# Reconfigurable unitary transformations of optical beam arrays

Aldo C. Martinez-Becerril  Department of Physics and Nexus for Quantum
Technologies, University of Ottawa, 25 Templeton St, Ottawa, ON, K1N 6N5
Canada  The Edward S. Rogers Department of Electrical and Computer
Engineering, University of Toronto, 10 King’s College Road, Toronto, ON M5S
3G4, Canada<EMAIL_ADDRESS>Siwei Luo  Department of
Physics and Nexus for Quantum Technologies, University of Ottawa, 25 Templeton
St, Ottawa, ON, K1N 6N5 Canada  Liu Li  Department of Physics and Nexus for
Quantum Technologies, University of Ottawa, 25 Templeton St, Ottawa, ON, K1N
6N5 Canada  Jordan Pagé  Department of Physics and Nexus for Quantum
Technologies, University of Ottawa, 25 Templeton St, Ottawa, ON, K1N 6N5
Canada
Lambert Giner  Département de Physique et d’Astronomie, Université de Moncton,
18 Ave. Antonine-Maillet, Moncton, NB E1A 3E9, Canada  Raphael A. Abrahao
Brookhaven National Laboratory, Upton, New York 11973, USA  Jeff S. Lundeen
Department of Physics and Nexus for Quantum Technologies, University of
Ottawa, 25 Templeton St, Ottawa, ON, K1N 6N5 Canada

###### Abstract

Spatial transformations of light are ubiquitous in optics, with examples
ranging from simple imaging with a lens to quantum and classical information
processing in waveguide meshes. Multi-plane light converter (MPLC) systems
have emerged as a platform that promises completely general spatial
transformations, i.e., a universal unitary. However until now, MPLC systems
have demonstrated transformations that are far from general, e.g., converting
from a Gaussian to Laguerre-Gauss mode. Here, we demonstrate the promise of an
MLPC, the ability to impose an arbitrary unitary transformation that can be
reconfigured dynamically. Specifically, we consider transformations on
superpositions of parallel free-space beams arranged in an array, which is a
common information encoding in photonics. We experimentally test the full
gamut of unitary transformations for a system of two parallel beams and make a
map of their fidelity. We obtain an average transformation fidelity of
$0{.}85\pm 0{.}03$. This high-fidelity suggests MPLCs are a useful tool
implementing the unitary transformations that comprise quantum and classical
information processing.

##  1 Introduction

The transformation of optical spatial modes in a reconfigurable and reliable
way would have a wide range of applications, ranging from fundamental studies
in optics to applications in telecommunications, classical computing, and
quantum information processing. Most passive optical information processing
tasks are unitary transformations $\bm{U}$ (i.e.,
$\bm{U}\bm{U^{\dagger}}=\bm{U^{\dagger}}\bm{U}=\bm{1}$) since they preserve
information in the field. A device that creates a reconfigurable unitary could
implement matrix multiplication [1], optical quantum gates [2, 3] and quantum
random walks [4, 5, 6]. Furthermore, adding a non-linear optical element to
such a device would allow one to obtain an optical neural network [7, 8, 9,
10, 11]. In this work, we experimentally implement arbitrary unitary
transformations in the space of a two-beam array (two parallel optical beams)
using a platform known as multi-plane light converter (MPLC) [12, 13, 14].

An N-dimensional unitary $\bm{U}$ possess $N^{2}$ independent real parameters
that need to be specified to fully determine $\bm{U}.$ To physically implement
a unitary on spatial modes such as optical beams, one typically needs two
elements: a mode coupler and the ability to impart phases to each mode. Two
approaches can be distinguished, one that uses two-mode beamsplitters as mode
couplers, and another that uses an optical Fourier transform or diffraction to
couple modes.

In optics, the first approach was introduced in a seminal paper [15], the Reck
scheme, which demonstrated a deterministic algorithm to decompose a general
unitary of dimension $N$ into phase shifters and beamsplitters. The result is
a ‘mesh’ of beamsplitters that composes a large interferometer with $N^{2}$
arms. Each arm length needs to be mechanically stabilized to much less than a
wavelength. All together, a stable mesh is a monumental challenge,
particularly for modes travelling in free-space [16]. Instead, almost all
demonstrations have used integrated optic waveguide meshes, with
reconfigurability provided by thermal phase-shifters. With the first such a
device, unitary transformations of dimension $N=6$ were used for implementing
quantum gates, boson sampling, and random unitary transformations [17]. This
approach has been extended to create reconfigurable circuits with $N=8$ [18],
$N=12$ [19] and $N=20$ [20] waveguides. Additionally, an alternative to the
Reck scheme was introduced by Clements et al. [21] that minimizes the depth of
the mesh. The beamsplitter mesh approach has proven to be enormously
influential in both quantum and classical information processing.

We now turn to the diffraction approach. An MPLC is composed of layers
alternating between a phase-mask plane and a diffraction layer. The latter is
either created by free-space propagation or an optical Fourier transform. The
phase-mask planes have been typically implemented using a spatial light
modulator (SLM). The MPLC was conceived and demonstrated in Refs. [12, 13,
14]. Initially, Refs. [12, 13] focused on the use of MPLCs to perform mode
multiplexing, and the phase-mask profiles were obtained by numerical
optimization. In Refs. [22, 23] a particular numerical optimization,
‘wavefront matching’, was adapted to MPLCs. They converted each beam in a two-
dimensional grid of beams to a different Laguerre-Gauss optical mode;
achieving the impressive result of converting $210$ optical modes using seven
phase-mask planes. In the recent years, there has been an increasing number of
works that use MPLCs to perform different tasks such as diffractive networks
[24], high-dimensional quantum gates using orbital angular momentum [2, 25],
processing entanglement in high-dimensions [26], spatial mode multiplexing
[27, 28], control of multiple degrees of freedom of light [29, 30], and
sorting states of light [31]. Other works are looking at different platforms
like dielectric slabs [32]. The MPLC has drawn much interest for both
applications and fundamental science.

An MPLC presents a series of advantages over the beamsplitter approach. To
mention a few: (i) in an MPLC one has more phase control channels, as there
are of the order of $10^{6}$ pixels on a single SLM. In the beamsplitter
approach, the number of phase controls is, in practice, limited to order of
$10^{2}$ [19]; (ii) An MPLC can be fully implemented in free-space, thus it
does not suffer from insertion losses to chip and waveguides; (iii) Rather
than being a microfabricated device, an MPLC is composed of common optical
elements, thereby making it implementable in a larger number of laboratories.
The simple design is also simpler to scale to higher dimension states.

One of the main drawbacks of the MPLC compared to the beamsplitter mesh is the
lack of a deterministic analytical algorithm for finding the phase-mask
profiles. Consequently, in an MPLC system, the minimum number of phase-mask
planes needed to create a given unitary of dimension $N$ is unknown. That
said, if there are $N$ pixels in each phase-mask, there must be at least $N$
planes so that one can set all the $N^{2}$ free parameters of the unitary.
This sets a bound for the minimum number of required phase-mask planes. The
seminal work [33] provided an existence proof that an MPLC could implement a
general unitary transformation. More recent work [34] gave an analytical
method to implement any unitary but it requires $6N$ phase-mask planes, far
from the minimum, albeit all but $N$ were fixed phase-mask profiles. At
present, there is no analytical MPLC design for an N-plane $N^{2}$ dimensional
unitary.

Compounding the lack of a analytical design is that, unlike the discrete space
theory mentioned above, a real MPLC acts in a continuous position space to
perform transformations. Moreover, in practice, diffraction replaces the
Fourier transform used in the above mentioned theory. In short, our analytical
understanding of the MPLC is far-removed from how it is used in practice. The
lack of an analytical algorithm means that the optimal transformation fidelity
and its dependence on the MPLC system parameters are unknown. With so much
unknown about MPLC design and performance it is important to characterize it
for a wide range of tasks. The origin of this complication is that an MPLC
uses a continuous position space to perform the transformations, and
diffraction is hard to analytically deal with under the restriction that the
MPLC works entirely on a discrete space.

Most previous implementations of MPLC systems have created a single chosen
transformation. Moreover, usually that transformation was special in some way,
e.g., Laguerre-Gauss modes are propagation eigenstates. It was unclear how
well an MPLC would perform for an arbitrary target unitary. In this work, we
experimentally demonstrate an MPLC’s versatility and reconfigurability by
implementing a set of transformations that densely samples and fully covers
the space of possible unitary transformations. In this way, we show that an
MPLC can be a universal unitary.

Like the original Reck scheme, we aim to create unitary transformations of a
linear array of parallel beams (a beam array). We consider a beam array
because it is a common information encoding in photonics for classical and
quantum information processing. Moreover, a linear array matches the square
grid of pixels in an SLM. Specifically, a linear beam array can be aligned
with the first column of the grid so that subsequent phase-mask planes use
subsequent SLM columns. Or, more abstractly, if the SLM has $N^{2}$ pixels,
that would be sufficient, in principle, to implement a unitary of dimension
$N$. In short, using a beam array on a line nominally makes an effective use
of the active area of the SLM, which could be helpful to scaling an MPLC to
higher-dimensional transformations.

We continue in the next section by introducing the MPLC in more detail,
formally defining a beam array, and describing the wavefront matching
algorithm that we use to find our phase-mask profiles for each unitary in our
full set. We then proceed to describe our experimental setup and to
characterize its performance using the design phase-mask profiles, thereby
creating a map of the transformation fidelity over all unitary
transformations. Finally, we summarize our work and point future research
directions.

##  2 Multi-plane light converter (MPLC)

A single phase-mask is insufficient to create a general spatial unitary. To
see this consider a general two-mode unitary transformation (Eq. (2)). Any
thin phase-grating will couple more than just these two modes [35]. For
example, a sinusoidal or blazed diffraction grating will diffract light into
new orders upon successive applications. This is Raman-Nath diffraction. On
the other hand, thick gratings are able to limit coupling to two and only two
modes through e.g., Bragg diffraction [35], as required by a two-mode unitary.
A thick grating can be pictured as a series of different phase-mask planes
interleaved with spatial propagation, i.e., diffraction. This arrangement, the
MPLC, is exactly what is needed to create a general spatial unitary.

Applying a phase-mask and diffraction are unitary transformations that we
denote by $\bm{\mathcal{P}}$ and $\bm{\mathcal{D}}$, respectively. An MPLC
effectively performs a unitary transformation $\bm{U}$ (assuming the numerical
aperture of the system is large enough) by concatenating $P$ blocks of
$\bm{\mathcal{D}}\bm{\mathcal{P}}$:
$\bm{U}=\bm{\mathcal{D}}_{P}\bm{\mathcal{P}}_{P}\ldots\bm{\mathcal{D}}_{1}\bm{\mathcal{P}}_{1}$.
A schematic of an MPLC is shown in Fig. 1. A flat mirror and an SLM parallel
to each other and $L$ distance apart form the MPLC system. That is, instead of
using a separate SLM for each phase-mask plane, we reflect the light back to
the same SLM. The light is inserted to the MPLC at an angle $\tau$, with
respect to the normal of the SLM, so that each incidence on the SLM is at a
different $x$ position, utilizing a different area on the SLM. In this figure,
we also show a linear beam array along $y$ (red circles) as the set of input
beams on which the transformation is being performed.

Figure 1: Schematic of a multi-plane light converter (MPLC), composed by a
spatial light modulator (SLM) and a flat mirror. These optical components are
separated by a distance $L$. In an MPLC, a set of $P$ phase-mask planes (PP)
(color indicates phase, red $=0$ and green $=\pi$) and diffraction implement a
unitary transformation of dimension $N$. A beam array (circles at PP1) is at
the input of the MPLC with insertion angle $\tau$. At the output plane after
the SLM, we show $\bm{U}\ket{m}$, i.e., the output state after the MPLC when
the input is beam $\ket{m}$ of the beam array.

###  2.1 Beam array

A beam array of dimension $M$ consists of a set of $M$ optical beams
$\ket{m}$, where $m$ is used as an index to label each beam $m=1,\ldots,M$. We
consider a beam array where the centers of each beam are located on a line
parallel to the $y$ direction. The beam spacing (center-to-center separation)
between two adjacent beams is $\Delta y$. We focus on the case where the
electric field $\psi_{m}\left(x,y\right)=\innerproduct{x,y}{m}$ of each beam
is Gaussian: $\psi_{m}\left(x,y\right)\propto
G\left(x\right)G_{m}\left(y\right)$, where
$G_{m}\left(y\right)=e^{-\left(y-y_{m}\right)^{2}/\omega^{2}\left(z\right)}$
and $G\left(x\right)=e^{-x^{2}/\omega^{2}\left(z\right)}$. Here, the beam size
$\omega\left(z\right)$ is the half-width at $1/e^{2}$ intensity of the beam,
which is a function of the propagation distance $z$. The overlap of two beams
exponentially decays with an increasing beam spacing $\Delta y$:
$\innerproduct{m}{m^{\prime}}=\int\psi_{m}^{*}\left(x,y\right)\psi_{m^{\prime}}\left(x,y\right)dxdy=e^{-\left(m-m^{\prime}\right)^{2}\Delta
y^{2}/\left(2\omega^{2}\left(z=0\right)\right)}$ where ∗ indicates complex
conjugation. Two beams are nearly orthogonal if $\Delta
y>\sqrt{2}\omega\left(z=0\right)$, and they become completely orthogonal in
the limit $\Delta y\gg\sqrt{2}\omega\left(z=0\right)$. Thus the beams form an
approximate orthonormal basis for the beam array Hilbert space. A general
state $\ket{\psi}$ in the beam array is a superposition of the $\ket{m}$
beams, i.e.,

$\ket{\psi}=\sum_{m=1}^{M}c_{m}\ket{m},$ (1)

where $c_{m}=\innerproduct{m}{\psi}$. For a single photon, this state can be
understood as the superposition of the amplitudes for the photon to be in each
beam. Or, from the viewpoint of a classical coherent field description, each
amplitude scales each beam’s Gaussian electric field distribution.

###  2.2 Unitary transformations in a beam array

The case of a general unitary transformation $\bm{U}$ on a two-beam array is
given by

$\bm{U}\left(\theta,\phi\right)=\begin{pmatrix}\cos\theta&ie^{-i\phi}\sin\theta\\\
ie^{i\phi}\sin\theta&\cos\theta\end{pmatrix},$ (2)

where $\theta\in[0,\pi/2]$ is the coupling parameter, and $\phi\in[-\pi,\pi)$
is a phase. The other two parameters for a general two-dimensional unitary
(see for example Ref. [36]) are omitted as one of them is a global phase, and
the other is a phase that can be factored as a diagonal matrix and can be
straightforwardly applied after the unitary by applying a different flat phase
shift to each beam. The unitary $\bm{U}\left(\theta,\phi\right)$ is a two-mode
beamsplitter transformation. In this context, one identifies $\theta$ with the
reflectivity $R$ of the beamsplitter as $|R|=\sin^{2}\theta$. The case
$\theta=\pi/4$ corresponds to a 50:50 beamsplitter. The case $\theta=\pi/4$
and $\phi=-\pi/2$ corresponds to the Hadamard transformation (with the beams
swapped) widely used in quantum information [37].

###  2.3 Optimization procedure

Now we describe the general idea behind the optimization procedure to find the
required phase-mask profiles for achieving a given unitary. We follow Ref.
[23] and use a wavefront matching algorithm [38, 22] to obtain each of the
phase-mask profiles in our system. In a general MPLC (as the one shown in Fig.
1), there are $P$ phase-mask planes equally separated by a distance $\Delta
z=2\times L$. The phase distribution at plane $p$ is
$\Phi_{p}\left(x,y\right)$. At the first plane, we have the $M$ input beams
${\ket{m}}$, where $m$ is used as an index to refer to each beam
$m=1,\ldots,M$. The beam array states $\ket{\psi_{t_{m}}}$ at the last plane
are of the form in Eq. (1) and are determined by the target unitary
$\bm{U}_{t}$ i.e., $\ket{\psi_{t_{m}}}=\bm{U}_{t}\ket{m}$. We call design
states ${\ket{\psi_{d_{m}}}}$ the theoretical output states of a given MPLC
design (i.e., a specific set of designed phase-mask profiles), thus
$\ket{\psi_{d_{m}}}=\bm{U}_{d}\ket{m}$, where
$\bm{U}_{d}=\bm{\mathcal{D}}_{P}\bm{\mathcal{P}}_{P}\ldots\bm{\mathcal{D}}_{1}\bm{\mathcal{P}}_{1}$.

The phase-mask profiles $\Phi_{p}$ are obtained through an inverse design
optimization. The goal of which is to minimize the phase difference between
input and target states at every propagation plane by adjusting the phase-mask
profiles $\Phi_{p}$ located at different propagation planes. The phase
mismatch of corresponding input-target states at phase-plane one is given by

$\int\int\text{arg}\left(\psi_{m}\left(x,y,\Delta
z\right)\psi_{t_{m}}^{*}\left(x,y,\Delta z\right)\right)dxdy,$ (3)

for each of the $m=1,\ldots,M$ pairs of input-output states. A superposition
of these phase errors is formed each of them being weighted by the overlap of
the corresponding input-target states of the MPLC system. The updated phase at
phase-mask plane one is given as the phase of such a superposition, this is
mathematically written as follows

$\Delta\Phi_{1}\left(x,y\right)=-\arg\left(\sum_{m}\psi_{m}\left(x,y,\Delta
z\right)\psi_{t_{m}}^{*}\left(x,y,\Delta
z\right)e^{-i\int\text{arg}\left(m\left(x,y,\Delta
z\right)\psi_{t_{m}}^{*}\left(x,y,\Delta z\right)\right)dxdy}\right).$ (4)

A similar procedure is used to update the other phase-mask profiles by
propagating the input and target states to the proper plane. For example, to
update the $p^{\text{th}}$ phase-mask profile, one needs to propagate the
input beams to this plane as
$\bm{\mathcal{P}}_{p}\ldots\bm{\mathcal{D}}_{1}\bm{\mathcal{P}}_{1}\ket{m}$. A
single iteration of the algorithm is conducted by repeating this phase update
for all $P$ phase-mask planes. This iteration is then repeated until the
figure of merit is reduced below a threshold value or a number of iterations
is achieved. The figure of merit is the gate fidelity $F_{G}$ between the
target unitary $\bm{U}_{t}$ from Eq. (2) and the one implemented by the design
MPLC $\bm{U}_{d}$,

$F_{G}(\bm{U}_{t},\bm{U}_{d})=\frac{1}{M}\sum_{m=1}^{M}\Big{\lvert}\bra{m}\bm{U^{\dagger}}_{t}\bm{U}_{d}\ket{m}\Big{\rvert}^{2}.$
(5)

Thus the gate fidelity can be obtained from the electric field of the target
and output design states as

$\displaystyle F_{G}(\bm{U}_{t},\bm{U}_{d})$ $\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{m=1}^{M}\Big{\lvert}\bra{\psi_{t_{m}}}\ket{\psi_{d_{m}}}\Big{\rvert}^{2}$
(6) $\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{m=1}^{M}\Big{\lvert}\int\int\psi_{t_{m}}^{{}^{*}}(x,y)\psi_{d_{m}}(x,y)dxdy\Big{\rvert}^{2}.$

For further details of the algorithm we refer the reader to Ref. [22].

###  2.4 Choice of parameters

In this subsection, we describe how we selected the geometric parameters of
the beam array and the MPLC system. As mentioned in Section 1, the realization
of an N-dimensional unitary requires at least $N$ phase-mask planes. We
estimated the number of possible reflections on an MPLC system, details can be
found in the Supplementary material. For our SLM (dimensions given in Section
3) we found there are up to ten reflections that can be achieved with
realistic parameters. We used five phase-mask planes for our MPLC. This number
is expected to achieve our target transformations from Eq. (2). We proceed to
perform an optimization of the system using the wavefront matching algorithm
as we explain next.

Figure 2:  Theoretical gate fidelity $F_{G}$ (shown in black markers and
obtained using Eq. (6)) between the target unitary
$\bm{U}\left(\theta=3\pi/20,\phi=\pi/2\right)$ and the one implemented by the
design MPLC system with different geometric parameters. The geometric
parameters are the following: distance between phase-mask planes (red), beam
waist (dark blue), and beam spacing (light blue). The range in which each
parameter is sampled is indicated by its maximum and minimum values on the
right. For each set of parameters (labeled by an integer index displayed in
the x axis) the wavefront matching algorithm was used to find the phase-mask
profiles to achieve the target unitary.

We used the wavefront matching algorithm [22, 23] to choose the parameters of
an MPLC system that implements the unitary transformations from Eq. (2). Fig.
2 shows the theoretical gate fidelity for the target unitary
$\bm{U}\left(\theta=3\pi/20,\phi=\pi/2\right)$ and the one implemented by
different MPLC systems (the different MPLC geometric parameters are given in
the caption of Fig. 2). We found the MPLC achieves a theoretical gate fidelity
higher than $0.9$ for a wide range of parameters. On the other hand, a spacing
between phase-mask planes smaller than $5\text{\,}\mathrm{cm}$ combined with a
beam waist greater than $500\text{\,}\mu\mathrm{m}$ leads to a gate fidelity
lower than $0.6$. We selected a spacing between phase-masks of
$10\text{\,}\mathrm{cm}$, for which the MPLC achieves a gate fidelity of $0.9$
with variations of $5\%$ across the sampled values of beam spacing and beam
waist, which suggests the system is robust to experimental imperfections. With
the chosen parameters, we verified the MPLC system achieved a similar
performance when changing the target unitary.

We now list the final design parameters. The wavelength is
$637\text{\,}\mathrm{nm}$ and the SLM’s pixel size is
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. We are using $P=5$ phase-mask
planes and a plane spacing of $2L=$10\text{\,}\mathrm{c}\mathrm{m}$$. Adjacent
input beams have a center-to-center separation of $\Delta
y=$704\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Each beam has a waist (half
width at $1/e^{2}$ intensity) of $161.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$
located $2.08\text{\,}\mathrm{cm}$ before the first phase-mask plane. The
overlap between both beams equals $7.5\text{\times}{10}^{-5}\text{\,}$, thus
they are nearly orthogonal. We apply a linear phase
$e^{iy\pi/\left(18\times$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$\right)}$ on
one of the beams to account for the beam tilt experimentally observed in our
setup. The output beam array is the same as the input one with the beam waist
and beam spacing demagnified by a factor of four. We observed that this
demagnification helps the MPLC to compensate for the diffraction of the beams
and reduces losses due to diffraction out of the light path.

###  2.5 Optimization results

The goal of the optimization is to design the phase-mask profiles to achieve
the target unitary transformations from Eq. (2). The parameters of the MPLC
are the ones listed in Subsection 2.4.

Figure 3:  Predicted performance of the design MPLC system. We plot the
theoretical gate fidelity
$F_{G}\left(\bm{U}\left(\theta,\phi\right),\bm{U}_{d}\right)$ (obtained using
Eq. (6)) between the target unitary $\bm{U}\left(\theta,\phi\right)$
transformation from Eq. (2), and the design one $\bm{U}_{d}$ obtained with the
wavefront matching algorithm. The plot samples the full space of unitary
transformations on a two-beam array.

We used the optimization algorithm to sample the whole space of unitary
transformations from Eq. (2) with the following grid: $\theta\in[0,\pi/2]$ in
steps of $\pi/120$, and $\phi\in[-\pi,\pi)$ in steps of $\pi/60$. Each design
unitary $\bm{U}_{d}$ is the result of an optimization with $100$ iterations.
Each optimization takes $53.806\text{\,}\mathrm{s}$ to run in a desktop
computer with an eight core Intel i7-10700 processor at
$2.9\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ with $16$ GB of RAM (this same
computer was used in the other data processing tasks).

The optimization results are shown in Fig. 3. Averaging $F_{G}$ over the
$\bm{U}\left(\theta,\phi\right)$ space, we found the theoretical average gate
fidelity to be $\bar{F}_{G}=0{.}90\pm 0.04$, where the uncertainty is the
standard deviation. However, regions with significantly higher and lower
fidelities than average can be distinguished. In particular, there is a strip
region in $\theta\approx 0$ which gives the lowest fidelity, which is
surprising since those transformations are close to the identity. The lower
fidelity regions are mostly due to phase mismatch in the output states, which
the wavefront matching algorithm was not able to correct even after increasing
the number of iterations. We now describe the experimental implementation of
unitary transformations in a two-beam array. Here we exploit the
reconfigurability of the MPLC.

##  3 Experiment

Figure 4: Experimental setup for the realization of an arbitrary unitary
transformation on a two-beam array using a multi-plane light converter (MPLC).
Beam array preparation. A polarizing beamsplitter (PBS) and a half wave plate
(HWP) produce light linearly polarized at $45$. A beam displacer shifts the
spatial mode of photons with vertical polarization, and transmits spatially
unshifted photons with horizontal polarization. The beam displacer creates the
two-beam array. After the beam displacer, a HWP and a PBS select the input
state. The transmitted light at the PBS is horizontally polarized, which is
the working polarization of the spatial light modulator (SLM). MPLC. The MPLC
is created with an SLM and a flat mirror. The beam array is inserted to the
MPLC by a non-polarizing 50:50 beamsplitter. Five reflections are achieved on
the SLM. After which, a second 50:50 beamsplitter is used to extract the
output states from the MPLC. The efficiency of the five reflections on the
MPLC (without considering the two beamsplitters) is $10\%$. Output state
reconstruction. A reference beam is created at the first PBS. The reference
beam propagates through a delay line which compensates the optical path
difference of the MPLC. A HWP at $45$ sets the reference beams’s polarization
to be horizontal. The output of the MPLC is imaged on a CMOS image sensor
(camera) by a 4f-lens pair (magnification $1{.}25$). The imaged output states
and the reference beam interfere after a beamsplitter and the interference
pattern is recorded by the camera. Output state reconstruction is performed
computationally from the interference pattern.

The experimental setup is shown in Fig. 4. Our light source is a laser diode
with a wavelength of $637.7\text{\,}\mathrm{nm}$ with a FWHM bandwidth of
$1.3\text{\,}\mathrm{nm}$ coupled to a single-mode fibre. We can describe the
setup in three stages: beam array preparation, MPLC and output state
reconstruction. The two-beam array is created with a birefringent beam
displacer and polarization optics (see caption of Fig. 4 for details). We
characterized the input beam array by measuring the beam size
$\omega\left(z\right)$ at several propagation planes $z$. The measured beam
characteristics were given in Subsection 2.4, where they were used for
defining the theoretical input beams. In the beam array preparation we have
the ability to choose the input beam ($\ket{1}$ or $\ket{2}$), to the MPLC.

Our MPLC implementation uses five reflections on the SLM as the $P=5$ phase-
mask planes. The SLM is a phase-only liquid crystal SLM (Hamamatsu X10468-07)
with an effective area of
$$15.8\text{\,}\mathrm{m}\mathrm{m}$\times$12\text{\,}\mathrm{m}\mathrm{m}$$
and a fill factor of $98\%$. The SLM size is $792\times 600$ pixels, with a
pixel pitch of $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The SLM is designed
to have high throughput efficiency from $620\text{\,}$ to
$1100\text{\,}\mathrm{nm}$. We measured a throughput efficiency of $63\%$ at
$637\text{\,}\mathrm{nm}$, and the manufacturer states a typical one of $82\%$
at $1064\text{\,}\mathrm{nm}$. Each pixel can be set to a value ranging from
$0$ to $256$. We use the calibration provided by the manufacturer, which gives
a linear response from pixel value to phase and a pixel value of $118$ for a
$2\pi$ modulation. We do not use the phase to correct aberrations provided by
the manufacturer. The phase on the SLM $\Phi_{SLM}$ is obtained as follows
$\Phi_{SLM}=\frac{118}{2\pi}\left(\Phi\mod{2\pi}\right)$ where $\Phi$ is the
phase at the five phase-mask planes directly obtained from the wavefront
matching algorithm.

We point out that we are not using the SLM in its most common mode of
operation, which we now describe. Typically, when creating or manipulating
optical modes with SLMs, the transformation is encoded on top of a modulation
of a regular saw-tooth phase pattern, a blazed grating [39, 40]. This is a
unitary process with each diffraction order carrying different information.
Particularly, the light diffracted into the first-order contains the desired
mode. Selecting a single diffraction order makes the process inherently non-
unitary. This mode of operation of the SLM is used to achieve amplitude and
phase modulation with a single hologram [39, 40]. It is also used as a
selection mechanism due to the fact that SLM’s diffraction efficiencies are
lower than $100\%$. In contrast, in the mode of operation we utilize, each
phase-mask the SLM imparts is directly given by the wavefront matching
algorithm. There is no underlying grating. While there may be technical
sources of loss (e.g., fill-factor), our mode of operation can be
fundamentally unitary.

While conducting the experiment, we noticed the last two of the five design
phase-mask profiles are the most important ones to realize the unitary
transformations $\bm{U}\left(\theta,\phi\right)$. To that point, Table 1 shows
the experimental gate fidelity between target transformations
$\bm{U}\left(\theta,\phi=\pi/2\right)$ from Eq. (2) and the experimentally
obtained ones. The performance is same or better using solely the last two
phase-mask profiles while setting the first three phase-mask planes to impart
no phase. When using all five of the design phase-mask profiles, we observed
that for many target unitary transformations the MPLC system produced a
complicated scattered intensity distribution that was greatly different than
the design output state. The physical mechanism for this behaviour is unknown
and a subject for future investigation, however in the Supplementary material
we provide an example of how the performance of an MPLC greatly decreases upon
the presence of a phase-mask profile not taken into account in the design. For
these reasons, the rest of the presented results use solely the last two of
the five design phase-mask profiles.

$\bm{U}\left(\theta,\phi=\pi/2\right)$ |  $F_{G}$ five PPs |  $F_{G}$ two PPs
---|---|---
$\theta=\pi/2$ | $0.91\pm 0.01$ | $0.914\pm 0.009$
$\theta=\pi/3$ | $0.859\pm 0.008$ | $0.85\pm 0.05$
$\theta=\pi/4$ | $0.69\pm 0.06$ | $0.86\pm 0.06$
$\theta=\pi/5$ | $0.83\pm 0.02$ | $0.85\pm 0.06$
$\theta=\pi/6$ | $0.83\pm 0.02$ | $0.82\pm 0.01$
Table 1:  Experimental gate fidelity
$F_{G}\left(\bm{U}\left(\theta,\phi=\pi/2\right),\bm{U}_{e}\right)$,
calculated using Eq. (6), between target unitary transformations
$\bm{U}\left(\theta,\phi=\pi/2\right)$ from Eq. (2) and the experimentally
obtained ones $\bm{U}_{e}$ using all five or only the last two of the five
phase-mask profiles (PP). The positions of the PPs were not changed when
reducing the number of planes. We found using solely two PPs give same or
better performance as five PPs. We indicate the standard deviation over five
trials for statistical error.

We exploit the reconfigurability of the SLM by experimentally sampling the
two-dimensional unitary space $\bm{U}\left(\theta,\phi\right)$. The sampling
used the following grid: $\theta\in[0,\pi/2]$ in steps of $\pi/120$ and
$\phi\in[-\pi,\pi)$ in steps of $\pi/60$. The gate fidelity for each
implemented transformation is obtained by using Eq. (6). As for data
acquisition, we set the phase-mask profiles on the SLM to perform a unitary
$\bm{U}\left(\theta,\phi\right)$. For every input state $\ket{\psi}=\ket{m}$
($m=1,2$), we record five images of the interference pattern. Each of these
images are used to numerically reconstruct the optical field of the output
state by off-axis holography (for a more detailed description of this method
see for example [41, 42, 43]). Acquiring this set of ten images takes
$25\text{\,}\mathrm{s}$. The off-axis holography routine takes
$44.12\text{\,}\mathrm{s}$ using MATLAB. Most of the time is taken by
geometric transformations such as magnifying and flipping that account for the
last 4f system in our experimental setup. This numerical implementation can be
optimized, see for example Ref. [43]. The alignment procedure, and data
acquisition are performed in LABVIEW. Further details of the alignment
procedure can be found in the Supplementary material.

##  4 Results

We first present the results of the method as described so far. We will then
describe how the use of an additional phase-mask plane improved the gate
fidelity considerably.

Figure 5:  Experimental gate fidelity
$F_{G}\left(\bm{U}\left(\theta,\phi\right),\bm{U}_{e}\right)$ (calculated
using Eq. (6)) between target unitary $\bm{U}\left(\theta,\phi\right)$ from
Eq. (2) and the unitary $\bm{U}_{e}$ experimentally implemented by the MPLC
system. These results correspond to Part I: No correcting phase-mask. We
sample the full space of the unitary transformations in a two-beam array given
by Eq. (2). This plot is the experimental analogue of Fig. 3.

###  4.1 Results - Part I: No correcting phase-mask

The results for Part I–No correcting phase-mask are shown in Fig. 5. We show
the experimental gate fidelity $F_{G}\left(\bm{U}_{t},\bm{U}_{e}\right)$,
obtained by using Eq. (6), between target unitary $\bm{U}_{t}$ and the
experimentally implemented by the MPLC $\bm{U}_{e}$. Averaging over all the
unitary space, we got an experimental average gate fidelity equal to
$\bar{F}_{g}=0{.}67\pm 0{.}17$. This plot shows our experiment is able to
implement a large set of unitary transformations with a gate fidelity greater
than $0.8$. However, there are regions of the unitary space implemented with
an experimental gate fidelity lower than $0.5$ in the center of the plot. This
contrast in gate fidelity is not expected from the design MPLC (Fig. 3), which
predicts values of gate fidelity greater than $0.77$. We next investigated
ways to improve the MPLC in order to obtain a better performance over the full
unitary space.

We now describe the approach we followed for improving the average gate
fidelity of our experiment. Observe that in Fig. 5 there is a region at
$\phi=\pi/2$ where the fidelity is higher than $0{.}8$ and independent of
$\theta$. Such a strip of transformations accounts for the value of $\theta$
of the transformation $\bm{U}\left(\theta,\phi\right)$. We use an additional
phase-mask plane to adjust the phase $\phi$ to create any target unitary
$\bm{U}\left(\theta,\phi\right)$. This is the method experimentally used that
we now report in the next subsection.

Figure 6: Performance of the design MPLC system plus a correcting phase-mask
plane as described in Subsection 4.2. We plot the theoretical gate fidelity
$F\left(\bm{U}\left(\theta,\phi\right),\bm{U}_{d}\right)$ (Eq. (6)) between
the target unitary $\bm{U}\left(\theta,\phi\right)$ from Eq. (2), and the one
implemented by the MPLC $\bm{U}_{d}$. This MPLC uses the set of
transformations $\bm{U}\left(\theta,\phi=\pi/2\right)$ from Fig. 3 and an
additional phase-mask plane to sample the unitary transformations of Eq. (2).

###  4.2 Results - Part II: With correcting phase-mask

Our experimental results presented in Fig. 5 show a non-ideal performance of
our MPLC over the unitary space. In other words, there are regions with
contrasting fidelities. To improve the performance of the MPLC, we use the
line of designs along $\bm{U}\left(\theta,\phi=\pi/2\right)$ and an additional
phase-mask profile located at a sixth phase-mask plane of the MPLC to achieve
any other transformation $\bm{U}\left(\theta,\phi\right)$. This phase-mask
applies a phase difference

$\Delta\phi=\left(\phi-\pi/2\right)/2,$ (7)

to the top beam and $-\Delta\phi$ to the bottom one. To test the correcting
phase-mask approach, we applied it to the design MPLC described in Section
2.5. The resulting theoretical gate fidelity of the design MPLC with
correcting phase-mask is shown in Fig. 6. It has an average gate fidelity
$\bar{F}_{G}=0{.}92\pm 0{.}04$, which has a percentage difference of $2.2\%$
respect to the one of Fig. 3. Thus our correcting phase-mask approach
preserves the performance of the original theoretical system design. The lower
performance of transformations with $\theta=0-6\pi/120$ is due to a phase
gradient of $\pi$ across each beam, which is not fixed by the wavefront
matching algorithm.

For the experimental implementation of the correcting mask approach, we used
an additional SLM located at the output plane of the MPLC system from Fig. 4.
This SLM is a Pluto-2 phase only SLM with a dielectric mirror at its back
plane to enhance reflectivity in the range of
$$730\text{\,}$-$950\text{\,}\mathrm{nm}$$. It achieves a maximum modulation
of $2{.}4\pi$ with a pixel value of $255\text{\,}$ at
$637\text{\,}\mathrm{nm}$. Implementing the correcting mask approach yielded
the results shown in Fig. 7. Averaging over the unitary transformation space,
we obtained an experimental average gate fidelity of $\bar{F}_{G}=0{.}85\pm
0{.}03$. This is a much improved result when compared to Fig. 5 (note the
change in scale).

Fig. 8 shows a sample of experimentally obtained output states for different
unitary transformations $\bm{U}\left(\theta,\phi\right)$ when the input state
is the beam $\ket{1}$. The amplitude and phase of the fields are shown in the
brightness and hue of the plot respectively. This figure explicitly
demonstrates the ability of an MPLC to implement the full gamut of unitary
transformations. The obtained output states cover the full range of amplitude
and phase a state in the beam array can have. As a result, we are also able to
generate any state in the two-beam array.

Figure 7: Experimental gate fidelity
$F_{G}\left(\bm{U}\left(\theta,\phi\right),\bm{U}_{e}\right)$ (Eq. (6))
between target unitary $\bm{U}\left(\theta,\phi\right)$ from Eq. (2) and the
unitary $\bm{U}_{e}$ experimentally implemented. These results correspond to
Part II: With correcting phase-mask. Figure 8: Field of experimentally
obtained output states for different transformations
$\bm{U}\left(\theta,\phi\right)$. The blue grid separates different states
corresponding to a different transformation. The values of $\theta$ and $\phi$
are indicated on the axes. In this sample, the input state is the beam
$\ket{1}$ of the beam array. The last column shows the target phase (TP) for
the fields on the same row. The field amplitude of each state was normalized
to have the same maximum amplitude. This was done for image visibility
purposes.

##  5 Discussion

As we mentioned earlier, the modeled performance of the design MLPC (Fig. 3)
is significantly higher than the observed experimental performance (Fig. 5).
We now outline an example of potential experimental imperfections that
effectively decrease the performance of an MPLC system. Examples of such
experimental imperfections include light scattering, misalignment of the
system, mismatch between design and experimentally available beams,
positioning of phase-mask planes, SLM’s surface curvature, inherent grid in
pixel structure of an SLM, etc. Modelling these imperfections is challenging
and they become more prominent with an increasing number of phase-mask planes.
We calculated the effect of a not unrealistic phase-mask distortion in an
MPLC. Such a phase-mask perturbation has an average (averaged over the five
locations of the reflections on the SLM) gradient equal to $0.058\pm 0.02$
$\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\text{SLM pixel}$. This
perturbation causes a $20\%$ drop in the theoretical gate fidelity compared to
the original design MPLC. The details of this calculation can be found in the
Supplementary material.

An MPLC system could be used as an auxiliary device in a quantum computer
e.g., for state preparation or in the implementation of gates. For such an
application an SLM with higher efficiency is needed. At present there are SLMs
with less than $3\%$ optical loss within a narrow band light spectral region.
This low loss is achieved by using a dielectric mirror behind the liquid
crystal array.

The realization of reconfigurable unitary transformations in a high-
dimensional beam array is the ultimate goal of an MPLC system. Such a device
would allow one to use linear optics in free-space for implementing and
studying boson sampling experiments, the implementation of quantum gates and
random walks which have many applications, see for example [44]. Another
interesting application is the use of an MPLC as a Meta-optic device,
implementing arbitrary spatial transformations such as imaging, spatial
filtering, or as a spaceplate [45].

##  6 Conclusions

As suggested by the experimentally obtained results, an MPLC system is a
challenging system to characterize and it is sensitive to experimental errors
which are hard to eliminate. On the other hand, the reconfigurability feature
of an MPLC allows experimentalists to adjust input states and phase-mask
profiles of the MPLC system, and to measure the output states after such
changes. Such an experimental information can potentially be used to optimize
the phase-mask profiles of the MPLC and account for experimental imperfections
that were not taken into account at the design stage. There are different
approaches to this experimentally optimized version of an MPLC system. For
example the phase-mask profiles could be optimized by machine learning
techniques e.g., neural networks, genetic algorithms, stochastic sampling or
by the optimization of an analytical model for the phase-mask profiles. This
is undergoing work in our lab and we leave details for future work.

In summary, we experimentally demonstrated a reconfigurable MPLC capable of
implementing arbitrary unitary transformations in a two-beam array. To do so,
we had to substantially modify the design created by the standard method,
wavefront matching optimization. Firstly, we had to omit all but the last two
of the five phase profiles output from the optimization. And secondly, we had
to use a second SLM to apply an additional phase profile after the nominal
MPLC. The latter phase profile was determined from the characterization of
that MPLC. Nonetheless, we demonstrate a gate fidelity of
$\bar{F}_{G}=0{.}85\pm 0{.}03$ that for the first time is found by testing and
averaging over all unitary transformations. Thus, this work demonstrates the
usefulness of MPLC systems and how to use them in a reconfigurable way,
leading to a novel photonics tool that can benefit both classical and quantum
optics.

## Acknowledgments

This work was supported by the Canada Research Chairs Program, the Natural
Sciences and Engineering Research Council, the Canada First Research
Excellence Fund (Transformative Quantum Technologies), the U.S. Department of
Energy QuantISED award, and the Brookhaven National Laboratory LDRD Grant No.
22-22.

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## Supplementary material

### Number of reflections in an MPLC system

In this section we calculate the number of reflections achievable in an MPLC
composed by a flat mirror and an SLM as shown in Fig. 1. The SLM-to-mirror
distance is $L$ and the insertion angle is $\tau$. The optical path length
between two consecutive reflections is $2L\sec\tau.$ We assume a Gaussian beam
with beam waist $\omega_{0}$ at the first plane. The beam size is
$\omega\left(z\right)=\omega_{0}\left(1+\left(z/z_{0}\right)^{2}\right)^{1/2}$
where $z$ is the propagation distance. Thus, the beam size at reflection $p$
on the SLM is given by

$\omega_{0}\left(1+\left(2Lp\sec\tau/z_{0}\right)^{2}\right)^{1/2}.$ (8)

Now we count the number of reflections on the SLM, such a number is
constrained by the SLM’s dimensions. The other constraint is avoiding overlap
between consecutive reflections. Given that $99\%$ of a Gaussian beam power
lies within a circle of radius $1{.}5\omega\left(z\right)$ [46]. We consider
there is no overlap if there is at least a distance equal to
$1{.}5\omega\left(z\right)$ (plus a few pixels to distinguish the reflections)
between two reflections.

Reflections on SLM in an MPLC
a) $L=$1\text{\,}\mathrm{cm}$$ b) $L=$10\text{\,}\mathrm{cm}$$

Figure 9: Number of reflections on an SLM (dimensions given in Section 3) in
an MPLC system as a function of the insertion angle $\tau$ and the beam waist
$\omega_{0}$ located at the first reflection plane. In a) the SLM-mirror
distance $L$ is set to $1\text{\,}\mathrm{c}\mathrm{m}$ and
$10\text{\,}\mathrm{c}\mathrm{m}$ in b). The regions with negative values
indicate overlap between reflections at that particular reflection. For
$L=$10\text{\,}\mathrm{cm}$$ the angle $\tau$ is restricted to be greater than
$0.7\text{\,}\mathrm{\SIUnitSymbolDegree}$ to achieve multiple reflections. As
$L$ decreases, the beam size is constrained to a few hundreds of microns.

We search the number of reflections numerically using Eq. (8) and above
mentioned constraints. Fig. 9 shows the number of reflections on the SLM (SLM
details given in Section 3) as a function of $\omega_{0}$ and $\tau$ for two
different distances $L$. The maximum number of reflections for different
values of $L$ is shown in Fig. 10. More reflections are achieved as the value
of $L$ decreases, as expected. As $L$ decreases, the amount of diffraction
also decreases. We anticipate this would have an effect in the phase-mask
profiles required to make a target unitary transformation.

Maximum number of phase-mask planes in an MPLC

Figure 10: Maximum number of phase-mask planes in an MPLC system (built with a
single SLM) as a function of the mirror-SLM separation $L$.

### Alignment of the MPLC

In what follows we give further information of the experimental implementation
of our MPLC system. The choice of parameters of the MPLC and beam array
characterization was detailed in Subsection 2.4. The experimental setup was
described in Section 3. We will now explain how to set the position of the
phase-mask planes on the SLM.

We use a ‘diffractive knife-edge’ method to locate the centers of the beam
reflections on the SLM. This method is a diffraction analogue of the widely
known ‘knife-edge method’ for Gaussian beam characterization [47]. The knife-
edge method records the transmitted light power after a knife-edge is moving
transversal to a beam. Such information is used to obtain the beam center and
beam size [47] at the knife plane. We now describe how this technique can be
emulated with an SLM. A phase-mask of random phase values (a random phase-
mask) diffracts light in all directions, which effectively creates loss in the
far field. Thus, a phase-mask with a translating rectangular pattern of random
phase values has the same effect as the translating knife-edge. In an MPLC,
the random phase-mask should be chosen so that it covers the full beam without
overlap between different reflections on the SLM. The transmitted power after
passing by each reflection is used for obtaining the center and beam size at
each phase-mask plane following the same computation from the knife-edge
method (see for example Ref. [47]). This computation requires post-processing
as the data needs to be noise filtered to be used for computing a derivative.

A rough estimate of the beam size and beam center can be done with a
modification of the diffractive knife-edge method. One can use a random phase-
mask of the expected beam size and scan it through the SLM. When the random
phase-mask overlaps with a beam reflection, the beam will disappear in the
far-field. Position and size of the random phase-mask can be adjusted for an
estimate of the beam size and center.

In the alignment of our experiment, described in Section 3, we used a random
phase-mask obtained from a Gaussian distribution with center in 60 and
standard deviation of ten. Each value of the random phase-mask is converted to
an unsigned integer to be displayed on the SLM.

For an MPLC, the centers of the phase-mask profiles need to be further
optimized. We use a trial transformation $\bm{U}\left(\theta,\phi\right)$ (Eq.
(2)) and use the intensity on the camera after we apply a new single phase-
mask at a time. The figure of merit to optimize is the ‘intensity fidelity’
$F_{I}\left(I_{d},I_{e}\right)$ defined by the following equation

$F_{I}\left(I_{d},I_{e}\right)=\int I_{d}\left(y\right)I_{e}\left(y\right)dy,$
(9)

where $I_{d}\left(y\right)$ is a theoretical intensity and
$I_{e}\left(y\right)$ is the experimental intensity we are optimizing. Such
intensities $I_{y}\left(y\right)$ along $y$ can be obtained from an x-y
intensity $I\left(x,y\right)$ by integrating along $x$, i.e.,
$I_{y}\left(y\right)=\int I_{y}\left(x,y\right)dx/\left(\int\left(\int
I(x,y)dx\right)^{2}dy\right)^{1/2}$. This intensity fidelity is a parameter
between zero (not matching) and one (perfect matching).

We start by the first phase-mask plane, the center of the first phase-mask is
scanned along the $y$ direction in steps of one pixel while the other phase-
masks are set to zero. The experimental intensity $I_{e}\left(x,y\right)$ is
measured on the CCD camera at the last plane of the MPLC system. This process
was previously simulated with the design phase-mask profiles giving us a
theoretical design intensity $I_{d}\left(x,y\right)$. The center of the first
phase-mask profile is scanned to maximize the intensity fidelity
$F_{I}\left(I_{d},I_{e}\right)$. We repeat the same procedure for the $x$
direction. We noticed the $y$ direction is more sensitive to changes compared
to the $x$ direction.

Having optimized the first phase-mask, we apply the first two phase-mask
profiles and repeat the fine tuning procedure only for the second phase-mask
plane. We repeat this procedure with the rest of the phase-mask planes.
Ideally the found positions are fixed and should work for any transformation
$\bm{U}\left(\theta,\phi\right)$.

Figure 11: Theoretical gate fidelity
$F_{G}\left(\bm{U}\left(\theta=\pi/4,\phi=\pi/2\right),\bm{U}_{pert}\right)$
between target unitary $\bm{U}\left(\theta=\pi/4,\phi=\pi/2\right)$ and the
unitary $\bm{U}_{pert}$ implemented by a perturbed MPLC system. The phase-mask
perturbation is given by $\alpha\Phi_{pert}$ with $\Phi_{pert}$ being the
correction phase pattern provided by the manufacturer of the SLM. This phase
was used as an example of a not unrealistic perturbation that can appear in an
MPLC system.

###  Effect of a perturbation on the phase-mask planes

We found a mismatch in the performance of the design MPLC (Fig. 3) and the
experimental one (Fig. 5). We now give an example of how a perturbation to the
imparted phase-mask profiles decreases the performance of an MPLC.

Phase-mask profiles for the perturbed MPLC with coefficient $\alpha=0.25$

Figure 12:  Phase-mask profiles for the design and perturbed MPLC systems.
First row shows the design phase-mask profiles at each of the $p=1,\ldots,5$
phase-mask planes for the unitary
$\bm{U}\left(\theta=\pi/4,\phi=\pi/2\right)$. Second row shows the
perturbation phase-mask $\alpha\Phi_{pert}$ with $\alpha=0.25$. The third row
shows the phase addition of rows one and two according to Eq. (10).

The procedure is as follows. We used the design phase-mask profiles
$\Phi\left(x,y\right)$ obtained in Subsection 2.5 to implement the unitary
transformations $\bm{U}\left(\theta,\phi\right)$ (from Eq. (2)). We added a
phase-mask perturbation $\alpha\Phi_{pert}\left(x,y\right)$. Here, $\alpha$ is
a coefficient and $\Phi_{pert}\left(x,y\right)$ is the phase perturbation. The
phase-mask profile imparted by the perturbed MPLC system is given by the sum
of $\Phi\left(x,y\right)$ and $\alpha\Phi_{pert}$ modulo $2\pi$, which is
mathematically written as follows

$\Phi\left(x,y\right)+\alpha\Phi_{pert}\left(x,y\right)\mod{2\pi}.$ (10)

We took $\Phi_{pert}$ to be the one provided by the manufacturer to correct
for the surface curvature of the SLM, i.e., the SLM applies an spatially
varying phase even when no phase-mask is being displayed on it; the correction
pattern $\Phi_{pert}$ is meant to cancel such an effect. Such a phase
perturbation has an average gradient equal to $0.058\pm
0.02\leavevmode\nobreak\
$\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\text{SLM pixel}$$. The
average was taken over the five reflections on the SLM.

We now assess the performance of the perturbed MPLC system. We propagate each
of the beams from the beam array ($\ket{1}$ or $\ket{2}$) through the
perturbed MPLC and use Eq. (6) to obtain a gate fidelity. Fig. 11 shows the
theoretical gate fidelity between the transformation implemented by the
perturbed MPLC (with varying coefficient $\alpha$) and a target unitary
$\bm{U}\left(\theta=\pi/4,\phi=\pi/2\right)$. The gate fidelity dramatically
drops with increasing values of $\alpha$. This trend also appears for
transformations with different values of $\theta$.

The phase-mask profiles obtained from this example are shown in Figs. 12 and
13 for $\alpha=0.25$ and $\alpha=1.25$ respectively. This example shows that
not unrealistic phase perturbations dramatically affect the performance of an
MPLC system. We finally mention the correction pattern $\Phi_{pert}$ was not
used in our experiment because it distorted the beams even when no other phase
was added to $\Phi_{pert}$.

Phase-mask profiles for the perturbed MPLC with coefficient $\alpha=1.25$

Figure 13:  Phase-mask profiles for the design and perturbed MPLC systems.
First row shows the design phase-mask profiles at each of the $p=1,\ldots,5$
phase-mask planes for the unitary
$\bm{U}\left(\theta=\pi/4,\phi=\pi/2\right)$. Second row shows the
perturbation phase-mask $\alpha\Phi_{pert}$ with $\alpha=1.25$. The third row
shows the phase addition of rows one and two according to Eq. (10).


# Regularity of linearly presented squarefree monomial ideals

Hailong Dao  Department of Mathematics, University of Kansas, Lawrence, KS
66045, USA<EMAIL_ADDRESS>and  Thanh Vu  Institute of Mathematics, VAST, 18
Hoang Quoc Viet, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

We prove a sharp bound for the regularity of a squarefree monomial ideal with
a linear presentation. This result also answers in positive a question on the
cohomological dimension of squarefree monomial ideals satisfying Serre’s
$S_{2}$-condition proposed by Dao and Takagi.

###### Key words and phrases:

Regularity, Green-Lazarsfeld index, squarefree monomial ideal, cohomological
dimension, Serre’s condition

###### 2020 Mathematics Subject Classification:

13D02, 13D05, 13H99

##  1\. Introduction

In this work, we establish bounds on the Castelnuovo-Mumford regularity of
squarefree monomial ideals with a linear presentation.

######  Theorem 1.1.

Let $I\subseteq S=\mathrm{k}[x_{1},\ldots,x_{n}]$ be a non-zero squarefree
monomial ideal generated in degree $d$. Suppose that $I$ has linear first
syzygies. Then

$\operatorname{reg}(I)\leq\max\left\\{d,\left\lfloor\frac{(d-1)n}{d+1}\right\rfloor+1\right\\}.$

Our bound is sharp for all odd $d>2$, and it was obtained via an elementary,
but quite intricate induction process. It is somewhat surprising that such a
simple statement is not known until now, despite the intense attention on the
topic especially in recent years, which we will describe shortly below.

To help explain fully the context of our work, it is worth restating our main
result via standard equivalences using the Alexander dual as follows:

######  Corollary 1.2.

$I\subseteq S=\mathrm{k}[x_{1},\ldots,x_{n}]$ be a non-zero squarefree
monomial ideal of height $c$. If $S/I$ satisfies Serre condition $(S_{2})$,
then

$\operatorname{pd}(S/I)=\operatorname{cd}(S,I)\leq\max\left\\{c,\left\lfloor\frac{(c-1)n}{c+1}\right\rfloor+1\right\\}.$

where $\operatorname{pd}$ denotes projective dimension and $\operatorname{cd}$
denotes the cohomological dimension.

Our result was first suggested in the study of the cohomological dimension.
Let $I$ be a non-zero ideal in a regular local ring $S$. Recall that
$\operatorname{cd}(S,I)$, the cohomological dimension of $I$ is defined as the
supremum of $i$ such that $H_{I}^{i}(M)\neq 0$ for some $S$-module $M$.
Finding bounds on this invariant was proposed by Grothendieck, and such
studies have been undertaken by many researchers, see for instance [PS, Ha,
HL, Va].

When $S$ contains a field $\mathrm{k}$, Faltings [F] proved that

$\operatorname{cd}(S,I)\leq
n-\left\lfloor\frac{n-1}{\operatorname{bigheight}(I)+1}\right\rfloor,$

where $\operatorname{bigheight}(I)$ is the big height of $I$. With no further
restriction on $I$, this is the best bound possible.

Dao and Takagi [DT] studied the cohomological dimension of ideals satisfying
Serre’s condition $(S_{i})$ and proposed the following strengthening of the
work of Huneke and Lyubeznik [HL], who proved such a bound for $I$ such that
$S/I$ is normal:

######  Question 1.3.

[DT, Question 3.10] Let $S$ be an excellent regular local ring containing a
field and $I\subset S$ be a proper ideal of height $c$. Assume that $S/I$
satisfies Serre’s $(S_{2})$ condition. Is it always true that

$\operatorname{cd}(S,I)\leq
n-\left\lfloor\frac{n}{c+1}\right\rfloor-\left\lfloor\frac{n-1}{c+1}\right\rfloor?$

When $I$ is a squarefree monomial ideal of height $2$, Dao, Huneke, and
Schweig [DHS] proved that $\operatorname{cd}(S,I)=O(\log(n))$. Besides that,
little is known even when $I$ is a squarefree monomial ideal. When $I$ is a
squarefree monomial ideal, by the results of Terai [T], Singh and Walther [SW]
we have that

$\operatorname{cd}(S,I)=\operatorname{pd}(S/I)=\operatorname{reg}(I^{\vee}),$

where $\operatorname{pd}(S/I)$ is the projective dimension of $S/I$ and
$\operatorname{reg}(I^{\vee})$ is the Castelnuovo-Mumford regularity of the
Alexander dual of $I$. Furthermore, by [Y, Corollary 3.7], the condition that
$I$ has height $c$ and satisfies Serre $S_{2}$-condition is equivalent to the
condition that $I^{\vee}$ is generated in degree $c$ and has linear first
syzygies. It is not hard to see that our Corollary 1.2 answers in affirmative
Question 1.3 for squarefree monomial ideals.

To give a broader context for our work from another perspective, we review
some literature on ideals with $k-1$ linear steps of resolutions, called
$N_{d,k}$ ideals. In algebraic geometry, such conditions when $d=2$ were
studied by Green and Lazarsfeld [GL], Eisenbud, Green, Hulek, and Popescu
[EHGP] and others under the name $N_{2,k}$. In combinatorics, the Alexander
dual of a square-free $N_{d,k}$ ideal gives a Stanley-Reisner ring satisfying
Serre’s condition $(S_{k})$, whose algebraic and combinatorial properties
display remarkable similarities to the Cohen-Macaulay situation, as shown by
the work of Murai and Terai [MT] and others. The diameter of the dual graph of
such rings has been studied in proposed modifications of the Hirsh conjecture
[AB]. Constantinescu, Kahle, and Varbaro [CKV] discovered a stunning
connection between such ideals and geometric group theory. They proved that
for a Coxeter group $W$, the virtual cohomological dimension of $W$ is equal
to the maximal of the regularity of the Stanley-Reisner ring of its nerve
complex $\mathcal{N}(W)$. The defining ideal of this Stanley-Reisner ring is
linearly presented and quadratic if and only if $W$ is a right-angled
hyperbolic Coxeter group. This connection allows them to build a series of
examples of such ideals whose regularity equals $O(\log(\log(n)))$, answering
a question raised in [DHS].

Recently, Dao and Eisenbud [DE] gave a sharp upper bound for the regularity of
$N_{d,k}$ primary monomial ideals. Studying ideals whose regularity achieves
the upper bound is an interesting topic. We refer to [DE] for further
information.

In the next section, we give the necessary background and prove our main
theorem.

##  2\. Linearly presented squarefree monomial ideals

Let $S=\mathrm{k}[x_{1},\ldots,x_{n}]$ be a standard graded polynomial ring
over a field $\mathrm{k}$. We first recall some definitions and results about
ideals with partial linear resolutions.

######  Definition 2.1.

We say that a homogeneous ideal $I\subseteq S$ satisfies property $N_{k}$
whenever the minimal free resolution of $I$ over the polynomial ring $S$ is
linear for $k-1$ steps. When $I$ satisfies property $N_{2}$, we also call $I$
linearly presented.

For a non-zero monomial $f\in S$, the support of $f$, denoted by
$\operatorname{supp}(f)$ is the set of all variables $x_{i}$ such that $x_{i}$
divides $f$. For the rest of the paper, we assume that $I$ is a monomial ideal
of $S$ with the unique monomial minimal generating set
$G(I)=\\{f_{1},\ldots,f_{r}\\}$. The support of $I$ is
$\operatorname{supp}(I)=\bigcup_{i=1}^{r}\operatorname{supp}(f_{i})$. For a
subset $U\subset[n]=\\{1,\ldots,n\\}$, we denote by $I(U)$ the restriction of
$I$ to $U$, i.e., $I(U)=(f_{i}\mid\operatorname{supp}(f_{i})\subseteq U)$.
First, we have

######  Lemma 2.2.

Let $I$ be a monomial ideal of $S$. Assume that $I$ satisfies $N_{k}$. Then
for any subset $U\subseteq[n]$, $I(U)$ satisfies $N_{k}$.

###### Proof.

The conclusion follows from [OHH, Corollary 2.5]. ∎

Now suppose that all generators of $I$ have degree $d$. We define the graph
$G_{I}$ whose vertex set is $G(I)$ and $\\{u,v\\}$ is an edge of $G(I)$ if and
only if $\deg(\operatorname{lcm}(u,v))=d+1$. For all $u,v\in G(I)$, let
$G_{I}(u,v)$ be the induced subgraph of $G_{I}$ with vertex set

$V(G_{I}(u,v))=\\{w\in G(I)\mid w\text{ divides }\operatorname{lcm}(u,v)\\}.$

The following criterion for a monomial ideal to satisfy property $N_{2}$ can
be deduced from the work of [GPW]. See [BHZ, Proposition 1.1] or [DE,
Proposition 2.2].

######  Lemma 2.3.

Let $I$ be a monomial ideal generated in degree $d$. Then $I$ satisfies
$N_{2}$ if and only if $G_{I}(u,v)$ is connected for all $u,v\in G(I)$.

Let $I$ be a monomial ideal and $f$ a monomial of $S$. We denote by $I_{f}$
and $\bar{I}_{f}$ the following monomial ideals

$I_{f}=(g\in I\mid f\text{ divides }g)\text{ and }\bar{I}_{f}=(g/f\mid g\in
I_{f}).$

######  Lemma 2.4.

Assume that $I$ satisfies $N_{2}$ and $f$ is a monomial of $S$. Then $I_{f}$
and $\bar{I}_{f}$ satisfy $N_{2}$.

###### Proof.

We may assume that $f$ divides a minimal generator of $I$ and that $f$ is not
a minimal generator of $I$ itself. By Lemma 2.3, we deduce that $I_{f}$
satisfies $N_{2}$. Now, for any minimal generators $u,v$ of $\bar{I}_{f}$, we
have $G_{\bar{I}_{f}}(u,v)$ is isormorphic to $G_{I_{f}}(fu,fv)$. The
conclusion follows from Lemma 2.3. ∎

Let $I\subseteq S$ be a squarefree monomial ideal and $d\in\mathbb{N}$ a
positive integer. We denote by $I_{[d]}$ the squarefree monomial ideal
generated by all squarefree monomials of degree $d$ in $I$.

We have the following key lemma to prove our main result.

######  Lemma 2.5.

Let $I$ be a squarefree monomial ideal generated in degree $d$. Let $f$ be a
squarefree monomial of $S$ such that $2\leq\deg(f)\leq d$,
$I\not\subseteq(f)$, and $(I+(f))_{[d]}$ satisfies $N_{2}$. Note that $f$
might be a minimal generator of $I$. Then there exists a minimal generator
$f_{1}$ of $I$ such that $\deg(\gcd(f_{1},f))=\deg(f)-1$ and
$(I+\gcd(f_{1},f))_{[d]}$ satisfies $N_{2}$.

###### Proof.

For simplicity of notation, we denote by $J=(I+(f))_{[d]}$. For ease of
reading, we divide the proof into several steps.

Step 1. Existence of $f_{1}$. Since $I\not\subseteq f$, there exists a minimal
generator $h$ of $I$ such that $f$ does not divide $h$. Let $g=\gcd(f,h)$ and
$f=gf_{1}$, $h=gh_{1}$. If $\deg(g)=\deg(f)-1$ then we are done. Thus, we may
assume that $\deg(f)-\deg(g)>1$. Let $h_{2}$ be any divisor of $h_{1}$ such
that $\deg(f)+\deg(h_{2})=\deg(h)=d$. Note that if $\deg(f)=d$ then $h_{2}=1$.
Since $\gcd(f,h_{1})=1$, $fh_{2}$ is a squarefree monomial and
$\deg(fh_{2})=d$. By assumption $J$ satisfies $N_{2}$, hence $G_{J}(fh_{2},h)$
is connected. Since $h_{2}|h$, $G_{J}(fh_{2},h)$ is isomorphic to
$G_{\bar{J}}(f,h/h_{2})$ where $\bar{J}=\bar{J}_{h_{2}}$. By Lemma 2.3 and
definition, there exists $u\in G_{\bar{J}}(f,h/h_{2})$ such that $u\in
G(\bar{J})$ and $\deg(\gcd(u,f))=\deg(f)-1$. Let $f_{1}=uh_{2}$, then
$f_{1}\in I$ and $\deg(\gcd(f,f_{1}))=\deg(f)-1$.

Now, we let $g=\gcd(f,f_{1})$ and $K=(I+(g))_{[d]}$. We prove by induction on
$d$ and $n$ that $K$ satisfies $N_{2}$. Let $u,v$ be any two minimal
generators of $K$. It suffices to show that there exists a path in
$G_{K}(u,v)$ that connects $u$ and $v$. By Lemma 2.3, we may assume that $u\in
I$, $u\notin(g)$, $v\in(g)$ and $v\notin I$.

Step 2. Reduction to the case $\gcd(u,g)=1$. Let $v=gv_{1}$ and
$g_{1}=\gcd(u,g)$. If $g_{1}\neq 1$, we may consider $\bar{J}_{g_{1}}$ and
$\bar{K}_{g_{1}}$ and by induction on $d$, we deduce that
$G_{\bar{K}_{g_{1}}}(\bar{u},\bar{v})$ is connected. Hence, we may assume that
$g_{1}=1$. Also, by Step 1, we may write $f=gx$ where $x$ is a variable of
$S$.

Step 3. Reduction to the case $v_{1}|u$. Let $u_{1}$ be any divisor of $u$
such that $\deg(u_{1})=\deg(v_{1})$. We have $gu_{1}|\operatorname{lcm}(u,v)$.
Hence, $gu_{1}$ is a vertex in $G_{K}(u,v)$. Clearly, we have a path in
$G_{K}(u,v)$ that connects $gu_{1}$ and $gv_{1}$. We may replace $gv_{1}$ by
$gu_{1}$ if necessary, so we may assume that $v_{1}|u$. In other words, we may
assume that $u=v_{1}h$.

Step 4. Reduction to the case where $x$ does not divide $h$. Assume that
$x|h$. Let $v_{2}$ be any divisor of $v_{1}$ such that
$\deg(v_{2})=\deg(v_{1})-1$. Then, the monomial $gxv_{2}\in J$. By assumption,
$G_{J}(u,gxv_{2})$ is connected. Note that since
$\operatorname{lcm}(u,gxv_{2})|\operatorname{lcm}(u,v)$, all vertices of
$G_{J}(u,gxv_{2})$ are also vertices of $G_{K}(u,v)$. Hence, $u$ is connected
to $gxv_{2}$, and $gxv_{2}$ is connected to $gv_{1}=v$ in $G_{K}(u,v)$. Hence,
$u$ is connected to $v$ in $G_{K}(u,v)$. Thus, we may assume that $x$ does not
divide $h$.

Step 5. Reduction to the case where $v_{1}$ is a variable. In recap, now we
have $u=v_{1}h$, $v=gv_{1}$ and $f=gx$ where
$\operatorname{supp}(v_{1}),\operatorname{supp}(g),\operatorname{supp}(h)$ and
$\\{x\\}$ are disjoint. Assume that $\deg(v_{1})>1$. We let $v_{2}$ be any
divisor of $v_{1}$ such that degree $\deg(v_{2})=\deg(v_{1})-1$. By Lemma 2.4,
$\bar{J}_{v_{2}}$ satisfies $N_{2}$ and
$\bar{K}_{v_{2}}=(\bar{J}_{v_{2}}+(g))_{[d-\deg(v_{2})]}$. By induction on
$d$, we deduce that $\bar{K}_{v_{2}}$ satisfies $N_{2}$. Hence, $u/v_{2}$ and
$v/v_{2}$ are connected in $G_{\bar{K}_{v_{2}}}(u/v_{2},v/v_{2})$. Thus, $u$
and $v$ are connected in $G_{K}(u,v)$.

Step 6. Conclusion step. By Step 5, we may assume that $\deg(v_{1})=1$. For
simplicity, we call $v_{1}=y$. Then we have $u=yh$, $v=gy$ and $f=gx$. Also,
$\operatorname{lcm}(u,v)=ygh$ and $\operatorname{lcm}(u,f)=xygh$. Let
$u_{1},\ldots,u_{r}$ be a shortest path in $G_{J}(u,f)$ that connects $u$ and
$f$. First, assume that there exists $u_{i}$ such that $x\not|u_{i}$. Then, we
have $u_{i}|\operatorname{lcm}(u,v)$. We may replace $u$ with $u_{i}$ and have
a shorter path from $u$ to $f$. Note that a path from $u_{i}$ to $v$ and a
path from $u$ to $u_{i}$ will give us a path from $u$ to $v$ in $G_{K}(u,v)$.
Hence, we may assume that $x|u_{i}$ for all $i=1,\ldots,r$. In particular, we
must have $u_{1}=xh$ as $\gcd(u,u_{1})=\deg(u)-1$. Let $w_{i}=u_{i}/x$. Then
$yw_{1},\ldots,yw_{r}$ is a path in $G_{K}(u,v)$ that connects $u$ and $v$.
That concludes the proof of the lemma. ∎

######  Remark 2.6.

In general, if $(I+(f))_{[d]}$ satisfies $N_{2}$ and $g$ divides $f$ then
$(I+(g))_{[d]}$ might not satisfy property $N_{2}$. The condition that $\deg
g=\deg f-1$ is crucial in Lemma 2.5. For example, Let
$S=\mathrm{k}[x_{1},\ldots,x_{8}]$ and
$I=(x_{3}x_{4}x_{7}x_{8},x_{3}x_{4}x_{5}x_{7},x_{3}x_{5}x_{6}x_{7},x_{1}x_{5}x_{6}x_{7},x_{1}x_{2}x_{5}x_{6})$.
Then $I$ has a linear free resolution. Let $f=x_{1}x_{2}x_{5}x_{6}$ is a
minimal generator of $I$ and $g=x_{1}x_{2}$, then $(I+(g))_{[4]}$ does not
satisfy property $N_{2}$.

For a finitely-generated graded $S$-module $M$, the (Castelnuovo-Mumford)
regularity of $M$ is defined to be

$\operatorname{reg}(M)=\max\\{j-i\mid\operatorname{Tor}_{i}^{S}(M,\mathrm{k})_{j}\neq
0\\}.$

We have the following simple result.

######  Lemma 2.7.

Let $J,K$ be non-zero homogeneous ideals of $S$. Then

  1. (i)

$\operatorname{reg}(J\cap
K)\leq\max\\{\operatorname{reg}(J),\operatorname{reg}(K),\operatorname{reg}(J+K)+1\\},$

  2. (ii)

$\operatorname{reg}(J+K)\leq\max\\{\operatorname{reg}(J),\operatorname{reg}(K),\operatorname{reg}(J\cap
K)-1\\}.$

###### Proof.

We have the short exact sequence

$0\to J\cap K\to J\oplus K\to(J+K)\to 0.$

The conclusion follows from a standard result on regularity along short exact
sequences. ∎

######  Lemma 2.8.

Let $I$ be a monomial ideal and $g=x_{1}\ldots x_{k}$ be a squarefree monomial
such that $g\notin I$. Then

$\operatorname{reg}(I,g)\leq\max\\{\operatorname{reg}(I,x_{i_{1}},\ldots,x_{i_{j}})+j-1\\}$

where $i_{1},\ldots,i_{j}$ are $j$ distinct indices of $\\{1,\ldots,k\\}$.

###### Proof.

We prove by induction on $k$. The case $k=1$ is obvious. Now assume that
$k\geq 2$. Write $(I,g)=(I,g_{1})\cap(I,x_{k})$ where $g_{1}=x_{1}\cdots
x_{k-1}$. By Lemma 2.7, we have

$\operatorname{reg}(I,g)\leq\max(\operatorname{reg}(I,g_{1}),\operatorname{reg}(I,x_{k}),\operatorname{reg}(I,g_{1},x_{k})+1).$

By induction, the conclusion follows. ∎

Assume that $d\geq 2$. Let $f(n,d)$ and $g(n,d)$ be defined as follows.

$f(n,d)=\begin{cases}0&\text{ if }n<d\\\ d&\text{ if }n=d\\\
\left\lfloor\frac{(d-1)n}{d+1}\right\rfloor+1&\text{ if }n>d,\end{cases}$

and

$g(n,d)=n-\left\lfloor\frac{n}{d+1}\right\rfloor-\left\lfloor\frac{n-1}{d+1}\right\rfloor.$

We now have some simple properties of $f(n,d)$ and $g(n,d)$.

######  Lemma 2.9.

Let $n,d,j$ be positive integers. Assume that $d\geq 2$. We have

  1. (i)

If $j\leq\left\lfloor\frac{d+1}{2}\right\rfloor$ and $n-j>d$ then
$f(n-j,d)+j-1\leq f(n,d).$

  2. (ii)

If $j\leq d+1$ and $n>d$ then $f(n-j,d)+j-2\leq f(n,d)$.

###### Proof.

For (i): by definition, it suffices to prove that

$\frac{(d-1)(n-j)}{d+1}+j-1\leq\frac{(d-1)n}{d+1}.$

Equivalently, $2j\leq d+1$.

For (ii): by definition, we may assume that $n-j>d$. Then the conclusion is
equivalent to $j-2\leq\frac{(d-1)j}{d+1}$, which is equivalent to $j\leq d+1$.
The conclusion follows. ∎

######  Lemma 2.10.

Let $n,d$ be positive integers. Assume that $d\geq 2$. Then $g(n,d)-1\leq
f(n,d)\leq g(n,d)$ and $f(n,d)=g(n,d)-1$ if and only if $n=(d+1)k+s$ for some
integers $k,s$ such that $1\leq k$ and
$\left\lfloor\frac{d+1}{2}\right\rfloor<s\leq d.$

###### Proof.

By definition, we may assume that $n\geq d+1$. Let $n=(d+1)k+s$ for integers
$k$, $s$ such that $0\leq s\leq d$. If $s=0$, then $f(n,d)=g(n,d)$. Thus, we
may assume that $1\leq s\leq d$. Then $g(n,d)=(d-1)k+s$ and

$f(n,d)=(d-1)k+\left\lfloor\frac{s(d-1)}{d+1}\right\rfloor+1.$

Since $(s-2)(d+1)<s(d-1)<s(d+1)$, we deduce that $g(n,d)-1\leq f(n,d)\leq
g(n,d)$. Furthermore, $f(n,d)=g(n,d)-1$ if and only if $s(d-1)<(s-1)(d+1)$.
Equivalently, $d+1<2s$. The conclusion follows. ∎

######  Remark 2.11.

Our bound is slightly better than the bound proposed by Dao and Takagi, as
shown in Lemma 2.10. For example, when $d=5$, the values of $f(n,d)$ and
$g(n,d)$ are as below

$n$ | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$
---|---|---|---|---|---|---|---|---|---|---|---|---
$f(n,5)$ | 5 | 5 | 5 | 6 | 7 | 7 | 8 | 9 | 9 | 10 | 11 | $\cdots$
$g(n,5)$ | 5 | 5 | 5 | 6 | 7 | 8 | 9 | 9 | 9 | 10 | 11 | $\cdots$

This small distinction is crucial for our induction argument though.

To prove the main result, we will prove by induction the bound for ideals of
the form $I+(f)$ where $f$ is a squarefree monomial of $S$ such that
$\deg(f)\leq d$ and $(I+(f))_{[d]}$ satisfies $N_{2}$. The initial phase of
the induction process has a certain subtlety, and we need to prove it directly
in the next two lemmas.

######  Lemma 2.12.

Let $I$ be a squarefree monomial ideal generated in degree $d$ and $f$ be a
non-zero squarefree monomial in $S=\mathrm{k}[x_{1},\ldots,x_{d+1}]$. Assume
that $1\leq\deg(f)\leq d$. Then $\operatorname{reg}(I+(f))\leq d$.

###### Proof.

We prove by induction on $d$. The base case $d=1$ is obvious. Now, assume that
$d\geq 2$ and $\deg(f)\geq 2$. We may assume that $x_{1}|f$. Write
$I=x_{1}I_{1}+I_{2}$, where $I_{2}=(g\mid g\in I,x_{1}\text{ does not divide
}g)$. Since $\deg(g)=d$ and there are only $d+1$ variables, so either
$I_{2}=(x_{2}\cdots x_{d+1})$ or $I_{2}=(0)$. When $I_{2}=(0)$, the conclusion
follows directly from induction. Now, assume that $I_{2}=(x_{2}\cdots
x_{d+1})$. Then we have

$I+(f)=x_{1}(I_{1}+(f_{1}))+I_{2}.$

Furthermore, $I_{2}\subseteq I_{1}+(f_{1})$, hence $x_{1}(I_{1}+(f_{1}))\cap
I_{2}=x_{1}I_{2}$. The conclusion follows from Lemma 2.7 and induction. ∎

######  Corollary 2.13.

Any ideal generated by squarefree monomials of degrees $d$ in $d+1$ variables
has a linear free resolution.

###### Proof.

The conclusion follows immediately from Lemma 2.12 by setting $f$ as one of
the minimal generators of the ideal itself. ∎

######  Lemma 2.14.

Let $I$ be a squarefree monomial ideal generated in degree $d$ and $f$ be a
non-zero squarefree monomial in $S=\mathrm{k}[x_{1},\ldots,x_{d+2}]$. Assume
that $1\leq\deg(f)\leq d$ and $I$ and $(I+(f))_{[d]}$ satisfy $N_{2}$. Then
$\operatorname{reg}(I+(f))\leq d$.

###### Proof.

We prove by induction on $d$. The base case $d=1$ is obvious. When
$\deg(f)=1$, the conclusion follows from induction and Lemma 2.2. Now, assume
that $d\geq 2$ and $\deg(f)\geq 2$. We may assume that $x_{1}|f$. Write
$f=x_{1}f_{1}$ and $I=x_{1}I_{1}+I_{2}$, where $I_{2}=(g\mid g\in
I,x_{1}\text{ does not divide }g)$. We may assume that $I_{2}$ is non-zero.
Let

$J_{2}=(g\mid g\text{ is a minimal generator of }I_{2}\text{ and }g\subseteq
I_{1}+(f_{1})).$

We will prove that $x_{1}(I_{1}+(f_{1}))\cap I_{2}=x_{1}J_{2}.$ Indeed, let
$h=x_{1}h_{1}$ be a minimal generator of $x_{1}I_{1}$ and $g$ be a minimal
generator of $I_{2}$. Since $x_{1}\notin\operatorname{supp}(g)$, we may assume
that $g=x_{2}\cdots x_{d+1}$. If $\operatorname{lcm}(h,g)=x_{1}g$, we are
done. Thus, we may assume that $\operatorname{lcm}(h,g)=x_{1}\cdots x_{d+2}$.
In other words, every minimal generator of $I$ will be a vertex in
$G_{I}(h,g)$. Let $h=u_{0},u_{1},\ldots,u_{r},g=u_{r+1}$ be a path in
$G_{I}(h,g)$. Let $i$ be the smallest index such that $x_{1}$ does not divide
$u_{i}$. Then $\operatorname{lcm}(u_{i-1},u_{i})=x_{1}u_{i}$. Since
$u_{i-1}\in x_{1}I_{1}$ and $u_{i}\in I_{2}$, we deduce that $x_{1}u_{i}\in
x_{1}I_{1}\cap I_{2}$. Hence, $\operatorname{lcm}(h,g)$ is not minimal in
$x_{1}I_{1}\cap I_{2}$. The intersection $(f_{1})\cap I_{2}$ can be done
similarly.

Now, by Lemma 2.7, we have

$\operatorname{reg}(I+(f))\leq\max\\{\operatorname{reg}(x_{1}(I_{1}+(f_{1}))),\operatorname{reg}(I_{2}),\operatorname{reg}(x_{1}J_{2})-1\\}.$

By induction and Lemma 2.12, the conclusion follows. ∎

######  Remark 2.15.

Alternatively, one can prove Lemma 2.14 as follows. It suffices to prove that
the square-free part of $(I+(f))_{[d]}$ has linear resolution, since if
$\operatorname{reg}(I+(f))>d$, there has to be a nontrivial homology
$\operatorname{Tor}_{i}(k,S/(I,f))_{m}$ at square-free monomial degree $m$
with $|m|>i+d$, as $I+(f)$ is a square-free monomial ideal. But the square-
free part of $(I+(f))_{[d]}$ still satisfies $N_{2}$, so it’s Alexander dual
$J$ is $(S_{2})$. However $J$ has codimension $d$ in a ring of $d+2$ variable,
so $S/J$ is Cohen-Macaulay, and thus the square-free part of $(I+(f))_{[d]}$
has linear resolution.

We are now ready for the proof of the main theorem.

###### Proof of Theorem 1.1.

We prove by induction on $n$ that $\operatorname{reg}(I+(f))\leq f(n,d)$ for
all squarefree monomial ideal $I$ and all squarefree monomial $f$ of
$S=\mathrm{k}[x_{1},\ldots,x_{n}]$ such that $I$ is generated in degree $d$,
and $I$ and $(I+(f))_{[d]}$ satisfies $N_{2}$. By Lemma 2.12 and Lemma 2.14,
we may assume that $n>d+2$. For ease of reading, we divide the proof into
several steps.

Step 1. $d+2<n\leq d+\left\lfloor\frac{d+1}{2}\right\rfloor+1$. We prove by
induction on the degree of $f$. Note that, in this range, we have
$f(n,d)=f(n-1,d)+1$. If $\deg(f)=1$, the conclusion follows from Lemma 2.2.
Now, assume that $\deg(f)\geq 2$. By Lemma 2.5, there exists $g$ such that
$f=xg$ for some variable $x$ of $S$ and $(I+(g))_{[d]}$ satisfies $N_{2}$. We
have

$(I+(f))=(I+(g))\cap(I+(x)).$

By Lemma 2.7, we deduce that

$\operatorname{reg}(I+(f))\leq\max\\{\operatorname{reg}(I+(g)),\operatorname{reg}(I+(x)),\operatorname{reg}(I+(g,x))+1\\}.$

By induction on the degree of $f$, $\operatorname{reg}(I+(g))\leq f(n,d)$. By
induction on $n$, $\operatorname{reg}(I+(x))$ and
$\operatorname{reg}(I+(g,x))\leq f(n-1,d)$. Since $f(n,d)=f(n-1,d)+1$, the
conclusion follows.

We now assume that $n>d+\left\lfloor\frac{d+1}{2}\right\rfloor+1$.

Step 2. $\deg(f)\leq\left\lfloor\frac{d+1}{2}\right\rfloor$. By Lemma 2.8, we
have

$\displaystyle\operatorname{reg}(I+(f))$
$\displaystyle\leq\max\\{\operatorname{reg}(I+(x_{i_{1}},\ldots,x_{i_{j}}))+j-1\mid\emptyset\neq\\{i_{1},\ldots,i_{j}\\}\subseteq\operatorname{supp}(f)\\}$
$\displaystyle\leq\max\\{f(n-j,d)+j-1\mid j=1,\ldots,\deg(f)\\}\leq f(n,d).$

The second inequality follows from induction and Lemma 2.2, and the third
inequality follows from Lemma 2.9.

Step 3. $\deg(f)>\left\lfloor\frac{d+1}{2}\right\rfloor$. By Lemma 2.5, there
exists $f_{1}\in I$ such that $\gcd(f_{1},f)=g$, $\deg(g)=\deg(f)-1$ and
$(I+(g))_{[d]}$ satisfies $N_{2}$. Write $f=xg$ and $f_{1}=gf_{2}$. We have
$\deg(f_{2})\leq\left\lfloor\frac{d+1}{2}\right\rfloor$ and

$I+(f)=(I+(g))\cap(I+(f_{2},x).$

By Lemma 2.7, we deduce that

$\operatorname{reg}(I+(f))\leq\max\\{\operatorname{reg}(I+(g)),\operatorname{reg}(I+(f_{2},x)),\operatorname{reg}(I+(g,f_{2},x))+1\\}.$

By induction on the degree of $f$, we have $\operatorname{reg}(I+(g))\leq
f(n,d)$. Since, $\deg(f_{2})\leq\left\lfloor\frac{d+1}{2}\right\rfloor$, as in
Step 1, we deduce that $\operatorname{reg}(I+(f_{2},x))\leq f(n,d)$. Note that
$\operatorname{supp}(g)$, $\operatorname{supp}(f_{2})$ and $\\{x\\}$ are
distinct and $\deg(g)+\deg(f_{2})=d$. Applying Lemma 2.8 twice to $g$ and
$f_{2}$, we deduce that

$\displaystyle\operatorname{reg}(I+(f_{2},g,x))+1$
$\displaystyle\leq\max\\{\operatorname{reg}(I+(x,x_{i_{1}},\ldots,x_{i_{s}},x_{j_{1}},\ldots,x_{j_{t}}))+(s+t-1)$
$\displaystyle\mid\\{i_{1},\ldots,i_{s}\\}\subseteq\operatorname{supp}(f_{2}),\\{j_{1},\ldots,j_{t}\\}\subseteq\operatorname{supp}(g)\\}.$

By induction, Lemma 2.2 and Lemma 2.9, we deduce that

$\operatorname{reg}(I+(g,f_{2},x))+1\leq\max\\{f(n-j-1,d)+j-1\mid
j=1,\ldots,\deg(f_{2})+\deg(g)\\}\leq f(n,d).$

The conclusion follows. ∎

Dao and Takagi [DT, Example 3.11] showed that the bound in Theorem 1.1 is
achieved when $d=2k+1$ and $n=t(k+1)$ for $t\geq 2$. We will now show that the
bound in 1.1 is achieved for all $n\geq d$ when $d$ is odd.

######  Lemma 2.16.

Assume that $d=2k+1\geq 3$. Then for all $n\geq d$, there exists an ideal $I$
generated in degree $d$ such that $I$ satisfies property $N_{2}$ and
$\operatorname{reg}(I)=f(n,d)$.

###### Proof.

We may assume that $n\geq d+1$. Let $n=(k+1)t+s$ for $0\leq s\leq k$ and
$t\geq 2$. We have

$f(n,d)=kt+\left\lfloor\frac{ks}{k+1}\right\rfloor+1=\begin{cases}kt+1&\text{
if }s=0,\\\ kt+s&\text{ if }s\geq 1.\end{cases}$

If $s=0$ or $s=1$, we let $J$ be the complete intersection of $t$ monomials of
degree $k+1$. If $2\leq s\leq k$, we let $J$ be the complete intersection of
$t$ monomials of degree $k+1$ and a monomial of degree $s$. Then, we have
$\operatorname{reg}(J)=f(n,d)$. Let $I=J_{[d]}$. By Lemma 2.3, [GPW, Theorem
2.1], and the truncation principle [EHU, Proposition 1.7], we have that $I$
satisfies property $N_{2}$ and
$\operatorname{reg}(I)=\operatorname{reg}(J)=f(n,d)$. The conclusion follows.
∎

######  Corollary 2.17.

Let $I$ be a squarefree monomial ideal of height $c$. Assume that $S/I$
satisfies Serre’s $S_{2}$ condition. Then

$\operatorname{cd}(S,I)\leq f(n,c)\leq
n-\left\lfloor\frac{n}{c+1}\right\rfloor-\left\lfloor\frac{n-1}{c+1}\right\rfloor.$

###### Proof.

The conclusion follows from the result of Terai [T], Singh and Walther [SW],
Yanagawa [Y], Lemma 2.10, and Theorem 1.1. ∎

######  Remark 2.18.

When $d=2\ell$ is even, the bound in Theorem 1.1 might not be tight yet. The
first case, where we are not able to construct a tight example, is when $d=4$
and $n=10$.

## Acknowledgments

A part of this paper was completed while the first author visited the Vietnam
Institute for Advanced Study in Mathematics (VIASM) in June 2024. He would
like to thank the VIASM for hospitality and financial support.

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