects had been previously classified elsewhere. I imported
those classifications to label "truth" objects to identify reliable swathes of
redshifts and less reliable ones.

  * •

Blank morphtype with photometry tended to fail for z<1.4, i.e., poor spectra.
Similarly, "PSF" morphtype with faint photometry (flux_z<1) tended to fail for
z<1.6. I rejected most of those, bookending the rejected swathes with
sequences of visually-inspected spectra which failed the inspection, plus
random inspections within such swathes.

  * •

Blank morphtype without photometry was excellent for z=3.082 to 4, i.e., all
spectra clear and decisive, also for most of z=2 to 2.233. Also its coverage
of the COSMOS X-ray and Andromeda Commissioning fields was excellent. Other
redshift ranges were heavily inspected and some rejected.

  * •

I visually checked all spectra of z$\geq$3.5 to be consistent with previous
handling of SDSS spectra in Milliquas. Those with unconvincing spectra were
rejected.

  * •

I visually checked or accepted all REX/EXP/DEV/SER/GGA morphtypes. There are
not many of these compared with the PSF/GPS morphtype for QSOs, but these AGN
spectral models were well-performed although not at high redshifts
(z$\geq$3.5).

  * •

There are 119 M-star/white-dwarf doublets DESI-classified as quasars in the
redshift range of 0.6433 to 0.6586, an example being
targetID=39627992170762093666viewable at
https://www.legacysurvey.org/viewer/desi-
spectrum/edr/targetid39627992170762093. To see any DESI spectrum, just replace
the DESI target ID at the end of the foregoing web address. Note that the
spectrum is displayed in red, whereas the the black lines show model fit(s)..
These are some crazy-looking spectra. I dropped them all, of course.

  * •

50 DESI sources were just star-glow artefacts offset 1-2 arcsec from their
true known centroid (thus a duplicate), an example being
targetID=39628526864829382 (z=1.4106) which does not exist and is offset 1.273
arcsec from quasar SDSS J094234.47+322035.6, z=1.409. These 50 artefacts were
dropped.

  * •

I was reluctant to accept spectra which had only negative or inverted lines.
They looked mostly random to me, and not like BAL-type lines. So I may have
missed valid quasars of that type.

  * •

I have assigned new redshifts to 3 DESI QSOs:

    * –

39633247998578429, DESI z=0.6528, changed to z=1.94, obvious Ly$\alpha$-SiIV-
CIV-CIII lines, dual qso-star.

    * –

39633161977597561, DESI z=1.6632, changed to z=0.747, 7 lines matched,
especially the Mg-II & O-II duo.

    * –

39632930619786218, DESI z=1.1321, changed to z=2.79, obvious Lyman forest,
emission lines match ok, anomalous absorption lines throughout the spectrum
are most mysterious.

  * •

I selected DESI-classified galaxies and stars which MORX shows to be radio-
associated, which is a good predictor of AGN activity. Of those, 41 DESI
galaxies and 6 DESI stars have been assigned new redshifts and "promoted" to
quasars, upon my inspection of those spectra, an example being
39628514877505752, DESI galaxy of z=1.0408, changed to z=2.376 with obvious
Ly$\alpha$ and other lines. Also there is 39628433285714226, a DESI star,
changed to z=2.235 with 7 lines matched, although you need to move the
"Gaussian Sigma Smooth" slider over to see them – showing how well the
smoother reveals faint lines otherwise concealed by noise. These 47 new
quasars can be identified in the Milliquas data by cite=’DESEDR’ and
zcite=’MQ’.

  * •

A further 101 DESI galaxies have been "promoted" to AGN, and 53 DESI galaxies
"promoted" to narrow-line (type-II) AGN by virtue of radio/X-ray association
and active spectral lines, especially when both Mg-II and O-II are prominent.
The DESI redshift is unchanged for these. An example is 39633332224397332.
These can be identified in the Milliquas data by cite=’DESEDR’ and
zcite=’DESEDR’ and classification of A or N, and the comment includes an ’a’
(host-dominated).

  * •

5 line poachers777”line poachers” are introduced with multiple examples in
Flesch (2021a), Section 5. were found, these are stars with phantom spectral
lines acquired via blended glare from a nearby quasar or galaxy; they are
easily recognized because they have very different optical colours compared to
the neighbouring quasar with which they putatively share the same redshift.
They are:

    * –

39632955013858840 with putative z=0.5763 at 1.295 arcsec offset from quasar
SDSS J075807.01+334955.9, z=0.576.

    * –

39628281896501832 with putative z=0.4250 at 2.089 arcsec offset from quasar 3C
47.0, z=0.425.

    * –

39628373638516357 with putative z=1.6683 at 1.108 arcsec offset from quasar
SDSS J125703.80+250457.5, z=0.821, which is also the correct redshift for the
DESI spectrum once the Mg-II line is moved to the obvious bump at 5100Å which
also aligns the O-II and other lines.

    * –

39633132038653380 with putative z=0.3977 at 1.075 arcsec offset from galaxy
SDSS J154931.53+425905.0, z=0.398. This object looks like a K star.

    * –

39633352331887198 with putative z=1.8645 at 5.291 arcsec offset from quasar
SDSS J120930.54+572050.6, z=1.867. Whilst this is quite far offset for a line
poacher, the DESI object spectrum shows no feature apart from the lines
faintly echoing the SDSS quasar; possibly it is a galaxy faintly lensing the
quasar.

These 5 line poachers were dropped from Milliquas processing.

As a final note on DESI pipeline redshifts, they are very good but sometimes
they wrongly match many minor lines while missing the "elephant in the room",
i.e., the Mg-II double-line which often presents as a large thick mound in the
spectral profile. Confirming its identity is the O-II double-line at a
standard offset from the Mg-II lines – the Mg-II wavelength is at 75.1% of the
O-II wavelength, and the O-II lines are typically used to refine the redshift
precisely. Thus this Mg-II/O-II combo can signal a precise mid-range redshift.
An example of the informative power of this combo is 39633413539368493 which
is an AGN but with a spectrum showing a strong galaxy continuum which conceals
the AGN lines888to see this clearly, go to
https://www.legacysurvey.org/viewer/desi-
spectrum/edr/targetid39633413539368493, set the Gaussian Sigma Smooth slider
to 4, and turn off the black pipeline model by pressing the ”pipeline fit” and
”other model” labels. The spectrum is now seen as basically featureless. for
which the DESI pipeline has settled on z=1.6366. Inspection shows a possible
swelling at 6100Å as a candidate Mg-II line, so the redshift sliders can be
used to bring the Mg-II lines over to it. Having done so, the O-II line is
seen to align pleasingly to a small spike in the spectrum, so tweaking the
redshift slider exactly there reveals a redshift of 1.169. Confirmation comes
at the left of the spectrum where the C-III line now aligns to a subtle
spectral mound.

The upcoming DESI Quasar Redshift VAC will provide an "Mg-II afterburner"
redshift in addition to the pipeline redshift, using the broad Mg-II line as
the anchor; presumably the O-II line will also be used to refine the redshift.
Also DESI will issue a QuasarNET VAC based on the hierarchical-heuristic
redshift-finding algorithm of Busca & Balland (2018), and use all these to
decide the quasar redshifts for their cosmological investigations. In lieu of
those future tools, I have used the above method of targetted visual
inspections to decide the redshifts of these DESI spectra.

##  5 Uptake of the SDSS-DR18Q

The Sloan Digital Sky Surveys DR18 (DR18: Almeida et al., 2023) was issued in
January 2023 and came with pipeline data and its "Black Hole Mapper" visual
VAC (DR18Q: Merloni et al., 2023) data for which, however, the VAC paper has
not yet been made available as of July 2023. The DR18 approach has changed
from earlier releases in that presented data is of newly observed objects
only, and the visual VAC covers all of the pipeline quasars. The DR18Q
consists of 3 files, one being the main file of visually inspected
spectroscopic redshifts and classifications, and the other two appear to be
working redshift data of uncertain status, with their paper unavailable to
clarify. Thus I used only the main visual file.

The file is of 13085 rows of which 6408 are classifed as quasars, all of which
are in the pipeline data also. Processing these yielded 4551 new type-I
quasars/AGN, 93 new type-II (because subclassed as STARRY and not BROADLINE),
and 1763 objects previously catalogued in Milliquas. There is also one new
line poacher: SDSS J083217.11+040403.9 with putative z=1.123 at 2.257 arcsec
offset from quasar SDSS J083216.99+040405.2, z=1.121. This line poacher was
dropped from Milliquas processing999For the record, there was also a new line
poacher in SDSS-DR17 (Abdurro’uf et al., 2022): SDSS J084856.10+011538.7 with
putative z=0.645 at 1.264 arcsec offset from quasar SDSS J084856.08+011540.0,
z=0.646..

There were 2 pipeline quasars not included in the visual file; why the
omission, is unclear. Those 2 objects have quasar-like photometry and pipeline
redshifts, and I have included them into Milliquas trusting that they are
valid quasars and that their absence from the visual file was inadvertent.
They can be identified in the Milliquas data by cite=’DR18’ and type=’Q’.

Table 2: Five new quasars presented in this paper # | new quasar ID |  new quasar ugriz | prev ID (doublet partner) |  previous ID ugriz | offset | redshift | comment
---|---|---|---|---|---|---|---
1 | SDSS J002620.66-034135.8 | 21.95 21.89 21.44 21.34 21.46 | SDSS J002620.72-034134.2 | 24.62 22.09 20.77 19.86 19.38 | 1.88" | 1.605 | bluish SDSS stellar PSF, prev
|  |  |  |  |  |  | ID is reddish SDSS "galaxy"
2 | SDSS J080212.45+285927.5 | 22.10 22.17 21.56 21.67 21.55 | SDSS J080212.35+285928.3 | 23.48 22.56 21.46 20.31 19.99 | 1.52" | 1.430 | radio ILT J080212.41+285927.0
3 | SDSS J105030.49+200420.9 | 20.26 19.57 20.16 19.41 19.34 | SDSS J105030.51+200422.1 | 21.13 20.64 18.76 18.31 18.00 | 1.23" | 1.650 |  SDSS-DR7 photometry, Gaia
|  |  |  |  |  |  |  3$\sigma$ proper motion for prev ID
4 | SDSS J134915.81+080120.3 | 19.60 19.49 19.36 19.19 18.83 | SDSS J134915.76+080122.5 | 24.10 22.35 20.46 19.56 19.14 | 2.41" | 2.038 | X-ray 4XMM J134915.8+080121
5 | SDSS J151135.04+495958.2 | 24.65 21.51 21.49 21.67 21.48 | SDSS J151135.19+495958.1 | 23.29 22.59 20.55 19.86 19.48 | 1.60" | 2.709 | radio ILD J151135.08+495958.0
|  |  |  |  |  |  |  (ILT Gaussian), Gaia-EDR3
|  |  |  |  |  |  |  20$\sigma$ proper motion for prev ID

##  6 Five new quasars presented by this edition

In Flesch (2021a) I presented 40 new quasars discovered by searching the total
SDSS-DR16 data, 28 of which had been confused with or concealed by stars in
close doublets on the sky. Here I present 5 more such new quasars revealed
within close star-quasar doublets for which SDSS identified the star as being
the quasar. Table 2 lists the 5 new quasars and their doublet partners,
showing the ugriz photometry for each. In each case, the new quasar shows
relatively flat photometry (with the bottom quasar also showing u-dropout
consistent with its redshift) whilst the previous ID shows colours typical of
a red star. The "comment" column shows radio/X-ray detections for 3 of the
doublets, each positionally better aligned to the new quasar than the previous
ID; also 2 of the previous IDs are assigned proper motion by Gaia-EDR3. The
3rd doublet photometry is sourced from SDSS-DR7 (Abazajian et al., 2009),
because SDSS-DR16 does not resolve this doublet. The 5 SDSS identifications
were basically "line poachers" with the true quasars not identified; but now,
this paper identifies the true quasars.

Figure 2 shows images of the 5 new quasars, with an explanatory caption. The
example spectrum shows typical features of such a combined spectrum; spectra
of the 4 other doublets are shown in Flesch (2021b) Figures 11 & 12 with
explanatory captions. In each case SDSS-DR16 processing had confused the star
with the quasar, and the current SDSS-DR18 has not updated the identification.

Figure 2: The 5 new quasars presented in this paper, on SDSS images with 15
arcsec sides. Each new quasar is at the exact centre of its image, with its
doublet partner (which was classified by SDSS as the quasar) alongside. For
each doublet, the merged spectrum shows quasar lines at the blue end where the
quasar’s flux dominates the star’s flux, and red stellar continuum dominating
the right side. The joint SDSS spectrum of the 4th doublet (displayed,
z=2.038) reveals the Ly$\alpha$ line dominating the blue end at left.

##  7 QSOs dropped since Milliquas v7.2

A recent prominent exercise to drop previously-included quasars from Milliquas
was the test against Gaia-EDR3 movers described in Section 3. However, that
test was not decisive on its own because there are always anomalies in large
data, so patterns need to be checked for. In this case some of these apparent
movers were found to be star-quasar superpositions, or a nearby moving star
which caused the observed optical PSF centroid of the quasar to subtly shift
within the changing star flux gradient, or that same outcome can happen if
either member of a close star-quasar doublet is optically variable. In such
cases the quasar can be valid even though flagged by Gaia-EDR3 as a mover.
Thus each object so flagged was inspected prior to deletion – if it had
neither a convincing spectrum nor radio/X-ray association, nor was in a close
doublet, it was dropped. Close doublets warranted additional checks to confirm
a valid quasar.

Also, Milliquas excludes low-confidence/quality or questionable objects (so
deemed by their researchers), but many such objects were inherited from VCV
(Véron-Cetty & Véron , 2010) which was more forgiving. They were removed as
encountered (after being checked over as described in the preceding paragraph)
but a residue remains. Sometimes such objects would be found to be duplicates
of nearby quasars, or positioned onto a faint galaxy with no quasar-like
object nearby nor at due N/S or E/W offsets (such offsets occasionally found
in 20th century publications). Table 3 gives 14 such individual deletions of
legacy "quasars" since Milliquas v7.2 (Flesch, 2021b), with explanations.

Table 3: 14 deletions of legacy quasars since Milliquas v7.2 # | Name | J2000 | rmag | z |  OA ref∗ | comment on deletion
---|---|---|---|---|---|---
1 | PHL 1222 | 01 53 53.89 +05 02 57.1 | 17.57 | 1.904 | 0282 |  a star, conflated with faint quasar SDSS J015354.03+050259.7 unseen by OA†.
2 | 1WGA J2204.9-1815 | 22 04 51.91 -18 15 36.4 | 21.08 | 0.210 | 0307 | duplicate of blazar 1REX J220451-1815.5 at 8.6 arcsec offset.
3 | NGP9 F324-1105786 | 13 38 43.10 +26 25 02.6 | 18.03 | 1.260 | 0424 | image shows galaxy, OA said "uncertain" redshift.
4 | Q J1943.9-1502 | 19 43 58.84 -15 02 48.0 | 18.79 | 3.300 | 0429 |  fuzzy, Gaia-EDR3 proper motion.
5 | Q 0057-2811 | 01 00 23.80 -27 55 46.1 | 19.61 | 2.120 | 0530 | image shows galaxy, redshift wrong for it.
6 | Q J08500+701 | 08 50 02.32 +70 18 06.6 | 14.08 | 1.900 | 0994 | obvious member of star group, X-ray is from NGC 2650.
7 | XFLS J17178+5847 | 17 17 50.73 +58 47 44.0 | 23.20 | 2.550 | 1041 | OA stated ’?’ for classification and redshift.
8 | SW002802.79-425957.0 | 00 28 02.79 -42 59 57.0 | 21.25 | 1.731 | 1042 | red object, OA stated obscured and that class & z were not secure.
9 | Abell 2690#075 | 23 59 56.60 -25 10 20.0 | 25.00 | 2.130 | 1145 | OA stated "tentative" classification & redshift.
10 | QNY1:29 | 12 36 27.94 -00 53 57.5 | 20.05 | 0.179 | 1534 | classified as galaxy by SDSS-DR17.
11 | 16V13 | 08 44 50.55 +44 35 53.9 | 22.53 | 1.833 | 1551 | absorption spectrum only, QSO "WEE 13" is 18 arcsec NW, possible confusion.
12 | 1RXS J104057.7+134216 | 10 40 57.68 +13 42 11.6 | 22.18 | 0.556 | 4LAC | duplicate of QSO SDSS J104058.37+134150.6, z=0.557, at 23 arcsec SE.
13 | LEDA 138501 | 02 09 34.59 +52 26 32.6 | 18.54 | 0.049 | BASS | duplicate of QSO 1ES 0206+522, z=0.049, at 26 arcsec NE.
14 | CGCG 70-200 | 12 38 22.39 +09 31 41.9 | 13.82 | 0.321 | PGC | SDSS finds non-AGN.
∗ references are indexed in the accompanying file "milliquas-references.txt".
† OA = original author(s) of discovery paper.

##  8 Radio/X-ray associations presented

Milliquas has always presented radio/X-ray associations for its objects. The
likelihoods for these associations are as calculated by a data-driven
algorithm which compares areal densities of photometry-association
combinations with backgound averages; this is documented in detail in my
previous papers, especially Flesch & Hardcastle (2004) which you would read at
your peril. The likelihoods of individual radio/X-ray associations are no
longer displayed in Milliquas (as the final Milliquas projects pQSO$\geq$99%
for all its objects) but are displayed in the MORX (Flesch, 2023) catalogue
which itemizes radio/X-ray associations for > 3 million optical objects over
the whole sky. All Milliquas v8 objects having radio/X-ray associations are
reported in MORX also; both final catalogues were extracted out of the same
underlying large database which was frozen after processing all data to 30
June 2023.

The MORX paper comprehensively describes the large radio and X-ray surveys
used there and in Milliquas. Here I list just a brief overview of changes
since Milliquas v7.2 (Flesch, 2021b):

  * •

Radio/X-ray association likelihoods are now calculated at a granularity of 0.1
arcsecond astrometric offsets for all radio/X-ray source catalogs. This
wouldn’t be necessary for low-resolution early catalogues such as those from
ROSAT but was done so that uniform onwards processing could be used for all.

  * •

VLASS (Gordon et al., 2020) has been reprocessed to include only Gaussian
detections, and its inclusion threshold lowered from S/N=5 to S/N=4. The
effects are better and more detections, respectively.

  * •

RACS (Hale et al., 2021) radio associations have been added.

  * •

LoTSS (Shimwell et al., 2022) radio associations have been added. This was a
very large input catalogue.

  * •

RASS (ROSAT All-Sky Survey) X-ray data has been dropped, as it has become
clear over time that its resolution is too coarse to confidently identify
optical sources in isolation.

##  9 Overview of Milliquas – data and structure

Table 4 shows the Milliquas data structure. Each object is displayed with
J2000 astrometry (usually from Gaia-EDR3 precessed to J2000 by
CDS101010https://cds.u-strasbg.fr/), red and blue photometry, redshift,
citations, and radio and X-ray associations where present. The intent is to
present simple data of one line per reliable object. The ReadMe elaborates on
the structure and indexes the values found in each field.

Table 4: Sample lines from Milliquas

Notes on columns (see ReadMe for full descriptions):

  * •

TYPE: 1st char is the object classification (if classified): Q=QSO, A=AGN,
N=type II AGN, see ReadMe for full list. Chars summarizing the associations
displayed: R=radio, X=X-ray, 2=double radio lobes.

  * •

COM: comment on photometry: p=POSS-I magnitudes, so blue is POSS-I O, j=blue
is SERC Bj, g=SDSS g & r, +=optically variable, G=Gaia-EDR3 astrometry,
a=faint nuclear activity, m=nominal proper motion.

  * •

PSF: for red & blue sources: ’-’=stellar, 1=fuzzy, n=no psf available, x=not
seen in this band.

  * •

CITE & ZCITE: citations for name and redshift; citations are indexed in
"milliquas-references.txt".

  * •

LOBE or EXTRA: if TYPE contains a ’2’ (=lobes), then double radio lobe
identifiers are displayed here. Otherwise, any additional radio and/or X-ray
identifiers are displayed here.

The full table can be downloaded from http://quasars.org/milliquas.htm.

Table 5 shows the top 25 contributing discovery papers ordered by numbers of
name citations. SDSS-DR16 catalogues dominate with 74% of all quasar
discoveries, but the large candidates catalogues also show prominently.

Table 5: Top 25 discovery papers for Milliquas v8 |  | # of | # of | # of |
---|---|---|---|---|---
# | ID | classified | candidates | redshifts | paper
1 | SDSS DR16Q visual | 716639 | 265 | 394077 | Lyke et al. (2020)
2 | DESI EDR | 60095 | 1 | 64603 | DESI Collaboration (2023)
3 | SDSS DR16 pipeline | 40898 | 96 | 373997 | Ahumada et al. (2020)
4 | 2QZ/6QZ | 27490 | 14 | 23260 | Croom et al. (2004)
5 |  PGC† | 20347 |  | 8 | Paturel et al. (2003)
6 | NBCKDE-v3 candidates | 231 | 17474 | 15257 | Richards et al. (2015)
7 | GAIA3 candidates |  | 16526 | 18962 | Gaia Collaboration (2022)
8 | XDQSO candidates |  | 13963 |  | Bovy et al. (2011)
9 | Milliquas v8 | 5 | 11349 | 20750 | data unique to Milliquas
10 | 2SLAQ | 10362 | 4 | 8259 | Croom et al. (2009)
11 | LAMOST QSO DR5 | 8131 |  | 7933 | Yao et al. (2019)
12 | LAMOST QSO DR3 | 7181 |  | 6846 | Dong et al. (2018)
13 | LAMOST QSO DR9 | 4875 |  | 4780 | Jin et al. (2023)
14 | SDSS DR18Q visual | 4731 |  | 4731 | Merloni et al. (2023)
15 | AllWISE candidates |  | 3558 |  | Secrest et al. (2015)
16 | NBCKDE candidates | 1 | 2208 | 1990 | Richards et al. (2009)
17 | SDSS DR7Q visual | 1969 |  | 200 | Schneider et al. (2010)
18 | AGES survey | 1849 | 2 | 1023 | Kochanek et al. (2012)
19 | 6dF | 1756 |  | 234 | Jones et al. (2009)
20 | DEEP2 Redshifts | 1544 |  | 1481 | Newman et al. (2013)
21 | SDSS DR14Q visual | 1528 |  | 1530 | Pâris et al. (2018)
22 | AAOz (XXL-South) | 1491 |  | 1498 | Lidman et al. (2016)
23 | OzDES2 | 1218 |  | 1239 | Lidman et al. (2020)
24 | HETDEX | 1144 |  | 1009 | Liu et al. (2022)
25 | DESI VI VAC | 1117 |  | 1391 | Alexander et al. (2023)
Candidates from papers of >1000 classified objects, are all radio/X-ray
associated objects which either have an
| insecure classification / redshift (ZWARNING=4 in the case of the SDSS
objects) or are classified stars
| with quasar-like photometry and so retained as QSO candidates.
† The PGC (Principal Galaxy Catalogue, a.k.a. LEDA) is not actually a
discovery paper, but is used to source
| names for AGN galaxies.

##  10 Conclusion

The Milliquas (Million Quasars) catalogue v8 is presented as a complete record
of published quasars to 30 June 2023. Milliquas presents 907 144 type 1 QSOs &
AGN, 66 026 high-confidence (pQSO=99%) radio/X-ray associated quasar
candidates, 2814 BL Lac objects, and 45 816 type 2 objects. Astrometry is 0.01
arcsecond accurate for most objects, and red-blue photometry is of 0.01
magnitude precision. Radio and X-ray associations for these objects are
presented as found, including double radio lobes. This is the final edition of
Milliquas.

In future, when quasar counts have gone over 2 million and Milliquas is
increasingly just a relic of the past, its enduring deliverable will be its
accurate positioning of all the legacy quasars. Any future cataloguing effort
should incorporate that.

##  11 A personal note

Born in the Netherlands, I made my way to New Zealand in 1984 with an
engineering degree with a physics minor from the University of Colorado. My
employment started in engineering, but settled into computer programming and
databasing. When I got onto the internet in 1995, I sought out physicists and
astronomers on Usenet, astronomy being my earliest interest. I learned a lot
from those discussions: facts, current thinking, and how to better present my
ideas. In those days I was interested in the cosmology of Halton Arp, and
needing data to test that, I waited impatiently for someone to combine the
optical, radio, and X-ray sky catalogues of the day. But this was not
forthcoming, and I realized that astronomers had their own specialties to
pursue, and combining catalogues was not one of them. So I undertook to do
that myself. I remember well my first download of an APM (Irwin & McMahon,
1992) optical file (which took an hour, and there were 958 of them), it was my
"alea jacta est" moment. I am grateful to Dave Monet for sending me CD’s of
his USNO-A1.0 and USNO-A2.0 optical catalogues which enabled me to process
over the whole sky. In the end, the new data did not support Arp’s
interpretation of quasar origins, so I abandoned that, but meanwhile was well
in-gear with my all-sky processing. I gradually added quasar cataloguing to my
repertoire, starting in Y2000 when Mira Véron-Cetty invited me to help check
over the pending VCV 9th edition data, which I did through to their final
(13th) edition. I have since published papers and catalogues, and can attest
that two key requirements for successful astronomy publication are a
mathematically-grounded science education, and perseverence.

## Acknowledgements

Thanks to my tolerant wife. This work was not funded.

## Data Availability

Milliquas can be downloaded from CDS at https://cdsarc.cds.unistra.fr/viz-
bin/cat/VII/294 or from its home page at https://quasars.org/milliquas.htm
which also provides a FITS file. Both sites also provide the ReadMe and the
references list. Query pages are provided by CDS and NASA HEASARC at
https://heasarc.gsfc.nasa.gov/W3Browse/all/milliquas.html.

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