# A hybrid-qudit representation of digital RGB images

Sreetama Das1,2 Filippo Caruso1,2,3 1Department of Physics and Astronomy,
University of Florence, Via Sansone 1, Sesto Fiorentino, I-50019, Italy
2European Laboratory for Non-Linear Spectroscopy (LENS), University of
Florence, Via Nello Carrara 1, Sesto Fiorentino, I-50019, Italy  3QSTAR and
CNR-INO, Largo Enrico Fermi 2 - 50125 Firenze - Italy

###### Abstract

Quantum image processing is an emerging topic in the field of quantum
information and technology. In this paper, we propose a new quantum image
representation of RGB images, which is an improvement to all the existing
representations in terms of using minimum resource. We use two entangled
quantum registers constituting of total 7 qutrits to encode the color channels
and their intensities. Additionally, we generalize the existing encoding
methods by using both qubits and qutrits to encode the pixel positions of a
rectangular image. This hybrid-qudit approach aligns well with the current
progress of NISQ devices in incorporating higher dimensional quantum systems
than qubits. We then describe the image encoding method using higher-order
qubit-qutrit gates, and demonstrate the decomposition of these gates in terms
of simpler elementary gates. We use the Google Cirq’s quantum simulator to
verify the image retrieval. We show that the complexity of the image encoding
process is linear in the number of pixels. Lastly, we discuss the image
compression and some basic RGB image processing protocols using our
representation.

##  I Introduction

State-of-the-art quantum computation, in both theoretical and experimental
grounds, largely uses two-dimensional quantum systems known as ‘qubits’. The
same applies for the present-day noisy intermediate-scale quantum (NISQ)
computing processors available for simulating real quantum circuits.
Intrinsically, all the quantum systems designed as qubits have more than two
energy levels, though the higher levels are maintained as non-functional. It
is in principle possible to exploit these levels for quantum computation.
Recently there has been a surge of interest to develop theoretical ideas and
experimental tools to realize quantum computation using $d>2$ dimensional
quantum systems, called ‘qudits’ [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16]. This is usually referred to as ‘multi-valued quantum
computation’. It is straightforward to observe that using qudits increases the
storage capacity and reduces the complexity of a quantum circuit, improving
the overall performance of a quantum information protocol. Experimental
platforms, e.g. photonic systems [13, 12], trapped ions [4], continuous spin
systems [11, 3], nuclear magnetic resonance [9, 10], and molecular magnets [2]
has been used to test qudit-based computation. The present literature on
qudits has shed light on to the construction of elementary gates for quantum
circuits where all the qudits have same dimension [1, 17, 18, 19, 20, 7, 21].
However, quantum circuit composed of qudits of different dimensions, often
termed as ‘hybrid-qudit system’, remain relatively unexplored [22, 23, 24].

Quantum computation has found its application in a plethora of technological
fields, quantum image processing being one of them. It is a rapidly developing
topic with potential applications in space science, medicine, automobile
engineering etc. Compared to classical bits, one needs logarithmically less
number of qubits to store the same amount of classical information.
Additionally, quantum properties like superposition and entanglement leads to
exponential quantum speed-up of the image processing algorithms [25, 26]. The
first step of an image processing algorithm is to encode the classical image
as a quantum state. A number of different forms of quantum encoding methods
have been proposed [27, 28, 29, 26, 30, 31, 32]. They differ in their
applicability for binary, greyscale or color images, the amount of resources
consumed, and the efficiency of the image retrieval process. For binary and
greyscale images, Flexible representation of quantum images (FRQI) [27] and
Novel-enhanced quantum representation of images (NEQR) [29] are the most
commonly used encoding methods in the current literature. Compared to FRQI,
NEQR employs higher number of qubits to encode the greyscale pixel values and
has a more accurate image retrieval process, which we will discuss in the
upcoming sections. For color images, specifically those having an RGB color
channel format, several quantum representations inspired from FRQI and NEQR
have been proposed. To mention some of them, [28] extends the FRQI method to
encode the color channel intensities using three qubits, though it suffers the
same drawback as the FRQI for image retrieval. On the other hand, the NEQR-
based color image encoding methods [30, 32] use 24 qubits to encode the
colors, and have accurate image retrieval. In [31], the authors show that the
number of necessary qubits can be reduced to 10 for encoding the colors, while
retaining the accurate image retrieval.

In this work, we aim to reduce the number of quantum units required to encode
an RGB image, while adhering to the accurate image retrieval process. As it
has been shown in [33], following the same approach as NEQR, and replacing the
qubits with qutrits, only 6 qutrits are required to encode the greyscale
values varying from 0 to 256. This means that an RGB image encoding should
take 16 qutrits, which is still pretty large. As we show here, it is possible
to encode an RGB image using only 7 qutrits. Using higher than three
dimensional qudits can logarithmically reduce the required resources.
Interestingly, in [34], it is argued that systems with three basis states
indeed construct the most cost-effective circuit compared to $n\neq 3$
dimensional systems. In the IBMQ platform, one can excite the third energy
level of a superconducting qubit and convert it to a qutrit. The Google Cirq
quantum computing platform [35] provides quantum simulation using qudits and
corresponding gates. Rigetti’s quantum computation supports the activation,
gate implementation and measurement on a real qutrit [36]. All the above imply
that the qutrit-based quantum computing is imminent, and these motivate us to
use qutrits in quantum representation of images.

Our encoding scheme also explores the possibility of using both qubits and
qutrits to encode the pixel position and color information respectively, thus
giving rise to a hybrid qubit-qutrit circuit. We call this a Hybrid Qudit
Quantum Representation (HQDQR) of RGB images. The elementary gates of a
hybrid-qudit system has been discussed in [22, 23]. In this manuscript, we
show the decomposition of higher-order hybrid qudit gates in terms of
elementary single-qudit and hybrid two-qudit gates. This in turn shows that,
the number of auxiliary qudits and elementary gates needed for our proposed
image encoding is much less compared to the other existing RGB image
representations. Thus, HQDQR uses minimum number of quantum units as well as
minimum number of gates to encode an RGB image. The complexity of the image
encoding process is linear in the number of pixels.

The paper is organized as follows. In section I we revisit some basic gates of
qutrit and hybrid qubit-qutrit quantum circuit, and show their decomposition
in terms of elementary single-qudit and two-qudit gates. In Sec. II we briefly
discuss the existing RGB image representations. Following that, we introduce
our hybrid-qudit representation in Sec. III, calculate the complexity of the
image encoding process and discuss the image compression. In Sec. IV, we
present some basic RGB image operations using our image representation.
Finally, we conclude in Sec. V.

sectionQuantum circuit using qudits

The gates used in a quantum circuit apply an unitary transformation on the
input state. It has been proved that any unitary transformation on a number of
qubits can be asymptotically achieved by repeatedly applying a set of single
and two-qubit elementary gates on those qubits, allowing a certain amount of
error. This set of elementary gates are called universal gates for qubits. The
idea of universality can be extended to qudits. In fact, a number of works
have proposed the universal gates in a qudit circuit [1, 17, 19, 7]. However,
the mathematical idea of universality may not always be suitable to apply for
the realization of a complex quantum circuit. In practice, the gates used to
achieve a unitary for any number of qudits, should be easy to realize
experimentally, so that it can be used in real quantum hardware. There has
been a number of experimental proposals to realize basic qudit gates in
laboratory.

All the above works assume that the all the qudits in a circuit has same
dimension $d$. In principal, it is possible to build a circuit with different
dimensional qudits. A handful of works discuss the basic gates in a hybrid
qudit circuit, and quantum computation using such systems [22, 23].

In the following subsections, we introduce the elementary gates of a qutrit
circuit and a hybrid qubit-qutrit circuit, which will be relevant for our work
discussed in this paper.

###  I.1 Quantum computation in qutrit systems

The Hilbert space of a three-level quantum system or “qutrit” is spanned by
the orthogonal basis vectors $\\{|0\rangle,|1\rangle,|2\rangle\\}$. The
elementary and universal qutrit gates, and the possibility of their physical
realization, have been investigated [1, 37, 17, 19, 38, 39, 40].

####  I.1.1 Single qutrit gates

Ternary bit-flip gates: In analogy to the qubit bit-flip or $X$ gate, the
ternary $X$ gate flips the basis states of a qutrit.

There are six ternary $X$ gates as listed below [37, 33],

$\displaystyle\sigma^{x}_{+0}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\hskip 8.5359pt\sigma^{x}_{+1}=\begin{pmatrix}0&0&1\\\
1&0&0\\\ 0&1&0\end{pmatrix},$ (1)
$\displaystyle\sigma^{x}_{+2}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{pmatrix},\hskip 8.5359pt\sigma^{x}_{01}=\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&0&1\end{pmatrix},$
$\displaystyle\sigma^{x}_{12}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\hskip 8.5359pt\sigma^{x}_{02}=\begin{pmatrix}0&0&1\\\
0&1&0\\\ 1&0&0\end{pmatrix}.$

The first one is identity matrix which of course does not change anything. The
operators $\\{\sigma^{x}_{+1},\sigma^{x}_{+2}\\}$ transform the basis
$|x\rangle$ by $|x\rangle\rightarrow|(x+1)\mod 3\rangle$ and
$|x\rangle\rightarrow|(x+2)\mod 3\rangle$ respectively, and work on all the
basis states simultaneously. Lastly,
$\\{\sigma^{x}_{01},\sigma^{x}_{12},\sigma^{x}_{02}\\}$ swaps the two basis
states $\\{|i\rangle,|j\rangle\\}$ in the subscript of $\sigma^{x}_{ij}$,
while leaving the third basis state unchanged. Following earlier works, we
will use a simple notation for the above gates when using them in our circuit
diagrams. The abbreviated form is shown in Fig. 1(a).

Ternary Hadamard gate: The Hadamard gate $H_{2}$ in a two-dimensional Hilbert
space $\mathcal{H}_{2}$ is the quantum Fourier transform from the
computational basis to the eigenbasis of Pauli matrix $\sigma^{x}$. In an
analogous way, the ternary Hadamard gate $H_{3}$ in the Hilbert space
$\mathcal{H}_{3}$ can be defined as the following,

$H_{3}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&1&1\\\
1&e^{i\frac{2\pi}{3}}&e^{-i\frac{2\pi}{3}}\\\
1&e^{-i\frac{2\pi}{3}}&e^{i\frac{2\pi}{3}}\end{pmatrix}.$ (2)

An experimental realization of this gate has been possible [38] using
superconducting qutrits.

####  I.1.2 Two-qutrit gates

Ternary controlled $X$ gates: A binary controlled $X$ gate is a two-qubit gate
such that the if the first qubit is in state $|1\rangle$, the second qubit
undergoes the bit-flip operation $X$. If the first qubit is in state
$|0\rangle$, nothing changes. The above is the standard notion, though a
controlled $X$ gate can be configured such that the target qubit is flipped
only when the control qubit is in state $|0\rangle$. In a qutrit system, there
are 18 different generalized controlled $X$ operations (three possible control
states and six target flips for each of them) possible. Each of them is a
$9\times 9$ unitary matrix. For example, if we want to flip the target qutrit
state from $|0\rangle$ to $|1\rangle$ when the control qutrit state is
$|2\rangle$, the corresponding unitary will be,

$\displaystyle U=(|0\rangle\langle 0|+|1\rangle\langle
1|)\otimes\mathbb{I}_{3}+|2\rangle\langle 2|\otimes\sigma^{x}_{01}$ (3)
$\displaystyle=\begin{pmatrix}1&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&1\\\ \end{pmatrix},$

where $\mathbb{I}_{d}$ is a $d$-dimensional identity operator. The rest 17
unitaries can be constructed in a similar way.

###  I.2 Quantum computation in hybrid qubit-qutrit systems

In a hybrid quantum system constituted of both qubits and qutrits, the single
qudit gates remain unchanged. However, there can exist two or multi-qudit
gates which act on a system of qubits and qutrits.

####  I.2.1 Two-qudit hybrid gates

Hybrid controlled $X$ gate: For this class of gates, the control can be a
qubit and the target can be a qutrit, or the vice versa. Suppose, if the
control qubit is in state $|1\rangle$, $\sigma^{x}_{12}$ is applied on the
target qutrit. The corresponding unitary is,

$\displaystyle U=|0\rangle\langle 0|\otimes\mathbb{I}_{3}+|1\rangle\langle
1|\otimes\sigma^{x}_{12}$ $\displaystyle=\begin{pmatrix}1&0&0&0&0&0\\\
0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&1&0\\\
\end{pmatrix}.$ (4)

On the other hand, if the control is on the qutrit and the target is a qubit,
and the qubit state flips when the control qutrit is in state $|2\rangle$,
then the unitary can be expressed as,

$\displaystyle U=(|0\rangle\langle 0|+|1\rangle\langle
1|)\otimes\mathbb{I}_{2}+|2\rangle\langle 2|\otimes\sigma^{x}$
$\displaystyle=\begin{pmatrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&1&0\\\ \end{pmatrix},$ (5)

where $\sigma^{x}$ is the qubit $X$ gate. This unitary is same as that in Eq.
4, but as we will see in the upcoming subsections, this may not always be the
case.

Figure 1: (a) The notation we use for different controlled X gates in our
circuit diagrams, where $i\in\\{+0,+1,+2,01,12,02\\}$. (b) Decomposition of a
three-qudit hybrid Toffoli gate. (Left) A hybrid Toffoli gate where the
control is on two qubits $\\{B_{1},B_{2}\\}$ and the target is a qutrit
$T_{1}$. (Right) The decomposition of the gate using an auxiliary qutrit
$A_{1}$ and hybrid controlled $X$ gates. (c) Decomposition of a four-qudit
hybrid Toffoli gate. (Left) A hybrid Toffoli gate where the control is on
$\\{B_{1},B_{2}\\}$ and $T_{1}$, and the target qutrit is $T_{2}$. (Right) The
decomposition of the gate using two auxiliary qutrits $\\{A_{1},A_{2}\\}$ and
hybrid controlled $X$ gates. In both figures, $``i"$ denotes an arbitrary
qutrit $X$ operation.

####  I.2.2 Multi-qudit hybrid gates:

Hybrid Toffoli gates: Now suppose that there are one control qubit and one
control qutrit, while the target is a qutrit. If the control qudits are
respectively in state $|1\rangle$ and $|2\rangle$, the target qutrit undergoes
the bit-flip operation $\sigma^{x}_{12}$. This is analogous to the Toffoli
gate or controlled controlled X gate for qubits, and we will call it hybrid
Toffoli gate in this paper. It can be decomposed using simpler single-qudit
and hybrid two-qudit gates. We show such a decomposition in Fig. 1(b), where
there are two control qubits and a target qutrit. The decomposition uses one
auxiliary qutrit. If there are $n$ control qudits, we will need $(n-1)$
auxiliary qutrits. The last two hybrid gates are applied on the auxiliary
qutrit to bring it back in the initial state $|0\rangle$, so that it can be
used for the decomposition of another gate. In Fig. 1(c), we show the
decomposition of a higher order hybrid Toffoli gate in which the control is on
two qubits and a qutrit, and the target is a qutrit.

######  Theorem 1.

The complexity of a hybrid Toffoli gate with $n$ number of control qubits and
qutrits is $4n-3$.

###### Proof.

There are $n$ controlled $X$ gates from $n$ controls qudits to the auxiliary
qutrits, and total $n-2$ controlled $X$ gates among all the adjacent pairs of
auxiliary qutrits. The same number of gates are applied to bring back the
auxiliary qutrits in state $|0\rangle$. There is one controlled $X$ gate
between the last auxiliary qutrit and the target qudit. So, the total number
of gates used is $2(n+(n-2))+1=4n-3$. ∎

Generalized hybrid Toffoli gates: For the case discussed above, we assumed
that for a control qubit and a control qutrit, the Toffoli gate is activated
when the control states are respectively $|1\rangle$ and $|2\rangle$, i.e.
when the control qudits are in their highest state. In principal, the gates
can be designed to be activated for any of the control states
$\\{|0\rangle,|1\rangle,...,|d-1\rangle\\}$ of a qudit. We call such gates
generalized hybrid Toffoli gate. The decomposition of a generalized hybrid
Toffoli gate is shown in Fig. 2, where $2k$ single qudit gates, $k$ being the
number of controls with generalized bit-values, and a higher order hybrid
Toffoli gate have been used. The later can be further decomposed in terms of
the hybrid two-qudit gates using auxiliary qutrits as shown in Fig. 1(c).

######  Lemma 0.1.

The complexity of a generalized higher order hybrid Toffoli gate with $n$
control qubits and qutrits is $4n-3+2k$. The maximum value is $6n-3$, reached
when $k=n$.

Figure 2: Decomposition of a higher order generalized Toffoli gate. (Left) The
control qudits are two qubits $\\{B_{1},B_{2}\\}$ and a qutrit $T_{1}$, and
the target is a qutrit $T_{2}$. The target qutrit undergoes $\sigma_{i}^{x}$
when the control qudits are in states $|b_{1}\rangle$, $|b_{2}\rangle$ and
$|t_{1}\rangle$ respectively. (Right) The decomposition of this gate in terms
of $\sigma^{x}_{i}$s and four-qudit hybrid Toffoli gate.

Figure 3: The improved gate decomposition using effective qutrits. (a) The
decomposition of the circuit in Fig. 1(b) where the qubit $B_{2}$ becomes
effective qutrit $T_{B_{2}}$. (b) The decomposition of the circuit in Fig.
1(c), where now one auxiliary qubit $A_{1}$ is required.

In [41], the authors demonstrate that the higher order Toffoli gates for
qubits can be synthesized without any auxiliary quantum system, if the third
energy levels of these qubits are activated. This extra energy level is used
only in the intermediate steps to store information, while the input and the
output of the circuit remains in the Hilbert space of a qubit. The authors
also show that such a circuit has depth $\log N$ for synthesizing a higher
order Toffoli gate acting on $N$ qubits. Clearly, the decomposition of a
hybrid qubit-qutrit gate can facilitate from a lower circuit depth and less
number of elementary gates by using effective qutrits in place of qubits. Such
a decomposition is shown in Fig. 3.

######  Theorem 2.

The complexity of a hybrid Toffoli gate with $n_{1}$ control qubits ( or,
$n_{1}-1$ effective qutrits ) and $n_{2}$ control qutrits is $2n+2n_{2}-1$,
where $n=n_{1}+n_{2}$.

###### Proof.

As evident from Fig. 3(b), the number of auxiliary qutrits needed in this case
is $n_{2}$. There are $n_{1}-1$ controlled $X$ gates between $n_{1}$ control
qubits, and 2 controlled $X$ gates for each of the $n_{2}$ qutrit-auxiliary
qutrit pair. Thus, the total number of gates is
$2(n_{1}-1+2n_{2})+1=2n+2n_{2}-1$. ∎

The improvement in the number of gates by using effective qutrits is
$4n-3-(2n+2n_{2}-1)=2n_{1}-2$.

######  Lemma 0.2.

The complexity of a generalized higher order hybrid Toffoli gate with $n_{1}$
control qubits ($n_{1}-1$ effective qutrits) and $n_{2}$ control qutrits is
$2n+2n_{2}-2+2k$.

##  II Quantum image representation

Equipped with the necessary tools to design hybrid qubit-qutrit circuit, in
this section, we describe our proposed RGB image representation. In the
beginning, we briefly discuss some of the existing RGB image representations,
to make a clear comparison of our work with the existing approaches.

###  II.1 Multi-channel representation for images on quantum computers (MCQI)

In this representation [28], a $2^{n}\times 2^{n}$ dimensional RGB image is
encoded using $2n+3$ qubits. The first 3 qubits encode the intensities of the
three channels $\\{\mathrm{R,G,B}\\}$ and the rest $2n$ qubits encode the
positions. The intensities of the three channels are encoded using the angles
$\theta_{i}$ ($i=R,G,B$). The quantum state corresponding to the image is the
following,

$\displaystyle|I\rangle=$
$\displaystyle\frac{1}{2^{n}}\sum\limits_{Y=0}^{2^{n}-1}\sum\limits_{X=0}^{2^{n}-1}(\cos\theta^{i}_{R}|000\rangle+\cos\theta^{i}_{G}|001\rangle+\cos\theta^{i}_{B}|010\rangle$
(6)
$\displaystyle+\sin\theta^{i}_{R}|100\rangle+\cos\theta^{i}_{G}|101\rangle+\cos\theta^{i}_{B}|110\rangle$
$\displaystyle+\cos 0|011\rangle+\sin 0|111\rangle)\times|YX\rangle,$

where $|X\rangle$ and $|Y\rangle$ are the $2^{n}$ basis states of an $n$-qubit
system. The 3-qubit register has 8 basis states, from which six are used to
encode the color information, and the coefficients corresponding to
$|011\rangle$ and $|111\rangle$ are set as constants $\sin 0$ and $\cos 0$, so
that they do not carry any information. To retrieve the images, one has to
perform repeated measurements on all the three color qubits to
probabilistically obtain the coefficients $\cos\theta_{i}$ and
$\sin\theta_{i}$. Because of the probabilistic nature of the output, such an
image retrieval is not accurate, and large number of measurements are required
to reach a particular accuracy.

###  II.2 Novel quantum representation of color digital images (NCQI)

This encoding method [30] is inspired from NEQR, where the intensities of each
channel is encoded using the basis vectors of 8 qubits. So, in total there are
24 qubits needed to encode the color of a pixel. For a $2^{n}\times 2^{n}$
dimensional image, the positions are encoded using $2n$ qubits. So, the total
number of qubits in $2n+24$. The quantum state corresponding to a $2\times 2$
image looks like the following,

$|I\rangle=\frac{1}{2^{n}}\sum\limits_{Y=0}^{2^{n}-1}\sum\limits_{X=0}^{2^{n}-1}|R_{XY}\rangle|G_{XY}\rangle|B_{XY}\rangle|YX\rangle,$
(7)

where $|C_{XY}\rangle=|C^{q-1}_{XY}...C^{1}_{XY}C^{0}_{XY}\rangle$,
$C=\\{R,G,B\\}$ and $q=8$. A projective measurement on the position and the
color qubits can accurately retrieve the pixel positions and corresponding
pixel colors and intensities. However, one should note that while using the
real quantum processors for retrieving the images, one still has to perform a
finite number of repeated measurements to obtain all the basis vectors
corresponding to the intensities.

The time complexity of preparing quantum images using NCQI is quadratically
less compared to that using MCQI. This advantage is similar to the advantage
of using NEQR over FRQI. Also, NCQI allows for more complex color
transformations, and solves the problem of probabilistic retrieval of pixel
values. A limitation of both the above representations is that they consider
only square images. To generalize this for rectangular images, an improved
encoding method has been proposed [32].

###  II.3 Optimized quantum representation of color images

In this encoding method, the color information is stored using two entangled
quantum registers. The first register with two qubits encodes the channel
information, while the second register with eight qubits encodes the intensity
of that channel. The pixel positions are again encoded using $2n$ qubits for a
$2^{n}\times 2^{n}$ classical image. The quantum image state is,

$\displaystyle|I\rangle=\frac{1}{2^{n}}\Big{(}|R_{XY}\rangle|00\rangle+|G_{XY}\rangle|01\rangle+|B_{XY}\rangle|10\rangle$
$\displaystyle+|S_{XY}\rangle|11\rangle\Big{)}\otimes|YX\rangle,$ (8)

where $|C_{XY}\rangle$ $(C=R,G,B)$ is the same as defined for NCQI. This
representation uses only 10 qubits to encode the colors, which is a
significant improvement over NCQI.

Figure 4: A $3\times 2$ dimensional RGB image and its corresponding quantum
image state $|I\rangle$ represented using HQDQR.

Figure 5: (a) The circuit for encoding the quantum image state $|I\rangle$ in
Fig. 4. The register $Q_{1}$ constituted of one qubit $B_{1}$ and one qutrit
$T_{1}$ encodes the pixel positions. The qutrit $Q_{2}$ encodes color
channels, and the register $Q_{3}$ with six qutrits $t_{i}$ ($i=0,1,..,5$)
encodes the intensity of the color channels. The six blocks with thin outlines
stand for encoding steps for six pixel positions. (b) The probabilities of the
outcomes of measurements on all the registers, when the circuit of (a) is
simulated using Google Cirq qudit simulator. The data is collected for 5000
instances of measurements. The $x$-axis shows the basis states in the same
order as in $|I\rangle$ in Fig. 4. (c) The same circuit simulated by
decomposing complex gates into single- and two-qudit gates, and adding
depolarizing noise to all the gates. The order of magnitude of the noise
strength has to be $\leq 10^{-4}$ for a successful image retrieval.

##  III Hybrid-qudit representation of color images (HQDQR)

In [33], the authors proposed a novel qutrit representation of greyscale
images, in which both the pixel positions and pixel values are encoded using
qutrits. Compared to 8 qubits in NEQR, it needs 6 qutrits to encode the 256
shades of grey. Since $3^{6}>256$, a number of the energy levels remain
redundant, which can be used for error correction. In this work, we show that
a total of 7 qutrits are required to encode the color information of an RGB
image. The color channel $\\{\mathrm{R,G,B}\\}$ can be encoded using the three
levels of a qutrit. With this qutrit, a register of 6 qutrits is entangled,
which encodes the information about the intensity of each color. Additionally,
we consider rectangular images instead of square ones, for which the pixel
positions can be encoded using quantum registers constituted of only qubits or
qutrits, or both where it is appropriate.

First, we consider the most general case of an $M\times N$ dimensional
classical image such that $M=3^{m}$ and $N=2^{n}$. The initial quantum state
of all the three registers is,

$|\Psi_{0}\rangle=\underbrace{|000000\rangle}_{\text{6
qutrits}}\otimes\underbrace{|0\rangle}_{\text{qutrit}}\otimes\underbrace{|000...0\rangle}_{\text{m+n
qudits}}$ (9)

Now, Hadamard operator $H_{3}^{\otimes m}\otimes H_{2}^{\otimes n}$ is applied
on the qudits of the last register to transform the state of this register to
a fully superposed state.

$\displaystyle|\Psi_{1}\rangle=\frac{1}{\sqrt{2^{n}3^{m}}}\sum\limits_{Y=0}^{2^{n}}\sum\limits_{X=0}^{3^{m}}|000000\rangle\otimes|0\rangle\otimes|YX\rangle.$
(10)

Now, $\mathcal{H}_{3}$ is applied on the second register to prepare the
following state,

$|\Psi_{2}\rangle=\frac{1}{\sqrt{2^{n}3^{m+1}}}\sum\limits_{Y=0}^{2^{n}}\sum\limits_{X=0}^{3^{m}}|000000\rangle\otimes(|0\rangle+|1\rangle+|2\rangle)\otimes|YX\rangle.$
(11)

We assume that the states $|0\rangle$, $|1\rangle$ and $|2\rangle$ represent
respectively the color channels R, G and B. In the next step, for each pixel
position $|YX\rangle$ and for each color channel, a number of controlled $X$
operations is used to flip the qutrits in the first register, where the
control lies on the qudits in the first and second register. We can express
this operation as,

$\displaystyle|\Psi_{3}\rangle=\frac{1}{\sqrt{2^{n}3^{m+1}}}\sum\limits_{Y=0}^{3^{m}}\sum\limits_{X=0}^{2^{n}}\Omega_{XY}\big{(}|000000\rangle\otimes(|0\rangle+|1\rangle$
$\displaystyle+|2\rangle)\otimes|YX\rangle\big{)}$
$\displaystyle=\frac{1}{\sqrt{2^{n}3^{m+1}}}\sum\limits_{Y=0}^{3^{m}}\sum\limits_{X=0}^{2^{n}}(|R_{XY}\rangle|0\rangle+|G_{XY}\rangle|1\rangle+|B_{XY}\rangle|2\rangle)$
$\displaystyle\otimes|YX\rangle,$ (12)

where $\Omega_{XY}=\bigotimes\limits_{i=0}^{5}\Omega_{XY}^{i,C}$,
$|C_{XY}\rangle=|C^{5}_{XY}C^{4}_{XY}..C^{0}_{XY}\rangle$, $C=\\{R,G,B\\}$,
and each $\Omega_{XY}^{i,C}$ is a higher order generalized hybrid Toffoli gate
flipping the $i^{\mathrm{th}}$ qubit of the color channel $C$, in position
$|YX\rangle$. Eq. 12 is our desired quantum representation. As an example, we
consider a $3\times 2$ dimensional RGB image shown in Fig. 4 and its
corresponding quantum image state $|I\rangle$.

The different steps for encoding this image are shown in Fig. 5(a), where each
block surrounded by thin rectangular border stands for one particular pixel
position. Thus, by the full use of quantum superposition and entanglement, in
place of 24 qubits needed to store the RGB color information, we need only 7
qutrits, when enough auxiliary qutrits are available.

To test the above image encoding and retrieval in a real quantum system, we
use Google Cirq’s quantum simulator, which provides an architecture of hybrid
qudits and corresponding gates. The simulator is a classical simulator which
mimics the behaviour of a quantum computer. We build the circuit of Fig. 5(a),
and finally measure all the registers to retrieve the image state. The result
is presented in Fig. 5(b). The data is collected over 5000 shots of
measurements. The probability amplitudes fluctuate around the expected value
$\big{(}\frac{1}{3\sqrt{2}}\big{)}^{2}\approx 0.577$. This fluctuation may
arise because of the random number generator used in the measurement
constructor in Cirq. The state where all the registers are in state
$|0\rangle$ does not correspond to the original image, though it appears in
the measurement outcome with negligible probability.

Next, we test the performance of the image encoding and retrieval under the
presence of noise. For this, we decompose each higher order gate into simpler
two-qudit gates. All the single- and two-qudit gates in the resulting circuit
are subject to depolarizing noise. The channels are constructed in Cirq using
the Kraus operators for single-qudit and two-qudit depolarizing channel, and
they are applied after each gate in the circuit. The measurement outcome of
this noisy circuit is presented in Fig. 5(c), for which the noise strength
corresponding to all the single- and two-qudit channels is $\lambda=10^{-4}$.
The 18 peaks corresponds to the 18 outcomes obtained in Fig. 5(b), and the
other smaller peaks appear due to the noise. We observe that increasing the
single-qudit noise strength by one or two orders of magnitude has little
effect on the measurement outcome, while increasing the two-qudit noise
strength to $10^{-3}$ can completely randomize the pattern in Fig. 5(c),
making the image retrieval process highly inefficient. Though the current
superconducting qubit based quantum computers typically have single qubit and
two-qubit gate errors of the order of $10^{-3}$, tracking the progress of
quantum hardware with time implies that in the near future the gate errors
will improve to the order of $10^{-4}$ [42, 41]. The image retrieval thus
remains feasible in the coming age quantum computers.

###  III.1 Special cases

Above we considered both qubits and qutrits to encode the pixel positions of a
rectangular image. However, depending on the dimension of the image, it might
be optimum to use only qubits or only qutrits to encode the positions, to
minimize the redundancy in the number of energy levels. In this following
subsection, we discuss the quantum representation for RGB images in such
cases.

####  III.1.1 All-qubit third register

Suppose the dimension of an image is $M\times N$ such that $M=2^{m}$ and
$N=2^{n}$. The quantum image state in this case becomes,

$\displaystyle|I\rangle=\frac{1}{\sqrt{2^{m+n}}}\sum\limits_{Y=0}^{2^{m}}\sum\limits_{X=0}^{2^{n}}(|R_{XY}\rangle|0\rangle+|G_{XY}\rangle|1\rangle$
$\displaystyle+|B_{XY}\rangle|2\rangle)\otimes|YX\rangle.$ (13)

The generalized higher order Toffoli gates now has only qubits as the control,
and a qutrit as the target. It is possible to decompose such gates using
auxiliary qutrits as shown in Fig. 1, and using effective qutrits as in Fig.
3. One can also use auxiliary qubits, the decomposition then can be obtained
in terms of Toffoli gates and hybrid Toffoli gates, as shown in Fig. 6. Of
course, one can use an extra auxiliary qubit to break the Toffoli gate in Fig.
6 into controlled-$X$ gates. We choose to use Toffoli gate since it is already
known that a higher order Toffoli gate with $p$ ($p>2$) controls, can be
decomposed into $4p-8$ Toffoli gates, when $p-2$ auxiliary qubits are present
[43].

Figure 6: Decomposition of a higher order hybrid Toffoli gate where the
control qudits $\\{B_{1},B_{2},B_{3}\\}$ are all qubits. The decomposition is
shown in terms of auxiliary qubit $A^{b}_{1}$, Toffoli gate, and hybrid
Toffoli gate.

####  III.1.2 All-qutrit third register

If the image dimension is such that $M=3^{m}$ and $N=3^{n}$, the quantum image
state becomes,

$\displaystyle|I\rangle=\frac{1}{\sqrt{3^{m+n}}}\sum\limits_{Y=0}^{3^{m}}\sum\limits_{X=0}^{3^{n}}(|R_{XY}\rangle|0\rangle+|G_{XY}\rangle|1\rangle$
$\displaystyle+|B_{XY}\rangle|2\rangle)\otimes|YX\rangle.$ (14)

The generalized higher order Toffoli gates acts only on qutrits in this case,
both as control and target. We do not show the decomposition of these gates,
because it can be obtained in the same way as shown in Fig. 1 and Fig. 2.

Representation | Elementary gates used
---|---
MCQI | $24\times 2^{4n}-9\times 2^{2n}+2n+2$
NCQI | $2n+24\times 2^{2n}\times 48(n-1)$
OCQR | $2n+2+24\times 2^{2n}\times 48n$
HQDQR | $m+n+1+18(2^{n}\times 3^{m})(6(m+n)+3)$
Table 1: The number of elementary gates (one and two-qudit gates) used for
different RGB image encoding methods. For the first three rows the number of
pixels in the image is $2^{2n}$, and for the last row the same is $2^{n}\times
3^{m}$.

###  III.2 Complexity of the encoding method

Here, we analyze the complexity of the quantum image encoding method in two
steps.

  1. 1.

There is total $m+n+1$ single qudit gates applied in the first step to encode
the pixel positions and color channels, the complexity of which is
$\mathcal{O}(m+n+1)$.

  2. 2.

There are $2^{n}\times 3^{m}$ pixels, each pixel having three channels. For
each channel, one needs to flip maximum 6 qutrits to encode the intensity. So
the maximum number of generalized higher order hybrid Toffoli gates required
for each channel is 6. Now, a generalized higher order hybrid Toffoli gate
with $m+n+1$ control qudits can be decomposed using maximum of $(6(m+n+1)-3)$
elementary gates. So the total complexity of this step is no more than
$(18(2^{n}\times 3^{m})(6(m+n+1)-3)$ which is $\mathcal{O}(N)$, i.e. linear in
the number of pixels.

So, the total complexity of the image encoding process is $\mathcal{O}(N)$. It
is to be noted that, though the NCQI and OCQR encoding method discussed
earlier has similar order of complexity, the number of of elementary gates
used for encoding is much less for HQDQR. A comparison between different RGB
image representations and the number of elementary gates used has been
presented in Table 1. For HQDQR, if we assume that the $3^{m}=2^{n}$, the
upper bound on the number of elementary gates is $2n+1+18\times
2^{2n}\times(12n+3)$, which is still much less than the number of gates for
all of the three other representations in Table 1.

Figure 7: (a) A $2\times 3$ dimensional greyscale image with pixel values
indicated on each pixel. (b)-(c) The six bitplanes of this image.

###  III.3 Image compression

The complexity of the encoding method, being linear in the number of pixels,
becomes significantly high for high-resolution images. It is possible to
drastically reduce the numbers of necessary gates for the encoding by using
the minimization of logical functions. Here, the logical functions correspond
to the bit values of pixel positions. This is a classical procedure, which has
been previously discussed in [27, 29, 33] for images encoded using only qubits
or qutrits. For our proposed representation, however, both qubits and qutrits
are used to encode the pixel positions. Hence, for the best-achieved
compression, we must employ an algebra applicable on a combination of binary
and ternary logical variables. In the following, we briefly explain the idea
of image compression for a simple exemplary image.

To start with, let us consider a $2\times 3$ greyscale image as shown in Fig.
7(a). In Fig. 7(b)-(g) we show the 6 bitplanes corresponding to 6 qutrits in
the first register, i.e. the $i^{\mathrm{th}}$ bitplane shows the state of the
$i^{\mathrm{th}}$ qutrit for all the pixel positions. Let us denote the
ternary states $\\{0,1,2\\}$ respectively by $\\{x_{+},x_{0},x_{-}\\}$, and
the binary states $\\{0,1\\}$ by $\\{x,\overline{x}\\}$. Now we consider Fig.
7(b), in which all the pixels along the $1^{\mathrm{st}}$ column has state
$|1\rangle$ for the first qutrit. The pixel positions along this particular
column are $|00\rangle$, $|10\rangle$ and $|20\rangle$ respectively. It
requires 3 generalized Toffoli gates to set the intensities of these three
pixels. However, if we consider the sum $S$ of the logical expressions of
these pixel positions, and apply the ternary logic algebra [44] as shown
below,

$S=x_{+}x+x_{0}x+x_{-}x=(x_{+}+x_{0}+x_{-})x=x,$ (15)

we see that instead of using three generalized Toffoli gates, we can encode
these pixels by using a single controlled-$X$ gate, where the control is only
on the position qubit being in state $|0\rangle$. Similarly, in the first row
of Fig. 7(b), the pixels positions are $|00\rangle$ and $|01\rangle$. We use
the minimization of the logical expression as following

$S^{\prime}=x_{+}x+x_{+}\overline{x}=x_{+}(x+\overline{x})=x_{+},$ (16)

so that these two pixels can be encoded using one ternary controlled $X$ gate,
where the control is on the position qutrit being in state $|0\rangle$. This
idea of image compression for greyscale intensities is translated into RGB
image by following the same procedure for each of the color channel
intensities. The best achievable compression depends on both the image as well
as the encoding method. For example, in [27] the authors achieve $90.63\%$
compression, whereas in [29] the compression obtained was $97.28\%$, and the
compression obtained using qutrits in [33] was $47.26\%$.

##  IV Basic color image processing

In this section, we demonstrate some common RGB image operation using HQDQR
encoding.

###  IV.1 Channel swapping

The color channel swapping operation $CSO$ is performed to swap the
intensities of any two color channels. For our quantum image representation,
it can be achieved by applying one of qutrit $X$ gates
$\\{\sigma^{x}_{01},\sigma^{x}_{12},\sigma^{x}_{02}\\}$, whichever applies.
For example, to swap the red and green channel, one needs to apply
$\sigma^{x}_{01}$ on the second register.

$\displaystyle
CSO_{RG}|I\rangle=\frac{1}{\sqrt{2^{n}3^{m+1}}}\sum\limits_{Y=0}^{3^{m}}\sum\limits_{X=0}^{2^{n}}(|R_{XY}\rangle|1\rangle$
$\displaystyle+|G_{XY}\rangle|0\rangle+|B_{XY}\rangle|2\rangle)\otimes|YX\rangle$
(17)

For swapping red and blue channel on the other hand, we have to apply
$\sigma^{x}_{02}$. The computational complexity of this operation is
$\mathcal{O}(1)$.

###  IV.2 One channel operation

The one channel operation $OCO$ performs a particular transformation to any
one of the color channel intensities. For example, the red channel can be
transformed by the following,

$\displaystyle
OCO_{R}|I\rangle=\frac{1}{\sqrt{2^{n}3^{m+1}}}\sum\limits_{Y=0}^{3^{m}}\sum\limits_{X=0}^{2^{n}}(|R^{\prime}_{XY}\rangle|0\rangle+|G_{XY}\rangle|1\rangle$
$\displaystyle+|B_{XY}\rangle|2\rangle)\otimes|YX\rangle,$ (18)

where $|R^{\prime}_{XY}\rangle=|R_{XY}^{\prime 5}..R_{XY}^{\prime 0}\rangle$
is the new intensity of the red channel. This is achieved by using a higher
order hybrid Toffoli gate with $m+n+1$ control qudits and a maximum of six
target qutrits.

##  V Conclusion

Quantum image processing is a promising venture towards achieving speed-up in
image processing, which in turn is an indispensable task in a plethora of
everyday applications. Standing in the era of NISQ devices, it is important to
optimize the number of quantum units as well as the depth of a quantum
circuit, in order to minimize the effect of noise in the output. In this work,
we proposed a quantum representation of RGB images, which uses only 7 qutrits
to encode the information about color channels and their intensity. When
compared with the existing encoding methods of RGB images, our representation
uses least number of quantum units to encode the color information. Moreover,
we considered both qubits and qutrits to encode the position information of
the pixels, which can be an optimum choice while encoding a general
rectangular image while keeping the number of redundant energy levels low. The
complexity of our image encoding algorithm is polynomial in the number of
pixels. We showed that our representation can be achieved by using much less
number of elementary gates compared to the existing encoding methods. The
complexity can be further improved by using compression of logical expression
corresponding to the pixel positions. Our representation naturally gives rise
to a hybrid qubit-qutrit circuit. We demonstrate decomposition of higher order
qubit-qutrit gates in terms of simpler single qudit and two-qudit gates in
these systems. The current trend of research on higher dimensional quantum
units and corresponding gates in these systems, indicates that quantum
processing units including qudits will soon become available to users all over
the world [35], and thus our work will be a strong candidate for processing of
RGB images.

## References

  * Muthukrishnan and Stroud [2000] A. Muthukrishnan and C. R. Stroud, Phys. Rev. A 62, 052309 (2000).
  * Leuenberger and Loss [2001] M. N. Leuenberger and D. Loss, Nature 410, 789 (2001).
  * Bartlett  _et al._ [2002] S. D. Bartlett, H. de Guise, and B. C. Sanders, Phys. Rev. A 65, 052316 (2002).
  * Klimov  _et al._ [2003] A. B. Klimov, R. Guzmán, J. C. Retamal, and C. Saavedra, Phys. Rev. A 67, 062313 (2003).
  * Ralph  _et al._ [2007] T. C. Ralph, K. J. Resch, and A. Gilchrist, Phys. Rev. A 75, 022313 (2007).
  * Gottesman [1998] D. Gottesman, in  _Selected Papers from the First NASA International Conference on Quantum Computing and Quantum Communications_ , QCQC ’98 (Springer-Verlag, Berlin, Heidelberg, 1998) p. 302–313.
  * Luo and Wang [2014] M. Luo and X. Wang, Science China Physics, Mechanics & Astronomy 57, 1712 (2014).
  * Luo  _et al._ [2014] M.-X. Luo, X.-B. Chen, Y.-X. Yang, and X. Wang, Scientific Reports 4, 4044 (2014).
  * Dogra  _et al._ [2014] S. Dogra, Arvind, and K. Dorai, Physics Letters A 378, 3452 (2014).
  * Gedik  _et al._ [2015] Z. Gedik, I. A. Silva, B. Çakmak, G. Karpat, E. L. G. Vidoto, D. O. Soares-Pinto, E. R. deAzevedo, and F. F. Fanchini, Scientific Reports 5, 14671 (2015).
  * Adcock  _et al._ [2016] M. R. Adcock, P. HØyer, and B. C. Sanders, Quantum Information Processing 15, 1361–1386 (2016).
  * Lu  _et al._ [2019] H.-H. Lu, Z. Hu, M. S. Alshaykh, A. J. Moore, Y. Wang, P. Imany, A. M. Weiner, and S. Kais, Advanced Quantum Technologies 3 (2019).
  * Gao  _et al._ [2020] X. Gao, M. Erhard, A. Zeilinger, and M. Krenn, Phys. Rev. Lett. 125, 050501 (2020).
  * Wang  _et al._ [2020] Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Frontiers in Physics 8, 10.3389/fphy.2020.589504 (2020).
  * Narvaez [2021] C. E. Q. Narvaez, Universality for sets of three-valued qubit gates (2021).
  * Su  _et al._ [2022] Q.-P. Su, Y. Zhang, L. Bin, and C.-P. Yang, Phys. Rev. A 105, 042434 (2022).
  * Kerntopf  _et al._ [2004] P. Kerntopf, M. Perkowski, and M. Khan, in  _Proceedings. 34th International Symposium on Multiple-Valued Logic_ (2004) pp. 68–73.
  * Bullock  _et al._ [2005] S. S. Bullock, D. P. O’Leary, and G. K. Brennen, Phys. Rev. Lett. 94, 230502 (2005).
  * Di and Wei [2011] Y.-M. Di and H. Wei, arXiv: Quantum Physics (2011).
  * Li  _et al._ [2013] W.-D. Li, Y.-J. Gu, K. Liu, Y.-H. Lee, and Y.-Z. Zhang, Phys. Rev. A 88, 034303 (2013).
  * Mischuck and Mølmer [2013] B. Mischuck and K. Mølmer, Phys. Rev. A 87, 022341 (2013).
  * Daboul  _et al._ [2003] J. Daboul, X. Wang, and B. C. Sanders, Journal of Physics A: Mathematical and General 36, 2525 (2003).
  * Khan and Perkowski [2006] F. S. Khan and M. Perkowski, Theor. Comput. Sci. 367, 336–346 (2006).
  * Dogra  _et al._ [2015] S. Dogra, A. Dorai, and K. Dorai, International Journal of Quantum Information 13, 1550059-394 (2015), arXiv:1503.06624 [quant-ph] .
  * Zhang  _et al._ [2015] Y. Zhang, K. Lu, and Y. Gao, Science China Information Sciences 58, 1 (2015).
  * Yao  _et al._ [2017] X.-W. Yao, H. Wang, Z. Liao, M.-C. Chen, J. Pan, J. Li, K. Zhang, X. Lin, Z. Wang, Z. Luo, W. Zheng, J. Li, M. Zhao, X. Peng, and D. Suter, Phys. Rev. X 7, 031041 (2017).
  * Le  _et al._ [2011] P. Q. Le, F. Dong, and K. Hirota, Quantum Information Processing 10, 63 (2011).
  * Sun  _et al._ [2011] B. Sun, P. Q. Le, A. M. Iliyasu, F. Yan, J. A. Garcia, F. Dong, and K. Hirota, in  _2011 IEEE 7th International Symposium on Intelligent Signal Processing_ (2011) pp. 1–6.
  * Zhang  _et al._ [2013] Y. Zhang, K. Lu, Y. Gao, and M. Wang, Quantum Information Processing 12, 2833 (2013).
  * Sang  _et al._ [2017] J. Sang, S. Wang, and Q. Li, Quantum Information Processing 16, 1–14 (2017).
  * Liu  _et al._ [2018] K. Liu, Y. Zhang, K. Lu, X. Wang, and X. Wang, International Journal of Theoretical Physics 57, 2938 (2018).
  * Su  _et al._ [2021] J. Su, X. Guo, C. Liu, S. Lu, and L. Li, Scientific Reports 11, 13879 (2021).
  * Dong  _et al._ [2022] H. Dong, D. Lu, and C. Li, Quant. Inf. Proc. 21, 108 (2022).
  * Richards [2021] R. Richards,  _Arithmetic Operations in Digital Computers_ (Creative Media Partners, LLC, 2021).
  * Developers [2022] C. Developers, Cirq (2022), See full list of authors on Github: https://github .com/quantumlib/Cirq/graphs/contributors.
  * [36] Rigetti computing, https://www.rigetti.com/.
  * Khan  _et al._ [2003] M. A. Khan, P. Kerntopf, and M. A. Perkowski, in  _2013 IEEE 43rd International Symposium on Multiple-Valued Logic_ (IEEE Computer Society, Los Alamitos, CA, USA, 2003) p. 146.
  * Yurtalan  _et al._ [2020] M. A. Yurtalan, J. Shi, M. Kononenko, A. Lupascu, and S. Ashhab, Phys. Rev. Lett. 125, 180504 (2020).
  * Goss  _et al._ [2022] N. Goss, A. Morvan, B. Marinelli, B. K. Mitchell, L. B. Nguyen, R. K. Nail, L. Chen, C. Jünger, J. M. Kreikebaum, D. I. Santiago, J. J. Wallman, and I. Siddiqi, High-fidelity qutrit entangling gates for superconducting circuits (2022).
  * Luo  _et al._ [2022] K. Luo, W. Huang, Z. Tao, L. Zhang, Y. Zhou, J. Chu, W. Liu, B. Wang, J. Cui, S. Liu, F. Yan, M.-H. Yung, Y. Chen, T. Yan, and D. Yu, Experimental realization of two qutrits gate with tunable coupling in superconducting circuits (2022).
  * Gokhale  _et al._ [2019] P. Gokhale, J. M. Baker, C. Duckering, N. C. Brown, K. R. Brown, and F. T. Chong, in  _Proceedings of the 46th International Symposium on Computer Architecture_, ISCA ’19 (Association for Computing Machinery, New York, NY, USA, 2019) p. 554–566.
  * Girvin [2014] S. M. Girvin, Quantum machines: measurement and control of engineered quantum systems , 113 (2014).
  * Yang  _et al._ [2006] G. Yang, X. Song, W. N. N. Hung, F. Xie, and M. A. Perkowski, in  _Theory and Applications of Models of Computation_ , edited by J.-Y. Cai, S. B. Cooper, and A. Li (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp. 365–374.
  * Vranesic  _et al._ [1970] Z. Vranesic, E. Lee, and K. Smith, IEEE Transactions on Computers C-19, 964 (1970).