These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Qiming Shao

[1]Hong Kong University of Science and Technology, Hong Kong

# Cloud-based Semi-Quantum Money

Yichi Zhang<EMAIL_ADDRESS>Siyuan Jin<EMAIL_ADDRESS>Yuhan Huang<EMAIL_ADDRESS>Bei Zeng<EMAIL_ADDRESS><EMAIL_ADDRESS>*

###### Abstract

In the 1970s, Wiesner introduced the concept of quantum money, where quantum
states generated according to specific rules function as currency. These
states circulate among users with quantum resources through quantum channels
or face-to-face interactions. Quantum mechanics grants quantum money physical-
level unforgeability but also makes minting, storing, and circulating it
significantly challenging. Currently, quantum computers capable of minting and
preserving quantum money have not yet emerged, and existing quantum channels
are not stable enough to support the efficient transmission of quantum states
for quantum money, limiting its practicality. Semi-quantum money schemes
support fully classical transactions and complete classical banks, reducing
dependence on quantum resources and enhancing feasibility. To further minimize
the system’s reliance on quantum resources, we propose a cloud-based semi-
quantum money (CSQM) scheme. This scheme relies only on semi-honest third-
party quantum clouds, while the rest of the system remains entirely classical.
We also discuss estimating the computational power required by the quantum
cloud for the scheme and conduct a security analysis.

###### keywords:

quantum money, post-quantum cryptography, cloud computation, quantum finance

##  1 Introduction

###  1.1 Research Background

In recent years, advancements in quantum computing and information theory have
renewed interest in quantum money. Researchers have developed new models, such
as public-key quantum money, which leverage the complexity of quantum problems
to ensure security. Progress in qubit technologies, including trapped ions,
superconducting circuits, and photonic systems, demonstrates the feasibility
of creating and controlling quantum states with high precision.

However, practical quantum money faces several challenges. Quantum states are
fragile and susceptible to decoherence, often leading to information loss.
Scalability is another major issue, as current quantum systems are small and
prone to errors.

The concept of semi-quantum money introduces a new type of quantum money that
keeps classical verifiability but changes the minting process. Unlike
traditional quantum money, where the bank mints the money using quantum
algorithms and sends it via quantum channels, semi-quantum money involves a
protocol where both the bank and the user collaborate using classical
resources.

In semi-quantum money, the user generates the quantum money state using
instructions provided by the bank. This process is designed to ensure that the
user cannot replicate the state to create additional money. The verification
of semi-quantum money is conducted using classical interactive protocols
between the user and the bank, avoiding the need for quantum communication or
computation during verification.

Table 1: Comparison of Quantum Money Protocol  Protocol |  | Bank
---
Node
| Client
---
Node
| Bank-Client
---
Channel
| Client-Client
---
Channel
Cloud
[1] | Quantum | Quantum | Quantum | Quantum | No
[2] | Quantum | Quantum | Quantum | Quantum | No
[3] | Classical | Quantum | Classical | Quantum | No
[4] | Classical | Quantum | Classical | Classical | No
Our Work | Classical | Classical | Classical | Classical | Yes

Table 1 shows the comparison of different quantum money protocols. Quantum
money have been an actively researched topic for several decades. Public-key
quantum money were first conceptualized by [1] in the 1960s, where there is a
bank that generates a secret-basis quantum state allowing easy verification of
the issued quantum money. The quantum money is unique and can not be
replicated without access to secret serial numbers. However, the system
requires that users have quantum computers and quantum channels with the banks
to verify each transaction.

Many researchers have subsequently suggested different quantum money schemes,
aiming to facilitate verification by any entity possessing a quantum device
[2, 5, 6]. Unfortunately, the methods proposed by [2] and [5] were completely
broken [7, 8]. For example, [6] sought to resolve these issues by proposing a
modification, which incorporated an indistinguishability obfuscator into the
protocol. Despite this, [9] highlighted that the protocol’s hardness
assumption was flawed.

In a subsequent response, [3] proposed that an ideal quantum token should rely
on local quantum computation and classical communication, obviating the need
for quantum communication. To this end, they integrated a homomorphic
encryption technique into the quantum money mining process, thereby
facilitating secure and feasible local mining.

Our approach provides a significant departure from purely quantum money
schemes, as it leverages classical resources for both minting and verification
while maintaining security properties comparable to quantum systems.

###  1.2 Motivation

At the current technological level, the cost of manufacturing and maintaining
quantum computers is extremely high, making it impractical for ordinary people
to own and use them directly. Looking ahead, as quantum computers gradually
enter practical use, cloud quantum computing services provided by a few large
institutions are likely to become mainstream. This trend is similar to the
evolution of classical computers: from early large centralized computers to
later personal computers, which took several decades. By analogy, using cloud
quantum computers to support the use of semi-quantum money could greatly
accelerate its practical adoption.

However, the high dependence of semi-quantum money on security poses new
challenges. Traditional security mechanisms may appear fragile in the face of
quantum computing. Therefore, ensuring that cloud-based semi-quantum money is
not compromised by unreliable third parties becomes an urgent issue.
Specifically, we need to explore whether there is a solution that, under the
premise of ensuring security, can utilize third-party quantum cloud services
along with classical users, banks, and communication channels to implement a
feasible cloud-based semi-quantum money system.

The core of this paper is to address this question. We will first review the
basic concepts of semi-quantum money and the existing technical challenges,
then analyze the historical experience of the evolution from centralized
computing to personal computing, and explore the potential of cloud quantum
computing in promoting the practical use of semi-quantum money. On this basis,
we will design and verify a secure cloud-based semi-quantum money scheme.
Through theoretical analysis and experimental verification, we aim to propose
a feasible solution to lay the foundation for the widespread application of
semi-quantum money.

###  1.3 Results

Our main result is the protocol of the cloud-based semi-quantum money (CSQM).
We first formally propose the specific steps of this protocol and use a small
4-qubit demonstration to help understand it. Then, we conducte a rigorous
analysis of the security of this protocol based on information-theoretic
security and estimate the feasibility of this protocol based on the current
hardware level.

##  2 Preliminary

This section introduces the cryptography primitives and notions borrowed from
the original papers [10, 4, 11].

###  2.1 Quantum Leveled Fully Homomorphic Encryption

  * •

$fhek\leftarrow QHE.KeyGen(1^{\lambda},1^{L})$: A classical probabilistic
polynomial-time (PPT) algorithm samples a classical secret key $fhek$ from
given security parameter $\lambda\in\mathbb{N}$ and circuit depth
$L\in\mathbb{N}$.

  * •

$Z^{z}X^{x}\ket{\Psi}:=\ket{\Psi}^{x,z}\leftarrow QHE.QOTP_{z,x}(\ket{\Psi})$:
A quantum polynomial-time algorithm given random string
$x,z\in\\{0,1\\}^{\lambda}$ and a $\lambda$-qubit quantum state $\ket{\Psi}$,
outputs $Z^{z}X^{x}\ket{\Psi}:=\ket{\Psi}^{x,z}$.

  * •

$x\oplus m\leftarrow QHE.OTP_{x}(m)$: A classical algorithm encrypts message
$m\in\\{0,1\\}^{*}$ with $x$ that $|m|=|x|$.

  * •

$ct_{x}\leftarrow QHE.Enc_{fhek}(x)$: A classical PPT algorithm encrypts
bit(s) in classical tensor $x$ bit-wisely and outputs the classical ciphertext
$ct_{x}$.

  * •

$(C\ket{\Psi})^{x^{\prime},z^{\prime}},ct_{x^{\prime},z^{\prime}}\leftarrow
QHE.Eval((\ket{\Psi}^{x,z},ct_{x,z}),C)$: A quantum polynomial-time algorithm
with input a general quantum circuit $C$, a quantum one-time-pad encrypted
state $\ket{\Psi}^{x,z}$ and the ciphertext $ct_{x,z}$. The algorithm will
output a new quantum state $(C\ket{\Psi})^{x^{\prime},z^{\prime}}$ and a
classical ciphertext $ct_{x^{\prime},z^{\prime}}$.

  * •

$x\leftarrow QHE.Dec_{fhek}(ct_{x})$: A PPT algorithm uses the secret key
$fhek$ to decrypt the ciphertext $ct_{x}$ bit-wisely.

###  2.2 Quantum Homomorphic Encryption Circuit

We use quantum fully-homomorphic encryption (QFHE) [10] as the cryptography
instantiation, which is based on the standard hardness assumption of learning
with errors (LWE). The classical sender sends a one-time-padded quantum state
(can be described classically) $\ket{\psi}^{(x,z)}$, the ciphertext
$ct_{(x,z)}$ of the one-time-pad and a general quantum circuit $C$ to a
receiver with a quantum computer. Then the receiver can output
$(C\ket{\psi})^{(x^{\prime},z^{\prime})}$ along with the ciphertext
$ct_{(x^{\prime},z^{\prime})}$. Since a general quantum circuit can be divided
into layers of Pauli, Clifford and Toffoli gates. If the circuit $C$ contains
only Pauli and Clifford gates, the transform from $ct_{x,z}$ to
$ct_{x^{\prime},z^{\prime}}$ is determinate [12]. And after applying Toffoli
gate, the transformation of $ct_{x,z}$ is random. Since the process is non-
trivial, we will introduce the QFHE Toffoli gate in the next section. Our
protocol will make use of the randomness to mint the anti-counterfeit quantum
state as semi-quantum money.

####  2.2.1 Encrypted CNOT Operation

Encrypted CNOT operation is the key module of QFHE Toffoli operation. The
sender samples a pair of injective functions $f_{0}(\mu,r),f_{1}(\mu,r)$ as
the encryption of a bit $s$ and a 2-qubit quantum state
$\ket{\psi}=\ket{a,b}$, then send them to the receiver. The pair of
$f_{0}(\mu,r),f_{1}(\mu,r)$ has below properties:

  1. 1.

Injective

  2. 2.

Easy to be convert with a trapdoor $t_{A}$

  3. 3.

Computationally difficult to find any pair of preimages
$(\mu_{0},r_{0}),(\mu_{1},r_{1})$ with the same image
$f_{0}(\mu_{0},r_{0})=f_{1}(\mu_{1},r_{1})=y$ without $t_{A}$

  4. 4.

For every $f_{0}(\mu_{0},r_{0})=f_{1}(\mu_{1},r_{1})=y$,
$\mu_{0}\oplus\mu_{1}=s$

Figure 1 shows the quantum circuit of the encrypted CNOT operation and the
abstract form that will be referred in further diagram. In the end of the
encrypted CNOT operartion, the receiver will output the quantum state
$(CNOT^{s}\ket{\psi})^{(x,z)}$ and $ct_{(x,z)}$, the ciphertext of the pad
$(x,z)$.

Figure 1: Top: the quantum circuit of encrypted CNOT gate defined in [10].
Lower left: the abstract form of the encrypted CNOT operation. Lower right:
the simplest form of the encrypted CNOT operation, which will be used in other
diagrams.

####  2.2.2 Toffoli Operation on Encrypted Quantum State

When applying a Toffoli gate directly on a OTP quantum state
$Z^{z}X^{x}\ket{\psi}$ we can get
$TZ^{z}X^{x}\ket{\psi}=(TZ^{z}X^{x}T^{\dagger})T\ket{\psi}$, where
$TZ^{z}X^{x}T^{\dagger}=TZ^{z_{1}}X^{x_{1}}\otimes Z^{z_{2}}X^{x_{2}}\otimes
Z^{z_{3}}X^{x_{3}}T^{\dagger}$.
And $TZ^{z_{1}}X^{x_{1}}\otimes Z^{z_{2}}X^{x_{2}}\otimes
Z^{z_{3}}X^{x_{3}}T^{\dagger}=CNOT^{x_{2}}_{1,3}CNOT^{x_{1}}_{2,3}\hat{Z}^{z_{3}}_{1,2}Z^{z_{1}+x_{2}z_{3}}X^{x_{1}}\otimes
Z^{z_{2}+x_{1}z_{3}}X^{x_{2}}\otimes Z^{z_{3}}X^{x_{1}x_{2}+x_{3}}$. Based on
the expansion, we apply the encrypted CNOT gates (Figure 2) to reveal the
Pauli layer $T\ket{\psi}^{(x^{\prime},z^{\prime})}$ along with the new
ciphertext $ct_{(x^{\prime},z^{\prime})}$.

Figure 2: Left: the quantum circuit of the QFHE Tofolli operation. Right: the
abstract form of QFHE Tofolli operation.

###  2.3 Indistinguishability Obfuscation

We call a PPT algorithm $iO$ is an indistinguishability obfuscator over a
circuit class $\\{C_{\lambda}\\}$ that satisfied:

  * •

For all security parameters $\lambda\in\mathbb{N}$, all $C\in C_{\lambda}$ and
all input $x$:

$Pr[O(x)=C(x):O\leftarrow iO(\lambda,C)]=1$

  * •

For any PPT distinguisher $D$, all pairs $C_{0},C_{1}\in C_{\lambda}$, there
exists a negligible function $negl$ that satisfied:

$\displaystyle|Pr[D(iO(\lambda,C_{0}))=1]$ $\displaystyle-
Pr[D(iO(\lambda,C_{1}))=1]|$ $\displaystyle\leq negl(\lambda)$

##  3 Main Result

###  3.1 Cloud-based Semi-quantum Money Protocol

Our paper aims to design a practical cloud-based semi-quantum money protocol.
We design the protocol in the scenarios of token issuance, redemption, and
circulation (shown in Figure 3). Token issuance and redemption involve a token
issuer, which we use a classical bank to represent.

The bank and users are classical, meaning they can only perform classical
computations and do not possess any quantum capabilities locally. This is a
realistic assumption in the NISQ era where quantum computers are limited. To
enable quantum functionalities, users and the bank can outsource quantum
computations to the untrusted quantum cloud server. The cloud provider has a
universal quantum computer but is not trusted. We assume it will follow the
protocol honestly but try to learn information. Our protocol uses quantum
homomorphic encryption to maintain privacy against the cloud. By clearly
defining the roles in our scenario, we can see both the bank and token holders
are classical without needing a local quantum computer.

(a) Mint

(b) Redeem

(c) Circulation

Figure 3: Quantum Token Process.

Algorithm 1 shows how banks can mint a quantum token. This algorithm presents
a protocol for minting a single quantum token [3]. It involves three parties:
a classical bank (Bank), a classical receiver (Rec), and a quantum third-party
cloud server (Cloud). The goal of the algorithm is for the Bank to output a
public key (pk), the Receiver to output a secret key (sk), and the Cloud to
output a quantum state representing the quantum token (referred to as the
shifted quantum state).

Algorithm 1 Quantum Token Minting

1:A classical bank $Bank$, a classical receiver $Rec$, a quantum third-party
cloud server $Cloud$, and a security paramater $\lambda$

2:In the end of the protocol, $Bank$ will output
$pk:=(O_{S_{0}+x},O_{S_{0}+x+w},O_{S^{\perp}+z})$, $Rec$ will output {$R$,
$X_{R}$, $Z_{R}$}, and the $Cloud$ will output
$(\ket{\$}_{pk})^{(R,0)}:=(\ket{\psi}^{(x,z)})^{(R,0)}$

3:$Bank$ samples a random $\frac{\lambda}{2}$-dimensional subspace, described
by matrix: $M_{S}\leftarrow\\{0,1\\}^{\frac{\lambda}{2}\times\lambda}$.

4:$Bank$ encrypts the matrix with random OTP key
$p_{x}\leftarrow\\{0,1\\}^{\frac{\lambda}{2}\times\lambda}$,
$M_{S}^{(p_{x})}=QHE.OTP_{p_{x}}(M_{S})$

5:$Bank$ encrypts the OTP key with $fhek\leftarrow QHE.Gen(1^{\lambda})$,
$ct_{p_{x}}\leftarrow QHE.Enc_{fhek}(p_{x})$. Then sends
$(M_{S}^{(p_{x})},ct_{p_{x}})$ to $Rec$

6:$Rec$ samples shift key $R\leftarrow\\{0,1\\}^{\lambda}$, shifts per row in
$M_{S}^{(p_{x})}$, outputs $(M_{S+R}^{(p_{x})}:=(M_{S}^{(p_{x})})_{ij}\oplus
R_{j})$

7:$Rec$ sends $(M_{S+R}^{(p_{x})},ct_{p_{x}})$ to $Cloud$

8:$Cloud$ homomorphically evaluates
$C:(\ket{\psi+R}^{(x,z)},ct_{(x,z)})\leftarrow
QHE.Eval((M_{S+R}^{(p_{x})},ct_{p_{x}}),C)$ and sends back the access of
$(\ket{\psi}^{(x,z)})^{(R,0)}$ and $ct_{x,z}$ to $Rec$

9:$Cloud$ appends an extra qubit $\ket{0}$ in the end of
$(\ket{\psi}^{(x,z)})^{(R,0)}$

10:$Rec$ sends back $ct_{x,z}$ to $Bank$

11:$Bank$ decrypts $(x,z)=QHE.Dec_{fhek}(ct_{x,z})$.

12:if $x\in S$ then $Bank$ terminates the interaction

13:else$Banks$ outputs the indistinguishability obfuscations
$(O_{S_{0}+x},O_{S_{0}+x+w},O_{S^{\perp}+z})$ where $w$ is the first row of
$M_{S}$, $S_{0}\in\\{0,1\\}^{(\frac{\lambda}{2}-1\times\lambda)}$ is the
remain matrix of $M_{S}$ without $w$.

14:end if

15:$X_{R}$ = $(R;0)$

16:$Z_{R}$ = $0^{\lambda+1}$

Digital token has high-security requirements. The shifted quantum state
practice ensures that the cloud can not know the token information and operate
the token in certain ways without the permission of token holders. We achieve
this by using a two-phase quantum homomorphic encryption algorithm so that the
cloud can conduct a quantum computation circuit along with encrypted data.
Besides, replicating the same quantum token in a polynomial time is impossible
because the generation process in the second step has a randomness that even
the user can not produce the same quantum token with the information. The bank
would only recognize the quantum token with $c_{x,z}$ received.

Algorithm 2 shows how token holders can verify the authentication of quantum
tokens. The receiver needs to know whether their token is valid and they can
use the verification algorithm to conduct quantum computation with Oracle $pk$
that is outputted by the bank in the mining algorithm. The receiver can use
the Oracle to delegate computational tasks to the cloud. If the result of the
computation is 1, the token is valid. Otherwise, the token is not valid.

Algorithm 2 Quantum Token Verification

1:A classical receiver $Rec$ holds {$X_{R}$, $Z_{R}$}, and a quantum third-
party cloud server $Cloud$ stores
$(\ket{\$}_{pk})^{(X_{R},Z_{R})}:=(\ket{\psi}^{(x,z)})^{(X_{R},Z_{R})}$.

2:In the end of the protocol, $Rec$ will output a new {$X_{R}$, $Z_{R}$}, and
the $Cloud$ will output a new
$(\ket{\$}_{pk})^{(X_{R},Z_{R})}:=(\ket{\psi}^{(x,z)})^{(X_{R},Z_{R})}$

3:$Rec$ encrypts $X_{R}$, $Z_{R}$ with $reckey\leftarrow
QHE.Gen(1^{\lambda})$, $ct_{(X_{R},Z_{R})}\leftarrow
QHE.Enc_{reckey}((X_{R},Z_{R}))$ and sends $ct_{(X_{R},Z_{R})}$,
$(O_{S_{0}+x},O_{S_{0}+x+w},O_{S^{\perp}+z})$ to $Cloud$.

4:$Cloud$ homomorphically evaluates one of the oracle $O$ of
$((O_{S_{0}+x}\vee O_{S_{0}+x+w})\otimes I,(H^{\otimes\lambda}\otimes
I)(O_{S^{\perp}+z})(H^{\otimes\lambda}\otimes I))$:
$((\ket{\$}_{pk})^{(X^{\prime}_{R},Z^{\prime}_{R})},ct_{(X^{\prime}_{R},Z^{\prime}_{R})})\leftarrow
QHE.Eval(((\ket{\$}_{pk})^{(X_{R},Z_{R})},ct_{(X_{R},Z_{R})}),O)$ and sends
the measurement result $\mu$ of the final qubit and
$ct_{(X^{\prime}_{R},Z^{\prime}_{R})}$ to $Rec$

5:$Rec$ updates $X_{R}\leftarrow QHE.Dec_{reckey}(ct_{X^{\prime}_{R}})$

6:$Rec$ updates $Z_{R}\leftarrow QHE.Dec_{reckey}(ct_{Z^{\prime}_{R}})$

7:$Rec$ decrypts result $m\leftarrow QHE.Dec_{reckey}(\mu)$

8:if $m=1$ then Verification pass

9:elseVerification fails

10:end if

11:$Cloud$ and $Rec$ repeat to test the other oracle $O$

12:if Both verifications passed then $Rec$ accepts the state

13:else$Rec$ rejects the state

14:end if

The quantum signing algorithm Algorithm 3 allows a receiver to obtain a
classical signature from a quantum state stored by a cloud provider while
maintaining privacy. The receiver encrypts their secret keys and sends them to
the cloud. The cloud uses homomorphic encryption to evaluate and sign the
quantum state with a bit provided by the receiver. This enables the signing of
quantum states without exposing the keys or the quantum state itself. The
receiver decrypts the result to obtain the signature. This allows verifiable
signing while keeping the state private.

The quantum transaction algorithm 4 enables two users with quantum tokens in
separate clouds to transact. One user has a valued set of quantum tokens, as
konw as cloud-base semi-quantum money, while the other has a set dummy tokens.
By having the dummy state user send a unique bit string to the valued token
holder, the real token holder can sign each quantum token and verify ownership
to the other party without exposing the tokens. This allows value to be
transferred from one user’s tokens to the other, facilitated by the signature
verification, without the need for the bank or direct access to the quantum
states. Overall, the protocols allow for privacy-preserving transactions with
quantum tokens in untrusted clouds via homomorphic encryption technology.

Algorithm 3 Quantum Signing

1:A classical receiver $Rec$ holds {$X_{R}$, $Z_{R}$}, a bit $b\in\\{0,1\\}$,
and a quantum third-party cloud server $Cloud$ stores
$(\ket{\$}_{pk})^{(X_{R},Z_{R})}:=(\ket{\psi}^{(x,z)})^{(X_{R},Z_{R})}$

2:In the end of the protocol, $Rec$ will get a classical string
$\sigma_{b}\in\\{0,1\\}^{\lambda}$.

3:$Rec$ encrypts $X_{R}$, $Z_{R}$ with $reckey\leftarrow
QHE.Gen(1^{\lambda})$, $ct_{(X_{R},Z_{R})}\leftarrow
QHE.Enc_{reckey}((X_{R},Z_{R}))$ and sends $ct_{(X_{R},Z_{R})}$,
$(O_{S_{0}+x},O_{S_{0}+x+w},O_{S^{\perp}+z})$ to $Cloud$.

4:$Cloud$ homomorphically evaluates
$((\ket{\$}_{pk})^{(X^{\prime}_{R},Z^{\prime}_{R})},ct_{(X^{\prime}_{R},Z^{\prime}_{R})})\leftarrow
QHE.Eval(((\ket{\$}_{pk})^{(X_{R},Z_{R})},ct_{(X_{R},Z_{R})}),O_{S_{0}+x+bw}\otimes
I)$ and sends the measurement result $\mu$ of the final qubit to $Rec$

5:$Rec$ updates $X_{R}\leftarrow QHE.Dec_{reckey}(ct_{X^{\prime}_{R}})$

6:$Rec$ updates $Z_{R}\leftarrow QHE.Dec_{reckey}(ct_{Z^{\prime}_{R}})$

7:$Rec$ decrypts result $m\leftarrow QHE.Dec_{reckey}(\mu)$

8:if $m=1$ then $Cloud$ measures the register $\ket{\$}_{pk})^{(X_{R},Z_{R})}$
to get measurement $(\sigma_{b})^{\prime}\leftarrow\sigma_{b}\oplus X_{R}$,
sends $(\sigma_{b})^{\prime}$ to $Rec$ and terminate.

9:elsePerform Algo. 2 of the oracle $(H^{\otimes\lambda}\otimes
I)(O_{S^{\perp}+z})(H^{\otimes\lambda}\otimes I)$ and return to step 4

10:end if

11:$Rec$ calculate $\sigma_{b}\leftarrow(\sigma_{b})^{\prime}\oplus X_{R}$

Algorithm 4 Quantum Transaction

1:A classical sender $A$ who can access a $\lambda$-length sequence of quantum
token
$Token_{A}:=(\ket{\$_{1}}_{pk_{1}})^{(X_{R_{1}},Z_{R_{1}})},\ket{\$_{2}}_{pk_{2}})^{(X_{R_{2}},Z_{R_{2}})},...,\ket{\$_{\lambda}}_{pk_{\lambda}})^{(X_{R_{\lambda}},Z_{R_{\lambda}})})$
stored in $Cloud_{A}$ and $x_{A},s_{A},\sigma_{A}$, where $x_{A}$ is the value
of $Token_{A}$, $s_{A}\in\\{0,1\\}^{\lambda}$ is a unique bit string and
$\sigma_{A}$ is the classical signature of
$((pk_{1},pk_{2},...,pk_{\lambda})_{A},x_{A},s_{A})$. A classical receiver $B$
can access a dummy $\lambda$-length sequence of quantum token $Token_{B}$
stored in $Cloud_{B}$, $x_{B}=0$.

2:In the end of the protocol, the value $x_{A}$ of $Token_{A}$ will be added
to the value $x_{B}$ of $Token_{B}$.

3:$B$ sends $A$ the series number $s_{B}\in\\{0,1\\}^{\lambda}$

4:$A$ signs each token in $Token_{A}$ with the bit from $s_{B}$ respectively:
$\sigma_{B\leftarrow
A}:=(\sigma_{s_{B_{1}}},\sigma_{s_{B_{2}}},...,\sigma_{s_{B_{\lambda}}})_{Token_{A}}$
and sends the signature $\sigma_{B\leftarrow A}$ to $B$

To assign actual value to QT. The user will pay a certain price to the bank to
obtain a string of QT sequence of length $\lambda$ issued by the bank to the
user, a unique random sequence number $s$, the value of this string of QT $x$,
and a classic signature to prove the integrity of this string of QT $\sigma$
(that is to say, $\sigma$ can guarantee that this string of QT can be used
normally if and only if its serial number $s$ and value $x$ are all consistent
with the state when they were issued).

###  3.2 Mint a Quantum Token Unit

In this section, we give out the mint circuit of the basic unit of the quantum
token along with a 4-qubit demonstration for clear.

####  3.2.1 Encrypted Space Expansion Circuit

In Algo. 1, the cloud $C$ receives $(M_{S+R}^{(r)},ct_{r})$ and the mint
circuit $C_{m}$ from the classical receiver $R$. The mint circuit $C_{m}$
(Figure 4) is constructed by $\frac{\lambda}{2}$ blocks. Each block will add
one new basis into the space (described by a quantum state) and in the end of
the circuit, it will output the OTP encrypted space row span of $M_{S+R}$,
also can be written as
$(\sum_{b\in\mathbb{Z}_{2}^{\frac{\lambda}{2}}}\ket{(\oplus
S_{i}^{b_{i}})\oplus R,b})^{(x,z)}=\ket{\$\oplus R}$, where $S_{i}$ is the
$i$-th row of $M_{S}$. In addtion, the classical ciphertext $ct_{(x,z)}$
transforms into $ct_{(x^{\prime},z^{\prime})}$.

Figure 4: Mint Circuit of a Quantum Token Unit

####  3.2.2 4-Qubit Demonstration

In this section, we suggest a 4-qubit demonstration for clear description
(Figure 5). We set the parameters as:

  1. 1.

$\lambda=4$

  2. 2.

$M_{S}=\begin{bmatrix}0&0&0&1\\\ 0&0&1&0\end{bmatrix}$

  3. 3.

$p_{x}=1000$

  4. 4.

$R=1001$

Let’s review Algo. 1. The Bank sends Receiver
$M_{S}^{p_{x}}=\begin{bmatrix}1&0&0&1\\\ 1&0&1&0\end{bmatrix}$. And the
Receiver applies $R$ on $M_{S}^{p_{x}}$ to earn
$M_{S+R}^{p_{x}}=\begin{bmatrix}0&0&0&0\\\ 0&0&1&1\end{bmatrix}$ and send it
to the cloud $C$. After homomorphically run the mint circuit
$(\ket{\$+R},ct_{(1000,0110)}\leftarrow
QHE.Eval(C_{m},\begin{bmatrix}0&0&0&0\\\ 0&0&1&1\end{bmatrix},ct_{1000}))$,
the cloud $C$ will hold the $R$-encrypted quantum token
$\ket{\$+R}=(\ket{0000\oplus 1001,00}+\ket{0001\oplus 1001,01}+\ket{0010\oplus
1001,10}+\ket{0011\oplus 1001,11})^{(1000,0110)}$ and $ct_{(1000,0110)}$.

Figure 5: Demonstration of a 4-Qubit Quantum Token Unit

####  3.2.3 Quantum Verification and Quantum Signing

After mint process, the bank will public three oracles
$(O_{S_{0}+x},O_{S_{0}+x+w},O_{S^{\perp}+z})$ to advise everyone how to verify
the quantum token $\ket{\$}=\sum_{s\in S}\ket{s}^{(x^{\prime},z^{\prime})}$ by
the measurement results through the mint process [4]. In other words, since
the banks holds the trapdoor $t_{r}$, it can reveal the exact value of
$(x^{\prime},z^{\prime})$. We define $S_{0}$ is a $(\frac{\lambda}{2}-1)$-dim
subspace of $S$ and $S_{0}+w$ is the coset of $S_{0}$. In the quantum signing
protocol (Figure. 6, the sender provide an arbitrary bit $b\in\\{0,1\\}$ and
the prover who holds the quantum token $\ket{\$}$ can sign the token with $b$.
Finally, the prover outputs $\sigma_{b}=y\in S_{0}+b\cdot w+x$. In Algo. 3,
the measurement is homomorphically executed by the cloud $C$ with the
assistant of $R$.

Figure 6: Quantum Signing

In this section, we delve into the analysis of the security aspects of our
delegation procedure in Cloud Quantum Computation. Our foundational assumption
is that the cloud, denoted as $C$, acts as a semi-honest third party that
possesses quantum capabilities.

##  4 Discussion

###  4.1 Claim on Quantum State Acquisition

Under the assumption of the cloud $C$ is fault tolerated. We assert that after
our mint protocol, the cloud $C$ successfully acquires a specific set of
quantum states. These states are particularly significant because they exhibit
a negligible difference, in terms of trace distance, from a defined quantum
state denoted as $\ket{\Psi+R}^{(x,z)}$.

The bedrock of this claim lies in the statistical security inherent in Quantum
Fully Homomorphic Encryption (QFHE). This security assurance leads to a high
probability - essentially certainty - that the cloud’s register will indeed
contain quantum states. These states, as previously mentioned, will have a
negligible trace distance from our specified state $\ket{\Psi+R}^{(x,z)}$.

In the quantum token verification and signing part, receiver $Rec$ sends
$ct_{(X_{R},Z_{R})}$, $(O_{S_{0}+x},O_{S_{0}+x+w},O_{S^{\perp}+z})$ to $C$.
$C$ homomorphically evaluates one of the oracle $O$ of $((O_{S_{0}+x}\vee
O_{S_{0}+x+w})\otimes I,(H^{\otimes\lambda}\otimes
I)(O_{S^{\perp}+z})(H^{\otimes\lambda}\otimes I))$:
$((\ket{\$}_{pk})^{(X^{\prime}_{R},Z^{\prime}_{R})},ct_{(X^{\prime}_{R},Z^{\prime}_{R})})\leftarrow
QHE.Eval(((\ket{\$}_{pk})^{(X_{R},Z_{R})},ct_{(X_{R},Z_{R})}),O)$ and sends
the measurement result $\mu$ of the final qubit and
$ct_{(X^{\prime}_{R},Z^{\prime}_{R})}$ to $R$. If $C$ performs QFHE correctly,
the output state will have a negligible trace distance from
$\ket{\$}_{pk})^{(X_{R},Z_{R})}$.

###  4.2 Claim on the Secrecy of Shift $R$

A pivotal aspect of our protocol is the confidentiality of the shift, denoted
as $R\in\\{0,1\\}^{\lambda}$. Our claim here is that this shift remains a
secret throughout the entire protocol.

We consider a hypothetical scenario where the cloud $C$ can manipulate another
quantum token, say $\ket{\Psi^{\prime}}$, in such a way that it passes the
quantum verification process (QV) in polynomial time. In such a case, $C$
would be able to decrypt a randomly selected secret $S$. This possibility,
however, leads to a contradiction in terms of the security of one-time-pad,
thereby underscoring our claim that the shift $R$ remains confidential
throughout the process.

###  4.3 Estimation of Quantum Resources

We set the key size $\lambda$ is 256 bits to ensure the security of the
quantum token. The plain space expansion circuit of a $\frac{\lambda}{2}$-dim
space requires $2.5\lambda$ qubits. The Toffoli gate of QFHE in the circuit
needs at least $\lambda$ qubits. Therefore our protocol requires quantum
computation resources with $~{}900$ qubits.

## References

  * [1] Stephen Wiesner.  Conjugate coding.  ACM Sigact News, 15(1):78–88, 1983.
  * [2] Scott Aaronson.  Quantum copy-protection and quantum money.  In 2009 24th Annual IEEE Conference on Computational Complexity, pages 229–242. IEEE, 2009.
  * [3] Omri Shmueli.  Public-key quantum money with a classical bank.  In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 790–803, 2022.
  * [4] Omri Shmueli.  Semi-quantum tokenized signatures.  In Annual International Cryptology Conference, pages 296–319. Springer, 2022.
  * [5] Scott Aaronson and Paul Christiano.  Quantum money from hidden subspaces.  In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 41–60, 2012.
  * [6] Mark Zhandry.  Quantum lightning never strikes the same state twice. or: quantum money from cryptographic assumptions.  Journal of Cryptology, 34(1):1–56, 2021.
  * [7] Andrew Lutomirski, Scott Aaronson, Edward Farhi, David Gosset, Avinatan Hassidim, Jonathan Kelner, and Peter Shor.  Breaking and making quantum money: toward a new quantum cryptographic protocol.  arXiv preprint arXiv:0912.3825, 2009.
  * [8] Marta Conde Pena, Jean-Charles Faugère, and Ludovic Perret.  Algebraic cryptanalysis of a quantum money scheme the noise-free case.  In IACR International Workshop on Public Key Cryptography, pages 194–213. Springer, 2015.
  * [9] Bhaskar Roberts.  Security analysis of quantum lightning.  In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 562–567. Springer, 2021.
  * [10] Urmila Mahadev.  Classical homomorphic encryption for quantum circuits.  SIAM Journal on Computing, (0):FOCS18–189, 2020.
  * [11] Sanjam Garg, Craig Gentry, Shai Halevi, Mariana Raykova, Amit Sahai, and Brent Waters.  Candidate indistinguishability obfuscation and functional encryption for all circuits.  SIAM Journal on Computing, 45(3):882–929, 2016.
  * [12] Andrew M Childs.  Secure assisted quantum computation.  arXiv preprint quant-ph/0111046, 2001.


# Closing Pandora’s Box – The deepest X-ray observations of Abell 2744 and a
multi-wavelength merger picture

Urmila Chadayammuri E-mail<EMAIL_ADDRESS>Max Planck Institut für
Astronomie, Königstuhl 17, 69121 Heidelberg, Germany  Center for Astrophysics
— Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Ákos Bogdán
Center for Astrophysics — Harvard & Smithsonian, 60 Garden St, Cambridge, MA
02138, USA  Gerrit Schellenberger  Center for Astrophysics — Harvard &
Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  John ZuHone  Center for
Astrophysics — Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA

###### Abstract

Abell 2744, also known as Pandora’s Cluster, is a complex merging galaxy
cluster. While a major merger is clear along the north-south axis, the
dynamical state of the northwest subcluster has been highly uncertain. We
present ultra-deep ($\approx$2.1 Ms) X-ray observations of Abell 2744 obtained
with the Chandra X-ray Observatory and reinterpret the multi-wavelength
picture with a suite of idealised simulations of galaxy cluster mergers. The
new data reveal in unprecedented detail the disruption of cool cores in the
three X-ray luminous subclusters and confirm the presence of a shock to the
NW. A position-velocity clustering of the cluster member galaxies shows a
clearly separated S2 component, with a $\Delta z$ implying a separation of 53
Mpc or a line-of-sight velocity of $4500\ \rm{km\ s^{-1}}$, or likely some
combination of the two. While binary simulations allow NW to have undergone a
gravitational slingshot after the first pericenter passage, triple merger
simulations rule out this scenario, because the two mergers would have had to
occur $\sim$0.5 Gyr apart, and the joint impact of the shocks from the two
mergers would completely disrupt the SE and NW cool cores; they only reform
after 1-2 Gyr, by which point the core separations greatly exceed
observations. The scenario that best describes Abell 2744 is a head-on N-S
merger $0.5-0.6$ Gyrs ago followed by a first infall of the NW subcluster.
Furthermore, we note that a model with three cluster-size halos, with masses
consistent with gravitational lensing constraints, nevertheless produces a
lensing convergence and surface brightness lower than observed in most of the
field of view, whereas the temperatures are consistent with observations. This
suggests the presence of a large-scale overdensity, which contributes to the
diffuse emission and total surface density without heating the densest gas.

###### keywords:

galaxies: clusters: individual: Abell 2744 – galaxies: clusters: intracluster
medium – X-rays: galaxies: clusters – methods: numerical

##  1 Introduction

Galaxy clusters represent the latest stages of structure formation in a
hierarchical cosmology, such as the benchmark $\Lambda$CDM. Their large masses
mean that gravity is the dominant force at play, and cluster statistics can be
used for cosmology (Jauzac et al., 2016; Natarajan et al., 2017; Schwinn et
al., 2017; Mao et al., 2018; Kimmig et al., 2022). Because structure assembles
hierarchically, galaxy clusters are always undergoing mergers; when the mass
ratios are large, we start to observe the effects of astroparticle and plasma
physics in addition to cosmology. Offsets between dark matter, galaxy and gas
peaks can constrain the amount of self-interaction in the dark matter (Massey
et al., 2015), whereas the evolution of shock and cold fronts in the
intracluster medium (ICM) depends on its detailed physical properties, such as
its magnetization, viscosity, and thermal conduction.(Markevitch et al., 2003;
Chadayammuri et al., 2022b). The distribution of magnetic fields and
relativistic electrons determines the radio emission (Govoni et al., 2001;
Pfrommer et al., 2008). Last but certainly not least, merging clusters are
also the most powerful gravitational lenses in the Universe. This produces
detailed maps of the projected mass within the cluster (Jauzac et al., 2015;
Furtak et al., 2022; Bergamini et al., 2023b, a), as well as magnifying the
light from objects up to 12 billion light years away, almost to the beginning
of time (Wang et al., 2023; Bogdán et al., 2024; Kovacs et al., 2024; Ananna
et al., 2024).

Abell 2744 is one of the most dramatic mergers of galaxy clusters in the
observable Universe. Due to its many observed substructures and complex non-
equilibrium features, the system has been referred to as “Pandora’s Cluster”.
Gravitational lensing studies report up to a dozen substructures (Jauzac et
al., 2016; Kimmig et al., 2022), with five strong lensing peaks coincident
with the Brightest Cluster Galaxies (BCGs). The galaxy velocity distribution
functions are also complex, modelled at best as composites of multiple
Gaussians (Boschin et al., 2006; Owers et al., 2011). Radio observations show
an extended central halo with multiple radio relics, several located up to 1
Mpc away from the cluster center (Govoni et al., 2001; Pearce et al., 2017;
Paul et al., 2019; Rajpurohit et al., 2021).

Galaxy clusters are particularly luminous and well-resolved at X-ray
wavelengths, where emission from the ICM peaks. X-ray observations reveal
shock fronts, whose intrinsic strength depends on the 3D relative velocity of
the merging subclusters, and which are obscured in projection based on the
inclination of the merger axis with respect to the plane of the sky. Cold
fronts lie at the boundary of stripped cores of subclusters; their distance
from the nearest shock depends on the infall velocity, merger phase, and
inclination angle. Initially, the gas is subjected to ram pressure, whereas
the collisionless dark matter moves unhindered; here, the lensing peak would
be ahead of the gas peak. After about a dynamical time, however, the gas will
once again trace the gravitational field, and the two peaks will coincide.
Given enough shock and cold fronts, one can reconstruct most of the pre-merger
properties of the subclusters as well as the geometry of the merger.
Constrained idealized simulations offer a powerful method to interpret multi-
wavelength observations of complex galaxy cluster mergers because they allow
us to forward model the non-equilibrium dynamics of a system, and interpret
features like shock and cold fronts, merger-driven radio emission, and
asymmetric gravitational potentials (Maurogordato et al., 2011; van Weeren et
al., 2011; Roediger et al., 2013; Hu et al., 2021; Chadayammuri et al.,
2022a).

In Abell 2744, the situation is further complicated because there are at least
three X-ray peaks, five BCGs, and numerous lensing peaks depending on the
lensing study. The most robust claims so far are that of a merger along the
north-south axis in the plane of the sky from X-ray features (Kempner & David,
2004), and of a line-of-sight merger in the southeast from velocity
distributions (Boschin et al., 2006). The inferred mass ratios are of order
unity and 3:1, respectively. On the other hand, there is almost no consensus
on the nature of the “northwest interloper”. Kempner & David (2004) thought it
was falling into the main cluster, producing a shock front ahead of it, while
Owers et al. (2011) failed to find this shock with additional X-ray data; they
also claimed that galaxy spectra were not consistent with this picture. Merten
et al. (2011) proposed that the substructure has already passed through the
main cluster from the south-east and reached its apocenter, and is now
returning for a second passage, with the gas getting “slingshotted” outwards
while the dark matter core returns faster. Medezinski et al. (2016) finds this
interpretation unlikely in the simulations of Molnar et al. (2012), where the
first core passage typically strips the interloper of most of its gas.
Instead, they propose a scenario where the interloper itself has two merging
components along the line of sight, corresponding to the two nearby BCGs, NW-1
and NW-2; this merging system, they suggest, is falling into the rest of the
cluster. Furthermore, since Merten et al. (2011) posits that the main merger
is along the SE-NW axis, they interpret the northern core as having passed by
a now highly stripped substructure to the west, which now appears only as a
sharp peak in strong lensing and the X-ray.

Assembling a merger picture for galaxy clusters like Abell 2744 is a crucial
step toward using them as laboratories for both astrophysics and cosmology.
The offset between the gas and lensing peaks, for example, is often used to
constrain the self-interaction cross-section of dark matter (Merten et al.,
2011; Dawson et al., 2012; Jauzac et al., 2016), although such a separation is
also found in simulations of merging clusters with completely collisionless
dark matter (ZuHone, 2011; Wittman et al., 2018; Chadayammuri et al., 2022a);
in other words, the mere existence of an offset is not a smoking gun of self-
interaction. The separation varies throughout the merger and depends on the
merger geometry as well as gas core density profiles, and would need to be
very well-constrained before claims can be made about dark matter self-
interaction. The rate of disruption of cold fronts can constrain thermal
conductivity in the intracluster plasma (Ettori & Fabian, 2000; Markevitch et
al., 2003; Wang et al., 2016; Richard-Laferrière et al., 2023), while the
suppression of Kelvin-Helmholtz instabilities constrains the plasma viscosity
(Roediger et al., 2013; Russell et al., 2014; Wang & Markevitch, 2018) or the
magnetic field strength (Vikhlinin et al., 2001). All observations currently
point to a very strong suppression of viscosity and thermal conduction
compared to the prediction from Coulomb-only interactions (Spitzer, 1956);
this is expected in a weakly magnetised plasma (Lyutikov, 2006; Dursi &
Pfrommer, 2008; ZuHone et al., 2011; Roberg-Clark et al., 2018) and has been
used to constrain the strength of magnetic fields in the ICM (Vikhlinin et
al., 2001; Su et al., 2017). All of these measurements rely on knowing the gas
densities, temperatures and velocities at the boundary layers.

In this paper, we present the ultra-deep Chandra X-ray observations of Abell
2744. We place these data in a multi-wavelength context to motivate possible
merger scenarios, and finally test these with tailored simulations of each of
the proposed mergers. Where relevant, we simulate triple mergers and assess
how well they can be described by the superposition of binary mergers.
Ultimately, we present a narrative of two mergers that successfully reproduce
all the observations.

This paper assumes the Planck Collaboration et al. (2020) cosmology, i.e. a
flat Universe with $\Omega_{m}$ = 0.315 and $h$ = 0.673. This means at the
redshift $z=0.308$ of Abell 2744, 1” = 4.7 kpc.

Figure 1: Top left: X-ray surface brightness utilizing ultra-deep, 2.1 Ms deep
Chandra observations of Abell 2744. Top right: Strong lensing map of Abell
2744 from Furtak et al. (2022). The BCGs are shown as stars in both maps;
BCG-W is shown in white on the surface brightness map to stand out in the
otherwise dim region. All the other BCGs are close to, but slightly offset
from, X-ray surface brightness peaks. Bottom left: Map of the Gaussian
Gradient Magnitude (GGM) of the X-ray surface brightness. GGM enhances sharp
features and allows us to identify key discontinuities - two to the southeast,
two to the northwest, and one to the north. Bottom centre: Contour binned maps
of the temperature of Abell 2744. The S, N and NW substructures all harbour
cool cores (CCs), although the one in N is weaker than the others; N also does
not show a metallicity peak characteristic of cool cores. There is a shock
front ahead of the southeast cold front, a smoking gun of a recent merger
close to the plane of the sky. N also has a very clear, elongated cold front
but no corresponding shock ahead of it, suggesting that this substructure had
its pericenter passage long enough ago that the shock has mostly dissipated.
The NW substructure shows a trail of cool gas behind the bright, cool core,
but there is no evidence of a shock in front of it. This favours a scenario
where it is still infalling. Bottom right: Contour-binned maps of the
metallicity. There are no clear trends with subcluster core or BCG position.

##  2 New X-ray observations of Abell 2744

Abell 2744 is an extraordinary cluster of galaxies with unique features that
justify long observing times across wavelengths. Many Chandra X-ray
observations have been performed over the last 20 years, summing up to an
observing time of $\approx 2.1$ Ms (101 ACIS-I and 1 ACIS-S observations). The
observations are summarised in Table 3 and the individual OBSIDs can accessed
through https://doi.org/10.25574/cdc.257. Our data reduction used the standard
tasks provided with the CIAO software package version 4.15 with CALDB 4.10.7,
and we followed the procedure described in Schellenberger et al. (2017). We
note that we use wavedetect to detect point sources and visually inspect and
edit the catalog of detections to ensure only cluster-related features enter
into our analysis. For the imaging analysis, we refill the regions of masked
point sources with Poisson noise from the surrounding area. Merged surface
brightness maps were created using merge_obs using a bin size of 4, which
corresponds to $\approx 2\arcsec$.

For the spectral maps (temperature, pressure, metallicity) we extract spectral
regions following the X-ray surface brightness (contour binning, Sanders,
2006), and connects regions of similar surface brightness until a signal to
noise threshold is reached (35 in our case). This method helps to identify
edges in the surface brightness, which might be smoothed over by the other
techniques, such as the adaptive binning (c.f. O’Sullivan et al., 2014; Kim et
al., 2019). The spectral background was subtracted by using blank sky
observations that were re-normalized by the count rate in the $9-12$ keV
energy range. The spectral fitting was performed using sherpa and we fit an
absorbed thermal plasma model ( $\texttt{phabs}\times\texttt{apec}$ ) to the
combined spectra. We freeze the column density at the line of sight value
($N_{\rm H}=1.46\times 10^{20}\,{\rm cm^{-2}}$) and the redshift ($z=0.308$),
and leave the temperature, abundance and normalization free to vary.

Surface brightness edges and discontinuities in the X-ray image may indicate
shocks or cold fronts in the cluster ICM. To detect these, we use the Gaussian
Gradient Magnitude (GGM) algorithm (Walker et al., 2016; Sanders et al., 2016)
which highlights these features. This method convolves the observed image with
Gaussian kernels of different sizes and computes the gradients thereof. The
images for each of the filtered scales are combined with a radius-based
weighting factor. We note that the features in the filtered images strongly
depend on the smoothing and weighting of each scale. The difference between
the two smoothed images then highlights sharp features on scales between the
two smoothings. We used Gaussian smoothing scales of $4"$ to $9"$, which are
an optimal set of smoothing scales for this cluster to most clearly display
the signatures of known discontinuities.

The top row of Figure 1 shows the X-ray surface brightness map of Abell 2744
on the left, with the strong lensing convergence map of Furtak et al. (2022)
for reference. The strong lensing peaks overlap precisely with the BCGs, which
are shown as black crosses in both the top panels. The bottom row then shows
the GGM maps, along with contour-binned maps of the temperature and
metallicity. The GGM map highlights the presence of two fronts to the
southeast, one in the centre, one to the north, and two to the northwest. The
key features are:

Figure 2: Profiles of X-ray surface brightness (center) and temperature
(right) across the three cold fronts highlighted in the left panels. Blue
lines in the surface brightness plots show the best-fit model, described as a
2-component power-law with a finite width. The N cold front has the sharpest
jump, suggesting that this merger axis is very close to the plane of the sky.
The NW front appears weak in the surface brightness profile but very clear in
the temperature, suggesting that this merger has a significant component along
the line-of-sight; this is independently corroborated by the galaxy velocities
in Figure 4. The temperature and surface brightness jumps are clear and well-
aligned to the SE. The front has a non-zero width, which could be due to
inclination or turbulence.

  * •

The average temperature of the cluster in the $0.1-2.4$ keV band is 8.77 $\pm$
0.7 keV. Following the scaling self-similar $\beta$-model of Evrard et al.
(1996), this corresponds to an $r_{500}=$ 1.63 Mpc. Within this aperture, the
luminosity is $L_{\rm(0.1-2.4~{}keV)}=1.50\times 10^{45}\ \rm{erg\ s^{-1}}$ ,
and the average metallicity is $Z/Z_{\odot}=0.21\pm 0.01$.

  * •

The main X-ray peak lies near the center of the field of view, offset from all
BCGs. For a simple scenario with well-separated components all near
hydrostatic equilibrium, this would correspond to the most massive cluster.
For the moment, however, we cannot assume this to be the case, as the central
emission may be the superposition of multiple sub-clusters of comparable mass.

  * •

There is a bright, bullet-like cold front to the SE with a temperature of
$\sim 7$ keV. The cold front, along with the clear South Shock (SS) $\sim 150$
kpc ahead of it, is a classic signature of a merger close to the plane of the
sky. SE lies near two BCGs, S1 and S2, and it is unclear which one is
associated with.

  * •

A substructure to the north (N) features an X-ray peak coinciding with a BCG
(BCG-N) and a strong lensing peak. This does not, however, appear in the
Subaru weak lensing maps of Medezinski et al. (2016). Its leading edge has a
temperature of $\sim 8.5$ keV, increasing slowly to $\sim 10$ keV in a plume
of bright gas fanning out behind it. The arched shape of this plume indicates
a core passage with a large impact parameter b. This is called the “tail” in
Owers et al. (2011).

  * •

To the northwest lies a relatively relaxed substructure close to two bright
galaxies, NW-1 and NW-2 (e.g. Jauzac et al., 2015; Furtak et al., 2022); both
BCGs are coincident with strong lensing peaks. At its brightest, this
substructure has a temperature of $\sim 6$ keV. As noted in the introduction,
there is no consensus on whether this NW “interloper” is infalling or being
slingshot on its way out, and why both nearby BCGs are offset from the X-ray
peak.

Equipped with these deep and high-resolution maps of the ICM, we address
several disagreements in the literature. First is the presence or absence of a
shock front ahead of the NW cold front; if present, this would favour the
infall scenario of (Kempner & David, 2004). The GGM image in Figure 1 does
show a discontinuity in this region, with a contour-binned temperature
slightly higher than its surroundings. However, this feature is oriented
almost orthogonally to the NW cold front, whereas a bow shock at infall would
be nearly concentric. If this shock detection is significant - which we assess
below - it cannot be associated with an infall of the NW subcluster.

Figure 3: Same as Figure 2, but for the two potential shock fronts now shown
on the left. The temperature jump is greater in the SE shock front than the
NW; the NW profile is further complicated by the presence of the NW interloper
along the radial section.

Second, there is disagreement about whether the North subcluster merged with
the subcluster in the South, or the lensing peak to the West, which Merten et
al. (2011) propose is a subcluster remnant almost entirely stripped of gas.
First, we note that there is no local peak in the X-ray emission or its GGM
around this lensing peak, even with $\approx 2$ Ms of additional exposure.
Second, we note that the N cold front has a long tail, arcing to the southwest
behind it. In the binary merger simulations of Chadayammuri et al. (2022a),
such cool trails traced the trajectory of the corresponding cool core and
indicated that N most likely merged with the SE substructure. We produce a
sample merger of this type below in §4.

###  2.1 Surface brightness and temperature profiles

Next, we produce profiles of the surface brightness and temperature across the
discontinuities identified in the GGM image. The surface brightness profile is
fit by a broken power law with a break ($r_{b}$) indicating the location of
the brightness jump. This $r_{b}$ is shown as a red line in the temperature
profiles. For the three fronts in Figure 2, the surface brightness jumps
coincide with sharp temperature drops, confirming that these features are cold
fronts, the disrupted cores of infalling subclusters. For the front to the
northwest, the temperature and surface brightness jumps appear offset;
however, we note that the surface brightness jump is weak and noisy, and
therefore abstain from interpreting this offset.

Figure 3 shows the profiles of the remaining fronts. Here, the surface
brightness jumps correspond to increases in temperature, identifying the
features as shock fronts. In fact, the profiles to the SE are consistent with
two shock fronts spaced $\sim 140~{}$kpc apart. The Mach numbers of all three
shocks, as computed from both surface brightness and temperature jumps, are
shown in Table 1.

Feature | MSB | MT
---|---|---
NW | $1.34^{+0.11}_{-0.10}$ | $1.17^{+0.29}_{-0.24}$
SE 1 | $1.39\pm 0.01$ | $1.46^{+0.25}_{-0.22}$
SE 2 | $1.44\pm 0.01$ | $1.69^{+0.40}_{-0.37}$
Table 1: Mach Numbers inferred from the surface brightness and temperature
jumps for the three shocks in Figure 3.

The profiles across the surface brightness discontinuities in our new images
therefore detect cool cores in the N, S and NW substructures, and find a shock
to the SE. The shock to the NW is still tentative and is weak at best. The
temperatures of the cold fronts inform our fiducial models for the central
properties of their pre-merger progenitors. The complex nature of the shock
fronts indicates the superposition of multiple shock waves and precludes us
from making inferences from any one of them in isolation.

##  3 Multi-wavelength picture

###  3.1 Gravitational lensing

Gravitational lensing models make various choices, e.g. on whether or not to
parameterize the lens model, how to parameterize it, and how to account for
noise in the galaxy data (Priewe et al., 2017; Raney et al., 2020). In
addition, the most powerful lensing clusters, including Abell 2744, are
inclined along the line of sight, so that the projected mass is maximized;
this leaves common assumptions about spherical symmetry poorly justified. The
density distributions of the merging subclusters vary over the course of a
merger, as described in detail in Lee et al. (2023). Most models of weak
gravitational lensing model each cluster component as an NFW profile; if the
observations do not have sufficient signal-to-noise, the concentration and
total mass of the halo are not fit independently, but instead are assumed to
follow concentration-mass relations that are in fact measured for relaxed
galaxy clusters (Duffy et al., 2008; Prada et al., 2012; Diemer & Joyce,
2019). This means that weak lensing masses can be overestimated by up to
60$\%$. Therefore, we use the weak lensing masses from Medezinski et al.
(2016), including $M_{\rm 200c}=(2.06\pm 0.42)\times 10^{15}\ \rm{M_{\odot}}$
for the entire system, as upper limits. Furthermore, the positions of weak
lensing peaks vary significantly between studies. The positions of the X-ray
peaks and BCGs, while known to be offset from the true cluster centres, are
nevertheless better traces of the subcluster positions than low-significance
weak lensing peaks.

Figure 4: Distribution of spectroscopically confirmed velocities of cluster
member galaxies using MUSE Furtak et al. (2022) and AAO (Owers et al., 2011).
BCGs are highlighted on the left by stars and on the right as grayscale lines.
Each row represents a different clustering of (RA, DEC, $z_{\rm spec}$) of the
cluster member galaxies, using $N_{\rm cluster}$ = 2, 3, and 4 respectively.
Using three clusters isolates the velocity peak around S1, which is also
spatially compact. Adding more than three clusters results in more spatial
categories without improving the representation of the velocity distribution.
The line-of-sight velocity components are therefore: N+S2, NW1+NW2, and S1.
Figure 5: Same as Figure 4, but excluding the galaxies associated with the S1
velocity structure. Now, there appears to be a meaningful separation between
galaxies to the north, south and northwest, with each distribution peaking
near the corresponding BCGs.

Strong lensing (e.g., Jauzac et al., 2015; Furtak et al., 2022; Bergamini et
al., 2023b) provides much tighter constraints on the mass of smaller regions
in the centers of the various subclusters, where the surface density exceeds
the critical value for strong lensing. These dense regions are often called
“substructures”, but do not necessarily represent the cores of merging cluster
components; Kimmig et al. (2022) found that analogous clusters in the
Magneticum Pathfinder simulation accreted such structures at $1<z<0.4$.
Ideally, as in Jauzac et al. (2016), strong lensing can be used to constrain
the mass in the cluster cores while weak lensing constrains the outskirts;
this method yields $M(<1~{}{\rm Mpc})=(1.85\pm 0.07)\times 10^{15}\
\rm{M_{\odot}}$, consistent with the weak-lensing-only mass from Medezinski et
al. (2016), which was integrated out to 2.5 Mpc. We use the locations of the
strong lensing peaks to trace the cluster potential minima in our simulations
in §4; reproducing the exact values, however, would require adding potentials
for the BCGs, which we do not do.

The contours in the right panel of Figure 1 show the strong lensing
convergence measured by Furtak et al. (2022) as part of the JWST-UNCOVER
program. The strongest signal is around S1-S2, followed by NW1-NW2 and then
around BCG-N. This alone challenges the model of Owers et al. (2011), who
claim that the most massive subcluster is centered on BCG-N. There is a small
strong lensing peak near BCG-W, which Merten et al. (2011) suggest is the
remnant of a merger with BCG-N. However, W does not show up in the X-ray map
nor in weak lensing maps (e.g. Merten et al., 2011; Medezinski et al., 2016).
When detected, the peak of the western substructure W varies widely between
studies, and Jauzac et al. (2016) argues that it is most likely a background
source. We therefore do not include it in our merger simulations.

###  3.2 Optical/NIR images and spectra

Galaxy spectra reveal the line-of-sight velocities of cluster member galaxies,
placing further constraints on the merger picture. These constraints are
stronger still when combined with the spatial information, which in turn lacks
the line-of-sight component. Self-supervised clustering algorithms like
K-means (Lloyd, 1982) offer a model-agnostic way of identifying structures in
such multi-dimensional data. Given a number of clusters, the algorithm seeds
cluster centers randomly in the space of the data. Each data point is assigned
to the nearest seed. At the end of each iteration, the new center of each
cluster is the N-dimensional mean of all its assigned data points. The process
is repeated until the cluster centers converge.

Figure 4 shows the results of K-means applied to the (RA, DEC, z) of the 364
galaxies with redshift measurements from either VLT-MUSE (Furtak et al., 2022)
or the Anglo-Australian Observatory (Owers et al., 2011); the latter has
better coverage of the entire cluster field. Since we do not a priori know how
many clusters are in this data, we try values between 2 and 5.

S1 and S2, despite their proximity in the plane of the sky, are separated in
the line-of-sight velocity space by $5030\ \rm{km\ s^{-1}}$; each of them sits
near the centre of a different Gaussian distribution of velocities. If their
velocity offset were entirely due to cosmological redshift, they would be 54
Mpc apart and not gravitationally bound at all. However, this is $2-3$ times
the characteristic infall velocity for two galaxy clusters of mass
$10^{14-15}\ \rm{M_{\odot}}$ (Li et al. (2020) and §4 below); ascribing 1/3
(2/3) of the observed redshift offset to an infall velocity still implies a
separation of 38 (21) Mpc. This means that S1, and the 146 galaxies in that
redshift cluster, are not part of the gravitationally bound system. They will,
nevertheless, contribute significantly to the gravitational lensing
convergence:

$\kappa\propto\frac{D_{ol}D_{ls}}{D_{os}}$ (1)

Name | $M_{200c,1}(M_{\odot})$ | Core 1 | $M_{200c,2}(M_{\odot})$ | Core 2 |  $d_{i}$ (Mpc) |  $b$ (kpc) |  $v_{i}$ ($\ \rm{km\ s^{-1}}$)
---|---|---|---|---|---|---|---
S2 - N | $8\times 10^{14}$ | CC | $6\times 10^{14}$ | CC | 2.36 | 225 | 2300
S2 - NW2 | $8\times 10^{14}$ | CC | $4\times 10^{14}$ | NCC | 3 | 1750 | 3478
S2 - NW1 | $8\times 10^{14}$ | CC | $4\times 10^{14}$ | NCC | 3 | 0 | 2000
Table 2: Initial conditions for the binary mergers inferred from X-ray and
galaxy velocity observations.

where $D_{ol}$, $D_{ls}$ and $D_{os}$ are the angular diameter distances
between the observer and lens, lens and source, and observer and source,
respectively. This ratio varies by only 0.8$\%$ for a lens at $z=0.30$ vs.
$z=0.32$ and sources at $z\sim 1$. Thus, the galaxy spectra tell us that the
mass inferred from gravitational lensing near the southeast cold front is much
greater than the mass of the components associated with the X-ray features.
Clusters are observed to follow a mass-richness relation (MRR, Simet et al.
(e.g. 2017)):

$\langle M|\lambda\rangle=M_{0}\frac{\lambda}{\lambda_{0}}^{\alpha}$ (2)

where the characteristic mass $M_{0}$ and richness $\lambda_{0}$ are largely
determined by the selection function. The slope $\alpha$ has been measured at
1.2-1.3 across a large range of cluster masses and redshifts. For a 2-cluster
model, the mass in S1 is therefore $\sim 1/3$ of the total mass, i.e.
$(5-6.7)\times 10^{14}\ \rm{M_{\odot}}$ within the uncertainties in the weak
lensing studies.

This leaves $(1-1.3)\times 10^{15}\ \rm{M_{\odot}}$ left to distribute between
S2, N and NW. Figure 5 similarly clusters these galaxies in the (RA, Dec, z)
space. The most physically meaningful clustering includes 3 components,
corresponding to the North, South and Northwest clusters. Thus, in addition to
the N-S merger in the plane of the sky, we infer a SE-NW merger slightly
inclined to the plane of the sky, to produce a line-of-sight velocity offset
as well as X-ray features in the plane of the sky.

##  4 Constrained idealised simulations

Figure 6: Initial density (left) and temperature (right) profiles for the five
simulated cluster components. The masses and gas profile parameters of these
components are detailed in Table 2.

Figure 7: Top: Best-matching simulation snapshot of the S2-N binary merger. It
occurs $0.5-0.6$ Gyr after pericenter passage. This image is perpendicular to
the merger plane so that the line-of-sight velocity of the two cluster
potential minima (marked by black crosses) is the same. Note that the cluster
centers - which, especially in cool core clusters, closely trace the BCG -
slightly lead the corresponding cool cores; this relative position varies over
the course of the merger. An impact parameter of $\sim 250$ kpc is required to
produce the arched trail. Center: Best-matching snapshot for a first infall of
NW into S, again orientated to match the line-of-sight velocity offset between
the BCGs. The images are almost identical in this merger phase for $0<b_{\rm
kpc}<900$; this scenario prefers $M_{200c,NW}\sim 4\times 10^{14}M_{\odot}$.
Bottom:  Best-matching snapshot for the scenario where NW is turning around
$\sim 2$ Gyr after first core passage. This scenario prefers $M_{200c,NW}\sim
2\times 10^{14}M_{\odot}$ in order to retain two compact cool cores.

The X-ray, gravitational lensing, and galaxy velocity observations presented
above together present a merger scenario, which is complex enough that its
validation requires targeted simulations. Therefore, we run a suite of cluster
merger simulations using the GPU-accelerated Adaptive MEsh Refinement
(GAMER-2) code (Schive et al., 2018). We start with binary simulations of the
N-S and NW-SE mergers inferred from the X-ray and galaxy velocity
observations. Then, we run triple mergers to assess whether the conclusions
from the binary simulations hold if the mergers are not isolated. We ran
dozens of simulations exploring each of the parameters in Table 2. Below, we
present only those that produced close analogues of the mergers observed in
Abell 2744.

###  4.1 Initial conditions

The defining parameters of each simulation are:

  * •

The virial masses $M_{1}$ and $M_{2}$ of the merging components, i.e. the
total mass enclosed with the radius $R_{200c}$ within which the mean density
is 200 times the critical density of the Universe. These can also be
parameterized as $M=M_{1}+M_{2}$ and $R=M_{1}/M_{2}$.

  * •

The initial gas density profile of each cluster, i.e. the size and density
gradient of its core. Cool-cores are steep and small in extent, while non-cool
cores are flat and extended.

  * •

The impact parameter $b$. This is the component of the distance between the
cluster centres that is perpendicular to their infall velocity. Larger $b$
corresponds to higher angular momentum, with a lower likelihood of core
disruption and a greater likelihood of sloshing.

  * •

Initial velocity $v$ of the cluster centres at the start of the simulation.

A characteristic velocity for a gravitationally bound system is its virial
velocity, $V_{h}=\sqrt{GM_{\rm vir}/R_{\rm vir}}$. For an NFW halo of mass
$10^{15}\ \rm{M_{\odot}}$, $V_{h}\sim$ $1761\ \rm{km\ s^{-1}}$. In $N$-body
simulations, Li et al. (2020) find that the characteristic infall velocities
of merging clusters are $10-20\%$ higher than this analytic estimate, with a
scatter of 0.2 dex. This motivates velocities of $1000-2000\ \rm{km\ s^{-1}}$
at initial separations of $4-5$ Mpc. We note, however, that systems like Abell
2744, with many merging components at a rich node of the cosmic web, are
likely to be outliers in this distribution. Torques from various components
can make a particular merger in the system faster or slower than if that
binary merger were isolated. We therefore explore this parameter space widely.

The particle files for the merging components are generated using the Python
package cluster$\\_$generator 111https://github.com/jzuhone/cluster_generator.
The gravitational potential of each cluster is modelled as a truncated NFW
potential (Baltz et al., 2009):

$\rho_{DM}(r)=\frac{\rho_{s}}{r/r_{s}\cdot(1+r/r_{s})^{2}\cdot(1+(r/r_{t})^{2})}$
(3)

The truncated profile, unlike the standard NFW prescription, ensures a
converging total mass. The tNFW profile agrees with the NFW profile out to a
sufficiently large radius that we can still use an NFW concentration-mass
relation (Diemer & Joyce, 2019) to determine the scale radius $r_{s}=r_{\rm
vir}/c$. The profile is normalised to yield a specified mass $M_{200c}$ within
the virial radius. The dark matter particle velocities with respect to the
cluster centre are initialized such that the cluster is in virial equilibrium.

The gas is initialised in hydrostatic equilibrium. This only yields the
gradient of the pressure profile, however, which is degenerate between the
density and temperature. We describe the gas density using the very flexible
formalism of Vikhlinin et al. (2006):

$\rho_{g}(r)=\frac{\rho_{0}}{[(r/r_{c})^{\alpha}(1+(r/r_{c})^{2})^{3\beta-\alpha/2}(1+(r/r_{s})^{\gamma})^{\epsilon/\gamma}]^{1/2}}$
(4)

Here, $\alpha$, $\beta$, and $\gamma/\epsilon$ determine the inner, central
and outer slopes of the gas profile. Crucially, $r_{s}$ is not necessarily the
same as the identically named parameters in the dark matter density profile
but instead represents the extent of the gas outskirts. We generate three
types of gas profiles: strong cool cores, where $r_{c}/r_{vir}=0.1$ and
$\alpha=2$, weak cool cores with $r_{c}/r_{vir}=0.3$ and $\alpha=1$, and non-
cool cores with $r_{c}/r_{vir}=0.5$ and $\alpha=0$. We fix $r_{s}=1.1r_{vir}$,
$\beta=2/3$ and $\epsilon=\gamma=3$, which described most of the galaxy
clusters in the sample of Vikhlinin et al. (2006).

Because the weak lensing masses and peak positions are so uncertain, we choose
to estimate the initial conditions of the clusters from the properties of
their remnant cores while making sure that the total mass of the components
does not exceed $2\times 10^{15}\ \rm{M_{\odot}}$. The fiducial initial
conditions (ICs) for each of the five observed sub-clusters are shown in
Figure 6. We emphasise that these are rough approximations, and are intended
only as informed guesses on the fiducial model of each subcluster.

GAMER-2 reads in the gas profiles, dark matter particle lists, as well as the
merger geometry, i.e. the 3-D position and velocity of each cluster centre. It
then adaptively refines a rectangular grid to a specified number of refinement
levels. The refinement criteria are based on the particle number and second
derivatives of density, pressure and temperature. The fiducial resolution of
the simulations is 13.7 kpc, or $3\arcsec$ at the redshift of Abell 2744.
Since the features we aim to replicate are tens of kpc to Mpc in size, this
resolution is sufficient. Higher resolution is warranted in studies that aim
to model small-scale features, such as unstable eddies along contact
discontinuities, or to reproduce profiles across shock fronts.

###  4.2 The S-N merger

The key features of the S-N merger are the bullet-like cold front to the
southeast, a bow shock ahead of it, and a cold front to the north with a long
“trail” of cool gas behind it. Both cold fronts are slightly offset from the
nearest BCGs. BCG-N and BCG-S2 have nearly the same redshift, as shown in
Figure 4, while that of BCG-S1 corresponds to a velocity offset of nearly
$6000\ \rm{km\ s^{-1}}$. Since the S-N merger shows all the features of a
merger close to the plane of the sky, we assume that BCG-S2 is associated with
the now southern component. We confirm in our simulations that both S2 and N
must have cool cores to produce their cold fronts today.

The top panel of Fig 7 shows the X-ray surface brightness (left), temperature
(middle) and lensing convergence (right) of a simulation snapshot that matches
the observed geometry of the N-S system. The parameters of this simulation are
listed in Table 2.

The temperatures are consistent with the observations, whereas the surface
brightness is lower by a factor of $\sim 5$ and the lensing convergence by a
factor of $5-10$, depending on the region. While the surface brightness and
lensing convergence could be boosted by doubling the mass, this would increase
the spatial scale of all the features; already the distance between the BCGs
is $\sim 1.6$ Mpc, more than twice as large as observed. Instead, we note that
the lensing signal will be boosted by the substructure around S1, and both
lensing and surface brightness quantities should change in the desired
direction by a third merging component. Therefore, we do not fine-tune these
properties at this point. Instead, we focus on the scale-free relative
morphology of the cool and shock fronts and subcluster potential minima, which
we treat as proxies for the BCG locations.

The two cold fronts with slightly arced trails behind them are indicative of a
recent pericenter passage with an impact parameter of $\sim 250$ kpc. The
greater mass of the S subcluster leads to a more bound and brighter core
remnant and a stronger lensing convergence.

The bow and reverse shocks from this merger are already starting to cool, so
that between each cold front and its corresponding shock front the temperature
pattern is not monotonic. We show the radial regions used to extract the
profiles in Figure 3 to highlight this difference in the structures of the
profiles; in particular, the biggest jumps in the two properties are not
coincident, as indeed is observed in Figure 3. Furthermore, we note that the
shock structure in mergers with such a moderate impact parameter is complex,
and produces additional discontinuities besides the primary (bow and reverse)
shocks. We highlight with an arrow a weak front in the surface brightness and
temperature maps on the top panel of Figure 7, which coincides with the
observed position of the NW front. Thus, this does not have to indicate that
the NW substructure is infalling. It is simply an evolution of the shock
fronts from the S-N merger, which likely occurred $\lesssim 0.5$ Gyrs ago; the
exact timing depends on the true masses of the substructures, which, as noted,
we do not constrain at this step.

Thus, the N-S merger explains the N and SE cold fronts and the complex
observed profiles of temperature and surface brightness across the SE and NW
fronts. The lensing maps indicate the need for over $5-10$ times more mass,
which, however, would worsen the agreement of the X-ray maps.

###  4.3 The SE-NW merger

The first step in constraining the properties of the SE-NW merger is
identifying the BCGs are the core of each substructure merging along this
axis. The two NW BCGs, NW1 and NW2, while at very similar redshifts, need not
be dynamically bound. The velocity offset between NW1 and NW2 on the one hand
and either S1 or S2 is exactly the same, so this does not give a preference to
any pair. As argued above, we consider the simplest model where S1 is not
dynamically bound to the rest of the system, and the NW component interacts
primarily with S2.

First off, we consider the simplest scenario, where NW is infalling for the
first time, as shown in the middle panel of Figure 7. Given the trail of cool
gas behind the NW core, this would have to be a wide-angle, i.e large impact
parameter fly-by. Such large impact parameters, however, are inefficient at
stripping gas from the subcluster before first pericenter passage; further, in
this scenario, the BCG is always well aligned with the subcluster X-ray peak.
Our binary simulations therefore disfavour a first-infall scenario for NW.

Next, we consider the “slingshot” scenario of Merten et al. (2011), where NW
is turning around after the first pericenter passage; a representative
snapshot is shown in the bottom panel of Figure 7. To disrupt the core of S2,
this merger has to have a very large impact parameter $\gtrsim 1$ Mpc. Such
high-angular-momentum mergers produce spiral motions in the gas, most visible
in the shape of the shock and cold fronts. Therefore, the cool core that is
currently to the SE must have initially come in from the SW to the NE and then
turned around. It is possible to produce a slingshot, where the gas cool of
what is now the NW subcluster is $150-250$ kpc further out than its BCG; we
find several such simulations when $700<b/{\rm kpc}<1200$. In this geometry,
NW-1 would be leading the gas core.

Figure 8: Three different scenarios for the triple merger. Top: NW is
infalling for the first time. Middle: The N-S merger occurs first; the S-NW
merger occurs shortly after. Bottom: The S-NW merger occurs first; the S-N
merger happens shortly after.

###  4.4 Triple merger simulations

In practice, the two mergers above do not occur in isolation. First, using the
same initial positions for each subcluster as in the binary simulations will
produce different trajectories, since there is now a third source of mass. The
lensing convergence will be higher in regions with significant contributions
from multiple substructures. The ICM temperature will be higher everywhere due
to multiple overlapping shocks. A cold front could be disrupted by a shock
generated by another substructure. The parameter space of triple mergers is
intractably large since motions can no longer be reduced to a single plane,
and because we now have to generate 3D positions and velocities such that all
three subclusters end up in a relative orientation consistent with
observations at some single snapshot in time. We therefore do not aim to
provide any quantitative constraints on the initial conditions of this triple
merger. Rather, this subset of the simulation suite is intended to test
whether the outcomes of the binary simulations hold in such a complex
environment, and potentially distinguish between the slingshot and primary
infall scenarios for NW.

The top row of panels in Figure 8 shows a snapshot where a subcluster of mass
$4\times 10^{14}\ \rm{M_{\odot}}$ is falling into a system that is already
undergoing a merger. This scenario results in a different picture than a first
infall of a binary merger along the SE-NW axis described above. The cooler
outskirts of the NW subcluster stand in contrast to the shock-heated ICM of
the N-S system. The temperature in its core also matches observations. This
core is only cool in comparison to the shock-heated ICM of the S-N merging
system; it is still hotter than the gas in the outskirts of the infalling
subcluster, creating the appearance of a cold tail to the south of the NW
X-ray peak. A cold front appears offset from the centre of NW, because of the
contrast with the shock-heated ICM.

Next, we consider the scenario where NW is past its first turnaround and
experiencing a gravitational slingshot. There are two options here - either
the N-S merger occurs first, or the S-NW merger does. These two scenarios are
shown in the middle and bottom panels of Figure 8, respectively. We note that
there is a very large parameter space to explore pertaining to the relative
timings of the two mergers and the relative impact parameters. We summarize
our extensive parameter search with two key examples here.

First, we consider case where the N-S core passage happens first, and the NW
subcluster approaches this system shortly afterwards. A key result here is
that temperatures and surface brightness are higher everywhere due to the
overlapping shocks. The temperatures could be lowered by lowering the masses
of all the subclusters, but this would also reduce the lensing convergence and
surface brightness, which are already lower than observed. Next, we find that
the shock from the first infall of NW completely destroys the SE cold front,
even if it has impact parameters of $\sim 1$ Mpc with respect to both the S
and N subcluster centres. This is not to be confused with the large impact
parameter required by the SE-NW binary merger above, which only ensures that
the core of NW remains relatively intact at the first turnaround. Rather, here
we are noting that a shock from a N-NW merger would destroy the S cold front.
This cold front is formed again by radiative cooling $\sim 0.5$ Gyr later, but
by this point, the N and NW cold fronts are both too disrupted and faint to be
observed.

Similarly, if the S-NW merger starts first, the additional shocks from the
infall of the N substructure disrupt all cool cores. We delayed the infall of
N until the CCs in S and NW were restored, but at this point, the NW cool is
too far away, diffuse, and cool to be observed at the dynamic range of the
existing observations. This is shown in the bottom panel of Figure 8.

In summary, we find that while binary simulations prefer a scenario where NW
has completed first pericenter passage and is in the turnaround phase, the
triple simulations show that the overlapping shocks from multiple mergers
would destroy all the cold fronts; cool cores are only restored $>0.5$ Gyr
later, at which point their separations greatly exceed the observed values.

###  4.5 On the tractability of constraining multi-component mergers with
idealised simulations

Idealised simulations of binary mergers have been very successful at
identifying analogues of observed systems, allowing us to infer the dynamics
and plasma physics from multi-wavelength observations (Asai et al., 2005;
Roediger et al., 2012; Bonafede et al., 2013; Machado & Lima Neto, 2013;
Machado et al., 2015; Albuquerque et al., 2024). This study shows, however,
that the conclusions do not extend trivially to systems where multiple mergers
occur at the same time.

First of all, the number of parameters to explore increases Even in a
simplified scenario, assuming a concentration-mass relation with no scatter,
describing each halo requires 1 total mass, and 2 parameters to describe the
gas core. In cases like Abell 2744, some shocks are far from the center of any
one subcluster, so that the outer profile of the gas density also matters.
Most crucially, however, we can no longer fix a single merger axis and align
the impact parameter with one of the orthogonal axes. Instead, we now have to
consider the relative positions and velocities (and therefore timings) of all
N components, i.e. 6N dimensions, where earlier 2 had sufficed. A binary
cluster merger study can thus perform reasonably well with 8 parameters,
whereas for N $\geq$ 3 components, we need to explore 9N parameters. We cannot
approximate multiple mergers as linear combinations of binary mergers, either,
because shock heating and radiative cooling interact with each other non-
linearly.

Last but certainly not least, we observe here that, even though the halo
masses were chosen to match the gravitational lensing constraints, the
resulting convergence and X-ray surface brightness maps are lower than
observed for most of the field of view. Meanwhile, the emission-weighted
temperatures do match observations. This suggests strongly that there is a
large scale component, more extended than the subcluster halos, which
contributes to the line-of-sight integrated quantities like surface density
and emission, without affecting the densest, hottest regions. As proposed by
Lee (2017), systems like Abell 2744 may be superclusters aligned with our line
of sight, and therefore not well-described as a sum of cluster halos.

A more appropriate technique for finding analogues to systems with $N>2$
components should therefore search large-volume cosmological boxes (Jauzac et
al., 2016; Kimmig et al., 2022). A particularly promising suite for this
purpose is TNG-Cluster (Nelson et al., 2023), which simulates 352 clusters
with $M_{200c}>10^{14}M_{\odot}$ at z=0; including snapshots out to z = 1,
this provides nearly 3000 clusters, each of which can be viewed from any
direction. We encourage future searches for simulated analogues of observed
cluster mergers to start with searching such cosmological suites.

##  5 Conclusions

We present the deepest X-ray observations of the complex galaxy cluster merger
Abell 2744. The 2 Ms Chandra Very Large Program (VLP, PI Bogdan), combined
with archival data, allows us to construct surface brightness, temperature and
metallicity maps in unprecedented detail.

  * •

The average temperature of the cluster in the $0.1-2.4$ keV band is 8.77 $\pm$
0.7 keV. Following the scaling self-similar $\beta$-model of Evrard et al.
(1996), this corresponds to an $r_{500}=$ 1.63 Mpc. Within this aperture, the
luminosity is $L_{\rm(0.1-2.4~{}keV)}=1.50\times 10^{45}\ \rm{erg\ s^{-1}}$,
and the average metallicity is $Z/Z_{\odot}=0.21\pm 0.01$.

  * •

We measure cold fronts to the southeast, north and northwest, a shock front to
the northwest and two concentric shocks to the southeast.

  * •

We combine spectroscopic measurements from Owers et al. (2011) and Furtak et
al. (2022) to perform a K-means clustering of the galaxies in the field of
Abell 2744 in the RA-DEC-z space. For $n_{\rm cluster}\geq 2$, a clear cluster
forms around S1, separated by $\Delta z\sim 0.2$ from the remaining galaxies.
This corresponds to a $\Delta v\sim 5030\ \rm{km\ s^{-1}}$ if the redshift is
entirely due to a velocity offset along the line of sight, or $\Delta
r_{z}\sim$ 54 Mpc if it is entirely cosmological. The reality is likely some
combination of the two, which nevertheless suggests that the S1 subcluster is
not dynamically bound to the rest of the cluster. Following the mass-richness
relation of Simet et al. (e.g. 2017), this accounts for $\sim 1/3$ of the
total projected mass in the area.

  * •

The BCG velocities further indicate that the N-S merger occurred very close to
the plane of the sky. The velocities of BCGs NW-1 and NW-2 could be at the
tail of the distributions of the N or S subclusters; they are better described
as sitting near the middle of a third velocity distribution. The velocities of
BCGs NW-1 and NW-2 indicate a merger along the SE-NW axis, with an inclination
such that $\Delta v_{z}\sim 2000\ \rm{km\ s^{-1}}$.

  * •

We use a suite of idealised simulations to assess the merger picture assembled
from the observations above. First, a simulation of a binary cluster merger
along the north-south axis with pericenter passage $0.2-0.3$ Gyr ago and a
modest impact parameter of $150\ \rm{km\ s^{-1}}$ reproduces the observed cold
and shock fronts of the N-S merger. They also produce a weak reverse shock at
the location of the observed NW shock. A binary merger with a much larger
impact parameter of $900\ \rm{km\ s^{-1}}$ and pericenter passage $\sim$2 Gyr
ago produces a gravitational slingshot as observed in NW, with a BCG position
close to the observed NW-2 turning around further than the cool-core ICM,
which instead matches the observed NW interloper.

  * •

Finally, we run a suite of triple-merger simulations, since the N, S, and NW
subclusters are all likely to affect each other in this complex merger. This
study reveals the gravitational slingshot scenario to be unviable because even
at a large impact parameter of $\sim 1$ Mpc from both the N and S subclusters,
the shock produced by the NW subcluster completely destroys the SE cold front.
In other words, the SE cold front cannot survive two mergers spaced $\sim
1~{}$Gyr apart, even if one of them is far from the SE core. This result holds
even for a mass ratio $R=M_{S}/M_{NW}$= 4, whereas observations suggest a much
more dramatic merger of R$\sim$2\. Therefore, the triple merger simulations
strongly favour the scenario where NW is infalling for the first time.

  * •

Lastly, we find that if we use the total mass measured within a $1~{}$Mpc
aperture around Abell 2744 and divide it between three NFW halos centered at
S2, N and NW, each of which is initialised at hydrostatic equilibrium, we
recover the correct temperature maps but significantly lower surface
brightness and lensing convergence throughout the observed field. This
corroborates the suggestion of Lee (e.g. 2017) that Abell 2744 is a
supercluster in the making, with large filaments feeding it along several
directions, including the line of sight. These filaments contribute to the
projected mass outside of the halo formalism; gas in these filaments
integrates to contribute to the surface brightness. The observed temperature,
on the other hand, is a spectral-weighted average, and as such dominated by
contributions from the densest gas in the halos. S1 would then be part of this
supercluster, as would the filament to the NE detected by Eckert et al.
(2016). It is not implausible that even NW-1 is associated with a structure
along the line of sight.

Abell 2744 is the most complex galaxy cluster merger observed to date. The
X-ray maps presented in this study can be used to constrain viscosity and
thermal conduction in the intracluster plasma. We show that an overly
idealised interpretation using binary merger simulations can be misguiding,
and further motivate the search for and study of more realistic clusters in
cosmological simulations such as C-EAGLE (Barnes et al., 2017), the Three
Hundred project (Cui et al., 2018) and the recent TNG-Cluster (Nelson et al.,
2023).

## Data Availability

The Chandra X-ray observations are available upon request. The gravitational
lensing and galaxy velocity maps were downloaded from the public data release
of the JWST UNCOVER program https://jwst-uncover.github.io/DR2.html, as
documented in Furtak et al. (2022). GAMER-2 is an open-source code, available
and documented at https://github.com/gamer-project/gamer/wiki. The initial
conditions and SLURM scripts used for our simulations are available at
https://github.com/milchada/Abell2744. This project is thus entirely
reproducible.

## References

  * Albuquerque et al. (2024) Albuquerque R. P., Machado R. E. G., Monteiro-Oliveira R., 2024, MNRAS, 530, 2146
  * Ananna et al. (2024) Ananna T. T., Bogdán Á., Kovács O. E., Natarajan P., Hickox R. C., 2024, arXiv e-prints, p. arXiv:2404.19010
  * Asai et al. (2005) Asai N., Fukuda N., Matsumoto R., 2005, Advances in Space Research, 36, 636
  * Baltz et al. (2009) Baltz E. A., Marshall P., Oguri M., 2009, J. Cosmology Astropart. Phys, 2009, 015
  * Barnes et al. (2017) Barnes D. J., et al., 2017, MNRAS, 471, 1088
  * Bergamini et al. (2023a) Bergamini P., et al., 2023a, arXiv e-prints, p. arXiv:2303.10210
  * Bergamini et al. (2023b) Bergamini P., et al., 2023b, A&A, 670, A60
  * Bogdán et al. (2024) Bogdán Á., et al., 2024, Nature Astronomy, 8, 126
  * Bonafede et al. (2013) Bonafede A., Vazza F., Brüggen M., Murgia M., Govoni F., Feretti L., Giovannini G., Ogrean G., 2013, MNRAS, 433, 3208
  * Boschin et al. (2006) Boschin W., Girardi M., Spolaor M., Barrena R., 2006, A&A, 449, 461
  * Chadayammuri et al. (2022a) Chadayammuri U., ZuHone J., Nulsen P., Nagai D., Felix S., Andrade-Santos F., King L., Russell H., 2022a, MNRAS, 509, 1201
  * Chadayammuri et al. (2022b) Chadayammuri U., ZuHone J., Nulsen P., Nagai D., Russell H., 2022b, MNRAS, 512, 2157
  * Cui et al. (2018) Cui W., et al., 2018, MNRAS, 480, 2898
  * Dawson et al. (2012) Dawson W. A., et al., 2012, ApJ, 747, L42
  * Diemer & Joyce (2019) Diemer B., Joyce M., 2019, ApJ, 871, 168
  * Duffy et al. (2008) Duffy A. R., Schaye J., Kay S. T., Dalla Vecchia C., 2008, MNRAS, 390, L64
  * Dursi & Pfrommer (2008) Dursi L. J., Pfrommer C., 2008, ApJ, 677, 993
  * Eckert et al. (2016) Eckert D., Jauzac M., Vazza F., Owers M. S., Kneib J. P., Tchernin C., Intema H., Knowles K., 2016, MNRAS, 461, 1302
  * Ettori & Fabian (2000) Ettori S., Fabian A. C., 2000, MNRAS, 317, L57
  * Evrard et al. (1996) Evrard A. E., Metzler C. A., Navarro J. F., 1996, ApJ, 469, 494
  * Furtak et al. (2022) Furtak L. J., et al., 2022, arXiv e-prints, p. arXiv:2212.04381
  * Govoni et al. (2001) Govoni F., Enßlin T. A., Feretti L., Giovannini G., 2001, A&A, 369, 441
  * Hu et al. (2021) Hu D., et al., 2021, ApJ, 913, 8
  * Jauzac et al. (2015) Jauzac M., et al., 2015, MNRAS, 452, 1437
  * Jauzac et al. (2016) Jauzac M., et al., 2016, MNRAS, 463, 3876
  * Kempner & David (2004) Kempner J. C., David L. P., 2004, MNRAS, 349, 385
  * Kim et al. (2019) Kim D.-W., James N., Fabbiano G., Forbes D., Alabi A., 2019, MNRAS, 488, 1072
  * Kimmig et al. (2022) Kimmig L. C., Remus R.-S., Dolag K., Biffi V., 2022, arXiv e-prints, p. arXiv:2209.09916
  * Kovacs et al. (2024) Kovacs O. E., et al., 2024, arXiv e-prints, p. arXiv:2403.14745
  * Lee (2017) Lee J., 2017, arXiv e-prints, p. arXiv:1701.04220
  * Lee et al. (2023) Lee W., et al., 2023, ApJ, 945, 71
  * Li et al. (2020) Li Z.-Z., Zhao D.-H., Jing Y. P., Han J., Dong F.-Y., 2020, ApJ, 905, 177
  * Lloyd (1982) Lloyd S., 1982, IEEE Transactions on Information Theory, 28, 129
  * Lyutikov (2006) Lyutikov M., 2006, MNRAS, 373, 73
  * Machado & Lima Neto (2013) Machado R. E. G., Lima Neto G. B., 2013, MNRAS, 430, 3249
  * Machado et al. (2015) Machado R. E. G., Monteiro-Oliveira R., Lima Neto G. B., Cypriano E. S., 2015, MNRAS, 451, 3309
  * Mao et al. (2018) Mao T.-X., Wang J., Frenk C. S., Gao L., Li R., Wang Q., Cao X., Li M., 2018, MNRAS, 478, L34
  * Markevitch et al. (2003) Markevitch M., et al., 2003, ApJ, 586, L19
  * Massey et al. (2015) Massey R., et al., 2015, MNRAS, 449, 3393
  * Maurogordato et al. (2011) Maurogordato S., Sauvageot J. L., Bourdin H., Cappi A., Benoist C., Ferrari C., Mars G., Houairi K., 2011, A&A, 525, A79
  * Medezinski et al. (2016) Medezinski E., Umetsu K., Okabe N., Nonino M., Molnar S., Massey R., Dupke R., Merten J., 2016, ApJ, 817, 24
  * Merten et al. (2011) Merten J., et al., 2011, MNRAS, 417, 333
  * Molnar et al. (2012) Molnar S. M., Hearn N. C., Stadel J. G., 2012, ApJ, 748, 45
  * Natarajan et al. (2017) Natarajan P., et al., 2017, MNRAS, 468, 1962
  * Nelson et al. (2023) Nelson D., Pillepich A., Ayromlou M., Lee W., Lehle K., Rohr E., Truong N., 2023, arXiv e-prints, p. arXiv:2311.06338
  * O’Sullivan et al. (2014) O’Sullivan E., David L. P., Vrtilek J. M., 2014, MNRAS, 437, 730
  * Owers et al. (2011) Owers M. S., Randall S. W., Nulsen P. E. J., Couch W. J., David L. P., Kempner J. C., 2011, ApJ, 728, 27
  * Paul et al. (2019) Paul S., Salunkhe S., Datta A., Intema H. T., 2019, MNRAS, 489, 446
  * Pearce et al. (2017) Pearce C. J. J., et al., 2017, ApJ, 845, 81
  * Pfrommer et al. (2008) Pfrommer C., Enßlin T. A., Springel V., 2008, MNRAS, 385, 1211
  * Planck Collaboration et al. (2020) Planck Collaboration et al., 2020, A&A, 641, A6
  * Prada et al. (2012) Prada F., Klypin A. A., Cuesta A. J., Betancort-Rijo J. E., Primack J., 2012, MNRAS, 423, 3018
  * Priewe et al. (2017) Priewe J., Williams L. L. R., Liesenborgs J., Coe D., Rodney S. A., 2017, MNRAS, 465, 1030
  * Rajpurohit et al. (2021) Rajpurohit K., et al., 2021, A&A, 654, A41
  * Raney et al. (2020) Raney C. A., Keeton C. R., Brennan S., Fan H., 2020, MNRAS, 494, 4771
  * Richard-Laferrière et al. (2023) Richard-Laferrière A., et al., 2023, MNRAS, 526, 6205
  * Roberg-Clark et al. (2018) Roberg-Clark G. T., Drake J. F., Reynolds C. S., Swisdak M., 2018, Phys. Rev. Lett., 120, 035101
  * Roediger et al. (2012) Roediger E., Lovisari L., Dupke R., Ghizzardi S., Brüggen M., Kraft R. P., Machacek M. E., 2012, MNRAS, 420, 3632
  * Roediger et al. (2013) Roediger E., Kraft R. P., Forman W. R., Nulsen P. E. J., Churazov E., 2013, ApJ, 764, 60
  * Russell et al. (2014) Russell H. R., et al., 2014, MNRAS, 444, 629
  * Sanders (2006) Sanders J. S., 2006, MNRAS, 371, 829
  * Sanders et al. (2016) Sanders J. S., Fabian A. C., Russell H. R., Walker S. A., Blundell K. M., 2016, MNRAS, 460, 1898
  * Schellenberger et al. (2017) Schellenberger G., Vrtilek J. M., David L., O’Sullivan E., Giacintucci S., Johnston-Hollitt M., Duchesne S. W., Raychaudhury S., 2017, ApJ, 845, 84
  * Schive et al. (2018) Schive H.-Y., ZuHone J. A., Goldbaum N. J., Turk M. J., Gaspari M., Cheng C.-Y., 2018, MNRAS, 481, 4815
  * Schwinn et al. (2017) Schwinn J., Jauzac M., Baugh C. M., Bartelmann M., Eckert D., Harvey D., Natarajan P., Massey R., 2017, MNRAS, 467, 2913
  * Simet et al. (2017) Simet M., McClintock T., Mandelbaum R., Rozo E., Rykoff E., Sheldon E., Wechsler R. H., 2017, MNRAS, 466, 3103
  * Spitzer (1956) Spitzer L., 1956, Physics of Fully Ionized Gases
  * Su et al. (2017) Su Y., et al., 2017, ApJ, 834, 74
  * Vikhlinin et al. (2001) Vikhlinin A., Markevitch M., Murray S. S., 2001, ApJ, 549, L47
  * Vikhlinin et al. (2006) Vikhlinin A., Kravtsov A., Forman W., Jones C., Markevitch M., Murray S. S., Van Speybroeck L., 2006, ApJ, 640, 691
  * Walker et al. (2016) Walker S. A., Sanders J. S., Fabian A. C., 2016, MNRAS, 461, 684
  * Wang & Markevitch (2018) Wang Q. H. S., Markevitch M., 2018, ApJ, 868, 45
  * Wang et al. (2016) Wang Q. H. S., Markevitch M., Giacintucci S., 2016, ApJ, 833, 99
  * Wang et al. (2023) Wang B., et al., 2023, ApJ, 957, L34
  * Wittman et al. (2018) Wittman D., Golovich N., Dawson W. A., 2018, ApJ, 869, 104
  * ZuHone (2011) ZuHone J. A., 2011, ApJ, 728, 54
  * ZuHone et al. (2011) ZuHone J. A., Markevitch M., Lee D., 2011, ApJ, 743, 16
  * van Weeren et al. (2011) van Weeren R. J., Brüggen M., Röttgering H. J. A., Hoeft M., 2011, MNRAS, 418, 230

##  Appendix A Chandra Observation Details

Cycle | Seq Num | PI | Exposure (ks)
---|---|---|---
2 | 800151 | David | 25
7 | 800633 | Kempner | 92
8 | 701380 | Garmire | 8
23 | 704323 | Bogdan | 2037
Table 3: Chandra observations used in this study.