[1]Jonas Bley

[1]Artur Widera

1]Department of Physics and Research Center OPTIMAS, RPTU Kaiserslautern-
Landau, Kaiserslautern, 67663, Germany

2]Department of Computer Science and Research Initiative QC-AI, RPTU
Kaiserslautern-Landau, Kaiserslautern, 67663, Germany

3]Embedded Intelligence, German Research Center for Artificial Intelligence,
Kaiserslautern, 67663, Germany

4]Faculty of Physics, Chair of Physics Education, Ludwig-Maximilians-
Universität München (LMU Munich), Munich, 80539, Germany

5]Department of Electrical and Computer Engineering, RPTU Kaiserslautern-
Landau, Kaiserslautern, 67663, Germany

# Multi-qubit state visualizations to support problem solving – a pilot study

<EMAIL_ADDRESS>Eva Rexigel<EMAIL_ADDRESS>Alda Arias
<EMAIL_ADDRESS>Lars Krupp<EMAIL_ADDRESS>Steffen Steinert
<EMAIL_ADDRESS>Nikolas Longen<EMAIL_ADDRESS>Paul Lukowicz<EMAIL_ADDRESS>Stefan Küchemann
<EMAIL_ADDRESS>Jochen Kuhn<EMAIL_ADDRESS>Maximilian
Kiefer-Emmanouilidis<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [
[  [  [

###### Abstract

In the rapidly evolving interdisciplinary field of quantum information science
and technology, a big obstacle is the necessity of understanding high-level
mathematics to solve complex problems. Visualizations like the (dimensional)
circle notation enable us to visualize not only single-qubit but also complex
multi-qubit states, entanglement, and quantum algorithms. Current findings in
educational research suggest that incorporating visualizations in settings of
problem solving can have beneficial effects on students’ performance and
cognitive load compared to solely relying on symbolic problem solving content.
In this pilot study, we aim to take a first step to identify in which contexts
students benefit from the presentation of visualizations of multi-qubit
systems in addition to mathematical formalism. We compare students’
performance, time taken, and cognitive load when solving problems using the
mathematical-symbolic Dirac notation alone with using it accompanied by the
circle notation or the dimensional circle notation in single- and multi-qubit
systems. Although little overall differences in students’ performance can be
detected depending on the presented representations, we observe that problem-
solving performance is student- and context-dependent. In addition, the
results indicate reduced cognitive load when participants are presented with
visualization. The results are discussed with respect to relevant design
aspects for future studies.

###### keywords:

Multi-qubit systems, Visualizations, Quantum operations, Entanglement,
Multiple external representations, Cognitive load

##  1 Introduction

The interdisciplinary nature of the field of Quantum Information Science and
Technology (QIST) necessitates endeavors to educate workforce with many
different backgrounds [1]. This implies that the quantum education community
needs to find ways to facilitate the introduction to QIST to a broad audience.
One way of facilitating such an introduction is the use of suitable
representations, including visualization, to convey nontrivial quantum
concepts [2].

Visualization is generally considered a useful tool in Science, Technology,
Engineering & Mathematics (STEM)education [3, 4, 5, 6, 7, 8] and in QIST
education in particular [9, 10, 2]. In educational psychology, the Design,
Functions, Tasks (DeFT)framework by Ainsworth [11] considers how and why to
use multiple external representations (MERs). Combined with Cognitive Load
Theory (CLT)[12], it offers theoretical predictions for possible benefits in
learning and testing using MERs. It is unknown whether and under which
conditions learning and problem solving are supported by providing
visualization in addition to the mathematical symbolic representation (e.g.,
the Dirac notation) in quantum computing education. With this study, our
objective is to take the first step in investigating whether visualizations
provide a benefit to solve problems in QIST, when provided in addition to the
Dirac notation, and under which circumstances possible benefits occur.

QIST is based on mathematics, in particular, linear algebra in complex Hilbert
spaces [13, 14]. This is because quantum states can be represented as vectors
in these Hilbert spaces, and quantum operations correspond to unitary
operations or projections. To combat the high level of abstraction in this
mathematical field and gain some intuition, a multitude of visualizations are
available. For example, there exist broad geometric depictions of quantum
states and entanglement as shown in [15, 16, 13, 17]. The Bloch sphere is a
popular representation for single-qubit systems, that can help make respective
quantum operations more intuitive by relating them to rotations and
projections in three-dimensional space [18]. It can also be used to show
multi-qubit states [19, 20, 21, 22]. The diagrammatic ZX-calculus [23, 24, 25,
26] can be used to describe specific quantum operations and whole algorithms
in a “LEGO-like” manner [27], simplifying the standard quantum circuit
notation. With so called color code, quantum error correction algorithms can
be visually depicted as acting on hypercubes or hypercube-like lattices [28,
29, 30, 31]. The DROPS representation utilises generalized Wigner functions to
visualize quantum states, entanglement and quantum operations in an operator-
basis [32, 33, 34].

All of these visualizations use other bases than the computational
(0,1)-basis. However, in this work, we focus on the computational basis as it
is often used for introductory purposes [35]. The Circle Notation (CN)(section
2.1) is a visual representation of the complex amplitudes of states in the
computational basis using circles with gauges [36]. While CN aligns the
amplitudes in a row as you would when writing out states in the Dirac
notation, qubits can also be assigned axes in space [37]. This dimensional
approach has the possible benefit of making quantum operations more intuitive
and can be combined with CN (then called Dimensional Circle Notation (DCN)).
It can be used to visualize the entanglement properties of qubit systems [38].

In this pilot study, we take a first step in exploring the possible benefits
of presenting students with CN or DCN, in addition to the Dirac notation.
Participants are tasked with predicting the effect of different quantum
operations and finding entanglement properties of quantum states of one-,
two-, and three-qubit systems. For this purpose, an array of test items
covering these domains was created.

The paper is structured as follows. First, we introduce the research
background. This includes visualization of quantum states and entanglement
with DCN in section 2 and the background of learning and testing with MERs and
Cognitive Load Theory (CLT)in section 3. We follow with the research questions
in 4 and describe the study design in section 5. The results are presented and
discussed in section 6. We conclude and provide an outlook for future research
in section 7.

##  2 Visualizing quantum states and entanglement

The Circle Notation (CN)is introduced in Section 2.1 and the Dimensional
Circle Notation (DCN)in section 2.2. How such dimensional notations can be
used to visualize entanglement is discussed in Section 2.3.

###  2.1 Circle notation

The Circle Notation uses filled circles with pointers to visualize the
amplitudes $c_{i}=re^{i\varphi}$ of a quantum state
$\ket{\psi}=\sum_{i}c_{i}\ket{i}$. We refer to qubit states using the
computational basis, which means that $i$ is written in binary form, with the
rightmost digit being the least significant and also the number corresponding
to the first qubit. The magnitude $r$ is represented by the radius of the
filling circle and the gauge at an angle $\varphi$, measured counterclockwise
from the vertical axis, as shown in figure 1. Because a quantum state is
normalized, that is, $\sum_{i}|c_{i}|^{2}=1$, the sum of the areas of the
inner circles is $\pi$. Each area can be interpreted as the measurement
probability of the corresponding basis state.

Figure 1: The state
$\ket{\psi}=\frac{1}{3}\ket{00}+\frac{1}{3}e^{i\pi/2}\ket{01}+\frac{1}{\sqrt{3}}e^{-i\pi/4}\ket{10}-\frac{2}{3}\ket{11}$
in CN [36].

###  2.2 Dimensional circle notation

In dimensional notations, every qubit is assigned an axis in space [37, 38].
DCN is such an extension of CN. For comparison, figure 2 shows the same state
as figure 1 in DCN. For further clarification, we also compare the use of a
Hadamard gate that creates or destroys superposition for CN and DCN in figure
3.

Figure 2: The same state as in figure 1,
$\ket{\psi}_{\\#2\\#1}=\frac{1}{3}\ket{00}+\frac{1}{3}e^{i\pi/2}\ket{01}+\frac{1}{\sqrt{3}}e^{-i\pi/4}\ket{10}-\frac{2}{3}\ket{11}$,
in DCN [37, 38]. Figure 3: Hadamard gate on the second qubit in the state
$\ket{\psi}_{\\#2\\#1}=\frac{1}{\sqrt{3}}\ket{00}+\frac{1}{\sqrt{3}}\ket{01}+\frac{1}{\sqrt{3}}\ket{11}$
in CN and DCN. The Hadamard gate creates and destroys superpositions, i.e., it
is defined as $H\ket{0}=1/\sqrt{2}(\ket{0}+\ket{1})$ and
$H\ket{1}=1/\sqrt{2}(\ket{0}-\ket{1})$. It can be seen as acting on pairs of
states where qubit #1 is constant, i.e., acting on the axis of qubit #2 in the
dimensional notation [37].

###  2.3 Entanglement

In the computational basis, separability of pure states can be seen as
symmetry apart from a complex factor with respect to every qubit of the system
[38]. This is described by the following theorem [39]:

######  Theorem 1.

Let $\alpha,\beta,c_{i}\in\mathbb{C}$. An $n$-qubit state
$\ket{\psi}=\sum_{i=0}^{2^{n}-1}c_{i}\ket{i}$ is 2-2n-1 separable into
$\ket{\psi}=(\alpha\ket{0}+\beta\ket{1})\otimes\sum_{i=0}^{2^{n-1}-1}c^{\prime}_{i}\ket{i}$
if and only if for all $i\in\\{0,\ldots,2^{n-1}-1\\}$ either $c_{2^{n-1}+i}=0$
or there exists a ratio $r\in\mathbb{C}$ such that $c_{i}=rc_{2^{n-1}+i}$.

This ratio $r$ is the complex factor that can be seen in separable states in
dimensional notations. To see this factor in two-qubit systems, one needs to
observe whether there is symmetry along the axis of one qubit, apart from a
complex factor, the ratio $r$. In the absence of symmetry, the system is
entangled. Similarly, in three-qubit systems, one can spot whether a qubit is
separable from the rest of the system, i.e., not entangled with it, if there
is symmetry in regard to the plane perpendicular to that qubit’s axis. A
system is called partially separable if only a subset of the qubits of that
system are separable from that system. This is shown in figure 4.

Figure 4: Partial separability of the state
$\ket{\psi}_{\\#1\\#2\\#3}=\frac{1}{2}\ket{000}+\frac{1}{2}e^{i\pi/2}\ket{001}+\frac{1}{2\sqrt{2}}e^{i\pi/2}\ket{010}-\frac{1}{2\sqrt{2}}\ket{011}-\frac{1}{2\sqrt{2}}\ket{110}+\frac{1}{2\sqrt{2}}e^{-i\pi/2}\ket{111}$
[38]. a) Separability in standard circle notation in terms of qubit #1 can be
seen by the fact that there exists a ratio $r_{1}=e^{i\pi/4}$ (in green) such
that for any coefficients $c_{xy0}$ and $c_{xy1}$,
$c_{xy1}=r_{1}e^{i\pi/4}c_{xy0}$ is the same everywhere. However, as is shown
in red, this is not the case for qubit #2. b) As depicted on the left hand
side, the state is separable in terms of qubit #1, because the state is
symmetric apart from a complex factor in terms of qubit #1. This is apparent
by the green symmetry plane. For the other two qubits, there exists no such
ratio, i.e., no such symmetry planes, as shown here for qubit #2 with the red
plane.

##  3 Problem solving with multiple external representations

In this section, we consider the effects of including multiple external
representations (MERs)in problem-solving tasks. In doing so, we refer to the
functions of MERs according to Ainsworth’s Design, Functions, Tasks
(DeFT)framework [11] in section 3.1, and the effects on cognitive load that go
along with the use of MERs in section 3.2.

###  3.1 Functions of multiple external representations

Based on educational research, it is well known that learning and problem
solving in STEM can be supported by focusing not only on text-based and
mathematical-symbolic representations (e.g., written text and formulas) but
also on visualizations (e.g., pictures and diagrams) [40, 41]. From a
theoretical perspective, the benefit of learning with text accompanied by
visualization can be explained by an efficient use of information processing
and the working memory dual-channel structure. When learning with text-based
or mathematical-symbolic representations alone, the processing load of
information is solely on the verbal channel of working memory. However, when
learning with text and visualization, both the verbal and visual channels of
working memory are activated and integrated in the construction of coherent
mental schemata [42, 43]. This distributed information processing across two
channels reduces the load on each single channel and, thus, the risk of
cognitive overload.

Within her Design, Functions, Tasks (DeFT)framework, Ainsworth [11] has
formulated three main functions of MERs to support learning. First, MERs can
complement each other by containing different information or supporting
different processes. Second, they can constrain each other, e.g., by
familiarity or inherent properties. Third, incorporating MERs can construct
deeper understanding as learners are confronted with the abstraction of the
underlying knowledge structures, the extension of knowledge to an unknown
representation or improving the understanding of the relations between
different representations.

Referring to the DeFT framework, we especially make use of the representations
complementary functions, by visualizing qubit characteristics and gate
operations additionally to the common mathematical-symbolic description. As
students are provided with both representations, they have further access to
QIST content knowledge compared to the mathematical description alone. In
addition to this, our aim is to facilitate the understanding of the
corresponding mathematical descriptions. When using both visual and
mathematical representations, students can be supported in extending existing
knowledge structures to mathematical descriptions. For example, in the context
of entanglement, dimensional visualizations like DCN provide new
representations of the separability of qubit systems or even subsystems [38].

However, incorporating additional representations imposes an extra cognitive
load on students, as they must not only understand how each representation
conveys information but also how to translate between different
representations. Therefore, the learning effectiveness of MERs does not only
rely on the learning material but also on the characteristics of the learner
[11, 44]. To learn and solve problems with MERs efficiently, students must
possess representational competence [45]. Representational competence is
divided into conceptual competencies (visual understanding, i.e., the ability
of connecting the representation to concepts and understanding of the main
features, and connectional understanding, i.e., the ability to connect and
compare the representation to other representations), perceptual competencies
(visual fluency, i.e., efficiency of connecting the representation to concepts
and the ability to work effectively in the representation, and connectional
fluency, i.e., efficiency and flexibility of connecting multiple
representations to each other and switching between them), and lastly meta-
representational skills (i.e., the ability to choose suitable representations)
[6].

To summarize, alternative representations of multi-qubit systems in QIST to
the Dirac notation can be used to visualize entanglement and the actions of
gate operations. They can also enable instructors to encode information in an
easily accessible representation to take advantage of MERs for these complex
concepts. Using MERs, however, only deems benefits if the learner possesses
representational competence.

###  3.2 Cognitive load

It is essential to consider the cognitive load imposed on students when
designing tasks and learning materials, as working memory capacity is a major
limiting factor in learning and problem solving [46]. In the original version
of Cognitive Load Theory (CLT)Sweller divided cognitive load into three
categories: extraneous cognitive load (ECL), intrinsic cognitive load (ICL)
and germane cognitive load (GCL) [47]. ECL is unnecessary cognitive load
imposed on the learner by design, i.e. the way the material is presented. ECL
should be reduced to improve the performance of students. ICL, on the other
hand, is cognitive load that is inherent to the task or the learning content
and cannot be reduced by extraneous factors. GCL is described as the effort to
construct “schemata”, that is, mental patterns and categorizations that allow
students to infer meaning from the learning material. To support students in
the construction of cognitive schemata, materials should be designed such that
ECL is low, but GCL is high.

In order to reduce ECL, there are various effects to consider when designing
tasks or learning materials [48]. Here, we summarize the effects that are
relevant to this study. The split-attention effect suggests that ECL is
increased, when MERs are spatially or temporally separated. In our case, when
presenting CN or DCN and Dirac notation, the representations of the same
quantum state should be close to each other. The redundancy effect suggests
that displaying unnecessary additional information can lead to an increase in
ECL. This effect could lead to additional ECL when CN or DCN is presented in
addition to the Dirac notation, if the representation does not offer
additional value to the learner. Whether this is the case should mainly depend
on the representational competence of the learner.

When incorporating a visualization in addition to the Dirac notation, learner
characteristics can determine whether the visualization helps or not. For
example, a novice might, in some cases, benefit from using the visual
representation of a complex number in Circle Notation, while an expert might
never need to use a second representation, and the additional representation
just adds to ECL. This effect is called the expertise reversal effect [49]. On
the other hand, it could be the case that someone who understands the action
of gates and the definition of separability in dimensional notation, i.e., has
representational understanding, benefits from DCN in these contexts. More
generally, these two effects are forms of aptitude-treatment interactions
[50].

###  3.3 Time taken

The time it takes a student to answer a question correctly depends on
different factors. The distance-difficulty hypothesis proposed in [51], and
revised in [52], predicts that the logarithm of the time taken to answer the
questions is proportional to the distance of the students’ ability to the
difficulty of the question. In [53], it was confirmed that in complex tasks,
students who took longer also performed better. In [54], it was found that
children with higher levels of ability take longer to incorrectly solve the
questions, so the distance-difficulty hypothesis holds in this context.
Considering the impact of providing a visualization on the question
difficulty, if one finds that participants take longer amount of times on
questions with a visualization, it could mean that the question was made less
difficult by providing a visualization, bringing the question difficulty
closer to the ability level of the student.

When only considering time taken for correctly solved questions, most of the
results where the question was above the skill level of the participant are
eliminated (except for those questions that were correctly solved by random
chance). Therefore, when considering time taken only for correctly solved
questions, one considers more of the students that are above or at matching
ability level to the question’s difficulty. If a question becomes trivial for
a student when presented with a visualization, they will take less time for
correctly solved questions. Therefore, when requiring that both the question
without visualization and with visualization is solved correctly, on average
the time taken should be lower if the question did, in fact, get easier by
providing a visualization.

In summary, the time taken for task solving is a multidimensional construct.
Clear results are expected to be very difficult to obtain, as there are so
many factors that play a role. However, according to the distance-difficulty
hypothesis [51, 52], the time taken could be used as a metric (albeit weak) to
determine how suitable the given tasks were to the level of ability of the
students. Here, all answers, correct or incorrect, should be considered. In
addition, a close analysis of the dependence of the difficulty of the question
on the time taken for correct answers could provide further details on the
benefits of providing a visualization.

##  4 Research Questions

Based on current educational research, we expect that the use of CN and DCN
can utilize the known advantages of learning with MERs in QIST by constructing
deeper understanding of entanglement and complementing the mathematical
symbolic Dirac notation by offering new strategies to predict the outcome of
quantum gate operations. In this study, our objective is to take a first step
towards verifying this expectation, by comparing the task solving abilities of
participants in the Dirac notation with those where the Dirac notation is
accompanied by CN or DCN. In addition, we take into account the
representational competence of the participants, more specifically, the
conceptual competencies of representational and connectional understanding and
the perceived cognitive load during tasks without and with visualization.
Here, we refer to representational understanding instead of visual
understanding (see [6]), including the competency with complex numbers in
mathematical symbolic notation. Because the supposed benefits, in addition to
being likely learner-dependent, are also likely to be context-dependent, we
investigate the effects of presenting a visualization in different contexts,
that is, measurements, various unitary operations, and entanglement &
separability.

We formulate the following research questions.

  1. 1.

In what contexts does incorporating CN and DCN support students’ answer
correctness and time taken in task solving on quantum operations and
entanglement?

  2. 2.

How does the inclusion of the visualizations affect cognitive load?

  3. 3.

How are performance and cognitive load affected by representational and
connectional understanding?

##  5 Methods

In order to gain precise insights into the effects and relevant conditions of
including CN and DCN in students’ problem solving, we decided to compare
students’ performance and cognitive load with and without visualization
intrapersonally for each participant and relate it to connectional
understanding. As is the nature of a pilot study, our aim is also to learn
about the suitability of the questions that are presented to the participants.

###  5.1 Structure of the study

The study consisted of three online multiple choice tests about one-, two-,
and three-qubit systems, respectively. The surveys took about 45 minutes each.
The participants received 10€ compensation per survey.

In each survey, except for the first, the participants were randomly assigned
to two groups, where one group was presented with CN and one group was
presented with DCN throughout the survey. Each survey was separated into two
equivalent parts, A and B, where (mostly) questions using the same contexts
(gate operations, measurements, entanglement, and separability) were asked in
part A and B. One group was presented with visualization in part A and without
visualization in part B (group vis-math) and one group was presented without
visualization in part A and with visualization in part B (group math-vis).

###  5.2 Participants

In the survey of the one-, two- and three-qubit systems, there were 17, 21 and
15 participants. We asked for demographic information at the end of the first
survey.

8 of the 17 participants in the first survey stated that they had been
introduced to the Dirac notation before and 9 of the 17 participants stated
that they were introduced to CN before the day of the survey. From the
participant group for the study on one-(two-)(three-)qubit systems, 1(1)(1)
was in the age range 18-21, 12(11)(9) were in the age range 22-25 and 3(2)(3)
were in the age range 26-30. 16(14)(13) were male participants and 1(1)(1)
female. The highest level of educational achievement was high school
graduation for 4(3)(4) participants, completed Bachelor’s degree for 9(7)(5)
participants, completed master’s degree for 3(3)(3) participants, and
completed Ph.D. for 1(1)(1) participant. There was 1(1)(1) participant in the
field of Economics, 8(8)(7) participants in Information/Computer Science,
3(3)(2) participants in Computer Science and Engineering, 4(4)(4) participants
in Mathematics, and 1(1)(1) participant in Physics.

###  5.3 Introductory slides

At the beginning of each survey, participants were given the following
introductory slides (in the corresponding visualization, CN or DCN, that they
were assigned):

  * •

Visualization of qubit states

  * •

Action of single-qubit gates in single-qubit systems

  * •

Action of multi-qubit gates in multi-qubit systems (only shown in the second
and third survey)

  * •

Measurement probabilities and the resulting state

  * •

Separability and Entanglement (only shown in the second and third survey)

An example slide is shown in figure 5. The introductory material is given in
the supplementary material.

Figure 5: Introductory example slide for the action of single-qubit gates in
two-qubit systems visualized in DCN.

###  5.4 Representational understanding

In the beginning of the first survey (one qubit), we asked participants if
they had previously been introduced to the Dirac notation and if they had been
introduced to CN before. Then, we asked them one question on the
multiplication of complex numbers and one on the Euler formula (e.g.,
translating between $e^{i\pi/2}$ and $1/\sqrt{2}+i/\sqrt{2}$), and similar
questions using only CN. An example is shown in figure 6. In addition,
participants were asked their confidence on each of these questions on a
Likert scale from 0 (random guess) to 4 (very sure). We did not repeat this
test in the other surveys due to time restrictions.

Figure 6: Example question for representational understanding in Circle
Notation for the calculation
$e^{-i\pi/2}\cdot(\frac{\sqrt{2}}{\sqrt{3}}-\frac{1}{\sqrt{3}}e^{i\pi/4})=\frac{\sqrt{2}}{\sqrt{3}}e^{-i\pi/2}-\frac{1}{\sqrt{3}}e^{-i\pi/4}$.
The correct answer is the bottom most.

###  5.5 Connectional understanding

To measure connectional understanding, we measure the translation competence
of participants between Dirac notation and visualization. After the
introductory slides, the participants were asked four questions (six questions
in the two-qubit survey) to translate between their assigned visualization and
the Dirac notation. two (three in the two-qubit survey) of these questions
were about translating from Dirac notation to visualization, and two (three)
were about translating back. One question was about translating phase, one was
about translating magnitude, and in the two-qubit survey, we added one
question that asked about both. An example task in the three-qubit survey is
shown in figure 7.

Figure 7: Example question for translating from Dirac notation to Circle
Notation. The correct answer is the top most.

###  5.6 Quantum Operations & entanglement

We tested the visualizations in one-, two-, and three-qubit systems. The most
standard quantum gates, the X, Z, and Hadamard gates, were tested for all
sizes of qubit systems. In addition, the measurement probabilities and the
resulting state after measurements were asked in all surveys. The phase gate
was only tested in the one-qubit survey, while in the two- and three-qubit
surveys the CNOT gate, (partial) separability, and (partial) entanglement were
also tested. All the questions used in the study are provided in the
supplementary material. Figure 8 shows an example task of the effect of a
Hadamard gate in a two-qubit system.

Figure 8: Example task of predicting the outcome of a Hadamard gate. The
correct answer is the one in the top left.

###  5.7 Cognitive load

The perceived cognitive load was measured after each part of the survey, A and
B. The following Cognitive Load test items were chosen, as suggested in [55]:

ICL question 1:

For the tasks of survey part A/B, many things needed to be kept in mind
simultaneously.

ICL question 2:

The tasks of survey part A/B were very complex.

GCL question 1:

For the tasks of survey part A/B, I had to highly engage myself.

GCL question 2:

For the tasks of survey part A/B, I had to think intensively what things
meant.

ECL question 1:

During the tasks of survey part A/B, it was exhausting to find the important
information.

ECL question 2:

The design of the tasks of survey part A/B was very inconvenient for learning.

ECL question 3:

During the tasks of survey part A/B, it was difficult to recognize and link
the crucial information.

Possible answers were on a Likert scale from 0 (completely wrong to 6 -
absolutely right.

The naive questionnaire of [55] was chosen to measure GCL, because we
concluded that the GCL test items of the first iteration fit our use case of
problem solving tasks better. Due to the limited effects of the naive GCL
items in [55], the GCL measurement should be critically viewed. However, the
items used measure the perceived mental effort of the participants (literally,
the perceived level of cognitive engagement and the intensity of thinking),
which is supposedly linked to GCL [56].

##  6 Results & Discussion

In the following, we refer to group math-vis as the group that was assigned
questions without visualization in survey part A and questions with
visualization in part B. Vice versa, the group vis-math was assigned questions
with visualization in survey part A and questions without visualization in
part B.

Due to the low number of participants, statistical analysis is limited. We
show the standard deviation, where applicable, to describe the variance of the
obtained data.

Table 1 summarizes the average translation competence, average scores, and
average times for correctly solved questions. In addition, it shows the
differences of these performance metrics with visualization minus without
visualization. Average ICL, GCL and ECL and the corresponding differences
(between with visualization and without visualization) are shown in 2. Figure
9 shows the average score per participant, and figure 10 the average time
taken for correctly solved questions. The influence of translation competence
on ICL, GCL and ECL as well as the average cognitive load is shown in figures
11, 12 and 13 for tasks with one, two, and three qubits.

Table 1: Average scores and time taken (in seconds) for correctly solved
questions, and the difference of the performance metrics with visualization
minus without.

number of qubits | one | two | three
---|---|---|---
visualization | CN | CN | DCN | CN | DCN
avg. transl. comp. | 0.85$\pm$0.23 | 0.85$\pm$0.14 | 0.71$\pm$0.24 | 0.85$\pm$0.17 | 0.9$\pm$0.12
score w/o vis. | 0.67$\pm$0.24 | 0.62$\pm$0.28 | 0.56$\pm$0.24 | 0.62$\pm$0.26 | 0.69$\pm$0.3
score with vis. | 0.68$\pm$0.24 | 0.63$\pm$0.25 | 0.53$\pm$0.29 | 0.68$\pm$0.25 | 0.62$\pm$0.27
score difference | 0.0$\pm$0.29 | 0.01$\pm$0.22 | -0.02$\pm$0.27 | 0.06$\pm$0.1 | -0.07$\pm$0.15
time w/o vis. | 104$\pm$74 | 82$\pm$47 | 121$\pm$58 | 105$\pm$32 | 148$\pm$136
time with vis. | 88$\pm$56 | 113$\pm$76 | 139$\pm$99 | 119$\pm$76 | 116$\pm$61
time difference | -16$\pm$65 | 32$\pm$51 | 17$\pm$67 | 14$\pm$57 | -32$\pm$87

00footnotetext: Means and standard deviations are calculated taking the
averages over the participants individual results.

Table 2: Average ICL, GCL and ECL with and without visualization, and the
corresponding differences of with visualization minus without.

number of qubits | one | two | three
---|---|---|---
visualization | CN | CN | DCN | CN | DCN
ICL w/o vis. | 3.75$\pm$0.97 | 4.2$\pm$0.6 | 4.7$\pm$1.49 | 4.62$\pm$0.99 | 4.4$\pm$1.36
ICL with vis. | 3.64$\pm$1.04 | 3.89$\pm$0.99 | 4.0$\pm$1.54 | 4.4$\pm$1.11 | 4.0$\pm$1.0
ICL difference | -0.08$\pm$1.07 | -0.22$\pm$0.79 | -0.5$\pm$1.28 | -0.25$\pm$1.2 | 0.0$\pm$1.22
GCL w/o vis. | 4.12$\pm$1.41 | 4.67$\pm$0.82 | 4.91$\pm$1.0 | 4.89$\pm$1.37 | 4.6$\pm$1.5
GCL with vis. | 3.94$\pm$1.26 | 4.11$\pm$0.74 | 4.73$\pm$1.05 | 4.4$\pm$1.36 | 4.0$\pm$1.41
GCL difference | -0.12$\pm$1.27 | -0.62$\pm$0.99 | -0.18$\pm$0.83 | -0.44$\pm$1.26 | -0.6$\pm$1.62
ECL w/o vis. | 2.88$\pm$1.67 | 2.67$\pm$0.75 | 3.95$\pm$1.42 | 3.05$\pm$1.17 | 3.3$\pm$1.6
ECL with vis. | 2.76$\pm$1.54 | 2.72$\pm$0.82 | 3.64$\pm$1.62 | 2.75$\pm$1.33 | 2.5$\pm$1.1
ECL difference | -0.12$\pm$2.21 | 0.06$\pm$1.34 | -0.32$\pm$1.5 | -0.3$\pm$1.33 | -0.8$\pm$2.11

00footnotetext: Means and standard deviations are calculated taking the
averages over the participants individual results.

Figure 9: Average scores for different numbers of qubits. Two points connected
by a line represent an individual participant, colored according to the
translation competence score of the participant in that survey. Mean and
standard error are shown in orange. Figure 10: Average time taken per
participant for correctly solved questions and translation competence. Two
points connected by a line represent an individual participant, colored
according to the translation competence score of the participant in that
survey. Mean and standard error are shown in orange. Times above 20 minutes
were excluded. Figure 11: Intrinsic Cognitive Load (ICL)by number of qubits,
without visualization (math) and with visualization. Two points connected by a
line represent single participants, colored according to the translation
competence score of the participant in that survey. Mean and standard error
are shown in orange. Figure 12: Germane Cognitive Load (GCL)by number of
qubits, without visualization (math) and with visualization. Two points
connected by a line represent single participants, colored according to the
translation competence score of the participant in that survey. Mean and
standard error are shown in orange. Figure 13: Extraneous Cognitive Load
(ECL)by number of qubits, without visualization (math) and with visualization.
Two points connected by a line represent single participants, colored
according to the translation competence score of the participant in that
survey. Mean and standard error are shown in orange.

Detailed results are described in the appendix A. In the following, we discuss
the results obtained regarding answer correctness and time taken for correct
answers for the one-qubit (section 6.1), two-qubit (section 6.2.1 and 6.2.2)
and three-qubit (section 6.3.1 and 6.3.2) surveys. We summarize the discussion
in section 6.4.

###  6.1 One-qubit systems

In the first survey, participants were asked to find

  * •

a state that is equal to another state apart from a global phase

  * •

the resulting state after application of an X gate (only in part A) or a Z
gate (only in part B), phase gate, and a Hadamard (H) gate

  * •

the probability of finding a particular result upon measurement of a given
state, and

  * •

the resulting state after measurement.

The average scores for single-qubit questions are shown in table 3 and the
average times for correctly solved questions are shown in table 4. Figure 19
shows the combined average score per question and figure 20 shows the combined
average time taken (for all answers, not only the correct ones).

During part A, group math-vis solved questions without visualization with an
average score of 0.67$\pm$0.13 and an average time of 2.09$\pm$1.27 minutes
for correctly solved questions, while group vis-math answered the same
questions with visualization with an average score of 0.5$\pm$0.24 and an
average time of 1.09$\pm$0.41 minutes. During part B, the average scores were
0.67$\pm$0.15 (group vis-math) and 0.83$\pm$0.12 (group math-vis),
respectively, at average times for correctly solved questions of 1.81$\pm$1.28
and 2.01$\pm$1.88 minutes. When combining parts A and B of the survey, the
average scores were 0.68$\pm$0.15 without visualization and 0.71$\pm$0.18 with
visualization, at 1.44$\pm$0.49 and 1.1$\pm$0.67 minutes.

Group math-vis scored higher than group vis-math in both parts of the survey,
regardless of whether visualization was presented (see table 3), especially
when asked about the relation of probabilities of measuring 0 or 1 in both
parts of the survey (0.62 to 0.25) and in the global phase questions of part B
(1.0 to 0.62). Knowing the probabilities and the state of the qubit after
measurement could be more closely related to the previous knowledge of the
students than whether CN is presented. In this domain, group math-vis appears
to have had higher previous knowledge than group vis-math. In the global phase
question, we see the largest score difference among all question types within
the group math-vis with an average score of 0.56 without visualization, as
compared to 1 with visualization. It could be the case that, here, the group
math-vis especially benefited from the visualization. This could be an
indicator of a beneficial effect of the visual depiction of a complex phase
with a gauge, depending on the learner.

Although group math-vis scored higher with visualization, the contrary is true
for group vis-math. These effects could be due to the characteristics of the
learners or the difference in difficulty of the questions in parts A and B of
the survey. Questions with visualization were solved correctly faster than
questions without visualization, except for the global phase question. This
could indicate that visualization made the questions easier to solve for those
students who already knew how to solve them, according to the distance-
difficulty hypothesis [51, 52].

##### Representational understanding

Participants were asked to calculate Euler formula identities and similar
questions to translate between CN to a complex number. In addition, they were
asked to multiply numbers written in Dirac notation and CN as shown in figure
6. Then they were asked to rate their confidence in the question from 0
(random guess), 1 (very unsure) to 4 (very sure). The correct answers were
weighted (multiplied by $x/4$, where $x$ is the confidence). The weighted
(unweighted) average mathematics representational understanding was
0.76$\pm$0.25 (0.87$\pm$0.23). The weighted (unweighted) average CN
representational understanding was 0.47$\pm$0.34 (0.69$\pm$0.32). The average
weighted (unweighted) difference of CN representational understanding minus
mathematics representational understanding was -0.29$\pm$0.23 (-0.18$\pm
0.19$).

Figure 14 shows, for each participant in the first survey, the difference in
scores (with visualization minus without visualization) and figure 15 the
difference in time, both plotted against the difference in representational
understanding between visualization and math. Figure 16 shows the dependency
of the difference in intrinsic, germane, and ECL on the representational
understanding difference.

Figure 14: Difference of average score (with CN minus without CN) plotted
against difference of representational understanding (with CN minus without
CN) in the one-qubit survey. The linear regression has slope $b=0.04$,
intercept $a=0.01$, Pearson $r=0.03$ at $p=0.91$, standard error
$\sigma=0.32$. The shaded areas represents a 95% confidence interval. Figure
15: Average time taken for correct answers (with CN minus without CN) plotted
against difference of representational understanding (with CN minus without
CN) in the one-qubit survey. The linear regression has slope $b=-26$,
intercept $a=-24$, Pearson $r=-0.09$ at $p=0.73$ and standard error
$\sigma=73$. The shaded areas represents a 95% confidence interval. Figure 16:
Perceived intrinsic, germane and ECL difference (with CN minus without CN) and
their dependency on the representational understanding difference (again, with
CN minus without CN) in the one-qubit survey. Shown are also the linear
regressions between the three variables and the representational understanding
difference with slope $b$ and intercept $a$. Pearson’s $r$, $p$ value and
standard error $\sigma$ are as follows. ICL: $b=-0.68$, $a=-0.27$, $r=-0.16$,
$p=0.59$, $\sigma=1.2$ GCL: $b=-0.83$, $a=-0.36$, $-0.16$, $p=0.56$,
$\sigma=1.4$ ECL: $b=-0.87$, $a=-0.37$, $r=-0.09$, $p=0.73$, $\sigma=2.5$. The
shaded areas represent 95% confidence intervals.

No one scored higher with CN than with the Dirac notation (weighted or
unweighted), suggesting that the representational understanding of most of the
participants with the visualization was lower than that of the D notation.
This could perhaps explain the rather low impact of the visualization, as the
results suggest that other factors play a bigger role. It is possible that
presenting the visualization to students with high representational
understanding with CN but low representational understanding with the Dirac
notation has greater benefits. However, such students might be difficult to
find.

All in all, there is no clear evidence that the visualization provides a
general benefit, but there is also no evidence that the visualization provides
no benefit. Possible effects are probably learner- and context-dependent. It
is worth investigating further under which circumstances benefits occur. In
one-qubit systems, the CN is merely a visual representation of complex
numbers, providing an aid, in theory, complementary to the Dirac notation. In
two- and three-qubit systems, DCN is an extension of CN that could provide
additional benefits due to the introduction of dimensionality.

##### Translation competence and cognitive load

On average translation competence, 12 students achieved perfect scores and 5
students achieved a score of 0.5. The students who did not score perfectly
also scored lower in the rest of the survey (0.5$\pm$0.21 to 0.75$\pm$0.21
without visualization and 0.63$\pm$0.19 to 0.69$\pm$0.25 with visualization)
and took a smaller amount of time on the correctly solved questions (82$\pm$74
to 114$\pm$71 seconds without visualization and 86$\pm$57 to 89$\pm$55 seconds
with visualization) than the students who solved the translation questions
perfectly. Obtaining only two average scores for translation competence does
not leave much room for interpretation of the correlation between translation
competence and score difference.

Average ECL (2.88$\pm$1.67 with visualization and 2.76$\pm$1.54 without) was
lower than the average ICL (3.75$\pm$0.97 with visualization and 3.64$\pm$1.04
without) and GCL (3.94$\pm$1.26 with visualization and 4.13$\pm$1.41 without).
In the difference of perceived ECL of survey parts supported by visualization
minus parts not supported by visualization, the numbers fluctuate much more
(as can be seen in the high variance, depicted in table 1). This high variance
could indicate a large dependency of ECL reduction due to visualization on
learner characteristics.

There is no apparent sign of cognitive load being linked to translation
competence, although it is difficult to say due to the coarse resolution of
translation competence, as most participants had a perfect score on the test
items. All in all, in the one-qubit system, due to the large fluctuations, one
cannot conclude that the visualization provides a benefit in terms of
cognitive load. This requires further investigation.

###  6.2 Two-qubit systems

In both parts of the second survey, participants were split into two groups,
one presented with CN and one presented with DCN, and were asked to find:

  * •

the resulting state after application of an X gate, a Z gate, and a Hadamard
(H) gate,

  * •

the probability of finding a particular result upon measurement of a given
state,

  * •

the resulting state after application of a CNOT gate,

  * •

a separable state and an entangled state (between non-separable, or non-
entangled states, respectively), and

  * •

a combination of two gates that were applied to transform one given state into
another given state (“guess-the-gates” (gtg) questions).

The average scores for two-qubit questions are shown in table 5 and the
average times for correctly solved questions are shown in table 6. In the
latter, the combined score of part A and part B show only cases where
participants managed to solve both questions, i.e. both questions on the same
concept with and without visualization, correctly. Figure 19 shows the
combined average score per question and figure 20 shows the combined average
time taken (for all answers, not only the correct ones).

####  6.2.1 Two-qubit systems with Circle Notation

During part A, participants assigned to the CN group solved the questions
without visualization with an average score of 0.72$\pm$0.14 with an average
time of 1.54$\pm$0.62 minutes for correctly solved questions, while the other
group answered the same questions with visualization with an average score of
0.65$\pm$0.26 with an average time of 2.18$\pm$1.02 minutes. During part B,
the mean scores with and without visualization were 0.55$\pm$0.17 and
0.62$\pm$0.12, respectively, at average times for correctly solved questions
of 1.46$\pm$0.67 and 1.68$\pm$1.12 minutes. When combining parts A and B of
the survey, the average scores were 0.64$\pm$0.13 without CN and 0.64$\pm$0.12
with CN, at 1.48$\pm$0.51 and 1.98$\pm$1.05 minutes.

In both survey parts, we see group math-vis scoring higher or the same as
group vis-math on all questions except the separability and entanglement
questions, where group vis-math scored higher. This could be explained with
learner characteristics, such as the specific competence to identify
entanglement. In addition, the vis-math group scored higher on these questions
when presented with visualization. Perhaps, the group used a strategy of
comparing circles similar to that shown in figure 4, to identify separability
and entanglement in CN, which the other group did not know. The H gate
question shows a large difference in score between the two groups, where group
math-vis scored 0.8 without visualization and group vis-math only 0.2. This,
again, could hint towards the Hadamard gate question performance being
especially prone to learner prerequisites. Using the average of all the
questions, there is no increase in the average score when participants solved
questions with visualization, with 64% being correctly solved in both cases.

Participants took longer when presented with the visualization when solving
questions correctly, except for only a few questions, namely the gtg questions
in both survey parts and the X gate question in survey part B. The H gate and
separability questions of part B were solved only marginally faster when
presented with CN. The overall longer time taken for questions with
visualization (113$\pm$76 seconds) than questions without visualization
(82$\pm$47 seconds) could be a result of visualization bringing the questions
further toward the level of the participant’s ability, according to the
distance-difficulty hypothesis [51, 52]. The high variance in the time taken
for correct answers could be a result of not controlling the survey
environments. In addition, for many questions, there is only a small number of
data points to consider since this number depends on the number of
participants that solved these questions correctly.

##### Translation competence and cognitive load

Similarly to the one-qubit survey, participants scored 0.85$\pm$0.14 in
translating between CN and Dirac notation. Again, we see that participants who
scored perfectly when solving these questions also scored higher in the rest
of the survey (0.88$\pm$0.15 without and 0.88$\pm$0.13 with visualization)
than participants without a perfect score (0.44$\pm$0.20 without visualization
and 0.46$\pm$0.17 with visualization). Translation competence therefore could
be a good predictor for student performance overall. The reason could be that
students more familiar with CN are also more familiar with basic quantum
computing concepts, since CN is usually used to teach such concepts.

The group of participants with perfect translation scores also took more
“reasonable” amounts of time to correctly solve questions, that is, the
variance in time taken was lower (1.67$\pm$0.65 seconds without visualization
and 2.03$\pm$0.67 with visualization) than in the group without a perfect
translation competence score (1.13$\pm$0.79 seconds without visualization and
1.78$\pm$1.52 with visualization). Presumably, students that don’t know the
answer and choose to take a guess will take a short amount of time, while
students that try hard to find an answer in tasks where this would not be
necessary for students with high previous knowledge, will take a long time.
Therefore, a higher variance in the time taken for task solving could be
expected for lower-performing students. This is expected according to the
distance-difficulty hypothesis [51, 52].

There is no observable correlation between translation competence and the
average score difference. This could be due to participants choosing one
representation over the other when solving the questions, such that
translation competence does not play a role for the performance. Translating
between the representations is not necessary to solve these tasks. In future
studies, eye tracking could be used to investigate whether participants
translate between representations or stay only in one.

The results are similar to the results of the one-qubit survey with respect to
perceived cognitive load. Average ICL (4.2$\pm$0.6 without and 3.89$\pm$0.99
with visualization) and GCL (4.67$\pm$0.82 without and 4.11$\pm$0.74 with
visualization) are higher than average ECL (2.7$\pm$0.75 without and
2.72$\pm$0.82 with visualization). It seems that on average, the questions
succeed in offering a high amount of intrinsic difficulty, while the design of
the questions does not contribute too much. However, it cannot be stated that
there is a significant difference in cognitive load when participants were
presented with CN as compared to when they only used the Dirac notation.

####  6.2.2 Two-qubit systems with Dimensional Circle Notation

During part A, group math-vis solved questions without visualization with an
average score of 0.56$\pm$0.22 with an average time of 2.4$\pm$0.59 minutes
for correctly solved questions, while group vis-math answered the same
questions with an average score of 0.6$\pm$0.24 with an average time of
3.57$\pm$2.66 minutes. During part B, the average scores were 0.55$\pm$0.24
and 0.48$\pm$0.26, respectively, at average times for correctly solved
questions of 2.56$\pm$1,73 and 3.62$\pm$2.12 minutes. When combining parts A
and B of the survey, the average scores were 0.56$\pm$0.2 without
visualization and 0.53$\pm$0.2 with visualization, at 2.96$\pm$2.22 and
5.24$\pm$5.93 minutes.

In all question types, one group performed better than the other group in both
part A and part B (regardless of whether visualization is presented). The
difference between the scores of the groups is particularly apparent in the H
gate, entanglement, and CNOT gate questions in favor of group vis-math (0.5 to
0.17, 0.7 to 0.42, and 0.9 to 0.69, respectively) and the gtg and separability
questions in favor of group math-vis (0.42 to 0.1 and 0.67 to 0.5,
respectively). This consistency can possibly be explained by learner
prerequisites that predict the performance in these questions better than
whether visualization is presented or not. In addition, this could be a
manifestation of the context dependency of student competence, i.e., a student
being competent with one quantum process does not necessarily mean that the
student is also competent with other processes. Group vis-math scored lower on
the gtg question in both parts of the survey than on all other questions. In
addition, group math-vis scored lower on the gtg question with visualization
than on the gtg question without visualization. This is consistent with the
lower performance of the group on the H gate question, because the H gate was
part of the gtg question in survey part B but not of the gtg question in
survey part A. Group math-vis appeared to have lower competence with the H
gate than group vis-math.

Considering the amount of time spent for correctly solved questions, we see
outliers of more than 6 minutes in the group vis-math on the Z gate A, H gate
A, and H gate questions of part B. Participants took, on average, longer when
presented with visualization than when they were not. This is largely due to
these outliers, as, when removing the outliers, the difference is only
marginal (1 minute 58 seconds without visualization to 2 minutes and 5 seconds
with visualization).

##### Translation competence and cognitive load

In the two-qubit survey, translation competence was lower in the DCN group at
0.71$\pm$0.25 than in the CN group at 0.85$\pm$0.23. In addition, no one in
the DCN group correctly answered all the translation questions. The average
scores in this survey were also lower than in all other surveys, at
0.56$\pm$0.2 without visualization and 0.53$\pm$0.2 with visualization.

For participants of the DCN group, ICL was slightly higher than for
participants of the CN group when solving math questions at 4.7$\pm$1.36, as
compared to 3.89$\pm$1.0, and we see indications of reductions in ICL and ECL
when participants are presented with DCN of -0.5$\pm$1.35 and -0.32$\pm$1.57,
respectively, with a rather large fluctuation of this difference. We do not
see a similar reduction in GCL as we do in the CN group. In total, these
results indicate that there could be reductions in ICL and ECL when students
are presented with DCN. The large fluctuation of this supposed reduction could
indicate that the possible benefit of DCN presentation in cognitive load would
depend on the learner.

ECL in general is high compared to all the other surveys at 3.95$\pm$1.42
without visualization and 3.64$\pm$1.62 with visualization. These results
could indicate an anti-correlation between translation competence and ECL that
future studies could look into further. students with higher representational
competence should experience less GCL when solving tasks where this
representational competence is necessary [56]. In addition, if students can
adopt new strategies to solve questions using DCN rather than the Dirac
notation, a reduction in ICL could be theoretically explainable, if these
strategies themselves make it easier to solve the task. These results warrant
further investigation into the exact conditions under which the presentation
of DCN has a positive impact on ICL and ECL.

###  6.3 Three qubits

In the third survey, participants were asked to find:

  * •

the resulting state after application of an X gate, a Z gate, and a Hadamard
(H) gate,

  * •

the probability of finding a particular result upon measurement of a given
state,

  * •

the resulting state after a measurement,

  * •

the resulting state after application of a CNOT gate,

  * •

a fully separable state and a fully entangled state (between partially
separable/entangled states), and

  * •

a combination of two gates that were applied to transform one given state into
another given state (“guess the gates” (gtg) questions).

The average scores for the three-qubit questions are shown in table 7 and the
average times for correctly solved questions are shown in table 8. In the
latter, the combined score of part A and part B show only cases where
participants managed to solve both questions, i.e., two questions of the same
topic, correctly. Figure 21 shows the combined average score per question and
figure 22 shows the combined average time taken (for all answers, not just the
correct ones). Cases where a participant took more than 20 minutes for one
question were excluded as we find that these are unreasonable times for these
tasks.

####  6.3.1 Three-qubit systems with Circle Notation

During part A, participants assigned to the math-vis group solved the
questions without visualization with an average score of 0.58$\pm$0.26 and an
average time of 1.77$\pm$0.94 minutes for correctly solved questions, while
the group vis-math solved the same questions with visualization with an
average score of 0.76$\pm$0.28 and an average time of 3.0$\pm$1.63 minutes for
correctly solved questions. During part B, the average scores were
0.67$\pm$0.31 and 0.6$\pm$0.25 for the two groups, respectively, with average
times for correctly solved questions of 2.43$\pm$1.26 and 1.25$\pm$0.45
minutes. When combining parts A and B of the survey, the average scores were
0.62$\pm$0.28 without visualization and 0.68$\pm$0.25 with visualization, at
2.38$\pm$1.33 and 2.37$\pm$1.51 minutes.

In general, the participants scored marginally higher when presented with CN
(0.68$\pm$0.25) than when presented with no visualization (0.62$\pm$0.28). The
results of the H gate and gtg questions could indicate that there was a
benefit in presenting the visualization to the participants in these contexts,
as both groups scored higher here when presented with visualization (0.9
compared to 0.6 for the H gate question and 0.6 compared to 0.1 for the gtg
question). In the entanglement question, group math-vis scored higher (0.6
without visualization and 0.4 with visualization) than group vis-math (0.2
with and without visualization) in both parts of the survey, while in the Z
gate and measurement probability questions, all participants in the vis-math
group correctly solved the questions in both parts of the survey, while 60% of
participants in the math-vis group correctly solved these questions in both
cases. This could indicate differences in context-dependent competence of the
two groups, similar to what we observed in the two-qubit survey. Group math-
vis appeared to have not benefited from visualization in the separability,
entanglement, and CNOT gate questions, scoring 0.2 and 0.4, 0.4 and 0.6, and
0.6 and 0.8, without and with visualization, respectively, while the vis-math
group scored the same with and without visualization on the X gate, Z gate,
measurement probability, separability, and entanglement questions (1.0, 1.0,
1.0, 0.4, and 0.2, respectively).

In summary, we see that the X gate, Z gate, CNOT gate, and measurement
probability questions. At the same time, the results warrant further
investigation in the regime of the more complex H gate, gtg, separability, and
entanglement questions to find the reasons for the increase in performance
with visualization in the H gate and gtg questions and the lack of performance
difference in the measurement state, separability, and entanglement questions.
To benefit from visualization in the context of separability and entanglement,
students may need a thorough introduction to the strategy to identify
separability and entanglement (figure 4).

##### Translation competence and cognitive load

In the three-qubit survey, participants presented with CN scored on average
0.85$\pm$0.17 on the translation competence questions, in line with the
results of the one- and two-qubit CN survey. Participants who scored perfectly
in translation competence also scored comparatively high in the rest of the
survey at 0.8$\pm$0.16 without visualization and 0.84$\pm$ 0.17 with
visualization, compared to 0.44$\pm$0.21 and 0.51$\pm$0.19 for the other
participants. However, we do not see expression of this translation competence
in an improvement of score when presented with visualization.

When participants are presented with CN, we observe indications of a reduction
in ICL, GCL and ECL of $-0.25\pm 1.28$, $-0.44\pm 1.33$ and $-0.3\pm 1.4$,
respectively. These results justify further investigation into the
circumstances under which a reduction in cognitive load occurs by presentation
of CN. We see a generally lower ECL (3.05$\pm$1.17 without and 2.75$\pm$1.33
with visualization) than ICL (4.63$\pm$0.99 without and 4.4$\pm$1.11 with
visualization) and GCL (4.89$\pm$1.37 without and 4.40$\pm$1.36 with
visualization), indicating that participants perceived the questions
themselves as difficult, but less so their design.

####  6.3.2 Three-qubit systems with Dimensional Circle Notation

During part A, participants assigned to the DCN group solved questions without
visualization with an average score of 0.89$\pm$0.21 with an average time of
4.75$\pm$1.77 minutes for correctly solved questions, while the same questions
with visualization were answered with an average score of 0.6$\pm$0.24 with an
average time of 2.32$\pm$0.63 minutes. During part B, the average scores were
0.56$\pm$0.16 and 0.72$\pm$0.34, respectively, at average times for correctly
solved questions of 2.0$\pm$1.34 and 2.29$\pm$1.33 minutes. When combining
parts A and B of the survey, the average scores were 0.69$\pm$0.15 without
visualization and 0.62$\pm$0.29 with visualization, at 3.03$\pm$0.78 and
2.44$\pm$0.9 minutes.

Group math-vis consists of two students and group vis-math of three
participants. Hence, the resolution on the average score value per question is
rather coarse. Due to the large impact a single persons performance can have
on the results, it is difficult to draw any conclusions. Future studies could
focus more on the impact of DCN in the domain of three qubits, in particular.
If there is any benefit of visualizing quantum states representing qubits as
axes in space, one would expect the benefit to be the largest here because the
difference between the dimensional approach and the approach of standard CN is
larger the higher the number of qubits in the system. Similarly to the results
for the answer correctness, it would be unreasonable to make statements about
time taken for correct answers due to the small number of participants.

##### Translation competence and cognitive load

The impact of the results of one participant seems too large to make any
assumptions about the translation competence and cognitive load in the three-
qubit DCN context.

###  6.4 Summary of discussion & Limitations

In general, we do not see indications of a large change in performance when
participants are presented with a visualization (see table 1). There are
slight indications of context-dependent performance increases with
visualization. This is apparent in the global phase question of the one-qubit
survey, the Z gate, the measurement probability, and the separability question
of the two-qubit CN survey, the H gate question of the two-qubit DCN survey,
and the H gate and gtg questions of the three-qubit CN survey. However,
context-dependent learner characteristics seem to have a greater impact on
performance than the presence or absence of a visualization. Some groups seem
to perform better than other groups specifically in certain types of
questions, as we see in the two-qubit CN survey in the X gate, the H gate,
separability and entanglement questions, all questions in the two-qubit DCN
survey, and in the Z gate, measurement probability, entanglement, CNOT gate,
and guess-the-gates questions in the three-qubit CN survey.

In order to find indicators of complexity and suitability of questions, we
considered the time taken for all answers, not just correct ones (see figure
18, figure 20 and figure 22 for the one-, two- and three-qubit surveys,
respectively). During the one-qubit survey, the global phase and the Hadamard
gate question took the participants the longest time. During the two-qubit
survey, the same was the case for the guess-the-gates and measurement
probability questions for participants in the CN survey, and the guess-the-
gates, H gate, and Z gate questions for participants of the DCN survey (see
figure 20). In the three-qubit survey, participants presented with both CN and
DCN took the longest in the guess-the-gates and H gate questions (see figure
22). Long times could in some cases be due to greater complexity and in some
cases to the distance-difficulty hypothesis [51, 52], which would state that
these questions were also the most suitable for the participants in terms of
level of ability.

We also measured the cognitive load imposed on the students during all parts
of all surveys. The generally high ICL and average scores between 0.5 and 0.7
indicate that the difficulty of the problems was adequate overall. We see
reductions in cognitive load when students are presented with visualization in
both CN and DCN surveys. ECL is mostly lower than ICL and GCL, except for the
two-qubit DCN survey. It could be the case that the amount of space DCN takes
up compared to CN leads to increased ECL, due to participants not being able
to see all answers at once (which means page scrolling becomes necessary),
resulting in a possible split-attention effect [57]. On the other hand, the
lower translation competence of participants in the two-qubit DCN survey could
also be a reason for the higher ECL.

In this study, we have seen how performance can be context-dependent. The
cognitive load might also be context dependent. The cognitive load results
give an overview of the cognitive load with and without visualization, but one
cannot make assumptions for specific contexts with these measurements. It can
also be taken into account that measurements of cognitive load after multiple
questions could be biased toward the most recently answered questions, which
would be the measurement questions in the one-qubit survey and the guess-the-
gates questions in the two- and three-qubit surveys.

In general, due to small sample sizes (especially in the three-qubit DCN
survey) and large fluctuations (possibly due to large deviations of learner
prerequisites), we can only suspect indicators of positive effects of
presenting a visualization regarding performance and cognitive load. The
circumstances and contexts in which these possible effects occur should be
investigated more thoroughly. Reevaluation of the test instrument can be based
on fine tuning to make the questions in parts A and B of the surveys more
similar. In addition, future studies could further investigate the competence
of students when presented with only a visualization and how this
representational competence impacts possible benefits of presenting a
visualization in addition to the Dirac notation.

##  7 Conclusion & Outlook

The results of this study, although inconclusive, open up a wide range of
questions to investigate in future studies. As we see some students benefiting
from the visualization (in terms of performance increase and cognitive load
reduction) and some not, it will be interesting to investigate further under
which circumstances exactly students benefit from the visualization.
Especially contexts within the three-qubit DCN setting should be investigated
more thoroughly, as the number of participants in this survey was low. In
addition, here DCN differs the most from the other two representations, since
the number of steps necessary to transform CN to DCN is larger the more qubits
the system has. Our results indicate a dependency of the results on context-
dependent learner characteristics. Therefore, future studies should control
for these characteristics in a more detailed way to draw conclusions regarding
the potential benefit of the visualizations, e.g., by constricting the context
by focusing on a specific type of question. If future studies find significant
benefits of visualization, the exact reasons for this could remain unclear, as
learners could have used the visualization as support, complementary to the
mathematical notation, or have worked exclusively in the visualization. Eye
tracking could be used to account for this [58].

Representational competence as a learner characteristic is theoretically the
deciding factor for whether a visualization supports learning. Therefore, it
could be considered to measure representational competence more thoroughly in
future studies. This could be done, for example, by increasing the number of
translation competence questions that measure connectional understanding, and
finding other ways to measure representational competence that also includes
visual understanding of CN or DCN and quantum operations in these
visualizations.

In general, we saw indications of reductions in perceived cognitive load by
presenting CN or DCN. Due to the consistency in which this occurs, it is
possible that there are beneficial effects regarding a reduction of cognitive
load when students are presented with a visualization. A decrease in ECL
should, in theory, lead to improved learning outcomes when learning with
visualization, if this decrease coincides with an increase in germane
cognitive load [59, 60]. If we find a decrease in all types of cognitive load,
it could be that the questions become inherently easier when asked and solved
in CN or DCN. Another point to consider is that the cognitive load induced on
the learner could be context-dependent, i.e., for some question types,
cognitive load change by presentation of a visualization might differ from
other question types. In addition, learning with visualization could also
improve the competency of participants when solving questions in
uncomplemented Dirac notation. Investigating further whether there is any
statistical significant benefit for students presented with visualization, and
under which circumstances and in which contexts this occurs, could therefore
be of great interest to the quantum education community.

Eye tracking studies can be used to control which representation is used
during task solving and can be combined with cognitive load measurements to
investigate the intricacies of cognitive processes during task solving [58].
Going beyond static multiple external representations, conveying processes in
DCN using video sequences and sound could be worth studying. Furthermore,
investigating learning outcomes by including interactivity in the teaching
process, for example, by gamification, could be an alternative path forward,
as there already exists a multitude of quantum games [61] where DCN could be
incorporated. Also, the possible beneficial effects of other visualization
techniques and their context- and learner-dependency remain to be investigated
in multi-qubit systems. In conclusion, the benefits of the use of
visualizations in quantum education have yet to be explored in a carefully
curated way. In this work, we took a first step towards answering the question
in which circumstances visualizations can support students in the field of
Quantum Information Science & Technology.

## Glossary

CLT

    Cognitive Load Theory
CN

    Circle Notation
DCN

    Dimensional Circle Notation
DeFT

    Design, Functions, Tasks
ECL

    Extraneous Cognitive Load
GCL

    Germane Cognitive Load
ICL

    Intrinsic Cognitive Load
MERs

    multiple external representations
QIST

    Quantum Information Science & Technology
STEM

    Science, Technology, Engineering & Mathematics

## Declarations

Availability of data and materials The datasets generated and analyzed during
the current study are available in the following repository:
https://doi.org/10.5281/zenodo.11258603.

Competing interests The authors declare that they have no conflict of
interest.

Ethical Approval All procedures performed in the study were in accordance with
the ethical standards of the institutional and/or national research committee
and with the declaration of Helsinki of 1964 and its subsequent amendments or
comparable ethical standards. The study involved data collection in three
online surveys. Participation in this study was voluntary and anonymous. The
data collected included participants’ age, gender, field of study, and highest
educational achievement. Data of participants is kept confidential and used
solely for research purposes.

Informed Consent Informed consent was obtained from all individual
participants included in the study, in each survey.

Authors’ contributions

J.B., E.R., A.A., S.K., J.K., M.K.-E., and A.W. designed the study. J.B.,
L.K., S.S., P.L., and M.K.-E. collected the data. J.B., E.R., A.A., and A.W.
analyzed and discussed the data. L.K. and S.S. programmed the questionnaire
website. L.K. and N.L. programmed and updated the DCN webtool. J.B., E.R.,
S.K., and M.K.-E. wrote the first draft of the manuscript. A.A and A.W.
reviewed the manuscript and provided feedback. P.L., J.K., M.K.-E., and A.W.
supervised the study. All authors have read and approved the final manuscript.

Acknowledgements

The authors thank all the study participants.

J.B., E.R., A.A., M. K-E., P.L., and A.W. acknowledge support by the QuanTUK
project at the RPTU in Kaiserslautern, supported by the Federal Ministry of
Education and Research (FKZ13N15995).

M.K-E., P.L., and A.W. acknowledge support from the Quantum Initiative
Rhineland-Palatinate (QUIP) and the Research Initiative Quantum Computing for
Artificial Intelligence (QC-AI).

L.K., S.S., N.L. and P.L. acknowledge support by the KI4TUK project at the
RPTU in Kaiserslautern, supported by the Federal Ministry of Education and
Research (BMBF) under grant number 16DHBKI058.

S.K. and J.K acknowledge support by the Quantum Lifelong Learning (QL3)
project at the LMU Munich, supported by the Federal Ministry of Education and
Research (BMBF) under grant number 13N16024, and the DigiQ project (EU),
supported by the European Union’s Digital Europe program under grant number
101084035.

##  Appendix A Detailed Results

We show detailed results for single-qubit (Section A.1), two-qubit (Section
A.2) and three-qubit studies (Section A.3). We describe the average
correctness per question with and without visualization and survey parts A and
B and the combined score of part A and B. Then, we do the same for average
time taken for correct answers, considering for the combination of parts A and
B only cases where both questions of each topic, with and without
visualization, were answered correctly.

###  A.1 one-qubit survey

The average scores for single-qubit questions are shown in table 3 and average
times for correctly solved questions in table 4. In the latter, the combined
score of part A and part B show only cases where participants managed to solve
both questions, i.e. both questions on the same concept with and without
visualization, correctly. Figure 17 shows the combined average score per
question and figure 18 shows the combined average time taken (for all answers,
not just the correct ones). Cases in which a participant took more than 20
minutes for one question were excluded.

During part A, group math-vis solved questions without visualization with an
average score of 0.67$\pm$0.13 and an average time of 2.09$\pm$1.27 minutes
for correctly solved questions, while group vis-math answered the same
questions with visualization with an average score of 0.5$\pm$0.24 and an
average time of 1.09$\pm$0.41 minutes. During part B, the average scores were
0.67$\pm$0.15 (group vis-math) and 0.83$\pm$0.12 (group math-vis),
respectively, at average times for correctly solved questions of 1.81$\pm$1.28
and 2.01$\pm$1.88 minutes. When combining parts A and B of the survey, the
average scores were 0.68$\pm$0.15 without visualization and 0.71$\pm$0.18 with
visualization, at 1.44$\pm$0.49 and 1.1$\pm$0.67 minutes.

In survey part A, group math-vis scored higher than group vis-math on the
global phase, X gate, phase gate, H gate, measurement probability, and
measurement state questions with an average score of 0.67$\pm$0.13 to
0.5$\pm$0.24 on these questions and did not score lower on any questions.
Group math-vis took longer on correct answers to the X gate, phase gate, H
gate, and measurement state questions with an average of 151$\pm$74 to
56$\pm$22 seconds while taking less time on the global phase and measurement
probability questions with 84$\pm$4 to 75$\pm$1 seconds.

In part B of the survey, group math-vis scored higher than group vis-math on
the global phase Z gate, H gate, measurement probability, and measurement
state questions, with an average of 0.84$\pm$0.13 to 0.62$\pm$0.14. Group vis-
math scored higher on the phase gate question, with 0.88 to 0.78. Group math-
vis took more time on the global phase, Z gate, and measurement state
questions with 178$\pm$120 to 140$\pm$85 seconds and group vis-math took
longer on phase gate, H gate and measurement probability questions with
77$\pm$23 to 63$\pm$19 seconds.

When presented with visualization, group math-vis scored higher on the global
phase, H gate, measurement probability, and measurement state questions with
average scores of 0.86$\pm$0.14 to 0.67$\pm$0.14. When presented questions
with visualization, group math-vis took longer on the global phase question
with 177 to 77 seconds and took a shorter amount of time on the phase gate, H
gate, measurement probability, and measurement state questions with 104$\pm$43
to 56$\pm$21 seconds than when the group was not presented with a
visualization. Group vis-math scored higher when presented without
visualization on the global phase, phase gate, H gate, and measurement
probability question with scores of 0.66$\pm$0.18 to 0.5$\pm$0.23. Group vis-
math took longer on the measurement probability and measurement state
questions when presented with visualization with 57$\pm$23 to 37$\pm$15
seconds and took less time on the global phase, phase gate, and H gate
questions with 118$\pm$43 to 63$\pm$20 seconds.

Table 3: Average scores for one-qubit questions, with and without
visualization (N=17).

survey part | A | B | combined
---|---|---|---
question | w/o vis.111N = 9 | with vis.222N = 8 | w/o vis.222N = 8 | with vis.111N = 9 | w/o vis.333N = 17 | with vis.333N = 17
global phase | 0.56 | 0.5 | 0.62 | 1.0 | 0.59 | 0.76
X gate | 0.56 | 0.25 | - | - | - | -
Z gate | - | - | 0.62 | 0.78 | - | -
phase gate | 0.78 | 0.75 | 0.88 | 0.78 | 0.82 | 0.76
H gate | 0.67 | 0.62 | 0.75 | 0.78 | 0.71 | 0.71
measurement probability | 0.56 | 0.12 | 0.38 | 0.67 | 0.47 | 0.41
measurement state | 0.89 | 0.75 | 0.75 | 1.0 | 0.82 | 0.88
Mean | 0.67$\pm$0.13 | 0.5$\pm$0.24 | 0.67$\pm$0.16 | 0.83$\pm$0.12 | 0.68$\pm$0.15 | 0.71$\pm$0.18
|  |  |  |  |  |

00footnotetext: Note: questions are shown in order of appearance in the study.

Table 4: Average time taken for correct answers to one-qubit questions, with
and without visualization in minutes (N=17).

survey part | A | B | combined
---|---|---|---
question | w/o vis.111N = 9 | with vis.222N = 8 | w/o vis.222N = 8 | with vis.111N = 9 | w/o vis.333N = 17 | with vis.333N = 17
global phase | 1.28$\pm$0.63(6) | 1.48$\pm$1.57(5) | 2.94$\pm$2.99(6) | 2.95$\pm$2.2(10) | 1.63$\pm$1.08(10) | 2.26$\pm$2.27(10)
X gate | 4.31$\pm$2.61(5) | 1.52$\pm$0.92(3) | - | - | - | -
Z gate | - | - | 3.69$\pm$5.56(5) | 5.43$\pm$4.11(7) | - | -
phase gate | 1.65$\pm$2.49(8) | 1.01$\pm$0.44(7) | 1.19$\pm$1.05(9) | 0.87$\pm$0.42(7) | 1.47$\pm$1.99(11) | 0.93$\pm$0.46(11)
H gate | 2.93$\pm$2.9(7) | 0.68$\pm$0.41(6) | 1.78$\pm$2.74(6) | 1.5$\pm$0.83(7) | 2.13$\pm$2.59(10) | 0.84$\pm$0.4(10)
measurement probability | 1.23$\pm$1.0(6) | 1.33(1) | 0.86$\pm$0.59(4) | 0.81$\pm$0.58(6) | 1.13$\pm$0.96(6) | 0.91$\pm$0.61(6)
measurement state | 1.14$\pm$1.39(9) | 0.55$\pm$0.24(7) | 0.38$\pm$0.17(6) | 0.54$\pm$0.68(9) | 0.83$\pm$1.14(14) | 0.55$\pm$0.55(14)
Mean | 2.09$\pm$1.27 | 1.09$\pm$0.41 | 1.81$\pm$1.28 | 2.01$\pm$1.88 | 1.44$\pm$0.49 | 1.1$\pm$0.67
|  |  |  |  |  |

00footnotetext: Note: The number of participants N that solved the question
correctly and are therefore included in the average is shown in brackets.
Standard deviations are shown where N $>1$. The confidence interval of the
mean is the mean of the standard deviations.

Figure 17: Average score for each question in minutes in the one-qubit survey,
for questions were participants were presented with CN (dark blue) and
questions were the participants weren’t presented with CN (light blue).
(N=17). Figure 18: Average time taken for each question in minutes in the one-
qubit survey, for questions were participants were presented with CN (dark
blue) and questions were the participants weren’t presented with CN (light
blue). (N=17).

###  A.2 two-qubit survey

Table 5 lists the average scores and table 6 lists the average time taken for
correctly solved one-qubit questions, with and without visualization, for
surveys that were taken supported by CN and DCN. Figure 19 shows the combined
average score per question and figure 20 shows the combined average time taken
(for all answers, not just correct ones). Cases where a participant took more
than 20 minutes for one question were excluded.

Table 5: Average scores for two-qubit questions, with and without
visualization (N=21).

| Circle Notation111N = 10
---|---
survey part | A | B | combined
question | w/o vis.222N = 5 | with vis.222N = 5 | w/o vis.222N = 5 | with vis.222N = 5 | w/o vis.111N = 10 | with vis.111N = 10
X gate | 1.0 | 0.4 | 0.6 | 0.8 | 0.8 | 0.6
Z gate | 0.6 | 0.6 | 0.4 | 0.8 | 0.5 | 0.7
H gate | 0.8 | 0.2 | 0.4 | 0.6 | 0.6 | 0.4
measurement probability | 0.6 | 0.6 | 0.4 | 0.6 | 0.5 | 0.6
Separability | 0.6 | 1.0 | 0.8 | 0.6 | 0.7 | 0.8
Entanglement | 0.8 | 1.0 | 0.6 | 0.4 | 0.7 | 0.7
CNOT gate | 0.8 | 0.8 | 0.8 | 0.6 | 0.8 | 0.7
gtg | 0.6 | 0.6 | 0.4 | 0.6 | 0.5 | 0.6
Mean | 0.72$\pm$0.14 | 0.65$\pm$0.26 | 0.55$\pm$0.17 | 0.62$\pm$0.12 | 0.64$\pm$0.13 | 0.64$\pm$0.12
| Dimensional Circle Notation111N = 10
survey part | A | B | combined
question | w/o vis.333N = 6 | with vis.222N = 5 | w/o vis.222N = 5 | with vis.333N = 6 | w/o vis.444Standard deviation is used for confidence intervals.Confidence intervals are the average of the standard deviations. | with vis.444N = 3
X gate | 1.0 | 0.8 | 0.8 | 0.83 | 0.91 | 0.82
Z gate | 0.5 | 0.4 | 0.6 | 0.67 | 0.55 | 0.55
H gate | 0.17 | 0.6 | 0.4 | 0.17 | 0.27 | 0.36
measurement probability | 0.5 | 0.6 | 0.6 | 0.33 | 0.55 | 0.45
Separability | 0.5 | 0.4 | 0.6 | 0.83 | 0.55 | 0.64
Entanglement | 0.5 | 0.8 | 0.6 | 0.33 | 0.55 | 0.55
CNOT gate | 0.67 | 1.0 | 0.8 | 0.5 | 0.73 | 0.73
gtg | 0.67 | 0.2 | 0.0 | 0.17 | 0.36 | 0.18
Mean | 0.56$\pm$0.22 | 0.6$\pm$0.24 | 0.55$\pm$0.24 | 0.48$\pm$0.26 | 0.56$\pm$0.2 | 0.53$\pm$0.2
|  |  |  |  |  |

00footnotetext: questions are shown in order of appearance in the study.

Table 6: Average time taken for correct answers to two-qubit questions in
minutes (N = 22).

| Circle Notation111N = 11
---|---
survey part | A | B | combined
question | w/o vis.222N = 6 | with vis.333N = 11 | w/o vis.333N = 5 | with vis.222N = 6 | w/o vis.111N = 11 | with vis.111N = 11
X gate | 0.71$\pm$0.16(5) | 1.34$\pm$0.06(2) | 2.05$\pm$1.06(3) | 0.8$\pm$0.35(5) | 0.95$\pm$0.39(6) | 1.08$\pm$0.23(6)
Z gate | 1.11$\pm$0.24(3) | 1.51$\pm$0.21(3) | 0.68$\pm$0.08(2) | 0.98$\pm$0.7(5) | 1.02$\pm$0.26(4) | 1.35$\pm$0.53(4)
H gate | 0.93$\pm$0.96(5) | 2.18(1) | 2.28$\pm$0.83(2) | 2.2$\pm$0.64(3) | 1.15$\pm$0.96(3) | 2.2$\pm$0.64(3)
measurement probability | 2.38$\pm$0.33(3) | 4.58$\pm$2.83(3) | 1.38$\pm$0.34(2) | 4.12$\pm$4.42(3) | 2.0$\pm$0.67(4) | 4.39$\pm$3.42(4)
Separability | 1.78$\pm$1.34(3) | 1.87$\pm$0.61(5) | 0.95$\pm$0.94(4) | 0.84$\pm$0.25(3) | 1.3$\pm$1.2(7) | 1.38$\pm$0.7(7)
Entanglement | 1.47$\pm$0.39(4) | 1.73$\pm$1.21(5) | 1.09$\pm$0.33(3) | 1.24$\pm$0.12(2) | 1.36$\pm$0.43(5) | 1.86$\pm$1.13(5)
CNOT gate | 1.59$\pm$1.77(5) | 2.14$\pm$1.31(4) | 0.92$\pm$0.57(4) | 2.08$\pm$0.77(3) | 1.65$\pm$1.41(7) | 2.11$\pm$1.11(7)
gtg | 2.34$\pm$2.01(3) | 2.08$\pm$0.61(3) | 2.34$\pm$2.29(2) | 1.17$\pm$1.04(3) | 2.41$\pm$2.37(4) | 1.48$\pm$0.91(4)
Mean | 1.54$\pm$0.62 | 2.18$\pm$1.02 | 1.46$\pm$0.67 | 1.68$\pm$1.12 | 1.48$\pm$0.51 | 1.98$\pm$1.05
| Dimensional Circle Notation111N = 11
survey part | A | B | combined
question | w/o vis.222N = 6 | with vis.333N = 5 | w/o vis.333N = 5 | with vis.222N = 6 | w/o vis.111N = 11 | with vis.111N = 11
X gate | 1.13$\pm$0.38(6) | 1.77$\pm$0.67(4) | 1.35$\pm$0.65(4) | 1.7$\pm$0.66(5) | 1.29$\pm$0.49(9) | 1.73$\pm$0.67(9)
Z gate | 2.42$\pm$1.12(3) | 9.83$\pm$7.95(2) | 3.25$\pm$0.98(3) | 2.95$\pm$2.69(4) | 3.0$\pm$0.93(3) | 8.84$\pm$6.83(3)
H gate | 2.73(1) | 6.12$\pm$7.98(3) | 6.17$\pm$1.65(2) | 5.88(1) | 7.82(1) | 17.4(1)
measurement probability | 3.22$\pm$1.7(3) | 2.91$\pm$1.62(3) | 1.71$\pm$0.26(3) | 5.13$\pm$3.27(2) | 1.86$\pm$0.34(4) | 2.65$\pm$1.48(4)
Separability | 2.43$\pm$1.85(3) | 3.08$\pm$1.99(2) | 2.48$\pm$2.22(3) | 1.68$\pm$0.9(5) | 2.83$\pm$2.03(5) | 2.14$\pm$1.51(5)
Entanglement | 2.27$\pm$0.49(3) | 1.39$\pm$1.4(4) | 1.41$\pm$0.95(3) | 2.77$\pm$2.48(2) | 1.81$\pm$0.95(5) | 2.09$\pm$2.04(5)
CNOT gate | 2.45$\pm$1.94(4) | 1.57$\pm$1.0(5) | 1.56$\pm$0.87(4) | 1.77$\pm$0.75(3) | 2.1$\pm$1.65(7) | 1.83$\pm$0.83(7)
gtg | 2.53$\pm$0.68(4) | 1.9(1) | (0) | 7.07(1) | (0) | (0)
Mean | 2.4$\pm$0.59 | 3.57$\pm$2.96 | 2.56$\pm$1.73 | 3.62$\pm$2.12 | 2.96$\pm$2.22 | 5.24$\pm$5.93
|  |  |  |  |  |

00footnotetext: Note: The number of participants N that solved the question
correctly and are therefore included in the average is shown in brackets.
Standard deviations are shown where N $>1$. The confidence interval of the
mean is the mean of the standard deviations.

Figure 19: Average score for each question in minutes in the two-qubit survey,
separated by participants who where presented with CN (blue, N=10) and
participants who where presented with DCN (green, N=11). Darker colors are for
questions where participants were not presented with a visualization and
lighter colors for questions where participants were presented with a
visualization. Figure 20: Average time taken for each question in minutes in
the two-qubit survey, separated by participants who where presented by CN
(blue, N=10) and participants who where presented with DCN (green, N=11).
Darker colors are for questions where participants were not presented with a
visualization and lighter colors for questions where participants were
presented with a visualization.

####  A.2.1 Circle Notation

During part A, participants assigned to the CN group solved questions without
visualization with an average score of 0.72$\pm$0.14 with an average time of
1.54$\pm$0.62 minutes for correctly solved questions, while the same questions
with CN were answered with an average score of 0.65$\pm$0.26 with an average
time of 2.18$\pm$1.02 minutes. During part B, the average scores with and
without visualization were 0.55$\pm$0.17 and 0.62$\pm$0.12, respectively, at
average times for correctly solved questions of 1.46$\pm$0.67 and
1.68$\pm$1.12 minutes. When combining survey parts A and B, the average scores
were 0.64$\pm$0.13 without CN and 0.64$\pm$0.12 with CN, at 1.48$\pm$0.51 and
1.98$\pm$1.05 minutes.

While group math-vis, that was assigned questions without CN first, scored
higher than group vis-math on the X gate and H gate questions in survey part A
with, on average, 0.9$\pm$0.1 to 0.3$\pm$0.1, the inverse is apparent for the
separability and entanglement questions, where group vis-math scored higher
than group math-vis 1.0 to 0.7$\pm$0.1. Group vis-math, with CN, took longer
to correctly solve the X gate, Z gate, H gate, measurement probability,
separability, entanglement, and CNOT gate questions with 132$\pm$61 to
86$\pm$32 seconds and took less time on the gtg questions with 140 to 125
seconds.

In survey part B, group math-vis (that was now given questions with CN) scored
higher than group vis-math on the X gate, Z gate, H gate, measurement
probability, and gtg questions with respective average scores 0.68$\pm$0.1 to
0.44$\pm$0.08 and group vis-math scored higher than group math-vis on the
separability, entanglement, and CNOT gate questions with 0.73$\pm$0.09 to
0.53$\pm$0.09. In this survey part, group math-vis, with CN, took longer on
the Z gate, measurement probability, entanglement, and CNOT gate questions
with, on average, 126$\pm$74 to 61$\pm$15 seconds while taking less time on
the X gate, H gate, separability, and gtg questions with 114$\pm$34 to
75$\pm$34 seconds on these questions on average.

Comparing within each group, group math-vis scored higher than itself on the Z
gate question with CN with 0.8 to 0.6, and scored worse with CN than without
CN on the X gate, H gate, entanglement, and CNOT gate questions with
0.85$\pm$0.09 to 0.6$\pm$0.14. When solving questions correctly, group math-
vis took longer on the X gate, H gate, measurement probability, and CNOT gate
questions when presented with CN, with 138$\pm$71 to 84$\pm$39 seconds, while
taking a shorter amount of time on the Z gate, separability, entanglement, and
gtg questions with 101$\pm$27 to 64$\pm$9 seconds.

When presented with CN, group vis-math scored higher than when not presented
with a visualization on the Z gate, measurement probability, separability,
entanglement, and gtg questions with, on average, with 0.76$\pm$0.2 to
0.52$\pm$0.16. The group scored higher when solving questions without CN on
the X gate and H gate questions with 0.5$\pm$0.1 to 0.3$\pm$0.1. They took
longer with CN when solving the Z gate, measurement probability, separability,
entanglement, and CNOT gate questions correctly, with an average of 142$\pm$68
to 60$\pm$14 seconds, and took less time on the X gate, H gate, and gtg
questions when presented with CN, with 134$\pm$8 to 112$\pm$22 seconds.

####  A.2.2 Dimensional Circle Notation

During part A, participants assigned to the DCN group solved questions without
visualization with an average score of 0.56$\pm$0.22 with an average time of
2.4$\pm$0.59 minutes for correctly solved questions, while the same questions
with visualization were answered with an average score of 0.6$\pm$0.24 with an
average time of 3.57$\pm$2.66 minutes. During part B, the average scores were
0.55$\pm$0.24 and 0.48$\pm$0.26, respectively, at average times for correctly
solved questions of 2.56$\pm$1,73 and 3.62$\pm$2.12 minutes. When combining
the survey parts A and B, the average scores were 0.56$\pm$0.2 without
visualization and 0.53$\pm$0.2 with visualization, at 2.96$\pm$2.22 and
5.24$\pm$5.93 minutes.

In both survey parts, on the X gate, Z gate, separability, and gtg questions,
group math-vis scored higher than group vis-math. In part A, this was
0.67$\pm$0.2 to 0.45$\pm$0.22 and in part B, with 0.62$\pm$0.27 to
0.5$\pm$0.3. On the other hand, group vis-math scored higher than group math-
vis on the H gate, measurement probability, entanglement, and CNOT gate
questions 0.75$\pm$0.17 to 0.46$\pm$0.18 in part A and 0.6$\pm$0.14 to
0.33$\pm$0.12 in part B.

In survey part A, group math-vis took longer on the measurement probability,
entanglement, CNOT gate, and gtg questions with 157$\pm$22 to 117$\pm$35
seconds and less time on the X gate, Z gate, H gate, and separability
questions with 312$\pm$186 to 131$\pm$37 seconds. In survey part B, group
math-vis took longer on the X gate, measurement probability, entanglement, and
CNOT gate questions with 170$\pm$83 to 90$\pm$9 seconds and group vis-math
took longer on the Z gate, H gate, and separability questions with 238$\pm$95
to 210$\pm$106 seconds.

Group math-vis had a higher score on the Z gate and separability questions
with DCN with, on average, 0.75$\pm$0.08 to 0.5$\pm$0.0 while performing worse
in this regard with DCN on the X gate, measurement probability, entanglement,
CNOT, and gtg questions with 0.67$\pm$0.18 to 0.43$\pm$0.23. The group took
longer with DCN when correctly solving the X gate, Z gate, H gate, measurement
probability, entanglement, and gtg questions with 255$\pm$114 to 143$\pm$38
seconds and took less time on the separability and CNOT questions with
146$\pm$0 to 103$\pm$3 seconds.

Supported by DCN, group vis-math scored higher on the H gate, entanglement,
CNOT, and gtg questions with 0.65$\pm$0.3 to 0.45$\pm$0.3 while scoring worse
on the Z gate and separability questions with 0.6$\pm$0.0 to 0.4$\pm$0.0. On
the X gate, Z gate, measurement probability, separability, and CNOT questions,
group vis-math took longer with DCN, with 230$\pm$184 to 124$\pm$42 seconds,
while taking longer without DCN on the H gate and entanglement questions with
227$\pm$143 to 225$\pm$142 seconds.

###  A.3 three-qubit survey

Table 7 lists the average scores and table 8 lists the average time taken for
correctly solved one-qubit questions, with and without visualization, for
surveys that were taken supported by CN and DCN. Figure 21 shows the combined
average score per question and figure 22 shows the combined average time taken
(for all answers, not just the correct ones). Cases in which a participant
took more than 20 minutes for one question were excluded.

Table 7: Average scores for three-qubit questions, with and without
visualization (N=15).

| Circle Notation111N = 10
---|---
survey part | A | B | combined
question | w/o vis.222N = 5 | with vis.222N = 5 | w/o vis.222N = 5 | with vis.222N = 5 | w/o vis.111N = 10 | with vis.111N = 10
X gate | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Z gate | 0.6 | 1.0 | 1.0 | 0.6 | 0.8 | 0.8
H gate | 0.6 | 0.8 | 0.6 | 1.0 | 0.6 | 0.9
measurement probability | 0.6 | 1.0 | 1.0 | 0.6 | 0.8 | 0.8
measurement state | 0.6 | 0.6 | 0.8 | 0.6 | 0.7 | 0.6
Separability | 0.4 | 0.4 | 0.4 | 0.2 | 0.4 | 0.3
Entanglement | 0.6 | 0.2 | 0.2 | 0.4 | 0.4 | 0.3
CNOT gate | 0.8 | 1.0 | 0.8 | 0.6 | 0.8 | 0.8
gtg | 0.0 | 0.8 | 0.2 | 0.4 | 0.1 | 0.6
Mean | 0.58$\pm$0.26 | 0.76$\pm$0.28 | 0.67$\pm$0.31 | 0.6$\pm$0.25 | 0.62$\pm$0.28 | 0.68$\pm$0.25
| Dimensional Circle Notation222N = 5
survey part | A | B | combined
question | w/o vis.333N = 2 | with vis.444N = 3 | w/o vis.444N = 3 | with vis.333N = 2 | w/o vis.222N = 5 | with vis.222N = 5
X gate | 1.0 | 0.67 | 0.67 | 1.0 | 0.8 | 0.8
Z gate | 1.0 | 0.67 | 0.33 | 1.0 | 0.6 | 0.8
H gate | 1.0 | 0.0 | 0.67 | 0.5 | 0.8 | 0.2
measurement probability | 1.0 | 0.67 | 0.67 | 1.0 | 0.8 | 0.8
measurement state | 0.5 | 1.0 | 0.33 | 1.0 | 0.4 | 1.0
Separability | 1.0 | 0.33 | 0.67 | 0.0 | 0.8 | 0.2
Entanglement | 1.0 | 0.67 | 0.67 | 1.0 | 0.8 | 0.8
CNOT gate | 1.0 | 0.67 | 0.33 | 0.5 | 0.6 | 0.6
gtg | 0.5 | 0.33 | 0.67 | 0.5 | 0.6 | 0.4
Mean | 0.89$\pm$0.21 | 0.56$\pm$0.27 | 0.56$\pm$0.16 | 0.72$\pm$0.34 | 0.69$\pm$0.15 | 0.62$\pm$0.29
|  |  |  |  |  |

00footnotetext: Standard deviations are shown, where N $>1$.

Table 8: Average time taken for correct answers to three-qubit questions in
minutes (N=15).

| Circle Notation111N = 10
---|---
survey part | A | B | combined
question | w/o vis.222N = 5 | with vis.222N = 5 | w/o vis.222N = 5 | with vis.222N = 5 | w/o vis.111N = 10 | with vis.111N = 10
X gate | 0.94$\pm$0.36(5) | 1.46$\pm$1.16(5) | 1.52$\pm$0.82(5) | 1.05$\pm$0.34(5) | 1.23$\pm$0.7(10) | 1.26$\pm$0.88(10)
Z gate | 1.03$\pm$0.04(3) | 2.02$\pm$1.18(5) | 1.53$\pm$0.87(5) | 1.17$\pm$0.29(3) | 1.35$\pm$0.73(8) | 1.7$\pm$1.04(8)
H gate | 3.72$\pm$1.74(3) | 6.82$\pm$5.99(4) | 3.64$\pm$1.59(3) | 1.85$\pm$0.53(5) | 3.68$\pm$1.67(6) | 4.79$\pm$5.26(6)
measurement probability | 1.42$\pm$0.66(3) | 2.7$\pm$1.17(5) | 1.27$\pm$0.87(5) | 0.44$\pm$0.2(3) | 1.32$\pm$0.8(8) | 1.85$\pm$1.43(8)
measurement state | 1.92$\pm$0.1(3) | 3.53$\pm$1.2(3) | 2.75$\pm$1.71(4) | 1.28$\pm$0.44(3) | 2.14$\pm$1.31(6) | 2.41$\pm$1.44(6)
Separability | 1.19$\pm$0.36(2) | 3.1$\pm$1.92(2) | 2.18$\pm$0.28(2) | 1.27(1) | 2.01$\pm$0.46(2) | 1.23$\pm$0.04(2)
Entanglement | 2.53$\pm$2.24(3) | 2.1(1) | 2.27(1) | 1.16$\pm$0.14(2) | 3.07$\pm$1.89(3) | 1.47$\pm$0.46(3)
CNOT gate | 1.41$\pm$0.46(4) | 1.68$\pm$1.72(5) | 1.6$\pm$0.94(4) | 1.06$\pm$0.04(3) | 1.53$\pm$0.8(7) | 1.48$\pm$1.49(7)
gtg | (0) | 3.59$\pm$1.35(4) | 5.13(1) | 1.98$\pm$0.42(2) | 5.13(1) | 5.13(1)
Mean | 1.77$\pm$0.94 | 3.0$\pm$1.63 | 2.43$\pm$1.26 | 1.25$\pm$0.45 | 2.38$\pm$1.33 | 2.37$\pm$1.51
| Dimensional Circle Notation222N = 5
X gate | 1.83$\pm$0.42(2) | 3.11$\pm$2.38(2) | 1.49$\pm$0.43(2) | 1.94$\pm$0.69(2) | 1.66$\pm$0.45(4) | 2.52$\pm$1.84(4)
Z gate | 1.23$\pm$0.55(2) | 0.97$\pm$0.1(2) | 1.95(1) | 1.88$\pm$1.25(2) | 1.47$\pm$0.56(3) | 1.61$\pm$1.09(3)
H gate | 8.02$\pm$3.47(2) | (0) | 1.79$\pm$0.51(2) | 1.32(1) | 4.55(1) | 1.32(1)
measurement probability | 2.43$\pm$1.0(2) | 1.12$\pm$0.04(2) | 0.52$\pm$0.09(2) | 2.35$\pm$1.9(2) | 1.48$\pm$1.19(4) | 1.74$\pm$1.48(4)
measurement state | 1.05(1) | 1.08$\pm$0.2(3) | 1.77(1) | 2.0$\pm$1.4(2) | 1.41$\pm$0.36(2) | 0.83$\pm$0.23(2)
Separability | 3.66$\pm$2.66(2) | 1.57(1) | 3.51$\pm$3.28(2) | (0) | 6.78(1) | 1.57(1)
Entanglement | 6.12$\pm$4.09(2) | 1.88$\pm$0.52(2) | 2.1$\pm$1.93(2) | 2.76$\pm$1.43(2) | 4.11$\pm$3.78(4) | 2.32$\pm$1.16(4)
CNOT gate | 0.89$\pm$0.18(2) | 1.55$\pm$0.62(2) | 2.68(1) | 4.52(1) | 1.88$\pm$0.81(2) | 2.72$\pm$1.79(2)
gtg | 17.48(1) | 7.28(1) | 2.18$\pm$1.78(2) | 1.52(1) | 3.97(1) | 7.28(1)
Mean | 4.75$\pm$5.37 | 2.32$\pm$2.12 | 2.0$\pm$0.82 | 2.29$\pm$1.01 | 3.03$\pm$1.91 | 2.44$\pm$1.92
|  |  |  |  |  |

00footnotetext: Note: The number of participants N that solved the question
correctly and are therefore included in the average is shown in brackets.
Standard deviations are shown where N $>1$. The confidence interval of the
mean is the mean of the standard deviations.33footnotetext: N = 2

44footnotetext: N = 3

Figure 21: Average score for each question in minutes in the three-qubit
survey, separated by participants who where presented by CN (blue, N=10) and
participants who where presented with DCN (green, N=5). Darker colors are for
questions where participants were not presented with a visualization and
lighter colors for questions where participants were presented with a
visualization. Figure 22: Average time taken for each question in minutes in
the three-qubit survey, separated by participants who where presented by CN
(blue, N=10) and participants who where presented with DCN (green, N=5).
Darker colors are for questions where participants were not presented with a
visualization and lighter colors for questions where participants were
presented with a visualization.

####  A.3.1 Circle Notation

During part A, participants assigned to the CN group solved questions without
visualization with an average score of 0.58$\pm$0.26 and an average time of
1.77$\pm$0.94 minutes for correctly solved questions, while the same questions
with visualization were answered with an average score of 0.76$\pm$0.28 and an
average time of 3.0$\pm$1.63 minutes. During part B, the average scores were
0.67$\pm$0.31 and 0.6$\pm$0.25, respectively, at average times for correctly
solved questions of 2.43$\pm$1.26 and 1.25$\pm$0.45 minutes. When combining
parts A and B of the survey, the average scores were 0.62$\pm$0.28 without
visualization and 0.68$\pm$0.25 with visualization, at 2.38$\pm$1.33 and
2.37$\pm$1.51 minutes.

In survey part A, group math-vis, that was presented questions without CN
first, scored higher than group vis-math on the entanglement question with 0.6
to 0.2. Group vis-math scored higher on the Z gate, the H gate, the
measurement probability, the CNOT, and the gtg questions with 0.92$\pm$0.1 to
0.52$\pm$0.27. In this survey part, group math-vis took longer when solving
the entanglement question correctly, with 152 to 126 seconds, while group vis-
math took longer for correct solutions to the X gate, Z gate, H gate,
measurement probability, measurement state, separability, and CNOT gate
questions, with 183$\pm$101 to 100$\pm$53 seconds.

In survey part B, group math-vis scored higher on the H gate, entanglement,
and gtg questions with 0.6$\pm$0.28 to 0.33$\pm$0.19, while group vis-math
scored higher on the Z gate, measurement probability, measurement state,
separability, and CNOT gate questions with 0.8$\pm$0.22 to 0.52$\pm$0.16.
Group vis-math took longer on all questions in survey part B with 146$\pm$71
to 75$\pm$26 seconds.

Group math-vis, that was presented questions without CN first, scored higher
on the H gate and gtg questions with 0.7$\pm$0.3 to 0.3$\pm$0.3 with CN. The
group scored worse with CN on the separability, entanglement, and CNOT gate
questions by a score difference of 0.6$\pm$0.16 to 0.4$\pm$0.16. With CN.
Participants in the group took longer when solving the X gate, Z gate and
separability questions correctly, 70$\pm$5 to 63$\pm$6 seconds, while taking
less time on the H gate, measurement probability, measurement state,
entanglement, CNOT gate, and gtg questions with 132$\pm$52 to 69$\pm$27
seconds.

Group vis-math, that was presented questions with CN first, scored higher on
the H gate, CNOT gate, and gtg questions with CN with 0.87$\pm$0.09 to
0.53$\pm$0.25. The group scored lower with CN on the measurement state
question with 0.8 to 0.6. When presented with CN, participants in the group
took longer on the Z gate, H gate, measurement probability, measurement state,
separability, and CNOT gate questions, with 198$\pm$101 to 130$\pm$49 seconds.
They took less time on the X gate, entanglement, and gtg questions, with
178$\pm$93 to 143$\pm$53 seconds.

####  A.3.2 Dimensional Circle Notation

During part A, participants assigned to the DCN group solved questions without
visualization with an average score of 0.89$\pm$0.21 with an average time of
4.75$\pm$1.77 minutes for correctly solved questions, while the same questions
with visualization were answered with an average score of 0.6$\pm$0.24 with an
average time of 2.32$\pm$0.63 minutes. During part B, the average scores were
0.56$\pm$0.16 and 0.72$\pm$0.34, respectively, at average times for correctly
solved questions of 2.0$\pm$1.34 and 2.29$\pm$1.33 minutes. When combining
parts A and B of the survey, the average scores were 0.69$\pm$0.15 without
visualization and 0.62$\pm$0.29 with visualization, at 3.03$\pm$0.78 and
2.44$\pm$0.9 minutes.

Group math-vis, that was assigned questions without DCN first, scored higher
than group vis-math on the X gate, Z gate, H gate, measurement probability,
separability, entanglement, CNOT, and gtg questions in survey part A, with
0.94$\pm$0.17 to 0.5$\pm$0.24. The inverse is apparent for the measurement
state question, where group vis-math scored higher than group math-vis with 1
to 0.5. Group vis-math, with DCN, took longer to correctly solve the Z gate,
measurement probability, separability, entanglement, and gtg questions, with
371$\pm$353 to 154$\pm$143 seconds, whereas group math-vis took longer for
correctly solving the X gate, measurement state, and CNOT gate questions,
115$\pm$52 to 76$\pm$25 seconds.

In survey part B, group math-vis (that now was presented questions with DCN)
scored higher than group vis-math on the X gate, Z gate, measurement
probability, measurement state, entanglement, and CNOT gate questions with
0.92$\pm$0.19 to 0.5$\pm$0.17 and group vis-math scored higher than group
math-vis on the H gate, separability, and gtg questions with 0.67$\pm$0.0 to
0.33$\pm$0.24. In this survey part, group math-vis, with DCN, took longer on
the X gate, measurement probability, measurement state, entanglement, and CNOT
gate questions, with 163$\pm$57 to 103$\pm$43 seconds, while taking less time
on the Z gate, H gate, and gtg questions, with 118$\pm$10 to 94$\pm$14
seconds.

Comparing within each group, group math-vis scored higher with DCN on the
measurement state question with 1 to 0.5, and scored lower with DCN than
without DCN on the H gate, Separability, and CNOT gate questions by averages
of 1.0$\pm$0.0 to 0.33$\pm$0.24. When solving questions correctly, group math-
vis took longer on the X gate, Z gate, measurement state, and CNOT gate
questions when presented with DCN, with 155$\pm$67 to 75$\pm$21 seconds, while
taking less time on the H gate, measurement probability, Entanglement, and gtg
questions, 511$\pm$333 to 119$\pm$35 seconds.

When presented with DCN, group vis-math scored higher than itself on the Z
gate, measurement state, and CNOT gate questions by, on average, 0.78$\pm$0.16
to 0.33$\pm$0.0. The group scored higher when solving questions without DCN on
the H gate, Separability, and gtg questions with 0.67$\pm$0.0 to
0.22$\pm$0.16. They took longer with DCN when solving the X gate, measurement
probability, and gtg questions correctly, 230$\pm$154 to 84$\pm$41 seconds,
and took less time on the Z gate, measurement state, separability,
entanglement, and CNOT gate questions when presented with DCN, on average
144$\pm$38 to 85$\pm$20 seconds.

##  Appendix B Data processing

During this study, some technical issues arose due to a short termed set up of
the surveys: On the first day of the study, 12 May 2024, we noticed that
answer IDs were saved randomly, could not scrape the data, and therefore lost
data of seven participants of the one-qubit survey. In addition, we had to
remove one guess-the-gates question from the analysis of the results of the
second survey because it was incorrectly incorporated into the survey, so the
participants got exactly the same question without DCN and with DCN in survey
part A and B. In addition, we had to remove the two-qubit question that asked
for the state after a measurement due to an error in the answer. Also, we had
some cases where participants randomly jumped within the survey, especially
during survey 1, due to server issues that we fixed later. This led to some
questions being answered twice by the same person. We corrected for all of
this in the data.

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