# QuBOBS, interactive objects and a visual representation
to explain quantum computing

Sophie Laplante IRIF, Université Paris Cité Loris Perez Musician and composer
Sylvie Tissot Software engineer, Anabole Lou Vettier Designer, louvettier.com

###### Abstract

We introduce a visual representation of qubits to assist in explaining quantum
computing to a broad audience. The representation follows from physical
devices that we developed to explain superposition, entanglement, measurement,
phases, interference, and quantum gates. We describe how this representation
can be used to explain teleportation and the Bennett-Brassard quantum key
agreement protocol.

##  1 Introduction

This project is the result of a collaboration between a computer scientist, a
designer, a software developer and a composer. Together, we are developing
tools to make quantum computing accessible to a wider audience, without
sacrificing (too much) mathematical correctness. In this paper, we describe
the mathematics behind the objects that we are developing. We assume that the
reader is familiar with the basics of quantum computing. Our goal is to
provide the reader with a representation that can be used for teaching at the
undergraduate and graduate level, to students with or without mathematical
background, as well as for outreach activities.

Most researchers in quantum information and algorithms agree that explaining
quantum computing to a wide audience without relying on mathematical machinery
is a difficult task. A recent article of Aaronson lays out the difficulties
particularly well [Aar21]. Many different approaches have been used to explain
quantum computing to a wide audience, and although there are far too many to
give an exhaustive survey here, we give a few examples.

For single qubit applications such as key agreement, Charles Bennett [Ben21]
and many others use double arrows to represent photon polarization. John
Preskill [Pre16] uses colored balls to represent 0 and 1. Qubits are
represented as boxes and gates with two doors. Putting in a qubit in the top
door and taking it out of the bottom door applies a change of basis. This
representation is used for single qubit protocols as well as to explain
entanglement.

Karl Svovil explains quantum cryptography using chocolate balls [Svo06] with a
0 or 1 symbol on each ball, in either green or in red to indicate the basis.
Special glasses (red and green) are used to explain measurement in the correct
basis.

The most complete representation we are aware of is proposed by Terry Rudolph
and coauthors [Rud17, ERB05, EB22]. This representation is intended to explain
quantum circuits and computation in full generality, without using
mathematical symbols. Qubits are presented as clouds of black and white balls,
in a proportion that corresponds to the probability of getting the
corresponding outcome when measuring the qubit. Blocks represent operations on
qubits. Using this representation, they explain several advanced applications,
such as Grover search [Gro96] and CHSH games [CHSH69].

For more advanced topics in quantum information, tensor networks have been
used as a graphical language to represent quantum states and processes [WBC15,
Bia19].

Our approach has been to construct simple physical devices to help the
audience develop a very concrete mental image of concepts that can be
difficult to grasp, such as randomness which is inherent to quantum computing,
and entanglement. Special care has gone towards distinguishing randomness and
truly quantum properties. Superposition is represented by two overlapping
paper disks. Measurement is represented by spinning a window over a disk and
observing the result through a small window. Correlation is made concrete by
having two windows spinning together as cog wheels. Once the physical
representation is clear, we move on to a slightly more abstract graphical
representation, on which we can apply various operations.

##  2 The QUBOBS representation

###  2.1 Qubits

Our main contribution is a simple, robust and surprisingly versatile
representation of qubits. A qubit $\alpha|0\rangle+\beta|1\rangle$ is
represented by a two-color disc. Blue represents $|0\rangle$ and orange
represents $|1\rangle$. The proportion of each color represents the
probability of measuring the corresponding outcome, hence the blue part takes
up an $|\alpha|^{2}$ fraction of the disc’s area (and circumference) and the
orange part takes up $|\beta|^{2}$.

Most elementary quantum algorithms use signed reals as amplitudes so to
complete the description, we use a negative sign to indicate when the phase is
-1.

Our qubit representation is made of colored paper discs. Two discs can be
interleaved to obtain superpositions with arbitrary amplitudes.

Figure 1:  An object that represents a quantum state $|\phi\rangle$. The
squared magnitude of $\left\langle\phi|0\right\rangle$ is blue, and the
squared magnitude of $\left\langle\phi|1\right\rangle$ is orange. On the
device on the right, the window spins around the edge of the qubit, When it
stops, the color under the window is the outcome of the measurement.

We illustrate measurements with a small device that spins around the qubit.
When it stops, a small round window reveals one of the colors. This is the
outcome of the measurement. The probability of each outcome is the part of the
disc that is the corresponding color, which is exactly the probability of
measuring the qubit in the standard basis.

Taking the modulus square of the amplitude sacrifices the phase, but we find
that this is a useful first step towards understanding the inherent randomness
of quantum measurements. When we use our devices, we show how this is also how
we could represent a biased coin. Flipping a coin and observing the outcome is
similar to measuring a qubit, but as we point out, a coin flip is not a
quantum phenomenon. This allows us to highlight the fact that the phase is a
crucial part of a quantum state, and that without phases, we are only
describing probabilistic binary states, and probabilistic computation.

Most elementary quantum algorithms use only positive and negative reals and do
not require the complex numbers in their full generality. We introduce the
phase as a sign, which is sufficient for quantum computing.

We write $|\phi\rangle\mapsto(P,Q)$ to mean that $(P,Q)$ is our representation
of $|\phi\rangle$. Using this notation, a qubit
$\alpha|0\rangle+\beta|1\rangle$ with $\alpha,\beta\in\mathbb{R}$ is
represented by $(sgn(\alpha)|\alpha|^{2},sgn(\beta)|\beta|^{2})$, and
conversely, $(A,B)$ is a representation of the qubit
$sgn(A)\sqrt{|A|}|0\rangle+sgn(B)\sqrt{|B|}|1\rangle$. When several qubits are
involved, the angle of the disk becomes important, and we will assume that,
unless indicated otherwise, the blue part starts at the top going clockwise.

###  2.2 Entanglement and partial measurements

The surprising expressiveness of our representation of qubits appears when we
consider two qubits. Separable states can be represented by two observation
windows spinning independently over two qubits. However, things become
interesting when we link the two spinning windows with a cog. so that they
spin synchronously. This allows us to represent entanglement as we explain
now.

####  2.2.1 Entanglement

Figure 2: Two ways of representing two entangled qubits. On the left and in
the middle, a cog makes both windows spin synchronously. On the right, the
qubits are stacked and a single window lets us observe both qubits. The state
represented on the right on both devices is
$\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$.

Consider an arbitrary two-qubit pure state
$|\phi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|11\rangle+\delta|10\rangle$.
(Notice that we have used a Gray code to order the basis elements.) If the
first qubit is measured with a standard measurement, the outcome is 0 with
probability $|\alpha|^{2}+|\beta|^{2}$ and 1 with probability
$|\gamma|^{2}+|\delta|^{2}$. Similarly, if the second qubit is measured, the
outcome is $0$ with probability $|\alpha|^{2}+|\delta|^{2}$ and 1 with
probability $|\beta|^{2}+|\gamma|^{2}$. This allows us to represent the two-
qubit state with one disc that has $|\alpha|^{2}+|\beta|^{2}$ fraction of blue
and $|\gamma|^{2}+|\delta|^{2}$ of orange, and the second disc with
$|\alpha|^{2}+|\delta|^{2}$ blue and $|\beta|^{2}+|\gamma|^{2}$ orange. They
key observation is that when the observation windows are lined up (say they
both start from the top of the disk and turn synchronously) it is always
possible to line up the two discs so that the probability of getting 00 when
both qubits are measured is $|\alpha|^{2}$, and similarly for 01, 11, and 10,
as illustrated below. To do this, line up the first qubit so that it is blue
starting from the top going clockwise, and line up the second qubit so that it
is orange starting at an angle $\theta=|\alpha|^{2}\cdot 2\pi$ from the top,
going clockwise. Then starting from the top and going clockwise, the two
qubits are blue for $|\alpha|^{2}\cdot 2\pi$, then the first qubit is blue and
the second is orange for $|\beta|^{2}\cdot 2\pi$ and so on.

We use a second device to visualize the four areas corresponding to each of
the outcome pairs 00,01, 11, and 10 (Figure 2, on the right). We stack the two
qubits on top of each other, with a single window going around the perimeter.
Going back and forth between these two representations, side by side and
stacked, gives us two complementary ways of seeing the effects of
entanglement.

When explaining protocols or algorithms, we set aside the physical devices
with the windows spinning around the perimeter, and work with the diagrams
illustrated in Figure 3.

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

Figure 3: Two representations of the same pair of entangled qubits,
$\frac{1}{\sqrt{3}}|00\rangle+\frac{1}{\sqrt{6}}|01\rangle+\frac{1}{\sqrt{3}}|11\rangle+\frac{1}{\sqrt{6}}|10\rangle$.
On top they are represented side by side like in the device with two windows
spinning synchronously. On the bottom the qubits are stacked the same way as
in the device with a single window.

Separable states can also be represented with synchronized observation
windows. For example, the state
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
can be represented with two half blue, half orange disks placed
perpendicularly.

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

Figure 4:  Our representation of the separable state
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$

This works well for two qubits. From three qubits onward we can no longer
represent arbitrary states with discs with two colors arranged into two
contiguous colored slices. However, it turns out that for many applications,
two qubits suffice, and furthermore, presenting qubits using more than two
color slices occurs naturally as a result of applying quantum gates.

####  2.2.2 Partial measurements

Partial measurements of a bipartite state and the residual state after a
measurement is made is generally viewed as difficult to explain without
appealing to a formal projection operator. Our representation is surprisingly
successful in conveying the math without writing out any equations.

Returning to our two-qubit state
$|\phi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|11\rangle+\delta|10\rangle$,
let us make a measurement on the first qubit. The blue and orange parts of the
first qubit split the second qubits into two areas. In Figure 5 we give an
example of how we can visualize the residual state of the second qubit given
that the first qubit has been measured. If the first qubit came out blue, then
the observation window could have fallen in any place where the first qubit
was blue. This part of the disk, on both qubits, since the measurement windows
are linked by a cog, is the residual state. The rest of the disks are no
longer accessible. Having this visual representation in mind makes it easier
for students to conceptualize what the projection does and what the
renormalization of the state accomplishes (not to mention what the norm
actually represents).

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

Measure 0  Measure 1

Figure 5: Visualizing the effect of a partial measurement. On the top, the
qubit before measuring the fist (outer) qubit. Below, the residual state when
the outcome is blue (on the left) or orange (on the right).

###  2.3 Phases and interference

In our basic representation, we restrict ourselves to +/-1 phases. In the
simpler quantum applications, such as key exchange or Deutsch’s and Deutsch-
Jozsa’s algorithm, when interference occurs, opposing amplitudes cancel
exactly. By this we mean that in the course of the computation, we obtain
states of the form
$\alpha|0\rangle-\alpha^{\prime}|0\rangle+\beta|1\rangle+\beta^{\prime}|1\rangle$
where $\alpha=\alpha^{\prime}$. This is a great stroke of luck since
$|\alpha-\alpha^{\prime}|^{2}=|\alpha|^{2}-|\alpha^{\prime}|^{2}$ = 0, and the
slices with opposing phases cancel exactly. Obviously this does not hold in
general.111The incorrect identity, $(x+y)^{2}=x^{2}+y^{2}$ has been coined
“freshman’s dream” following Saunders Mac Lane [Lan40] who attributes it to
Stephen Kleene. However, the fact that this happens to be correct in some key
cases of interest allows us to convey how interference plays a role in quantum
computation, in a meaningful and (almost) mathematically correct way. To more
mathematically advanced audiences, we always point out that we are taking a
sinificant shortcut here, and in a classroom situation we can return to the
more conventional mathematical representation, and address other instances of
interference later on, when discussing Grover’s algorithm, or quantum
strategies for Bell inequality violations for instance.

$|0\rangle$

$|1\rangle$

Figure 6: The qubit $\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$ represented as
$(1/2,-1/2)$.

When explaining interference, we have found that referring to colors (blue,
orange) makes it easier to grasp what is meant by interference than when using
numerical values 0 and 1. We have often been asked what it means for +0 and -0
to cancel, since $+0-0=0$. This problem of confusing the state with its label
does not occur when talking about colors. It also offers an opportunity to
discuss the fact that blue and orange can be labels for many different
propoerties, such as spin, or polarization, as long as these properties are
fully distinguishible.

To help explain phases and interference, we appeal to the audience’s knowledge
that light behaves like a wave. Positive and negative phases can be
illustrated with sine waves that can be shifted by half a wavelength.222For
more advanced audiences, one could generalize to arbitrary shifts between 0
and half of the wavelength but we have not found this to be necessary to
explain basic quantum algorithms and protocols. For younger audiences, we have
illustrated this with two people pulling and pushing on two ends of a table in
a synchronized cyclical motion. If both have the same phase, one pushing as
the other pulls, then the amplitudes add up. If they have opposite phases,
they cancel each other and the table doesn’t move. Pushing on a blue table
does not interfere with pushing on an orange table. Many audience members are
familiar with this phenomenon in the realm of sound waves and have pointed out
that this is how noise-canceling headphones work.

###  2.4 Single qubit gates and linearity

We illustrate quantum gates by flipping over a disk to reveal the effect of a
gate on each of the basis states.333Many of the gates we consider are self-
adjoint, and flipping over twice gets us back to the initial state.

In order to explain simple quantum circuits, it would be nice to apply quantum
gates on our representation of qubits by appealing to linearity. This works
well in some cases, like the X gate, or the Z gate.

However, we have to proceed with caution with other gates, such as the
Hadamard gate. Hadamard applied to zero is the state composed of half blue and
half orange, which we denote by $(1/2,1/2)$. Similarly Hadamard applied to
orange is $(1/2,-1/2)$. If we apply Hadamard on $(1/2,1/2)$, we apply it to
both parts. The blue half becomes $(1/2,1/2)$ and the orange half becomes
$(1/2,-1/2)$, so we obtain four parts, which we will write
$(1/4,1/4,1/4,-1/4)$. Up to this stage, our representation accurately reflects
the effect of the gate (up to normalization). What can get us into trouble is
interference.

In some cases we can add the parts of same sign and let parts of opposite
signs cancel. In the case of Hadamard, we are tempted to say that the orange
parts with opposite signs cancel, leaving an all-blue disk. In this case, this
correctly mimics what happens to the amplitudes, and it allows us to convey a
message that is essentially correct: the fact that applying Hadamard twice to
basis states is identity, and this occurs because of interference. It is also
all that is needed to explain how and why some simple applications, such as
the BB84 key exchange protocol, or Deutsch’s algorithm, work.

In general, however, we refrain from adding together and rearranging the
slices in our representation. The first reason is that positive and negative
interference does not usually behave well in our representation. But there is
a second very imoirtant reason which is that when multiple qubits are
entangled, moving parts around would affect the entanglement between the
qubits. We explain this is more detail in Section 2.5.

We consider the effect of single qubit gates $G$, with
$G|0\rangle=a|0\rangle+b|1\rangle$ and $G|1\rangle=c|0\rangle+d|1\rangle$ We
will say that in our representation, $G(P,Q)=(A,B,C,D)$ where

$A=P\cdot\mathrm{sgn}(a)|a|^{2}\quad B=P\cdot\mathrm{sgn}(b)|b|^{2}\quad
C=Q\cdot\mathrm{sgn}(c)|c|^{2}\quad D=Q\cdot\mathrm{sgn}(d)|d|^{2}$

In the schematic representation, we obtain four parts, starting from the top
going clockwise with blue, and alternating blue and orange parts.

$|0\rangle$

$|1\rangle$

$\stackrel{{\scriptstyle H}}{{\longrightarrow}}$

$|0\rangle$

$|1\rangle$

$|0\rangle$

$|1\rangle$

=

$|0\rangle$

$|1\rangle$

Figure 7: Applying Hadamard to $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
behaves well since the positive and negative parts cancel completely.

$|0\rangle$

$|1\rangle$

$\stackrel{{\scriptstyle H}}{{\longrightarrow}}$

$|0\rangle$

$|1\rangle$

$|0\rangle$

$|1\rangle$

= ???

Figure 8: Hadamard applied to other states does not behave well. Applying
Hadamard to $\frac{\sqrt{2}}{\sqrt{3}}|0\rangle+\frac{1}{\sqrt{3}}|1\rangle$,
we get
$\frac{1}{\sqrt{3}}|0\rangle+\frac{1}{\sqrt{3}}|1\rangle+\frac{{1}}{\sqrt{6}}|0\rangle-\frac{{1}}{\sqrt{6}}|1\rangle$.
Our representation suggests that there should be $\frac{1}{3}-\frac{1}{6}$
orange (1) left, whereas the correct fraction is
$(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{6}})^{2}$.

Henceforth we will use the notation $(A,B,C,D)$ to describe a disc with four
areas, whose colors are blue, orange, blue, orange. For example, if Hadamard
is applied to $(1/2,1/2)$, we get $(1/4,1/4,1/4,-1/4)$. Unless the state is
entangled with another, we start from the top with a blue slice of size $A$,
and so forth, going clockwise.

###  2.5 Applying gates on two qubits

Applying two-qubit gates works similarly to the one-qubit case.

Recall that two qubits in our representation, $(P,Q)$ and
$(P^{\prime},Q^{\prime})$, do not fully determine the two-qubit state, since
the angle at which they are placed determines how they are correlated. If the
first qubit is blue going clockwise from the top, and the second disk is
orange starting from an angle of $\theta\cdot 2\pi$, going clockwise, then
(ignoring for the sake of simplicity any possible phases) the state
represented is

$|\psi\rangle=\sqrt{\theta}|00\rangle+\sqrt{P{-}\theta}|01\rangle+\sqrt{1{-}P{-}P^{\prime}{+}\theta}|11\rangle+\sqrt{P^{\prime}{-}\theta}|10\rangle.$

We will use the notation
$|\psi\rangle\mapsto(P,Q)\theta(P^{\prime},Q^{\prime})$.

When we apply a single qubit gate to the first qubit of a two-qubit state, we
apply it while preserving the angle between the two qubits. To ensure that
entanglement is preserved, we apply it to all four areas corresponding to the
basis elements $|00\rangle,|01\rangle,|11\rangle,|10\rangle$. These four areas
are easily seen in the stacked representation. Each of the four areas will be
further subdivided by gates such as Hadamard.

Similarly, when applying a two-qubit gate, we apply it to all four areas.
Control gates are particularly easy to visualize. (See Figure 11.) In the area
where the control bit is 0, we do nothing, and in the area where the control
bit is 1, we apply the gate as usual.

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

Figure 9: The CNOT gate is applied to the qubit on the left. The control bit
is the outer bit. When the outer bit is blue, the inner bit remains the same.
When the outer bit is orange, the colors of the inner bit are flipped.

##  3 Applications

###  3.1 Key exchange

We have used our representation to explain the BB84 key agreement protocol
[BB84] to high school students and members of the general public. We
distribute to participants a kit containing

  * •

Qubits with $|0\rangle$ (blue) on one side and $H|0\rangle$ on the other,

  * •

Qubits with $|1\rangle$ (orange) on one side and $H|1\rangle$ on the other,

  * •

Two dice, one to pick a bit, and one to apply, or not, Hadamard.

  * •

Eight numbered black envelopes in which qubits can be placed and observed on
either side using small cutout flaps.

  * •

A grid to mark the results: what value was selected, what basis was used, and
what measurement was made, and what the outcome was.

Figure 10: Our BB84 kit comes with eight envelopes and two dice. Alice picks 8
qubits and places each them in a numbered envelope. Qubits can be measured by
opening a small flap in the envelope. Flipping a qubit over changes the basis.
Flipping over the envelope changes the measurement basis.

Participants playing Alice can prepare qubits in the standard or diagonal
basis by placing them in envelopes with the correct side facing up. A second
participant playing Bob can measure a qubit in the standard basis by opening a
flap on the front of the envelope, or in the Hadamard basis by turning over
the envelope (and hence the qubit inside) and opening a flap on the back of
the envelope. All the results are marked on a grid. Participants can see for
themselves that any time the basis chosen for the qubit and the basis for the
measurement coincide, the outcome of the measurement made by Bob equals the
bit that was chosen by Alice. We can then explain that any discrepancy would
be due to an observation made by a third party.

###  3.2 Teleportation

We like to explain quantum teleportation [BBC+93] in two steps. First we give
a simplified and fully classical protocol to teleport a coin with an arbitraty
bias. This protocol only uses a CNOT gate, and Alice measures only one of her
qubits and sends the outcome to Bob. This is enough for Bob to obtain the bias
of Alice’s coin.

Splitting the protocol into two steps, a fully classical step with CNOT and
the truly quantum part with Hadamard demystifies how and why the protocol
works, and allows us to clearly identify what part of the magic is due to
classical correlations and what part is attributable to quantum.

####  3.2.1 Teleporting a biased coin

At the start of the protocol, Alice has a biased coin, and Alice and Bob share
one pair of perfectly correlated unbiased coins. This is similar to Alice
having a qubit without any phase and the two players sharing an EPR pair.

In Figure 11, the first stacked pair of qubits belongs to Alice and is
composed, on the inner disk, of the qubit (biased coin) that Alice wishes to
transmit, and on the outer disk, half of an EPR pair (or correlated coins).
The rightmost qubit is Bob’s half of the EPR pair.

In Step 2 of the simplified protocol, Alice applies a CNOT operation on her
qubits, using the inner disk as the control qubit.

In Step 3, Alice measures her outer qubit. Since the qubits are entangled
(correlated), this also affects Bob’s part of the EPR pair. In Step 3a, if
Alice measures 0 (blue) on the outer disk, Bob gets the correct qubit (without
the phase). In Step 3b, if Alice measures 1 (orange) on the outer disk, Bob
needs to apply a correction to reverse the values.

Alice Bob

Step 1.

Setup

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

$|0\rangle$

$|1\rangle$

Step 2.

CNOT

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

$|0\rangle$

$|1\rangle$

Step 3a.

Measure 0

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

$|0\rangle$

$|1\rangle$

Step 3b.

Measure 1

$|00\rangle$

$|01\rangle$

$|11\rangle$

$|10\rangle$

$|0\rangle$

$|1\rangle$

Figure 11: Classical teleportation of a qubit without a phase. Alice’s two
qubits (or coins) are represented on the left, and Bob’s qubit (or coin) is on
the right.

When the measurement outcome is 0 (blue), the residual state on Bob’s side has
the correct bias. When the outcome is 1, the bias is correct provided he flips
the colors. Notice that we have given a fully classical protocol that
transmits one coin flip with an arbitrary bias, using a pair of correlated
unbiased coins plus the transmission of an unbiased coin flip.

####  3.2.2 Full teleportation protocol

In order for Alice to transmit the phase of her qubit, she adds the truly
quantum part of the protocol by applying Hadamard to her inner qubit. This
will further split all the slices into two as we illustrate in Figure 12.

Alice Bob

Step 3’.

Hadamard

$|00\rangle$

$|01\rangle$

\- $|11\rangle$

$|10\rangle$

$|00\rangle$

\- $|01\rangle$

$|11\rangle$

$|10\rangle$

$|0\rangle$

$|1\rangle$

Figure 12: Applying the Hadamard gate to the inner qubit before measuring.

Measure 00

$|0\rangle$

$|1\rangle$

Measure 01

$|0\rangle$

\- $|1\rangle$

Measure 10

$|0\rangle$

$|1\rangle$

Measure 11

$|0\rangle$

\- $|1\rangle$

Figure 13: The residual state on Bob’s side for the four possible measurement
outcomes.

The four possible measurements oucomes and corresponding residual state on
Bob’s side are given in Figure 13. Here we can see that in two of the cases
(on top), the proportions are correct, and in the remaining two cases (on the
bottom), the colors need to be flipped. Similarly, we can see that in two of
the cases (on the left), the signs are correct and in the remaining two (on
the right), the signs need to be flipped.

##  4 Conclusions

We have presented new physical devices and a visual representation of qubits
that can be used for teaching or for outreach activities. We have found it
particularly helpful in conveying concepts such as entanglement and
interference, concepts that key to understanding quantum algorithms, but can
be difficult to grasp at an intuitive level, especially for audiences who are
not accustomed to using mathematical formalism.

We have used our devices and representation in the classroom in addition to
the usual mathematical notation and have found that it lends itself
particularly well to walking through quantum circuits to see how and why they
work.

Our representation has its drawbacks, the main one being that in general,
applying linearity in the usual way on the squared-amplitudes vector does not
yield the same result as on the amplitudes. However, by being cautious with
interference, and checking whether a particular operation behaves well on the
states that they are being applied to, it does work in sufficiently many cases
so as to allow us to explain some basic algorithms and protocols.

In forthcoming work, we plan to extend our applications to include more
algorithms as well as nonlocality with the CHSH game. Also in the works is an
interactive interface and a musical representation of qubits so that it can be
more widely accessible. More information on our project can be found on the
website https://qubobs.irif.fr.

## Acknowledgements

We would like to thank the students and participants in our outreach
activities for their feedback. We thank Maris Ozols for references to tensor
diagrams and Serge Massar for references to some of the other representations.
We are grateful to the many colleagues who served as sounding boards
throughout the different versions of this work.

This project was funded by IRIF, by a grant from IDEX Université Paris Cité,
and the ANR grant SAPS-RA-MCS QUBOBS.

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