Parameter Transfers for Warm-Started QAOA

Julian Obst

, Felix Truger

, Johanna Barzen

and Frank Leymann

Institute of Architecture of Application Systems, University of Stuttgart, Universit

atsstraße 38, Stuttgart, Germany

Keywords:

Quantum Approximate Optimization Algorithm, Warm-Start.

Abstract:

Variational Quantum Algorithms (VQAs) run shallow parameterized quantum circuits on quantum devices and

are thus suitable for current limited hardware. However, adverse effects like barren plateaus and local optima

may hinder the classical parameter optimization. Warm-starting techniques attempt to alleviate such problems

by utilizing pre-computed approximations or known solutions to initialize VQAs. In this work, we analyze a

combination of two different kinds of warm-starts, based on biased initial states and parameter transfers, re-

spectively. In particular, we investigate a warm-started variant of the Quantum Approximate Optimization Al-

gorithm (WS-QAOA) applied to the MAXCUT problem and analyze the transferability of optimized parameter

values between random graphs. We leverage their decomposition into subgraphs and regularities among these

subgraphs to facilitate parameter transfers for WS-QAOA and demonstrate a transfer for a random graph.

1 INTRODUCTION

Quantum computing is an emerging ﬁeld in com-

puter science with promising applications in various

domains, such as chemistry and optimization (Cao

et al., 2018; Harrigan et al., 2021). However, this

potential cannot be fully realized yet, as current

Noisy Intermediate-Scale Quantum (NISQ) devices

are error-prone and the size of quantum circuits that

can be executed successfully is limited (Preskill,

2018; Leymann and Barzen, 2020). Errors can accu-

mulate during lengthy computations on quantum de-

vices, making circuit depth a particular issue. Thus,

circuits executed on current hardware should be shal-

low. Until fault-tolerant quantum computers become

available, Variational Quantum Algorithms (VQAs)

bridge the gap and make use of current, limited quan-

tum hardware (Cerezo et al., 2021).

VQAs resort to shallow parameterized quan-

tum circuits, frequently referred to as Ans

atze,

that are executed on the quantum computer, while

their parameters are adjusted by an optimizer that

is executed on classical hardware. A prominent

example of such algorithms is the Quantum Approx-

imate Optimization Algorithm (QAOA) (Farhi et al.,

2014), which is designed to solve various combi-

https://orcid.org/0000-0002-1898-2167

https://orcid.org/0000-0001-6587-6431

https://orcid.org/0000-0001-8397-7973

https://orcid.org/0000-0002-9123-259X

natorial optimization problems like the MAXCUT,

MINIMUMVERTEXCOVER, and TRAVELINGSALES-

PERSON problem (Farhi et al., 2014; Zhang et al.,

2022; Ruan et al., 2020).

However, VQAs may run into problems like

barren plateaus and local minima (Wang et al., 2021;

Bittel and Kliesch, 2021), which are detrimental

effects in the optimization that may lead to slow con-

vergence of the algorithm. One class of approaches

that tries to circumvent these problems and speed

up the computation are so-called warm-starts (Egger

et al., 2021; Galda et al., 2021; Truger et al., 2024a).

The general idea is to give the algorithm an advantage

by utilizing known or efﬁciently generated results,

such as optimized parameter values from related

problem instances and classically approximated

parameterizations or solutions for the problem at

hand. Several techniques can be used to warm-start

quantum algorithms (Truger et al., 2024a). In

particular, there are two major groups of approaches

for warm-starting VQAs. The ﬁrst group encodes an

approximate solution into the algorithm itself. In this

sense, Egger et al. propose a QAOA variant called

warm-started QAOA (WS-QAOA) for the MAXCUT

problem, that uses an approximate solution to prepare

a biased initial state and improve upon the approx-

imation (Egger et al., 2021). Similarly, Tate et al.

propose a biased initial state directly based on a re-

laxation of MAXCUT to warm-start the QAOA (Tate

et al., 2023). The second group focuses on obtaining

Obst, J., Truger, F., Barzen, J. and Leymann, F.

Parameter Transfers for Warm-Started QAOA.

DOI: 10.5220/0013539300004525

In Proceedings of the 1st International Conference on Quantum Software (IQSOFT 2025), pages 37-48

ISBN: 978-989-758-761-0

good initial parameter values that are improved upon

in the optimization. One way to get good parameter

values is to identify similar problem instances and

use their optimized parameters for another instance.

This transfer of parameters has been investigated,

e.g., by Galda et al. for QAOA (Galda et al., 2021).

This work combines the two approaches and in-

vestigates the possibility of parameter transfers be-

tween instances of the MAXCUT problem for Egger

et al.’s WS-QAOA. Therefore, we analyze the algo-

rithm’s performance on subgraphs of random graphs

up to degree 5 to explore regularities and a structure

for potential parameter transfers. Moreover, we de-

scribe how the insights gained from this analysis are

utilized to determine good parameter values for larger

graphs when warm-starting is employed. We demon-

strate this by ﬁnding good initial parameter values for

a 50-node random graph instance with a maximum

degree of 5 that are very close to the actual optimum.

The remaining sections of this paper are organized

as follows. Section 2 introduces the necessary back-

ground and related work. Section 3 describes how the

investigation was performed. The insights and results

as well as their utilization are covered in Section 4.

Section 5 discusses limitations and possible extension

of the method. Finally, Section 6 summarizes our

work and gives an outlook on possible future work.

2 BACKGROUND AND RELATED

WORK

The QAOA, one of the most prominent examples

of VQAs, was introduced in a seminal work by

Farhi et al. (2014). They illustrate that the algorithm

can be applied to approximate solutions to the MAX-

CUT problem. In short, the parameterized circuit that

is executed on the quantum computer consists of an

initialization to a uniform superposition and two sub-

circuits parameterized by

γ and

β, respectively. These

subcircuits are referred to as the problem circuit and

mixer. The problem circuit is constructed such that

its ground state is an optimal solution to the problem

at hand, while the mixer’s ground state is the initial

state of the overall circuit. The problem circuit and

mixer are repeated p times, which is referred to as

the QAOA depth of the Ansatz. Thus, the overall cir-

cuit implements a trotterized approximation of an adi-

abatic evolution that aims to achieve a transition from

the known ground state of the mixer into the unknown

ground state of the problem circuit, representing an

optimal solution. The algorithm converges towards

an optimum with the increase of the QAOA depth p.

A partitioning of the nodes of an undirected graph

into two sets is called a cut, with the number of edges

between the two sets being the value of the cut. A

MAXCUT or Maximum Cut is thus a partitioning that

maximizes the number of edges between the two sets.

In the case of weighted graphs, the goal is to maxi-

mize the sum of the edge weights between both sets.

The problem is well known to be NP-hard, but can

be approximated. Executing the QAOA for a set of

parameters creates a distribution of cuts, with the ex-

pected value of the cut being referred to as the energy.

The objective of the QAOA is to optimize this energy,

thereby producing good cuts.

Egger et al. (2021) adapted the QAOA to uti-

lize approximations to the MAXCUT problem as

a starting point by encoding them into the initial

state of the quantum circuit. The adaptation, called

Warm-Started QAOA (WS-QAOA), makes use of the

best known classical approximation algorithm for

the MAXCUT problem by Goemans and Williamson

(1995). Thereby the QAOA is enhanced in the sense

that less optimization may be necessary if the ini-

tial state is biased towards a good solution as a start-

ing point or, conversely, lower QAOA depth suf-

ﬁces to reach a good result quality, making WS-

QAOA a promising algorithm for current NISQ hard-

ware. Speciﬁcally, instead of the uniform superpo-

sition, each qubit in the initial state of WS-QAOA

is more likely to be in either of the states |0⟩ or |1⟩

as prescribed by the approximation obtained from the

Goemans-Williamson algorithm (GW). The deviation

of the initial state from the uniform superposition and

the bias towards the approximation is controlled by a

regularization parameter ε. In addition, the mixer in

WS-QAOA is adapted for the modiﬁed initial state.

Egger et al. also propose a modiﬁed mixer that does

not have the initial state as a ground state, however, as

a ﬁrst step we focus on the version that is in line with

the adiabatic evolution behind QAOA.

Another way to warm-start VQAs, is by ﬁnding

a good initialization for the Ansatz parameters, e.g.,

by pre-computing viable parameter values or transfer-

ring optimized parameters from related problem in-

stances (Truger et al., 2024a). The advantage here

is, that all VQAs can be warm-started by parameter

initializations without having to change the underly-

ing algorithm. The difﬁculty lies in ﬁnding viable pa-

rameter values that yield good results. One approach

is to optimize parameters for a smaller problem in-

stance and then transferring the optimized parame-

ter values to the actual target problem. To this end,

Galda et al. (2021) investigated the transfer of opti-

mal QAOA parameter values for the MAXCUT prob-

lem between problem instances. In particular, they

IQSOFT 2025 - 1st International Conference on Quantum Software

considered regular and random graphs up to a degree

of 8 and 6, respectively. This analysis showed that a

transfer between instances is possible and primarily

depends on the degree of subgraphs. For instance, a

transfer between k- and d-regular instances is likely

to be successful, provided that k and d are both even

or both odd. Our work extends this analysis to WS-

QAOA, for which the degree alone is not enough to

guarantee a successful transfer anymore.

There are several works similar to the ones intro-

duced above. Tate et al. (2023) successfully warm-

start the QAOA for MAXCUT with low-rank Burer-

Monteiro relaxations, while Cain et al. (2022) show

that a warm-start that naively initializes the circuit

with an approximation fails. Shaydulin et al. (2021)

introduce a repository for optimized QAOA parame-

ters that enables parameter transfers. Brandao et al.

(2018) show that the landscapes of large 3-regular

graphs are instance independent, whereas Akshay

et al. (2021) demonstrated a concentration of opti-

mal QAOA parameters for p ∈ {1, 2}. Shaydulin

et al. (2023) explain how to transfer parameters for

weighted MAXCUT. More generally, Truger et al.

(2024b) document warm-starting patterns for quan-

tum algorithms, including warm-starts via biased ini-

tial states and warm-starts via parameter transfers.

However, none of the work that we are aware of com-

bines a warm-start via a biased initial state with a pa-

rameter transfer.

The Quantum Modelling Extension (QuantME)

introduces a warm-starting task that conceptually

facilitates parameter transfers for WS-QAOA in

workﬂow-based quantum applications (Beisel et al.,

2023). In particular, this task provides conﬁguration

attributes for warm-starting quantum executions uti-

lizing biased initial states and parameter initializa-

tions. However, it does not provide guidance on how

to effectively combine the two.

3 PARAMETER TRANSFER FOR

WS-QAOA

This section introduces our approach to parameter

transfers for WS-QAOA for the MAXCUT problem.

We ﬁrst give an overview of the decomposition of

such parameter transfers based on subgraphs, before

we provide more details on the concrete experimental

setup for our analysis.

3.1 Overview

On a high level, our approach considers the transfer-

ability of parameters for subgraphs to derive a param-

eter transfer for a targeted problem instance. As was

shown in the original QAOA paper, the energy values

for a particular graph can be evaluated as the sum over

subgraph contributions (Farhi et al., 2014). We inves-

tigate the QAOA at p = 1, where only two parameters

need to be optimized. To expand the investigation to

p > 1 one must conduct the analysis on larger and

more subgraphs as will be discussed in Section 5. In

general, the investigation is centered around energy

landscapes of subgraphs. Energy landscapes are cre-

ated by systematically evaluating the QAOA circuit in

its parameter space and computing the corresponding

energy values. They allow for the comparison of en-

ergy extrema within the parameter space to draw con-

clusions about the transferability of parameter values.

To determine the transferability of parameters for

WS-QAOA, we investigate the energy landscapes for

different subgraphs while considering different possi-

ble initial states. Speciﬁcally, we assess whether op-

timal parameter values coincide, i.e., whether good

values for one instance are also suitable for another.

For our purposes, optimal parameters are located at

maxima, since we investigate the MAXCUT problem.

For a ﬁxed set of parameters, the energy is evaluated

according to Equation (1) (cf. Galda et al. (2021)).

⟨C⟩

γ,

β⟩

= ⟨

γ,

β|C |

γ,

β⟩

= ⟨

γ,

β|

∑

(i, j)∈E

(1−Z

γ,

β⟩

|E|

−

∑

(i, j)∈E

⟨

γ,

β|Z

γ,

β⟩

|E|

−

∑

(i, j)∈E

i j

(

γ,

β)

(1)

C in Equation (1) is the problem-speciﬁc operator in

QAOA and e

i j

is the contribution of an individual

edge in the set of edges E. This means, instead of

looking at the graph as a whole, we can break down

the problem by investigating how each edge inﬂu-

ences the energy. The contribution e

i j

can be deter-

mined by looking at the subgraph that is induced by

the edge. The induced subgraph consists of the edge

itself, called the central edge that connects the central

nodes, and all nodes (and connecting edges) that are

at most distance p away. Therefore, the worst case is

that we have to consider as many subgraphs, as there

are edges in a graph. In reality, however, many sub-

graphs appear repeatedly. For example, for k-regular

graphs, there are k different subgraphs that need to be

investigated. Additionally, in WS-QAOA we need to

consider different initializations due to different pos-

sible approximations encoded into the initial state as

Parameter Transfers for Warm-Started QAOA

warm-starts. Thus, the number of energy landscapes

that need to be taken into account increases in the con-

text of WS-QAOA compared to standard QAOA.

To facilitate the identiﬁcation of individual sub-

graphs we introduce the following notation, which is

adapted from Galda et al. (2021): A subgraph is re-

ferred to as (x-y)-z, where x ≤y are the number of

neighbors of the two central nodes and z is the num-

ber of neighbors shared by both nodes. For example,

the graph in Figure 1 is called (3-3)-0.

3.2 Experimental Setup

Assuming a ﬁxed conﬁguration of WS-QAOA, our

analysis extends over every possible initialization for

every subgraph of regular graphs up to degree 5. After

accounting for isomorphisms, this yields 214 different

landscapes. Expanding the scope of the investigation

to random graphs up to degree 5 yields 35 subgraphs

and increases the total number of landscapes to 620.

For reasons discussed in Section 4.2, this number can

be reduced to 558.

To create landscapes for a subgraph, one ﬁrst

needs to construct the QAOA circuit, where each node

corresponds to one qubit. As discussed previously, we

adopt WS-QAOA from Egger et al. (2021) in its ba-

sic form. The ﬁrst consideration is the depth p that

we choose as p = 1 resulting in a circuit with two pa-

rameters (γ, β) to be optimized and two-dimensional

energy landscapes that can be illustrated. Addition-

ally, WS-QAOA can be adjusted by its regulariza-

tion parameter ε ∈ [0,0.5], where ε = 0.5 leads to the

original QAOA and lower values increase the inﬂu-

ence of the approximate solution utilized as a warm-

start. To put more emphasis on the warm-start, we

choose ε = 0.1. To prepare the biased initial state,

a rotation R

by an angle θ

is applied to each qubit

, where θ

= 2arcsin

√

ε if the corresponding node

is initialized with 0, and θ

= 2arcsin

√

1−ε if it is

initialized with 1. Thus, a single qubit reproduces

its initialization with probability 1 −ε if we were to

measure it. For example, a qubit of a node initial-

ized with 0 measures |0⟩ with probability 1 −ε. An

example for the WS-QAOA circuit is given in Fig-

ure 1. After constructing the circuit, it is executed

on a simulator where we measure the qubits of the

central edge, which are q

in all our experiments.

The portion of measurements, where q

̸= q

is the

probability of the edge being cut, i.e., the energy con-

tribution e

i j

of Equation (1). We create the land-

scape by sampling the energy in the parameters space

(γ,β) ∈[0, 2π] ×[0, π]. To this end, we sample 30 pa-

rameter values for each dimension to obtain the two

sets Γ and B, resulting in a total of |Γ| ·|B| = 900

equidistant samples. Since we are dealing with un-

weighted graphs, the landscapes are periodic, and it

is sufﬁcient to analyze the parameter space within the

given range.

4 RESULTS

This section presents the results of the analysis for

subgraphs and the transferability of parameters be-

tween graphs in WS-QAOA for MAXCUT. In partic-

ular, we ﬁrst look at the landscapes of regular graphs,

before moving on to random graphs. Next, we intro-

duce a transferability map to analyze the transferabil-

ity in more detail. Finally, we utilize the insights to

ﬁnd good parameters for a larger graph instance. For

veriﬁcation and reproducibility, the results can be ac-

cessed online (Obst and Truger, 2024). This includes

a visualization of all subgraphs with their correspond-

ing landscapes and numeric data.

(2.5)

•

ZZ(-γ)

• •

(-2.5) R

(2β) R

(2.5)

(0.6)

•

ZZ(-γ)

• •

(-0.6) R

(2β) R

(0.6)

•

ZZ(-γ)

(-0.6) R

(2β) R

(0.6)

(2.5)

ZZ(-γ)

•

(-2.5) R

(2β) R

(2.5)

(0.6)

•

ZZ(-γ)

(-0.6) R

(2β) R

(0.6)

•

(-0.6) R

(2β) R

(0.6)

Figure 1: Example subgraph and its circuit consisting of initialization, problem circuit, and mixer separated by dashed lines.

The subgraph is depicted with its initialization where purple and yellow nodes correspond to an initialization of 0 and 1

respectively. Each qubit corresponds to one node of the graph, where the ﬁrst two qubits q

are the central nodes, and each

ZZ gate corresponds to an edge in the graph. Since ε = 0.1, the parameters of the initial R

gates are 2arcsin

√

ε ≈ 0.64 if

the initialization of the corresponding node is 0, and 2arcsin

√

1−ε ≈ 2.50 if the initialization is 1. The parameter values are

rounded to one decimal place.

IQSOFT 2025 - 1st International Conference on Quantum Software

2π

low high

Figure 2: Graph (3-3)-0, one of the three subgraphs of a 3-

regluar graph, with all its initializations up to isomorphism.

An initialization with 0 is represented by a purple node, an

initialization with 1 by a yellow node. The central edge that

is being measured is highlighted. Values of the landscape

are normalized, as we only care about maxima and minima,

not their exact value. The graphs are ordered such that all

initializations on the left have central nodes that are initial-

ized equally. Additionally, the landscapes are paired with

their mirrored version on the opposite side.

4.1 Landscapes for Regular Graphs

In the following, we analyze landscapes for regular

graphs and describe regularities found between the

landscapes.

4.1.1 3-Regular Graphs

Examining the landscapes for subgraphs on their own

helps us gain a better understanding of the reasons for

good transferability and identify regularities among

the results. First, we focus on the subgraphs of regular

graphs, as their regularities also appear in other sub-

graphs. Figure 2 shows the most general subgraph in

the sense that there are no common neighbors among

the nodes of the central edge. The graph is depicted

with every possible initialization indicated by a color

coding: Nodes initialized to 0 are colored in purple,

nodes initialized to 1 are colored in yellow. Isomor-

phisms are not shown, as they produce identical re-

sults. Moreover, we also consider it an isomorphism

if all initializations of the nodes are inverted, as such

conﬁgurations would also result in the same energy

values. When checking for isomorphisms, it is im-

portant to label the central edge, as it is the one be-

ing measured as described in Section 3.2. This makes

no difference for d-regular graphs with d ≥ 3, but is

relevant for small random graphs. That way, all 12

different initializations and their corresponding land-

scapes are presented in Figure 2. All landscapes are

normalized for the visualization. The maximum is the

brightest spot and the minimum the darkest. For the

transfer of parameters, the actual energy values are

less relevant than the locations of extrema, particu-

larly the global maxima, as these mark the optimal

parameter values. Therefore, we can disregard that

scale for transfers and in the visualizations. For a bet-

ter illustration of regularities among the landscapes,

they are grouped in two categories: The ﬁrst (left) half

shows landscapes where outskirts have a low energy,

while the center generally shows a high energy. In the

second (right) half, it is the other way around. The

difference between the two groups lies in the initial-

ization of the central nodes. In the ﬁrst category, the

nodes are initialized identically, i.e., the central edge

is not part of the cut. For the second category, the

edge is part of the cut, i.e., nodes are initialized dif-

ferently. Intuitively, this is explained by the fact that

we only look at the energy of the central edge. Given

a good initialization, parameters that favor the initial

state, i.e., parameters around zero, should be expected

to result in high energy values.

The chosen arrangement of Figure 2 highlights an-

other relationship between the landscapes. The six

landscapes on the right-hand side appear to be in-

verted mirror images of those on the left-hand side.

More precisely, the landscapes on the opposite side

have the same shape but appear to be ﬂipped verti-

cally/horizontally and inverted, i.e., minima of one

landscape become maxima in the other and vice versa.

Landscapes that are related this way also have a con-

nection in the respective graphs: Taking one of the

central nodes and inverting the initialization of each

adjacent node, including the other central node, re-

sults in an isomorphic graph. Future work may

provide a better understanding of this phenomenon

through a deeper mathematical analysis.

4.1.2 Subgraphs with Common Neighbors

As mentioned before, we can understand the sub-

graphs (x-y)-z with z ≥ 1 as a version of the

graph (x-y)-0 where the central nodes have z com-

mon neighbors. Analogous to the observations of

Galda et al. (2021), the landscapes of k-regular sub-

graphs are very similar for a ﬁxed k when their ini-

tialization is equivalent, as explained in the follow-

ing. This fact is demonstrated in Figure 3 with equiv-

alent initializations for the subgraphs (3-3)-0, (3-3)-1,

Parameter Transfers for Warm-Started QAOA

2π

Figure 3: Subgraphs of 3-regular graphs with similar land-

scapes. From left to right, the subgraphs are (3-3)-0, (3-

3)-1, (3-3)-1, (3-3)-2. Similar landscapes arise when identi-

cally initialized neighbors are merged into a common neigh-

bor of both central nodes. The MAD between the normal-

ized landscapes is at most 0.03. The second and third land-

scape look alike, because splitting the common neighbor

would result in isomorphic graphs.

and (3-3)-2, which induce resembling landscapes. To

quantify this similarity, we determine the Mean Ab-

solute Difference (MAD). For two normalized energy

landscapes E

and sets of parameter values Γ, B it

is calculated as follows:

MAD(E

) =

|Γ|·|B|

∑

(γ,β)∈Γ×B

(γ,β) −E

(γ,β)

Since the energy is in the interval [0, 1], identical land-

scapes have an MAD of 0 and opposite landscapes

have an MAD of 1. In Figure 3, the graph (3-3)-1

is shown twice with different initialization, yet their

landscapes look identical. To be precise, their MAD

is 0.02. While they are in fact non-isomorphic, the

similarity appears because of their relation to (3-3)-

0: In both cases, replacing the common neighbor of

the central nodes with two identically initialized sep-

arate nodes would create the same version of (3-3)-

0, i.e., the initialization of these subgraphs is equiva-

lent to the one depicted for (3-3)-0. Hence, (3-3)-0 is

the most general of the three subgraphs, as it allows

for the largest number of initializations, and “split-

ting” common neighbors by replacing them with sep-

arate nodes returns one of these instances. Since both

graphs are descendants of the same parent, we con-

sider only one of them in further analyses.

Increasing z, i.e., introducing common neighbors,

does not induce additional initializations to consider

over those known for the most general subgraph.

Conversely, there are initializations for (3-3)-0 for

which there is no equivalent initialization for (3-3)-1

or (3-3)-2. For example, in the top right graph de-

picted in Figure 2, the neighbors of the central nodes

are initialized both with 1 and both with 0, respec-

tively. Thus, no pair of these nodes could be merged

into a common neighbor and there is no equivalent

initialization for (3-3)-1.

2π

Figure 4: Different types of graphs that produce similar

landscapes. Neighbors that are initialized oppositely (one

is 0, the other is 1) cancel each other out. Removing them

does not substantially affect the landscape, i.e., the MAD

between them is smaller than 0.04.

4.1.3 Node Canceling

We brieﬂy extend the analysis to subgraphs of

5-regular graphs, whose landscapes exhibit similari-

ties with those of 3-regular graphs. Namely, all land-

scapes of graph (3-3)-0 are also present among those

for (5-5)-0. An example is depicted in Figure 4.

Again, the similarity is due to a connection between

the underlying graphs: The central nodes of the (5-5)-

0 instance each have two neighbors that are initialized

differently. Disregarding such pairs of neighbors and

removing them from the graph on both sides of the

central edge results in the graph (3-3)-0 which pro-

duces the same landscape. In that sense, nodes can

cancel each other out and some initializations for (5-

5)-0 can be reduced to an equivalent initialization for

(3-3)-0. We refer to this as node canceling.

4.2 Random Graphs

The analysis can also be extended to random graphs.

In particular, the observations regarding common

neighbors and node canceling as introduced above

also apply to subgraphs of random graphs up to a

degree of 5. There are 35 subgraphs with a total

of 620 initializations. However, due to equivalent

initializations of graphs with common neighbors,

these can be reduced to 558 that need to be ana-

lyzed. For subgraphs of random graphs, the number

of neighbors of the central nodes can differ. For

example, the subgraph (3-5)-0 has a central node that

is missing two neighbors compared to (5-5)-0. The

rightmost example in Figure 4 illustrates that node

canceling also leads to resembling landscapes for

these types of subgraphs.

However, subgraphs of random graphs also intro-

duce landscapes with shapes that are not present for

regular graphs. Speciﬁcally, these landscapes appear

when the central nodes have a mixed parity, i.e., one

central node has an even and the other has an odd de-

gree. Still, subgraphs of random graphs exhibit the

IQSOFT 2025 - 1st International Conference on Quantum Software

same regularities as regular graphs, such as the mir-

rored relationship, merged nodes, and node cancel-

ing. Node cancelling is particularly interesting, as we

can also go in the opposite direction. Instead of re-

moving nodes, we can also add them to a graph, until

both central nodes have the same number of neigh-

bors. That way, subgraphs of random graphs can be

related to those of regular graphs. Clearly, this is only

possible if both central nodes have the same parity,

i.e., the degree of the central nodes is either even or

odd for both, as we add two nodes at a time. There-

fore, subgraphs of random graphs fall in one of two

categories: Either their central nodes have the same

parity, or they have mixed parity.

4.3 Transferability Map

This section examines the parameter transferability

between subgraphs of random graphs up to a degree

of 5 in more detail. To this end, we create a transfer-

ability map that visualizes the quality of such trans-

fers. We describe the construction of the transferabil-

ity map, the key observations, and the inﬂuence of the

degree of central nodes on the transferability.

4.3.1 Constructing the Transferability Map

To quantify the transferability between different sub-

graphs we determine a transferability coefﬁcient τ. It

is calculated by taking the sampled optimal parame-

ters of one subgraph, the donor graph, and evaluating

their performance for another subgraph, the acceptor

graph. Determining the transferability is straightfor-

ward given the sampled landscapes. The transferabil-

ity coefﬁcient τ can be computed as follows:

τ = avg

max

,β

max





acc

(γ

max

,β

max

)

max

γ,β

acc

(γ,β)}





(2)

We take the locations of global maxima in the sam-

pled landscape, (γ

max

,β

max

), of the donor subgraph

(there can be multiple maxima), evaluate the energy

acc

of the acceptor graph at these locations, normal-

ize by the maximum energy of the acceptor graph and

average the values. The coefﬁcient τ is a value be-

tween 0 and 1, where 0 is the worst possible transfer-

ability and 1 is the optimal transferability.

Figure 5 shows the transferability coefﬁcient for

every pair of two subgraphs and every initialization.

As Galda et al. noted, transferability is a directional

property. Therefore, the map is not symmetric along

its diagonal axis. The appearance of the map depends

on the arrangement of the subgraphs and their initial-

izations. While there are multiple possibilities to or-

der the graphs and their initializations, we decided for

an arrangement that is beneﬁcial for the illustration of

some key observations. In particular, the graphs are

ordered by increasing minimum degree x and increas-

ing maximum degree y, i.e., ﬁrst all the graphs (1-∗),

then (2-∗), etc. In addition, within each category, the

number of merged nodes z increases. The initializa-

tions of a single subgraph are divided into two groups,

the ﬁrst having identically initialized central nodes,

the second having central nodes initialized oppositely.

The complete ordering of initializations of a subgraph

can be found online (Obst and Truger, 2024) together

with all landscapes.

4.3.2 Key Observations

As the transferability map contains a substantial

amount of information, we draw attention to key ob-

servations that unveil its utility. For one, the map

facilitates ﬁnding a suitable donor graph for a given

subgraph by scanning the respective row to identify

good candidates for a parameter transfer. Moreover,

the map helps to identify certain patterns and regular-

ities, to estimate the success of parameter transfers,

and to gain insights into properties that can be used to

make more general statements about transferability.

An important observation is, that the initialization

introduced with WS-QAOA signiﬁcantly inﬂuences

the transferability. To be precise, the transferabili-

ties described by Galda et al. (2021) no longer apply

when a biased initial state is introduced, as these con-

ditions relied on graph structure alone. For example,

optimal parameters of a subgraph were trivially trans-

ferable to the same subgraph. However, for the same

instance in WS-QAOA with different initializations,

such transfers may fail. In particular, this is the case

across initializations where the central edge is cut and

others where the central edge is not cut, i.e., the two

groups described earlier. This restriction is visible in

the map by the fact that almost all subgraphs have

four distinct sectors that appear like a checkerboard

pattern (see Figure 5 box A). Note that such a section

contains all possible initializations for a single sub-

graph. The four sectors reﬂect all possible transfers

for the different initializations of the central nodes of

the donor and acceptor graph. They illustrate, that a

transfer is most successful, when the central nodes are

either initialized identically in both graphs or initial-

ized differently in both.

The next observation is, that the number of com-

mon neighbors has little inﬂuence on the transferabil-

ity (B), which is reﬂected in the similarity of sections

separated by dotted lines in Figure 5. This occurs be-

cause splitting common neighbors results in very sim-

ilar landscapes, as discussed above and shown in Fig-

ure 3. This phenomenon is already present when no

Parameter Transfers for Warm-Started QAOA

(1-∗) (2-∗) (3-3) (3-4) (3-5) (4-4) (4-5) (5-5)

Donor subgraph

(1-∗)

(2-2)

(2-3)

(2-4)

(2-5)

(3-3)

(3-4)

(3-5)

(4-4)

(4-5)

(5-5)

Acceptor subgraph

A B C

0.0

0.2

0.4

0.6

0.8

1.0

Transferability Coefﬁcient

Figure 5: Transferability coefﬁcients between every pair of subgraphs forming the Transferability Map. Each pixel is a

coefﬁcient τ ∈ [0,1]. Dark colors indicate a bad transferability, bright colors indicate a good transferability. Each row is an

acceptor graph, each column a donor graph. Thick white lines distinguish graphs with differing minimum degree. Dashed

white lines separate graphs with different degrees of the central nodes, dotted lines further divide the number of common

neighbors. Overall, there are 35 subgraphs with a total of 558 initializations, starting with low degrees up to degree 5.

For clarity, some of the labels are omitted: (1-∗) stands for (1-1) through (1-5), (2-∗) stands for (2-2) through (2-5). The

highlighted sections and labels A, B, C serve as references for the observations described in Section 4.3.2.

biased initial state is involved (Galda et al., 2021). For

our purposes, this means that we need to prioritize the

initialization over the exact type of graph when look-

ing for a suitable donor graph.

Distinctive features of the map are absolutely per-

fect and disastrous transferabilities in some cases (C).

These become apparent as yellow and black patches

in Figure 5, most notably in the sections of (4-5).

Here we have optimal and worst case transferability

with coefﬁcients of 1 and 0, respectively. This is due

to the shape of the respective landscapes, which ex-

hibit either exactly one maximum or one minimum

that is located right at the center of the landscape, re-

sulting in either a perfect transfer or the worst pos-

sible. Interestingly, these patches only appear for

donors whose central nodes are initialized the same.

However, the other sector, where the central nodes of

donor and acceptor are initialized oppositely, also ex-

hibits relatively high coefﬁcients. This sector is par-

ticularly important for a parameter transfer of WS-

QAOA, as we will discuss in Section 4.4.

4.3.3 Inﬂuence of Node Degrees

Another important characteristic is the degree of the

central nodes, particularly, their parity. Galda et al. al-

ready described this and noted that a transfer between

IQSOFT 2025 - 1st International Conference on Quantum Software

k-regular and d-regular graphs is likely to be success-

ful, provided k and d are either both even or both odd

(Galda et al., 2021). When taking the initialization of

the central edge into account, this also seems to hold

with warm-starts in place. For example, (5-5) appears

to be a better donor for (3-3) than (4-4).

They also found that there is a good transfer-

ability between graphs ( j-k) and (l-m) provided that

{j, k, l, m} are all odd or all even. This also seems

true for WS-QAOA. For example, (3-5) as an accep-

tor has brighter sections when receiving parameters

from (5-5) rather than (4-4). Conversely, a transfer

from odd to even parity is also unlikely to be success-

ful. Notably, for (4-4) as an acceptor, the graph (2-4)

appears to be a better donor than (3-3). Interestingly,

graphs with even degrees do not show a checkerboard

pattern when receiving parameters from mixed parity

donors. An example for this would be (4-4) accepting

parameters from (4-5).

Moreover, Galda et al. showed cases of good

transferability between subgraphs with mixed parity.

With WS-QAOA, there seems to be good transferabil-

ity if both donor and acceptor graph have a mixed par-

ity, since the aforementioned sections with yellow and

black patches appear in exactly these cases.

4.4 Application for Transferability

This section describes, how we can use the above

insights to obtain viable parameter values for larger

graphs composed of subgraphs. To this end, we de-

compose a larger graph into its subgraphs and try to

reduce the number of subgraphs that we have to con-

sider. We illustrate this application by determining vi-

able parameter values for the 50-node random graph

with 75 edges and a maximum degree of 5 shown in

Figure 6. A suitable initialization for the MAXCUT

problem on this graph was determined using the GW

algorithm (Goemans and Williamson, 1995).

The following procedure is based on the fact, that

the total energy is composed of contributions of the

subgraphs as described in Section 3.1. We can lever-

age the insights described above to obtain an approx-

imation of the landscape of the target graph shown in

Figure 6, from which we can derive a good parame-

terization. Speciﬁcally, subgraphs can be transformed

into a standardized format to increase the number of

isomorphic subgraphs, thereby reducing the number

of subgraphs that need to be considered and minimiz-

ing the effort required to determine good parameter

values. The ﬁrst simpliﬁcation is to consider only

subgraphs of the form (x-y)-0 by splitting common

neighbors of the central nodes. Since the landscapes

are very similar before and after the split, we do not

lose much information. Further, a graph falls in one

of three categories: Both central nodes have even

degrees, both have odd degrees or their degrees are

mixed. To transform these subgraphs further, we can

utilize the node canceling effects, i.e., add and remove

nodes that cancel out. In case that both central nodes

have even degrees, we can add two oppositely initial-

ized nodes at a time to a central node until the graph

takes the form of (4-4). If their degrees are odd, we

can add as many pairs as necessary to end up with (5-

5), and for mixed parity we can transform them into

instances of (4-5). In total, these 3 graphs allow for

90 different initializations.

Adding up the energy values of the subgraphs

with the respective initializations for each sampled

point results in the true energy landscape of the tar-

get graph. This true energy landscape is illustrated on

the left-hand side of Figure 6. Transforming the sub-

graphs before applying this procedure generates the

approximate landscape next to it. The target graph

consists of 38 different subgraphs, with the 3 most

frequent appearing 8, 7, and 5 times, respectively.

The simplifying transformations halve the number of

different subgraphs to 19, with the 3 most frequent

ones appearing 11, 10, and 9 times, respectively.

These 3 most frequent subgraphs produce the land-

scape shown on the right-hand side of Figure 6.

2π

True Energy

2π

Approximate Energy

2π

Energy of Top 3

Figure 6: Approximating the landscape of the graph on the left. The 50-node graph is pictured with its initialization, calculated

by the GW algorithm. Landscapes from left to right are the true energy landscape, its approximation after transforming all the

subgraphs to an equivalent of the form (4-4), (4-5), or (5-5), and the approximation of the landscape by only considering the

3 most frequent graphs after the simplifying transformations.

Parameter Transfers for Warm-Started QAOA

As a ﬁrst observation, the landscapes show low

energy values in the center and high energy values

near the edges. Thus, they fall into the second group

of landscapes, where the central edge is cut (cf. Sec-

tion 4.1.1). This is expected because the initialization

is derived from a good approximation utilized for the

warm-start, resulting in subgraphs where many cen-

tral edges are already cut, i.e., initialized oppositely.

Further, it is expected that most of the subgraphs will

have many such edges, similar to the top right graph

of Figure 2. Therefore, the overall landscape resem-

bles the landscape of these subgraphs. Thus, an even

simpler transfer would be taking optimized parame-

ters from one of these graphs.

Evidently, the landscapes of Figure 6 appear very

similar despite the simpliﬁcations described above. In

particular, the MAD of true and approximate energy

is smaller than 0.02. If we consider only the three

most frequent graphs, the landscape changes notice-

ably, but not substantially. Since we only considered

the energy of 30 subgraphs for the latter, the total en-

ergy values are generally lower than in the original

landscape. However, it is not the total energy value

that is crucial for obtaining good parameter values

as mentioned in Section 4.1.1, but rather the location

of the maxima, that need to remain close. Keep in

mind, that the parameters (γ, β) were sampled in the

range of [0,2π] ×[0,π] with 30 samples per dimen-

sion. For the ﬁrst approximation, the maximizing pa-

rameters are off by only one sample in the direction

of β, i.e., we are off by

from the sampled maxi-

mum. Surprisingly, the second approximation from

the top three subgraphs retains the maximum at the

exact same sample position as the true energy.

What the experiments have shown is, that we may

only focus our attention towards subgraphs that have

a central edge that is cut, as these are the most fre-

quent ones. Additionally, it can sufﬁce to consider

only three types of graphs, one for each of the cate-

gories of even, odd, and mixed parity.

5 DISCUSSION AND

LIMITATIONS

While we analyzed many different subgraphs, and

demonstrated how to successfully transfer parame-

ters, there still remain open questions. Here, we dis-

cuss limitations of our study and how these shortcom-

ings may be addressed.

For one, it is not always feasible to calculate

the entire landscape, e.g., when larger QAOA depths

p > 1 are involved. In such cases, it is beneﬁcial to re-

sort to the simpliﬁcations introduced above, identify

the most frequent subgraphs and optimize the energy

of their central edges. The parameter values can then

be used as an initialization for the instance at hand.

We suspect, that there is a concentration of good pa-

rameters in proximity to the optimal parameters of our

example. However, further research is required to ver-

ify whether this is the case.

As mentioned in Section 3.2, the choice of ε = 0.1

puts more emphasis on the initialization. However,

this parameter can also be selected differently or even

optimized, depending on the objective function in

use (Truger et al., 2022). Such alternative objective

functions for WS-QAOA may lead to different re-

sults for the landscapes and regularities for parameter

transferability.

The number of possible initializations grows

rapidly with the size of the graph. For this reason, we

only analyzed graphs up to degree 5, which may be

sufﬁcient for some use cases. While we suspect that

the regularities described above generalize to larger

degrees, it is unclear whether they actually hold. It

would be interesting to see the transferability in WS-

QAOA for graphs up to degree 8 as analyzed by Galda

et al. Since many of their ﬁndings for standard QAOA

are consistent with ours for WS-QAOA, we suspect

that the regularities also hold for degrees up to 8.

However, an analysis could provide further insights

into the impact of the degree on transferability.

To calculate the transferability coefﬁcient, we

sampled landscapes, which are limited to the grid of

points that were evaluated. Therefore, there may be

even better parameterizations than those correspond-

ing to the maxima in the sampled landscapes, which

could be obtained by additional optimization. Nev-

ertheless, the sampled landscapes appear smooth and

thus sufﬁcient for the analysis of parameter transfers.

As noted above, our analysis is centered on a

QAOA depth of p = 1. While we suspect that some of

the regularities may also be present for p > 1 and that

techniques for determining good parameters could

still be applied, further study is required for conﬁr-

mation. Such an investigation would involve larger

subgraphs, since edges that are distance p away from

the central edge form the subgraph and contribute to

its energy. This implies that more types of subgraphs

need to be analyzed, and more initializations have to

be considered. The rapid growth of initializations that

comes with increasing subgraph size may render an

analysis as thorough as for p = 1 impractical. More

than two circuit parameters further complicate an in-

vestigation, requiring the sampling of signiﬁcantly

more data points and limiting the visualization.

Thus far, our analysis is limited to unweighted

graphs. Investigating weighted graphs requires a dif-

IQSOFT 2025 - 1st International Conference on Quantum Software

ferent approach, as the enumeration of every subgraph

is impossible with weights. Previous work explored a

transfer of QAOA parameters for the MAXCUT prob-

lem on weighted graphs (Shaydulin et al., 2023). It is

unclear how weighted graphs affect the transferability

of optimal parameters for WS-QAOA.

Our analysis was conducted based on noise-

less quantum simulation, however, using error-prone

NISQ hardware may pose additional challenges and

potentials for the approach of parameter transfers in

(WS-)QAOA. In particular, it was shown that apply-

ing the circuit cutting concept to QAOA for MAX-

CUT by breaking down the graph at hand can sig-

niﬁcantly reduce the effects of noise on NISQ de-

vices (Bechtold et al., 2023). Circuit cutting focuses

on partitioning circuits into multiple smaller circuits,

whose individual execution results are post-processed

to obtain the original circuit’s result. On a higher

level, the approach of parameter transfers for WS-

QAOA based on subgraph decomposition is similar

to circuit cutting, since the subgraph decomposition

in our approach also results in an effective breakdown

on the circuit level. Therefore, parameter transfers

for (WS-)QAOA based on subgraphs likely allow for

similar advantages on current hardware.

6 SUMMARY AND FUTURE

WORK

In this paper, we investigated a combination of two

warm-starting techniques for the QAOA applied to

the MAXCUT problem. The techniques in question

are starting the computation with a biased initial state

following Egger et al.’s WS-QAOA (Egger et al.,

2021) and initializing parameters by transfers from

subgraph instances (Galda et al., 2021). To this end,

we analyzed energy landscapes of subgraphs of ran-

dom graphs up to a degree of 5. We documented a

number of regularities among the graphs and their ini-

tializations that can be leveraged to derive suitable ini-

tial parameters for larger graphs. The application of

these insights was demonstrated on a 50-node graph

instance initialized with a suitable approximation en-

coded into the biased initial state. We found, that a

few small subgraph instances sufﬁce to acquire good

initial parameter values for the target graph.

For future work, it would be interesting to increase

the scope of the approach, e.g., to graphs of higher

degrees or increased QAOA depth p to see whether

the regularities continue and to observe potential new

ones. Further, some questions remain open, for ex-

ample why the landscapes of two categories exhibit

a mirror relationship. Additionally, tackling different

algorithms or variations like the modiﬁed mixer pro-

posed by Egger et al. (2021) would be an interesting

expansion. Another idea would be the investigation of

similar parameter transfers for WS-QAOA in the con-

text of the MAXCUT problem for weighted graphs, as

discussed in Section 5.

ACKNOWLEDGMENTS

This work was funded in part by the BMWK

projects SeQuenC (01MQ22009B) and EniQmA

(01MQ22007B)

REFERENCES

Akshay, V., Rabinovich, D., Campos, E., and Biamonte, J.

(2021). Parameter concentrations in quantum approx-

imate optimization. Physical Review A, 104(1).

Bechtold, M., Barzen, J., Leymann, F., Mandl, A., Obst, J.,

Truger, F., and Weder, B. (2023). Investigating the ef-

fect of circuit cutting in QAOA for the MaxCut prob-

lem on NISQ devices. Quantum Science and Technol-

ogy, 8(4).

Beisel, M., Barzen, J., Bechtold, M., Leymann, F., Truger,

F., and Weder, B. (2023). QuantME4VQA: Modeling

and Executing Variational Quantum Algorithms Using

Workﬂows. In Proceedings of the 13

International

Conference on Cloud Computing and Services Science

(CLOSER 2023), pages 306–315. SciTePress.

Bittel, L. and Kliesch, M. (2021). Training Variational

Quantum Algorithms Is NP-Hard. Physical Review

Letters, 127:120502.

Brandao, F. G. S. L., Broughton, M., Farhi, E., Gutmann, S.,

and Neven, H. (2018). For Fixed Control Parameters

the Quantum Approximate Optimization Algorithm’s

Objective Function Value Concentrates for Typical In-

stances. arXiv:1812.04170.

Cain, M., Farhi, E., Gutmann, S., Ranard, D., and Tang, E.

(2022). The QAOA gets stuck starting from a good

classical string. arXiv:2207.05089.

Cao, Y., Romero, J., and Aspuru-Guzik, A. (2018). Poten-

tial of quantum computing for drug discovery. IBM

Journal of Research and Development, 62(6):6:1–

6:20.

Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S. C.,

Endo, S., Fujii, K., McClean, J. R., Mitarai, K.,

Yuan, X., Cincio, L., and Coles, P. J. (2021). Vari-

ational quantum algorithms. Nature Reviews Physics,

3(9):625–644.

Egger, D. J., Mare

cek, J., and Woerner, S. (2021). Warm-

starting quantum optimization. Quantum, 5:479.

Farhi, E., Goldstone, J., and Gutmann, S. (2014).

A Quantum Approximate Optimization Algorithm.

arXiv:1411.4028.

Parameter Transfers for Warm-Started QAOA

Galda, A., Liu, X., Lykov, D., Alexeev, Y., and Safro, I.

(2021). Transferability of optimal QAOA parameters

between random graphs. In 2021 IEEE International

Conference on Quantum Computing and Engineering

(QCE), pages 171–180. IEEE Computer Society.

Goemans, M. X. and Williamson, D. P. (1995). Improved

approximation algorithms for maximum cut and sat-

isﬁability problems using semideﬁnite programming.

Journal of the ACM (JACM), 42(6):1115–1145.

Harrigan, M. P., Sung, K. J., Neeley, M., Satzinger, K. J.,

Arute, F., Arya, K., Atalaya, J., Bardin, J. C., Barends,

R., Boixo, S., Broughton, M., Buckley, B. B., Buell,

D. A., Burkett, B., Bushnell, N., Chen, Y., Chen, Z.,

Chiaro, B., Collins, R., Courtney, W., Demura, S.,

Dunsworth, A., Eppens, D., Fowler, A., Foxen, B.,

Gidney, C., Giustina, M., Graff, R., Habegger, S., Ho,

A., Hong, S., Huang, T., Ioffe, L. B., Isakov, S. V.,

Jeffrey, E., Jiang, Z., Jones, C., Kafri, D., Kechedzhi,

K., Kelly, J., Kim, S., Klimov, P. V., Korotkov, A. N.,

Kostritsa, F., Landhuis, D., Laptev, P., Lindmark, M.,

Leib, M., Martin, O., Martinis, J. M., McClean, J. R.,

McEwen, M., Megrant, A., Mi, X., Mohseni, M.,

Mruczkiewicz, W., Mutus, J., Naaman, O., Neill, C.,

Neukart, F., Niu, M. Y., O’Brien, T. E., O’Gorman,

B., Ostby, E., Petukhov, A., Putterman, H., Quintana,

C., Roushan, P., Rubin, N. C., Sank, D., Skolik, A.,

Smelyanskiy, V., Strain, D., Streif, M., Szalay, M.,

Vainsencher, A., White, T., Yao, Z. J., Yeh, P., Zal-

cman, A., Zhou, L., Neven, H., Bacon, D., Lucero, E.,

Farhi, E., and Babbush, R. (2021). Quantum approx-

imate optimization of non-planar graph problems on

a planar superconducting processor. Nature Physics,

17(3):332–336.

Leymann, F. and Barzen, J. (2020). The bitter truth

about gate-based quantum algorithms in the NISQ era.

Quantum Science and Technology, 5(4):044007.

Obst, J. and Truger, F. (2024). Parameter Transfer

WS-QAOA GitHub Repository. https://github.com/

UST-QuAntiL/Parameter-Transfer-WS-QAOA.

Preskill, J. (2018). Quantum computing in the NISQ era

and beyond. Quantum, 2:79.

Ruan, Y., Marsh, S., Xue, X., Liu, Z., and Wang, J. (2020).

The Quantum Approximate Algorithm for Solving

Traveling Salesman Problem. Computers, Materials

& Continua, 63(3):1237–1247.

Shaydulin, R., Lotshaw, P. C., Larson, J., Ostrowski, J., and

Humble, T. S. (2023). Parameter Transfer for Quan-

tum Approximate Optimization of Weighted Max-

Cut. ACM Transactions on Quantum Computing,

4(3):1–15.

Shaydulin, R., Marwaha, K., Wurtz, J., and Lotshaw, P. C.

(2021). QAOAKit: A Toolkit for Reproducible Study,

Application, and Veriﬁcation of the QAOA. In 2021

IEEE/ACM 2

International Workshop on Quantum

Computing Software (QCS), pages 64–71.

Tate, R., Farhadi, M., Herold, C., Mohler, G., and Gupta,

S. (2023). Bridging classical and quantum with SDP

initialized warm-starts for QAOA. ACM Transactions

on Quantum Computing, 4(2):1–39.

Truger, F., Barzen, J., Bechtold, M., Beisel, M., Ley-

mann, F., Mandl, A., and Yussupov, V. (2024a).

Warm-Starting and Quantum Computing: A System-

atic Mapping Study. ACM Computing Surveys, 56(9).

Truger, F., Barzen, J., Beisel, M., Leymann, F., and Yus-

supov, V. (2024b). Warm-Starting Patterns for Quan-

tum Algorithms. In Proceedings of the 16

Interna-

tional Conference on Pervasive Patterns and Applica-

tions (PATTERNS 2024), pages 25–31. Xpert Publish-

ing Services (XPS).

Truger, F., Beisel, M., Barzen, J., Leymann, F., and Yus-

supov, V. (2022). Selection and Optimization of Hy-

perparameters in Warm-Started Quantum Optimiza-

tion for the MaxCut Problem. Electronics, 11(7).

Wang, S., Fontana, E., Cerezo, M., Sharma, K., Sone, A.,

Cincio, L., and Coles, P. J. (2021). Noise-induced bar-

ren plateaus in variational quantum algorithms. Na-

ture Communications, 12(1).

Zhang, Y., Mu, X., Liu, X., Wang, X., Zhang, X., Li, K.,

Wu, T., Zhao, D., and Dong, C. (2022). Applying the

quantum approximate optimization algorithm to the

minimum vertex cover problem. Applied Soft Com-

puting, 118:108554.

IQSOFT 2025 - 1st International Conference on Quantum Software