# Simulation of open quantum system dynamics based on the generalized quantum
master equation on quantum computing devices

Yuchen Wang  Department of Chemistry, Department of Physics and Purdue Quantum
Science and Engineering Institute, Purdue University, West Lafayette, Indiana
47907, USA  Ellen Mulvihill  Department of Chemistry, Yale Quantum Institute,
Yale University, New Haven, CT 06511, USA  Zixuan Hu  Department of Chemistry,
Department of Physics and Purdue Quantum Science and Engineering Institute,
Purdue University, West Lafayette, Indiana 47907, USA  Ningyi Lyu  Department
of Chemistry, Yale Quantum Institute, Yale University, New Haven, CT 06511,
USA  Saurabh Shivpuje  Department of Chemistry, Department of Physics and
Purdue Quantum Science and Engineering Institute, Purdue University, West
Lafayette, Indiana 47907, USA  Yudan Liu  Department of Chemistry, University
of Michigan, Ann Arbor, MI 48109, USA  Micheline B. Soley  Department of
Chemistry, Yale Quantum Institute, Yale University, New Haven, CT 06511, USA
Department of Chemistry and Department of Physics, University of Wisconsin-
Madison, Madison, WI 53706, USA  Eitan Geva<EMAIL_ADDRESS>Department of
Chemistry, University of Michigan, Ann Arbor, MI 48109, USA  Victor S. Batista
<EMAIL_ADDRESS>Department of Chemistry, Yale Quantum Institute, Yale
University, New Haven, CT 06511, USA  Sabre Kais<EMAIL_ADDRESS>Department of
Chemistry, Department of Physics and Purdue Quantum Science and Engineering
Institute, Purdue University, West Lafayette, Indiana 47907, USA

###### Abstract

The simulation of open quantum system dynamics, namely the reduced dynamics of
a quantum system coupled to a quantum bath, is the cornerstone of quantum rate
theory, optical response theory and decoherence science, which are central
concepts in modern physics and chemistry. The generalized quantum master
equation (GQME) formalism provides a universal framework for simulating the
dynamics of open quantum systems. Using this framework allows one to derive a
formally exact equation of motion, i.e., the GQME, for the reduced density
matrix that describes the state of a system coupled to a bath, without
employing commonly made restrictive assumptions such as weak system-bath
coupling and Markovity. Within this GQME, the eﬀect of the bath on the time
evolution of the system’s reduced density matrix is fully captured by a memory
kernel superoperator. In this work we develop a general-purpose GQME-based
quantum algorithm for simulating open quantum system dynamics. Starting out
from the memory kernel as the input, we solve the GQME for the system’s non-
unitary time evolution superoperator. We then use dilation techniques to
convert the non-unitary time evolution superoperator into a unitary time
evolution superoperator in an extended Hilbert space, which can be implemented
on quantum circuits. The GQME-based quantum algorithm is demonstrated with the
spin-boson benchmark model, including implementations on the IBM QASM quantum
simulator and IBM quantum computers.

##  1 Introduction

Real-life quantum systems are rarely perfectly isolated, which makes it
necessary to account for the effect of their surrounding, or bath, on the
system’s dynamics. Furthermore, many important phenomena, such as rates of a
wide range of chemical and physical processes, lifetimes of excited states,
spectral diffusion, and line-broadening, either rely on or are driven by
interactions between the system of interest and its surrounding. The
simulation of open quantum system dynamics, namely the reduced dynamics of a
system coupled to a bath, is therefore central to many science and engineering
disciplines, including physical chemistry, molecular physics, condensed-phase
physics, nanoscience, molecular electronics, quantum optics, nonequilibrium
statistical mechanics, spectroscopy and quantum information science [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30]. Open quantum system dynamics therefore serves as a
unifying platform that brings together different disciplines and bridge the
gap between theory and experiment. Notable examples include electronic energy
and charge transfer and dephasing, vibrational relaxation and dephasing,
nonadiabatic dynamics, photochemistry, and a variety of other light-induced
physical and chemical processes [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 23, 22, 45, 46, 28, 47, 48, 49].

Although numerous quantum algorithms have been developed for calculating
electronic structure [50, 51, 52, 53] and simulating quantum dynamics of
closed quantum systems [54, 55, 56, 57], relatively few studies have targeted
simulating open quantum system dynamics [58, 59, 60, 61, 62, 63]. Furthermore,
the majority of those previous studies were based on Lindblad-type quantum
master equations (QMEs), which are guaranteed by construction to satisfy
complete positivity and conservation of probability, but also assume
Markovity, which, strictly speaking, is only valid in the weak system-bath
coupling limit or the singular bath limit [9]. It is therefore highly
desirable to develop quantum algorithms for simulating open quantum system
dynamics based on a less-restrictive equation of motion (EoM). In this paper,
we propose such a quantum algorithm based on the generalized quantum master
equation (GQME), which corresponds to the formally exact EoM for an open
quantum system.

A major challenge facing the quantum simulation of open quantum system
dynamics is the fact that the time evolution operators are non-unitary whereas
quantum gates are unitary. To this end, we have previously developed a quantum
algorithm for open quantum dynamics based on the Sz.-Nagy unitary dilation
theorem, which converts non-unitary operators into unitary operators in an
extended Hilbert space. This algorithm was originally applied to simulating a
Markovian two-level model on the IBM quantum computer [64]. Later, the same
method was applied to simulating a non-Markovian two-level model (Jaynes-
Cummings model) on the IBM quantum computer [65]. In a recent work, the same
quantum algorithm was generalized to also work with the Lindblad QME and
applied to simulate the dynamics of the Fenna-Matthews-Olson complex, which
includes five quantum levels and seven elementary physical processes [66].
Thus far, this quantum algorithm has been used to simulate the dynamics of
open quantum systems described by the operator sum representation or Lindblad-
type QMEs.

Figure 1: A comparison of the workflows for simulating the dynamics of a
closed quantum system governed by the quantum Liouville equation vs. an open
quantum system governed by the GQME. 1\. The EoM is established; 2. the time
evolution superoperator is generated from the EoM; 3. A unitary dilation is
required in order to convert the GQME-based non-unitary time evolution
superoperator into a unitary superoperator in an extended Hilbert space; 4.
Translation of the unitary matrix into a quantum gate sequence.

However, these approaches are not entirely general: the Lindblad QME used in
Ref. 66 has several restrictive approximations including Markovian dynamics
and the ensemble of Lindbladian trajectories method in Ref. 65, while capable
of describing non-Markovian dynamics, involves user selection of ad-hoc
system-bath parameters, therefore limiting the range of applications.
Furthermore, while the operator sum representation of open quantum system
dynamics is general, it requires knowledge of the Kraus operators, which to
the best of our knowledge are only known in closed form for systems whose
dynamics can be described by Lindblad-type QMEs.

Extending the range of quantum simulation of open quantum systems therefore
calls for describing the dynamics by a less restrictive theoretical framework.
The GQME formalism introduced by Nakajima [67] and Zwanzig [68] represents
such a general framework since the GQME it leads to corresponds to the
formally exact EoM of the open quantum system, as opposed to the Lindblad-type
QMEs which correspond to approximate EoMs of the open quantum system.

A comparison of the workflows for simulating the dynamics of a closed quantum
system governed by the quantum Liouville equation vs. an open quantum system
governed by the GQME is shown in Fig. 1. The derivation of the GQME involves
projecting out the bath degrees of freedom (DOF) to obtain the EoM of the
system’s reduced density matrix. Within this EoM, which is referred to as the
GQME, the memory kernel superoperator, $\cal{K}(\tau)$, fully accounts for the
impact of the bath on the system’s dynamics. Thus, the GQME replaces the
Liouville equation as the formally exact EoM of the system when we transition
from a closed quantum system to an open quantum system, with the memory kernel
playing a similar role in the open system to that of the Hamiltonian or
Liouvillian in the closed system.

In this work, we develop a GQME-based quantum algorithm for simulating the
dynamics of an open quantum system. Firstly, we provide a route from the
memory kernel to the non-unitary time evolution superoperator that describes
the dynamics generated by the GQME for that memory kernel. Then the Sz.-Nagy
unitary dilation theorem is used to convert the GQME-based non-unitary time
evolution superoperator into a unitary superoperator in an extended Hilbert
space. Given this dilated and now unitary time evolution superoperator and the
initial state of the system, we can evolve the dynamics for any open quantum
system on quantum computers. Given the fact that the GQME is the exact EoM of
the open quantum system, this quantum algorithm greatly extends the range of
possible systems that can be simulated on a quantum computer, including
complex non-Markovian photosynthetic and photovoltaic systems[69, 28],
molecular electronics[48], linear and nonlinear spectroscopy[70], systems with
inter-system crossing[71], and conical intersections[72]. Thus, this GQME-
based quantum algorithm provides an essentially universal protocol for
simulating open quantum system dynamics on quantum computing platforms. Given
a powerful enough quantum computer, this algorithm opens the door for
simulating open quantum system dynamics of large and complex molecular
systems, which are currently beyond the reach of classical computers.

##  2 Results

###  2.1 The GQME-based time evolution superoperator

In this sub-section, we describe the first step in the workflow outlined in
Fig. 1, namely obtaining the time evolution superoperator of an open quantum
system starting from its formally exact EoM in GQME form.

For the sake of concreteness, we will focus on molecular systems with an
overall Hamiltonian of the following commonly encountered form

$\hat{H}=\sum_{j=1}^{N_{e}}\hat{H}_{j}|j\rangle\langle
j|+\sum_{\begin{subarray}{c}j,k=1\\\ k\neq
j\end{subarray}}^{N_{e}}\hat{V}_{jk}|j\rangle\langle k|~{}~{}$ (1)

and an overall system initial state of the following commonly used single-
product form

$\hat{\rho}(0)=\hat{\rho}_{n}(0)\otimes\hat{\sigma}(0)~{}~{}.$ (2)

It should be noted that the GQME approach is not limited to this form of
Hamiltonian and initial state and that the choice to focus on them is solely
motivated by clarity of presentation. The system and bath in this case
correspond to the electronic and nuclear DOF, respectively. In Eqs. (1) and
(2), $\hat{H}_{j}=\hat{\bf P}^{2}/2+V_{j}\left(\hat{\bf R}\right)$ is the
nuclear Hamiltonian when the system is in the diabatic electronic state
$|j\rangle$, with the index $j$ running over the $N_{e}$ electronic states;
$\hat{\bf R}=\left(\hat{R}_{1},...,\hat{R}_{N_{n}}\right)$ and $\hat{\bf
P}=\left(\hat{P}_{1},...,\hat{P}_{N_{n}}\right)$ are the mass-weighted
position and momentum operators of the ${N_{n}}\gg 1$ nuclear DOF,
respectively; $\left\\{\hat{V}_{jk}|j\neq k\right\\}$ are the coupling terms
between electronic states (which can be either nuclear operators or
constants); and $\hat{\rho}_{n}(0)$ and $\hat{\sigma}(0)$ are the reduced
density operators that describe the initial states of the nuclear (bath) and
electronic (system) DOF, respectively. Throughout this paper, boldfaced
variables, e.g., ${\bf A}$, indicate vector quantities; a hat over a variable,
e.g., $\hat{B}$, indicates an operator quantity; and calligraphic font, e.g.,
${\cal L}$, indicates a superoperator.

Using projection operator techniques, one can then derive the following
formally exact EoM, or GQME, for the reduced electronic density operator,
$\hat{\sigma}(t)$ [27, 28, 29, 30]:

$\frac{d}{dt}\hat{\sigma}(t)=-\frac{i}{\hbar}\langle\mathcal{L}\rangle_{n}^{0}\hat{\sigma}(t)-\int_{0}^{t}d\tau\
\mathcal{K}(\tau)\hat{\sigma}(t-\tau)~{}~{}.$ (3)

The open quantum system dynamics of the reduced electronic density matrix
described by this GQME is generated by the two terms on the R.H.S. of Eq. (3).
The first term is given in terms of the projected overall system Liouvillian
$\langle{\cal
L}\rangle_{n}^{0}\equiv\text{Tr}_{n}\left\\{\hat{\rho}_{n}(0){\cal
L}\right\\}$ [where ${\cal L}(\cdot)=[\hat{H},\cdot]$ is the overall system
Liouvillian and $\text{Tr}_{n}\\{\cdot\\}$ is the trace over the nuclear
(bath) Hilbert space], which is represented by a $N_{e}^{2}\times N_{e}^{2}$
time-independent matrix. The second term is given in terms of the memory
kernel ${\cal K}(\tau)$, which is represented by a $N_{e}^{2}\times N_{e}^{2}$
time-dependent matrix. Further details about the GQME approach, including the
derivations of Eq. (3), and the explicit form of the memory kernel are
provided in Sec. 4.

For the purpose of this paper, it is more convenient to rewrite the GQME in
Eq. (3) in terms of the system’s time evolution superoperator, ${\cal G}(t)$,
which is defined by:

$\hat{\sigma}(t)={\cal G}(t)\hat{\sigma}(0)~{}~{}.$ (4)

Substituting Eq. (4) into Eq. (3) and noting that the GQME should be satisfied
for an arbitrary choice of $\hat{\sigma}(0)$, it is straightforward to show
that ${\cal G}(t)$ satisfies the same GQME as $\hat{\sigma}(t)$:

$\displaystyle\frac{d}{dt}{\cal
G}(t)=-\frac{i}{\hbar}\langle\mathcal{L}\rangle_{n}^{0}{\cal
G}(t)-\int_{0}^{t}d\tau\ \mathcal{K}(\tau){\cal G}(t-\tau)~{}~{}.$ (5)

Thus, given the projected Liouvillian and memory kernels
[$\langle\mathcal{L}\rangle_{n}^{0}$ and ${\cal K}(\tau)$, respectively],
${\cal G}(t)$ can be obtained by solving Eq. (5).

###  2.2 The GQME-based quantum algorithm for simulating open quantum system
dynamics

In this sub-section, we describe the next step in the workflow outlined in
Fig. 1, namely using the Sz.-Nagy’s unitary dilation procedure [73] to convert
the non-unitary quantum open system time evolution superoperator ${\cal G}(t)$
[see Eqs. (4) and (5)] into a unitary time evolution superoperator in an
extended Hilbert space.

We start out by calculating the operator norm of ${\cal G}(t)$ to determine if
it is a contraction. For ${\cal G}(t)$ to be a contraction, the operator norm
of ${\cal G}(t)$ needs to be less than or equal to $1$, i.e., $||{\cal
G}(t)||_{O}=\sup\frac{||{\cal G}(t)\bm{v}||}{||\bm{v}||}\leq 1$. In the case
where the original ${\cal G}(t)$ is not a contraction, we introduce a
normalization factor $n_{c}=||{\cal G}(t)||_{O}$ in order to define a
contraction form of ${\cal G}(t)$, namely ${\cal G}^{\prime}(t)={\cal
G}(t)/n_{c}$.

In the next step, we apply a $1$-dilation procedure to ${\cal G}^{\prime}(t)$
to obtain a unitary ${\cal U}_{{\cal G}^{\prime}}(t)$ in an extended Hilbert
space of double the dimension of the original system Hilbert space:

${\cal U}_{{\cal G}^{\prime}}(t)=\begin{pmatrix}{\cal G}^{\prime}(t)&{\cal
D}_{{\cal G}^{\prime\dagger}}(t)\\\ {\cal D}_{{\cal G}^{\prime}}(t)&-{\cal
G}^{\prime\dagger}(t)\end{pmatrix}~{}~{}.$ (6)

Here, ${\cal D}_{{\cal G}^{\prime}}(t)=\sqrt{I-{\cal
G}^{\prime\dagger}(t){\cal G}^{\prime}(t)}$ and ${\cal D}_{{\cal
G}^{\prime\dagger}}=\sqrt{I-{\cal G}^{\prime}(t){\cal G}^{\prime\dagger}(t)}$,
where ${\cal D}_{{\cal G}^{\prime}}(t)$ is the so-called defect superoperator
of ${\cal G}^{\prime}(t)$. The 1-dilation procedure generates a unitary
superoperator ${\cal U}_{{\cal G}^{\prime}}(t)$ that operates in the extended
Hilbert space and replicates the effect of the contraction form of the
original time evolution superoperator, ${\cal G}^{\prime}(t)$, when the input
and output vectors are both restricted to the original smaller Hilbert space.

In the original system Hilbert space, the system reduced density operator
$\hat{\sigma}(t)$ is represented by an $N_{e}\times N_{e}$ matrix:

$\hat{\sigma}(t)\doteq\left(\begin{matrix}{{\sigma}_{11}}(t)&\ldots&{{\sigma}_{1N_{e}}}(t)\\\
\vdots&\ddots&\vdots\\\
{{\sigma}_{N_{e}1}}(t)&\cdots&{{\sigma}_{N_{e}N_{e}}}(t)\\\
\end{matrix}\right).$ (7)

Alternatively, the same system reduced density operator can also be
represented by an $N_{e}^{2}$-dimensional vector in Liouville space:

$\hat{\sigma}(t)\doteq{{\left({{\sigma}_{11}}(t),...,{{\sigma}_{1N_{e}}}(t),{{\sigma}_{21}}(t),...,{{\sigma}_{2N_{e}}}(t),...\,...,{{\sigma}_{N_{e}1}}(t),...,{{\sigma}_{N_{e}N_{e}}}(t)\right)}^{T}}~{}~{}.$
(8)

Since the GQME formalism is given in terms of superoperators, it is convenient
to work in Liouville space, which we will do from this point on. We also
define the norm of the vector representing $\hat{\sigma}(t)$ in Liouville
space as the Frobenius norm:
$\left\|{{\sigma}(t)}\right\|_{F}=\sqrt{\sum\limits_{ij}{{{\left|{{\sigma}_{ij}}\right|}^{2}}}}$
and divide $\hat{\sigma}(t)$ by $\left\|{{\sigma}(t)}\right\|_{F}$ to
normalize $\hat{\sigma}(t)$. [64]

Given the dilated unitary operator ${\cal U}_{{\cal G}^{\prime}}(t)$ and the
initial quantum input state ${\hat{\sigma}}(0)$, operating with the non-
unitary ${\cal G}^{\prime}(t)$ on $\hat{\sigma}(0)$ has now been converted
into a unitary transformation as follows:

${\cal G}^{\prime}(t)\hat{\sigma}(0)\xrightarrow{\text{unitary dilation}}{\cal
U}_{{\cal G}^{\prime}}(t)\left(\hat{\sigma}(0)^{T},0,\cdots,0\right)^{T}.$ (9)

The $0$s in the input vector on the R.H.S. are added to match the dimension of
the input vector with that of ${\cal U}_{{\cal G}^{\prime}}(t)$. The unitary
process can then be simulated on a quantum circuit with unitary quantum gates.
The electronic populations, $\\{\sigma_{jj}(t)\equiv\langle
j|\hat{\sigma}(t)|j\rangle|j=1,...,N_{e}\\}$ can be retrieved by taking the
square roots of the probability of measuring each basis state
$P_{j}(t)=|\sigma^{\prime}_{jj}(t)|^{2}$ and multiplying by the $n_{c}$
factor.

Finally, we perform a complexity analysis of the quantum algorithm. Given that
${\cal G}(t)$ in its most general form is represented by an matrix of
$N_{e}^{4}$ non-zero elements, the defect superoperators ${\cal D}_{{\cal
G}^{\prime}}(t)$ as well as $-{\cal G}^{\dagger}(t)$ as shown in Eq. (6) all
have $N_{e}^{4}$ non-zero elements. Generally speaking, the number of the two-
level unitaries necessary to decompose a unitary gate is comparable to the
number of non-zero elements in the lower-triangular part of the gate [74, 75].
Therefore, the gate complexity to simulate this specific ${\cal U}_{{\cal
G}^{\prime}}(t)$ is $O(N_{e}^{4})$. If the two-level unitaries are further
decomposed into $1$-qubit and $2$-qubit elementary gates commonly used to
design conventional quantum circuits, they need to be transformed to the Gray
code sequences and some multi-control gate sequences, adding another factor of
complexity logarithmic in $N_{e}^{2}$, and the total complexity becomes
$O(N_{e}^{4}\log^{2}N_{e}^{2})$ [74]. This means that the maximum total
complexity of a GQME-based simulation of a open quantum system dynamics is
comparable to classical methods [64]. However, as demonstrated in previous
simulations of certain dynamical models, our quantum algorithm can take
advantage of the case when the ${\cal G}(t)$ is a sparse matrix, and thus the
gate complexity scaling for ${\cal G}(t)$ can be reduced to
$O(\log^{2}N_{e}^{2})$ instead of $O(N_{e}^{4})$ [64, 66].

###  2.3 A demonstrative application to the spin-boson model

In this sub-section, we demonstrate the applicability of the quantum algorithm
outlined in the previous sections on the spin-boson benchmark model. This
model has a wide range of applicability to chemical and physical systems,
including electron, proton, energy, and charge transfer processes; polaron
formation and dynamics in condensed phase environments; vibrational
relaxation, impurity relaxation in solids, and spin-lattice relaxation; and
dissipation effects in qubits in quantum computing [76, 23, 22, 77]. It should
also be noted that quantum-mechanically exact memory kernels can be calculated
for this model [78, 79, 80].

The spin-boson Hamiltonian has the form of Eq. (1) with $N_{e}=2$ and
$\\{\hat{H}_{j},\hat{V}_{jk}\\}$ given by:

$\begin{split}\hat{H}_{0}&\equiv\hat{H}_{D}=\epsilon+\sum_{k=1}^{N_{n}}\frac{\hat{P}_{k}^{2}}{2}+\frac{1}{2}\omega_{k}^{2}\hat{R}_{k}^{2}-c_{k}\hat{R}_{k},\\\
\hat{H}_{1}&\equiv\hat{H}_{A}=-\epsilon+\sum_{k=1}^{N_{n}}\frac{\hat{P}_{k}^{2}}{2}+\frac{1}{2}\omega_{k}^{2}\hat{R}_{k}^{2}+c_{k}\hat{R}_{k},\\\
\hat{V}_{01}&\equiv\hat{V}_{DA}=\hat{V}_{10}\equiv\hat{V}_{AD}=\Gamma.\end{split}$
(10)

Here, the two electronic states are designated as the donor and acceptor
($|D\rangle$ and $|A\rangle$, respectively), $2\epsilon$ is the shift in
equilibrium energy between the donor ($D$) and acceptor ($A$) states, and
$\Gamma$ is a positive constant describing the electronic coupling between the
donor and acceptor states. Since $\Gamma$ is a constant, this system is
assumed to satisfy the Condon approximation.

The results shown below were obtained for the case where the nuclear modes’
frequencies and coupling coefficients $\\{\omega_{k},c_{k}\\}$ are sampled
from an Ohmic spectral density with exponential cutoff:

$J(\omega)=\frac{\pi}{2}\sum_{k=1}^{N_{n}}\frac{c_{k}^{2}}{\omega_{k}}\delta(\omega-\omega_{k})~{}\stackrel{{\scriptstyle\raisebox{1.0pt}{\text{\footnotesize$N_{n}\rightarrow\infty$}}}}{{\xrightarrow{\hskip
21.33955pt}}}~{}\frac{\pi\hbar}{2}\xi\omega e^{-\omega/\omega_{c}}~{}~{}.$
(11)

Here, $\xi$ is the Kondo parameter and $\omega_{c}$ is the cutoff frequency.
The reader is referred to Appendix C of Ref. 27 for a description of the
procedure used to obtain a discrete set of $N_{n}$ nuclear mode frequencies
$\\{\omega_{k}\\}$ and coupling coefficients $\\{c_{k}\\}$ from the spectral
density in Eq. (11).

Table 1: Spin-boson model and simulation parameters.  | Model Parameters | Numerical Parameters
---|---|---
Model $\\#$ | $\epsilon$ | $\Gamma$ | $\beta$ | $\xi$ | $\omega_{c}$ |  $\omega_{\text{max}}$ |  $N_{n}$ | $\Delta t$
1 | 1.0 $\Gamma$ | 1.0 | 5.0 $\Gamma^{-1}$ | 0.1 | 1.0 $\Gamma$ | 5 $\Gamma$ | 60 | 1.50083 $\times 10^{-3}\,\Gamma^{-1}$
2 | 1.0 $\Gamma$ | 1.0 | 5.0 $\Gamma^{-1}$ | 0.1 | 2.0 $\Gamma$ | 10 $\Gamma$ | 60 | 1.50083 $\times 10^{-3}\,\Gamma^{-1}$
3 | 1.0 $\Gamma$ | 1.0 | 5.0 $\Gamma^{-1}$ | 0.4 | 2.0 $\Gamma$ | 10 $\Gamma$ | 60 | 1.50083 $\times 10^{-3}\,\Gamma^{-1}$
4 | 0.0 $\Gamma$ | 1.0 | 5.0 $\Gamma^{-1}$ | 0.2 | 2.5 $\Gamma$ | 12 $\Gamma$ | 60 | 4.50249 $\times 10^{-3}\,\Gamma^{-1}$

The initial state is assumed to be of the form of Eq. (2), with the initial
electronic (system) reduced density operator given by
$\hat{\sigma}(0)=|D\rangle\langle D|$ and the initial nuclear (bath) reduced
density operator given by

$\hat{\rho}_{n}(0)=\frac{e^{-\beta(\hat{H}_{D}+\hat{H}_{A})/2}}{\text{Tr}_{n}\Big{\\{}e^{-\beta(\hat{H}_{D}+\hat{H}_{A})/2}\Big{\\}}}.$
(12)

Calculations were carried out for four different sets of parameter values (see
Table 1). Models 1 and 2 correspond to systems with an energy bias between the
donor and acceptor states ($\epsilon\neq 0$) and differ in their cutoff
frequencies, with model 2 having a higher cutoff frequency. Model 3
corresponds to a biased system with the same parameters as model 2 except for
a larger Kondo parameter. Model 4 corresponds to a symmetric system with zero
energy bias between the donor and acceptor states ($\epsilon=0$). The results
reported in this paper were obtained with a time step of $\Delta
t=1.50083\times 10^{-3}\,\Gamma^{-1}$ for models 1-3 and a time step of
$\Delta t=4.50249\times 10^{-3}\,\Gamma^{-1}$ for model 4.

Starting with the quantum-mechanically exact memory kernels (adopted from Ref.
80), the time evolution superoperator for the electronic reduced density
matrix ${\cal G}(\tau)$ was generated for the four models given in Table 1 by
solving the corresponding GQME, Eq. (5). Details on the method used to obtain
${\cal G}(\tau)$ are given in Sec. 4.

(a)

(b)

(c)

(d)

Figure 2: The spin-boson model simulated by the GQME-based quantum algorithm
as implemented on the IBM QASM quantum simulator, showing the electronic
population difference between the donor state and acceptor state
$\sigma_{z}(t)=\sigma_{DD}(t)-\sigma_{AA}(t)$ as a function of time for (a)
model 1, (b) model 2, (c) model 3, and (d) model 4 as given in Table 1, with
units scaled to the electronic coupling, $\Gamma$. Each figure shows the
comparison between the GQME-generated exact results represented by the black
curves and the QASM-simulated results represented by the yellow dots. The time
step for both the exact and simulated results is $\Delta t=1.50083\times
10^{-3}\Gamma^{-1}$ for models 1-3 and $\Delta t=4.50249\times
10^{-3}\Gamma^{-1}$ for model 4. Each model is simulated for $4000$ time
steps. The number of projection measurements applied by the QASM simulator to
simulate a single time step is $2000$ shots.

(a)

(b)

Figure 3: The spin-boson model simulated by the GQME quantum algorithm as
implemented on the IBM quantum computers ibmq quito and ibmq lima, showing the
electronic population difference between the donor state and acceptor state
$\sigma_{z}(t)=\sigma_{DD}(t)-\sigma_{AA}(t)$ as a function of time for (a)
model 3 and (b) model 4 as given in Table 1, with units scaled to the
electronic coupling, $\Gamma$. Each figure shows the comparison between the
GQME-based results represented by the black curves and quantum-computer-
simulated results represented by the red dots with error bars. The time step
for the real machine simulation is $\Delta t=0.150083\,\Gamma^{-1}$ for model
3 and $\Delta t=0.450249\,\Gamma^{-1}$ for model 4. The experiments of both
models take 40 evenly-spaced time steps out of the 4000 time steps used in the
QASM simulator runs and the error bars represent the standard derivations of
the $10$ separate runs on the ibmq quito and ibmq lima for models 3 and 4. The
number of projection measurements applied by both the devices to simulate a
single time step is 2000 shots.

The GQME-based quantum algorithm for simulating the electronic dynamics within
the spin-boson model was implemented on the IBM quantum platforms via the
Qiskit package [81]. The quantum implementation involved the translation at
each time step of ${\cal G}^{\prime}(t)$ into ${\cal U}_{{\cal
G}^{\prime}}(t)$, followed by the construction of a quantum circuit based on
${\cal U}_{{\cal G}^{\prime}}(t)$, and lastly the use of the quantum circuit
to simulate the time evolution of the reduced electronic density matrix. To
build the circuit, we dilate the $4\times 4$ ${\cal G}^{\prime}(t)$ into a
unitary $8\times 8$ ${\cal U}_{{\cal G}^{\prime}}(t)$ using a $1$-dilation
procedure [see Eq. (6)]. The unitary ${\cal U}_{{\cal G}^{\prime}}(t)$ is then
transpiled into a $3$-qubit quantum circuit composed of three elementary
quantum gates: $R_{Z}$, $\sqrt{X}$, and $CX$. Examples of ${\cal U}_{{\cal
G}^{\prime}}(t)$ and details of the elementary quantum gates and circuits are
shown in the supplementary information (SI). The initial electronic state is
set to ${{\left(1,0,0,0,0,0,0,0\right)}^{T}}$, where the last four $0$s are
the extra dimensions from the dilation procedure. The QASM simulator and the
real quantum devices initialize the input state
${{\left(1,0,0,0,0,0,0,0\right)}^{T}}$ and apply the unitary operation ${\cal
U}_{{\cal G}^{\prime}}(t)$ to the input state followed by projection
measurements to retrieve the probability distribution of all the $8$ basis
states. Each circuit runs 2000 shots and the resulting probabilities
$P_{000}(t)$ of measuring the state $|000\rangle$ and $P_{011}(t)$ of
measuring $|011\rangle$ correspond to the diagonal elements of the modified
density matrix $|\sigma^{\prime}_{00}(t)|^{2}$ and
$|\sigma^{\prime}_{11}(t)|^{2}$. To retrieve the populations of the donor
state $\sigma_{00}(t)$ and acceptor state $\sigma_{11}(t)$ we have

$\sigma_{00}(t)=\sqrt{P_{000}(t)}\times
n_{c}\;\mathrm{and}\;\sigma_{11}(t)=\sqrt{P_{011}(t)}\times n_{c}~{}~{}.$ (13)

In what follows, we report results in terms of the difference between the
donor and acceptor populations, $\sigma_{z}(t)=\sigma_{00}(t)-\sigma_{11}(t)$.

The comparison between the exact results obtained by solving the GQME on a
classical computer and results obtained by performing the quantum algorithm on
the QASM simulator is shown in Fig. 2. The QASM simulator results are in
excellent agreement with the exact results for all four models under
consideration. The small amplitude oscillations of the QASM-based results
around the exact results can be traced back to the inherent uncertainty
associated with projection measurements. These results validate the GQME-based
quantum algorithm and demonstrate its ability to reproduce results obtained
via the GQME-based classical algorithm.

To test the performance of the quantum algorithm on real quantum devices, we
also performed the simulations on the quantum computers provided by IBM
Quantum (IBM Q). The simulations were performed for models 3 and 4 on ibmq
quito and ibmq lima, respectively. Both devices are equipped with $5$ qubits
that have the same qubit connectivity and use IBM’s Falcon r4T processor with
the same architecture. In each simulation of a given model, three qubits were
used and 10 repeated experiments were performed. In a single experiment, 40
time steps are chosen at an equal spacing out of the 4000 time steps used in
the QASM simulations, i.e., the time step in each experiment is $100$ times
greater than the time step used in the QASM simulations as listed in Table 1.
The average $CX$ gate error and readout error are $(1.191\times
10^{-2},5.194\times 10^{-2})$ for the ibmq quito and $(1.032\times
10^{-2},2.834\times 10^{-2})$ for the ibmq lima as of the time of the
experiments. The quantum circuits are the same in both the QASM simulations
and the real machine simulations. The transpiled quantum gate counts for each
of the ${\cal U}_{{\cal G}^{\prime}}(t)$ superoperators are $153$ $R_{Z}$
gates, $98$ $\sqrt{X}$ gates, and $41$ $CX$ gates. The transpiling process is
done internally by the Qiskit package and examples of the quantum circuits can
be found in the SI.

The comparison between the GQME-generated exact results and real machine
simulations is given in Fig. 3. In the figure, the red dots are the average of
the 10 experiments and the error bars represent standard derivations of the 10
experiments. The simulations with the IBM Q quantum computers are seen to be
in qualitative agreement with the exact results, which demonstrate that the
GQME-based quantum algorithm can capture the essential features of the
dynamics even with the noise and errors within real quantum devices.

##  3 Discussion

The GQME-based quantum algorithm proposed herein substantially expands the
range of open quantum systems that can be simulated on a quantum computer. We
demonstrated the applicability and versatility of the algorithm with the
implementations on the IBM QASM quantum simulator and IBM quantum computers.

The results obtained via the QASM simulator were found to be highly accurate,
with the error that is inherently associated with the quantum projection
measurements giving rise to very slight deviations from the exact results. The
results obtained by implementing the simulation on the IBM Q quantum computers
were also found to be in reasonable qualitative agreement with the exact
results, although clearly the noise and errors inherent to the NISQ era
quantum machines negatively affected the accuracy [82]. For model 3, the
results obtained on the IBM quantum computer capture the general dynamical
trends [see Fig. 3(a)]. The fact that the quantum result vanishes
asymptotically rather than relaxes to the actual non-zero equilibrium value is
likely due to decoherence caused by the gate errors. It should be noted that
the quantity plotted is the donor-acceptor population difference. Thus, the
gate errors and system noise cause the computational basis states to be more
uniformly populated, so that the difference between them is diminished.
Indeed, the agreement with the exact results is better for model 4, which is
symmetrical and therefore corresponds to zero population difference at
equilibrium [see Fig. 3(b)].

There are several future directions of research that have the potential to
improve the performance of the quantum algorithm on the NISQ quantum machines.
One direction is to optimize the quantum circuit design and reduce the circuit
depth. In general, the shallower the circuit depth (in terms of the number of
elementary gates), the less the overall error in simulations on real quantum
devices. To this end, there have been many proposals for possible ways to
optimize the decomposition of unitary operations into elementary gate
sequences [83, 84, 85, 86]. One particularly interesting idea is to reduce the
circuit depth by adding additional qubits [87], which may be applicable to the
current model, as we have only used 3 qubits out of the 5 currently available
on freely accessible IBM quantum computers. Another method to mitigate the
errors on the IBM machine is dynamical decoupling (DD) [88, 89, 90, 91, 92].
This method employs pulses to suppress the system’s coupling with the
environment. Some recent implementations of DD on IBM machines improve the
fidelity of the overall performance [93, 94, 95]. Yet another direction is to
implement the circuit on high-dimensional qudit systems. Quantum computers
based on three-dimensional circuit quantum electrodynamics (3D cQED) microwave
cavities are particularly promising, as they feature unique quantum error
correction schemes [96, 97, 98] and longer coherence times [99, 100] than
standard superconducting quantum computers. Bosonic quantum computing
algorithms have also been recently shown to significantly reduce the number of
quantum gates required for the calculation of the Franck-Condon factors [101]
and dynamics of rhodopsin near conical intersections [102]. Lossless 3D cQED
systems have not yet been employed to simulate open quantum system dynamics.
An adaptation of the algorithm presented here to bosonic quantum computing
could therefore provide another way to efficiently simulate open quantum
system dynamics and demonstrate how qudit-based quantum architectures can
reduce the computational cost and enhance the accuracy of quantum simulations.

##  4 Methods

The GQME formalism provides a general framework for deriving the exact EoM for
any quantity of interest. The derivation begins with the Nakajima-Zwanzig
equation [67, 68], which describes the dynamics of a projected state ${\cal
P}\hat{\rho}(t)$, where ${\cal P}$ is a projection superoperator and
$\hat{\rho}(t)$ is the density operator of the overall system:

$\displaystyle\frac{d}{dt}\mathcal{P}\hat{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\mathcal{P}\mathcal{L}\mathcal{P}\hat{\rho}(t)-\frac{1}{\hbar^{2}}\int_{0}^{t}d\tau\mathcal{P}\mathcal{L}e^{-i\mathcal{QL}\tau/\hbar}\mathcal{Q}\mathcal{L}\mathcal{P}\hat{\rho}(t-\tau)$
$\displaystyle\quad-\frac{i}{\hbar}\mathcal{P}\mathcal{L}e^{-i\mathcal{QL}t/\hbar}\mathcal{Q}\hat{\rho}(0).$

Here, ${\cal L}$ is the overall system-bath Liouvillian and ${\cal
Q}=\mathit{1}-{\cal P}$ is the complimentary projection superoperator to
${\cal P}$. Importantly, the only requirements are that ${\cal L}$ is
Hermitian and $\cal{P}$ satisfies ${\cal P}^{2}=\cal{P}$. Otherwise, there is
complete flexibility in the choice of ${\cal L}$ and ${\cal P}$, with each
choice leading to a different GQME for a different quantity of interest [30].

Following Ref. 27 we focus an overall system-bath Hamiltonian of the form of
Eq. (1) and the following choice of projection operator which gives rise to
the GQME for the system reduced density matrix, $\hat{\sigma}(t)$:

$\mathcal{P}(\hat{A})=\hat{\rho}_{n}(0)\otimes\text{Tr}_{n}\\{\hat{A}\\}.$
(15)

Plugging Eq. (15) into Eq. (4) and tracing over the nuclear (bath) Hilbert
space leads to the GQME in Eq. (3). The memory kernel in Eq. (3) is given by

$\mathcal{K}(\tau)=\frac{1}{\hbar^{2}}\text{Tr}_{n}\Big{\\{}\mathcal{L}\,e^{-i\mathcal{Q}\mathcal{L}\tau/\hbar}\mathcal{Q}\mathcal{L}\hat{\rho}_{n}(0)\Big{\\}},$
(16)

and can be obtained by solving the following Volterra equation [27]:

$\displaystyle\mathcal{K}(\tau)$
$\displaystyle=i\dot{\mathcal{F}}(\tau)-\frac{1}{\hbar}\mathcal{F}(\tau)\langle\mathcal{L}\rangle_{n}^{0}+i\int_{0}^{\tau}d\tau^{\prime}\mathcal{F}(\tau-\tau^{\prime})\mathcal{K}(\tau^{\prime}).$
(17)

Here, $\mathcal{F}(\tau)$ and $\dot{\mathcal{F}}(\tau)$ are the so-called
projection-free inputs (PFIs), which are given by

$\begin{split}\mathcal{F}(\tau)&=\frac{1}{\hbar}\text{Tr}_{n}\Big{\\{}\mathcal{L}e^{-i\mathcal{L}\tau/\hbar}\hat{\rho}_{n}(0)\Big{\\}},\\\
\dot{\mathcal{F}}(\tau)&=-\frac{i}{\hbar^{2}}\text{Tr}_{n}\Big{\\{}\mathcal{L}e^{-i\mathcal{L}\tau/\hbar}\mathcal{L}\hat{\rho}_{n}(0)\Big{\\}}.\end{split}$
(18)

The memory kernels for the spin-boson model used in this paper were adopted
from Ref. 80, where they were obtained from quantum-mechanically exact PFIs
calculated via the tensor-train thermo-field dynamics(TT-TFD) method.

The derivation of Eq. (5) begins with the definition of the time evolution
operator, Eq. (4). Plugging Eq. (4) in Eq. (3), we obtain:

$\displaystyle\frac{d}{dt}{\cal
G}(t)\hat{\sigma}(0)=-\frac{i}{\hbar}\langle\mathcal{L}\rangle_{n}^{0}{\cal
G}(t)\hat{\sigma}(0)-\int_{0}^{t}d\tau\ \mathcal{K}(\tau){\cal
G}(t-\tau)\hat{\sigma}(0).$ (19)

Then, by eliminating the $\hat{\sigma}(0)$ on both sides, we reach Eq. (5),
which was then solved for ${\cal G}(t)$ numerically via a Runge–Kutta fourth-
order (RK4) algorithm [28].

##  5 Acknowledgement

We acknowledge the financial support of the National Science Foundation under
award number 2124511, CCI Phase I: NSF Center for Quantum Dynamics on Modular
Quantum Devices (CQD-MQD). We acknowledge the use of IBM Quantum services for
this work. The views expressed are those of the authors and do not reflect the
official policy or position of IBM or the IBM Quantum team.

##  Appendix A Supplementary Information

###  A.1 Quantum circuit examples

In this section, we include further details concerning the quantum algorithm,
including the dilation process, circuit transpiling, QASM simulations, and
simulations running on the IBM quantum computers ibmq quito and ibmq lima. The
normalized time evolution operator of the electronic reduced density operator
${\cal G}^{\prime}(t)={\cal G}(t)/n_{c}$ (where ${\cal G}(t)$ is generated
from the GQME formalism) is dilated into a unitary operator ${\cal U}_{{\cal
G}^{\prime}}$(t). We start with ${\cal G}_{3}$, which corresponds to the
${\cal G}(t)$ of the $1500^{\mathrm{th}}$ time step from model 3, and ${\cal
G}_{4}$, which corresponds to the ${\cal G}(t)$ of the $1500^{\mathrm{th}}$
time step from model 4. The matrix of ${\cal G}_{3}$ and ${\cal G}_{4}$ are,
respectively:

${\cal
G}_{3}=\begin{pmatrix}0.38-$3.76\text{\times}{10}^{-10}$j&0.04+$2.90\text{\times}{10}^{-02}$j&0.04-$2.90\text{\times}{10}^{-02}$j&0.06-$1.88\text{\times}{10}^{-10}$j\\\
-0.13+$7.04\text{\times}{10}^{-02}$j&0.28-$2.63\text{\times}{10}^{-02}$j&0.02+$2.37\text{\times}{10}^{-02}$j&-0.15-$3.06\text{\times}{10}^{-02}$j\\\
-0.13-$7.04\text{\times}{10}^{-02}$j&0.02-$2.37\text{\times}{10}^{-02}$j&0.28+$2.63\text{\times}{10}^{-02}$j&-0.15+$3.06\text{\times}{10}^{-02}$j\\\
0.62+$3.77\text{\times}{10}^{-10}$j&-0.04-$2.90\text{\times}{10}^{-02}$j&-0.04+$2.90\text{\times}{10}^{-02}$j&0.94+$1.87\text{\times}{10}^{-10}$j\end{pmatrix},$

(A.1)

and

${\cal
G}_{4}=\begin{pmatrix}0.54+$4.7\text{\times}{10}^{-11}$j&$-1.7\text{\times}{10}^{-06}$+$5.7\text{\times}{10}^{-02}$j&$-1.6\text{\times}{10}^{-06}$-$5.6\text{\times}{10}^{-02}$j&0.46+$7.1\text{\times}{10}^{-11}$j\\\
-0.46+$5.7\text{\times}{10}^{-02}$j&$3.6\text{\times}{10}^{-02}$+$6.1\text{\times}{10}^{-05}$j&$-1.6\text{\times}{10}^{-02}$-$5.7\text{\times}{10}^{-05}$j&-0.46-$5.7\text{\times}{10}^{-02}$j\\\
-0.46-$5.7\text{\times}{10}^{-02}$j&$-1.6\text{\times}{10}^{-02}$+$5.7\text{\times}{10}^{-05}$j&$3.7\text{\times}{10}^{-02}$-$6.1\text{\times}{10}^{-05}$j&-0.46+$5.7\text{\times}{10}^{-02}$j\\\
0.54-$4.7\text{\times}{10}^{-11}$j&$1.6\text{\times}{10}^{-06}$-$5.6\text{\times}{10}^{-02}$j&$1.6\text{\times}{10}^{-06}$+$5.6\text{\times}{10}^{-02}$j&0.54-$7.1\text{\times}{10}^{-11}$j\end{pmatrix}.$

(A.2)

The normalization factors used for model 3 and model 4 are $n_{c3}=1.376$ and
$n_{c4}=1.376$.

Following the $1$-dilation process, the $4\times 4$ ${\cal G}^{\prime}(t)$
[derived from corresponding ${\cal G}(t)$ divided by the $n_{c}$ factor] is
converted into a unitary $8\times 8$ ${\cal U}_{{\cal G}^{\prime}}(t)$. We
show ${\cal U}_{{\cal G}^{\prime}_{3}}$ and ${\cal U}_{{\cal G}^{\prime}_{4}}$
in the form of heat maps in Fig. 4.

(a)

(b)

(c)

(d)

Figure 4: Heat map illustrations of the dilated $8\times 8$ unitary matrix
${\cal U}_{{\cal G}_{3}}$ and ${\cal U}_{{\cal G}_{4}}$ for the
$1500^{\mathrm{th}}$ ${\cal G}(t)$ matrix, ${\cal G}_{3}$ at
$t=2.25\Gamma^{-1}$ for model 3 and ${\cal G}_{4}$ at $t=6.75\Gamma^{-1}$ for
model 4.

The unitary operation ${\cal U}_{{\cal G}^{\prime}}(t)$ is transpiled into a
$3$-qubit quantum circuit composed of three elementary quantum gates: $R_{Z}$,
$\sqrt{X}$, and $CX$, which have the matrix form:

$\displaystyle\begin{split}R_{Z}(\lambda)&=\exp\left(-i\frac{\lambda}{2}Z\right)=\begin{pmatrix}e^{-i\frac{\lambda}{2}}&0\\\
0&e^{i\frac{\lambda}{2}}\end{pmatrix}\end{split}$ (A.3)
$\displaystyle\begin{split}\sqrt{X}&=\frac{1}{2}\begin{pmatrix}1+i&1-i\\\
1-i&1+i\end{pmatrix}\end{split}$ (A.4) $\displaystyle\begin{split}CX\
q_{0},q_{1}&=I\otimes|0\rangle\langle 0|+X\otimes|1\rangle\langle
1|=\begin{pmatrix}1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
0&1&0&0\end{pmatrix}\end{split}$ (A.5)

The full quantum circuits for ${\cal U}_{{\cal G}_{3}}$ and ${\cal U}_{{\cal
G}_{4}}$ are shown in Fig. 6 and 7. The probability distribution of the
projection measurement results of the two circuits are shown in Fig. 5. Both
the QASM simulator results and the real machine simulated results are
recorded.

(a)

(b)

(c)

(d)

Figure 5: Probability distribution of the quantum state after the projection
measurement applied to the circuit for ${\cal U}_{{\cal G}_{3}}$ and ${\cal
U}_{{\cal G}_{4}}$ on the QASM and real quantum devices. The $|000\rangle$
state corresponds to the population squared of the donor state
$\sigma_{DD}(t)$ and the $|100\rangle$ state corresponds to the population
squared of the acceptor state $\sigma_{AA}(t)$. The last four states are
ancilla states. Figure 6: Transpiled quantum circuit of the dilated ${\cal
U}_{{\cal G}_{3}}$ matrix at $1500$ time steps for model 3. Each horizontal
black line denotes a qubit. The $\sqrt{X}$ gate (blue square) is the square
root of $X$ gate; the $R_{z}$ gate (magenta square) is the rotation $Z$ gate.
The two-qubit gates are the controlled-NOT gate, where the dot denotes the
controlled qubit and $\bigoplus$ denotes the target qubit. The black gates at
the end of the circuit denote the projection measurements. The number of
required $R_{z}$, $\sqrt{X}$, and CNOT gates are $153,98$, and $41$,
respectively. Figure 7: Transpiled quantum circuit of the dilated ${\cal
U}_{{\cal G}_{4}}$ matrix at $1500$ time steps for model 4. Each horizontal
black line denotes a qubit. The $\sqrt{X}$ gate (blue square) is the square
root of $X$ gate; the $R_{z}$ gate (magenta square) is the rotation $Z$ gate.
The two-qubit gates are the controlled-NOT gate, where the dot denotes the
controlled qubit and $\bigoplus$ denotes the target qubit. The black gates at
the end of the circuit denote the projection measurements. The number of
required $R_{z}$, $\sqrt{X}$, and CNOT gates are $153,98$, and $41$,
respectively.

###  A.2 Amplitude-damping model

In this section, we will show that the method of flattening the density matrix
for dilation, which was outlined in the supplementary information in our
previous publication [64], can be implemented on the quantum device for the
simple amplitude damping model. The same method is verified by the
implementation described in this and the following section. This verification
allows us to confidently incorporate the general algorithm with GQME.

The general algorithm for open quantum system dynamics is applicable to the
time-evolution of density matrices governed by Kraus operators [64]. The time-
evolution representation for such open systems is given by
$\hat{\rho}(t)=\sum\limits_{k}{{{\cal{M}}_{k}(t)}\hat{\rho}{{{\cal{M}}_{k}^{\dagger}}(t)}}$.
For simplicity, the notation of time dependency is omitted hereafter for
superoperators ${\cal{M}}$ and superoperators derived from it. In the first
step, the density matrix is flattened to vector form:
$\hat{\rho}\to{{\mathbf{v}}_{\rho}}={{\left({{\rho}_{11}},...,{{\rho}_{1n}},{{\rho}_{21}},...,{{\rho}_{2n}},...\,...,{{\rho}_{n1}},...,{{\rho}_{nn}}\right)}^{T}}$.
We calculate the Frobenius norm of $\mathbf{v}_{\rho}$ as
$\left\|{{\mathbf{v}}_{\rho}}\right\|_{F}=\sqrt{\sum\limits_{ij}{{{\left|{{\rho}_{ij}}\right|}^{2}}}}$
and divide $\mathbf{v}_{\rho}$ by $\left\|{{\mathbf{v}}_{\rho}}\right\|_{F}$
to normalize $\mathbf{v}_{\rho}$. Next, for every $k$, the ${{\cal{M}}_{k}}$
is transformed into ${{\cal{\tilde{M}}}_{k}}={{\cal{M}}_{k}}\otimes{I}$;
similarly, the ${\cal{M}}_{k}^{\dagger}$ is transformed into
${{\cal{\tilde{N}}}_{k}}={I}\otimes{{\cal{\bar{M}}}_{k}}$. The $\otimes$
stands for the Kronecker product and the bar over ${{\cal{M}}_{k}}$ indicates
complex conjugation. The new equivalent form for the Kraus representation is:

${{\cal{M}}_{k}}\hat{\rho}{\cal{M}}_{k}^{\dagger}\overset{\text{equivalent}}{\longleftrightarrow}{{\tilde{\cal
N}}_{k}}{{\tilde{\cal M}}_{k}}{{\mathbf{v}}_{\rho}}.$ (A.6)

The input state is initialized to the normalized ${{\mathbf{v}}_{\rho}}$ in
the execution. In Ref. 64, it is shown that the Kraus operator
${{\cal{M}}_{k}}$ is a contraction. Therefore ${{\tilde{\cal
M}}_{k}}={{\cal{M}}_{k}}\otimes{I}$ and ${{\tilde{\cal
N}}_{k}}={I}\otimes{{\cal{\bar{M}}}_{k}}$ are also contractions as per the
norm property of the Kronecker product. To build the quantum circuit of
${{\tilde{\cal N}}_{k}}{{\tilde{\cal M}}_{k}}{{\mathbf{v}}_{\rho}}$ with
unitary gates, we need two separate 2-dilations:

${{\tilde{\cal N}}_{k}}{{\tilde{\cal
M}}_{k}}{{\mathbf{v}}_{\rho}}\xrightarrow{\text{unitary
dilation}}{{\mathbf{\cal{U}}}_{{{\cal{N}}_{k}}}{\mathbf{\cal{U}}}_{{{\cal{M}}_{k}}}}{{\left(\mathbf{v}_{\rho}^{T},0,...,0\right)}^{T}}.$
(A.7)

For ${{\cal{M}}_{k}}$ of dimension $n\times n$, ${{\tilde{\cal M}}_{k}}$ and
${{\tilde{\cal N}}_{k}}$ are ${{n}^{2}}\times{{n}^{2}}$; and consequently, the
2-dilations ${{\mathbf{\cal{U}}}_{{{\cal M}_{k}}}}$ and ${{\mathbf{\cal
U}}_{{{\cal N}_{k}}}}$ are $3{{n}^{2}}\times 3{{n}^{2}}$. The ${{\mathbf{\cal
U}}_{{{\cal M}_{k}}}}$ and ${{\mathbf{\cal U}}_{{{\cal N}_{k}}}}$ are
fragmented into sequences of two-level unitary gates and tallied to compute
the gate complexity. To realize ${{\tilde{\cal N}}_{k}}{{\tilde{\cal
M}}_{k}}{{\mathbf{v}}_{\rho}}$, the total gate complexity is
$3{{n}^{3}}+{{n}^{2}}$ for each $k$. The classical complexity to realize
${{{\cal M}}_{k}}\hat{\rho}{\cal M}_{k}^{\dagger}$ based on a naive algorithm
is higher, though of same order as quantum algorithm, namely
$4{{n}^{3}}-2{{n}^{2}}$.

All the evolved density matrices in the circuit calculated at each timestep
are obtained as the output vector
${{\mathbf{v}}_{k}}\left(t\right)={{\tilde{\cal N}}_{k}}{{\tilde{\cal
M}}_{k}}{{\mathbf{v}}_{\rho}}$. The desired information to be collected from
the density matrix is extracted by applying projection measurements on
${{\mathbf{v}}_{k}}\left(t\right)$ using an optical setup [103]. The detailed
procedure for obtaining information located at both diagonal and off-diagonal
elements of ${{\hat{\rho}}_{k}}\left(t\right)$ from final
${{\mathbf{v}}_{k}}\left(t\right)$ is described in the supplementary
information of Ref. 64.

###  A.3 Simulation of the amplitude damping model with Kraus operators

We tested the theory mentioned in the previous section for spontaneous
emission of a 2-level atom modeled by amplitude-channel damping. The
corresponding Lindblad master equation is:

${{\mathbf{\dot{\hat{\rho}}}}}\left(t\right)={\gamma}\left[{\sigma}^{+}\hat{\rho}(t){\sigma}^{-}-\frac{1}{2}\\{{\sigma}^{-}{\sigma}^{+},\hat{\rho}(t)\\}\right],$

where the spontaneous emission rate is
$\gamma=1.52\times{{10}^{9}}\;{{\text{s}}^{-1}}$, and the
${\sigma}^{+}=|0\rangle\langle 1|$ and
${\sigma}^{-}={({\sigma}^{+})}^{\dagger}$ are Pauli raising and lowering
operators, respectively. The density matrix $\rho(t)$ in the Kraus
representation is as follows:

$\displaystyle\begin{array}[]{ccc}\hat{\rho}(t)&=&{{\cal{M}}}_{0}(t)\hat{\rho}{{\cal{M}}}_{0}{(t)}^{\dagger}+{{\cal{M}}}_{1}{(t)}\hat{\rho}{{\cal{M}}}_{1}{(t)}^{\dagger},\\\
{{\cal{M}}}_{0}{(t)}&=&\frac{1+\sqrt{{e}^{-\gamma
t}}}{2}{\bf{I}}+\frac{1-\sqrt{{e}^{-\gamma
t}}}{2}{\sigma}_{z}=(\begin{array}[]{cc}1&0\\\ 0&\sqrt{{e}^{-\gamma
t}}\end{array}),\\\ {{\cal{M}}}_{1}{(t)}&=&\sqrt{1-{e}^{-\gamma
t}}{\sigma}^{+}=(\begin{array}[]{cc}0&\sqrt{1-{e}^{-\gamma t}}\\\
0&0\end{array}).\end{array}$ (A.15)

For ${{{\cal M}}_{k}}$ of dimension $2\times 2$, ${{\tilde{\cal M}}_{k}}$,
${{\tilde{\cal N}}_{k}}$, and ${\cal D}_{A}$ are $4\times 4$ matrices, as
given below in Eq. (LABEL:eq:MNDmats). In this way, the 2-dilations
${{\mathbf{\cal U}}_{{{\cal M}_{k}}}}$ and ${{\mathbf{\cal U}}_{{{\cal
N}_{k}}}}$ are $12\times 12$ following the $k$-dilation [64, 73]. Note that,
though these superoperators are time dependent, only for simplicity we omitted
the notation of time dependency. However, realization of the dilated matrices
using quantum gate is of dimension of form ${2}^{n}\times{2}^{n}$. We append
the dilated matrix with an ancillary $12\times 4$ zero matrix on the right and
$4\times 12$ at the bottom, and an $4\times 4$ identity matrix along the
diagonal. The resulting dilated superoperator matrix is $16\times 16$,
requiring 4 qubits for quantum implementation. Quantum implementation is
accomplished with Qiskit as mentioned in the main text.

$\displaystyle{{\tilde{\cal M}}_{0}}=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\sqrt{{{e}^{-\gamma t}}}&0\\\ 0&0&0&\sqrt{{{e}^{-\gamma t}}}\\\
\end{matrix}\right),\quad{{\tilde{\cal N}}_{0}}=\left(\begin{matrix}1&0&0&0\\\
0&\sqrt{{{e}^{-\gamma t}}}&0&0\\\ 0&0&1&0\\\ 0&0&0&\sqrt{{{e}^{-\gamma t}}}\\\
\end{matrix}\right),\quad$ $\displaystyle{{\mathbf{\cal D}}_{{{\tilde{\cal
M}}_{0}}}}=\left(\begin{matrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&\sqrt{1-{{e}^{-\gamma
t}}}&0\\\ 0&0&0&\sqrt{1-{{e}^{-\gamma t}}}\\\
\end{matrix}\right),\quad{{\mathbf{\cal D}}_{{{\tilde{\cal
N}}_{0}}}}=\left(\begin{matrix}0&0&0&0\\\ 0&\sqrt{1-{{e}^{-\gamma t}}}&0&0\\\
0&0&0&0\\\ 0&0&0&\sqrt{1-{{e}^{-\gamma t}}}\\\ \end{matrix}\right).$
$\displaystyle{{\tilde{\cal
M}}_{1}}=\left(\begin{matrix}0&0&\sqrt{1-{{e}^{-\gamma t}}}&0\\\
0&0&0&\sqrt{1-{{e}^{-\gamma t}}}\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{matrix}\right),\text{ }{{\tilde{\cal
N}}_{1}}=\left(\begin{matrix}0&\sqrt{1-{{e}^{-\gamma t}}}&0&0\\\ 0&0&0&0\\\
0&0&0&\sqrt{1-{{e}^{-\gamma t}}}\\\ 0&0&0&0\\\ \end{matrix}\right),\text{ }$
$\displaystyle{{\mathbf{\cal D}}_{{{\tilde{\cal
M}}_{1}}}}=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\sqrt{{{e}^{-\gamma
t}}}&0\\\ 0&0&0&\sqrt{{{e}^{-\gamma t}}}\\\ \end{matrix}\right),\text{
}{{\mathbf{\cal D}}_{{{\tilde{\cal N}}_{1}}}}=\left(\begin{matrix}1&0&0&0\\\
0&\sqrt{{{e}^{-\gamma t}}}&0&0\\\ 0&0&1&0\\\ 0&0&0&\sqrt{{{e}^{-\gamma t}}}\\\
\end{matrix}\right).$

For an initial density $\hat{\rho}(0)=\frac{1}{4}\left(\begin{matrix}1&1\\\
1&3\\\ \end{matrix}\right)$, we calculate the populations in the basis
$\\{|0\rangle,|1\rangle\\}$ from $t=0$ to $t=1000\text{ ps}$ with a time step
of 10 ps. With ${{\left\|\hat{\rho}\right\|}_{HS}}=\frac{\sqrt{3}}{2}$, the
input state is:

${{\mathbf{v}}_{0}}=\frac{1}{{{\left\|\hat{\rho}\right\|}_{HS}}}{{\left(\mathbf{v}_{\rho}^{T},\overbrace{0,...,0}^{m}\right)}^{T}}=\frac{1}{2\sqrt{3}}{{\left(1,1,1,3,\overbrace{0,...,0}^{m}\right)}^{T}},$
(A.17)

where $m=12$ for the vector $\mathbf{v}_{\rho}^{T}$ to be of length 16. After
extracting the output ${{\mathbf{v}}_{k}}\left(t\right)$, the ground state and
excited state populations are obtained as the first and fourth entry of the
vector, respectively. The Fig. 8 result manifests the consistency with the
result in Ref. 64.

Figure 8:  Population of ground state and excited state for the amplitude-
damping model obtained by the quantum implementation on the IBM Qiskit
simulator.

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#  $\infty$-CATEGORY AND SOME APPLICATIONS ON ORBIFORDS

JIAJUN DAI<EMAIL_ADDRESS>
##  1 Introduction

This paper in progress is motivated by a question about a given orbispace $X$
whether there exists a global presentation $X=Y/G$ for $G$ a compact Lie group
(such an orbispace is usually called a global quotient).
Lück–Oliver [LO01] made that if $X=Y/\Gamma$ for a discrete group $\Gamma$,
then $X$ is a global quotient; Henriques–Metzler [HM04] gave that every
paracompact smooth effective $n$-dimensional orbifold is a global quotient (of
its orthonormal frame bundle by $O(n)$), and a sufficient condition for a
general (not necessarily effective) orbifold to be a global quotient;
Henriques [Hen05] conjectured that every compact orbispace is a global
quotient; the analogous question for algebraic stacks has been studied by
Edidin–Hassett–Kresch–Vistoli [EHKV01] and Totaro [Tot04]; Kresch–Vistoli
([KV04] theorem2) ([Kre09] theorem4.4) (using an unpublished result of Gabber)
proved that smooth separated Deligne–Mumford stacks are global quotients;
Pardon [Par19] showed that all orbispaces satisfying very mild hypotheses are
global quotients and particularly that all compact orbispaces are global
quotients which verifies Henriques’ conjecture.
Since an orbifold can be viewed as a groupoid which is also the fundamental
groupoid of an $\infty-category$, a naive idea to investigate orbifold lies in
$\infty-category$ theory [Lur21] [Lur09]; on the other hand, as an orbifold is
also a geometric/topological stack which takes values in groupoid, we could
even extend higher stack and derived stack theories respectively to higher
orbifold to obtain higher structure in orbifold and to derived orbifold to
approximate an arbitrary but not too bad orbifold. Moreover, derived orbifold
would become a stable $\infty-category$, allowing us to study its stability
via either stable $\infty-category$ theory or stable homotopy theory or even
equivariant homotopy theory.
Here is mainly about an early result that orbifold stack is globally
representable via some $\infty-categorical$ techniques.

##  2 Preliminary $\infty-category$ theory

In the setting of the category of topological spaces, homotopy could be
intuitively viewed as a “path” between two continuous maps $f_{1}$ and $f_{2}$
between two topological spaces, recording that $f_{1}$ could be “deformed” to
$f_{2}$ or the other way around, but not being able to tell us to what extent
the “deformation” is. To obtain such information, a try could be made to
regard the homotopy as a “path space”, and consider the “path”, which could
also be viewed as homotopy and be called $2-homotopy$ to distinguish from the
formal homotopy that we would call it $1-homotopy$ or $2-map$ (ordinary
continuous map be called $1-map$), between two “path spaces” $p_{1}$ and
$p_{2}$ which are literally homotopies, thus $2-homotopy$, called $3-map$
inductively, could form a new “2-path spaces”, encoding more information about
the “deformation or derivation” between two $1-homotopies$ $p_{1}$ and
$p_{2}$. Inductively, we could define $n-homotopy$ which could be viewed as
“n-path spaces” between two $(n-1)-maps$ $h_{1}$ and $h_{2}$ to record the
“derivation” between $h_{1}$ and $h_{2}$, where we call it the derivation of
“degree $n-1$” in an extremely non-strict way. When $n$ goes to $\infty$, we
might get all homotopies involved.
As for approaches to realize the trial mentioned above, the Homotopy
Hypothesis proposed by Grothendieck has been nominated that the $\infty-
groupoids$ are equivalent to the topological spaces (considered modulo weak
homotopy equivalence), and $\infty-groupoids$ are equivalent to homotopy
$n-types$ for all extended natural numbers and moreover this equivalence is
induced by the fundamental $\infty-groupoid$ construction. A well-known model
for $\infty-groupoids$ in the context of higher category is $Kan$ complex:
specifically, $Kan$ complex is homotopy equivalent to the singular complex of
a topological space. Another approach to develop higher category is based on
simplicial sets which serve as models for homotopy types: every $CW$ complex
is homotopy equivalent to the geometric realization of a simplicial set.
Therefore, it is convenient to relate $Kan$ complex with simplicial set to
deal with homotopy theories combinatorically.

######  Definition 2.1.

A $Kan$ complex is a simplicial set $K$ which has the following property: for
any $0\leq i\leq n$, any map $f_{0}:\Lambda_{i}^{n}\rightarrow K$ admits an
($Kan$) extension $f:\Delta^{n}\rightarrow K$ or equivalently
$K_{n}\rightarrow\Lambda_{i}^{n}(K)$ is surjective for $0\leq i\leq n$.

######  Example 2.1.

If $X$ is a topological space, then its singular complex $Sing(X)$ is a $Kan$
complex; $\mathcal{G}$ is a groupoid object, if and only if $N(\mathcal{G})$
is a $Kan$ complex where $N$ is the simplicial nerve functor.

For a category $\mathcal{C}$, its simplicial nerve is characterized by the
following formulation:
$N(\mathcal{C})_{n}:\operatorname{Fun}(\overbrace{\cdot\rightarrow\cdot\rightarrow...\rightarrow\cdot}^{n+1\>\text{copies}},\mathcal{C})=\left\\{X_{0}\rightarrow
X_{1}\rightarrow...\rightarrow X_{n},X_{i}\in\mathcal{C}\right\\}$, whose
$i$th face map is obtained by composing the adjacent two morphisms in the ith
position $X_{i-1}\rightarrow X_{i}\rightarrow X_{i+1}$ and $i$th degeneracy
map by inserting the identity $X_{i}\rightarrow X_{i}$. It is well known that
there is a pair of adjoint functors $|-|\dashv N$, where $|-|$ is the
geometric realization functor.

######  Remark 2.1.

$Kan$ complex is a simplicial model category which is enriched over the
category of simplicial sets $\text{Set}_{\Delta}$ endowed with Joyal model
structure. To illustrate the model structure of $Kan$ complex, the three
distinguished classes of morphisms are:

  * •

cofibration: monomorphism which has the left lifting property with respect to
all trivial $Kan$ fibrations and whose collection is weak saturated (closed
under pushout, retract and transfinite composition); acyclic/trivial
cofibration: anodyne morphism which is defined to have the left lifting
property with respect to all $Kan$ fibrations.

  * •

fibration: $Kan$ fibration which has the right lifting property with respect
to all horn inclusion $\Lambda_{i}^{n}\rightarrow\Delta^{n}$ for all $n>0$ ,
$0\leq i\leq n$, and is closed under pullback, retract, composition and
filtered colimit; acyclic/trivial fibration: trivial $Kan$ fibration which has
the right lifting property with respect to all boundary inclusion
$\partial\Delta^{n}\rightarrow\Delta^{n}$ for all $n>0$.

  * •

weak equivalence: weak homotopy equivalence which could be easily checked to
possess the ‘two out of three’ property.

Thus, the weak factorization system of $Kan$ complex is (anodyne morphisms,
$Kan$ fibrations) and (monomorphisms, trivial $Kan$ fibrations).
This model structure implies that the fibrant objects of $\text{Set}_{\Delta}$
are precisely the $\infty-categories$.

######  Definition 2.2.

An $\infty-category$ is a simplicial set $K$ which has the following property:
for any $0<i<n$, any map $f_{0}:\Lambda^{n}_{i}\rightarrow K$ admits an (weak
$Kan$) extension $f:\Delta^{n}\rightarrow K$ or equivalently
$K_{n}\rightarrow\Lambda^{n}_{i}(K)$ is surjective for $0<i<n$.

Particularly, objects of an $\infty-category$ are the $0-simplices$; the
morphisms are the $1-simplices$; the horn-fillers exhibit not only the
homotopies, i.e. horn-fillers of higher degree represent higher homotopies,
but also the associator and unit of the composition law in higher category.
Furthermore, for any objects $x,y\in\mathcal{C}$, the mapping space
$\operatorname{Map}_{\mathcal{C}}(x,y)$ is the $Kan$ complex whose
$n-simplices$ are maps $\Delta^{n}\times\Delta^{1}$ to $\mathcal{C}$ which
sends $\Delta^{n}\times\left\\{0\right\\}$ to the vertex $x$ and
$\Delta^{n}\times\left\\{1\right\\}$ to the vertex $y$.

######  Remark 2.2.

$\infty-category$, as named by Lurie, is literally, or more precisely a model
for, $(\infty,1)-category$ (whose $k-morphisms$ are invertible for $k\geq 2$)
which was called quasi-category by Joyal and first referred to as weak $Kan$
complex by Boardman-Vogt. Other models for $(\infty,1)-category$ like Segal
category, Rezk category (also known as complete Segal category) are equivalent
to $\infty-category$. These models are employed to keep track of all
homotopies as demonstrated before.

######  Example 2.2.

Obviously, any $Kan$ complex is an $\infty-category$, so is $Sing(X)$ for a
topological space $X$; for an ordinary category $\mathcal{C}$,
$N(\mathcal{C})$ is an $\infty-category$, moreover, $S$ is isomorphic to the
nerve of an ordinary category $\mathcal{C}$ if and only if, for each $0<i<n$,
the map $S_{n}\rightarrow\Lambda^{n}_{i}(S)$ bijective; the Eilenberg-Mac Lane
space $K(A,n)$ for an abelian group $A$ is an $\infty-category$ for $n\geq 1$.

######  Proposition 2.1 ([Lur21]).

For an $\infty-category$ (respectively $Kan$ complex) $\mathcal{C}$, for a
simplicial set ⋅, $\operatorname{Fun}(S_{\cdot},C)$ is again an $\infty-
category$(respectively $Kan$ complex).

In particular, if $\mathcal{D}$ is an $\infty-category$ (respectively $Kan$
complex),
$\operatorname{Fun}(\mathcal{D},\mathcal{C}):=\operatorname{Map}_{\text{Set}_{\Delta}}(\mathcal{D},\mathcal{C})$
is an $\infty-category$ (respectively $Kan$ complex), which is referred to as
the $\infty-category$ of functors and whose set of $n-simplices$ are by
definition
$\operatorname{Hom}_{\text{Set}_{\Delta}}(\mathcal{D}\times\Delta^{n},\mathcal{C})$.
This property ensures that $\infty-category$ and $Kan$ complex are internal.

######  Definition 2.3.

For an $\infty-category\>\mathcal{C}$, an object $X\in\mathcal{C}$ is  final
(respectively initial) if for any $Y\in\mathcal{C}$,
$\operatorname{Map}_{\mathcal{C}}(Y,X)$ (respectively
$\operatorname{Map}_{\mathcal{C}}(Y,X)$) is contractible.

######  Definition 2.4.

For simplicial sets $S,S^{\prime}$, their join $S\star S^{\prime}$ is
characterized by the formulation $(S\star S^{\prime})_{n}:S_{n}\bigcup
S^{\prime}_{n}\bigcup\cup_{i+j=n-1}S_{i}\times S^{\prime}_{j}$.

It can be easily seen that $\Delta^{i}\star\Delta^{j}$ is isomorphic to
$\Delta^{i+j+1}$. Moreover, for two ordinary categories $\mathcal{C}$ and
$\mathcal{C^{\prime}}$, $N(\mathcal{C}\star\mathcal{C^{\prime}})$ is
isomorphic to $N(\mathcal{C})\star N(\mathcal{C^{\prime}})$.

######  Proposition 2.2 (Joyal).

The join of two $\infty-categories$ is an $\infty-category$.

######  Notation 2.1.

For simplicial set $K$, the left cone $K^{\vartriangleleft}$ is defined to be
the join $\Delta^{0}\star K$ . Dually, the right cone $K^{\vartriangleright}$
is defined to be the join $K\star\Delta^{0}$. Either cone contains a
distinguished vertex (belonging to $\Delta^{0}$ ), which will be referred to
as the cone point.

######  Proposition 2.3 (Joyal).

If $p:K\rightarrow S$ is a map of simplicial sets, then there exists a
simplicial set $S_{/p}$ with the following universal property:
$\operatorname{Hom}_{\text{Set}_{\Delta}}\left(Y,S_{/p}\right)=\operatorname{Hom}_{p}(Y\star
K,S)$, where the subscript on the right hand side means that the hom subset
consists of only morphisms $f:Y\star K\rightarrow S$ such that $f|_{K}=p$.

######  Notation 2.2.

For above map $p$, if S is an $\infty-category$, $S_{/p}$ denotates an
overcategory of $S$ or the $\infty-category$ of objects of $S$ over $p$;
dually, replacing $Y\star K$ by $K\star Y$, $S_{p/}$ denotes an undercategory
of $S$ or the $\infty-category$ of objects of $S$ under $p$.

######  Definition 2.5 (Limits and colomits).

Let $\mathcal{C}$ be an $\infty-category$ and let $p:K\rightarrow\mathcal{C}$
be an arbitrary map of simplicial sets. A colimit for $p$ is an initial object
of $\mathcal{C}_{p/}$ , and a limit for p is a final object of
$\mathcal{C}_{/p}$ .

A colimit diagram is a map
$\bar{p}:K^{\vartriangleright}\rightarrow\mathcal{C}$ which is a colimit of
$P=\bar{p}|_{K}$ . In this case, $\bar{p}(\infty)\in\mathcal{C}$ is referred
to as a colimit of $p$ where $\infty$ denotes the cone point of
$K^{\vartriangleright}$; similarly, a limit diagram is a map
$\bar{p}:K^{\vartriangleleft}\rightarrow\mathcal{C}$ which is a limit of
$P=\bar{p}|_{K}$, notated $\bar{p}(-\infty)\in\mathcal{C}$ where $-\infty$
denotes the cocone point of $K^{\vartriangleleft}$.

##  3 Orbifolds

An orbifold is locally the quotient of a manifold by a finite group. A more
general notion is the orbispace which is locally the quotient of a space by a
group. There are two conventional approaches to orbifolds: one is via
(topological) groupoid objects, and the other is via (topological) stacks.
Both orbifold groupoid and orbifold stack can be added new ingredients of and
reinterpreted by $\infty-category$ theory.
We assume that every orbifold $\mathcal{X}$ of concern has finite stablizers
in $\mathcal{F}$ and fix a family $\mathcal{F}$ of allowed isotropy groups
that is an essentially small class of paracompact topological groups which are
closed under isomorphisms.

######  Definition 3.1.

A (left) action of groupoid $\mathbb{G}$ on a space
$X\rightarrow\mathbb{G}_{0}$ over $\mathbb{G}_{0}$ is exhibited by a map a:
$\mathbb{G}_{1}\times_{\mathbb{G}_{0}}X\rightarrow X$ satisfying the
associative and unital conditions that the following diagrams commute

$\begin{array}[]{ccc}\mathbb{G}_{1}\times_{\mathbb{G}_{0}}\mathbb{G}_{1}\times_{\mathbb{G}_{0}}X&\stackrel{{\scriptstyle
m\times 1}}{{\longrightarrow}}&\mathbb{G}_{1}\times_{\mathbb{G}_{0}}X\\\
\downarrow^{1\times a}&&\downarrow^{a}\\\
\mathbb{G}_{1}\times_{\mathbb{G}_{0}}X&\stackrel{{\scriptstyle
a}}{{\longrightarrow}}&X\end{array}$
$\begin{array}[]{ccc}\mathbb{G}_{0}\times_{\mathbb{G}_{0}}X&\stackrel{{\scriptstyle
i\times 1}}{{\longrightarrow}}&\mathbb{G}_{1}\times_{\mathbb{G}_{0}}X\\\
\downarrow^{\simeq}&&\downarrow^{a}\\\
X&\stackrel{{\scriptstyle=}}{{\longrightarrow}}&X\end{array}$

We can briefly define this action groupoid as $\mathbb{G}\ltimes
X:=(\mathbb{G}_{1}\times_{\mathbb{G}_{0}}X\rightrightarrows X)$. Such space
$X\rightarrow\mathbb{G}_{0}$ (or briefly referred to as $X$) equipped with a
(left) action of $\mathbb{G}$ is called a $\mathbb{G}-space$.
Roughly speaking, spaces over $\mathbb{G}_{0}$ could be viewed as a continuous
$\mathbb{G}_{0}-indexed$ family of spaces and a left action of $\mathbb{G}$ on
a space over $\mathbb{G}_{0}$ as a continuous contravariant functor from
$\mathbb{G}$ to spaces.
For a space $U\rightarrow\mathbb{G}_{0}$, the restriction $\mathbb{G}_{U}$ of
$\mathbb{G}$ to $U$ consists of object space $U$ and arrow space
$U\times_{\mathbb{G}_{0}}\mathbb{G}_{1}\times_{\mathbb{G}_{0}}U$.

######  Definition 3.2.

A cellular groupoid $\mathbb{G}$ is a topological groupoid of the form
$\text{colim}\>\mathbb{G}_{\alpha}$ whose structure is built as follows:

$\begin{array}[]{ccc}S^{n-1}\times\mathbb{O}&\longrightarrow&D^{n}\times\mathbb{O}\\\
\downarrow^{f}&&\downarrow\\\
\mathbb{G}_{\alpha-1}&\longrightarrow&\mathbb{G}_{\alpha}\end{array}$

where $\mathbb{G}_{n}:=\operatorname{map}(\Delta^{n},\mathbb{G})$; $\alpha$
runs through the set of ordinals smaller than a given ordinal; $\mathbb{O}$ is
an orbit groupoid with the action groupoid of the form $G\ltimes(G/H)$ for
$G/H$ being a $G-orbit$ with allowed isotropy group; $\mathbb{G}_{\alpha}$ is
obtained from $\mathbb{G}_{\alpha-1}$ via the pushout of the cell groupoid
$D^{n}\times\mathbb{O}$ along an attaching map $f$ if $\alpha-1$ exists, and
as the colimit $\varinjlim_{\beta<\alpha}\mathbb{G}_{\beta}$ otherwise.

######  Definition 3.3.

Given a topological groupoid $\mathbb{G}$ , a subspace
$Z\subset\mathbb{G}_{0}$ is called saturated if there are no arrows from $Z$
to its complement. A saturated inclusion of topological groupoids is a map of
the form $\mathbb{G}_{Z}\rightarrow\mathbb{G}$ such that
$Z\subset\mathbb{G}_{0}$ is a saturated subspace and $\mathbb{G}_{Z}$ is the
restriction of $\mathbb{G}$ to $Z$.

The map $S^{n-1}\times\mathbb{O}\rightarrow D^{n}\times\mathbb{O}$ associated
with a cellular groupoid is a closed saturated inclusion which can easily be
checked to be compatible with cofibrations in simplicial model category
$\text{Set}_{\Delta}$.
Define a cofibrant groupoid to be a retract of a cellular groupoid, it is
still cellular.

######  Proposition 3.1 (Gepner-Henriques[GH07]).

Given topological $\mathbb{H}$ and $\mathbb{G}$ , there is a natural
equivalence of groupoids

$\\{Principal\mathbb{G}-bundleson\mathbb{H}\\}\simeq\operatorname{hocolim}_{U\in\operatorname{Cov}(\mathbb{H}_{0})}\operatorname{\mathbb{H}om}(\mathbb{H}_{U},\mathbb{G})$

, where $\operatorname{Cov}(\mathbb{H}_{0})$ denotes the cover of
$\mathbb{H}_{0}$.

This proposition tells us that we could recover the principal bundle on a
certain groupoid $\mathbb{H}$ that has a global section by gluing local
sections of this principal bundle with respect to a well-refined(cofinal)
cover of the underlying space $\mathbb{H}_{0}$ of $\mathbb{H}$.

######  Definition 3.4.

A topological groupoid $\mathbb{G}$ is fibrant if
$\mathbb{G}_{1}\rightarrow\mathbb{G}_{0}$ is a universal $\mathbb{G}-bundle$.
A retract of a fibrant groupoid is still fibrant.

######  Definition 3.5.

The fibrant replacement $\text{fib}(\mathbb{G})$ of a topological groupoid
$\mathbb{G}$ is the gauge groupoid of the universal principal
$\mathbb{G}-bundle\>||\mathbb{EG}||\rightarrow||\mathbb{G}||$; that is,
$\text{fib}(\mathbb{G}):=\text{gauge
}(||\mathbb{EG}||)=((||\mathbb{EG}||\times_{\mathbb{G}_{0}}||\mathbb{EG}||)/\mathbb{G}\rightrightarrows||\mathbb{G}||)$
, where $||-||$ is the fat geometric realization functor, $\mathbb{EG}$ the
translation groupoid associated to $\mathbb{G}$ with the action groupoid
$\mathbb{G}\ltimes\mathbb{G}_{1}$.

Fibrant replacement preserves pushouts along closed saturated inclusions which
means it behaves well in the operation of gluing cells. Specifically, the
fibrant replacement of a cellular groupoid is cellular.

######  Proposition 3.2 (Gepner-Henriques[GH07]).

Let $\mathbb{G}$ be any topological groupoid. Then $\text{fib}(\mathbb{G})$ is
fibrant.

Furthermore, we can adapt cellular groupoid to be simplicial groupoid.

######  Definition 3.6.

Let
$\mathbb{G}^{n}:=\overbrace{\mathbb{G}_{1}\times_{\mathbb{G}_{0}}\mathbb{G}_{1}\times_{\mathbb{G}_{0}}..._{\mathbb{G}_{0}}\mathbb{G}_{1}}^{n\>\text{copies}},\mathcal{C})$
, then $\mathbb{G}^{n}\hookrightarrow\mathbb{G}^{n+1}$ are saturated
inclusions. In this case, we can define
$\mathbb{G}:=\text{colim}(\mathbb{G}^{1}\hookrightarrow\mathbb{G}^{2}\hookrightarrow...)$
to be a simplicial groupoid.

A simplicial groupoid is naturally cellular, thus its cofibrant object and
fibrant replacement could be defined in the same fashion as those of a
cellular groupoid and inherit the latters’ properties.
On the other hand, since a simplicial groupoid is by defibition an $\infty-
category$, it is fibrant subsequently.

######  Definition 3.7.

A lax presheaf of groupoids $\mathcal{X}:=\operatorname{Fun}(Top^{op},Grpd)$
on $Top$, the category of topological spaces, is a topological stack if, for
each object $T$ of Top and each cover $U\rightarrow T$ of $T$, the natural map

$\mathcal{X}(T)\rightarrow\operatorname{holim}\left\\{\mathcal{X}(U)\rightrightarrows\mathcal{X}\left(U\times_{T}U\right)\leavevmode\hbox
to9.94pt{\vbox to7.49pt{\pgfpicture\makeatletter\hbox{\hskip
1.19998pt\lower-1.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{8.07591pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.07591pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{8.07591pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.07591pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{5.69046pt}\pgfsys@lineto{8.07591pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.07591pt}{5.69046pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\mathcal{X}\left(U\times_{T}U\times_{T}U\right)\right\\}$

is an equivalence of groupoids.

Let a functor $\mathcal{M}:\\{\text{Topological
groupoids}\\}\rightarrow\\{\text{Topological stacks}\\}$ send the topological
groupoid $\mathbb{G}$ to the stack
$\mathcal{M}_{\mathbb{G}}:\\{\text{Topological
spaces}\\}\rightarrow\\{\text{Groupoids}\\}$ which is defined by
$\mathcal{M}_{\mathbb{G}}(T):=\\{\text{Principal }\mathbb{G}\text{ -bundles on
}T\\}$.

It can be easily seen that The functor $\mathcal{M}_{\mathbb{G}}$ is the
stackification of the presheaf of groupoids $\operatorname{Hom}(-,\mathbb{G})$
and that the functor $\operatorname{Hom}(-,\mathbb{G})$ classifies trivial
principal $\mathbb{G}-bundles$ up to isomorphism. What’s more, if $\mathbb{G}$
is fibrant, then $\operatorname{Hom}(-,\mathbb{G})$ is a stack.

######  Proposition 3.3 (Gepner-Henriques[GH07]).

Let $\mathbb{H}$ be a topological groupoid with paracompact object space. Then
for any topological groupoid $\mathbb{G}$, there is a natural equivalence of
groupoids
$\operatorname{Hom}(\mathbb{H},\text{fib}\mathbb{G})\simeq\mathcal{M}_{\mathbb{G}}(\mathbb{H})$.

Roughly speaking, we can simplify this proposition with a slogan that fibrant
topological groupoids are stacks. For simplicial groupoids are fibrant, we can
soon get the result:

######  Corollary 3.1.

Let $\mathbb{H},\mathbb{G}$ be a pair of simplicial groupoids where
$\mathbb{H}$ is paracompact,
$\operatorname{Hom}(\mathbb{H},\text{fib}\mathbb{G})\simeq\mathcal{M}_{\mathbb{G}}(\mathbb{H})$.

######  Theorem 3.1.

For $\mathbb{G}$ an orbifold groupoid, $\mathcal{M}_{\mathbb{G}}$ is globally
representable.

Sketch of the proof
Step 1: Slice
For a stabilizer $H_{x_{i}}$ in $FlowerF$, where $x_{i}$ is the corresponding
fixed point, there is an action groupoid $\mathbb{G}\ltimes H_{x_{i}}$,
denoted ${}_{i}\mathbb{H}$. Then ${}_{i}\mathbb{H}$ can be constructed by
gluing cells one by one in finite times for $H_{x_{i}}$ is finite thus can be
obtained in a simplicial way, i.e. it is a simplicial groupoid. By corollary
3.1, for each action groupoid ${}_{i}\mathbb{H}$ , there is a (sub)stack
$\mathcal{M}_{\mathbb{G}}(_{i}\mathbb{H})$.
Step 2: Join
By definition, $\mathcal{M}_{\mathbb{G}}(_{i}\mathbb{H})$is principal
$\mathbb{G}-bundles$ on $(_{i}\mathbb{G})_{0}$ with an (left) action
$\mathbb{G}\ltimes Hx_{i}$, thus can be recovered with respect to a collection
of cofinal covers of $(_{i}\mathbb{H})_{0}$ by proposition 3.1. Since
${}_{i}\mathbb{H}$ is simplicial, we can reason in two directions: simplicial
groupoids ${}_{i}\mathbb{H}$ for all $i$ can be joined to obtain a new
simplicial groupoid $\mathbb{H}$; such covers
$\operatorname{Cov}((_{i}\mathbb{H})_{o})$ need to be simplicial thus cofinal
covers become $\infty-categories$. Join
$\operatorname{Cov}((_{i}\mathbb{H})_{o})$ for all $i$ whose number is finite
to get a cover $\operatorname{Cov}$ of the underlying object space
$\mathbb{H}_{0}$, then $\operatorname{Cov}$ would be again $\infty-category$
by proposition 2 to be cofinal. Hence, we can join substacks
$\mathcal{M}_{\mathbb{G}}(_{i}\mathbb{H})$ to obtain a new stack
$\mathcal{M}_{\mathbb{G}}(\mathbb{H})$ by proposition 3.1.

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