3.1.

Interactive Tools

A number of different computer-based quantum tools have been developed for educational purposes. Typically, these tools are geared toward the visualization of quantum phenomena. Early efforts by the Consortium for Upper Level Physics Software released quantum visualization software packages in the mid-1990s;^67,68 some of the authors of these software packages also released a book on quantum mechanics simulation.⁶⁷ Later efforts included a series of books by Thaller^69,70 and Zollman et al.⁷¹

In the early 2000s, Paul Falstad released a series of Java applets simulating physics,⁷² including 12 on quantum mechanics and many others on waves physics. In particular, the one-dimensional quantum mechanics applet allows one to draw any potential and any wavefunction, as well as explore the position space, momentum space, energies, and eigenstates simultaneously (Fig. 2). The ripple tank applet allows for the exploration of interference—crucial for both classical wave physics and quantum phenomena. Unlike other simulations, a user can draw the interference setup in a sandbox mode. Paul Falstad’s applets gained enough popularity to be ported as mobile apps and rewritten in JavaScript to be usable with modern browsers.

Fig. 2

1-D quantum mechanics applet (a single-particle quantum mechanics states in one dimension) by Paul Falstad.⁷³ An early simulation gives a user the ability to simultaneously see (and edit): 1-D potential, eigenstates with eigenenergies, wavefunction as a function of position and time, wavefunction as a function of momentum, and relative amplitudes of each eigenstate.

Other web-based applications include the Physlet simulations⁷⁴ and a set of simulations developed by Joffre.⁷⁵ The ubiquitous PhET simulations⁷⁶ have developed a set of quantum-specific simulations; PhET researchers have published a wide array of research on the design and use of their simulation.^77–80 Other such research-validated quantum simulation tools include the QuILTs⁸¹ developed at the University of Pittsburgh,⁸² the QuViS tutorials⁸³ developed at St. Andrew’s University,^84,85 and the Quantum Composer simulation tool⁸⁶ developed at Aarhus University,^87,88 which is described in more detail later in this paper.

With the exception of the node-based programming in Quantum Composer, the tools developed here are typically set modules exploring a specific quantum phenomenon (sometimes including supplementary material that instructors can use to develop exercises), and interactive tools (e.g., the PhET, QuViS, and QuILT simulations) allow students to easily modify parameters and explore the simulation in more detail. However, none of these tools can be classified as games as they lack the required elements in the interface and design that engage the students outside their traditional educational goals. In Sec. 4, we explore more recent and ongoing efforts in accomplishing this task, but first, we introduce the concept of science-based games and explorable explanations as an example.

3.2.

Science-Based Games

The traditional way of teaching includes textbooks, lectures, and projects. This teaching method is performed top-down, with clear instructions and expectations. Students are tested via grading of projects, homework assignments, and exams. Usually fun, or the ability to explore on one’s own, is not the key focus. Often students’ motivation is extrinsic—to pass a course with a good grade. These activities can be gamified, provided with supplementary goals, scores, and challenges to make them more engaging.

In contrast, science-based games²⁹ approach teaching from the opposite direction: to primarily focus on creating an experience sparking intrinsic motivation,⁸⁹ that is, students play for fun, but learn in the process, as their gaming experience requires learning concepts to proceed or provides an explorative pathway through the game that promotes learning⁹⁰ (even if it is not strictly necessary).

There are several existing games showing different aspects of physics—special relativity theory,⁹¹ electromagnetism,⁹² classical physics, and orbital mechanics.⁹³ There are games that teach computing—creating circuits from the simplest blocks—NAND gates⁹⁴ and low-level programming in assembly. Although a lot of games are standalone one-off projects, some game developers specialize in science-based games, e.g., Zachtronics (algorithmics) and Test Tube Games (physics). Some science-based games go way beyond the niche – SpaceChem and Kerbal Space Program sold over 1 million and over 2 million copies on Steam,⁹⁵ respectively.

3.3.

Explorable Explanations

Explorable explanations⁹⁶ (or “explorables”) come at educational games from the opposite direction: instead of “games, but with science communication added,” they are “science communication, but with interactivity added,” although we should note that interactivity is a necessary but not sufficient condition for something to be considered a game. Most explorables are educational articles, with embedded simulations instead of static media such as images and videos. The term “explorable explanation” was coined in 2011 by interaction designer Bret Victor,⁹⁷ with an ambitious goal of creating a two-way communication between the author and the reader. With an open-ended creative environment, readers can challenge the author’s assumptions by going beyond what the author ever imagined. This ambition is explained more in the 2019 essay⁹⁸ by former Khan Academy designer Andy Matuschak and quantum physicist Michael Nielsen. These approaches are tightly related to exploring new visual languages of communication (see Fig. 3).

Fig. 3

A self-descriptive formula of the discrete Fourier transform: a web-based MathJax implementation⁹⁹ of an equation color-coding scheme designed by Stuart Riffle from 2011.¹⁰⁰ Earlier approaches of color-coding go back to the Byrne’s Euclid from 1847¹⁰¹ subtitled “In which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners.”

Deep learning shares many similar concepts with QT—both hyped, fast-growing technologies, are heavily based on linear algebra with tensor products and have a nontrivial entry barrier. The abundance of online tools that help users to learn and directly use deep learning already gets traction in gaming¹⁰² and can provide guidance on how to present quantum technologies in an accessible, ready-to-use way. There are already numerous explorable explanations in machine learning and AI,¹⁰³ including general introductions to probability and statistics,¹⁰⁴ decision trees and validation in machine learning,¹⁰⁵ training of artificial neural network classifiers,¹⁰⁶ and interpretability of convolutional neural networks.^107,108 In 2016 to 2021, a dedicated peer-reviewed journal Distill aimed at providing interactive explanations of novel research.¹⁰⁹

Although the interactive environment of Jupyter Notebooks is a standard way of providing introductions to quantum software frameworks,^110,111 there is only a handful of other quantum explorable explanations, e.g., a spaced-repetition-based introduction to quantum computing,¹¹² an introduction to quantum Fourier transform,¹¹³ and an exploration of single-qubit gates.¹¹⁴

3.4.

Evaluation of Quantum Games

A common question when considering any tool is whether it truly works as intended, and quantum games are no exceptions. Currently, neither common design methodology¹¹⁵ nor universal evaluation tools for quantum games are available. This is largely due to fragmented game-development processes¹¹⁶ (i.e., by professionals, by university researchers, or in hackathons) and the diverse game goals (i.e., research, education, outreach, and/or fun). Indeed, some of the games described in Sec. 4 have been evaluated individually. For example, the developers of Quantum Moves 2, with its focus on citizen science, published their results in a peer-reviewed journal.¹¹⁷ Similarly, the games Particle in a Box⁶¹ and the Quantum Odyssey¹¹⁸ have articles published in peer-reviewed journals containing evaluation sections. However, a proper and comprehensive evaluation and assessment scheme that can be applied to a variety of quantum games is missing in the literature. Validation of these quantum games, especially when ensuring that educational games serve to teach students relevant concepts, is important and should be further developed as activities in the field are progressing. Although the development and discussion of a universal quantum game evaluation tool is beyond the scope of this paper, we offer some suggestions here.

First, we need to categorize quantum games in terms of their basic information such as on which platform they run and their licenses, their types, and the concepts that they are drawing upon. Efforts to do this are underway,¹¹⁶ but the task difficulty grows with the impressive growth rate of the quantum games landscape and with the rate at which games become obsolete because of, e.g., operating-system compatibility and security or operating system development leading to the acquisition of game-breaking bugs. We believe that it could be useful to develop a quantum games version of Physport,¹¹⁹ which is a repository of research-based recommendations, assessments, tools, and practices for physics education, where games can be rated. For example, the Force Concept Inventory¹²⁰ received a gold star rating from Physport for its extensive research-based evaluation.

In parallel, researchers should work to develop best practices for the evaluation of quantum games (similar to those developed for quantum software¹²¹), focusing on the common goals of research, education, and fun; these best practices can be collected in the same repository as the games themselves. Also, within the repository, we can denote whether the game is being actively developed, is in a usable state, or has been deprecated in some way. This can be an invaluable tool for tracking how quantum games, their goals, and their use cases change over time.

Evaluating the effectiveness and efficiency of a video game for education requires a number of steps,¹²² which as a pilot we are completing. To start with, we need to work out a storytelling format, in which the use of the purpose of the quantum game or interactive tool is made explicit. This has been the subject of theoretical work introducing the concept of culture-scientific storytelling with special attention to quantum science and technologies,¹²³ which is being tested in outreach events such as the workshops devoted to high-school students around the Quantum Jungle interactive installation.¹²² Then, suitable evaluation tools must be designed in correspondence with on-purpose didactic experiments. For example, in one such experiment performed in a high school in Treviso (Italy), the Quantum TiqTaqToe video game was used as a central tool in a highly compact didactic intervention lasting just two afternoons (including a TiqTaqToe tournament) to educate students on the concepts of quantum state and measurement, quantum superposition, and entanglement.¹²⁴ Suitable evaluation tools in the form of questionnaires were designed for this purpose and analyzed; they showed a high degree of effectiveness of the quantum game in supporting the students’ ability to develop an operational understanding of the aforementioned concepts.¹²⁴ Although much systematic research needs to be performed along similar directions, these first steps appear quite promising and encouraging.

A preliminary effort to categorize the games presented in our paper is provided in Table 1.

Table 1

A summary of games presented in Sec. 4.

Finally, these different approaches introduced above can be supported by a criteria-based assessment, utilized by the quantum games hackathon juries organized by the Quantum AI Foundation to assess the submissions. These criteria are listed as correctness, playability, originality, and completeness for that particular hackathon (further information on the hackathon is provided in the Sec. 4.8.1):

To sum up, we are in the early stages of quantum games being accepted as a concept of their own, instead of just games designed to teach quantum physics. Therefore, development of evaluation tools and practices should also be considered to be an important research direction.

4.1.

Hello Quantum/Hello Qiskit

The IBM Quantum Composer was released in 2016 and provides a simple and graphical way to write quantum circuits and run them on prototype quantum computers. This service was (and is) available for free to the general public, allowing them unprecedented access to quantum computer hardware.¹²⁹ However, without the knowledge of what quantum circuits are or how quantum gates work, there was still a significant barrier to entry.

The Hello Quantum project, a collaboration between IBM Research and the University of Basel, was intended to alleviate this issue. The idea was to visualize quantum states and the effects of gates, so people could gain an intuitive understanding of simple quantum circuits. This interactive and visual method for learning was presented as a puzzle game, with the player being challenged to find the circuit that would take an input state to a desired output state.

The resulting products were released in four stages:

Fig. 4

(a) Screenshot of the Hello Quantum app and (b) a Jupyter widget-based puzzle from the Hello Qiskit section of the Qiskit textbook.

In each case, the learning material was presented by providing interactive puzzles, intended to build intuition and provide context, as well as text providing an explanation of how aspects of the game relate to quantum computing.

For “Hello Quantum,” the puzzles and text were delivered entirely separately. The idea was that the player could play through the puzzles as a short and informal game. Then, once they had mastered the game mechanic, interested players could go on to read about how their new in-game skills translate to quantum computation. This text was not limited to the app itself, but instead represented the inaugural posts of the Qiskit blog. These posts were entitled “Visualizing bits and qubits,”¹³² which explained the visualization used in the app, and “Getting started with the IBM Q Experience,”¹³³ which explained how to transfer the learned skills into what is now called the IBM Quantum Composer.

It was expected that the readers would be led to the former blog post by the app and to the latter by a link in the former. However, view statistics show that only around 1% of views for the former can be traced to following the in-app link. Instead, the majority (around 58%) came from search engines. For the latter post, 81% of views come from sources external to the blog, such as 40% through email, instant message (IM), and direct and around 27% from search engines.

Given these results, we can conclude that the blog posts were a more successful learning resource than the app. This is because every app user who went on to fully understand the game will have followed the link to the app, but these app users are only a small percentage of the total readership.

For the “Hello Qiskit” section of the Qiskit textbook, the puzzles were implemented within an interactive Jupyter notebook. This was located in a chapter specifically for interactive demos. An analysis of views for the textbook was conducted for the period March to November, 2020.⁶⁰ It was found that the view share of this section is very comparable to those of the first two chapters. In fact, the average number of views for sections in chapters 1 and 2 during the period is very similar to the number of views for the “Hello Qiskit” section. During this time, 71% of views for the chapter come from within the textbook. These statistics suggest that it serves an integrated role as part of the introductory materials within the textbook. This motivated the more prominent usage of the visualization in the “Visualizing Entanglement” section of the new interactive Qiskit introduction course.

These results demonstrate an additional benefit to the design of quantum games and interactive tools. By finding visual and interactive representations of the mathematics behind quantum computing, we make it easier to describe. This allows us to create more engaging and understandable explanations in text. It is therefore worth considering making fully blog-based companions to any educational quantum game or interactive tool.

It is worth noting briefly that a similar insight was gained from the game Battleships with partial NOT gates. This was one of the first games to use a quantum computer and was designed primarily so that the source code could be explained in a blog post.¹³⁴ There is therefore little evidence of it being played, but the blog post remains the most viewed on the Qiskit blog by some margin.

4.2.

Particle in a Box/Psi and Delta

“Particle in a Box” and “Psi and Delta” are digital games designed to support high school and undergraduate students in learning introductory quantum physics. The games were designed and developed as part of an interdisciplinary team of electrical engineers, media scholars, and HCI researchers comprised of undergraduate and graduate students as well as faculty at Georgia Tech. “Particle in a Box” aims to support learning quantum physics by allowing students to virtually explore, compare, and experiment with worlds that follow the laws of classical and quantum physics.^61–63,135 Building upon Particle in a Box, Psi and Delta aims to teach quantum physics through collaborative inquiry by requiring two players to cooperate, develop, and employ concepts of quantum mechanics together to resolve challenges involving quantum physics.¹³⁶ Both games can be downloaded and played at Ref. 137.

Both games integrate quantum physics into the game mechanics using a concept map of the infinite square well. For example, to reify the concept of light color affecting energy levels, Particle in a Box requires students to find and collect light bulbs of the appropriate color and bring them to a lamp to shine light onto an electron. Similarly, to promote engagement with the relationship between “area under the curve” (the curve referring to the probability distribution of the electron) and the probability of the electron, Psi and Delta uses platforms with different widths and measurement counters that count how many times an electron is measured under each platform to express probability. The concept map of the square well problem used by both games is provided in Fig. 5.

Fig. 5

The Concept Map of the square well problem.

4.2.1.

Particle in a Box

Particle in a Box is a 2-D single-player platformer game built in Unity in which the player plays the game through a virtual avatar who travels through a 2-D rendition of the classical and quantum worlds. The classical world, set in a lab-like environment, (Fig. 6) involves challenges based on Newtonian physics such as jumping over a ball and increasing its total energy. After playing the classical world, players shrink down to the quantum world (Fig. 7) which is based on the infinite square well problem. It involves an electron that is trapped in a 1-D quantum wire and exhibits quantum behavior such as superposition and energy quantization. The challenges here are based on understanding introductory quantum physics. Experiencing both worlds affords students the opportunity to compare their similarities and differences—an approach that has shown to be effective in learning quantum physics.

Fig. 6

The classical world in Particle in a Box. The player must reach the energy bolts, carry them, and place them in the path of the moving ball.

Fig. 7

The quantum world in Particle in a Box. The player must carry the colored bulbs to the lamp (left) that will shine light and increase the electron’s energy level (top). The blue dot is the electron, and the bright line it is on is the wire. The curved white line illustrates the wavefunction of the electron.

Both worlds have similar objectives: to raise a particle’s energy. In the classical world (Fig. 6), the objective is to raise the energy of a rolling ball so that it pushes a lever located higher up, which opens the door to the next level. In the quantum world (Fig. 7), the objective is to increase an electron’s energy up 3 levels by gathering light bulbs of the correct color and shining light on the electron’s quantum wire. Having similar objectives enables direct comparison between concepts such as position and energy, which manifest differently in classical and quantum physics.

Further, in both worlds, if the player hits the particle, they faint and must restart. In the classical world, this means coming into contact with the ball. In the quantum world, this means coming into contact with the measurement of an electron particle. The electron by default is in a state of superposition. However, it is also measured regularly and “appears” each time as a particle. The position where this electron particle appears is random but follows a probability distribution (curved white line in Fig. 6). Through repeated measurements, the game simulates an “experience” of probability as the player learns to identify the locations where the electron is likely to be measured and therefore where they are more likely to be hit. Because players faint if they are hit by the particle in either world, they must learn to carefully analyze the behavior of those particles in both worlds, thereby promoting further comparison.

Particle in a Box was evaluated with undergraduate electrical engineering students at Georgia Tech and showed an improvement in students’ conceptual understanding of probability.⁶¹ Students further reported an increase in comfort level with the key concepts taught in the game. The game was awarded the Student’s Choice Award at the Serious Games Showcase and Challenge in 2015.

4.2.2.

Psi and Delta

Psi and Delta builds on our findings of Particle in a Box by students adopting the role of two robots, with the aim of defeating opposing robots in a world governed by the laws of quantum mechanics. Students accomplish this task using concepts of quantum mechanics to lure and “shock” the opposing bot. If a student’s bot touches the opposing bot or gets “shocked,” it loses part of its health. If any player’s bot loses all of their health, the level restarts.

The game is divided into two parts. In part one, the students develop models of the concepts of superposition and probability (Fig. 8). As discussed earlier, when an electron is confined in a small area, it will enter a superposition state, i.e., it will exist in multiple positions simultaneously. To break this superposition, one needs to take a “measurement.” In the game, the electron is confined to a small blue quantum wire. In contrast to the automatic measurements, in Particle in a Box, it is the players who take measurements by pulling a lever. Each time a player pulls the lever, a measurement is taken, which collapses the electron from its superpositions state to an unpredictable position on the wire for a brief moment. Any robot (player or enemy) standing on a platform directly above the collapsed electron will get “shocked” and lose some health (see Fig. 8). The position where the electron collapses is probabilistic, i.e., some positions are more likely than others. The relative probability of these positions is illustrated by the electron’s probability distribution, the orange curve in Fig. 8. The longer the platform is and the higher the curve above it is, the more likely the electron will be measured under it. After each measurement, the electron returns to superposition.

Fig. 8

Taking a measurement in Psi and Delta. One player (left) takes the measurement by pulling a lever, while the other (middle) lures the enemy bot to a higher probability area. Here, the enemy bot is shocked by the measured electron as it stands on a platform under which the electron was measured.

In part two, students develop models of energy levels (Fig. 9). Here, the opposing bot has a shield that protects it from getting shocked. To break the shield, the electron needs more energy. Electrons can only have a discrete amount of energy such as 1 eV (electron-volt) or 3 eV in the case shown in Fig. 9, but nothing in between. Energy can be supplied to an electron in the form of light, which consists of discrete energy packets (photons) with energy that depends on their color. To excite an electron from a lower energy (say 1 eV) to a higher energy (3 eV), one must shine photons with the exact energy as the gap (2 eV). In the game, students can shine light using a lamp and change its color using a spectrum.

Fig. 9

Shining light to increase the electron’s energy level in Psi and Delta. The horizontal lines on top indicate the electron’s energy levels. The spectrum (bottom-left) lets players change the color of the light shined.

This collaborative inquiry approach encourages students to share their ideas and work together to understand basic quantum physics concepts.¹³⁶ Qualitative evaluations of Psi and Delta with undergraduate physics and astronomy students showed that it was effective at engaging them with the basic concepts of quantum physics and had potential to be integrated more formally into traditional lectures.¹⁸ The game was also presented at the Smithsonian Creativity and Innovation Festival held at the National Museum of American History in Washington DC.

4.3.

QPlayLearn

QPLayLearn¹³⁸ is an online platform conceived to explicitly address, within a global strategic vision, the challenges and opportunities of widespread quantum literacy. QPlayLearn was born from a group of scientists passionate about quantum science and firmly believing that everyone can learn about quantum physics and its applications. QPlayLearn’s mission is to provide multilevel education on quantum science and technologies to anyone, regardless of age and background. To this aim, interactive tools enhance the learning process effectiveness, fun, and accessibility, while remaining grounded with scientific rigor. As a strategy for cultural change in a wide range of social strata, QPlayLearn offers diversified content for different target groups, from primary school to university physics students to quantum science enthusiasts. It is also addressed to companies interested in the emergent quantum industry, journalists, and policy makers needing to grasp what quantum technologies are about.

Inspired by the theory of multiple intelligences,¹³⁹ QPlayLearn’s holistic perspective stems from the recognition that different types of intelligence dominate the learning process of each person. Therefore, the platform is conceived to accompany users via different paths aimed at (1) building intuition and engagement through games and videos; (2) understanding physical concepts through accessible and scientifically accurate descriptions, graphics, animations, and experiments; and (3) acquiring formal understanding through mathematics. Such a three-step method is used in Quest, the QPlayLearn quantum dictionary, in which each entry is composed of various sections exploiting different kinds of learning-approaches and resources. For each concept in the dictionary, each user can begin from the approach that feels easier to them and then possibly explore the others. Eventually, it is the combination of the different pathways that shifts and expands the understanding of quantum physics and technologies.

In this context, games are one of the tools, embodied in the section Play, to stimulate the interactive participation of the users to grasp the counterintuitive features of quantum physics. In addition to games, short animations (Quantum Pills) explain concepts at different depths of understanding, which is useful for experts and nonexperts. In the Discover section, specialists explain concepts with short videos, using metaphors, experiments, or deductive examples. The Learn section enters the formal core of quantum theory, devoted to a more expert audience. In the Apply section, concepts can be practiced by running code samples on real-world quantum devices. Finally, there is also a space dedicated to art and science projects, as art can help bring people closer to science and quantum technology in a way that goes beyond academic learning or scientific popularization.

Because of the foundational concept, QPlayLearn is aimed at wide and diverse (free) use, to tailor education processes on quantum science on the many aptitudes of different users, operate a diffuse and massive cultural change, and build up literacy and awareness. Because of this flexibility, QPlayLearn can be incorporated by educators in a widespread context. The effectiveness of the QPlayLearn approach needs to be assessed by means of physics education research tools in on-purpose designed experiments. Meanwhile, a number of developments are underway; these include completing the quest environments, adding a multiple language version, and extending the class of beneficiaries to 0 to 99 years old users, together with pedagogy experts. In fact, the latter represents an authentic priority and an unprecedented challenge.¹⁸

4.4.

Virtual Lab by Quantum Flytrap

Quantum Flytrap is a company founded in 2019 that is focused on bringing quantum technologies closer to users by developing user-friendly graphical interfaces. In addition to on-going commercial projects involving developing a no-code Integrated Development Environment (IDE) for quantum hardware, Quantum Flytrap provides educational tools to demonstrate capabilities of interfaces and real-time simulations.

Virtual Lab by Quantum Flytrap (2019)¹⁴⁰ is a real-time online simulation of an optical table, supporting up to three entangled photons. The Lab uses a drag-and-drop interface for positioning optical elements, such as photon sources, mirrors, polarizing and nonpolarizing beam splitters, and Faraday rotators. The inspiration comes from ease of use and capabilities of construction systems, such as LEGO Bricks, the Goldberg machine video game The Incredible Machine,¹⁴¹ and grid-based games such as Chromatron and Minecraft. Most elements have configurable parameters such as phase shift (for reflections, delays, and wave plates), polarization rotation (for Faraday rotators and sugar solutions), and absorption rate. The Lab supports both destructive measurements and nondemolition POVM measurements.

Virtual Lab is a successor to Quantum Game with Photons (2016),¹⁴² an open-source puzzle game with 34 levels and a sandbox. Quantum Game with Photons simulates a single photon and its polarization. Although most setups could be understood within the framework of classical wave optics, the binary character of photon detection shows a few quantum phenomena. For example, the game demonstrates the Elitzur–Vaidman bomb tester—an experiment in which the quantum measurement plays a crucial role.

The original Quantum Game with Photons had over 100k gameplays and was cited in seven peer-reviewed publications by unrelated researchers.^18,143–148 It was selected as the top pick in gamifying quantum theory in The Quantum Times.¹⁴⁹ Based on in-person and email feedback, we learned that it was possible to solve a large fraction of levels without prior exposure to quantum physics, e.g., for high school and undergraduate students. At the same time, a few levels provided a challenge to PhD students and established quantum optics professors.

Virtual Lab is focused on reinventing the interface, design, and numerical performance. Our simulation supports multiparticle quantum physics. Unlike most other simulations, it goes beyond qubits. Virtual Lab simulates pure quantum states with photon polarization, four cardinal directions, and a discrete position of a photon on a grid. Consequently, for a typical grid size, a single photon requires around 1000-dimensional vector space, so a three-photon simulation requires a billion dimensions. Therefore, Quantum Flytrap needed to develop a custom high-performance sparse array simulation in the low-level programming language Rust—as none of the off-the-shelf solutions were suitable.

Virtual Lab offers a set of example experimental setups demonstrating key experiments in quantum measurement, quantum cryptography, quantum communication, and quantum computing. Example setups involve single photon experiments, such as nonorthogonal state discrimination, the Elizur–Vaidman bomb tester, the BB84 quantum key distribution protocol, and the concept of nondemolition measurement affecting interference. The two-particle setups include the Ekert quantum key distribution protocol, the Deutsch–Jozsa algorithm, and the CHSH Bell inequality violation. With three particles, we can demonstrate quantum teleportation and Greenberger–Horne–Zeilinger correlation (Fig. 10).

Fig. 10

A three-particle Greenberger–Horne–Zeilinger entangled state, visualized simultaneously as (A) waves on the board, (B) entanglement between particles, and (C) the quantum state, in bra-ket form.

The measurement results can be subject to logical operations (such as AND or XOR), used to alter other elements (e.g., to perform quantum teleportation), treated as a goal (for puzzles and course assignments), correlated (for Bell inequality violation), or saved as a CSV table for arbitrary processing.

Moreover, a laser mode allows us to explore classical wave phenomena, such as the three-polarizer paradox (generalizing to the quantum Zeno effect), interferometers (Mach–Zehnder, Michelson–Morley, Sagnac), optical activity, and magneto optic effect (with applications such as an optical diode).

Virtual Lab provides multiple ways to investigate an experiment, its outcomes, and the current quantum state. The multiverse tree feature allows users to explore the whole experiment, including all possible measurement scenarios. In the Copenhagen interpretation, all branches are related to probabilistic outcomes. Within the Everettian many-world interpretation, each branch is a coexisting part of the quantum state, representing worlds that are unlikely to interfere with each other.

In each time step, it is possible to explore the exact state (ket) with a preferred choice of the basis and coordinates of complex numbers. Furthermore, we visualize entanglement of every particle with the system.

Virtual Lab has been used for quantum information courses at Stanford University and the University of Oxford, as well as by Qubit for Qubit’s Introduction to Quantum Computing online course. Virtual Lab features over 450 custom, user-created setups. It has around 70 users on a regular working day and peaks at over 700 during events.

Quantum Flytrap released Quantum Tensors¹⁵⁰ is an open-source JavaScript library for in-browser quantum information processing. It goes beyond qubits and supports physical multiparticle physical systems (polarization of light and spin), as well as qudits. It allows for the creation of states, changing bases, performing unitary operations, measurements (destructive and nondemolition POVMs), and entanglement entropy. It has sparse linear algebra operations (including with tensor products) and support for quantum circuits.

Virtual Lab allows for exploration of quantum states at each single time step (see Fig. 11) and matrices for unitary operation and projections (see Fig. 12). Quantum Flytrap open-sourced BraKetVue (or $⟨\phi |\psi ⟩$.vue)¹⁵¹ is a visualization of quantum states and operations in the popular web-development framework Vue.js. It allows for the creation of interactive quantum explorable explanations (e.g., in a blog post with an introduction to quantum gates¹¹⁴) and can be used in other web-based environments, such as presentations with Reveal.js (created in RISE in Jupyter Notebook or with R Markdown).

Fig. 11

An experimental setup for the CHSH Bell Inequality violation. Note that the experiment depends on two random variables (Alice’s and Bob’s basis), provides a table of all measurements (downloadable as a CSV file), and presents all correlation in real time.

Fig. 12

A matrix visualization for a beamsplitter with BraKetVue. The matrix values are presented with a circle size (radius for the amplitude, area for the probability) and color (hue for the phase). Index labels incorporate the tensor structure.

Matrix visualizations allow us to show the magnitude and phase of transition in a visual way, dynamically change the basis, and explore the tensor structure (e.g., change the order of particles).

4.5.

Quantum Odyssey

Quantum Odyssey is a privately funded, large software project, still in development by Quarks Interactive,¹⁵² but the alpha version is already available for the public. Currently, the software comes with over 20 h of fully narrated, self-paced education ranging from quantum optics phenomena to coding quantum algorithms. The software also contains a vast encyclopedia that verbosely explains key quantum mechanics concepts. People working on the software have various backgrounds (researchers in education, quantum physicists, artists, and game developers).

Behind Quantum Odyssey is the concept of quantum literacy, as coined by Nita et al.¹⁹ The aim of Quantum Odyssey is to make quantum physics and computing knowledge accessible to all audiences, regardless of their background, by making learning about and working with quantum algorithms exciting and fun. Quantum Odyssey is a fully visual and software-assisted method for learning how to create new and optimize quantum algorithms for quantum computers. Equivalent to everything that can be achieved using the gate model framework, users can design quantum algorithms for any purpose (ranging from designing physical interactions in optics to more advanced quantum computing algorithms) without requiring any background knowledge in STEM. Quantum Odyssey shows in real time the dynamics of the quantum state vector, including its evolution as the user adds quantum gates to build quantum algorithms (Fig. 13).

Fig. 13

Images 2, 3, and 4 are snapshots of the evolution of a quantum state vector in visual form for two particles that start in the ground state 00 and reach the Bell state 00 and 11. The balls at the bottom show the desired output solution, and the user is required to place a Hadamard gate to achieve it as shown in 1.

The software was tested in schools and found to be suitable for teaching quantum physics and computing from the age of 12 and above. The results of the research study¹¹⁸ show that the Quantum Odyssey visual methods are efficient in portraying counter-intuitive quantum computational logic in a visual and interactive form. This enabled untrained participants to quickly grasp difficult concepts in an intuitive way and solve problems that are traditionally given today in master’s level courses in a mathematical form. The results also show an increased interest in quantum physics after play, with a higher openness and curiosity to learn the mathematics behind computing on quantum systems. Participants developed a visual, rather than mathematical, intuition that enabled them to understand and correctly answer entry level technical quantum information questions. Another interesting find is that the design of the visuals of the software makes it equally appealing to both female and male participants, that is, both genders were found to be equally able to solve complex mathematical problems if they were given in a visual puzzle form within Quantum Odyssey.

In future releases, the following new features will be introduced:

4.6.

ScienceAtHome

ScienceAtHome¹⁵³ was founded in 2011 at Aarhus University with the goal of developing gamified tools to enable citizen scientists to participate in cutting-edge quantum research. Today, ScienceAtHome boasts a suite of games and tools for education and citizen science in a variety of academic fields.

ScienceAtHome’s flagship game is Quantum Moves 2¹⁵⁴ (Fig. 14), a game in which citizen scientists can play through a variety of scenarios, including three scientific problems relevant to cutting-edge research in quantum technologies. To date, at least 250,000 players have played Quantum Moves 2. Research has shown that player-generated solutions could have some advantages over random seeding when applied to optimization of the scientific problems, even when the solution landscapes of these problems are complex.¹¹⁷ In particular, players explore the solution landscape more efficiently and can capture solution strategies that algorithms may miss. This is not to say that players are better than algorithms, but rather that the insights of citizen scientists can be useful when deciding how to solve a given quantum problem.

Fig. 14

The Quantum Moves 2 interface, showing the Shake Up level. The goal is to move the potential (green line) left and right within the cyan dashed box to excite the quantum fluid (green shaded curve with red-yellow outline) from its initial, ground state (as shown) to the final, first excited state (yellow outline). This must be done within a certain amount of time, and the goal is to achieve the highest overlap (fidelity) between the final state and the yellow-outlined state. For more information, see Ref. 117.

In 2017, researchers at ScienceAtHome opened up their quantum laboratory to the general public in the Alice Challenge.¹⁵⁵ The goal of the Alice Challenge was for users to optimize the number of rubidium-87 atoms in a Bose-Einstein condensate, or BEC,^156,157 a cloud of extremely cold atoms that can then be used for a variety of quantum technology experiments. We found that citizens operating the experiment remotely¹⁵⁸ were able to create larger BECs than the local experimental experts,¹⁵⁹ a difference that we attribute largely to interfacing and gameplay (Fig. 15), that is, we built a simple and intuitive interface that allowed citizens to collaborate (in that any player could copy a previous solution and modify it) to solve the problem-at-hand; such intuitive interfacing is key to success when considering citizen science projects, quantum or otherwise.

Fig. 15

The interface for the Alice Challenge, showing (a) the spline curves that the users could manipulate to control the time-dependent laser powers and magnetic field coil currents that controlled the final BEC atom number, (b) a top score list, (c) a list of the latest submissions and their scores, and (d) a control toolbar, including the estimated wait time. Players could take any other player’s solution [as shown in (b) and (c)] and modify it. Figure reproduced from Ref. 158.

ScienceAtHome has also developed a number of educational tools, including the Quantum Composer,⁸⁶ which is built on the QEngine C++ library;¹⁶⁰ both tools are freely available for download and use. The QEngine is a research tool for the simulation and control of 1-D quantum mechanics,¹⁶¹ and the Quantum Composer wraps the functionality of the QEngine into a flow-based tool for quantum visualization. Thus, the Quantum Composer (Fig. 16) is aimed at introducing students to quantum mechanics and quantum research via a modular interface that can be used to simulate a wide range of problems in introductory and advanced quantum mechanics, including those that are not analytically tractable. ^87,88

Fig. 16

The Quantum Composer interface. Users can drag-and-drop nodes (representing space, the potential, a plot, etc.) from the left-hand tray into the simulation space (called a flowscene). Connections are then made between the nodes to direct information from one node to another. This figure shows how one can set up a simulation that shows the first three eigenenergies and eigenstates of the quantum harmonic oscillator. Figure reproduced from Ref. 87.

ScienceAtHome researchers and educators have used these tools in a number of educational and outreach settings, including the aforementioned week-long Alice Challenge in 2017 in which 600 citizen scientists around the world took part. Quantum Moves 2 featured heavily in the 2018 ReGAME cup,¹⁶² with more than 2000 students from 200 high school classes around Denmark playing a set of citizen science games as they worked through educational content that scaffolded the games, and the planned Project FiF¹⁶³ (Forskningsspil i Folkeskolen, or research games in high schools) will expand this to 150 schools around Denmark. Quantum Composer has also been widely used as a part of ScienceAtHome outreach efforts in Denmark and around Europe, including a 3-year-long outreach training effort coordinated as a part of the quantum sensing and control (QuSCo¹⁶⁴) Marie Skłodowska-Curie interactive training network.¹⁶⁵ The outreach training program was aimed at teaching PhD students to communicate their research via a variety of methods, including videos, blog posts, and the development of inquiry-based learning sessions.

4.7.

Virtual Quantum Optics Laboratory

The Virtual Quantum Optics Laboratory (VQOL) is a graphical web-based software tool for designing and simulating quantum optics experiments^166,167 developed by the Applied Research Laboratories at the University of Texas at Austin. The VQOL was developed initially as an educational tool for introducing high school and early college students to quantum information science, but it has also proven to be useful as a research tool for analyzing real-world experiments.¹⁶⁸ The graphical interface presents a top-down view of a notional optics table with a menu of components that can be placed and configured as desired. To aid visualization, six base colors are used to represent polarization and interpolate these across the Poincaré sphere (see Fig. 17). A variety of components are available; these include passive linear optics (waveplates, polarizers, beam splitters, etc.), light sources (LEDs, lasers, and Gaussian entangled states), and measuring devices (power meters and detectors).

Fig. 17

Screen shot of the VQOL interface showing a delayed-choice experiment. A laser (left) produces horizontally polarized light (red) that is attenuated by a neutral density filter before entering a Mach–Zehnder interferometer. The random polarizations (multiple colors) are due to vacuum noise. The upper arm has a half-wave plate (blue disk) that can provide which-way information by changing the polarization to vertical (blue). The two detectors have polarizers (green disks) to detect horizontal light.

In an educational setting, VQOL may be used to introduce optics concepts (either classical or quantum) or to explore quantum phenomena and applications. Over the past 2 years, we have used VQOL as part of a full-year high school course on quantum computing¹⁶⁹ as well as part of a two-semester introduction to quantum computing for university freshmen and sophomores. A typical introduction to qubits begins with the concept of polarization, which can easily be understood in terms of transverse waves on a string. We then introduce projective measurements through an initial investigation of Malus’s law using a laser, polarizer, and power meter. We transition to the Born rule by introducing a neutral density filter, which attenuates the laser light to the single-photon regime, and replacing the power meter with a detector for photon counting. Students perform guided experiments and explore the phenomena before drawing inferences to uncover the physical laws behind them. Upon this foundation of physical intuition, a more concise abstract mathematical description may then be introduced. One important aspect of these experiments is that students are challenged to consider appropriate data analysis methods for conceptually connecting “counts” with probabilities. Because, from an experimental point of view, there is not a well-defined number of “trials,” this exercise raises interesting interpretive questions. Students must consider how to normalize their data and account for dark counts, just as they would need to do in a physical optics lab.

Beyond introducing elementary concepts, we have also used VQOL for deeper explorations into quantum physics. Over the past 2 years, we have offered a virtual experimental quantum optics program to our university students. The program aims to connect theory to experiment by exploring quantum information science in an experimental optics setting. Students work in groups of two to four individuals and are tasked with exploring about 10 different experiments over a 9-week period. Students select which experiments to investigate and are given only general guidance on the goals and phenomena to be explored. At the end of each week, the groups get together to present and discuss their findings. Possible experiments include quantum state tomography, an optical implementation of Deutsch’s algorithm, delayed choice and quantum eraser experiments, quantum teleportation, entanglement swapping, and violations of Bell’s inequality.¹⁶⁶ We have found through post-program surveys that students generally respond well to the open-ended nature of the experiments and feel that it gives them a more authentic research experience than traditional, more structured undergraduate laboratories.

4.8.

Activities in Creating Quantum Games

The first recorded hackathon was organized by developers of the OpenBSD¹⁷⁰ software in June 1999, and the term was coined by its organizers. It is commonly used to refer to “a usually competitive event in which people work in groups on software or hardware projects, with the goal of creating a functioning product by the end of the event.”¹⁷¹ The first (or the zeroth depending on the perspective) game jam was organized by The Indie Game Jam community in March 2002.¹⁷² It usually refers to a more relaxed and noncompetitive event compared to a hackathon, similar to a jam session of a musical band. The created games can have educational or research value.¹⁷³ Moreover, a game jam can foster creativity, learning, and collaboration of the game developers to explore technologies and develop skillsets.¹⁷⁴ Since their emergence, both terms have been adopted by software communities and beyond, and the number of hackathons and jams have grown to hundreds per year.¹⁷⁵ Therefore, their adoption by the community of quantum game developers was not a surprise. Some further readings on game jams can be found here.^176–180

The first quantum game jam was organized in Finland in 2014 as a part of the Finnish Game Jam,¹⁸¹ and the first quantum hackathon (not focused on games) was organized in April 2018 by Rigetti,¹⁸² where two quantum games were also developed. Following these, several quantum game jams¹⁸³ and hackathons focused on game development were organized. Reference 183 describes the organization of five quantum game jams and rates for each the quantum game developed based on how enjoyable and informative on quantum concepts that it was.

In what follows, we provide a recent example of a quantum games hackathon organized in September 2021. This is followed by the description of a quantum computer games project-based course at the University of Texas at Austin, organized by the developers of the previously introduced Virtual Quantum Optics Laboratory tool. Both examples are described by their organizers in detail, to be used as templates by any interested readers.