††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: Corresponding Author:
<EMAIL_ADDRESS>
# Ultrashort electron wavepackets via frequency-comb synthesis

Matteo Aluffi  Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel,
38000 Grenoble, France  Thomas Vasselon  Université Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France  Seddik Ouacel  Université
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Hermann Edlbauer  Université Grenoble Alpes, CNRS, Grenoble INP, Institut
Néel, 38000 Grenoble, France  Clément Geffroy  Université Grenoble Alpes,
CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France  Preden Roulleau
Université Paris-Saclay, CEA, CNRS, Gif sur Yvette, 91191, France  D.
Christian Glattli  Université Paris-Saclay, CEA, CNRS, Gif sur Yvette, 91191,
France  Giorgos Georgiou  James Watt School of Engineering, Electronics and
Nanoscale Engineering,
University of Glasgow, Glasgow, G12 8QQ, United Kingdom  Christopher Bäuerle
Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble,
France

###### Abstract

Single-electron sources are an essential component of modern quantum
nanoelectronic devices. Owing to their high accuracy and stability, they have
been successfully employed for metrology applications, studying fundamental
matter interactions and more recently for electron quantum optics. They are
traditionally driven by state-of-the-art arbitrary waveform generators that
are capable of producing single-electron pulses in the
sub-$100\text{\,}\mathrm{ps}$ timescale. In this work, we use an alternative
approach for generating ultrashort electron wavepackets. By combining several
harmonics provided by a frequency comb, we synthesise Lorentzian voltage
pulses and then use them to generate electron wavepackets. Through this
technique, we report on the generation and detection of an electron wavepacket
with temporal duration of $27\text{\,}\mathrm{ps}$ generated on top of the
Fermi sea of a 2-dimensional electron gas – the shortest reported to date.
Electron pulses this short enable studies on elusive, ultrafast fundamental
quantum dynamics in nanoelectronic systems and pave the way to implement
flying electron qubits by means of Levitons.

## Introduction

Electronic excitations in solid-state, nanoelectronic systems provide a unique
platform for investigating fundamental matter interactions, such as many-body
interactions, the fractional quantum Hall effect, Fermion statistics as well
as Abelian and non-Abelian statistics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The
ability to control such excitations at the individual electron level opens
exciting perspectives in terms of quantum information processing and real-time
control of topologically protected excitations [4, 10, 11]. In this respect,
the advent of highly efficient and precise single-electron sources was found
to be technologically very important for metrology and more specifically for
the definition of the quantum current standard [12, 13, 14]. In addition to
metrology applications, the quality of these sources has made them very
popular in the field of quantum nanoelectronics. Over the last few years they
have been the driving force for the development of electron quantum optics,
where – in analogy to quantum optics – quantum information is encoded in the
charge or spin degree of freedom of single propagating electrons [15, 16, 17,
18, 19, 20, 21, 22].

There are currently several types of single-electron sources, with the
majority of them being based on isolating electrons in well-defined regions
through electrostatic barriers, also known as electron pumps. These pumps, are
essentially quantum dots that can accommodate single electrons, which are then
pumped into nanoelectronic circuits [23, 24, 25, 26, 18]. The main application
for these quantum-dot-based sources is in the field of metrology, where
current advances report electron pumping accuracy of as high as
$0.92\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ (parts per million) [27]. In
addition to metrology, their use for electron quantum optics is still an open
research topic [17, 18]. Indeed, these quantum-dot-based sources inject
electrons well above the Fermi energy, which makes them fragile and more prone
to decoherence mechanisms [28].

An alternative technique for generating single electrons relies on temporally
short voltage pulses, also known as Levitons. Typically, any voltage signal
that is applied on an electron reservoir can induce the motion of free
carriers (electrons and holes). It was however postulated by Levitov et al.
[29, 30], that voltage pulses with Lorentzian profile enable pure electron
excitation without any holes. The amount of excited electrons can be
controlled by the amplitude of the pulses and can be tuned such that only a
single electron is excited from the reservoir. In contrast to quantum-dot-
based sources, Levitons are excited with an energy slightly above the Fermi
energy and therefore have fundamentally longer coherence lifetime [28]. The
first successfully implemented Leviton source [5] enabled experiments on
electron tomography [31, 32] and time-resolved reconstruction of electron
wavefunctions [33]. In addition, recent theoretical works predict that
Levitons can be used to study entanglement and non-locality in Mach-Zehnder
interferometers [34, 35] or to generate an effective fractional charge e/2
[36]. In the form of ultrashort wavepackets, Levitons are also discussed as
promising approach to generate electron flying qubits [15, 16].

The generation of Levitons has been mainly driven by state-of-the-art
arbitrary waveform generators (AWGs) that have the capacity of generating
single-electron pulses with temporal duration as short as
$42\text{\,}\mathrm{ps}$ [32]. Although using an AWG for generating single-
electron pulses is a powerful technique, one cannot create very short
Lorentzian voltage pulses with sufficient precision due to limited bandwidth
and sampling rate. As demonstrated in a recent tomography experiment [32],
these AWG limitations degrade the purity of single-electron wavepackets
resulting in unwanted formation of holes. With the shortest timescales
provided by state-of-the-art AWGs, access to elusive dynamics of low-
dimensional systems – such as crystallisation and fractionalisation [37],
dynamically control of interference patterns [38] and spectroscopy of quantum-
coherent circuits [39] – is so far not possible.

An alternative to the generation of Levitons through AWGs is to synthesise
Lorentzian-shaped voltage pulses using frequency combs. The discovery of
frequency combs has revolutionised optical spectroscopy and metrology in the
visible and ultraviolet region of the electromagnetic spectrum [40]. Although
very popular in optics, frequency combs have not been utilised extensively for
radio-frequencies (RF).

In this paper, we use RF frequency combs to synthesize ultrashort Lorentzian
voltage pulses that enable stable generation of single-electron Levitons. By
controlling the amplitude and phase of each harmonic of the comb, we show the
capability to finely tune the extent of the charge excitation enabling Leviton
formation and – more importantly – the precise correction of dispersion from
the coaxial line bringing the voltage pulse to the sample. Additionally, we
demonstrate that the temporal duration of Levitons is tuneable from the
continuous limit (sinusoidal waves) down to electron wavepackets with
$20\text{\,}\mathrm{ps}$ duration. Finally, we inject the output from our
frequency-comb generate into a quantum nanoelectronic device and perform a
time-resolved measurement of synthesized Lorentzian voltage pulses in-situ on
a cryogenic setup. This work sets strong foundations for exploring time-
resolved ultrafast electron dynamics in solid-state systems and opens
possibilities for a competitive electron flying qubit technology.

## Arbitrary voltage pulse generation

Figure 1:  Principle of Fourier synthesis. (a) and (b) The principle of
Fourier synthesis explained through a Lorentzian (blue) and square (orange)
voltage pulses. A periodic pulse (a) in the time domain can be decomposed into
a series of harmonics in the frequency domain (b), whose amplitude and phase
(not shown here) varies as a function of frequency. Figure 2:  The arbitrary
voltage pulse synthesiser block diagram. A 3 GHz fundamental frequency is fed
through a commercially available frequency comb, which generates higher
harmonics. Each harmonic is then selected by a band-pass filter (BPF),
attenuated by a voltage-controlled attenuator (VCA) and delayed by the
voltage-controlled phase-shifter (VCPS). The output is measured in real-time
with a sampling oscilloscope, which provides feedback to a homemade control
software to automatically tune the VCAs and VCPSs using a programmable digital
to analog converter (DAC).

Any periodic signal can be represented by a Fourier series, which is a linear
superposition of orthonormal functions, such as sinus waves. As shown in
Figure 1 (a) and (b), a square pulse (orange curve) is for instance described
by a series of odd harmonics with a 1/n decaying amplitude as a function of
frequency. In the same manner, a Lorentzian pulse (blue curve) can be
decomposed into a series of harmonics whose amplitude falls exponentially. The
frequency-comb synthesizer that we demonstrate in this work is now a precise
experimental implementation of this mathematical concept.

Figure 3:  Voltage pulses generated by the frequency comb synthesiser. The
black solid line corresponds to the pulse measured with a fast oscilloscope,
and the blue line to the theoretical voltage pulse fed into the optimisation
algorithm. (a) and (b) are Lorentzian voltage pulses with duration
$80\text{\,}\mathrm{ps}$ and $50\text{\,}\mathrm{ps}$, respectively. (c)
Impulse response of the synthesiser, where the input to the optimisation
algorithm is a delta-like function and (d) $25\text{\,}\mathrm{ps}$ Lorentzian
voltage pulse used for the measurement in Figure 5 (b).

Multiple harmonic microwave signals with proper adjustment of amplitude and
phase are mixed to obtain a Lorentzian target pulse – the central ingredient
for Leviton formation. As illustrated by the block diagram of Figure 2, a
fundamental sinusoidal signal with a frequency of $3\text{\,}\mathrm{GHz}$ and
a power of $19\text{\,}\mathrm{dBm}$ is fed into a frequency comb of the type
MARKI NLTL-6026. This microwave component is a Schottky-diode-based
transmission line with the ability to generate several harmonics from any
fundamental frequency up to $50\text{\,}\mathrm{GHz}$. After generation, the
harmonics are spread into different paths by an eight-way RF splitter. A
$200\text{\,}\mathrm{MHz}$ bandwidth band-pass filter selects for each path a
specific harmonic frequency. The power of each harmonic signal is determined
by the selection of microwave components, such as splitters or band-pass
filters, giving for the current implementation $-22.8\text{\,}\mathrm{dB}$ at
$3\text{\,}\mathrm{GHz}$ to $-53.1\text{\,}\mathrm{dB}$ at
$24\text{\,}\mathrm{GHz}$. After filtering, each harmonic passes through a
voltage-controlled attenuator (VCA) and a voltage-controlled phase-shifter
(VCPS) that automatically adjust the amplitude and phase for each harmonic
signal. Having set the desired harmonic composition, the recombined signal
provides the targeted voltage pulse on the output that is finally amplified
with a Marki AMZ-40 and measured with a sampling oscilloscope.

The automatic adjustment of the amplitudes and phases is accomplished by a
feedback-loop. As shown in Figure 2, the output of our frequency-comb
synthesizer is monitored by a digital sampling oscilloscope. The continuously
captured data is Fourier transformed and the thus-obtained discrepancies to
the target are compensated via VCA and VCP modifications until the desired
voltage pulse is obtained. For a Lorentzian-shape voltage pulse, the
optimisation algorithm usually converges within a couple of minutes. It is
worthwhile to note that adjusting the attenuators and phase-shifters is only
needed once and that, as it is shown below, the generated pulses are stable
over days.

In order to generate Leviton pulses in a quantum nanoeletronic device, it is
necessary to form Lorentzian charge excitation that carry exactly one or
multiple electrons. Accordingly, the voltage pulse generation must allow for
amplitude control that is extremely precise. Figure 3 shows real-time
oscilloscope data (black curves) from various synthesised Lorentzian pulses.
The traces of Figure 3 (a) and (b) show the output for target pulses with
duration of $80\text{\,}\mathrm{ps}$ and $60\text{\,}\mathrm{ps}$. Remarkably,
they are nearly identical to the theoretical target (blue lines) and have
minimum distortion, which allows generating Levitons with more than
$99.93\text{\,}\mathrm{\char 37\relax}$ purity (see supplementary material).
It is worthwhile to note that voltage pulses with similar duration, as the one
illustrated here (Figure 3 (a) and (b)), have been used in the past to
generate ultrafast Levitons, using state-of-the-art AWGs [33, 32]. Those
voltage pulses resembled more a gaussian-like pulse rather than the Lorentzian
shape required for the generation of Leviton pulses as demonstrated in Ref.
[33].

Analog Fourier synthesis – as demonstrated in this work – enables on the other
hand the generation of nearly ideal ultrashort Lorentzian pulses that go
beyond what state-of-the-art AWGs can achieve. To characterise the limit of
our device, we have programmed the synthesiser to output a Dirac delta-like
pulse (Lorentzian pulse with a width of $0.1\text{\,}\mathrm{ps}$). This
allows to measure the impulse response of the system, which can be used to
predict the output waveform for any requested input. The output can then be
simply computed as a convolution product between the requested input and the
impulse response. In our case, the impulse response, shown in Figure 3 (c), is
a cardinal sinus with a Full Width at Half Maximum (FWHM) of
$22\text{\,}\mathrm{ps}$.

Figure 3 (d) illustrates a $25\text{\,}\mathrm{ps}$ Lorentzian pulse, which
appears to be almost identical to its theoretical line shape (blue curve) and
with minimal distortion. Even for this case, the generated hole excitations
due to imperfect Lorentzian line shape is less than
$0.55\text{\,}\mathrm{\char 37\relax}$ (see supplementary material). The
slight distortion observed to either side of the main peak is due to the
absence of harmonics higher than $24\text{\,}\mathrm{GHz}$. It is worth noting
that any limitations imposed by the cutoff frequency of standard SMA
connectors used in our setup as well as any dispersion due to the RF cables is
automatically compensated by our optimisation algorithm.

Figure 4:  Output stability over time. The synthesiser output was monitored
every 10 minutes for 50 hours. (a) Relative time drift of each harmonic,
referenced to the $3\text{\,}\mathrm{GHz}$ fundamental frequency. (b)
Amplitude variation for each harmonic.

One of the main advantages of using a frequency comb based generator, as
opposed to individual phase locked frequency generators [5], is that all the
harmonics are generated from the same source signal. Even if the phase of the
fundamental frequency shifts over time, the phases of all higher harmonics
will shift in the same manner, and therefore the generated signal will not be
distorted. The stability of the phase between the harmonics of the
synthesiser, as well as their amplitude, is of paramount importance as any
phase or amplitude variation can distort the shape and duration of the
generated pulses. This means that for a frequency-comb based system, an ultra-
stable picosecond-duration pulse can be synthesised once at the beginning of
an experiment and then be kept the same over several days.

The stability of this setup is apparent from the phase and amplitude variation
of each harmonic, as shown in Figure 4. To generate these data, a Lorentzian
pulse was measured through a fast oscilloscope, once every ten minutes for 50
hours. Then, the time-domain data were Fourier transformed, in order to
extract the phase and amplitude information for each harmonic. Figure 4 (a),
shows the phase stability of this synthesiser. In order to have a more
meaningful representation of the data, Figure 4 (a) represents the relative
time delay of each harmonic with respect to the master frequency
($3\text{\,}\mathrm{GHz}$). As all higher harmonics are generated from the
same master frequency, their relative delay depends only on the passive
components used in our setup, such as the filters, attenuators and phase-
shifters. The $\sim$ $0.2\text{\,}\mathrm{ps}$ delay variation between the
harmonics is mainly due to temperature variations in the laboratory over the
course of a 2-day period. This relative delay is 2 orders of magnitude smaller
than the shortest Lorentzian pulse ($25\text{\,}\mathrm{ps}$) and can be
regarded as negligible.

The amplitude variation of each harmonic is also very stable, as illustrated
in Figure 4 (b). The amplitude of each harmonic, besides being important for
determining the duration and shape of the final pulse, is also crucial for
defining the overall amplitude of the Leviton pulses. In order to generate
single-electron Levitons, it is important to control the amplitude of the
generated pulse down to the microvolt level, where indicatively
$1\text{\,}\mathrm{\SIUnitSymbolMicro V}$ corresponds roughly to the addition
of $1\text{\,}\mathrm{\char 37\relax}$ of an extra electron. In this setup,
the observed amplitude variation is less than $1\text{\,}\mathrm{\char
37\relax}$ over the 50-hour acquisition time, and therefore the generated
Levitons can contain a single electron with a very small uncertainty.

## Generation of electron wavefunctions in a quantum conductor

Figure 5: Time-resolved measurement of an electron wavepacket using the
frequency comb synthesiser. (a) Scanning tunneling micrograph of the quantum
device made from a GaAs/AlGaAs heterostructure with mobility
$\mu_{\rm{e}}=$1.77\text{\times}{10}^{6}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{V}}^{-1}\text{\,}{\mathrm{s}}^{-1}$$
and electron concentration
$\rm{n_{e}}=$1.92\text{\times}{10}^{11}\text{\,}{\mathrm{cm}}^{-2}$$. The
voltage pulse signal generated with the frequency comb synthesiser is split
into two. One voltage pulse is sent to the upper left Ohmic contact to launch
the electron wavepacket while the second voltage pulse is sent to the QPC
(highlighted in yellow) via a mechanical delay line. The QPC is operated as an
ON/OFF switch, following the procedure detailed in [33]. Changing the time
delay between the two voltage pulses allows reconstructing in a time-resolved
manner the waveform of the generated electron wavepacket. The generated
current is detected through a $10\text{\,}\mathrm{k\SIUnitSymbolOhm}$
resistor, amplified at room temperature and measured by a lock-in amplifier
(LIA) using a demodulation technique. (b) Time resolved measurement of a
Leviton pulse. The inset shows the Fourier transform of the pulse, which
follows an exponential trend and therefore confirming the Lorenztian nature of
the generated electron wavepacket. (c) Calculated excitation spectrum of the
electron wavepacket generated in (b). $\lvert p_{l}\rvert^{2}$ denotes the
probability of an electron to absorb or emit $l$-photons of an energy quanta
$h\nu$ (see supplementary material).

The generation of single-electron Leviton pulses relies on voltage pulses with
Lorentzian shape, applied directly on the Fermi sea of a quantum conductor. By
perturbing the Fermi sea with these pulses we can generate pure electronic
excitations, without any holes accompanying the electrons [30, 5, 31, 32], in
stark contrast to other popular electron sources [23, 24]. This unique
property of Levitons makes them ideal candidates for electron flying qubit
technologies [16], as well as an attractive platform for studying fundamental
time-resolved electron-electron interactions.

To showcase the full potential of our novel voltage pulse synthesiser, we
demonstrate the generation and time-resolved detection of an electron
wavepacket with a record duration of $27\text{\,}\mathrm{ps}$ in a two-
dimensional electron gas (2DEG). A dilution refrigerator with a base
temperature of $20\text{\,}\mathrm{mK}$ was used to cool down our 2DEG to
cryogenic temperatures. The quantum device consists of a Mach-Zehnder
interferometer realized by depositing electrostatic surface gates on top of a
GaAs/AlGaAs heterostructure as shown in Figure 5 (a).

Applying a set of negative gate voltages to these electrostatic gates allows
engineering of the trajectory of the propagating electron wavepacket. For the
time-resolved measurements only the upper path of the interferometer is used,
that means the gate voltages on the central island as well as the adjacent
split wires are sufficiently negative such that the electron wavepacket
follows the path indicated by the white dashed line.

To perform time-resolved measurements and reconstruct the electron waveform in
the time domain, we split the voltage pulse output of our synthesiser into two
parts using a power divider. The first part, denoted as “Injection” on Figure
5 (a), is sent through a computer controlled delay line and then into the
injection Ohmic contact of our sample (left white square box in Figure 5 (a)).
The second part, denoted as “QPC”, is directly fed into the RF line, which is
connected to a Quantum Point Contact (QPC) positioned along the electron wave
guide. Both, the injection Ohmic contact and the sampling QPC line is equipped
with a high bandwidth Bias Tee SHF BT45R in order to be able to apply a DC and
AC component. For detecting the electrons, we measure the voltage generated
across a $10\text{\,}\mathrm{k\SIUnitSymbolOhm}$ resistor, which is then
amplified and measured by a LockIn Amplifier (LIA). To enable LIA
measurements, the “master” source of the synthesiser is modulated at
$11.95\text{\,}\mathrm{kHz}$.

The QPC line is employed to realise the time-resolved measurement. In line
with Roussely et al. [33], to reconstruct the time-domain profile of an
electron wavepacket, the QPC is negatively biased with a sufficient DC
voltage, such that it blocks all transmission through the waveguide. Then, on
top of that DC bias we apply an ultrashort voltage pulse to the QPC, time-
delayed by $\Delta\tau$ with respect to the injection pulse. This voltage
pulse will effectively act as a switch, enabling transmission for a very short
amount of time. By appropriately choosing the negative DC bias, the switching
time can be much shorter than the duration of the electron wavepacket,
therefore enabling accurate reconstruction.

Figure 5 (b) is the main result of this paper, showing the shortest ever
measured electron wavepacket propagating on top of the Fermi sea with
$27\text{\,}\mathrm{ps}$ duration. In order to benchmark the dispersion of our
system, the pulse was generated with a relatively high voltage amplitude and
contains around $\sim$ 20 electrons. The dispersion appears to be minimal, as
the injected pulse (Figure 3 (d)) had a similar width of
$25\text{\,}\mathrm{p}\mathrm{s}$. To better demonstrate the time-resolved
measurement technique, we have used a $6\text{\,}\mathrm{GHz}$
($166\text{\,}\mathrm{ps}$) fundamental frequency for the synthesiser. This
allowed us to measure two pulses within the same scanning window, since the
scanning range of our delay line is limited to $230\text{\,}\mathrm{ps}$. The
Fourier transform of the generated electron wavepacket, shown in the inset of
Figure 5 (b)), exhibits an exponential behaviour, which is distinctive of a
Lorentzian electron excitation.

Finally, we analyze the purity of the generated Lorentzian voltage pulses as
well as the generated electron wavepackets by calculating their excitation
spectrum in the framework of Floquet scattering theory [41, 42] (see
supplementary material). The excitation spectrum for the experimentally
measured electron wavepacket of $27\text{\,}\mathrm{ps}$ time duration is
shown in Figure 5 (c)). We observe a hole excitation that is less than,
$0.43\text{\,}\mathrm{\char 37\relax}$ which is quite remarkable. This has to
be compared to what was obtained with standard AWG generators ($\approx$ 3 %)
[32]. Similar analysis (see supplementary material) for the generated voltage
pulses display an extremely clean excitation spectrum with less than
$0.22\text{\,}\mathrm{\char 37\relax}$ hole excitations when the voltage
pulses have a duration of $50\text{\,}\mathrm{ps}$ or more. For the shortest
voltage pulses ($25\text{\,}\mathrm{ps}$), we obtain a maximum of
$0.55\text{\,}\mathrm{\char 37\relax}$ hole excitations. Adding additional
harmonics in future will further allow improving the signal quality as well as
shortening the generated voltage pulses. With a fundamental frequency of 3 or
6 GHz, the maximum of harmonic frequencies lies at $36\text{\,}\mathrm{GHz}$
due to limited frequency bandwidth of the radio-frequency components. Within
this range, it is possible to generate voltage pulses of a duration down to
$14.6\text{\,}\mathrm{ps}$.

It is possible, on the other hand, to create even shorter voltage pulses, in
the range of $1\text{\,}\mathrm{ps}$, using an optoelectronic approach [43],
but unfortunately their integration at cryogenic temperatures [44] still
remains a challenge.

To conclude, in this paper we have demonstrated a novel technique for
generating Levitions (single-electron Lorentzian electron wavepackets) based
on a frequency-comb synthesiser. The synthesized Levitons are the shortest
ever generated electron wavepackets riding on the Fermi sea, with a record
duration of $27\text{\,}\mathrm{ps}$. We evaluate the spurious hole
contribution to be below $0.43\text{\,}\mathrm{\char 37\relax}$ which make
these Levitons ideal carriers of quantum information in the field of electron
quantum optics, as well as for studying fundamental matter interactions in the
time domain.

## Acknowledgements

This project received funding from the European Union’s H2020 research and
innovation program under Grant Agreement No. 862683. C. B. and D. C. G.
acknowledge funding from the French Agence Nationale de la Recherche (ANR),
Project Fully Quantum ANR-16-CE30-0015. T. V. acknowledges funding from the
French Laboratory of Excellence project "LANEF" (ANR-10- LABX-0051).

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# Synthesis and upper bound of Schmidt rank of the bipartite controlled-
unitary gates

Gui-Long Jiang1, Hai-Rui Wei1111Corresponding author<EMAIL_ADDRESS>Guo-
Zhu Song2, and Ming Hua3 1 School of Mathematics and Physics, University of
Science and Technology Beijing, Beijing 100083, China
2 College of Physics and Materials Science, Tianjin Normal University, Tianjin
300387, China
3 Department of Applied Physics, School of Physical Science and Technology,
Tiangong University, Tianjin, 300387 China

###### Abstract

Quantum circuit model is the most popular paradigm for implementing complex
quantum computation. Based on Cartan decomposition, we show that $2(N-1)$
generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the
$y$\- and $z$-axes, and $N+5$ single-partite $y$\- and $z$-rotation-types
which are defined in this paper are sufficient to simulate a controlled-
unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$. In the scenario of the unitary gate
$\mathcal{U}_{cd(M\otimes N)}$ with $M\geq 3$ that is locally equivalent to a
diagonal unitary on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$, $2M(N-1)$ GCX gates
and $2M(N-1)+10$ single-partite $y$\- and $z$-rotation-types are required to
simulate it. The quantum circuit for implementing $\mathcal{U}_{cu(2\otimes
N)}$ and $\mathcal{U}_{cd(M\otimes N)}$ are presented. Furthermore, we find
$\mathcal{U}_{cu(2\otimes 2)}$ with $A$ controlling has Schmidt rank two, and
in other cases the diagonalized form of the target unitaries can be expanded
in terms of specific simple types of product unitary operators.

Keywords: quantum circuit, controlled-unitary gate, quantum Schmidt rank,

###### pacs:

03.67.Lx, 03.65.Ud, 03.67.Mn

##  I Introduction

Tremendous progress has been made in quantum computationcomput1 ; comput2 ;
comput3 ; comput4  and communicationcommun1 ; commun2 ; commun3 ; commun4 ;
commun5  in recent years. Unitary operations play a central role in many
quantum information processing tasks: quantum circuits,universal1 ; universal2
; universal3 ; universal4 ; universal5 ; universal6 ; universal7  quantum
algorithm,algorithm1  creating quantum entangled states,cluster1 ; cluster2 ;
cluster3 ; state4  quantum state fusion,fusion1 ; fusion2
cryptography,cryptography  entanglement purification and concentration,purifi1
; purifi2 ; purifi3  etc. There are local and nonlocal unitary operations. The
local unitary operations, known as the tensor product operators locally acting
on subsystems, can be deterministically implemented by local operations and
classical communication (LOCC).Soeda  Unfortunately, the local unitary
operations alone cannot create entanglement. The nonlocal unitary operations,
which can not be implemented by LOCC, have a more complex structure and play a
more powerful role than local unitary operations in quantum information
processing.Cao1 ; Cao2 ; Cao3  So far, the properties and implementations of
nonlocal unitary operations, even the simplest and the most popular
controlled-unitary operations, are still far from complete. Hence it is
interesting and important to find simple ways to implement nonlocal unitary
operations.

There are two ways to study complex unitary operations. One approach is to
decompose a unitary operation into a sum of product operations:
$\mathcal{U}=\sum_{j=1}^{m}A_{j}\otimes B_{j}\otimes C_{j}\otimes\cdots\otimes
R_{j}$, where $A_{j}$, $B_{j}$, etc. are linearly independent local operations
acting on respective parties. The smallest possible $m$ is defined as Schmidt
rank $\text{Sch}(\mathcal{U})$,rank  which can be used for quantifying the
nonlocality of $\mathcal{U}$ defined by
$K_{\text{Har}}(\mathcal{U})\equiv\log_{2}[\text{Sch}(\mathcal{U})]$,Power2003
; Yu2016-1 ; Yu2016-2  providing a sufficient condition for when $\mathcal{U}$
is a controlled-unitary operation,Yu2013 ; Yu2014-1 ; Yu2014-2 ; 2-qubit-Rank1
; Yu2015 ; Yu2021 ; Yu2010  and optimizing the synthesis of quantum
computationgate  and quantum transistorsCao2 . Another common approach, called
synthesis or quantum circuit, is to factorize $\mathcal{U}$ into fewer
achievable simple local and nonlocal operations from a universal library,
which may simplify such physical implementations:
$\mathcal{U}=X_{1}X_{2}\cdots X_{m}$. The complexity of the quantum circuit is
characterized by assessing the number of entangled operations involved in the
implementation. Quantum circuit is the dominant paradigm for implementing
efficiently complex quantum computation.

One of the central problems in quantum computing is to minimize the number of
one- and two-partite gates required to implement a desired quantum gate.
Utilizing Cartan decomposition,Cartan  Vatan et al.optimal  designed a
controlled-NOT (CNOT)-optimized general two-qubit quantum circuit in 2004,
Shende et al.lower  presented the highest known lower bound on asymptotic CNOT
cost required to implement an unstructured $n$-qubit quantum computing, Di and
WeiDi1 ; Di2  synthesized universal multiple-valued quantum circuits. Using
higher-dimensional Hilbert spaces, Lanyon et al.comput1  reduced the cost of a
Toffoli gate from six CNOTs to three CNOTs in 2009, Li et al.Liwendong
further optimize the $n$-qubit universal quantum circuit, Liu and
Weiuniversal2  decreased the complexity of a Fredkin gate to three entangling
gates in 2020. Nowadays, many works have been devoted to multiple-valued
quantum circuitDi1 ; Di2 . Nonetheless, the synthesis of multi-valued quantum
gates is still far from complete, and multiple-valued unitary operations still
are open problems.

Bipartite controlled-unitary operation is one of the most easily understood,
extensively studied, and widely used quantum operations, such as CNOT gate. In
this paper, we study the synthesis and possible Schmidt rank of the
controlled-unitary operation on bipartite Hilbert space. By analogy with
single-qubit rotations, we give the concepts of single-partite $y$\- and
$z$-rotation-types in higher-dimensional system, and present the synthesis of
arbitrary single-partite unitary gate. Then, utilizing Cartan decomposition
technique, we present a program for synthesizing a bipartite controlled-
unitary gate with $A$ controlling on $\mathbb{C}^{2}\otimes\mathbb{C}^{N}$
with $2(N-1)$ generalized controlled-$X$ gates (GCX), $6$ single-qubit
rotations about the $y$\- and $z$-axes, and $N+5$ single-partite $y$\- and
$z$-rotation-types in the worst case. And generally, the gate on
$\mathbb{C}^{M}\otimes\mathbb{C}^{N}(M\geq 3)$ locally equivalent to a
diagonal gate, which is the subset of controlled-unitaries with $A$ or $B$
controlling, was synthesized by $2M(N-1)$ GCX gates and $2M(N-1)+10$ single-
partite $y$\- and $z$-rotation-types in the worst case. Furthermore, the
possible Schmidt rank of the bipartite controlled-unitary operation are
presented in detail. The results indicate that upper bound of Schmidt rank of
the bipartite controlled-unitary operation with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ is $2N$, and the one on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ is two. In the scenario of the unitary
operation on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$ locally equivalent to a
diagonal unitary, the upper bound is $MN$.

##  II Cartan decomposition of Lie group U(N)

Cartan decomposition technique is a extremely valuable tool employed to obtain
the possible Schmidt rank and design compact quantum circuits.Power2003 ;
optimal ; synthesis2 ; epl  Cartan decomposition of real semi-simple Lie
algebra $\mathfrak{g}$ is defined as

$\displaystyle\mathfrak{g}=\mathfrak{l}\oplus\mathfrak{p}.$ (1)

The Lie subalgebra $\mathfrak{l}$ and the complement subspace
$\mathfrak{p}=\mathfrak{l}^{\perp}$ satisfying the commutation relations

$\displaystyle[\mathfrak{l},\mathfrak{l}]\subseteq\mathfrak{l},\;\;[\mathfrak{l},\mathfrak{p}]\subseteq\mathfrak{p},\;\;[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{l}.$
(2)

Let $G$ be a compact Lie group with a real semi-simple Lie algebra
$\mathfrak{g}$, then

$\displaystyle G=K_{1}\cdot A\cdot K_{2},$ (3)

with $K_{1}$, $K_{2}\in e^{\mathfrak{l}}$ and $A\in e^{\mathfrak{h}}$ for the
Lie group exponential. $\mathfrak{h}$ is the maximal Abelian subalgebra of
($\mathfrak{g}$, $\mathfrak{l}$) contained in $\mathfrak{p}$.

It is well known that the generators of the Lie algebra $u(2)$ can be given by

$\displaystyle
u(2)=\textrm{span}\\{\sigma_{x},\sigma_{y},\sigma_{z},I_{2}\\}.$ (4)

Here

$\displaystyle\begin{split}\sigma_{x}=T_{12}^{(2,1)}=\left(\begin{array}[]{cccc}0&1\\\
1&0\\\
\end{array}\right),\;\;\sigma_{y}=T_{12}^{(2,2)}=\left(\begin{array}[]{cccc}0&-\mathrm{i}\\\
\mathrm{i}&0\\\
\end{array}\right),\;\;\sigma_{z}=T_{12}^{(2,3)}=\left(\begin{array}[]{cccc}1&0\\\
0&-1\\\ \end{array}\right),\;\;I_{2}=\left(\begin{array}[]{cccc}1&0\\\ 0&1\\\
\end{array}\right).\end{split}$ (5)

The basis for $u(3)$ takes the form

$\displaystyle\begin{split}&T_{12}^{(3,1)}=\left(\begin{array}[]{cccc}0&1&0\\\
1&0&0\\\ 0&0&0\\\
\end{array}\right),\;\;\;\;T_{13}^{(3,1)}=\left(\begin{array}[]{cccc}0&0&1\\\
0&0&0\\\ 1&0&0\\\
\end{array}\right),\;\;\;T_{23}^{(3,1)}=\left(\begin{array}[]{cccc}0&0&0\\\
0&0&1\\\ 0&1&0\\\ \end{array}\right),\\\
&T_{12}^{(3,2)}=\left(\begin{array}[]{cccc}0&-\mathrm{i}&0\\\
\mathrm{i}&0&0\\\ 0&0&0\\\
\end{array}\right),\;\;T_{13}^{(3,2)}=\left(\begin{array}[]{cccc}0&0&-\mathrm{i}\\\
0&0&0\\\ \mathrm{i}&0&0\\\
\end{array}\right),\;T_{23}^{(3,2)}=\left(\begin{array}[]{cccc}0&0&0\\\
0&0&-\mathrm{i}\\\ 0&\mathrm{i}&0\\\ \end{array}\right),\\\
&T_{12}^{(3,3)}=\left(\begin{array}[]{cccc}1&0&0\\\ 0&-1&0\\\ 0&0&0\\\
\end{array}\right),\;T_{13}^{(3,3)}=\left(\begin{array}[]{cccc}1&0&0\\\
0&0&0\\\ 0&0&-1\\\
\end{array}\right),\;I_{3}=\left(\begin{array}[]{cccc}1&0&0\\\ 0&1&0\\\
0&0&1\\\ \end{array}\right).\end{split}$ (6)

Lie algebra $u(N)$ is spanned by $N^{2}$ matrices $\\{T_{ab}^{(N,1)}$,
$T_{ab}^{(N,2)}$, $T_{1a}^{(N,3)},I_{N}\\}$ $(a<b)$, where $I_{N}$ is the
$N\times N$ identity matrix, $T_{ab}^{(N,1)}$, $T_{ab}^{(N,2)}$, and
$T_{1a}^{(N,3)}$ are defined as

$\displaystyle(T_{ab}^{(N,1)})_{cd}=\left\\{\begin{array}[]{ccc}1,&\;c=a,d=b&\\\
1,&\;c=b,d=a&\\\ 0,&\text{others}&\\\ \end{array}\right.$ (10)
$\displaystyle(T_{ab}^{(N,2)})_{cd}=\left\\{\begin{array}[]{ccc}-\mathrm{i},&\;c=a,d=b&\\\
\mathrm{i},&\;c=b,d=a&\\\ 0,&\text{others}&\\\ \end{array}\right.$ (14)
$\displaystyle(T_{1a}^{(N,3)})_{cd}=\left\\{\begin{array}[]{ccc}1,&c=d=1&\\\
-1,&c=d=a,&2\leq a\leq N.\\\ 0,&\text{others}&\\\ \end{array}\right.$ (18)

The subscript $a$ and $b$ are order index of the basis. The $c$ and $d$
represent the $c$th row and $d$th column in matrix $T_{ab}^{(N,1)}$,
$T_{ab}^{(N,2)}$, and $T_{1a}^{(N,3)}$. The $b,c,d\in[1,2,\cdots,N]$, and $N$
represents the $N$ dimensional system.

Based on Equations (1-18), it is easily to find that the Cartan decomposition
of Lie algebra $u(N)$ may has the form
$u(N)=\mathfrak{l}_{u(N)}\oplus\mathfrak{p}_{u(N)}$, where

$\displaystyle\mathfrak{l}_{u(N)}=\textrm{span}\\{T_{ab}^{(N,2)},I_{N}\\},$
(19)
$\displaystyle\mathfrak{p}_{u(N)}=\textrm{span}\\{T_{ab}^{(N,1)},T_{1a}^{(N,3)}\\}.$
(20)

The Cartan subalgebra $\mathfrak{h}_{u(N)}$ is generated by

$\displaystyle\mathfrak{h}_{u(N)}=\textrm{span}\\{T_{1a}^{(N,3)}\\}.$ (21)

Then, the associated Cartan decomposition of group $\mathcal{U}_{u(N)}\in
U(N)$ is of the form

$\displaystyle\begin{split}\mathcal{U}_{u(N)}=&e^{\mathrm{i}\theta_{N}}\cdot
M_{1}\cdot
e^{\mathrm{i}(\theta_{1}T_{12}^{(N,3)}+\theta_{2}T_{13}^{(N,3)}+,\cdots,+\theta_{N-1}T_{1N}^{(N,3)})}\cdot
M_{2},\end{split}$ (22)

with $M_{1},\;M_{2}\in e^{\textrm{span}\\{T_{ab}^{(N,2)}\\}}$. For instance,
any two-dimensional unitary operator can be expanded into

$\displaystyle\mathcal{U}_{u(2)}=$ $\displaystyle
e^{\mathrm{i}\theta_{2}}\cdot e^{\mathrm{i}\vartheta_{1}\sigma_{y}}\cdot
e^{\mathrm{i}\theta_{1}\sigma_{z}}\cdot
e^{\mathrm{i}\widetilde{\vartheta}_{1}\sigma_{y}}.$ (23)

Any three-dimensional unitary operator can be expanded into

$\displaystyle\mathcal{U}_{u(3)}=$ $\displaystyle
e^{\mathrm{i}\theta_{3}}\cdot
e^{\mathrm{i}(\vartheta_{1}T_{12}^{(3,2)}+\vartheta_{2}T_{13}^{(3,2)}+\vartheta_{3}T_{23}^{(3,2)})}\cdot
e^{\mathrm{i}(\theta_{1}T_{12}^{(3,3)}+\theta_{2}T_{13}^{(3,3)})}\cdot
e^{\mathrm{i}(\widetilde{\vartheta}_{1}T_{12}^{(3,2)}+\widetilde{\vartheta}_{2}T_{13}^{(3,2)}+\widetilde{\vartheta}_{3}T_{23}^{(3,2)})}.$
(24)

Local single-qubit gates $e^{\mathrm{i}\vartheta\sigma_{y}}$
($e^{\mathrm{i}\widetilde{\vartheta}\sigma_{y}}$) and
$e^{\mathrm{i}\beta\sigma_{z}}$ are single-qubit rotations about the $y$\- and
$z$-axes, respectively. That is, two single-qubit rotations about the $y$-axes
and one single-qubit rotation about the $z$-axes are sufficient to simulate a
generic single-qubit gate. The global phase factor $e^{\mathrm{i}\theta_{2}}$
is missing here because it is irrelevant for quantum information processing.
The matrices in the set $\\{T_{ab}^{{}^{(N,2)}}\\}$ $(a<b)$ do not commute
with each other.

Definition: Similar to the definition of single-qubit rotations, we define
$e^{\mathrm{i}(\vartheta_{1}T_{12}^{(N,2)}+\vartheta_{2}T_{13}^{(N,2)}+\cdots+\vartheta_{N(N-1)/2}T_{(N-1)N}^{(N,2)})}$
($e^{\mathrm{i}(\widetilde{\vartheta}_{1}T_{12}^{(N,2)}+\widetilde{\vartheta}_{2}T_{13}^{(N,2)}+\cdots+\widetilde{\vartheta}_{N(N-1)/2}T_{(N-1)N}^{(N,2)})}$)
as single-partite $y$-rotation-type, and
$e^{\mathrm{i}(\theta_{1}T_{12}^{(N,3)}+\theta_{2}T_{13}^{(N,3)}+,\cdots,+\theta_{N-1}T_{1N}^{(N,3)})}$
as single-partite $z$-rotation-type. When $N=2$, they are compatible with
single-qubit rotations about the $y$\- and $z$-axes.

Based on Equations (19-22), one can see that arbitrary single-partite multi-
valued unitary $\mathcal{U}_{u(N)}$ can be implemented by two single-partite
$y$-rotation-types and one $z$-rotation-type. Each $y$-rotation-type and
$z$-rotation-type has $N(N-1)/2$ and $N-1$ free parameters, respectively. We
omit the overall phase change $e^{\mathrm{i}\theta_{N}}$.

##  III Synthesis and quantum Schmidt rank of controlled-unitary gate with $A$
controlling on $\mathbb{C}^{2}\otimes\mathbb{C}^{N}$

It is known that, if the controlled-unitary gate $\mathcal{U}_{cu(M\otimes
N)}$ on $\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ is controlled from $A$ side,
then $\exists U_{A}$, $V_{A}$ unitaries such that

$\displaystyle\begin{split}\mathcal{U}_{cu(M\otimes N)}=(U_{A}\otimes
I_{B})\cdot(\sum_{i=0}^{M-1}|i\rangle_{A}\langle i|\otimes
U_{i})\cdot(V_{A}\otimes I_{B}).\end{split}$ (25)

That is, up to local unitaries, $\mathcal{U}_{cu(M\otimes N)}$ with $A$
controlling is equivalent to $\widetilde{\mathcal{U}}_{cu(M\otimes
N)}=\sum_{i=0}^{M-1}|i\rangle_{A}\langle i|\otimes U_{i}$.

Using the well-known commutation relations

$\displaystyle[A\otimes B,C\otimes D]=[A,C]\otimes(B\cdot D)+(C\cdot
A)\otimes[B,D],$ (26)

one can verify that the Cartan decomposition of
$\widetilde{\mathcal{U}}_{cu(2\otimes N)}$ has the form
$\widetilde{u}_{cu(2\otimes N)}=\mathfrak{l}_{u(2\otimes
N)}\oplus\mathfrak{p}_{u(2\otimes N)}$, where

$\displaystyle\mathfrak{l}_{u(2\otimes N)}=\textrm{span}\\{I_{2}\otimes
u(N),\sigma_{z}\otimes I_{N}\\},$ (27) $\displaystyle\mathfrak{p}_{u(2\otimes
N)}=\textrm{span}\\{\sigma_{z}\otimes su(N)\\}$ (28)

with $su(N)=\textrm{span}\\{T_{ab}^{(N,1)},T_{ab}^{(N,2)},T_{1a}^{(N,3)}\\}$.
And the Cartan subalgebra is

$\displaystyle\mathfrak{h}_{u(2\otimes N)}=\textrm{span}\\{\sigma_{z}\otimes
T_{1a}^{(N,3)}\\}.$ (29)

As a consequence, $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling can be
decomposed as

$\displaystyle\mathcal{U}_{cu(2\otimes N)}=(U_{A}\otimes
U_{B})\cdot\wedge(\triangle_{2\otimes N})\cdot(V_{A}\otimes V_{B}),$ (30)

where $\wedge(\triangle_{2\otimes N})=e^{\textrm{span}\\{\sigma_{z}\otimes
T_{1a}^{(N,3)}\\}}.$

Based on Equation (22), it is easily verified that each of local unitary
operators $U_{A}$ and $V_{A}$ acting on subsystem $A$ can be implemented by
three single-partite rotations about the $y$\- and $z$-axes. Each of local
unitary operators $U_{B}$ and $V_{B}$ acting on subsystem $B$ can be
implemented by three single-partite $y$\- and $z$-rotation-types.

_Synthesis and Schmidt rank of $\mathcal{U}_{cu(2\otimes 2)}$ with $A$
controlling_. When $\mathcal{U}_{cu}$ with $A$ controlling is acting on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, it can be decomposed as

$\displaystyle\begin{split}\mathcal{U}_{cu(2\otimes 2)}&=(U_{A}\otimes
U_{B})e^{\mathrm{i}\theta\sigma_{z}\otimes\sigma_{z}}(V_{A}\otimes V_{B})\\\
&=(U_{A}\otimes U_{B})\cdot\text{CNOT}\cdot e^{\mathrm{i}\theta
I_{2}\otimes\sigma_{z}}\cdot\text{CNOT}\cdot(V_{A}\otimes V_{B}).\end{split}$
(31)

Here each of single-qubit gates $U_{A}$, $V_{A}$, $U_{B}$, and $V_{B}$ can be
expanded in the form of $e^{\mathrm{i}\alpha\sigma_{y}}\cdot
e^{\mathrm{i}\beta\sigma_{z}}\cdot
e^{\mathrm{i}\widetilde{\alpha}\sigma_{y}}$, that is, each one is the product
of three single-qubit rotations about the $y$\- and $z$-axes, see Equation
(23). CNOT represents a controlled-NOT gate, and it is given by the matrix

$\displaystyle\text{CNOT}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 0&0&1&0\\\ \end{array}\right).$ (36)

Therefore, as shown in Figure 1, two CNOT gates together with $13=3\times 4+1$
single-qubit rotations about the $y$\- and $z$-axes are sufficient to simulate
a controlled-unitary $\mathcal{U}_{cu(2\otimes 2)}$ with $A$ controlling.

Based on Equation (31), we may expand $\wedge(\triangle_{2\otimes
2})=e^{\mathrm{i}\theta\sigma_{z}\otimes\sigma_{z}}$ as

$\displaystyle\begin{split}\wedge(\triangle_{2\otimes
2})=c_{\theta}I_{2}\otimes
I_{2}+\mathrm{i}s_{\theta}\sigma_{z}\otimes\sigma_{z},\end{split}$ (37)

where $c_{\theta}\equiv\cos(\theta)$, $s_{\theta}\equiv\sin(\theta)$.
Therefore, $\mathcal{U}_{cu(2\otimes 2)}$ with $A$ controlling only has
Schmidt rank 2 when $c_{\theta}$, $s_{\theta}\neq 0$.

Figure 1: Synthesis of a controlled-unitary gate with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Each of local operations $U_{A}$,
$V_{A}$, $U_{B}$, and $V_{B}$ is the product of three single-qubit rotations
about the $y$\- and $z$-axes. The overall phase change is ignored here because
it has no effect on the present program.

_Synthesis and Schmidt rank of $\mathcal{U}_{cu(2\otimes 3)}$ with $A$
controlling_. When $\mathcal{U}_{cu}$ with $A$ controlling is acting on
$\mathbb{C}^{2}\otimes\mathbb{C}^{3}$, it can be decomposed as

$\displaystyle\begin{split}\mathcal{U}_{cu(2\otimes 3)}=&(U_{A}\otimes
U_{B})e^{\mathrm{i}(\theta_{1}\sigma_{z}\otimes
T_{12}^{(3,3)}+\theta_{2}\sigma_{z}\otimes T_{13}^{(3,3)})}(V_{A}\otimes
V_{B}).\end{split}$ (38)

We find that each $e^{\mathrm{i}(\theta\sigma_{z}\otimes T_{1a}^{(3,3)})}$ can
be synthesized by

$\displaystyle\begin{split}e^{\mathrm{i}(\theta\sigma_{z}\otimes
T_{1a}^{(3,3)})}=\text{GCX}(1\rightarrow X^{(0,a-1)})\cdot e^{\mathrm{i}\theta
I_{2}\otimes T_{1a}^{(3,3)}}\cdot\text{GCX}(1\rightarrow
X^{(0,a-1)})).\end{split}$ (39)

Here the universal generalized controlled-$X$ gate,Di1 ; Di2
GCX$(m\rightarrow X^{(ij)})$, implements the operation

$\displaystyle X^{(ij)}=|i\rangle\langle j|+|j\rangle\langle i|+\sum_{k\neq
i,j}|k\rangle\langle k|,$ (40)

on the target particle if and only if the control particle is in the state
$|m\rangle$, and has no effect otherwise. Hence, Equation (38) can be
rewritten as

$\displaystyle\begin{split}\mathcal{U}_{cu(2\otimes 3)}=&(U_{A}\otimes
U_{B})\cdot\text{GCX}(1\rightarrow X^{(01)})\cdot
e^{\mathrm{i}\theta_{1}I_{2}\otimes
T_{12}^{(3,3)}}\cdot\text{GCX}(1\rightarrow X^{(01)})\\\
&\cdot\text{GCX}(1\rightarrow X^{(02)})\cdot
e^{\mathrm{i}\theta_{2}I_{2}\otimes
T_{13}^{(3,3)}}\cdot\text{GCX}(1\rightarrow X^{(02)})\cdot(V_{A}\otimes
V_{B}).\end{split}$ (41)

That is to say, a $\mathcal{U}_{cu(2\otimes 3)}$ with $A$ controlling can be
synthesized by four GCX gates, $6=3\times 2$ single-qubit rotations about the
$y$\- and $z$-axes, and $8=3\times 2+2$ single-partite $y$\- and $z$-rotation-
types in the worst case, see Figure 2. Note that each above $y$-rotation-type
and $z$-rotation-type has 3 and 2 free parameters, respectively, see Equation
(24). By the same argument as that made for
$\mathbb{C}^{2}\otimes\mathbb{C}^{4}$, we can find that as shown in Figure 3,
the quantum circuit for implementing $\mathcal{U}_{cu(2\otimes 4)}$ with $A$
controlling contains six GCX gates, $6=3\times 2$ single-qubit rotations about
the $y$\- and $z$-axes, and $9=3\times 2+3$ single-partite $y$\- and
$z$-rotation-types.

Based on Equation (38), we may expand $\wedge(\triangle_{2\otimes
3})=e^{\mathrm{i}(\theta_{1}\sigma_{z}\otimes
T_{12}^{(3,3)}+\theta_{2}\sigma_{z}\otimes T_{13}^{(3,3)})}$ as

$\displaystyle\begin{split}\wedge(\triangle_{2\otimes
3})&=\frac{1}{3}(c_{(\theta_{1}+\theta_{2})}+c_{\theta_{1}}+c_{\theta_{2}})I_{6}+\frac{1}{3}(c_{(\theta_{1}+\theta_{2})}-2c_{\theta_{1}}+c_{\theta_{2}})I_{2}\otimes
T_{12}^{(3,3)}\\\
&+\frac{1}{3}(c_{(\theta_{1}+\theta_{2})}+c_{\theta_{1}}-2c_{\theta_{2}})I_{2}\otimes
T_{13}^{(3,3)}+\frac{1}{3}\mathrm{i}(s_{(\theta_{1}+\theta_{2})}-s_{\theta_{1}}-s_{\theta_{2}})\sigma_{z}\otimes
I_{3}\\\
&+\frac{1}{3}\mathrm{i}(s_{(\theta_{1}+\theta_{2})}+2s_{\theta_{1}}-s_{\theta_{2}})\sigma_{z}\otimes
T_{12}^{(3,3)}+\frac{1}{3}\mathrm{i}(s_{(\theta_{1}+\theta_{2})}-s_{\theta_{1}}+2s_{\theta_{2}})\sigma_{z}\otimes
T_{13}^{(3,3)},\end{split}$ (42)

where $s_{(\theta_{1}+\theta_{2})}\equiv\sin(\theta_{1}+\theta_{2})$,
$c_{(\theta_{1}+\theta_{2})}\equiv\cos(\theta_{1}+\theta_{2})$. That is to
say, the upper bound of Schmidt rank of $\mathcal{U}_{cu(2\otimes 3)}$ with
$A$ controlling is six. By calculating similarly to Equation (42), we can
further find $\mathcal{U}_{cu(2\otimes 4)}$ with $A$ controlling has Schmidt
rank at most eight. The upper bound on the Schmidt rank is not tight. A tight
upper bound for the Schmidt rank is only two for generic diagonal unitaries on
$2\times N$ space. But the expansion given above is interesting in that it is
in terms of product unitary operators, where the local operators are either
the identity operator or with support in two-dimensional subspaces.

Figure 2: Synthesis of a controlled-unitary gate with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{3}$. Both local operations $U_{B}$ and
$V_{B}$ are the product of three single-partite $y$\- and $z$-rotation-types
from Equation (24). We omit the overall phase change. Figure 3: Synthesis of a
controlled-unitary gate with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{4}$. The global phase factor is missing
here.

_Synthesis and Schmidt rank of $\mathcal{U}_{cu(2\otimes N)}$ with $A$
controlling_. When $\mathcal{U}_{cu}$ is acting on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$, it can be decomposed as

$\begin{split}\mathcal{U}_{cu(2\otimes N)}=&(U_{A}\otimes U_{B})\cdot
e^{\mathrm{i}(\theta_{1}\sigma_{z}\otimes
T_{12}^{(N,3)}+\theta_{2}\sigma_{z}\otimes
T_{13}^{(N,3)}+\cdots+\theta_{N-1}\sigma_{z}\otimes
T_{1N}^{(N,3)})}\cdot(V_{A}\otimes V_{B})\\\ =&(U_{A}\otimes
U_{B})\cdot\text{GCX}(1\rightarrow X^{(01)})\cdot
e^{\mathrm{i}\theta_{1}I_{2}\otimes
T_{12}^{(N,3)}}\cdot\text{GCX}(1\rightarrow X^{(01)})\\\
&\cdot\text{GCX}(1\rightarrow X^{(02)})\cdot
e^{\mathrm{i}\theta_{2}I_{2}\otimes
T_{13}^{(N,3)}}\cdot\text{GCX}(1\rightarrow X^{(02)})\\\ &\cdot\cdots\\\
&\cdot\text{GCX}(1\rightarrow X^{(0,N-1)})\cdot
e^{\mathrm{i}\theta_{N-1}I_{2}\otimes
T_{1N}^{(N,3)}}\cdot\text{GCX}(1\rightarrow X^{(0,N-1)})\cdot(V_{A}\otimes
V_{B}).\end{split}$ (43)

We note that there are $N-1$ items in the exponential
$e^{\mathrm{i}(\theta_{1}\sigma_{z}\otimes
T_{12}^{(N,3)}+\cdots+\theta_{N-1}\sigma_{z}\otimes T_{1N}^{(N,3)})}$, and
each of them can be synthesized by two GCX gates and one single-partite
$z$-rotation-type. Hence, the quantum circuit for implementing
$\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling contains $2(N-1)$ GCX
gates, $6=3\times 2$ single-qubit rotations about the $y$\- and $z$-axes, and
$N+5=3\times 2+(N-1)$ single-partite $y$\- and $z$-rotation-types, see Figure
4. It is noted that each $y$-rotation-type and $z$-rotation-type has
$N(N-1)/2$ and $N-1$ free parameters, respectively.

Based on Equation (43), we may expand $\wedge(\triangle_{2\otimes
N})=e^{\mathrm{i}(\theta_{1}\sigma_{z}\otimes
T_{12}^{(N,3)}+\theta_{2}\sigma_{z}\otimes
T_{13}^{(N,3)}+\cdots+\theta_{N-1}\sigma_{z}\otimes T_{1N}^{(N,3)})}$ as

$\begin{split}\wedge(\triangle_{2\otimes
N})=&\gamma_{1}I_{2N}+\delta_{1}\sigma_{z}\otimes
I_{N}+\sum_{k=2}^{N-1}(\gamma_{k}I_{2}\otimes
T_{1k}^{(N,3)}+\delta_{k}\sigma_{z}\otimes T_{1k}^{(N,3)}).\end{split}$ (44)

Here parameters $\gamma_{k}$ and $\delta_{k}$ can be expressed by following:

$\gamma_{k}=\left\\{\begin{array}[]{cc}\displaystyle\frac{1}{N}[c_{(\theta_{1}+\theta_{2}+\cdots+\theta_{N-1})}+\displaystyle\sum\limits_{n=1}^{N-1}c_{\theta_{n}}]&k=1,\\\
\displaystyle\frac{1}{N}[c_{(\theta_{1}+\theta_{2}+\cdots+\theta_{N-1})}+\displaystyle\sum\limits_{n=1}^{N-1}c_{\theta_{n}}-Nc_{\theta_{k-1}}]&k=2,\cdots,N-1.\\\
\end{array}\right.$ (45)
$\delta_{k}=\left\\{\begin{array}[]{cc}\displaystyle\frac{\mathrm{i}}{N}[s_{(\theta_{1}+\theta_{2}+\cdots+\theta_{N-1})}-\displaystyle\sum\limits_{n=1}^{N-1}s_{\theta_{n}}]&k=1,\\\
\displaystyle\frac{\mathrm{i}}{N}[s_{(\theta_{1}+\theta_{2}+\cdots+\theta_{N-1})}-\displaystyle\sum\limits_{n=1}^{N-1}s_{\theta_{n}}+Ns_{\theta_{k-1}}]&k=2,\cdots,N-1.\\\
\end{array}\right.$ (46)

Therefore, the Schmidt rank of a $\mathcal{U}_{cu(2\otimes N)}$ with $A$
controlling is no more than $2N$.

Figure 4: Synthesis of a controlled-unitary gate with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$. The global phase is omitted.

##  IV Synthesis and quantum Schmidt rank of the gate locally equivalent to
diagonal unitary on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$

It is well known that, if $\mathcal{U}_{cd(M\otimes N)}$ is a unitary gate
locally equivalent to a diagonal unitary on
$\mathbb{C}^{M}\otimes\mathbb{C}^{N}$, then it can be viewed as a controlled
unitary controlled from $A$ side or $B$ side, and $\exists U_{A}$, $U_{B}$,
$V_{A}$, $V_{B}$ unitaries such that

$\displaystyle\mathcal{U}_{cd(M\otimes N)}=(U_{A}\otimes
U_{B})\cdot(\sum_{i=1}^{d_{A}}\sum_{j=1}^{d_{B}}|i\rangle_{A}\langle
i|\otimes|j\rangle_{B}\langle j|)\cdot(V_{A}\otimes V_{B}).$ (47)

Here we use the symbol $cd$ in the subscript, rather than simply using $d$, in
order to distinguish them from the diagonal unitaries
$(\sum_{i=1}^{d_{A}}\sum_{j=1}^{d_{B}}|i\rangle_{A}\langle
i|\otimes|j\rangle_{B}\langle j|)$. For the same unitary of this type, either
party may act as the control, and the choice would not affect the
decomposition.

As $(\sum_{j=1}^{d_{B}}|i\rangle_{A}\langle i|\otimes|j\rangle_{B}\langle
j|)\in e^{\text{span}\\{T_{1\tilde{a}}^{(M,3)}\otimes
T_{1a}^{(N,3)},I_{M}\otimes T_{1a}^{(N,3)},T_{1\tilde{a}}^{(M,3)}\otimes
I_{N}\\}}$ and the matrices in the set $\\{T_{1\tilde{a}}^{(M,3)}\otimes
T_{1a}^{(N,3)},I_{M}\otimes T_{1a}^{(N,3)},T_{1\tilde{a}}^{(M,3)}\otimes
I_{N}\\}$ do not commute with each other, local single-partite $z$-rotation-
type $e^{\text{span}\\{I_{M}\otimes T_{1a}^{(N,3)}\\}}$ will be “absorbed” by
their neighboring gate $U_{B}$ or $V_{B}$, and
$e^{\text{span}\\{T_{1\tilde{a}}^{(M,3)}\otimes I_{N}\\}}$ will be “absorbed”
by $U_{A}$ or $V_{A}$. Then $\mathcal{U}_{cd(M\otimes N)}$ can be rewritten as

$\displaystyle\begin{split}\mathcal{U}_{cd(M\otimes N)}=&(U_{A}\otimes
U_{B})\cdot\wedge(\triangle_{M\otimes N})\cdot(V_{A}\otimes
V_{B}),\end{split}$ (48)

where $\wedge(\triangle_{M\otimes
N})=e^{\text{span}\\{T_{1\tilde{a}}^{(M,3)}\otimes T_{1a}^{(N,3)}\\}}$ with
$\tilde{a}\in[2,3,\cdots,M]$ and $a\in[2,3,\cdots,N]$.

_Synthesis and Schmidt rank of $\mathcal{U}_{cd(3\otimes 3)}$ on
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}$_. When the $\mathcal{U}_{cd}$ is acting
on $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$, it can be decomposed as

$\begin{split}\mathcal{U}_{cd(3\otimes 3)}=&(U_{A}\otimes
U_{B})e^{\mathrm{i}(\theta_{1}T_{12}^{(3,3)}\otimes
T_{12}^{(3,3)}+\theta_{2}T_{13}^{(3,3)}\otimes
T_{12}^{(3,3)}+\theta_{3}T_{12}^{(3,3)}\otimes
T_{13}^{(3,3)}+\theta_{4}T_{13}^{(3,3)}\otimes T_{13}^{(3,3)})}(V_{A}\otimes
V_{B}).\end{split}$ (49)

One can verify that each $e^{\mathrm{i}(\theta T_{1\tilde{a}}^{(M,3)}\otimes
T_{1a}^{(N,3)})}$ can be synthesized by

$\begin{split}e^{\mathrm{i}(\theta T_{1\tilde{a}}^{(M,3)}\otimes
T_{1a}^{(N,3)})}=&\wedge(0\rightarrow e^{\mathrm{i}\theta
T_{1a}^{(N,3)}})\cdot\wedge(\tilde{a}-1\rightarrow e^{-\mathrm{i}\theta
T_{1a}^{(N,3)}}),\end{split}$ (50)

where $\wedge(m\rightarrow e^{\mathrm{i}\theta T_{1a}^{(N,3)}})$ is to
implement the operation $e^{\mathrm{i}\theta T_{1a}^{(N,3)}}$ on the target
particle if and only if the control particle is in the state $|m\rangle$, and
has no effect otherwise. Given verification, $\wedge(m\rightarrow
e^{\mathrm{i}\theta T_{1a}^{(N,3)}})$ can be simulated by two GCX gates and
two single-partite $z$-rotation-types as described in Figure 5, that is,

$\begin{split}\wedge(m\rightarrow e^{\mathrm{i}\theta
T_{1a}^{(N,3)}})=&e^{\mathrm{i}\frac{\theta}{2}I_{M}\otimes
T_{1a}^{(N,3)}}\cdot\text{GCX}(m\rightarrow X^{(0,a-1)})\cdot
e^{-\mathrm{i}\frac{\theta}{2}I_{M}\otimes
T_{1a}^{(N,3)}}\cdot\text{GCX}(m\rightarrow X^{(0,a-1)})\\\
=&\text{GCX}(m\rightarrow X^{(0,a-1)})\cdot
e^{-\mathrm{i}\frac{\theta}{2}I_{M}\otimes
T_{1a}^{(N,3)}}\cdot\text{GCX}(m\rightarrow X^{(0,a-1)})\cdot
e^{\mathrm{i}\frac{\theta}{2}I_{M}\otimes T_{1a}^{(N,3)}}.\end{split}$ (51)

Hence, Equation (49) can be rewritten as

$\begin{split}\mathcal{U}_{cd(3\otimes 3)}=&(U_{A}\otimes
U_{B})\cdot\wedge(0\rightarrow
e^{\mathrm{i}(\theta_{1}+\theta_{2})T_{12}^{(3,3)}})\cdot\wedge(0\rightarrow
e^{\mathrm{i}(\theta_{3}+\theta_{4})T_{13}^{(3,3)}})\cdot\\\
&\wedge(1\rightarrow
e^{-\mathrm{i}\theta_{1}T_{12}^{(3,3)}})\cdot\wedge(1\rightarrow
e^{-\mathrm{i}\theta_{3}T_{13}^{(3,3)}})\cdot\wedge(2\rightarrow
e^{-\mathrm{i}\theta_{2}T_{12}^{(3,3)}})\cdot\wedge(2\rightarrow
e^{-\mathrm{i}\theta_{4}T_{13}^{(3,3)}})\cdot(V_{A}\otimes V_{B}).\end{split}$
(52)

Note that the rightmost gate of the circuit of Figure 5(a) will be “absorbed”
by its neighboring gate $U_{B}$ of Figure 6, and the leftmost gate of the
circuit of Figure 5(b) will be “absorbed” by its neighboring gate $V_{B}$ of
Figure 6. Therefore, according to Equations (51) and (52), 12 GCX gates and
$22=3\times 4+(12-2)$ single-partite $y$\- and $z$-rotation-types are
sufficient to simulate a $\mathcal{U}_{cd(3\otimes 3)}$ locally equivalent to
a diagonal unitary as in Figure 6.

Based on Equation (52), we may expand $\wedge(\triangle_{3\otimes 3})$ as

$\begin{split}\wedge(\triangle_{3\otimes 3})=&CI_{3}\otimes
I_{3}+[C-\frac{1}{3}(c_{(\theta_{1}+\theta_{2})}+c_{\theta_{1}}+c_{\theta_{2}})+\frac{\mathrm{i}}{3}(s_{(\theta_{1}+\theta_{2})}-s_{\theta_{1}}-s_{\theta_{2}})]I_{3}\otimes
T^{(3,3)}_{12}\\\
&+[C-\frac{1}{3}(c_{(\theta_{3}+\theta_{4})}+c_{\theta_{3}}+c_{\theta_{4}})+\frac{\mathrm{i}}{3}(s_{(\theta_{3}+\theta_{4})}-s_{\theta_{3}}-s_{\theta_{4}})]I_{3}\otimes
T^{(3,3)}_{13}\\\
&+[C-\frac{1}{3}(c_{(\theta_{1}+\theta_{3})}+c_{\theta_{1}}+c_{\theta_{3}})+\frac{\mathrm{i}}{3}(s_{(\theta_{1}+\theta_{3})}-s_{\theta_{1}}-s_{\theta_{3}})]T^{(3,3)}_{12}\otimes
I_{3}\\\
&+[C-\frac{1}{3}(c_{(\theta_{2}+\theta_{4})}+c_{\theta_{2}}+c_{\theta_{4}})+\frac{\mathrm{i}}{3}(s_{(\theta_{2}+\theta_{4})}-s_{\theta_{2}}-s_{\theta_{4}})]T^{(3,3)}_{13}\otimes
I_{3}\\\
&+[C-\frac{1}{3}(c_{(\theta_{1}+\theta_{2})}+c_{(\theta_{1}+\theta_{3})}-c_{\theta_{1}}+c_{\theta_{2}}+c_{\theta_{3}})+\frac{\mathrm{i}}{3}(s_{(\theta_{1}+\theta_{2})}+s_{(\theta_{1}+\theta_{3})}+s_{\theta_{1}}-s_{\theta_{2}}-s_{\theta_{3}})]T^{(3,3)}_{12}\otimes
T^{(3,3)}_{12}\\\
&+[C-\frac{1}{3}(c_{(\theta_{1}+\theta_{3})}+c_{(\theta_{3}+\theta_{4})}+c_{\theta_{1}}-c_{\theta_{3}}+c_{\theta_{4}})+\frac{\mathrm{i}}{3}(s_{(\theta_{1}+\theta_{3})}+s_{(\theta_{3}+\theta_{4})}-s_{\theta_{1}}+s_{\theta_{3}}-s_{\theta_{4}})]T^{(3,3)}_{12}\otimes
T^{(3,3)}_{13}\\\
&+[C-\frac{1}{3}(c_{(\theta_{1}+\theta_{2})}+c_{(\theta_{2}+\theta_{4})}+c_{\theta_{1}}-c_{\theta_{2}}+c_{\theta_{4}})+\frac{\mathrm{i}}{3}(s_{(\theta_{1}+\theta_{2})}+s_{(\theta_{2}+\theta_{4})}-s_{\theta_{1}}+s_{\theta_{2}}-s_{\theta_{4}})]T^{(3,3)}_{13}\otimes
T^{(3,3)}_{12}\\\
&+[C-\frac{1}{3}(c_{(\theta_{2}+\theta_{4})}+c_{(\theta_{3}+\theta_{4})}+c_{\theta_{2}}+c_{\theta_{3}}-c_{\theta_{4}})+\frac{\mathrm{i}}{3}(s_{(\theta_{2}+\theta_{4})}+s_{(\theta_{3}+\theta_{4})}-s_{\theta_{2}}-s_{\theta_{3}}+s_{\theta_{4}})]T^{(3,3)}_{13}\otimes
T^{(3,3)}_{13},\end{split}$ (53)

where

$\begin{split}C=&\frac{1}{9}[c_{(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4})}+\sum^{4}_{n=1}c_{\theta_{n}}+c_{(\theta_{1}+\theta_{2})}+c_{(\theta_{1}+\theta_{3})}+c_{(\theta_{2}+\theta_{4})}+c_{(\theta_{3}+\theta_{4})}+\mathrm{i}(s_{(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4})}\\\
&+\sum^{4}_{n=1}s_{\theta_{n}}-s_{(\theta_{1}+\theta_{2})}-s_{(\theta_{1}+\theta_{3})}-s_{(\theta_{2}+\theta_{4})}-s_{(\theta_{3}+\theta_{4})})].\end{split}$
(54)

Hence, the Schmidt rank of $\mathcal{U}_{cd(3\otimes 3)}$ is no more than
nine.

Figure 5: Synthesis of $\wedge(m\rightarrow e^{\mathrm{i}\theta
T_{1a}^{(N,3)}})$ in terms of the $\text{GCX}(m\rightarrow X^{(0,a-1)})$.
Figure 6: Synthesis for implementing a gate on
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ which is locally equivalent to a
diagonal gate. Each of local operations $U_{A}$, $V_{A}$, $U_{B}$, and $V_{B}$
is the product of two single-partite $y$-rotation-types and one $z$-rotation-
type from Equation (24), and the overall phase change is missing here.

_Synthesis and Schmidt rank of $\mathcal{U}_{cd(M\otimes N)}$ on
$\mathbb{C}^{M}\otimes\mathbb{C}^{N}$_. When the $\mathcal{U}_{cd}$ is on
$\mathbb{C}^{M}\otimes\mathbb{C}^{N}$, it can be decomposed as

$\begin{split}\mathcal{U}_{cd(M\otimes N)}=&(U_{A}\otimes
U_{B})\cdot\prod_{\stackrel{{\scriptstyle\tilde{a}=2,3,\cdots,M}}{{a=2,3,\cdots,N}}}e^{\mathrm{i}(\theta_{i}T_{1\tilde{a}}^{(M,3)}\otimes
T_{1a}^{(N,3)})}\cdot(V_{A}\otimes V_{B}).\end{split}$ (55)

Putting all the pieces together, and taking the trick “absorbtion” into
account, we find that $2M(N-1)$ GCX gates and $2M(N-1)+10$ single-partite
$y$\- and $z$-rotation-types are sufficient to implement a
$\mathcal{U}_{cd(M\otimes N)}$ locally equivalent to a diagonal unitary. It is
noted that each $y$\- or $z$-rotation-type has many free parameters in it when
$M$, $N$ are larger.

We note that there are $MN-1$ linearly independent diagonal bases in $u(MN)$,
and $\mathcal{U}_{cd(M\otimes N)}$ is equivalent to
$\wedge(\triangle_{M\otimes
N})=\prod_{\stackrel{{\scriptstyle\tilde{a}=2,3,\cdots,M}}{{a=2,3,\cdots,N}}}e^{\mathrm{i}(\theta_{i}T_{1\tilde{a}}^{(M,3)}\otimes
T_{1a}^{(N,3)})}$. Hence, $\wedge(\triangle_{M\otimes N})$ can be expanded as

$\begin{split}\wedge(\triangle_{M\otimes N})=&\varepsilon
I_{MN}+\sum_{k=1}^{N-1}\gamma_{k}I_{M}\otimes
T_{1k}^{(N,3)}+\sum_{j=2}^{M-1}\delta_{j}T_{1j}^{(M,3)}\otimes
I_{N}+\sum_{j=2}^{M-1}\sum_{k=2}^{N-1}\tau_{jk}T_{1j}^{(M,3)}\otimes
T_{1k}^{(N,3)}.\end{split}$ (56)

Based on Equation (56), one can see that the Schmidt rank of a
$\mathcal{U}_{cd(M\otimes N)}$ is no more than $MN$.

##  V Conclusion

Quantum circuit is the dominant paradigm for implementing and characterizing
complex quantum computation, and the works are mainly focused on two-valued
systems. Utilizing Cartan decomposition technique, we have presented compact
quantum circuits for implementing controlled-unitary gates with $A$
controlling on $\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ and controlled-diagonal
gates on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$, respectively. We first showed
the Cartan decomposition of Lie algebra $u(N)$ and reported that an arbitrary
single-partite unitary can be implemented by three single-partite $y$\- and
$z$-rotation-types which are defined in this paper, and their appearance in
decomposition of nonlocal gates is not previously studied, to our best
knowledge. Subsequently, we designed compact quantum circuits for implementing
controlled-unitary gates with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, $\mathbb{C}^{2}\otimes\mathbb{C}^{3}$,
$\cdots$, $\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ in terms of GCX gates and
local single-partite rotation-types in detail. The results indicate that
$2(N-1)$ GCX gates together with $6$ single-qubit rotations about the $y$\-
and $z$-axes, and $N+5$ single-partite $y$\- and $z$-rotation-types are
sufficient to implement a controlled-unitary gate with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$. Lastly, we extended the program to
gates on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$ which are locally equivalent to
diagonal gates, and the quantum circuit is comprised of $2M(N-1)$ GCX gates
and $2M(N-1)+10$ single-partite $y$\- and $z$-rotation-types.

Based on Cartan decomposition, we have verified that expanding the diagonal
form of the unitary is a way to get the upper bound of Schmidt rank of the
controlled-unitary gate with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ or the gate locally equivalent to a
diagonal gate on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$, and such upper bound
may be not tight. The results showed that the controlled-unitary gate with $A$
controlling on $\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ has Schmidt rank at most
$2N$, whereas the one on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ reaches the
lower bound of 2, and the gate locally equivalent to a diagonal gate on
$\mathbb{C}^{M}\otimes\mathbb{C}^{N}$ has Schmidt rank at most $MN$. Although
the bounds are not tight in general, the form of our decomposition involve
special types of operators: the local operators in each term are either the
identity or unitaries in low-dimensional subspaces. This explains that our
decomposition in general has more terms than the Schmidt rank.

It is expected that the algorithm and technique employed here to implement
controlled-unitary gates with $A$ controlling on
$\mathbb{C}^{2}\otimes\mathbb{C}^{N}$ and the gates locally equivalent to
diagonal gates on $\mathbb{C}^{M}\otimes\mathbb{C}^{N}$ could be useful for
studying arbitrary multi-partite quantum computation in multi-valued systems.

## Acknowledgements

This work is supported by the Fundamental Research Funds for the Central
Universities under Grants FRF-TP-19-011A3, the National Natural Science
Foundation of China under Grant No. 12004281 and 11704281.

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