# A Framework for Curriculum Transformation in Quantum Information Science and
Technology Education

Simon Goorney1,2,†, Jonas Bley3,†, Stefan Heusler4, Jacob Sherson1,2,∗ 1
Department of Management, School of Business and Social Science, Aarhus
University  2 Niels Bohr Institute, University of Copenhagen, Denmark  3
Department of physics and Research Center OPTIMAS, RPTU Kaiserslautern-Landau,
67663 Kaiserslautern, Germany  4 Institute for Didactics of physics, Münster
university, Wilhelm-Klemm-Str. 10, D-48149 Münster, Germany  † These authors
contributed equally to this work and share first authorship  ∗ Correspondence:
<EMAIL_ADDRESS>
###### Abstract

The field of Quantum Information Science & Technology (QIST)is booming. Due to
this, many new educational courses and university programs are needed in order
to prepare a workforce for the developing industry. Owing to its specialist
nature, teaching approaches in this field can easily become disconnected from
the substantial degree of science education research which aims to support the
best approaches to teaching in Science, Technology, Engineering & Mathematics
(STEM)fields. In order to connect these two communities with a pragmatic and
repeatable methodology, we have synthesised this educational research into a
decision-tree based theoretical model for the transformation of QIST
curricula, intended to provide a didactical perspective for practitioners. The
Quantum Curriculum Transformation Framework (QCTF)consists of four steps: 1.
choose a topic, 2. choose one or more targeted skills, 3. choose a learning
goal and 4. choose a teaching approach that achieves this goal. We show how
this can be done using an example curriculum and more specifically quantum
teleportation as a basic concept of quantum communication within this
curriculum. By approaching curriculum creation and transformation in this way,
educational goals and outcomes are more clearly defined which is in the
interest of the individual and the industry alike. The framework is intended
to structure the narrative of QIST teaching, and with future testing and
refinement it will form a basis for further research in the didactics of QIST.

History: Received: 21st August 2023, Revised: 6th December 2023

Keywords: Quantum Education, Interdisciplinarity, Didactical Framework,
Quantum Teleportation, Curriculum Development

##  1 Introduction

The field of Quantum Information Science & Technology (QIST), although
originating in physics, is gaining increasing interdisciplinary popularity
[1]. This is presently inducing a significant rise in the number of degree
programs and courses in this field [2, 3], which requires a didactical
perspective which is currently significantly lacking, due to its rapid
development and specialist nature. As a result, it is of great importance that
the results of didactical research are incorporated into quantum curriculum
creation and transformation at the present time. There are also several
immense challenges that the field presents: Its interdisciplinarity and
complexity makes content preparation difficult as many concepts require
perspectives from different fields. Additionally, many current teaching
methods are suited to learners of particular academic backgrounds, stemming
from disciplines such as physics and computer science. However, learners come
to the QIST classroom from many fields which means that these approaches must
be adapted.[4].

Given the importance of widespread, progressive education and training in
QIST, the European Quantum Technology Education community (QTEdu) have
developed practical guidelines for educators in non-formal contexts, and in
the high school environment [5, 6]. However, the most urgent need for quality
education is in the university context, where graduates of Bachelor and Master
programs in STEM fields will make up the emerging quantum workforce [3]. More
advanced topics in QIST are particularly challenging to teach, due to the
specialist knowledge required [7]. As a result, the majority of teaching
faculty are based in physics, Information Sciences, or Engineering, without
access to significant pedagogical support [4].

Many of these practitioners are not grounded in knowledge of those core
educational principles which may be background for high school teachers. For
this reason, and in order to promote dialogue between didactics and educators,
there needs to exist advice that is easy to follow. Additionally, this should
be industry-needs oriented as preparing the quantum workforce for work in the
industry is a major intent of many quantum courses [8, 9, 10]. For this
purpose, in this work we present a didactical teaching framework, the Quantum
Curriculum Transformation Framework (QCTF), based on a decision tree which can
be used to guide practitioners through teaching design, from topic selection
to the specific teaching approach based on existing didactical theory. It
should be emphasised that subsections of the framework may also be used in
order to support specific aspects of pedagogical preparation, even if the
complete framework is not applicable in every case. In addition, it may be
extended and modified when presented with new research results. The framework
is summarised in Fig. 1.

In the level of school education, there exists a storied tradition of research
on the use of pedagogical content mapping, [11, 12], as a means of aligning
learning material with teaching methods which are most optimal for promoting
different kinds of learning outcomes. However, this approach is rarely taken
at the university level, and never in the field of QIST. There are many
differences between higher education and secondary school in the complexity
and the interdisciplinarity of content, in the time scopes and time available
to study specific subjects and, perhaps most importantly, in the learners.
Although at the university level, QIST has attracted attention toward research
into student understanding [13], and development of novel teaching methods
[14], there is little research available into how best to build up curricula
from these components. This article, and the QCTF we propose, aims to address
this.

The paper is structured as follows: In Sect. 2, we summarise the basis for
choosing educational content (the “what”), which makes use of the European
Competence Framework (CF)for Quantum Technologies [15, 16]. Then, three
targeted skills areas are introduced in Sect. 3, as both a motivation and a
means of communicating the content chosen to teach. In Sect. 4, a modified
Bloom’s taxonomy is described for use in defining learning goals. In Sect. 5,
Ainsworth’s Designs, Functions, Tasks (DeFT)framework is summarised and
applied to the QIST context in Sect. 6. Lastly, we show a possible explicit
teaching approach to quantum teleportation based on our framework in Sect. 7.
Throughout this work, we will stick to examples to ground the QCTF and convey
the meaning of the didactical theory in the QIST context. Quantum
teleportation is chosen as a topic specifically because it makes use of
various concepts central to QIST: entanglement, measurements and unitary
operations in three-qubit systems and has many applications [17]. In order to
make the ordering of the framework’s sections more explicit and intuitive for
practitioners, we roughly structure it into a “what” - “why” - “how” format,
in which we first choose content to be taught, then formulate specific
teaching goals for the interaction, and then consider the specific teaching
approaches which best address these goals while also respecting the diversity
of student backgrounds. As such, our approach is akin to a backwards design
process [18].

Figure 1: The Quantum Curriculum Transformation Framework (QCTF), a decision-
tree based methodology for designing and transforming educational material and
methods in QIST. It is based on a What - Why - How - approach, starting with
the European Competence Framework for Quantum Technologies [15] and
concretising towards specific teaching approaches that are based on didactical
research.

##  2 Competence Framework

The European Competence Framework (CF)for Quantum Technologies is a tool
designed by the Quantum Flagship, the EU funding program for QIST, intended as
a common language for communicating skills in the field. The framework has
utilised a Delphi method to identify key content according to industry needs
[10], and is presently available in version 2.0 [19], providing a structured
overview of the field. It may be used to describe skills required for jobs,
learning outcomes from courses, and content of entire degree programs. The
QCTF builds on the CF as a backbone, providing a means to determine content
selection when designing educational initiatives. Each topical domain is
assigned a number 1-8, covering Quantum Background (domains 1 and 2), Core
Device Technologies (domains 3 and 4), and QT systems and applications
(domains 5-8). Within each domain, topics are assigned numbers such that a
complete map can be generated of knowledge in QIST. An overview of the CF
domains is shown in Fig. 2.

Figure 2: The 8 domains of the European Competence Framework for Quantum
Technologies, covering Quantum Background (domains 1 and 2), Core Device
Technologies (domains 3 and 4), and QT systems and applications (domains 5-8).
Adapted from [15].

For use in designing curricula, we present an application-oriented example of
a small curriculum with topics chosen from the CF, focusing on the basics of
quantum technologies and quantum communication, in Fig. 3.

Figure 3: Application oriented example curriculum coloured using the
Competence Framework [15]. These three modules could be part of, e.g. a one-
or two-week workshop covering the basics of QIST, quantum communication and
giving an outlook into quantum sensing and computing.

That plain knowledge transfer, e.g. in the form of purely a traditional
lecture is not effective is well known [20, 21]. Therefore, planning courses
should not stop after it has been decided which concepts to teach but rather
should go further, considering first the skills that the learners should
achieve in the context of the chosen concepts. We classify these skills in the
following.

##  3 Targeted Skills

In the context of higher education on quantum technology, we see possible
targeted skills in three main areas: Theory & Analytics, Computation &
Simulation and Experiment & Real World. We emphasise that these targeted
skills are those engaged in different approaches to teaching topics in all
eight domains of the CF. We do not present these as a classification scheme
for concepts, which is in fact the role of the CF itself [10]. Rather it is
possible for many concepts to engage several or all of the “targeted skills”.
Quantum Teleportation, for example, is a rich topic which can be approached
with a view to developing each of these three targeted skills, while other
topics may be limited to developing a subset of these skills.

QIST, in many cases, allows or even necessitates an interdisciplinary view.
For example, quantum computing algorithms are significantly restricted by the
hardware that they run on. Gate times and fidelities limit circuit depth and
general complexity of possible algorithms, which can be summarised by
performance benchmarks [22]. This leads to the practical problems of finding
the right quantum computer for a specific algorithm [23] and, in addition,
determining which algorithms work well on a given hardware architecture [24,
25]. This means that expertise in both areas is required to make justified
decisions. As is also pointed out in [10], generally one would address
industry needs by having skills in all three of these skill areas.

###  3.1 Theory & Analytics

When targeting theoretical skills, one targets the skill of derivation and
proof of theoretical ideas within QIST like the no-cloning-theorem or the no-
communication theorem. Instead of just being able to recite these theorems,
learners ideally should understand the process of arriving at them. By doing
so, they gain skills required to expand the given set of ideas, the
theoretical framework of QIST, built up on complex euclidean spaces or so-
called Hilbert spaces [26, 27], to generate new knowledge. In Sect. 4, we
concretise this using Bloom’s taxonomy.

Analytical skills refer to calculations involving numbers or variables. In
QIST, e.g. the Dirac ket notation is popular to represent vectors and linear
transformations in these complex vector spaces. As an example, the skill of
calculating the effects of various quantum logic gates (linear
transformations) within this notation or the equivalent vector notation can be
targeted. We refer to, e.g. the textbooks [26, 27, 28] for more details on
QIST from this perspective.

###  3.2 Computation & Simulation

When targeting computational skills, a focus lies on programming within QIST.
This can be done in various different languages and libraries, e.g. in Python
using a library like Qiskit [29] or Pennylane [30], in C++, e.g. with the
Intel Quantum Software Development Kit (QSDK) [31] or in the standalone
languages Open Quantum Assembly Language (OpenQASM) [32], Q# [33] or Silq
[34], but there are also many more that are reviewed in, e.g. [35]. These
languages can be deployed on real quantum hardware or simulators. Real quantum
hardware tends to be noisy, especially for large systems of qubits and
simulating them is exponentially difficult in the number of qubits [36, 37].
Gate errors can also be simulated and analysed within these simulations [38]
as these are and will be much more relevant than in classical computing. This
is where the field of quantum error correction starts [39].

We refer to textbooks such as [40, 41] for more detail on QIST from this
perspective.

###  3.3 Experiment & Real World

Last but not least, QIST can be approached from an experimental perspective.
Skills in this area can cover a vast array of planning and preparation of
experimental setups, a deep understanding of the functioning of the devices
that are used, practical skills like calibration and adjustment of these
devices and data acquisition and analysis [42]. In the context of quantum
information science and technology, qubit platforms and types are of special
interest for transport of quantum information (quantum communication), precise
measurement of real world quantities like temperature or magnetic fields
(quantum sensing) or universal and analogue quantum computing. Knowledge and
skills required to get into one of these fields in these areas can be targeted
with textbooks like [43, 44] and in practical lab courses. Typical challenges
that are discussed in more detail in [42] include evoking excitement and
making quantum effects “visible” with these challenges being tied to another.
How to achieve the latter and further benefits, like gaining intuition and
familiarity of quantum mechanics, gaining conceptual understanding and getting
a sense for possible applications in the field of quantum technologies, are
discussed in [45].

All of these three categories are vast in that they each define multiple
fields of research. The sheer endless amount of content to choose from and
skills that could be acquired is a challenge for educators in the field of
QIST education. Due to this complexity, theoretical frameworks like the CF are
needed to aid in the selection process. With our framework, we aim to go
beyond the level of pure content towards a practical guideline that is applied
to curriculum transformation and creation by educators. This is shown further
in the following sections.

##  4 Learning Objectives & Bloom’s taxonomy

The more specific question of which cognitive learning goals within these
areas of expertise should be accomplished can be asked next. A classification
of these learning goals can help for various reasons. First of all, it
encourages aiming for learning goals beyond just remembering and understanding
concepts. Secondly, the structure can help align the educator’s purposes with
that of the learners, aligning the teaching process with valid performance
tests [46]. In [47], the authors discuss this alignment for the purpose of
designing empirical studies, arguing that if learning goals beyond
memorisation should be achieved, ways to test this also have to be found. In
QIST education specifically, there is a lack of standardisation of teaching
material and measurement of learning outcomes [4].

Bloom’s taxonomy is a classification scheme of such learning goals widely used
in high school education. It comes in variations, but often six levels are
defined based on increasing cognitive demand which represent lower order
thinking skills to higher order thinking skills [48, 49]. In the traditional
taxonomy, higher order skills can only be reached if the previous skill is
attained, defining a theoretical progression. This formulation of Bloom’s
taxonomy is displayed in Fig. 4.

Bloom’s taxonomy is used, e.g. in computer science education [50] and is also
explored in physics education [51, 52]. Its wide use may be attributed to its
simplicity and practicality, particularly in the high school setting where the
hierarchical nature of the 6 levels can be used to set clear learning
objectives for students with a variety of backgrounds and interests. Despite
this, the taxonomy in its traditional form may come with some limitations,
particularly when applied to the context of higher education. In [53], Bhaw
and Kriek point out that while cognitive demand associated with the six levels
may indeed be hierarchical, learning situations in science education rarely
involve only individual levels. Rather, they are inherently connected, for
example when applied to problem-solving. In an evaluation problem for example,
comparing two different approaches to address a problem requires understanding
them both.

Similarly, in the context of QIST, we further re-interpret the taxonomy, for
the following reason. The foundational concepts in QIST are well known to be
abstract and disconnected from everyday life, far more so than many other
areas of science [53]. As a result, the lower-order skills: understanding, and
even basic knowledge, of concepts in QIST actually demand application of the
higher-order skills. It is impossible to understand ideas such as
superposition and entanglement, off which quantum technologies are based,
without creative thought and imagination, as any possible realistic
representation of these concepts are inherently non-everyday objects. This
abstract nature also demands an inherent comparison to the classical world,
and associated evaluative and analytical thinking. Our re-interpretation of
Bloom’s taxonomy, as applicable in fields such as QIST in which the
relationship between the cognitive skills involved is not purely hierarchical,
is shown in Fig. 4.

Figure 4: Revised Classical Bloom’s Taxonomy appropriate for use in describing
learning outcomes in QIST, adapted from [48]. Since creative abstraction is
required in order to understand basic quantum concepts, the relationship
between Bloom’s thinking skills is non-hierarchical.

In the following, we provide example tasks in each of the three targeted skill
areas for every level of the original Bloom’s taxonomy in the context of
quantum teleportation. These don’t constitute an exhaustive list of such
questions. Nevertheless, one can assume that the concept of quantum
teleportation and its connection to QIST is thoroughly understood if all these
tasks are accomplished by a learner.

###  4.1 Bloom’s taxonomy applied to Quantum Teleportation

In the following, we show example tasks in the context of quantum
teleportation for each of the three targeted skill areas (Sect. 3 and for
every level of Bloom’s taxonomy) and cite references that can be consulted to
approach these tasks. In this way, Bloom’s taxonomy enables the creation of
tasks for topics within QIST and structuring of the educator’s intentions for
student’s learning outcomes. Bloom’s taxonomy is often implemented using verb
lists, where task-defining verbs are mapped to a corresponding level in
Bloom’s taxonomy, see, e.g. [54, 55], with some overlap of verbs into multiple
levels, which is discussed in [56]. In the following, we use selected verbs
from these lists.

#### Theory & Analytics

Remember

Name the quantum gates used in the quantum teleportation algorithm. [28]

Understand

Describe the process of quantum teleportation abstractly (independent of the
implementation) [28]. Describe why the no-cloning-theorem is not violated
[57]. Discuss why no faster than light communication is happening. [57]

Apply

Use the Bell basis to describe the quantum teleportation algorithm
mathematically. [58]

Analyse

Examine the effect of decoherence on the resulting state in the quantum
teleportation algorithm. [59, 60]

Examine the effect of quantum teleportation if it is applied to a qubit that
is itself entangled in a multi-qubit system. [61]

Evaluate

Argue whether Quantum Teleportation is an inherent advantage of quantum
physics in respect to classical physics or if it is more of a workaround
necessary due to no-cloning. [62]

Create

Design a quantum teleportation algorithm for states of two qubits. [63, 64]
Design a quantum teleportation algorithm such that three parties can exchange
quantum information with each other. [65]

#### Computation & Simulation

Remember

List the functions in the Qiskit library that are used in the quantum
teleportation algorithm. [31]

Understand

Add comments to explain each line of code in the quantum teleportation
algorithm. [66]

Apply

Translate the Qiskit code to another Python library, like Pennylane. [67]

Analyse

Compare the resulting fidelities of teleported states in various quantum
computers for different qubit choices. [60, 66]

Evaluate

Select the best hardware to perform quantum teleportation on (being
restricted, e.g. to the free IBM hardware). [31, 68]

Create

Program a Python notebook with the Qiskit library, implementing a quantum
network using a starting node, two middle nodes and an end node, where quantum
information is teleported from one qubit to the last using three auxiliary
qubits and a stationary qubit at every node [69]. Modify the teleportation
algorithm using quantum error correcting code and/or substituting quantum
gates for gate combinations with smaller errors. [70] Improve error correction
using quantum teleportation. [71]

#### Experiment & Real World

Remember

List the necessary optical components used in all-optical quantum
teleportation. [72]

Understand

Describe the function of the optical components within the quantum
teleportation algorithm. [72]

Apply

Sketch an experimental design of quantum teleportation incorporated into a
quantum cryptography protocol to extend the range of this protocol. [73]

Analyse

Differentiate Quantum Teleportation from other quantum optical algorithms
shown in, e.g. [74]. What are the differences and similarities in experimental
setups?

Evaluate

Summarize the advantages and challenges of different areas of application
(urban areas, ground-satellite, …) to weigh these areas against each other in
the context of quantum communication. [75, 76, 77, 78, 79, 80]

Create

Design an experiment to teleport photon states to atomic particles. [81] Given
an algorithm for teleportation of two-qubit states, extend the single qubit
experimental design to teleport two-qubit states. [82]

Having decided on tasks that should be covered according to Bloom’s taxonomy,
now is the time to answer how exactly this is to be done, a question that can
be answered by the educator and given as requirement to the students or by the
students themselves. For example, the task “Describe the process of quantum
teleportation abstractly” can be answered descriptively (using words)
symbolically (using mathematics), using graphical illustrations or any
combination of these. Alternatively, students could be asked to “translate”
between these representations, for example to explain a schematic of the
teleportation protocol. This is discussed in the following section.

In addition, some of these tasks may seem to difficult to answer straight
away, especially some of the tasks in the Create category. In these cases,
proper support should be given to students (a practice commonly referred to as
scaffolding [83] that we also briefly discuss in Sect. 7.3). With this support
and given enough time, it is possible that students find satisfying solutions
for these tasks.

##  5 Representations

An important part of the question of why and how to teach is why and how to
represent the chosen concepts. By representation, here we mean anything
external (as opposed to internal or mental), visual or auditory that is used
to convey a concept [84]. Most of the time, multiple external representations
(MER s) will be used in science education as, for example, text will be
combined with equations or figures. In the QIST context, MER s are used in
order to convey scientific ideas and concepts, like the Dirac bra-ket
notation, vector and matrix notations, circuit diagrams or diagrams of
experimental set ups. Understanding and working with these representations is
part of the scientific workflow and necessary to be able to communicate
concepts within the QIST community and can therefore be seen as an essential
learning goal of QIST curricula.

The Designs, Functions, Tasks (DeFT)Framework [85] is a theoretical framework
for the use of MER s that will be summarised here for our purpose. It consists
of three layers to consider when combining representations: Designs, functions
and tasks. While designs and tasks consider the how of representing
information, the function layer constitutes the why. Therefore, in this work,
we will firstly cover the function layer in order to line up the DeFT
framework with the taxonomy presented here. The DeFT framework is visualized
in 5.

Figure 5: The DeFT framework [86] adapted for the purpose of this work for the
use of MER s. Firstly, the functions describe various reasons for why to use
MER s. Then, the MER s should be designed (utilising the five design aspects)
such that ECL is reduced, considering learner characteristics and the listed
effects [87]. How this is best done in practice in various QIST contexts is
mostly an open research question.

###  5.1 Functions

The Function layer within this framework describes three main roles that MER s
can play: Firstly, to complement, which means to provide various perspectives
and information on the same concept so that learners can benefit from the
specific advantages of each representation and to support different processes,
i.e. strategies and individual differences of learners. For example, the Bloch
sphere provides learners with an additional strategy of reasoning about the
effect of quantum gates geometrically, as rotations on the sphere [88],
complementing calculating in the computational basis typical Dirac notation.
Another possible function MER s is to constrain interpretation i.e. to specify
ambiguous or complex information, e.g. coming from text, for example with
symbols or mathematical formulas. This can be an inherent property of the
representation, or can be by familiarity of the learner with the
representation that is used. And lastly to construct deeper understanding,
i.e. to support learners in achieving new insights on a concept which would be
difficult with only one representation [85]. The Bloch sphere would be another
example of this as it visualises two complex numbers, utilising the redundancy
of the global phase, to enable new visual insights on single-qubit quantum
systems, sometimes even leading to scientific advances, e.g. in the context of
Rabi oscillations [89, 90]. In these various ways, different MER s should be
chosen carefully and given as tools to support learners in achieving the
levels in Bloom’s taxonomy that the educators want them to achieve. For
example, in order to select the best hardware for a given quantum algorithm,
one could analyse coupling maps with gate errors like the one in [91, Fig. 5],
while, on the other hand, this representation is not suitable to describe the
mathematics of any quantum algorithm.

###  5.2 Designs

The Design layer constitutes the how of the use of MER s. It describes the
number of representations used, the way the information is distributed, the
form of the MER s (i.e. pictures, text, animations, sound, equations or
graphs), the sequence (i.e. the order in which the representations are shown),
and the translation (i.e. indications of relations between the
representations, for example visual cues using colouring or spatial
proximity). MER s should be designed to reduce unnecessary cognitive tasks as
described in the following.

###  5.3 Tasks

Last but not least, the Task layer describes the cognitive tasks that learners
need to overcome in order to benefit from the MER s used. Cognitive tasks
depend on characteristics of the representations (e.g. addressed sensory
channel, modality, level of abstraction, type or dimensionality [86]) and on
characteristics of the learners. For example, learners should gain
representational competence, i.e. they need to understand the various Design
aspects of the depicted MER s, in order to benefit from them [92]. Cognitive
tasks are analysed within the Cognitive Load Theory (CLT)[93, 87]. The core
idea is that “Learning, reflected by performance change, requires working-
memory capacity” [93]. Psychology research has uncovered that the human brain
works on a limited amount of such temporary working memory, a short term
storage, that is used to store information acutely necessary to perform a task
[94, 95]. The working memory is often seen as consisting of two channels: a
channel for visual and spatial information and a channel for audio information
[96, 97]. Further, it was shown that it is beneficial for learning to make use
of both these memory channels by giving visual and audio information at the
same time, for example in a typical presentation, as opposed to giving
information in written words and pictures, i.e. using the same working memory
channel for different information [98]. This is referred to as the modality
principle.

Another known effect of CLT is the redundancy effect suggesting that redundant
material can hinder learning rather than having positive effects [99]. The
redundancy effect, e.g. suggests that experts, i.e. learners with high
previous knowledge, could be hindered by the use of MER s when the different
representations don’t provide them with additional information or processes.
This is referred to as the expertise-reversal effect [100] as one form of the
more general aptitude treatment interaction [101]. The split attention effect
[102] suggests that MER s should be integrated to one single source of
information rather than split either spatially or temporally. Another point to
consider when using MER s is that learners need to translate between these
representations. This translation will be harder the more the representations
differ in format and operators like various levels of abstraction or supported
strategies [85]. In QIST, a typical example is that of the use of vector
notation and Dirac notation in course content design. If learners aren’t good
at translating between these representations, even if they are somewhat
familiar with both, the use of both at the same time may lead to increased
cognitive load and may therefore worsen learning outcomes [86].

All of these effects have an influence on Extraneous cognitive load (ECL)
which is the cognitive load imposed on the learner by the way the information
is provided, i.e. the Design, and not the complexity of the information itself
which is referred to as Intrinsic cognitive load (ICL) [103]. When ECL is
reduced, learners can focus more on learning the content itself. Especially if
the ICL of a given task is high, the negative effects of high ECL can become
stronger (element-interactivity effect [104]). Even though ECL is usually seen
as not desirable, some representations might be necessary to learn even though
they are not the most efficient ECL-wise, because they are widely accepted by
the academic community and broadly used for communicating scientific concepts,
an example of this being the Dirac notation. In the following, the DeFT
framework is applied to the QIST context in more detail.

##  6 Applying the DeFT-framework to QIST by the example of quantum
teleportation

Quantum information science and technology is a field with a rich variety of
forms of representations. To show this and how to apply the DeFT framework to
this space, we again refer to quantum teleportation and show examples of
descriptive (i.e. with text), symbolic (i.e. mathematical) and graphical (i.e.
with pictures and diagrams) visual representations of the process within the
three different approaches. This is chosen as a practical classification of
visual external representations, rather than a comprehensive one. As pointed
out in, e.g. [105], creating a comprehensive classification of representations
is a non-trivial endeavour that has been approached in, e.g. [106], a
philosophical approach to models in language and maths, and [107], where a
general differentiation is made between verbal, mathematical and
diagrammatic/pictorial models, similar to what we do in this paper. However,
the work by Gilbert goes into much more detail.

Models and representations are often defined using the other [108, 109] and
further exploring the semantic differences is beyond the scope of this paper.
Rather than doing so or giving a comprehensive characterisation of such
representations, this application of the DeFT framework serves to show how
possible representations can be categorised into the three targeted skill
areas and point towards the thought processes that come with choice or
creation of multiple suitable external representations using the DeFT
framework. As is laid out in [110], learners need to understand all the
representations that they work with, all the given support on the syntactic
and/or semantic level and how these representations connect to each other in
order to gain deep understanding of the content. Some typical representations
used in QIST are also discussed in [111, 112] in the context of problem
solving in Quantum Computing.

###  6.1 Visual Representations of the Theory & Analytics of Quantum
Teleportation

#### Descriptive

We provide a text/descriptive representation of the Theory & Analytics of
quantum teleportation in A. This is an abstract text describing the quantum
teleportation protocol as a transfer of quantum information made possible due
to the transitivity of entanglement. This description leaves some room for
interpretation. For example, it describes a connection between entanglement
and measurement that is not necessarily immediately visible. Therefore, it
could benefit learners to constrain interpretation by providing symbolic
and/or graphical representations to complement the description.

#### Symbolic

There are different ways to represent quantum teleportation symbolically. It
is often represented in Dirac ket notation [28]. Within this notation, one can
use the computational basis $\left|0\right\rangle$ and $\left|1\right\rangle$.
A more abstract formulation uses Bell states and Bell measurements, i.e.
measurements in the Bell basis [113]. Both are also shown in A. There is a
more general formulation with density matrices that can be suitable for a
formal mathematical approach to the topic and can also be used for
generalisations like the teleportation of mixed states and quantum gates
[114]. Within the DeFT framework, the mathematical representation serves the
function of complementing, i.e. offering multiple perspectives on the same
concept and providing learners with more strategies to explain these concepts
to themselves or others by calculation. In addition, the mathematical language
can constrain interpretation by being less ambiguous than the text. Lastly,
the formulation using Bell states and Bell measurements could construct deeper
understanding by abstraction which, e.g. could also be used in other contexts
that benefit from a Bell state formulation like the investigation of Bell’s
inequality. On the other hand, learners could benefit from pre-existing
familiarity with Bell states. Thus, it can be hypothesised that the text
supported by mathematical representations helps learners understanding and
problem solving in this context and that the abstraction in particular can
help learners by enabling connections to other related contexts. Symbolic
descriptions of quantum teleportation using the Dirac ket notation can be
found in A.

#### Graphical

The theory of quantum teleportation can be visualised in abstract schematics
(Fig. 6) [115, 116]. Abstract representations like this allow for re-design
and generalisation of processes like quantum teleportation and could provide
benefits even for experts in that regard by supporting higher levels in
Bloom’s taxonomy. A graphical representation providing an example for the
creative abstraction needed to illustrate Bell-state measurements is provided
by Fig. 4/5 in [58]. The typical ket notation can pose problems to students
[4]. To aid in understanding, the explicit analytics can, e.g. be graphically
visualised using circle notation [114] (see Fig. 9) which complements the
mathematical notation by providing the information of complex numbers in a
graphical form. This is done with circles representing the basis states, inner
circles of which the radius represents the absolute value of the corresponding
coefficient and a line which represents the phase. Another possibility is
introducing dimensionality, assigning every qubit an axis in space and
attaching the amplitudes visually [117].

The SpinDrops program [118] offers an alternative representation to the ones
above, visualizing quantum states of up to three qubits in a different basis
(akin to generalises Wigner functions [119]) than the usual computational
basis, putting a larger focus on quantum correlations, i.e. entanglement. An
alternative way of visualizing entanglement properties in the computational
basis is shown in [120], utilising the circle notation combined with
dimensionality.

Figure 6: Schematic representations of Quantum Teleportation. These
illustrations support a high level explanation of quantum teleportation and
describe the flow of information, whether that information is quantum or
classical as well as the role of the qubits in the protocol. To the left, the
description leans more towards the mathematical perspective with states and
classical information written as variables. The illustration to the right
might be more suitable for learners that are more reluctant towards the
mathematical symbols. Both combine multiple representations as both are arrow
schematics supported by text or symbolic representations. The schematic to the
left is adapted from [116] while the schematic to the right is adapted from
[72].

###  6.2 Visual Representations of the Computation & Simulation of Quantum
Teleportation

#### Descriptive

We provide computer code of Quantum Teleportation using the Python library
Qiskit as a supplementary file. The code is annotated with description of what
the functions do (complementing the Python code/providing support on the
syntactic level [105]), which step of quantum teleportation is being executed
(providing support on the semantic level [110]) and some other important notes
like the impossibility to “read out” the quantum information of a qubit in the
real world (providing deeper understanding).

#### Symbolic

As the gate-based language is often used in contexts describing the
mathematical procedure, the same can be used for the purpose of supporting the
computer code. Various functions and variables used within the code can be
seen as the symbolic representation of the computation/simulation of quantum
teleportation. This is a concise representation, albeit not easily
understandable for someone not as familiar with the Python functions and the
Qiskit library. Therefore, comments within the computer code are usually used
to support the more concise formulation, improving readability. In addition,
educators should make sure that learners have the necessary representational
competence or know how to gain it using, e.g. the Qiskit documentation.

#### Graphical

A common visual representation of the code of quantum teleportation is the
corresponding quantum circuit showing a line for every qubit and “boxes” for
different gates. This picture can be annotated with the creation of the Bell
state, Bell state measurement and final unitary operations (a question of
design within the DeFT framework). It is a more concise representation of the
computer code and can therefore serve to constrain interpretation. In
addition, the Bloch sphere can be used to describe the state
$\left|\psi_{1}\right\rangle$ of the qubit to be teleported and compare it to
the state $\left|\psi_{3}\right\rangle$ of the qubit that the state is
teleported onto in order to visualise whether the algorithm has worked. It can
be used to more easily check whether the code works or not, giving access to
the otherwise hidden information to provide learners with another strategy of
bug fixing (in practice, this information is, ofcourse, actually hidden and
can not be accessed by any means in a single measurement), complementing the
Dirac ket notation output. On the Bloch sphere, it can be seen for example
whether the X and Z gates are assigned to the wrong classical registers which
could be more difficult when comparing the Dirac ket state output. An example
of a Jupyter notebook that incorporates the quantum circuit representation and
the Bloch sphere is shown in Fig. 10. This could also be designed in the form
of a computational essay, combining descriptive Educators need to make sure
that learners have the necessary representational competence to use these
tools.

###  6.3 Visual Representations of Experimental Implementation & Real World
Application of Quantum Teleportation

#### Descriptive

We provide a text representation of experiment and real world application of
quantum teleportation in C. It is described how Bell states between photons
are created and how quantum teleportation can be verified in a proof-of-
concept experiment by measuring correlations. This is a surface-level
description with the aim of introducing quantum teleportation broadly while
leaving open questions, for example how exactly spontaneous parametric down
conversion [121] or Hong-Ou-Mandel interference [122] function. Explanations
of these could be referenced, providing semantic support for the learners, but
possibly increasing cognitive load significantly.

#### Symbolic

The underlying theory can be represented with polarisation letters
$\left|H\right\rangle$ and $\left|V\right\rangle$ or arrows
$\left|\leftrightarrow\right\rangle$ and $\left|\updownarrow\right\rangle$ as
is done in, e.g. [72]. This means that the representation is designed such
that the Dirac ket notation is related to the representation of horizontal and
vertical polarisation, providing a connection between the two different
representations and possibly benefiting learners in understanding of how the
qubit is implemented optically.

#### Graphical

To describe real world quantum teleportation graphically, one can use a
diagram including Alice and Bob to illustrate the process more abstractly, or
show the more explicit optical experimental setup, both of which are commonly
done, e.g. [28, 53]. The more abstract representation can complement the
information provided in a corresponding text on the overall process while the
graphic of the optical experimental setup (see Fig. 11) is more suitable to
support a text describing the optical components and experimental setup and
procedure.

##  7 Teaching Approach

Much of modern pedagogy is underpinned by Vygotsky’s constructivist theory of
learning [123]. While in this article we will not dig into its development,
which arose from the field of psychology, we will instead consider its
implications for designing Quantum physics curricula. Put most simply, the
chief implication of constructivism for the educator is that learners
construct their own understanding, knowledge, and skills, through interaction
with their environment. Crucially, this includes the social surroundings [124]
the core tenet of social constructivism.

We argue that teaching must take into account this nature of learning, and
thus propose three key considerations for educators which we call “teaching
approaches”. These influence the environment in which learning is constructed.
These are the chosen level of inquiry, and the degree of cooperative learning,
and the degree of scaffolding included. These are heavily intertwined as will
become clear in this section.

Furthermore, all three pedagogical approaches are tunable factors which may be
used by educators to adapt their teaching for QIST courses, which can be
difficult to teach due to the diversity of student’s disciplinary backgrounds.
This includes different levels of prior competences in Mathematics and various
MERs, programming, and many other areas. We describe the three approaches
below.

###  7.1 Level of Inquiry

Inquiry based education is a pedagogical approach in which learning is driven
interactively [125]: by asking, and attempting to answer, questions. Inquiry
is deeply embedded in science education because it is the nature of scientific
study itself. There are many models of Inquiry-based learning (IBL) [126,
127], all sharing a cyclical and phasic nature. For example, the 5E’s cycle
[126] labels phases with Engagement, Exploration, Explanation, Elaboration,
and Evaluation. These can be mapped onto the phases of the scientific inquiry
cycle, as shown in Fig 7. For this reason, IBL may be not only beneficial, but
in fact necessary for students to gain the skills needed for scientific
thought. It should be noted that there exist other models of scientific
inquiry, such as modelling-based inquiry [128] which describes the process by
which a complex phenomena can be understood through constructing a scientific
model. Regardless of the specific inquiry model referred to, the core process
of attempting to uncover an answer to a question, in a cyclical manner, is the
same. This gives rise to a class of inquiry based learning approaches.

Figure 7: The scientific method (left) and the inquiry-based learning cycle
(right). Both are cyclical processes, with a common structure (exploration,
hypothesis-generation, testing/evaluation, and refinement.) As a result IBL is
a powerful means of teaching QIST concepts whilst also engaging scientific
thinking skills.

A core benefit of IBL is that, as engaging the very process of inquiry is the
goal of any interaction, it respects diversity of student backgrounds and
prior learning to a greater extent than more teacher-led teaching methods. We
thus advocate that in the interdisciplinary QIST environment, some degree of
IBL is essential. One may then ask what strategies can be taken to implement
it in a teaching scenario. Pragmatism dictates a distinction between different
“levels of inquiry”, the most frequently used model of which is Bell and
Banchi’s (Confirmation, Structured, Guided, Open) [129].

Research suggests that conceptual understanding and subject knowledge are best
fostered in more closed, structured inquiry based tasks [125], whilst
scientific thinking and inquiry skills are best fostered with more open,
unstructured tasks [127]. This may be explained by the greater cognitive load
associated with the more open levels of inquiry [127, 130], leaving less
cognitive ability to develop conceptual understanding or commit ideas to long-
term memory. For this reason, educators must choose carefully the level of
inquiry in order to match their learning goals. In the field of QIST, there
have been many recent examples of interactive tools which create possibilities
for teaching approaches which span the four levels of inquiry [131]. For
example, the visual and gamified simulation tool for Quantum Computing,
Quantum Odyssey [132], is designed with a coherent structure of stand-alone
scenarios addressing individual concepts, such as particular gate operations
and communication protocols. These can be used to create more structured and
confirmatory inquiry scenarios in which participants are both introduced to
concepts and walked through them towards attaining particular targeted
knowledge and understanding around the concepts. In addition, Odyssey has an
editor mode in which users can create quantum algorithms from scratch using
custom gates, therefore allowing for more open inquiry. The Quantum Composer
tool is also effective for open inquiry, as it provides a blank slate
interface in which staff and students may construct any one-dimensional
quantum system and investigate its properties. In fact, the tool has been used
to support research in real problems in QIST [133, 134], which demonstrates
how this form of inquiry can support scientific thinking as a learning
outcome. Both of these tools provide a means of visual representation of
concepts in QIST, but it should be noted that IBL models need not only use
visualisation, but other forms of representation as described in Sect. 5.

###  7.2 Cooperative Learning

Cooperative learning is an educational practice where learners work in small
groups in order to achieve learning goals under the instruction of an
educator. There is a large amount of empirical evidence that cooperation is
more effective than competition and individualistic efforts in enabling higher
achievement and productivity and that therefore, cooperative learning improves
learning outcomes [135, 136, 137]. This is only true, however, if cooperative
learning is implemented correctly. In [136], five basic elements of effective
cooperative learning are identified: (1) positive interdependence, i.e.
everyone is needed and contributes to the group’s goal, (2) face-to-face
promotive interaction, i.e. enabling constructive help and criticism within
the group, (3) individual accountability and personal responsibility (e.g. to
reduce diffusion of responsibility), (4) frequent use of interpersonal and
small group social skills that should be promoted and rewarded and (5)
frequent, regular group processing of current function, i.e. conjoint
reflection on the group work. Findings also suggest that this depends on the
level of inquiry, because more open ended questions are an effective tool to
promote group discussion [138]. Therefore, the decision of how to incorporate
cooperative learning into teaching is intertwined with the decision on the
level of inquiry discussed previously.

There are findings that suggest that cooperative learning has advantages
especially in an interdisciplinary context for tasks requiring input from
multiple perspectives [139]. Therefore, cooperative or team-working skills
should be fostered as they are especially needed not only in general, but also
more specifically in the interdisciplinary context of QIST [2, 140, 141].
Findings suggest that diverse teams perform better than homogeneous teams
under the right conditions [142, 143]. One such condition is psychological
safety, reducing dysfunctional social barriers within the group. However, the
exact impact of cognitive diversity on team performance, i.e. the differences
of the way people think which is supposedly influenced by academic background,
is still an open research question [144, 145]. Part of promoting functional
team work is improving student’s understanding that their individual previous
knowledge and skills and also that of others are relevant to the group and the
common goals that the group wants to achieve [146, 147] and therefore a
“diversity mindset” [148]. Diversity in academic background is one dimension
of diversity such that one can argue that cooperative learning might be
especially beneficial in the field of QIST with its diverse range of students’
academic backgrounds [4].

In practice, more open tasks that require multiple perspectives are often
tasks reaching into higher levels of Bloom’s taxonomy. These tasks might
benefit more from cooperative learning methods than tasks on the lower levels
of Bloom’s taxonomy, because they allow cognitive diversity to be utilised
more effectively. However, there is no clear evidence of this and it is very
dependent on the specific context [69]. Further, there exists evidence that
teaching techniques like peer-to-peer instruction are beneficial also when
trying to achieve lower levels of Bloom’s taxonomy [149, 69] and when applying
Bloom’s taxonomy in general [150]. Nevertheless, making sure teams are diverse
in academic background helps foster cooperative skills and improve group
performance and learning when solving tasks with high interdependence [151].
Examining this effect is a possible direction of future empirical research in
QIST education.

###  7.3 Scaffolding

Scaffolding is a well known teaching practice that can be defined as
interactive, temporary support given to learners for tasks that they otherwise
would not be able to complete [152, 153]. Here, the contingency, i.e.
interactivity or responsiveness, is what differentiates it from non-
scaffolding support [152]. Scaffolding techniques include promoting student
interest, controlling frustration, providing feedback, helping with
task/subtask prioritisation, offering models of expert processes and
questioning/encouragement of reflection [152, 153]. Scaffolding can be seen as
essential to IBL as a transfer of responsibility from educator to student is a
key characteristic [152, 154]. Examples of how scaffolding can be applied in
IBL in various STEM context include [155, 156, 157, 158, 159] and there exists
extensive theory on how to do so [160, 161, 162]. Scaffolding can be offered
one-to-one, by peers; it can be computer-based or AI generated [83, 163]. In
many different contexts, it was shown that students perform better when
scaffolding techniques were applied [164, 165, 166, 167], also when applying
cooperative learning techniques and in the context of quantum physics [168].
In addition, scaffolding techniques were shown to reduce anxiety and to have
beneficial effects on motivation and reading comprehension [169].

Especially in the QIST context, we argue that various scaffolding techniques
could be beneficial, because they consider individual academic backgrounds by
definition. In addition, the concepts in QIST are quite abstract while the
methods to deal with these concepts are often new to students, with possibly
discouraging effect. Lastly, the field of QIST is new and therefore, knowledge
acquisition often requires students to do their own research, reading
scientific research articles that are not designed with them as a target
audience in mind. Scaffolding could, e.g. be provided in the form of providing
on demand suitable visualisations, encouraging peer-to-peer instruction and
cooperativity (as covered in the previous section), providing help with
representations and how to perform calculations, concept maps (e.g. connecting
content with content from different targeted skill areas or different domains
of the CF) aiding in research tasks by providing strategies and foundational
material for understanding or close interactions with researchers (e.g.
interviews).

###  7.4 Teaching Approach to Quantum Teleportation

In order to show how to put the framework to use, we present an example lesson
on quantum teleportation using our framework. It is structured using the what
- why - how approach and the corresponding steps in the framework.

#### What

This example teaching unit could be part of the example curriculum presented
in Fig. 3. The goal of this curriculum is to introduce learners to the basics
of quantum technologies, targeting primarily applications in the real world.
Previous to the lesson, learners have been introduced to single-photon states
and optical equipment like beam splitters and wave plates. They know how to
use single photons to decrypt messages using the BB84 key distribution
protocol. Then, they were introduced to the creation of entangled photon pairs
via spontaneous parametric down conversion and a possible use of entanglement
for efficient transport of classical information using superdense coding.
During the latter, they learned the concepts of Bell states and Bell state
measurements. Learners understand both the experimental implementation and
description of these concepts using qubits and Dirac ket notation. During this
example teaching unit, learners are introduced to quantum teleportation. To
start the lesson, quantum teleportation can be introduced as a quantum
protocol where – in contrast to superdense coding that is designed to send
classical information – quantum information is transported over arbitrary
distances.

#### Why

After this teaching unit, learners should understand the theory behind quantum
teleportation and its experimental realisation in quantum optics. They are
able to apply quantum teleportation in quantum networks in practice and can
discuss applications, e.g. in quantum key distribution, comparing it to
classical networks and classical key distribution in these networks. Learners
can also evaluate the challenges that quantum networks pose and discuss
possible ways of coping with these challenges. In terms of Bloom’s taxonomy,
this means that the learners should reach the level of Understanding in the
Theory & Analytics area of expertise and reach the Evaluate level in the
Experiment & Real world area of expertise. This should enable them to
understand practical problems within quantum communication and make justified
decisions regarding various aspects of implementation of the quantum
teleportation algorithm.

#### How

There are infinitely many possible ways to teach quantum teleportation with
these prerequisites and goals in mind. However, the process of first deciding
on learning goals and then thinking about how to achieve them, also called a
backwards design process, is still much more constrained than an upwards
design process, where you start with textbook contents or activities and
figure out the learning goals from there [18]. We will discuss only one
possibility that incorporates the ideas laid out in this paper. For this, we
propose five steps in which the following inquiry is to be investigated: What
is quantum teleportation and what role could it play in quantum networks? What
are the advantages and challenges of such quantum networks in comparison to
classical networks? The teaching steps, the corresponding targeted skill
areas, levels of Bloom’s taxonomy and activities are summarised in Tab. 1 and
described in more detail in the following, including proposals for possible
scaffolding techniques.

Table 1:  The five steps of our example teaching approach to quantum teleportation summarised: In various skill areas, learners achieve the corresponding levels in Bloom’s taxonomy (and the levels before) in the activities while supported by (optional) scaffolding techniques. Step |  Skill Area |  Up to this level in Bloom’s taxonomy |  Activity
---|---|---|---
(i) |  Theory |  Understand |  Listening to short, conceptual presentation on quantum teleportation and discussion
(ii) |  Analytics |  Apply |  Calculate necessary gates, choose proper representations
(iii) |  Experiment |  Apply |  Plan an experiment and sketch an experimental setup
(iv) |  Real World Application |  Evaluate |  Discuss real world application of quantum teleportation in quantum networks in a presentation
(v) |  Computation & Simulation or Experiment |  Remember |  Carry out experiments or outlook on further use cases of quantum teleportation in quantum computing & programming the algorithm

  1. 1.

Learners are introduced to the quantum teleportation protocol with a
description and a schematic like the one in Fig. 6 via presentation. The
general idea of a transfer of quantum information by transfer of entanglement
is introduced. The presenter should keep principles and effects of cognitive
load theory summarised in Sect. 5 in mind. The learners are asked why the
protocol does not utilise faster than light communication and why it does not
contradict the no-cloning theorem in group discussion. To end the
presentation, possible applications in quantum computing (entanglement
swapping) and quantum communication (quantum repeaters) are touched on. The
educator-student interaction can be scaffolded by the educator, including
giving appropriate hints and direction and giving substantive feedback [170].

  2. 2.

Learners are instructed to find out the explicit quantum gates that are needed
for the quantum teleportation protocol (see A). For this, they know that a
Bell pair $\left|\text{EPR}\right\rangle$ has to be created, and
$\left|1\right\rangle$ is part of a Bell state measurement by Alice with one
of the two qubits in the Bell pair. The aim is to find the right unitary
operations that Bob has to apply to the third qubit. For this, e.g. circle
notation or dimensional circle notation could be used or just the Dirac ket
notation. The choice of representation can be left to the learners (which
representation is the most helpful for this purpose is an open didactical
research question). Examples of possible graphical representations are shown
in Fig. 9. This task is probably not suitable for cooperative approaches,
because it is very closed inquiry-wise. Here, further scaffolding can be
deployed by providing suitable representations and possibly giving advice on
how to utilise them.

  3. 3.

The third task is to plan an optical experiment to test whether quantum
teleportation actually works in reality. For this, a schematic like in Fig.
11, a list of necessary optical components and a descriptive text should be
created, as well as a methodology and a prediction for the experiment. The
learners are told to use the ideas of how to create an entangled photon pair
and how to perform a Bell measurement. Therefore, they need to have or develop
skills in the Apply level of Bloom’s taxonomy regarding the previous topics of
Bell state creation and measurements. They need to figure out how to make sure
that it is clear that an entangled photon pair is created and how to prepare
different initial states. The learners can try and predict Bob’s measurement
outcome probabilities depending on Alice’s measurement outcomes. A problem in
this experiment is that Bob has little time to adjust his experimental setup.
To solve this, one could use ways of prolonging the path of Bob’s photon and
finding ways to apply conditional optical gates as in, e.g. [79] or, as in
[72], only consider the Bell state $\psi^{-}$ (coincidences of Alice’s two
detectors) and discard all other cases. In that case, Bob’s measurement
outcome is certain and he can check whether that is the case, referring to
measurements at detector $p$. This task can be done in groups as its openness
and the possibility to incorporate many technical details into the experiment
can promote group discussion. Possible scaffolding techniques include
promoting positive interdependence within these group discussions, providing a
list of optical components that can be used with descriptions and providing
feedback and hints in case of problems.

  4. 4.

Preparing and holding a presentation of the finished work as is usually done
also in science and industry is a considerable final task. In this
presentation that is again prepared and done in a group, learners discuss how
to implement quantum teleportation into quantum networks, how these networks
could look like and what these networks could be used for. They are tasked
with discussing quantum key distribution over large distances where multiple
network nodes are necessary. In addition, learners discuss the challenges of
building such networks, especially decoherence, quantify these errors from the
literature and explain what could be possible measures to cope with these
challenges. They then evaluate the advantages and disadvantages in comparison
to classical networks that are secured with classical key distribution. Groups
should make a plan and split up, completing subtasks individually or in pairs
and staying in contact. For example, one person could learn about classical
networks and inform the others about how they function while another person
learns about the details of spontaneous parametric down conversion, such that
this information can be incorporated into the final presentation. This
procedure makes sure that the group members are positively interdependent.
Communication and social interaction should play a role in the working process
and regular group meetings and discussions about the current status should be
encouraged [136]. Learners are also tasked to find or create suitable
representations for conveying their thoughts and ideas. Scaffolding techniques
that can be used here are, again, promoting positive interdependence by
providing instruction on effective group work, propose possible structures and
inquiries to deal with in the presentation (e.g. how would quantum networks
function, what would they look like abstractly, or what challenges do they
face), recommend literature and provide reading aid and encouraging the use of
suitable representations.

  5. 5.

Outlook and possible next steps: If sufficient lab equipment is available,
preparing this protocol and testing it in a real lab could be a possible next
step. The amount of time this would take, however, can only be estimated by
the educators. It could be a difficult and time consuming experiment to make
work, depending on the available equipment and preparation done.
Alternatively, the learners could be tasked with programming quantum
teleportation in a programming language of their/the educators choice. An
example of such a program is shown in Fig. 10, using a Python notebook
provided in the supplementary files. Different levels of fill-in code could be
prepared for the students such that the different backgrounds in coding Python
and Qiskit are considered (a form of scaffolding [153]). The code could be in
the form of computational essays [171, 172] which can be combined with
scaffolding [173]. Then, learners would have reached at least the level of
Understanding in Bloom’s taxonomy from each of the three different
perspectives that QIST offers.

##  8 Outlook and Discussion

Given the substantial expansion in the number of courses and degree programs
attempting to educate a workforce in the specialist field of QIST, it is
crucial that careful attention is paid to how to do this most effectively. The
development of the QCTF approach described in this article is intended as a
guiding tool to be used for educators, in a pragmatic and digestible manner,
to reduce an overwhelming library of educational research down to a single
framework which may be used when designing educational material and classroom
approaches.

The framework is structured in the manner of “what” - “why” - “how”, and thus
breaks down curriculum construction into individual elements, which may be
addressed with the support of didactical theory and tools. The first - which
content to include - may be addressed using the Competence Framework, designed
as a common language for content in QIST, and a major EU policy initiative
which will become widespread in years to come. One must also select which
skills are intended to be targeted in their education: be they theoretical,
computational, or experimental. These skills are part of both the “what” and
the “why”, as they are in themselves a learning outcome as skills to be
developed through the education, but equally they are skills used akin to the
knowledge used in addressing the areas of the CF representing field-specific
content.

Educators must then select the learning-outcome specific thinking skills they
wish to address. Foundational to this approach is Bloom’s Taxonomy, which we
argue in the field of QIST requires re-interpretation due to the strongly
counter-intuitive nature of quantum concepts, but is still a useful guiding
principle, albeit with less hierarchy in skill development. Given the content
and targeted skills, educators designing material must next decide on the kind
of representation they use to present concepts or explain to students such
that they can represent concepts themselves. The DeFT framework is the tool
intended to support this, and we emphasise that the framework is a support
structure which may be used to several degrees of depth. In the most pragmatic
and simple application, one may choose tasks based on the representational
competences they wish learners to acquire, for example using Dirac’s bra-ket
notation when intending a mathematical approach, or using a simulation tool
for a visualisation approach.

The final aspect of how the framework is applied is in choosing the teaching
approach applied in the classroom. Here we consider the level of inquiry,
ranging from more open and exploratory approaches, addressing more scientific
thinking skills, to more guided ones appropriate for more directly targeting
acquisition of field-specific knowledge. One must also consider how students
interact with others in their environment, and cooperation in learning can be
a key factor in determining the exact activities used for teaching, beyond
traditional “chalk and talk” approaches which lack the benefit of a social
underpinning.

We emphasise that the framework is a guiding structure, intended to distil the
results of educational research into a pragmatic approach for educators.
Whilst most effectively applied in full, educators may also pick aspects from
the framework to address individually - for example refining primarily their
level of inquiry, or choosing topics using the CF. The framework is also open
to refinement and iteration based on community feedback, as more and more
educational material is developed in this field, and in particular with an eye
to a didactically-sound approach. One particular refinement which may be
beneficial in the future is the inclusion of proficiency levels in the “what”
stage, while choosing content. Suggested proficiency levels were included in
the Competence Framework version 2.0 [19], providing a rough equivalency
between content difficulty and qualification level. Since then, the framework
was refined on the basis of interviews with industry representatives, adding 9
qualification profiles that could further guide development of curricula [16].
The refinement of the proficiency levels, as the framework itself, we
recommend for future work in the course of the flagship European Quantum
Technology education projects, DigiQ [174] and QTIndu [175].

## Acknowledgements

We thank Franziska Greinert from the TU Braunschweig for valuable discussion.

J.B. acknowledges support by the project QuanTUK at the RPTU in
Kaiserslautern, supported by the Federal Ministry of Education and Research
(FKZ13N15995).

##  Appendix A Visual Representations of the Theory & Analytics of Quantum
Teleportation

Here, we categorise typical representations in QIST in three categories, but
point out that this is neither a comprehensive categorisation of
representations [105] nor a comprehensive list of possible representations of
quantum teleportation. These representations are suitable for introductory
QIST courses targeting different skills in the three skill areas (Sect. 3).

###  A.1 Descriptive

Quantum teleportation can be seen as a transfer of quantum information made
possible by a transfer of entanglement [28]. Qubit #1 is in an arbitrary state
that is to be teleported to qubit #3. For this, qubit #3 is entangled with
qubit #2 in a Bell state. Upon performing a Bell measurement on qubit #1 and
qubit #2, one will get one of four possible results with equal probability. By
doing so, quantum information of the qubit #1 is transferred to qubit #3.
However, by definition of entanglement, the exact state of qubit #3 after
measurement depends on the measurement result. This way, quantum information
is transported over arbitrary distances, but can only be retrieved after the
classical information has been transmitted. This information is not copied,
because the quantum information contained in qubit #1 collapsed due to the
measurement.

###  A.2 Symbolic

The quantum teleportation algorithm can be seen as a gate-based computing
algorithm and is explained similarly to here in many textbooks. The system
starts in the state $\left|\psi_{i}\right\rangle$ that consists of a Bell
state $\left|\psi_{23}\right\rangle$ and the state
$\left|\psi_{1}\right\rangle$ of qubit #1 to be teleported:

$\displaystyle\left|\psi_{i}\right\rangle$
$\displaystyle=\left|\psi_{1}\right\rangle\otimes\left|\psi_{23}\right\rangle$
(1)
$\displaystyle=(\alpha\left|0\right\rangle+\beta\left|1\right\rangle)\otimes(\frac{1}{\sqrt{2}}\left|00\right\rangle+\frac{1}{\sqrt{2}}\left|11\right\rangle)$
(2)
$\displaystyle=\frac{\alpha}{\sqrt{2}}\left|000\right\rangle+\frac{\beta}{\sqrt{2}}\left|001\right\rangle+\frac{\alpha}{\sqrt{2}}\left|011\right\rangle+\frac{\beta}{\sqrt{2}}\left|111\right\rangle.$
(3)

Alice entangles qubit #1 and #2 using a CNOT-gate and applies a Hadamard gate
on qubit #1 to prepare a Bell state measurement of qubit #1 and #2 to obtain
the final state $\left|\psi_{f}\right\rangle$ before measuring:

$\displaystyle\left|\psi_{f}\right\rangle$
$\displaystyle=H_{1}CNOT_{12}\left|\psi_{i}\right\rangle$ (4)
$\displaystyle=H_{1}(\frac{\alpha}{\sqrt{2}}\left|000\right\rangle+\frac{\alpha}{\sqrt{2}}\left|011\right\rangle+\frac{\beta}{\sqrt{2}}\left|110\right\rangle+\frac{\beta}{\sqrt{2}}\left|101\right\rangle)$
(5)
$\displaystyle=\frac{1}{2}\left(\alpha(\left|000\right\rangle+\left|100\right\rangle)+\alpha(\left|011\right\rangle+\left|111\right\rangle)+\beta(\left|010\right\rangle-\left|110\right\rangle)+\beta(\left|001\right\rangle-\left|101\right\rangle)\right).$
(6)

To show what happens upon measuring qubit #1 and #2, the state can be
rewritten as

$\displaystyle\left|\psi_{f}\right\rangle=$
$\displaystyle\left|00\right\rangle\otimes(\alpha\left|0\right\rangle+\beta\left|1\right\rangle)$
(7) $\displaystyle+$
$\displaystyle\left|01\right\rangle\otimes(\beta\left|0\right\rangle+\alpha\left|1\right\rangle)$
(8) $\displaystyle+$
$\displaystyle\left|10\right\rangle\otimes(\alpha\left|0\right\rangle-\beta\left|1\right\rangle)$
(9) $\displaystyle+$
$\displaystyle\left|11\right\rangle\otimes(\beta\left|0\right\rangle-\alpha\left|1\right\rangle).$
(10)

Upon measuring, Bobs qubit #3 will be in one of these four possible states and
Bob needs to apply gates depending on the measurement result in order to
obtain the initial state of qubit #1:

$\displaystyle 00:\;$
$\displaystyle\alpha\left|0\right\rangle+\beta\left|1\right\rangle$ (11)
$\displaystyle 01:\;$
$\displaystyle\beta\left|0\right\rangle+\alpha\left|1\right\rangle\xrightarrow{X}\alpha\left|0\right\rangle+\beta\left|1\right\rangle$
(12) $\displaystyle 10:\;$
$\displaystyle\alpha\left|0\right\rangle-\beta\left|1\right\rangle\xrightarrow{Z}\alpha\left|0\right\rangle+\beta\left|1\right\rangle$
(13) $\displaystyle 11:\;$
$\displaystyle\beta\left|0\right\rangle-\alpha\left|1\right\rangle\xrightarrow{XZ}\alpha\left|0\right\rangle+\beta\left|1\right\rangle.$
(14)

One can alternatively represent the initial state of quantum teleportation
using the Bell basis:

$\displaystyle\left|\psi\right\rangle_{1}\left|\phi^{+}\right\rangle_{23}=$
$\displaystyle\frac{1}{\sqrt{2}}(\alpha\left|000\right\rangle+\alpha\left|011\right\rangle+\beta\left|100\right\rangle+\beta\left|111\right\rangle)$
(15) $\displaystyle=$
$\displaystyle[\left|\phi^{+}\right\rangle_{12}(\alpha\left|0\right\rangle+\beta\left|1\right\rangle)_{3}$
(16) $\displaystyle=$
$\displaystyle\left|\phi^{-}\right\rangle_{12}(\alpha\left|0\right\rangle-\beta\left|1\right\rangle)_{3}$
(17) $\displaystyle=$
$\displaystyle\left|\psi^{+}\right\rangle_{12}(\beta\left|0\right\rangle+\alpha\left|1\right\rangle)_{3}$
(18) $\displaystyle=$
$\displaystyle\left|\psi^{-}\right\rangle_{12}(\beta\left|0\right\rangle-\beta\left|1\right\rangle)_{3}]$
(19)

Then, a Bell measurement is performed by Alice to achieve the desired result.
This representation supports the more abstract explanation of quantum
teleportation using the Bell basis [115, 58] (see also Fig. 8). Another
alternative is the use of the vector and matrix representation [28], but,
considering these three-qubit vectors would have eight entries, this seems
like an impractical solution.

###  A.3 Graphical

Here we provide graphical representations of the application of Alice’s gates
in the quantum teleportation algorithm. In the more abstract Fig. 8, Alice’s
shift to the Bell basis (”looking” at the system differently) transforms the
system such that Bob can receive the appropriate information, while the
representations in Fig. 9 are more explicit and only represent the step in the
computational basis.

Figure 8: Creative abstraction of the mathematical relations relevant for
quantum teleportation: There are two equivalent manners to represent the
three-photon state
$\left|\psi_{1}\right\rangle\left|\phi^{+}_{23}\right\rangle$ (see B). In this
visualisation of this abstract relation, Bob throws a pyramodial “dice” with
four possible outcomes, corresponding to four possible Bell-states at Alice
site, at the corresponding four possible teleported photon states at his side
[58]. Once Alice informs Bob about her result, he knows which local unitary
transformation he has to perform at his photon to teleport Alice original
state $\left|\psi_{1}\right\rangle$ to his side. This approach is more
abstract than the one in Fig. 8, focussing on the basis transformation from
computational basis to the Bell basis that the explicit circle and dimensional
circle notations don’t represent.

Figure 9: Graphical representations of an explicit mathematical example of the
application of Alice’s gates within the quantum teleportation algorithms (see
B), adapted from [114, 117, 120]. This step is laid out more explicitly in all
of these works. The illustrations could help learners gain understanding of
the theory behind the protocol, provided they have the necessary
representational competence. This can be done by visually laying focus on the
information transfer between qubit #1 and that of qubit #3 or, in the case of
the representation on the bottom, the information transfer from one axis to
the other represented by the green and yellow ellipses [120].

##  Appendix B Visual Representations of the Computation & Simulation of
Quantum Teleportation

In this section, we provide examples of descriptive, symbolic and graphical
representations of the quantum teleportation algorithm or parts of the
algorithm that can be used to target the Computation & Simulation area of
expertise.

###  B.1 Descriptive

Considering the Computation & Simulation area of expertise, one can describe
the algorithm in terms of the quantum gates that are used, which is also done,
e.g. in [28]. The first and second qubit are initialised in the state
$\left|0\right\rangle$, while the third qubit is in an arbitrary state to be
teleported. Then, the first and second qubit are entangled via a Hadamard gate
on the first qubit and a CNOT gate with the first as control and the second as
target. The same is done to entangle the second with the third qubit. Then,
the first and second qubit are measured to a classical register. In order for
the first qubit to be in the original state of the third qubit, quantum gates
have to be applied depending on the measurement result, using classical if-
operations. If the measurement result of the second qubit is 1, a $Z$-gate has
to be applied. If the measurement result of the first qubit is 1, an $X$-gate
has to be applied. The order matters only by a global phase.

###  B.2 Symbolic

As symbolic representation, the same as in A can be chosen.

###  B.3 Graphical

Computer code is also a form of using Multiple External Representations.
Python functions can be called symbolic, while comments and pictures are
supportive representations useful for understanding the code and bug fixing.
Some excerpts from a Python notebook for quantum teleportation are depicted in
Fig. 10.

Figure 10: Excerpts from a Python notebook for quantum teleportation in
Qiskit. It utilises the circuit representation and the Bloch sphere for
visualisation of the computer code and the underlying processes. The first
Bloch sphere can be compared to the Bloch sphere of qubit 2 below the code to
show that the computer code has functioned properly. Note that this depiction
of a quantum state is not possible in reality. Many measurements have to show
that a quantum teleportation has worked successfully. Comments are used to
make the code more readable. The whole notebook can be found in the
supplementary files. Comments in Python code as shown in Fig. 10 are also a
form of a descriptive representation of the underlying processes that supports
the rather symbolic representation of Python functions.

##  Appendix C Visual Representations of Experimental Implementation & Real
World Application of Quantum Teleportation

In this section, we provide examples of descriptive, symbolic and graphical
representations of the quantum teleportation algorithm or parts of the
algorithm that can be used to target the Experiment & Real World area of
expertise.

###  C.1 Descriptive

One possible experimental approach to quantum teleportation is described in
[72] in the context of quantum optics, i.e. photons. Initiated by a light
pulse in the ultraviolet (UV) range, a pair of entangled photons is created
via a process called spontaneous parametric down conversion (SPDC) [121] in a
non-linear crystal in such a way that the polarisations are anti-correlated.
When the pair now passes through a polarising beam splitter (PBS), they are
spatially separated. One can be sent to Bob (photon 3), the other to Alice
(photon 2). The initial UV pulse is sent back through the crystal to create a
second pair of entangled photons, one of which will be the photon 1 to be
teleported for the sake of the experiment, the other of which will serve as an
idler photon, indicating that the others are on the way. Alice superposes
photon 1 and photon 2 on another beam splitter. If they constructively
interfere at that beam splitter, the result will be that each of two detectors
will trigger, showing that they were in an anti-correlated state (due to a
phenomenon called Hong-Ou-Mandel interference [122]). As Photon 1 and 2 were
anti-correlated as well as photon 2 and 3, we know (apart from a phase) that
photon 3 is now in the original state of photon 1. This phase is detected by
Bob using another PBS.

###  C.2 Symbolic

The same symbolic representation as in A can be chosen, but
$\left|0\right\rangle$ and $\left|1\right\rangle$ can be represented as
$\left|\leftrightarrow\right\rangle$, $\left|\updownarrow\right\rangle$ to
represent the experimental implementation of qubits as done in, e.g. [72].

###  C.3 Graphical

Fig. 11 shows the explicit experimental setup of quantum teleportation and
refers to some process and components that can be used for optical quantum
teleportation, namely a nonlinear crystal for SPDC, mirrors, polarisors,
(polarising) beam splitters and detectors.

Figure 11: Illustration of experimental quantum teleportation as introduced in
[72]. An ultraviolet pulse is shot at a nonlinear crystal (blue) and back
through the crystal to create four photons labelled 1, 2 and 3 and an idler
photon that is used to indicate that the photons were created. Photon 2 and 3
are entangled. Photon 1’s state is manipulated using a polarisor and Alice
interferes photon 1 and 2 at a beam splitter. Upon detection of photon 1 and 2
at different detectors, Photon 3 has, apart from a phase, the same
polarisation as photon 1 initially had which Bob can confirm using detectors
and a polarising beam splitter.

## Acronyms

CF

    Competence Framework
CLT

    Cognitive Load Theory
DeFT

    Designs, Functions, Tasks
ECL

    Extraneous Cognitive Load
IBL

    Inquiry-Based Learning
ICL

    Intrinsic Cognitive Load
MER

    Multiple External Representations
QCTF

    Quantum Curriculum Transformation Framework
QIST

    Quantum Information Science & Technology
QT

    Quantum Technologies
STEM

    Science, Technology, Engineering & Mathematics

## Glossary

aptitude treatment interaction

    A dependency (interaction) between the learner characteristics (aptitude) and the instruction (treatment) [101]
backwards design

    A backwards design process is a process where one starts with the learning goals and then deduces from there how these learning goals can be achieved [18]
element-interactivity effect

    For tasks with high intrinsic cognitive load (high element-interactivity), a high extraneous cognitive load can be especially harmful for learning and task solving [100]
expertise-reversal effect

    A form of an aptitude treatment interaction, where experts benefit less from certain treatments that are beneficial for beginners, e.g. due to the redundancy principle [100]
extraneous cognitive load

    The cognitive load imposed on the learner due to the design of the task rather than the task itself (which would be intrinsic cognitive load), usually undesirable [176]
inquiry-based learning

    A class of pedagogical approaches in which learning is driven by asking and attempting to answer, questions, in a cyclical manner of inquiry
interdependence

    in a collaborative work environment, participants are influenced by each other. This interdependence should be positive, i.e. success of one person benefits the others. Positive interdependence can be further broken down into different types of interdependence: goal, reward, resource, role, identity, environmental, fantasy, task and outside enemy [177]
intrinsic cognitive load

    Cognitive Load imposed on the learner due to the nature of the task itself [176]
modality principle

    Presenting both visual and audio information to learners can reduce cognitive load as opposed to using just one of the two types [103]
redundancy principle

    Redundant material can hinder learning due to increased extraneous cognitive load [99]
scaffolding

    Interactive, temporary support given to learners for tasks that they otherwise would not be able to complete [152, 153]
split attention effect

    s should be integrated to one single source of information rather than split either spatially or temporally to reduce extraneous cognitive load [102]

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