Update: check out more pictures taken by Max De Martino with the new XF 18-135mm lens here!
Today Fujifilm announced its first weather-sealed zoom lens and X Photographer Max De Martino got a sample right before he started his new photography project. Max generously shared his impressions with us and allowed us to publish the English version of his post as well as some pictures. The same thing happened three months ago when Max shot the first wedding with the Fuji X-T1.
The XF 18-135mm f/3.5-5.6 is water and dust resistant thanks to more than 20 sealed areas, making it the perfect lens for the weather-sealed X-T1. Even more interesting is a new version of their O.I.S. (Optical Internal Stabilisation) that claims to gain 5 stops of equivalent exposure thanks to two high precision gyro sensors. These sensors are installed with quartz oscillators which detect movement from high frequency to low frequency bandwidths. These are very interesting specs that we will likely see in the upcoming f/2.8 constant aperture zooms. According to Max, the new optical stabilisation works better than the other O.I.S. lenses such as the 55-200mm f/4.8-5.6. The lens seems to be technically advanced also thanks to its built-in air ventilation system that is placed on the inside of the base of the lens barrel. Its role is to prevent dust particles and moisture from getting into the lens.
Be sure to head over to Max's personal blog to see more images taken with the 18-135mm. The post will be updated with more images throughout today and tomorrow so make sure to visit often.
You can also follow him on Twitter and Facebook. At the end of the article there are three extra images coming from the official press kit that are available at full resolution.
A couple of months ago, Fujifilm Italia informed me that they wanted to let me try a new lens, specifically the 18-135mm f/3.5-5.6 (a 27-200mm equivalent on full frame), seeing that my photos from Chernobyl had made an impression on the Japanese staff of Fujifilm.
Naturally, I accepted without hesitation, despite feeling a twinge of doubt about the image quality of any lens with a zoom that long: in the past, I had used a similar lens on a Nikon D7000 and I wasn't overly impressed. Yes, it was a lens that covered all the bases but it usually wasn't very sharp.
This time however I discovered yet another lens that lives up to the quality of the Fuji lens line-up: it is as sharp as a knife.
Starting from the build, I have to say that the feel of the 18-135mm, like the rest of the XF series that came before it, is very good. The lens has been completely constructed out of metal if we exclude the lens hood and obviously the glass. This gives a sensation of true sturdiness.
I decided to use the 18-135mm to tell a story that I will continue to explore for the next few months: the life of the largest monastic community in Italy, L'Abbazia di Praglia (The Praglia Abbey). I will subsequently post many pictures of this project on my blog.
For the occasion, Fujifilm decided to create backstage video (coming soon) to show you just how the 18-135mm behaves out in the field.
Coming back to the lens itself, it has to be said that it is the ideal companion for the X-T1. It is the first weather resistant lens, like the X-T1 itself and the VG X-T1 battery grip. This means having a set-up that allows you to cover the most commonly-used focal lengths for travel, from wide to telephoto, with only 930 grams of weight to carry around and zero worries about the weather. It isn't an underwater camera of course but it wouldn't have any trouble withstanding a good soaking. And this, for the most anxious of us, is a feature not to be underestimated!
This functionality proved very useful when I wanted to set my ISO to Auto with a limit of 6400, and concentrate on the monks while they were praying in the church in lighting conditions that were challenging to say the least. Moreover, the temperature had soared over 35 degrees, a factor that pushed me to work with only one camera and one lens. It was worth it for the reduced weight and fatigue!
After the abbey, I then used the 18-135mm as my main lens at two weddings. Despite the fact that the fastest aperture cannot match a fixed lens, the bokeh was still very soft and the subject never lost its sharpness as it might have with the 18-55mm kit lens that comes with the X-T1 and X-E2.
Go to Max De Martino's first impression article (in Italian) to see more images taken with the 18-135mm inside the abbey.
Below are some extra images from the official Fuji press release kit taken by X Photographer Issaque Foujita. Click on the images to see them at their full size.


Not only a riveting account of a remarkable life, The Discovery of Slowness is also a profound and thought-provoking meditation on time.The result is an extraordinary reading experience that justifies the novel's reputation as one of the classics of contemporary German literature.... slowness Download slowness or read online here in PDF or EPUB. Please click button to get slowness book now. All books are in clear copy here, and all files are secure so don't worry about it.
The British discovery of Buddhism The British discovery of Buddhism Philip C. Almond Reader in Studies in Religion Uni... The Spread of Buddhism The Spread of Buddhism Handbook of Oriental Studies Handbuch der Orientalistik Section Eight Central Asia Edited by... Dedicated June 25, 2000, at the University of Michigan in Ann Arbor, Michigan. Commemorative Booklet (PDF) Moses Gomberg, a chemistry professor at the University of Michigan, discovered an organic free radical in 1900 and affirmed what had been thought impossible.


Steven A. McKean is a marketing and technology entrepreneur who partners with companies to build their product, distribution and customer acquisition strategies. He has expertise in Business Development, Operations, Executive and Board roles. Steven is a CEO with proven success delivering impressive contributions to business growth and sustained profitability with top Fortune 500 companies. His career has encompassed both executive leadership positions in large corporations, Big 4 Public Accounting, as well smaller entrepreneurial companies. His passion is to Create – Grow – and Contribute. Steven specializes in starting and scaling technology companies. He has broad experience in Customer Acquisition, Distribution, Product Development, and advising numerous companies on go-to-market strategy.
In his most recent position as Founder and CEO of Acceller, he was the driving visionary behind the company's unique comparison shopping and information platform for selling digital lifestyle services to consumers. Under Steven's leadership, the company grew to over 300 employees, secured digital service partnerships with top national providers and was recognized as Best Place to Work in South Florida in 2009 and 2010. Acceller merged with Bridgevine in June 2014 and Steven assumed the Vice-Chairman role of the combined company. He is also co-founder Animusoft, a software platform provider to the robotics industry. Additionally, Steven advises numerous start-ups on building teams, culture, financing and marketing.
The foundation of Steven's career began as CEO of Dade Confections, franchisor for International Dairy Queen in Miami, Florida. Steven and his team grew the installed base of stores from 8 to 30 in 3 years. He began his career with Arthur Anderson as a senior auditor. Whether for technology brands or non-profits, Steven's career has been distinguished by delivering strong financial returns for stakeholders while having a solid commitment to developing and building teams that are engaged, increase employee performance and improve operating results based on a performance management program.
He currently sits on the Board of Miami Children's Hospital, as their Chair of Technology and has served on their finance, audit and marketing committees. He also sits on the Orange Bowl Committee, University of Miami Citizen's Board, the Young Presidents Organization (YPO), and the Zell Lurie Entrepreneurial Institute Advisory Board. He earned his BBA and MAcc from the University of Michigan. Steven enjoys golf, poker, coaching basketball and spending time with his family.


Are You Looking for an "Actionable" Definition of Leadership?
Are You a Self-Differentiated Leader?
I teach leadership in both undergraduate and graduate programs at my university. Additionally, I provide leadership development and management consulting in the private and public sectors. So much of what happens in both settings revolves around providing tools and specialized knowledge that facilitate leadership development and effective leadership and management execution. Without exception, students and/or clients are looking for tools, resources, strategies, and specialized knowledge which will allow them to help resolve barriers, facilitate, growth, and maximize performance.
Clearly, there is a burgeoning hunger to know what it takes to be the type of leader that effects change in people, performance, and systems. This is certainly worthy and critically important. We are in constant need of women and men who have the knowledge, expertise, and imagination required to effectively lead our complex organizations and the people who serve within those organizations.
What does an Afghani Fulbright Scholar have to say about western-based leadership practices? Plenty!
Nang Attal is a graduate student enrolled in my fall course entitled, Leadership and Management Excellence. From Afghanistan, Attal grew up in a rural tribal community often influenced by insurgents and extremists. What made and makes Attal unique is that he was and continues to be committed to the education of girls in a region that does not look kindly on such efforts. Attal has remained committed to this effort, despite threats and intimidation, not only because he wanted his sisters to receive an education but also because he cares deeply about equal access to an educational system for all Afghani boys and girls.
Attal's efforts have been recognized nationally and globally. In October 2014, he was named the United Nations Youth Courage Award Recipient which was the same award given to Malala Yousafzai in 2013 for her work in advocating for the education of young girls in Pakistan. The recognition and notoriety of Attal's accomplishments continue to mount.
And here he sits in my course in the very heart of San Francisco.


text:
8,268 Dubai Tennis Stadium Premium High Res Photos
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Victoria Azarenka of Belarus hits a smash during her semi final match against Agnieszka Radwanska of Poland during day six of the WTA Barclays Dubai...
Maria Sharapova of Russia practises on the center court of the Dubai Tennis Stadium on February 19, 2006 in Dubai. The Dubai Women's Tennis...
Carla Suarez Navarro of Spain returns the ball to Karolina Pliskova of the Czech Republic during day two of the WTA Dubai Duty Free Tennis...
Roger Federer of Switzerland in action during his first round victory over Ivo Minar of Czech Republic in the Dubai Duty Free Men's Open Tennis...
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Nicolas Kiefer of Germany winces during his victory over Hyung-Taik Lee in their second round match in the Dubai Duty Free Men's Open Tennis...
Amanda Anisimova of the US serves the ball to Barbora Strycova of Czech Republic on the first day of the WTA Dubai Duty Free Tennis Championship, at...
Ivan Ljubicic of Croatia in action during the Singles Final against Roger Federer of Switzerland in the Dubai Duty Free Men's Open Tennis...
Roger Federer of Switzerland in action during the Singles Final against Ivan Ljubicic of Croatia in the Dubai Duty Free Men's Open Tennis...
Alize Cornet of France in action against Venus Williams of the USA during the final of the WTA Dubai Dury Free Tennis Championship at the Dubai...
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Richard Gasquet of France in action against Igor Andreev of Russia during the ATP Barclays Dubai Tennis Championships at the Dubai Tennis Stadium on...
Serena Williams of the USA in action in her match against Jie Zheng of China during day four of the WTA Barclays Dubai Tennis Championship at the...
Samantha Stosur of Australia plays a volley during her first round match against Tathiana Garbin of Italy during day two of the WTA Barclays Dubai...
Svetlana Kuznetsova of Russia looks on during her first round doubles match with Victoria Azarenka of Belarus against Ipek Senoglu of Turkey and...
Shahar Peer of Israel serves during her first round match against Yanina Wickmayer of Belgium during day two of the WTA Barclays Dubai Tennis...
Roger Federer of Switzerland holds the trophy after victory over Ivan Ljubicic of Croatia in the Singles Final of the Dubai Duty Free Men's Open...
Svetlana Kuznetsova of Russia celebrates winning a point in her semi-final against Flavia Pennetta of Italy during day six of the Dubai Duty Free...
Jennifer Brady of the US returns the ball to Garbine Muguruza of Spain during the WTA Dubai Duty Free Tennis Championship, at the Dubai Tennis...
Roger Federer of Switzerland and Andy Murray of Great Britain pose for photos before their match during the ATP Barclays Dubai Tennis Championships...
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Frustrated Roger Federer of Switzerland in his match against Andy Murray of Great Britain pose for photos before their match during the ATP Barclays...
Andy Murray of Great Britain smiles before his match against Roger Federer of Switzerland during the ATP Barclays Dubai Tennis Championships at the...
Tommy Haas of Germany in action against David Ferrer of Spain during the ATP Barclays Dubai Tennis Championships at the Dubai Tennis Stadium on March...
Fans show there support for Roger Federer of Switzerland in his match against Andy Murray of Great Britain pose for photos before their match during...
Elena Dementieva of Russia receives the trophy after defeating Svetlana Kuznetsova of Russia in the final during day six of the WTA Barclays Dubai...
Venus Williams of the USA hits a forehand during her final match against Victoria Azarenka of Belarus during day seven of the WTA Barclays Dubai...
Venus Williams of the USA poses with her winners trophy after her final match against Victoria Azarenka of Belarus during day seven of the WTA...
Venus Williams of the USA celebrates after winning match point during her final match against Victoria Azarenka of Belarus during day seven of the...
Victoria Azarenka of Belarus looks dejected after falling over during her final match against Venus Williams of the USA during day seven of the WTA...
Svetlana Kuznetsova of Russia falls over during her third round match against Regina Kulikova of Russia during day four of the WTA Barclays Dubai...
Svetlana Kuznetsova of Russia looks dejected after missing a shot during her third round match against Regina Kulikova of Russia during day four of...
Karolina Pliskova of the Czech Republic plays a shot in her match against Su-Wei Hsieh of Chinese Taipei during day five of the WTA Dubai Duty Free...
Su-Wei Hsieh of Chinese Taipei serves in her match against Karolina Pliskova of the Czech Republic during day five of the WTA Dubai Duty Free Tennis...
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Andy Murray of Great Britain in action in a training session during previews for the ATP Barclays Dubai Tennis Championships at the Dubai Tennis...
Elise Mertens of Belguim in action against Catherine Bellis of USA during day two of the WTA Dubai Duty Free Tennis Championship at the Dubai Tennis...
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redpajama_set_name: RedPajamaCommonCrawl


text:
Mar 5, 2019, 12:07pm EST
Brazil Bounces Back With A Slew Of New Billionaires
Nov 9, 2018, 01:14pm EST
Fortune Magazine's New Owner Is Member Of Thailand's Richest Family
Nov 2, 2018, 06:42pm EDT
Southeast Asia's Richest Woman Signs $6.5 Billion Deal For 50 Airbus Jets
Oct 30, 2018, 10:27am EDT
The Indonesian Tycoons Behind Lion Air, Whose Plane Crashed Off Jakarta on Monday
Oct 22, 2018, 12:58pm EDT
How This Video Gaming Billionaire Invests His More Than $2 Billion
Oct 3, 2018, 07:02am EDT
LinkedIn Cofounder Reid Hoffman On His Billion-Dollar Impact Investing Bet
Sep 4, 2018, 12:45pm EDT
Jeff Bezos Is $1.8 Billion Richer As Amazon's Market Cap Briefly Hits $1 Trillion
Aug 28, 2018, 01:45pm EDT
The $4 Billion Crypto Billionaire Who No One Has Heard Of
Meet Warren Buffett's New Business Partner: India's Youngest Billionaire
Nov 5, 2014, 06:00am EST |
Four Cargill Heirs Emerge As New Billionaires Behind World's Largest Food Company
Dan Alexander
Senior editor at Forbes, covering Donald Trump's business.
This story appears in the November 23, 2014 issue of Forbes. Subscribe
W. Duncan MacMillan died in 2006 as the 216th-richest man in America, leaving his $1.6 billion fortune to four daughters: Sarah Chaney, Lucy Stitzer, Katherine Tanner and Alexandra Daitch. Their estimated 3.75% stakes in Cargill--the $135 billion Minnesota-based food giant that has more than doubled its revenues since MacMillan's death--give each heiress a personal fortune of $1.3 billion.
The daughters join six of their relatives in the billionaire ranks--all descendants of W.W. Cargill, who started the company in a single warehouse in 1865 and today sits atop a family tree worth some $36 billion. The newest ten-figure family members are just as private as their richer kin. "They want to draw the curtain down," says Cargill CEO David MacLennan. "A lot of rich people want to be on TV, want you to know who they are and that they own this and that. Not these people."
Stitzer served on the board of directors at Cargill from 1992 to 2010 and currently chairs Waycrosse, the family investment company. Her sisters have remained entirely out of the public eye. FORBES reached Chaney by phone when reporting this issue's piece on Cargill. "I don't have time to chat," she said. "I can't really help you with this article."
Follow me on Twitter or LinkedIn. Check out some of my other work here. Send me a secure tip.


text:
using System;
using System.Linq;
using Xamarin.Forms.CustomAttributes;
using Xamarin.Forms.Internals;

#if UITEST
using NUnit.Framework;
using Xamarin.UITest;
using Xamarin.Forms.Core.UITests;
#endif

namespace Xamarin.Forms.Controls.Issues
{
	[Preserve (AllMembers = true)]
	[Issue (IssueTracker.Github, 764, "Keyboard does not dismiss on SearchBar", PlatformAffected.Android)]
	public class Issue764 : TestContentPage
	{

		protected override void Init ()
		{
			Title = "Issue 764";

			var searchBar = new SearchBar {
				Placeholder = "Search Me!"
			};

			var label = new Label {
				Text = "Pending Search"
			};

			searchBar.SearchButtonPressed += (s, e) => label.Text = "Search Activated";

			var layout = new StackLayout {
				Children =  {
					searchBar,
					label
				}
			};

			Content = layout;
		}

		// Issue 416
		// NavigationBar should be visible in modal

#if UITEST
		[Test]
		[Category(UITestCategories.ManualReview)]
		public void Issue764TestsKeyboardDismissedForEnter ()
		{
			Assert.Inconclusive ("Needs test");
		}
#endif

    }
}

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}
In recent years, the field of
quantum machine learning has attracted much attention. There, quantum circuits are employed as central processing units for data-driven applications~\citep{biamonte_quantum_2017,schuld_supervised_2018,dunjko_machine_2018}.
While it is currently not clear whether or not quantum processing can provide a benefit on practical machine learning problems~\citep{schuld_is_2022,schuld_supervised_2021}, there has been some evidence that quantum machine learning models can outperform classical models in certain tasks~\citep{liu_rigorous_2021,sweke_quantum_2021,gyurik_establishing_2022, gyurik_towards_2022}.
While most studies, theoretical~\citep{bharti_noisy_2022,montanaro_quantum_2016} as well as experimental~\citep{graham_multi-qubit_2022, pino_demonstration_2021, kjaergaard_superconducting_2020}, focus on quantum systems consisting of two-level quantum systems, i.e., quantum bits (qubits), quantum computing hardware and algorithms can also be based on $d$-level systems, which are typically
called qudits~\citep{wang_qudits_2020,ringbauer_universal_2022}. Such qudit systems were shown to have advantages in specific contexts~\citep{cozzolino_highdimensional_2019,sheridan_security_2010} and have already been applied to several tasks~\citep{bravyi_hybrid_2022,deller_quantum_2022, weggemans_solving_2022}. However, a full evaluation on application-relevant tasks is still lacking.

Currently, it is not clear whether there is a fundamental advantage or disadvantage of utilizing qudit systems over their qubit counterparts for quantum machine learning or other application domains such as quantum optimization. However, qudits provide an complementary route to increasing the Hilbert space size in pursuit of better computation performance.
This is an alternative direction to qubit systems, where the route to larger Hilbert spaces is to increase the number of qubits.
Among other challenges, increasing the number of qubits comes with increased difficulty of engineering high-fidelity interactions between separate qubits, which are technologically challenging and typically involve larger gate errors than single qubit operations.
In that regard, expanding the local Hilbert space dimension by utilization of qudit systems seems very promising, since certain long-range qubit operations could instead be mapped to local qudit operations, which might prove to be more efficient.
We discuss several of such  possible advantages of qudit system at various parts in this work.

We explore the usage of qudit systems in the context of the data re-uploading~\citep{perez-salinas_data_2020} quantum machine learning approach to build regression and multi-class classification models. In this scheme, a single qudit provides a natural way of encoding multiple classes by representing each class label as an orthogonal basis state.

This paper is structured as follows: In Sec.~\ref{sec:qudits}, we introduce the mathematical description of qudits. Sec.~\ref{sec:data_re-uploading} illustrates the implementation of the data re-uploading algorithm \citep{perez-salinas_data_2020} with qudits and shows the circuit structure as well as the chosen loss functions used during training.
Sec.~\ref{sec:experimental_setup} presents the training procedure which is done numerically and on IBMQ hardware. To be able to run our model on qubit based quantum hardware, we present a way to encode qudits with multiple qubits. In Sec.~\ref{sec:results}, we verify the expressivity of our model by testing it on a simple regression problem. We then go on and test our model on multi-class classification problems where we show an intrinsic bias between the qudit state representation and the data structure. Finally, we present numerical results of the model when being trained on the MNIST handwritten digits data set~\citep{lecun_mnist_2005}. We additionally investigate an equivalent qubit-based implementation on IBMQ hardware and the effect of entangling operations on the performance of the model.

\section{Qudits}\label{sec:qudits}

$d$ level quantum systems, typically called qudits, are a generalization of qubits to $ d>2 $  and can serve as a basis for quantum information processing.
The  Hilbert space is spanned by $d$ orthonormal basis vectors, denoted by $ \ket{0}, \ket{1}, ... \ket{d-1} $  and
arbitrary qudit states can be represented by the supersposition $ \ket{\psi} = \sum_{k=0}^{d-1}c_k\ket{k} $ with
the normalization condition  $ \sum_{k=0}^{d-1}\vert c_k\vert^2 = 1 $.

Inspired by cold atom systems \citep{kasper_universal_2022}, we interpret a $d$-level qudit as a spin with total angular momentum $\ell=\tfrac{d-1}{2}$, such that the basis state $\ket{k}$ corresponds to the spin eigenstate with angular momentum $m=\tfrac{2k-d+1}{2}$.
We employ the angular momentum operators $\{L_x, L_y, L_z\}$, which generate rotations around the corresponding axes and which obey the canonical commutation relations of the special unitary group $SU(2)$, $[L_i,L_j]=i \epsilon_{ijk}L_k$.
The action of the angular momentum operators on the qudit basis state are given by
\begin{align}
\label{eq:angularmometum}
 L_x\ket{k} & =  \tfrac12\big(\gamma_{d,k+1}\ket{k+1}+\gamma_{d,k-1}\ket{k-1}\big) \\
 L_y\ket{k} & = \tfrac1{2i}\big(\gamma_{d,k+1}\ket{k+1}-\gamma_{d,k-1}\ket{k-1}\big) \\
L_z\ket{k} & = \tfrac{2k-d+1}{2}\ket{k}
\end{align}
where $k\in[0,d-1]$ and with $\gamma_{d,k}=\sqrt{(d-k-1)(k+1)}$.

There are various  ways to define a universal gate set to realize actions in the qudit Hilbert space, see e.g.\ \cite{wang_qudits_2020} and \cite{luo_geometry_2014}.
For a single qudit, as it is considered in this work, we choose the two angular momentum operators $L_x$ and $L_z$ and the nonlinear squeezing  or one-axis twisting operator $L_{z^2}=L_z^2$. This set is sufficient to generate any state by (possibly many) repeated finite rotations as detailed in \cite{kasper_universal_2022} and \cite{giorda_universal_2003}.
Equivalent to qubits, the gates for qudit circuits are then generated by exponentiation of the basic operators which implements the rotations
\begin{align}
    R_j(\theta) = e^{-i \theta L_j} ,
\end{align}
with $j \in \{x, z, z^2\}$ and where $\theta\in\mathbb R$ are free parameters.

\begin{figure*}
\centering
   a\hspace*{0.3\linewidth}b\hspace*{0.5\linewidth} \\[-9mm]
   \includegraphics[width=0.24\linewidth]{Fig1a.pdf} \includegraphics[width=0.5\linewidth]{Fig1b.png}\\
   c\hspace*{0.4\linewidth}\hspace*{0.41\linewidth} \\[-6mm]
     \includegraphics[width=0.9\linewidth]{Fig1c.png}
\caption{a: Schematic representation of $d=7$ qudit states on a generalized Bloch sphere. b: Schematic illustration of the action of the three operators $R_x$ (left), $R_z$ (middle) and $R_{z^2}$ (right) on a qudits state. c: Illustration of the data re-uploading circuit structure}\label{fig:data_reuploading}
\end{figure*}
In Fig.~\ref{fig:data_reuploading} we illustrate the qudit states (panel A)  and the action of the elementary operators by showing the Husimi-Q quasi-probability distribution \citep{husimi_formal_1940} on a generalized  Bloch sphere in panel B. The pure rotation does not deform the probability distributions, but applying the squeezing operation leads to a deformation of the state, indicating its nonlinear action.

\section{
Data re-uploading with a single qudit
}\label{sec:data_re-uploading}

We utilize a single $d$-level qudit and implement the quantum machine learning model as a data re-uploading quantum circuit \citep{perez-salinas_data_2020,jerbi_quantum_2023}.
The quantum circuit in its general form is build up from $L$ layers and encodes a quantum state, which depends on the input data $\mathbf{x}$ as
\begin{align}\label{eq:state_unitary}
    \ket{\mathbf{x},\boldsymbol{\omega},\boldsymbol{\theta}}  &
=\prod_{l=1}^LU(\mathbf{x}, \boldsymbol{\omega}^{(l)},\boldsymbol{\theta}^{(l)}) \ket{0}\,,
\end{align}
where $\ket0$ is the initial state. The characteristic of the data re-uploading architecture is that the unitary operation of each layer $l$ encompasses data dependent unitaries, which are parametrized by the scaling parameters $\boldsymbol{\omega}^{(l)}$ and data independent operations with free parameters $\boldsymbol{\theta}^{(l)}$.

We tested several layer structures and found that they all produce similar results. In the following we report results for two architectures.  The first one is inspired by classical Euler rotations, where the unitaries of each layer have the structure:
\begin{align}
\label{eq:DRUL_euler1}
    U(\mathbf{x}, \boldsymbol{\omega}^{(l)},\boldsymbol{\theta}^{(l)}) = W(\boldsymbol{\theta}^{(l)}) S(\mathbf{x}, \mathbf{\omega}^{(l)}) \,.
\end{align}
The data encoding block $S$ of each layer consists of alternating $x$ and $z$ rotations,
\begin{align}
\label{eq:DRUL_euler2}
    S(\mathbf{x}, \bm\omega^{(l)}) = R_\alpha(x_D\omega_D^{(l)} )\cdots    {R}_z(x_2\omega_2^{(l)}){R}_x(x_1\omega_1^{(l)})
\end{align}
where the number of rotations is determined by the dimensionality $D$ of the input data and consequently $\alpha=x$ ($\alpha=z$) in case $D$ is odd (even).
The alternating $x$ and $z$ rotations is chosen to ensure that these are non-commuting. This is necessary to distinguish between the individual dimensions  of the data vector. If only $x$ (or $z$) rotations were chosen to encode the data, the classifier would only be able to learn the sum of the input data.

The data-independent block in each layer is composed of a sequence of three rotations followed by a squeezing gate $R_{z^2}$ as the last operation, i.e.\
\begin{align}\label{eq:DRUL_euler3}
     W(\boldsymbol{\theta}^{(l)}) = {R}_{z^2}(\theta^{(l)}_4){R}_{x}(\theta^{(l)}_3){R}_{z}(\theta^{(l)}_2){R}_{x}(\theta^{(l)}_1)
\end{align}
 The first three operations give rise to Euler angles, starting from $\ket{0}$, and thus the freedom to create overlap with an arbitrary qudit state.
The total number of adjustable parameters of a $L$-layer model  is  given by $(4+D) L$.
The action of this circuit structure is illustrated in Fig.~\ref{fig:data_reuploading} C.

The second architecture considered here is inspired by the simplified form presented by \cite{perez-salinas_data_2020} and has the structure:
\begin{align}
    \label{eq:simleCirc}
    U(\mathbf{x}, \boldsymbol{\omega}^{(l)},\boldsymbol{\theta}^{(l)}) = e^{-i\sum_{j}^D (\theta_j^{(l)}+\omega_j^{(l)} x_j) L_{c(j)} -i \theta_{D+1}^{(l)} L_{z^2} }\,.
\end{align}
Here the first term in the exponent is the sum over the three angular momentum operators, i.e.\ the generators of the $SU(2)$,  and the function $c(j)=(j\mod 3)$ selects one of them. The second term in the exponent is a generalization of the original simplified architecture by including the squeezing operator in each layer.           The total number of adjustable parameters for an $L$-layer model of this structure  is  given by $(2D+1) L$.

In our work we approach supervised classification and regression tasks by using qudit-based quantum circuits.
In a supervised learning setting, there exists a set of $N$ training samples $(\mathbf{x}_i,y_i)$ with $i=1,\dots, N$, which consist of pairs of input samples $\mathbf{x}$ with corresponding output values $y$.
The input samples are real-valued $D$-dimensional vectors $\mathbf{x}\in\mathbb{R}^D$ while the output values are either real numbers or a finite set of integers for regression and classification tasks, respectively.

For classification problems the output values $y\in(0, 1,\dots,d-1)$ indicate to which of the $d$ classes the data sample $\textbf{x}$ belongs.
In the quantum formulation, each class label is represented by a basis state of the $d$-level qudit $\ket{y}$ with $y=0,\dots,d-1$. The model prediction, i.e., the probability that a data sample $\mathbf{x}$ belongs to class $y$, can then conveniently be calculated from the overlap of the label state and the qudit wave function obtained from the  quantum circuit with input $\mathbf{x}$,
\begin{align}
\label{eq:overlap}
    P(y\vert \mathbf{x},\boldsymbol{\omega},\boldsymbol{\theta}) = \vert \braket{ y\vert \psi(\mathbf{x},\boldsymbol{\omega},\boldsymbol{\theta}) }\vert ^2\,.
\end{align}
Training the quantum circuit is achieved by minimizing a loss function over the given training data set $\mathcal{D} = \{ (\textbf{x}_i,y_i)\}_{i=1,\dots, N}$.
The overlap of Eq.~\eqref{eq:overlap} is the basis for formulating the  mean squared error (MSE) loss function,
\begin{align}\label{eq:mse}
    \mathcal{L}_{\text{MSE}}(\boldsymbol{\omega},\boldsymbol{\theta})  &=\frac{1}{N}\sum_{i=1}^N \big(\braket{ \bar{y}_i}- y_i\big)^2
 \end{align}
 with the average predicted label of the quantum model
\begin{align}\label{eq:averageoutput}
\braket{ \bar{y}_i} = \sum_{y=0}^{d-1} y \,P(y \vert \mathbf{x}_i,\boldsymbol{\omega},\boldsymbol{\theta})\,.
 \end{align}
 Another popular choice is the overlap loss as used in \cite{perez-salinas_data_2020},
\begin{align}
    \mathcal{L}_\text{overlap}(\boldsymbol{\omega},\boldsymbol{\theta} )  =
      \sum_{i=1}^N \big(1- P(y_i \vert  \mathbf{x}_i,\boldsymbol{\omega},\boldsymbol{\theta} )\big).
\end{align}
The learning procedure amounts to adjusting the parameters of the quantum circuit $(\boldsymbol{\omega},\boldsymbol{\theta})$ in order to minimize the loss function, which is done by running a classical optimization algorithm.

After the quantum circuit has been trained its accuracy is evaluated by analyzing its predictions of the output variables on a test data set, i.e.\ a set of  data samples not used during the training procedure.
For a classification task the predicted output labels is given by the basis state with the highest probability in the quantum state with the corresponding input data sample, i.e.
\begin{align}
\label{eq:predictedclassification}
    y_i^\m{predicted}=\m{argmax}_yP(y\vert \mathbf{x}_i,\boldsymbol{\omega}_\m{opt},\boldsymbol{\theta}_\m{opt})\,,
\end{align}
where  $(\boldsymbol{\omega}_\m{opt},\boldsymbol{\theta}_\m{opt})$ are the optimized values of the circuit parameters.
The accuracy of the trained model can then be evaluated by calculating the fraction of correctly predicted labels in the test data set,
 \begin{align}
    \text{Accuracy} = \frac1{N_\m{test}}\sum_{i\in\mathcal{D}_\text{test}} \delta_{y_i^\m{predicted},y_i}
    \,,
\end{align}
where  $\mathcal{D}_\text{test}$ denotes the test data set which contains $N_\m{test}$ data samples and $\delta_{a,b}$ is the Kronecker delta.

The MSE loss of Eq.~\eqref{eq:mse} is also suitable to learn regression tasks. In that case the output variables are finite-range continuous variables, $y\in[0,d-1]$. The resulting qudit state of the trained quantum circuit is then a superposition of basis states and the predicted output value for input $\mathbf{x}_i$ is calculated as the expectation value of Eq.~\eqref{eq:averageoutput},
\begin{align}\
\label{eq:predictedregression}
    \braket{\bar {y}_i}^\m{predicted}=\sum_{y=0}^{d-1} y \,P( y \vert \mathbf{x}_i,\boldsymbol{\omega}_\m{opt},\boldsymbol{\theta}_\m{opt})\,.
\end{align}

For simulations running on actual quantum hardware, the probability distribution of Eq.~\eqref{eq:overlap} is estimated by performing a  finite number of measurement shots and recording the measurement results as a histogram. This approximative distribution is then used to calculate the predicted output values of Eqs.~\eqref{eq:predictedclassification} and  \eqref{eq:predictedregression}.

\section{Experimental setup}\label{sec:experimental_setup}
We perform the training of the quantum circuit utilizing exact numerical simulations of the qudit states and quantum gates.
The parameter values of $\boldsymbol{\omega}$ and $\boldsymbol{\theta}$ are initialized randomly in the range $[-\pi, \pi]$. As classical optimization algorithms  for minimizing the loss function  we employ the ADAM optimizer~\citep{kingma_adam_2014} paired with the exact gradients calculated by the automatic differentiation package JAX~\citep{bradbury_jax_2018}. Additionally, we also used L-BFGS-B and Powell optimization approaches from the \texttt{scipy} library \citep{virtanen_scipy_2020}, where we do not utilize automatic differentiation. Apart from differences in the run time for the training, the results obtained with all approaches were always comparable.
If not stated otherwise, all models are trained on randomly selected data sets with size $N=750$. The performance is evaluated on a separate randomly selected test data set, which contains $\tfrac N3=250$ data samples.
In order to obtain reliable statistics of the results, we run 60 different simulations for each setting, each time randomly varying the data set and the model initialization. To visualize the decision boundaries of the trained classifiers, we utilize the visualization library \textit{orqviz}~\citep{rudolph_orqviz_2021}.

In addition to the numerically exact simulation, in \ref{sec:ibmq} we re-train and evaluate the qudit quantum circuit on the IBM \texttt{ibmq\_lima} hardware using the qiskit \cite{treinish_qiskitqiskit_2023} framework. This allows us to estimate the impact of gate errors and noise of actual NISQ hardware on the learning performance.
As the IBM hardware naturally operates on qubits, we employ a mapping of the $d$-level qudit Hilbert space to $d-1$ qubits, which is inspired by cold atom systems~\citep{kasper_universal_2022}.
The qudit basis state $\ket{k}$ is represented by the qubit Dicke-state $\ket{D^{d-1}_k}$, i.e.\  $\ket k \to\ket{D^{d-1}_k}$ \citep{gasieniec_deterministic_2019}. The $k^{th}$ Dicke state $\ket{D^{d-1}_k}$ of $d-1$ qubits is given by the equal superposition of all states which have $k$ qubits in the state $\ket{1}$ and $d-1-k$ in state $\ket 0$, i.e.\
\begin{align}
     \ket{D^{d-1}_k}=\begin{pmatrix}d-1\\k\end{pmatrix}^{-\tfrac{1}{2}}\sum_{x\in\{0,1\}^{d-1}, hw(x)=k } \ket{x}\,.
\end{align}
where $hw(x)$ indicates the Hamming weight of string $x$, i.e.\ the number of 1's in $x$.
The angular momentum operators for the $d-1$ qubit states are defined as the sum over the single-qubit operators,
$L^\text{tot}_a=\sum_{j=0}^{d-1} L_a^j$ with $a=\{x,y,z\}$, and they act as described by Eqs.~\eqref{eq:angularmometum} on the Dicke states.
In particular, the Dicke state $\ket{D^{d-1}_k}$ and the qudit state $\ket{k}$ have the same $z$-component of  the angular momentum, i.e.\ $ L^\text{tot}_z\ket{D^{d-1}_k}=\tfrac{2k-d+1}{2} \ket{D^{d-1}_k}$.

In this representation the squeezing operation  consists of all pairwise qubit interactions,
\begin{align}
L^\text{tot}_{z^2}=\big(L^\text{tot}_{z}\big)^2=\sum_{i,j} L_z^i L_z^j
\end{align}
which is implemented as two-qubit $zz$-rotation gates on the hardware~\citep{treinish_qiskitqiskit_2023}.
On the qubit device, we start from the optimized parameters found by the training with the exact simulation and re-train the model on the actual hardware. There, we perform 512 measurement shots for each qiskit circuit evaluation.

For comparison we also report the accumulated results on classification tasks of a standard classical machine learning approach, namely the \texttt{scikit-learn} \citep{pedregosa_scikit-learn_2018} implementation of the random forest (RF) classifier with 100 estimators, a $k$-nearest neighbor classifier with $k=3$ ($k$nn) and a support vector classifier (SVC). The performance of these approaches were always comparable and we report the cumulative results of running each algorithm 50 times on randomized data stets.

\section{Results}\label{sec:results}
\begin{figure}
\centering
 \includegraphics[width=0.23\textwidth]{Fig2a.png}
  \includegraphics[width=0.23\textwidth]{Fig2b.png}
      \caption{Results of trained data re-uploading models with structure of Eqs.~\eqref{eq:DRUL_euler1}-\eqref{eq:DRUL_euler3} using a $d=3$ qutrit with one (left) and two (right) layers for the regression task of learning the function in Eq.~\eqref{eq:simple1D} (blue line).
      The purple dots indicate the results of the numerically exact model simulation obtained and the green dots the results from  re-training and evaluation on the IBM \texttt{ibmq\_lima} hardware
    }\label{Fig:03}
\end{figure}

\subsection{Expressivity in one dimension}
First, we examine the expressivity of the data re-uploading circuit with a single qudit on a continuous one-dimensional regression problem  analogous to ~\cite{schuld_effect_2021}.
We train a model with one qutrit, i.e.\ a $d=3$ qudit, to learn the simple function
\begin{align}\label{eq:simple1D}
    f(x)=\tfrac12 \big(\cos{(1.5 x)} + \cos{(2.5 x)}\big)
\end{align}
with \(x \in [-\pi, \pi]\).
As the model output we use the expectation value of Eq.~\eqref{eq:averageoutput} but shift it to match the data range, i.e.\  $f_i^\m{predicted}=\braket{\bar{y_i}}-1$. For training the quantum circuit we employ the
MSE loss function of Eq.~\eqref{eq:mse} and use a training set of 100 linearly distributed samples.

We show the results of a $L=1$ layer model in the left panel of Fig.~\ref{Fig:03}.
It is evident that this model is not able to learn the function properly. This confirms the previous insight for qubit circuits~\citep{schuld_effect_2021}, that data re-uploading models learn truncated Fourier series, and that a circuit containing only one angle parametrized by the input can only encode one Fourier component.
In the right panel we show the results of a model with $L=2$ layers.  As expected, this model is now able to learn the function with two Fourier coefficients exactly. The derivation is analogous to the qubit case~\citep{schuld_effect_2021}, but is not shown in this work.
Apart from the results obtained by numerically exact simulations, we also include results from the evaluation on the IBM \texttt{ibmq\_lima} hardware (green dots in the plot). These predicted values are generally noisy which result from the inherent errors in the quantum circuit, as well as the shot noise from the finite number of samples used to estimate the probability distribution of Eq.~\eqref{eq:overlap}.

\subsection{Two-dimensional classification tasks}
\begin{figure*}
    \includegraphics[width=0.95\textwidth]{Fig3.png}
   \caption{Results of the seven-class horizontal stripe classification problem with circuit structure of Eqs.~\eqref{eq:DRUL_euler1}-\eqref{eq:DRUL_euler3}.
   In the upper row the qudit states are aligned with the class labels, while in the lower row the class labels are randomly assigned to the qudit states as illustrated in the pictures on the left.
   The first stripe pattern shows the  ground truth and the training data samples.
   The middle and right stripe patterns show the classification regions obtained after training  with three and six layers, respectively.
   The rightmost plots show the statistics of classification accuracy, obtained from 50 independent training runs, as a function of layers in the quantum circuit with (blue) and without (green) the squeezing operation in the $W$ blocks.
   Colored boxes indicate 25\% and 75\% percentiles and horizontal whiskers the minimal and maximal values.
   In the right panels, cML indicates the results of the classical machine learning approaches
   }\label{fig:StraightLines}
\end{figure*}

Inspired by the original benchmark of the data re-uploading algorithm~\citep{perez-salinas_data_2020}, we investigate the  performance of the qudit circuit with the structure of Eqs.~\eqref{eq:DRUL_euler1}-\eqref{eq:DRUL_euler3} on various two-dimensional multi-class classification problems.
The data samples are located on a two-dimensional square $\mathbf{x}=(x_1,x_2) \in [-1, 1]^2$, where each sample is associated with one out of $d$ classes.
In the first problem setting, the classes are arranged in  parallel horizontal stripes.
Fig.~\ref{fig:StraightLines} shows the results for seven classes, where we used a  $d=7$ qudit to represent the classes in the quantum circuit.

In the upper row of the figure, we show results from the case where we align the qudit states with the labels in such a way, that the $z$-components of the spin are in order with the class label. In that case, adjacent classes are represented by qudit states with adjacent $z$-spin values. Under these circumstances, the model has a strong inductive bias towards the data set.
This is illustrated in the leftmost figure where we draw the qudit states on the generalized Bloch sphere.
The lowest qudit state $\ket 0$ is associated with the class of the lowest (blue) stripe, the second lowest qudit $\ket 1$ with the second lowest stripe, and so on.
The corresponding results show that the data re-uploading circuit can predict this data set almost perfectly with three or more layers, $L\geq3$ (blue graph in top right plot).
Remarkably, the performance of the data re-uploading circuit is even better than the classical machine learning models shown on the far right of the right panel.
However, the learned classes and decision boundaries, as shown in the middle panels of the Figure, can still differ from the ground truth. This is due to the small size of the training data set, which necessarily leads to small random variations of the learned  decision boundaries, depending on the precise location of the training data close to the decision boundaries.

The importance of the squeezing operation $R_{z^2}$ in the $W$ operator of Eq.~\eqref{eq:DRUL_euler3} is highlighted by observing the massively degraded performance of the same circuits without this gate (green graph in the top right panel of Fig.~\ref{fig:StraightLines}). Without squeezing, the median accuracy saturates at around $0.7$ while with squeezing the median accuracy reaches $0.95$ and higher.
This performance difference is attributed to the fact, that the squeezing operator is necessary to allow for the representation of arbitrary unitary operations in the qudit Hilbert space.

 \begin{figure*}
  \includegraphics[width=0.99\linewidth]{Fig4.png}\\[-3mm]
       \caption{Learned classification regions of a single qudit classifier circuit with $L=5$ layers  trained on the first two (a), three (b) and five (c) classes (i.e.\ digits) of the MNIST data set, which was compressed down to two dimensions using PCA.
       Each background color indicates one region associated with the same label. The training data samples are shown in the plots and colored according to their true label.
       (d) Accuracy of the qudit classifier as a function of the number of layers for various numbers of classes.  RF denotes the  random forest classifier for comparison
       }\label{fig:orqviz_mnist_pca}
\end{figure*}

The second row of Fig.~\ref{fig:StraightLines} shows the performance when the class labels are randomly assigned to the qudit states.
This removes the inductive bias of the model and makes the problem much harder to learn for the qudit quantum circuit. Consequently, the accuracy drops significantly and many more layers are necessary to recover the performance level of the scenario with aligned labels.

The plots of the accuracies also show the accumulated results of the three classical machine learning classifiers RF, SVC and $k$nn for comparison. The performance of the quantum circuits including squeezing with randomized assignment of qudit states to labels is comparable to the classical approaches.  For aligned labels the quantum circuit even slightly outperform the classical approaches.

We also considered several other two-dimensional multi-class classification problems and trained data re-uploading models on several horizontal stripe data sets with varying number of stripes (i.e.\ classes), as well as data sets where the stripes are not horizontal but rotated by an angle.
We also investigated the models on data sets where the class regions are given by concentric rings with approximately  the same width and where the center of the rings is somewhere in the plane.
The  results (not shown) on those models  were always qualitatively similar to the ones presented here. Squeezing is always necessary to achieve good performance and when the label order is aligned with the qudit states, the performance is consistently higher.
The overall performance was slightly reduced for the tilted stripes and the concentric rings and the variance in the results was also slightly larger as compared to the horizontal stripes cases presented here.

\subsection{Classifying MNIST data}
As the next application example, we train the model of Eqs.~\eqref{eq:DRUL_euler1}-\eqref{eq:DRUL_euler3} on subsets of a \texttt{scikit-learn}~\citep{pedregosa_scikit-learn_2018} version of the MNIST handwritten digits data set~\citep{lecun_mnist_2005}.
This version includes down-sampled images with  $8$x$8$ pixels instead of the  $28$x$28$ pixels of the original MNIST data set.
Since it is very computationally demanding to encode $8\times8=64$ dimensional data samples into the quantum circuit, we reduce the input dimension further using the principal component analysis (PCA)~\citep{jolliffe_principal_2016}.

In Fig.~\ref{fig:orqviz_mnist_pca}(a)-(c) we visualize the classification boundaries of a $L=5$ layer circuit for two, three and five classes, where the input dimension is reduced to $D=2$ using a PCA. The plots show that the data re-uploading classifier is able to learn highly non-linear decision boundaries. It can also be seen that the classifier tends to produce disconnected classification regions due to the oscillatory nature of parametrized quantum circuits. The statistics of the accuracy for various classes is shown in Fig.~\ref{fig:orqviz_mnist_pca} (d) as a function of the quantum circuit layers.
It can be observed that the accuracy,  as expected, increases with more layers, and that increasing the number of classes in the problem reduces the accuracy. For more classes, the reduction to two dimensions using PCA leads to more spacial overlap between classes, which makes the problems inherently noisy and limits the overall achievable accuracy. The overlapping classes can be directly observed in panel (c).
Additionally, panel (d) shows the result from 50 runs of a random forest (RF) classifier.
The accuracy of the quantum-based classifier approaches the values of the RF model with increasing number of layers.
However, the variance in the result is significantly larger for the data re-uploading circuit. One reason for this is found in the increasingly more complex classical optimization problem when increasing the number of parameters in the quantum circuit.

\subsection{Qudit vs Qubit}

The quantum circuits we employed up to this point used a single $d$-level qudit to solve  classification problems  with $d$ classes, where each basis state encodes one class label.
While it may appear less natural, multi-class classification problems can also be learned with a data re-uploading circuit operating with a single qubit~\citep{perez-salinas_data_2020}.
In those approaches, the different classes are represented by single-qubit quantum states  which are chosen to be maximally orthogonal. Unless a two-class classification problems is considered, where the labels states are $\ket{0}$ and $\ket{1}$, these states cannot be fully orthogonal to each other. As a $d=6$ class example, we choose the eigenstates of the three spin operators, i.e.\
$\ket{y} \in  \{\ket{0}, \ket{1}, \tfrac{1}{\sqrt{2}}(\ket{0}+\ket{1}),\tfrac{1}{\sqrt{2}}(\ket{0}-\ket{1}),\tfrac{1}{\sqrt{2}}(\ket{0}+i\ket{1}),\tfrac{1}{\sqrt{2}}(\ket{0}-i\ket{1})\}$,
as the maximally orthogonal label states.
Note that in this situation, the overlap of Eq.~\eqref{eq:overlap} as a function of the labels is no longer a proper probability distribution since $\sum_y P(y)>1$.

The natural question which arises is whether there is any difference when using qubit or qudit data re-uploading circuits for multi-class classification problems. Therefore, we study and compare the performance of a single qubit as well as a single qudit on several problems.
For the single qubit approach, we utilize the simplified data re-uploading structure of Eqs.~\eqref{eq:simleCirc}.
Here, the squeezing operator can be removed from the circuit since it is proportional to the identity, $L_{z^2}=\tfrac{\mathds{1}}{4}$, and therefore only applies a global phase.

\begin{figure}
\includegraphics[width=0.23\textwidth]{Fig5a.png}
     \includegraphics[width=0.23\textwidth]{Fig5b.png}
       \caption{
       Comparison of $d=6$ multi-class classification performance of qubit and qudit data re-uploading circuits with the simplified structure of Eq.~\eqref{eq:simleCirc}.
       Left: tilted $D=2$ dimensional stripes data set with a rotation angle of $27^\circ$.
       Qudit accuracies are shown for aligned and randomized label and state assignement.
       Right: six randomly selected digits of the MNIST data set which are reduced to $D=2$ input dimensions using a PCA
       }\label{fig:qubitVSqudit}
\end{figure}

Fig.~\ref{fig:qubitVSqudit} shows the results where the qubit and qudit models were trained on two different classification problems with $d=6$ classes and $D=2$ input dimensions. The left panel of Fig.~\ref{fig:qubitVSqudit} shows the classification accuracy as a function of the circuit layers for the tilted stripes data set, where the horizontal stripes from Fig.~\ref{fig:StraightLines} are rotated by an angle of $27^\circ$.
It can be seen that the accuracy of the qubit and qudit circuits are comparable for the case where the class labels were randomly assigned to the qudit basis states.
However, for labels aligned with the qudit states, the performance is consistently higher for the qudit circuits.

The right panel shows the results of the qubit and qudit data re-uploading circuits for the  MNIST handwritten digits data set limited to six randomly selected digits (i.e.\ classes) and where each image is reduced  to two input dimensions using a PCA.
Here, qubit and qudit approaches are comparable in their performance, while the qubit architectures seem to outperform the qudit ones for smaller circuit depths.
These results highlight the importance and potential strength of the bias exhibited by the qudit-based model. When applied mindfully, this bias can be leveraged to enhance training performance and possibly generalization capabilities.

In our simulations, the loss function and the classification accuracy are calculated  numerically exact with full access to the quantum state.
When running on real quantum hardware, this cannot be done and one needs to prepare and sample, i.e.\ measure, the quantum state multiple times in order to estimate the probabilities/overlaps of Eq.~\eqref{eq:overlap}.
For the qudit system, where each class label is associated with an orthogonal basis state in the measurement basis, this is straight forward since the statistics of the measurement outcomes directly translate to estimating the probabilities.
However, for qubit systems with non-orthogonal label states (i.e. for $d>2$), this involves the necessity for a full quantum state tomography, or techniques such as classical shadows~\citep{huang_predicting_2020} with appropriately chosen measurements. Therefore, more measurement samples will typically be required for qubit systems.

Additionally and more importantly, when the different label states are overlapping, the difference in overlap between the true label state to be estimated as compared to an undesired label state is less pronounced.
For qudits, this difference is always 1 since $P_\text{qudit}(y\vert y')=\delta_{y,y'}$ for label states $y,y'$ (here, basis states). In the qubit representation, the difference is $\leq1$ since $P_\text{qubit}(y\vert y')=\delta_{y,y'} + \sum_{y''\neq y'} c_{y''} \delta_{y,y''}$ with $c_{y}\geq 0$.
For example, in the case of $d=6$ classes considered here, $P_\text{qubit}(0\vert y')=\vert\braket{ 0\vert y' }\vert^2=\tfrac{1}{2}$ for $y'\in\{2,3,4,5\}$.
This leads to a significantly reduced training signal in the overlap loss function in Eq.~\eqref{eq:overlap}, since even contributions from incorrect label states have non-zero overlap and therefore reduce the loss.
Obtaining the overlaps using a finite number of measurement samples necessarily adds noise to the estimates,
which in turn makes the problem of discriminating between  correctly and wrongly predicted labels more difficult.
As a consequence,
the shot noise
is expected to have a more pronounced negative effect on the training performance for multi-class problems when overlapping label states are used.

\subsection{Circuit structure and basic operators}
For qubit circuits, the spin-$\frac12$ Pauli matrices  allow to represent arbitrary single-qubit unitary operations inside each layer.
For $d$-level qudits, one instead needs to include all $d^2-1$ generators of the special unitary group in $d$ dimensions $SU(d)$, to achieve the same arbitrary control.
However, as indicated above and shown in \cite{kasper_universal_2022} and \cite{giorda_universal_2003}, the three operators $L_x$, $L_z$ and $L_{z^2}$ are sufficient to represent any unitary operation by repeated finite rotations with multiple layers.
This leads to the question whether there is a benefit when more than this reduced set of three operators are used in the data re-uploading circuit. We test this hypothesis by adding the following two types of operators,
\begin{align}
\label{eq:extOps}
    X_{j}&=\ket{0}\bra{0} - \ket{j}\bra{j} \qquad  (1\leq j\leq d-1) \nonumber\\
    Y_{j}&=\ket{0}\bra{j} + \ket{j}\bra{0} \, .
\end{align}
The motivation behind this choice is that these operators directly couple the initial state $\ket{0}$ to all other states $\ket{j}$ and thus may allow for a more efficient learning. We then use the simplified structure of Eq.~\eqref{eq:simleCirc} and replace the squeezing operator by the sum over all operators in Eq.~\eqref{eq:extOps}.

The results of the three types of circuit structures on the six-class reduced MNIST data set are shown in Fig.~\ref{fig:extOps}.
The left panel shows the accuracies as a function of the number of circuit layers. First, one can observe that there is no significant difference between the simplified structure of Eq.~\eqref{eq:simleCirc}  and circuit structure of Eqs.~\eqref{eq:DRUL_euler1}-\eqref{eq:DRUL_euler2}. On the other hand, the performance of the circuits with the extended set of operators is significantly better.
At first glance this seems to support the hypothesis that adding more operators to the set generators does in fact enhance the trainability of qudit quantum circuits.
However, adding more operators also introduces more free parameters to the quantum circuit for a given number of layers, which allows for more flexibility in the trainable circuit.
When comparing the accuracies as a function of number of free parameters, as shown in the right panel of Fig.~\ref{fig:extOps}, one can observe that the performance is statistically the same for all circuit structures and number of operators.
Therefore, we conclude that it is not the number of operators which is determining the trainability, but rather the number of trainable parameters.

This reveals a tradeoff between the complexity of each layer, i.e.\ the number of operators and free parameters in each layer, and the total number of layers which are necessary to achieve a certain accuracy.
Including more operators in each layer allows to achieve good  performance with fewer number of layers. On quantum hardware, where gate errors play an important role, the freedom to choose the elemental operations provides additional flexibility. And because different hardware may natively support different gate sets, this allows to choose configurations which result in running variational quantum circuits with less errors.
For qubits, this is not possible since the three Pauli matrices form a basis and no additional operators can be constructed.

\begin{figure}
\includegraphics[width=0.23\textwidth]{Fig6a.png}
     \includegraphics[width=0.23\textwidth]{Fig6b.png}
       \caption{
       Comparison of results of qudit data re-uploading circuits of different structures on the $d=6$ class MNIST data set with input dimension $D=2$.
       The left panels shows the results as function of the number of layers, while the right panel shows it as function of the number of free parameters in the circuit (notice: the $x$-axis is not to scale in the right plot)
       }\label{fig:extOps}
\end{figure}

\subsection{Re-training on IBM hardware}\label{sec:ibmq}

Finally, we compare the classification performance of the model of Eqs.~\eqref{eq:DRUL_euler1}-\eqref{eq:DRUL_euler3} with and without the squeezing operation between numerically exact results and the case where the re-training and evaluation is done on actual \texttt{ibmq\_lima} hardware. For the qubit implementation of the qudit circuits we use the Dicke-state encoding described in Sec.~\ref{sec:experimental_setup}.
The left panel of Fig.~\ref{fig:MNIST} shows the accuracy as a function of the quantum circuit layers for the first five digits, i.e.\ qudit dimension $d=5$,  and with input dimension $D=5$.
The numerically exact simulations show the same trend as previously described for two input dimensions and approach the values of the traditional RF classifier with an increasing circuit depth. However, when re-trained and evaluated on actual hardware the performance is only comparable for the first two layers, after that it saturates and then decreases strongly.
This is due to the noise and infidelities  in the actual hardware realization of the entangling gates used to implement the squeezing operations.
Removing the squeezing operations from the circuit reveals this explicitly, since the accuracies of both approaches are then comparable as shown in the right panel of Fig.~\ref{fig:MNIST}.
\begin{figure}
    \includegraphics[width=0.95\linewidth]{Fig7.png}
       \caption{
       Results for the classification of the first five digits of the MNIST data set with input dimension  $D=5$ with (left) and without (right) squeezing as a function of the layers in the quantum circuit operating with a $d=5$ qudit. The numerical exact simulations are shown in blue, whereas the results from \texttt{ibmq\_lima} simulations are in green.
        The performance of a random forest (RF) model is displayed to the right in each plot
       }\label{fig:MNIST}
\end{figure}

This is an explicit example of a situation where is more efficient to increase the local Hilbert space and work with qudits, instead of increasing the number of qubits.

\section{Discussion}

In this work, we demonstrated that multi-level qudit systems are well-suited to be applied to multi-class classification problems, as each qudit basis state can naturally encode one class of the data. We implemented data re-uploading quantum circuits, where we used the angular momentum and the squeezing operators to build a universal gate set. We illustrated the capabilities of the qudit-based approach on regression and classification benchmarks. Owing to the ability to learn highly non-linear classification boundaries, the models were able to successfully learn on various data sets and achieve performances comparable to standard classical machine learning models.
Interestingly, the achievable performance was strongly dependent on the qudit states representing the class labels and their relation to the structure of the labels in the data set.
This intrinsic bias due to the label alignment can boost the performance of qudit circuits  substantially, which might be beneficial for certain types of application problems. One may therefore conclude that there exist tasks where variational circuits based on qudits might be the better choice compared to their qubit counterparts.

We also studied the influence of the choice of the elementary operations and the layer structure in qudit quantum circuits. There
we found, that the structure, i.e.\ the particular sequence and types of rotations, does not appear to have significant influence on the performance, as long as the number of parameters was accounted for.
This reveals a trade-off between the number of elementary operators in the gate set and the number of layers to achieve the same performance. In some situations it might be advantageous to utilize the minimum set of operators and to employ more layers, while in others, more operators and less layers might be the better choice.
This is especially interesting on quantum hardware where different gates typically have different error rates, and the freedom to choose between several configurations may allow to minimize the influence of errors on the resulting performance.

It should be noted that we did not employ any of the more sophisticated techniques to improve the performance of  single-qudit data re-uploading circuits, for example, employing different classical optimizers~\citep{deller_quantum_2022,lavrijsen_classical_2020}, fine-tuning hyperparameter settings~\citep{pascal_hyperparameter_2022}, and finding better parameter initializations~\citep{sack_quantum_2021,grant_initialization_2019,egger_warm-starting_2021}. All techniques have been shown to be very beneficial in related contexts and can be used in the future to improve the current approach.

A necessary next step is to investigate the performance of multi-qudit circuits, since here we only investigated single-qudit circuits.
Interesting research questions include how the intrinsic bias of a single qudit is influences a multi-qudit circuit and its learning performance, and what the role of the set of elementary operators is in these cases. Crucially, the effect of multi-qudit entangling gates needs to be elucidated.

In summary, our results and discussion support the conclusion that qudit systems offer a promising alternative quantum computing architecture. There are several differences to qubit-based systems which could be leveraged and thus provide a practical benefit for quantum algorithms and quantum machine learning tasks in particular.

\bmhead{Acknowledgments}
NLW acknowledges funding from the Honda Research Institute Europe GmbH  for attending the conference  on Quantum Techniques in Machine Learning (QTML) 2022. NLW would like to thank the Van der Waals-Zeeman Institute in Amsterdam for the extended hospitality.
SS acknowledges funding by the European Union under Horizon Europe Programme -- Grant Agreement 101080086 -- NeQST.
Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or European Climate, Infrastructure and Environment Executive Agency (CINEA). Neither the European Union nor the granting authority can be held responsible for them.

\section*{Declarations}
\bmhead{Conflict of interest} The authors declare no competing interests.

redpajama_set_name: RedPajamaArXiv


text:
Абу-Губейха () — город на юге Судана, расположенный на территории штата Южный Кордофан. Административный центр одноимённого округа.

Географическое положение
Город находится в восточной части штата, на плато Кордофан, на высоте 542 метров над уровнем моря.
Абу-Губейха расположен на расстоянии приблизительно 168 километров к востоко-северо-востоку (ENE) от Кадугли, административного центра штата и на расстоянии 457 километров к юго-юго-западу (SSW) от Хартума, столицы страны.

Климат

Демография
По данным переписи 2008 года численность населения Абу-Губейхи составляла 47 188 человек.
Динамика численности населения города по годам:

См. также
 Города Судана

Примечания

Города Судана

redpajama_set_name: RedPajamaWikipedia


Testers are needed to help investigate Civil Rights violations in Clinton, Eaton, Ingham, Jackson, Lenawee, Livingston, Monroe and Washtenaw Counties.
The FHC is seeking women and men to assist in the investigation of housing discrimination complaints ("testing"). "Testers" pose as home seekers in a variety of housing rental and sales situations. Testers will be part-time as-needed employees ($24 an hour plus mileage reimbursement), or may choose to volunteer. Hours vary and are flexible. Attention to details is a must. Access to personal transportation is helpful, but not required.
The FHC currently needs African, African American, Asian, Caucasian, and Latino/a/x people of all genders and ages. We also need wheelchair users.
Fill out, SAVE, and submit the following FHC Tester Application (please save the form with a new name to your computer before emailing or faxing it over). Once you've faxed (734-340-6598) or emailed (info@fhcmichigan.org) your application, we may contact you to discuss your application and potentially set up an interview.
If you would like more information on becoming a tester please call the Fair Housing Center using our toll free number 877-979-FAIR.
As a tester you are playing the role of a home seeker – a person who is looking for a housing unit in which to live. Testers must be objective, careful observers, accurate reporters, and truthful witnesses. Besides visiting many interesting homes and apartment complexes, you may have an opportunity to serve as a witness in a court case. Most importantly, you become a key part of the FHC's efforts to end discrimination in housing.
Testing is a term that refers to a process of checking the housing market to determine the nature and extent of discriminatory treatment accorded different home seekers. Testers take careful notes of what transpires.
discrimination. Fair housing litigation filed with the help of the FHC has been based, in large part, on the evidence provided by FHC testers. Both the sworn oral testimony and the actual written reports of testers are routinely accepted by courts as evidence of discriminatory conduct by a defendant.
Watch this fun video out our Testers/Superheroes!
Help end housing discrimination: print and post this these fliers for your school, office, neighborhood, etc. Thank you!


text:
Теорема Кэли о числе деревьев — теорема, утверждающая, что число деревьев с  пронумерованными вершинами равно .

История
Теорема названа в честь Артура Кэли, который доказал её в 1889 году.
Сам Кэли признавал, что то же утверждение было доказано раньше Карлом Борхардом и в эквивалентной формулировке ещё раньше в статье Джеймса Джозефа Сильвестра 1857 года.

В своей статье Кэли по сути доказывает более общее утверждение. Если раскрыть скобки выражения

то коэффициент при одночлене вида  будет равен числу деревьев, у которых степени вершин равны степеням переменных данного терма: .

Кэли подробно разбирает случай  и заявляет, что доказательство легко обобщается.

Формулировки
Две эквивалентные формулировки:

Число различных деревьев на  вершинах, пронумерованных числами от  до , равно .

Число остовных деревьев в полном графе  равно .

Связанные утверждения
Количество деревьев на  пронумерованных вершинах оказывается также равным числу разложений -цикла  в произведение  транспозиции.

Количество деревьев на  пронумерованных вершинах оказывается также равным числу (соответствующим образом нормированных) многочленов степени  с заданными  критическими значениями общего положения.
Наконец, это последнее является частным случаем топологической классификации  сферы Римана — тем самым, подсчёт числа деревьев оказывается частным случаем вычисления чисел Гурвица, соответствующим случаю накрывающей поверхности рода 0.

О доказательствах
Формула Кэли немедленно следует из свойств кода Прюфера — способа однозначного кодирования -вершинного помеченного дерева упорядоченной последовательностью из  номеров его вершин.

Формула Кэли также легко выводится из матричной теоремы о деревьях.

Одно из доказательств строится на следующем соотношении

на экспоненциальную производящую функцию

где  обозначает число корневых деревьев на  данных вершинах. По теореме Лагранжа об обращении рядов, из этого соотношения следует, что . Последнее влечёт формулу Кэли поскольку для каждого остовного дерева есть ровно  способов выбрать корневую вершину.

Вариации и обобщения
 Количество способов связывания графа, состоящего из  несвязных компонент, каждая размером  вершин, равно

Здесь   — общее количество вершин графа.
Если каждая компонента состоит из одной вершины , то , и формула дает исходное число Кэли .

Число остовных деревьев в полном двудольном  графе  равно

 Матричная теорема о деревьях даёт выражение числа остовных деревьев графа как определитель лапласиана (матрицы Кирхгофа) графа.

Примечания

Литература
 Ю. М. Бурман, записки курса «Критические значения многочленов»: , , , .
 М. Э. Казарян, записки курса «Геометрия, топология и комбинаторика разветвленных накрытий сферы».

 T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein. Hurwitz numbers and Hodge integrals.

К
Комбинаторика


Description: Citalopram is a selective serotonin reuptake inhibitor, balances levels of serotonin in the brain and heals major depression associated with mood disorders.
Generic Celexa has active ingredient(s) Citalopram. We guarantee exceptional quality of Citalopram medication that is sold via our website.
Generic Celexa contains the same active ingredients as a brand pill. Generic Celexa is manufactured by the certified producer but not by the original drug's producer. If you have any questions please use Live chat.


text:
Be sociable!
AI content writers, revisited
23rd August 2021 Content creationTags: AI writer
What difference has 12 months made?
It's been almost a year since I wrote my blog post 'AI content writers: Threat or ally?'
In the post, I trialled Google's top ranking AI writer to find out what a bot might be capable of. I wanted to know whether it would help or harm content writers like me. And whether it could actually write like a human.
If you want to see how it did, you can read the original post here.
SPOILER ALERT: it was pretty terrible!
As technology is evolving at an alarming rate, this is an experiment I always planned to revisit, so here I am, a year later, doing it all over again.
Am I nuts?
Probably. But let's see what happens.
Choosing an AI writer
In the last experiment, I tried out AI Writer. At the time, this was the highest organic-ranking AI content writer on Google — and it still is. So no change there. But this time I've decided to try a couple of different ones.
Well, on first inspection, a year doesn't seem to have changed AI Writer too much. It's still asking for a subject and promising an article with sources in two minutes.
This didn't bode well last time and it seems unlikely that's changed.
This time, I'll be trying two of the best rated AI writers for 2021, as recommended by MyWPLife.
This is the number one ranking article on Google and covers 10 of the best AI writers available.
We'll be starting with Writesonic, then moving on to Copysmith. I chose these two AI writers because they both offer something different — and they both came with a free trial*.
Writesonic is a writer that actually writes articles. Copysmith is more of a guide that gives you an outline to work from.
*It's worth noting that some AI writers on the list didn't seem to offer free trials or guarantees. I don't understand how anyone would sign up for something like this without seeing what it could do first — but maybe that's just me.
In the last experiment, I asked AI Writer to rewrite my last three blog articles, which it did — badly.
For this experiment, I tested the AI with another copywriting brief I undertook recently. This time, a piece of content with the title 'How to choose more sustainable toilet paper'. I thought this might be an easier task that would give better results.
My human response to the brief
The article I wrote was 1673 words and had the following title and heading structure:
Title: How to choose more sustainable toilet paper
The world's most expensive toilet paper
Not all countries use toilet paper
The environmental impact of toilet paper
Choose plastic-free toilet paper
What is plastic-free toilet paper?
Choose recycled toilet paper
How is recycled toilet paper made?
How to choose recycled toilet paper
Look for FSC labelling
What if there's no labelling?
Choose unbleached toilet paper
Processed Chlorine-Free (PCF)
Elemental Chlorine-Free (EFC)
Choose bamboo toilet paper
How is bamboo toilet paper made?
Does bamboo toilet paper break down?
Is bamboo toilet paper better?
Ready to buy more sustainable toilet paper?
The AI writers' response to the brief
So how did the AI writers respond to the brief and did they do a better job than me?
Writesonic
I decided to try out Writesonic's AI writer and see how it handled my brief. But that didn't go quite to plan. It seems you can only specify a subject for your article and not the whole title.
Choosing your subject
I typed in 'toilet paper' and got this list of nine suggested articles it could write for me:
The Best Toilet Paper for Your Home: Which One Should You Buy?
Why You Should Be Using Charmin Toilet Paper: Your Guide to a Cleaner, Fresher Home
What's the Best Toilet Paper?
Best Toilet Paper: Which Is The Most Durable, Affordable?
5 Tips for Finding the Best Toilet Paper: Picking Your Favorite Brand
How To Buy Toilet Paper: The Best Toilet Papers for Your Needs
12 Ways To Use Toilet Paper That You Didn't Know About
Toilet Paper: The Best Toilet Paper Brands For Every Budget
Why Toilet Paper is Great for the Environment, and What You Can Do to Help
I was fascinated by number 2. Who knew toilet paper could make your whole home cleaner and fresher?!
And I was intrigued by number 7. What uses could it have — other than than the obvious?
But to make this a fair experiment, I had to follow the environmental angle and go for number 9 — even if the title is weird, nonsensical and factually incorrect!
Writing the introduction
Having chosen the title 'Why Toilet Paper is Great for the Environment, and What You Can Do to Help', I was asked to write an article intro of between 100–150 words.
Here's what I wrote, which follows the process I went through with my own article:
Most UK consumers don't know that their toilet paper is made from virgin wood pulp, which means trees are cut down to make it. How can we choose toilet paper that's more sustainable and kinder to the environment? In this article, we'll discuss the environmental impact of toilet paper and explain why choosing more sustainable toilet paper matters. We'll show you how alternative options such as recycled and bamboo toilet papers help make a difference. And we'll explain what to look for on the packet, including what the FSC accreditations mean and how to avoid papers that have been bleached in manufacturing.
Before you say anything, I know this isn't my best work — and it isn't meant to be. After all, this is supposed to be for people who can't write. What the intro does is outline my thought process for the article I wrote and give as many cues as possible.
Generating outlines for the article
From the intro I wrote, Writesonic developed five different outlines for me to choose from.
Outline 1
Section 1: Understanding the environmental impacts of toilet paper
Section 2: Sustainable options for your bathroom
Section 3: What to look for on a toilet paper packet
Section 4: Conclusion
Section 1: The Environmental Impact of Toilet Paper
Section 2: Sustainable Options for Toilet Paper
Section 3: What to look out for on the packet
Section 2: Choosing Sustainable Toilet Paper
Section 3: What to Look for on the Packet
Section 1: Toilet Paper Is Made Of Trees
Section 2: Why Sustainability Matters
Section 3: What's The Problem With Manufacturing?
Section 4: Alternative Options For Sustainable Living
Section 5: How To Help The Environment With Your Toilet Paper Choices
Subsection 5.1: Choose Papers That Are Free In Bleaching Process- Easy!
Subsection 5.2: Choose Papers That Are Free From Artificial Softeners- They Don't Help!
Section 1: Why toilet paper matters
Section 4: What are you waiting for?
Choosing an outline
This was a no-brainer for me and I obviously chose Outline 4. This is the most fleshed out article of the five and even has subheadings.
Writing the article
Next, Writesonic writes the article for you. This is reasonably quick, taking around 30 seconds per section.
And I know you're just dying to read it, so here it is — exactly as it came out:
Toilet Paper Is Made Of Trees
Most of us are aware that toilet paper is a type of paper product, so why does it matter whether we choose the more sustainable ones? Using a disposable newspaper or a tissue or tissues instead of a wipe is a waste of resources. These tissues, which come into contact with your hands and face every day, often include bleached pulp. This means that the paper ends up in a landfill, or worse, burning in an incinerator or generating heat-trapping emissions in our homes. Paper towels are a very similar product, but their paper is virgin wood pulp, and they are thus completely biodegradable. Most UK consumers don't know that their toilet paper is made from virgin wood pulp, which means trees are cut down to make it.
Why Sustainability Matters
Sustainable products can have a positive effect on the environment and human health. They can also make your money go further because you're buying more-efficient products. Of course, buying environmentally-friendly toilet paper doesn't make you automatically do the right thing with your other purchases. It's just one of many things you can do to help the environment. And, when buying new toilet paper, you'll often find that many eco-friendly brands will be more expensive than traditional brands, and more packaging is used. So, if you're short on budget or just want to go green, think about the price and how much you can pay for what you buy. Reducing the amount of toilet paper you use is another way to help the environment and improve your health.
What's The Problem With Manufacturing?
The problem with toilet paper is that it's made from trees. In fact, all modern toilet paper has to be made from virgin wood pulp. Wood pulp is produced by mixing wood fibers with lignin in the form of glue, which is then heated and dried. The resulting sheets are used to make paper. Each sheet is made from the fibres that make up two-thirds of the tree and the glue that binds them together. The toilet paper we use might be pretty — and might even have cute cartoon characters on the packaging, but this is just paper and cardboard with a fresh lick of paint. Some recycled paper, but still paper. The major problem with paper is that it's made from trees. In 2016, this waste sector was responsible for 7.4 million tons of carbon dioxide emissions. This is the same amount produced by 3.
Alternative Options For Sustainable Living
How does toilet paper get made? We all have a certain number of tissues to clean ourselves, and what we need is the right type for the job. There are three main kinds of fibre used in making toilet paper: cotton, wood pulp and the remaining fibres from other materials that can't be recycled into toilet rolls. Wood pulp is made from wood waste. Cotton and paper are man-made. Cotton is produced from seeds or by trimming the plants. Paper is made from recycled paper and a kind of starch, both from corn. The three major types of toilet paper that you can buy are cotton, bamboo and recycled paper. Each has its own advantages and disadvantages: Cotton This type of toilet paper uses cotton fibre for its padding, and is often marketed as 'recycled' as it is made from recycled waste paper.
How To Help The Environment With Your Toilet Paper Choices
The Tree Nature paper labels are a good place to start if you want to find a more sustainable toilet paper. The two 'best' products are the 'best of Britain' toilet paper, and the 'greenest toilet paper', both of which are made from 100% organic cotton. They are also made in the UK, and as a result of these two simple measures, their carbon footprint is around 11.5 and 6.6 tonnes respectively. These choices will give your toilet paper a good carbon footprint, in contrast to some other toilet papers, which have a bigger carbon footprint than their main ingredients. The sustainable toilet paper manufacturers are following guidance issued by the UK government and, for instance, avoiding the production of pulp products with bleached pulp.
Choose Papers That Are Free In Bleaching Process- Easy!
Making a plastic toilet paper roll is incredibly energy-intensive: all the chemicals, water and energy used in the bleaching process are an added cost to the manufacturer. The paper itself is also not free of bleaching chemicals: some brands are sold by the kilogram, while others are made in bulk and sold by the ton. Fortunately, there are two ways to make eco-friendly toilet paper at home. The first way is to ask your local supermarket to stop selling bleached toilet paper. Alternatively, you can buy environmentally friendly toilet paper and find out which brands are free of bleaching agents in the manufacturing process.
Choose Papers That Are Free From Artificial Softeners- They Don't Help!
The water-repellent natural paper is one of the main reasons why we love it, but it's also made from wood pulp. Both are natural products, and to get the softness you want from these natural products, they often contain chemicals to make them smoother and softer. These chemicals are called softeners. Fortunately, the most popular natural toilet paper contains no softeners and is often made from virgin cellulose. So, if you want softness, but also want to minimise the damage you're causing to forests, then you're best looking for this kind of toilet paper. Look for FSC (Forest Stewardship Council) papers- they don't use softeners and are often made from renewable sources.
Choosing a more sustainable toilet paper makes a big difference, but you should still be able to buy a product that's fairtrade, with low- or zero-tarifs, or better for the environment. However, there's nothing more important than choosing toilet paper that is good for your health too. So why not try something more traditional and embrace the joys of a good old fashioned bar of green soap.
Sources: http://www.mercosmeticsuk.com/green-bath-soaps-apr2015/ http://www.ultras-int.com/health-and-beauty/soaps-and-hair-creams/soap-choices/seal-tear-soap-scrubs-chinese-green-scrubs/chinese-green-scrubs/ https://www.ecosystems.wiley.com/doi/10.1002/eop.1233/full http://www.entrepreneur.
My evaluation
What the actual…?! 😧
Where do I even start with this?
At 1,129 words, it's a respectable length. But this is the only positive thing I have to say.
It scored 58.6 on the Flesch Reading Ease test, which is fairly difficult to read. And it had multiple issues including too much passive voice and too many long sentences of more than 20 words.
It makes no sense
The whole article is nonsense, from the title to the conclusion. Here are some choice excerpts:
If toilet paper was great for the environment, we wouldn't need this article!
"Using a disposable newspaper or a tissue or tissues instead of a wipe is a waste of resources."
Err… what?!
"The toilet paper we use might be pretty — and might even have cute cartoon characters on the packaging, but this is just paper and cardboard with a fresh lick of paint."
I thought we were going to get to a point here, but I was mistaken!
"So why not try something more traditional and embrace the joys of a good old fashioned bar of green soap."
This is the conclusion! Where does green soap come into it?!
It waffles — a lot
One of the golden rules of content writing is not to waffle. You should trim off the fat and make every sentence count. Writesonic seems to have missed this memo.
"Of course, buying environmentally-friendly toilet paper doesn't make you automatically do the right thing with your other purchases. It's just one of many things you can do to help the environment."
A masterclass in how to bulk up your word count!
"We all have a certain number of tissues to clean ourselves, and what we need is the right type for the job."
Again… what?!
It's repetitive
"Most UK consumers don't know that their toilet paper is made from virgin wood pulp, which means trees are cut down to make it." ~ Start of intro
"Most UK consumers don't know that their toilet paper is made from virgin wood pulp, which means trees are cut down to make it." ~ End of first AI paragraph
"Toilet Paper Is Made Of Trees" ~ Section 1 heading
"The problem with toilet paper is that it's made from trees." ~ Section 3 body
"The major problem with paper is that it's made from trees." ~ Section 3 body
It's factually incorrect
"…all modern toilet paper has to be made from virgin wood pulp."
This isn't true and the whole purpose of the article is to show it doesn't have to be!
"…many eco-friendly brands will be more expensive than traditional brands, and more packaging is used."
It's not true that eco-friendly brands use more packaging — or that they're more expensive.
The thinking isn't joined up
"These tissues, which come into contact with your hands and face every day, often include bleached pulp. This means that the paper ends up in a landfill, or worse, burning in an incinerator or generating heat-trapping emissions in our homes."
The first sentence gives absolutely no meaning to the second. And the second sentence is… surreal!
"The three major types of toilet paper that you can buy are cotton, bamboo and recycled paper. Each has its own advantages and disadvantages: Cotton This type of toilet paper uses cotton fibre for its padding, and is often marketed as 'recycled' as it is made from recycled waste paper."
It sets up to talk about the advantages and disadvantages of the three different types of toilet paper, starting with cotton, which it then wrongly describes as recycled. Then it gives up on that completely and moves on to the next paragraph!
It's just not good content
This is badly researched, badly written and not optimised for search engines at all.
You can't use it as it is and no amount of editing will make it better. It doesn't even have the bones of a decent article for a writer to flesh out.
So what's the point of it? And why are people paying for it?
Well, I'm still at a loss with that one.
So let's move on…
Copysmith
Copysmith uses a different approach. Rather than typing in a title, pressing a button and — hey presto — getting an article, you get an article outline that you can use to write the article yourself.
Here's a lowdown on the process.
Your title and description
With this AI writer, you start by giving your post a title and a description of what it's about.
I went with:
How to choose more sustainable toilet paper
We all have a duty to be more environmentally sustainable, so how can we choose toilet paper that's better for the environment?
Getting the article outlines
When you input your title and description, you get a list of 12 article outlines with headings. And you can generate more of these by clicking a button.
The idea is that you choose the article outline that's most relevant to what you're looking for, then use it as template to write the article yourself.
Let's have a look at the top three article outlines I was given.
Section 1: Check your toilet paper packaging
Section 2: Check your toilet paper roll
Section 3: Check your toilet paper brand
Takeaway: This guide will give you the information you need to choose more sustainable toilet paper. By checking for recycled content and SFI certification, you can make a positive difference.
Section 1: What is tea tree oil and why is it so important?
Section 2: The benefits of tea tree oil for acne and other skin conditions.
It's quite basic, but not a bad start.
I covered some of these in my own article.
No idea where I asked about tea tree oil though — that's a complete mystery!
Section 1: How to choose the most sustainable toilet paper
Section 2: How to pick the right type of toilet paper
Section 3: What's in your toilet paper?
Section 4: Using less toilet paper
Takeaway: Choosing sustainable toilet paper is easy and can help reduce your impact. This guide will explain how to do it and why it's important.
On this one, the first two headings are too similar.
I covered these themes in my article — with the exception of using less toilet paper, which I felt didn't really answer the brief.
The ordering of the four headings seems odd. If you want to change people's behaviour, explaining what's in their toilet paper (and why that makes it bad for the environment) would be better as an introduction.
Section 1: The pros and cons of recycled toilet paper
Section 2: Learn how to properly dispose of your used toilet paper
Section 3: Look for toilet paper with less packaging
Section 4: Choose a brand with a good environmental message
Section 5: Buy in bulk to reduce packaging waste
Takeaway: There are lots of options when it comes to choosing toilet paper, but it can be difficult to gauge which option is best for the environment. This guide will help you make an informed decision.
You may have noticed a pattern here. Outline 1 has three sections, Outline 2 had four — and this one has five. But it's not actually a pattern and Outline 4 doesn't have six. The rest of the outlines have a mixture of three, four and five sections.
Section 2 seems to be an introduction to composting toilets, which may not be an option for most people.
And section 3 would seem to refer to buying in bulk, which is repeated in section 5.
In my own article, I looked at choosing paper packaging rather than plastic.
Would I recommend Copysmith?
From my experience of the trial, no.
The article outlines don't provide enough substance for a detailed article that answers all the questions.
In fact, I'm confident that any human content writer — and even some non-content writers — could come up with a better and more thorough article outline than these.
But aren't the headings useful?
When I first saw all the headings, I thought maybe picking and choosing a selection from each outline would help to inspire an article — and I guess it might. But as the article outlines were all covering the same subject, a lot of the headings were very similar to each other.
Some didn't seem too relevant to the brief. Like these, for example:
Sustainable options for your bathroom
Sustainability in other areas of your life
Compost your waste or recycle it, but don't throw it away.
And some were completely bizarre, like these:
Making your own toilet paper from scratch
Gourmet toilet paper: Advice from toilet paper experts.
Making your own toilet paper sounds above and beyond what most people would be prepared to do.
And I hope this is the first and only time I'll see the word 'gourmet' used to talk about toilet roll!
Should we be concerned?
Content writers are constantly being told we need to fear AI and that it's going to take over our jobs.
But based on this experiment, I don't think we have anything to worry about just yet!
I'm Jenny Lucas, a freelance content writer and copywriter based in Leicester, UK.
If you need interesting, helpful and engaging content for your business blog, you'll find all the information you need on the Content Writing page of my website.
About Jenny
For copywriters
101 Blog post ideas to benefit your business
15 Cringeworthy web design mistakes
Why you're not investing in SEO content (and why you should be)
Content ideas for Winter 2021/2022
Jenny Lucas Copywriting
Sunnyside, Leicester Lane
Desford, Leicester LE9 9JJ
© Jenny Lucas Copywriting 2011 to 2022
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