There is something decidely cool about the vibraphone even though it's not the type of instrument you hang out with or stick under your arm like Chet Baker or Lester Bowie. Pascal Schumacher (L), a promising representative of the new generation of vibraphone players is a jazz musician, based in Brussels, where he surrounds himself with the astonishing piano player Jef Neve (B), real acrobatic poet on the keyboard, and one of the strongest rhythm sections in Europe composed of Christophe Devisscher (B) on bass and Jen Düppe (D) on drums. Yet over two albums, Pascal Schumacher and his Quartet have given the vibe a new burst of interest. They combine cool competency with a patina of blues, which is all the more surprising as they come from tiny Luxembourg. Lyrical, stylish and yet still cool, this is jazz we could vote for.


MONTCLAIR, N.J., Feb. 7, 2017- Communities Promoting Animal Welfare NJ (CPAW NJ) is a new nonprofit organization serving Montclair and surrounding towns. CPAW NJ is going dedicate their time to preventing animals from being surrendered to shelters, as well as assist in controlling the community cat population.
The organization aims to keep animals out of shelters by preserving the bond between humans and their pets. By providing their knowledge, expertise and resources to anyone considering giving up their pet, they will be able to keep cats out of shelters. They will do whatever they can to find a solution, whether it is just assistance with cat food or finding a new home for the animal.
CPAW NJ will also be working to control the population of community cats. They will share their advice with the people who care for the community cats in their neighborhoods. They will also be partnering with other organizations that make it their mission to trap, neuter, vaccinate and return (TNVR) in order to prevent overpopulation.
By educating the people that care for community cats and controlling the population, the organization will decrease the burden placed on local animal shelters. "We believe that by collaborating with various parts of the community, we will be able to effectively accomplish our humane goal of promoting animal welfare," said Karen Shinevar, one of the founders of CPAW NJ.
Anyone interested in getting involved with CPAW NJ's efforts is encouraged to become a Solution Advocate. These Solution Advocates have various roles. They can be storytellers, social media experts or donation solicitors. The organization is always seeking donations from anyone able to help out monetarily.
Shinevar, Marie-Christine Lochot and Rafik Tawadrous founded CPAW NJ with a mission to keep animals out of shelters and control the population of community cats. The organization values compassion, collaboration and celebration in order to promote animal welfare. For more information or to get involved, visit www.cpawnj.org.


text:
<?php namespace SunMailer;

interface MailerInterface {

    /**
     * To send an email
     *
     * @param      $email
     * @param null $name
     * @param      $subject
     * @param      $body
     * @param null $attachment
     * @param null $bcc
     *
     * @return mixed
     */
    public static function send( $email, $name = null, $subject, $body, $attachment = null, $bcc = null);
}

redpajama_set_name: RedPajamaGithub


You can use Adobe Dreamweaver CC to create new websites or redesign existing ones. In Dreamweaver CC, which stands for Creative Cloud, you find the familiar Welcome page plus new tools for managing Cascading Style Sheets (CSS) and web fonts. Enhance Dreamweaver by installing plug-ins, or finesse your website by perusing a list of links for web designers.
Even experienced web designers refer to online resources to keep up with their skills and add special features to their websites. The following websites will make it easy for you to find some resources that can help you with your web design projects.


text:
Anindita Banerjee speaks on Aelita, Queen of Mars in Radiant Futures keynote speech
Natasha Bluth
On April 8, 2016, the NYU Jordan Center for the Advanced Study of Russia hosted a conference entitled "Radiant Futures: Russian Fantasy and Science Fiction." After the first panel, NYU Professor of Russian and Slavic Studies Eliot Borenstein introduced keynote speaker Anindita Banerjee. "If we think of our conference and our field in terms of science fiction, then she is Queen of Mars, our Aelita," Borenstein said. Banerjee, a professor of comparative literature at Cornell University, centered her talk on Aelita, Queen of Mars, a 1924 Soviet silent film directed by Yakov Protazanov based on Alexei Tolstoy's eponymous novel.
Banerjee's talk, titled "The Telescope and the Bioscope: Science Fiction Spectacle on the Early Soviet Screen," targeted the lack of critical attention to Soviet and post-Soviet science fiction media—something particularly ironic, given the prominence of the Soviet space project in the Russian global imaginary. Banerjee explained that her project grew out of her book, We Modern People, which argues that science fiction was not just a literary phenomenon, but an aesthetic as well as ideological platform for negotiating a uniquely Russian vision of being, thinking and acting modern. "I realized even then that science fiction on the screen deserved a sustained study of its own, which attends to its mediatic specificity," she said. Instead of the traditional focus on production, Banerjee made a point to approach Soviet science fiction cinematography from the side of consumption.
Banerjee started her presentation with a clip from Aelita, which introduces the martian telescope-bioscope, a device that enables observation of life on other planets. Calling the film "the first Soviet experiment in cinematic fiction," she mentioned that critical work on Aelita has been limited to the role of politics in influencing how the film was made, rather than the study of the film itself. For Banerjee this is peculiar, given that "the cosmos was a unique frontier that was completely mediated through the conjuncture between technology and the imagination." What's more, the fascination with outer space emerged almost simultaneously with the advent of cinema.
Looking back at history, Banerjee considered the effect that the astronomical theater had on the Russian public in the early 20th century. By 1901, the three observatories in Moscow, St. Petersburg and Kazan transformed a previously restricted scientific domain into a commercialized exhibition space. Spectators could attend screenings of silent documentaries accompanied by an orator. Technological advances such as spectroscopic lenses made images of space into discernible images, which were often displayed in magazines. "Suddenly clear views across unimaginable distances radically changed the reading public's mode of locating the self in the world," Banerjee said.
The self-reflexive power possessed by the cosmos also plays out on the movie screen. Arguing that scientific cinema and literature are more than simply mimetic representational devices for what lies outside their pages and frames, Banerjee said that science fiction in cinema "transform[ed] the modes of seeing, being, thinking and acting encompassed by both ideology and the practices of everyday life." Aelita, she said, represents a case study of how cinema provided a new interface between science, spectatorship and subjectivity.
Banerjee then screened a second clip from the film, in which Aelita looks through the telescope-bioscope and sees a movie theater on an alternate planet – Earth. Instead of showing Earth as an inanimate object, the device "enables Aelita to inhabit the everyday spaces of everyday life on it," Banerjee said. With its attention to the apparatus, the scene marks the convergence of the telescopic fantasy of space and the bioscopic fantasy of life. It also is a moment of estranged reversal. Aelita views life on Earth from Mars, and at the same time, the Earthly spectators at the movie theater watch as their own lives unfold on-screen. "By placing the telescope-bioscope front and center, both in the frame and the plot of Protazanov's movie, it also draws the viewer's attention to what the device actually visualizes for both the protagonist and the [movie theater] audience," Banerjee said.
In the Q&A session that followed, Banerjee was able to expand on elements of Aelita and her book, We Modern People. She mentioned that while snippets of early film shown in Aelita through the telescope-bioscope may seem old-fashioned to us in the 21st century, scenes in the film portray modern industrialized cities, tanks, and other war machines that are overpowering and technologically advanced for the time period. Banerjee also referenced her book, drawing a comparison between the national exhibitions that began in the 19th century and the emergence of cinematography. Both media are built on what she called "circuits of contact and contamination," and invite the self-construction or self-imagination inherent to the modern age. Like cinema, technology was integral national expositions and fairs, providing a similarly immersive and futuristic experience.
Tags: cinema, Film, Literature, Russia, science fiction, Soviet Union
Category: Event Recaps

redpajama_set_name: RedPajamaCommonCrawl


text:
Ellen DeGeneres Meets Her 'SNL' Impersonator Kate McKinnon
On the latest episode of Ellen DeGeneres' talk show, the comedian met Kate McKinnon, the woman who has been receiving praise for her impression of Ellen on 'Saturday Night Live'.
"It's been such a journey. I'm trying not to cry because I have loved you my whole life, truly," McKinnon said. "My mother has come up with schemes to try to meet you."
"I used to paint, I was an oil painter, and she would always say, 'If you would just paint a portrait of Ellen and send it to her, I'm sure that she would have you as a guest on the show. I don't know why you don't just paint a portrait.'"
"I never did and I don't know why. I guess it's because I was meant to appear in a different capacity on the show."
"It's like I've met Jesus, if you believe in that sort of thing," she joked. "I've arrived at the gates and here you are."
"I hope I don't disappoint you," DeGeneres quipped. "That's a lot of pressure for me."
DeGeneres then asked McKinnon to change out of her blue dress and put on a matching outfit.
"I will treasure this forever," McKinnon joked. "Thank you!"
The jeans had a big hole in them but that didn't stop McKinnon from dancing, stretching and enjoying herself.
"It's time to say goodbye now," McKinnon said. "Be kind to one another. Adios, arrivederci, avocados. I'm vegan. Just kidding! I'm Ellen."
Ellen DeGeneres Pays Tribute To Late Father: 'Not One Bone of Judgment'
Sofia Vergara Tries To Say 'Discombobulated' and Tells Story From Nude Shoot
Ellen Doesn't Want Trump On Her Show: 'He's Against Everything I Stand For'
Ellen DeGeneres Recalls Coming-Out Episode As 'Hardest Thing I Ever Had to Do'
Ellen DeGeneres Explains Why She Canceled Kim Burrell's Appearance
Kim Burrell Will Not Appear On Ellen DeGeneres' Show After Comments
Ellen DeGeneres Adds Dancing To Classic Christmas Movies
Ellen DeGeneres Teaches Britney Spears How To Be A Diva At Mall
Ellen DeGeneres Gives Teen Who Returned Chris Hemsworth's Wallet $10,000
Kate McKinnon Shows Off Her Gal Gadot Impression On 'Tonight Show'
SNL Makes Kellyanne Conway Pennywise the Dancing Clown
Gal Gadot & Kate McKinnon Kiss in 'Wonder Woman' Parody on SNL
Kate McKinnon Hammered Holes into SNL's Office Walls After Season
Kate McKinnon To Voice Ms. Frizzle in 'Magic School Bus' Reboot
Kate McKinnon, Josh Brolin & Fallon Do Celebrity Impressions In 'First Impressions'
Melissa McCarthy, Kristen Wiig, Leslie Jones & Kate McKinnon In 'Ghostbusters'
Kate McKinnon On Keys To Justin Bieber and Ellen Impressions With Conan


Mercurial repos inside Dropbox are not a great idea after all. But I had one problem: I sometimes forget to push changes to some files, and Dropbox comes to the rescue; at the same time I need history for such files, and I don't want problems to arise.
Create your dropbox-aware folder, e.g. "somefolder"
create a .hg folder inside it, e.g. "somefolder/.hg"
now use your dropbox client, be it the CLI or the GUI, enable Selective Sync and remove such .hg folder from being synced on your current machine. You must take this step on all machines that you want to use with such technique.
disabling sync for .hg should remove it from the somefolder directory.
Now you can use hg init inside "somefolder", and you'll obtain a non-synced hg repository.
What for? Whatever! I just create a repo on bitbucket and push/pull changes to/from here.
to obtain such .hg directory.
Dropbox syncs the actual working copy, letting you have the latest version always 100% synced without manual intervention.


text:
Home News Nigerian Airforce Advices the Public Residence aroud Arepo Area of Ogun State...
Nigerian Airforce Advices the Public Residence aroud Arepo Area of Ogun State not to Panic
It has come to Notice of the Nigerian Military Operation that Panic has widely Spread out in the Pipe Line Are of Lagos-Ibadan Express way called Arepo Town in Ibafo due to Activities carried out by the NAF recently to destroy the Hideouts of Oil Bunker and alleged Niger Delta Militant whom has been terrorizing the Area for too long.
In an official Statement Released it reads:
The attention of the Nigerian Air Force (NAF) has been drawn to media report on apprehension and panic by members of the public living around the area of ongoing air operations at AREPO and its environs. Thus, it has become necessary to allay the fear of the people with a view to sustaining public confidence and support for the ongoing operations against pipeline-vandals code named OPERATION AWATSE in the South West region of the Country. Therefore, the NAF wishes to assure the general public, within the affected areas, of their safety as the ongoing air operation is carefully planned to avoid collateral damage to the civil populace and property.
READ ALSO : Nollywood actress Sola Sobowale shares adorable Photos of her Twin Daughter has they turned a year older
The NAF is careful to ensure that only legitimate targets, such as observation post, anti-aircraft gun position, boats laden with suspected stolen petroleum products and armed combatants, are selected for strike. In aligning with this global best practice, the NAF usually undertakes surveillance flights before and after each strike to mitigate incidental harm to civilians and property. While pre-strike assessment missions are undertaken to help identify and select legitimate military targets for neutralisation, post-strike or Battle Damage Assessment missions are meant to ascertain whether the desired degree of impact was achieved or not, and if a follow-up re-attack is required.
It may also be necessary to state that NAF pilots are adequately trained, and as professionals, would not engage in human right violations and abuses. The successes recorded in the past 5 days since the NAF commenced strike mission in AREPO, in support of surface forces under OPERATION AWATSE, attest to this display of absolute professionalism by NAF pilots involved in the operation as no single incident of collateral damage has been recorded.
Meanwhile, it is necessary to add that the NAF, working in collaboration with men of the Nigerian Navy Special Boat Services (SBS), has set on fire the hideouts and camps used by the pipeline vandals in AREPO. As a result, Community leaders and members of the public with useful information on the whereabouts of the fleeing vandals and other criminal elements are please requested to oblige such information to the Security Agencies. Furthermore, the civil populace is please advised not to panic but to go about their legitimate businesses.
#Tas2017 : Feyi Olubodun Will Be Coming To The Academic Summit 8.0!
Arepo in Ogun State
Operation Awatse
Previous articleNollywood actress Sola Sobowale shares adorable Photos of her Twin Daughter has they turned a year older
Next articleOlori Wuraola Ogunwusi Lovely in New Photo
You're my ride or die – BBNaija's Prince pens romantic message to Dorathy on...
Violette - November 4, 2020 0
Big Brother Naija star, Prince have nothing but sweet words for Dorathy on her 25th birthday as he noted how blessed he his to...
S.A celebrity Bonang Matheba goes braless in mesh outfit
South African celebrity Bonang Matheba was all shades of stunning as she stepped out for an event yesterday. rocking a white mesh outfit, the beauty...
Popular Skit Maker, Isbae U buys a luxury mansion in Lagos (video)
Ifeoma Ngozichukwu - December 31, 2021 3
What a wonderful way to end the year as Popular skit maker, Isbae U acquires a luxurious mansion in Lagos. His friends took to their...
#BBNAIJA: Tacha Announce Tiwa Savage Ex Husband Teebillz CEO of Billz Vizion as her...
Violette - November 10, 2019 0
Things seems to be going in the right direction for former big brother naija housemate, Tacha as she announce her new management. Former Big Brother...
I have the right to threesome as long as it's legal – Naira Marley...
Controversial singer and songwriter, Naira Marley has sent a strong message to fans criticising him on social media over a post he shared on...


text:
\section{Introduction}

The combination of new ideas from machine learning and recent developments in quantum computing has lead to an impressive array of new applications. \cite{harrow2009quantum,wiebe2012quantum,lloyd2014quantum,wittek2014quantum,wiebe2014quantum,rebentrost2014quantum,biamonte2017quantum,mcclean2018barren,schuld2019quantum,tang2019quantum,havlivcek2019supervised,liu2021rigorous,Liu:2021wqr}. Prominent examples of interplay are the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) \cite{mcclean2016theory,cerezo2021variational, Farhi_2014}, which are  considered  among the most promising quantum machine learning approaches in the Noisy Intermediate-Scale Quantum (NISQ) \cite{preskill2018quantum} era. VQEs and QAOA have shown tremendous promise in quantum simulation and quantum optimization \cite{peruzzo2014variational,mcardle2020quantum,Yuan:2020xmq,Liu:2021otn, Farhi_2021, Hadfield_2019}.

One of the most important neural network architectures in classical machine learning are Convolutional Neural Networks (CNNs) \cite{lecun1998gradient, krizhevsky2012imagenet,simonyan2014very,szegedy2015going,lecun2015deep}. In recent years, CNNs have also found applications in condensed matter physics and quantum computing. For instance, authors in~\cite{Lukin_2019} proposes a quantum convolutional neural network with $\log N$ parameters to solve topological symmetry-protected phases in quantum many-body systems, where $N$ is the system size. One of the key properties of classical CNNs is equivariance, which roughly states that if the input to the neural network is shifted, then its activations translate accordingly. Equivariance is one of the main reasons behind the unreasonable success of CNNs. However, when working with discrete representational spaces used to describe finite-dimensional quantum systems, generalizing the classical notion of equivariance becomes challenging.

In quantum systems, there is a discrete set of translations corresponding to permuting the qudits as well as a continuous notion of translation corresponding to spatial rotations by elements of $\operatorname{SU}(d)$. Combining these two is the basis of so-called Permutational Quantum Computing (PQC) \cite{pqc}. Therefore, a natural starting point for realizing convolutional neural networks in quantum circuits is to look for {\it permutational equivariance}. In the most general setting this is somewhat of a daunting task however, since by Cayleigh's Theorem, it contains as special cases equivariance to any finite group.

In recent years, there has been a surge of interest in designing neural networks that are equivariant to the action of various symmetry groups \cite{Cohen2016}. In particular, it has been recognized that by constructing neural networks that operate on the basis of irreducible representations of the group (so-called Fourier space neural networks), and
group equivariant convolution is easy to implement because it simply reduces to matrix multiplication \cite{KondorTrivedi2018}. Such Fourier-space CNNs have seen applications in spherical image recognition \cite{Cohen2017spherical,SphericalCNNNeurIPS2018}, chemistry \cite{TensorFieldNetworks2018arxiv,CormorantNeurIPS} and particle physics \cite{Kondor_2020} (e.g. Lorentz group equivariance).
Permutation equivariant networks have been used in the context of learning set-valued functions \cite{Zaheer2017a} and learning on graphs \cite{maron2018invariant,thiede2020general}.
Most applications of permutation-equivariant neural networks work with a subset of representations of $S_n$. In contrast, in physical and chemical models where the Hamiltonian exhibits global $\operatorname{SU}(d)$ symmetry, such as the Heisenberg model, it is necessary to consider \emph{all} the $S_n$ irreps. However, even the best classical Fast Fourier Transform (FFT) over the symmetric group $S_n$ requires at least $\mathcal{O}(n!n^2)$ operations \cite{Maslen_1998, Clausen_1993}, which dashes any hope of calculating the Fourier coefficients even for relatively small $n$. Indeed, despite increasing realization of the importance of enforcing $\operatorname{SU}(2)$ symmetry, none of the neural-network quantum state (NQS) ans{\"a}tze are able to respect $\operatorname{SU}(2)$ symmetry for all $\operatorname{SU}(2)$ irreps, due to the super-polynomial growth of the number of multiplicities of irreps and the super-exponential cost to compute Fourier coefficients over $S_n$. Finding variational ans{\"a}tze respecting continuous rotation symmetry is highly desirable because it not only helps to gain important physical insights about the system but also leads to more efficient simulation algorithms  \cite{Vieijra_2021}.

There have been several attempts to introduce theoretically sound analogs of convolutional neural networks to quantum circuits, but they have generally been somewhat heuristic. The major difficulty is that the key ingredient of the success of CNNs -- translation invariance -- lacks a mathematically rigorous quantum counterpart due to the discrete spectrum of spin-based quantum circuits. For example, \cite{Lukin_2019} uses the quasi-local unitary operators to act  vertically across all qubits.

In this paper, we argue that the natural form of equivariance in quantum circuits is permutation equivariance  and proposing a broad class of  $k$-local $S_n$ equivariant quantum convolutional ans{\"a}tze. We introduce a theoretical framework to incorporate group-theoretical CNNs into the quantum circuits, building on and generalizing the PQC's framework.  As a feature, the \emph{Schur-Weyl} duality between $S_n$ and $\operatorname{SU}(d)$ appears naturally and will be used throughout the paper. We rigorously demonstrate the equivalence of labeling Schur basis by YJM-elements and by Casimir operators, first conjectured in Harrow's thesis \cite{harrow2005PhD}, (see also discussions by Bacon, Chuang, and Harrow \cite{Harrow06} and Krovi \cite{krovi}), using the Okounkov-Vershik approach \cite{Okounkov1996}. Using the refined duality we show that the quantum circuits are the natural Fourier space in the sense that one does not need to encode Fourier coefficients to perform the \emph{Fourier space $G$-equivariance convolution}, with the corresponding initial states, having been prepared using constant-depth qudit circuits. An immediate consequence of this is the existence of a super-exponential speed-up when evaluating matrix elements of $S_n$ Fourier coefficients and on performing $G$-equivariant convolution via simulating $k$-local $\mathbb{C}[S_n]$ Hamiltonian on qudits. This quantum super-exponential speed-up naturally circumvents the need to perform classical Fourier space activation, suggesting that quantum circuits are well-suited to implement the group-equivariant CNNs.

Motivated by the class of problems with a global $\operatorname{SU}(d)$ symmetry, we construct the variational $S_n$-equivariant Convolutional Quantum Alternating ans\"{a}tze ($S_n$-CQA). We show that $S_n$-equivariant CQA is dense in every $S_n$-irrep, hence it acts as universal variational model for problems that possess global SU(2) symmetry. A direct consequence of this density result is that  $S_n$-CQA  can be applied to a wide array of machine learning and optimization tasks that exhibit global SU($d$) symmetry or require explicit computation of high dimensional $S_n$-irreps, thus presenting a quantum super-exponential speed-up. The proof techniques in the density result of $S_n$-CQA are of independent interest and can be applied to the study of computational power of Quantum Adiabatic Optimization Algorithm. The latter is designed to solve combinatorial optimization problems and can be implemented in near-term quantum hardware~\cite{Farhi_2014}. It is shown in \cite{Lloyd_2018, Morales_2020} that QAOA ans\"{a}tze generated by simple local Hamiltonians are universal. We use our techniques to derive the universality for a different but related class of QAOA ans{\"a}tze with a richer set of mixer Hamiltonians. In addition, we show that the $S_n$-CQA ans{\"a}tze correspond to a new class of Quantum Adiabatic Optimization Algorithms under the Schur basis, with path dependent coupling strengths for the mixer Hamiltonian.

We illustrate the potential of this framework by applying it to the problem of finding the sign structure of the ground state of $J_1-J_2$ antiferromagnetic Heisenberg magnets.  This is a paradigmatic case where classical neural networks struggle to discover the sign structure to an admissible accuracy due to violation of global $\operatorname{SU}(2)$ symmetry \cite{Choo_2019,Castelnovo_2020}.  We include numerical simulation to show the effectiveness of the $S_n$-CQA ans{\"a}tze in finding the ground state's sign structure in the frustrated region using only $\mathcal{O}(pn^2)$ parameters for $p$ alternating layers.

All statements and theorems discussed in the main text are proved in full detail in Supplementary Material (SM).

\section{Refined Schur-Weyl Duality} \label{s2}

Schur-Weyl duality is widely used in quantum computing, quantum information theory and high energy physics. In particular, in Quantum Chromodynamics it was used to decompose the $n$-fold tensor product of $\operatorname{SU}(3)$ representations. In this context, standard Young tableaux are referred to as Weyl-tableaux and labeled by the three iso-spin numbers ($u,d,s$). The underlying Young diagrams contain three rows $\lambda = (\lambda_1,\lambda_2,\lambda_3)$ that also correspond to irreps of $S_n$ via Schur-Weyl duality (Fig.\ref{YoungDiagram} \& Fig.\ref{SchurWeyl} $(a), (b)$). In physics, the convention is to use Young diagrams $\lambda' = (\lambda_1',\lambda_2')$ labeled by Dynkin integers via highest weight vectors. Thus Young diagrams are one row shorter than their mathematical counterparts (for any $\operatorname{SU}(d)$). We will henceforth adhere to the mathematical notation for the rest of this paper.

\begin{figure}[H]
	\centering
	\includegraphics[width=2in]{Figures/YoungDiagram.pdf}
	\caption{Mathematical (left) and physical (right) notations of Young diagrams $\lambda = (6,3,1), \lambda' = (5,2)$ representing to the same $\operatorname{SU}(3)$-irrep of highest weight $(3,2)$. The mathematical notation also corresponds to an $S_{10}$-irrep in Schur-Weyl duality.}
	\label{YoungDiagram}
\end{figure}

\subsection{Background on the Representation Theory of the Symmetric Group} \label{s2.A}

In this section we define some of the mathematical concepts and notations used in the rest of the paper. Further details can be found in~\cite{Sagan01,Goodman2009,Tolli2009}.

Let $V$ be a $d$-dimensional complex Hilbert space with orthonormal basis $\{e_1,...,e_d\}$. The tensor product space $V^{\otimes n}$ admits two natural representations:
the \textit{tensor product representation} $\pi_{\operatorname{SU}(d)}$ of $\operatorname{SU}(d)$ acting as
\begin{align*}
	\pi_{\operatorname{SU}(d)}(g) (e_{i_1} \otimes \cdots \otimes e_{i_n}) \vcentcolon = g \cdot e_{i_1} \otimes \cdots \otimes g \cdot e_{i_n},
\end{align*}
where $g \cdot e_{i_k}$ is the fundamental representation of $\operatorname{SU}(d)$, and  \textit{permutation representation} $\pi_{S_n}$ of $S_n$ acting as
\begin{align*}
	\pi_{S_n}(\sigma) (e_{i_1} \otimes \cdots \otimes e_{i_n}) \vcentcolon =
	e_{i_{\sigma^{-1}(1)}} \otimes \cdots \otimes e_{i_{\sigma^{-1}(n)}}.
\end{align*}

\begin{theorem*}[Schur-Weyl Duality]
	For all Young diagrams $\mu,\lambda$ of size $n$ with at most $d$ rows, $\pi_{\operatorname{SU}(d)}$ and $\pi_{S_n}$ can be decomposed as
	\begin{align*}
		& \pi_{\operatorname{SU}(d)} \cong \bigoplus_\mu W_\mu \otimes \operatorname{1}_{m_{\operatorname{SU}(d),\mu}}, \quad  \pi_{S_n} \cong \bigoplus_\lambda \operatorname{1}_{ m_{S_n,\lambda}} \otimes S^\lambda ,
	\end{align*}
	where $W_\mu$ denotes an $\operatorname{SU}(d)$-irrep, $S^\lambda$ denotes the dual $S_n$-irrep and $m_{\operatorname{SU}(d),\mu},m_{S_n,\lambda}$ are multiplicities. With respect to the same Young diagram, $\dim W_\lambda = m_{S_n,\lambda}$ and $\dim S^\lambda = m_{\operatorname{SU}(d),\lambda}$ (see for example, Fig.\ref{SchurWeyl} $(a)$ and $(b)$).
\end{theorem*}

\begin{figure*}[ht]
	\centering
	\includegraphics[width=0.9\textwidth]{Figures/SchurWeyl.pdf}
	\caption{$(a)$ is the decomposition of $(\mathbb{C}^2)^6$ with respect to $\operatorname{SU}(2)$ while $(b)$ is for $S_6$ by Schur-Weyl duality. $(c)$ respects both permutation modules and irreps of $S_6$.}
	\label{SchurWeyl}
\end{figure*}

One can easily verify that $\pi_{\operatorname{SU}(d)}$ and $\pi_{S_n}$ commute (further properties are described in the SM). Consider the \textit{symmetric group algebra} $\mathbb{C}[S_n]$ consisting of all formal finite sums $f = \sum_i c_i \sigma_i$. Its representation is then $\tilde{\pi}_{S_n}(f) = \sum_i c_i \pi_{S_n}(\sigma_i)$. When there is no ambiguity, we denote by $U_\sigma$ or simply $\sigma$ the representations $\pi_{S_n}(\sigma)$.

Another decomposition that will be important is the decomposition of the so-called \textit{permutation module} $M^\mu$. Fortunately there is an accessible way to understand $M^\mu$ in the tensor product space $V^{\otimes n}$. To make things simpler, consider the $d=2$ case of $\operatorname{SU}(2)-S_n$ duality on $(\mathbb{C}^2)^{\otimes n}$. Only two-row Young diagrams $\lambda = (\lambda_1, \lambda_2)$ appear in this duality and the half of difference $\frac{1}{2}(\lambda_1 - \lambda_2)$ between the lengths of the two rows gives the total spin of the $\operatorname{SU}(2)$-irrep $W_\lambda$. The permutation module $M^\mu$ is isomorphic with the linear span of all computational basis vectors with $z$-spin components equal to $\frac{1}{2}(\mu_1 - \mu_2)$.

Note that, $(\mathbb{C}^2)^{\otimes n} = \bigoplus_\mu M^\mu$ and each $M^\mu$ is invariant under $S_n$ permutation. Furthermore, $M^\mu$ can be further decomposed into $S_n$-irreps. For two-row Young diagrams ($\operatorname{SU}(2)-S_n$ duality), the decomposition is easy: $M^\mu = \bigoplus_{\lambda \geq \mu} S^\lambda$ where $\lambda, \mu$ have the same size $n$ and we use the \textit{dominance order} $\lambda \unrhd \mu$ if $\lambda_1 \geq \mu_1$. In summary, we have
\begin{align*}
	(\mathbb{C}^2)^{\otimes n} = \bigoplus_\mu M^\mu = \bigoplus_\mu \bigoplus_{\lambda \unrhd \mu} S^\lambda \cong \bigoplus_\lambda \operatorname{1}_{ m_{S_n,\lambda}} \otimes S^\lambda.
\end{align*}
Isomorphic copies of $S^\lambda$ come from different permutation module $M^\mu$. The largest permutation module contains all distinct $S_n$-irreps in $(\mathbb{C}^2)^{\otimes n}$ (see e.g., Fig.\ref{SchurWeyl} $(c)$). For general Young diagrams, decompositions of $M^\mu$ would have nontrivial multiplicities \cite{Sagan01,Tolli2009}.

\begin{figure}[H]
	\centering
	\includegraphics[width=3.3in]{Figures/Bratteli.pdf}
	\caption{Bratteli diagram for $S_6$. Upper Young diagrams connecting by arrows to lower ones arise from the decomposition. Orange arrows form a path. Note that diagrams with more than two rows cannot appear in $\operatorname{SU}(2)-S_6$ duality.}
	\label{Bratteli}
\end{figure}

Each $S^\lambda$ can be decomposed further with respect to $S_{n-1} \subset S_n$ as $S^\lambda = \bigoplus_{S_{n-1},\rho} S^{\lambda\rho}$. The so-called \textit{branching rule} guarantees that the decomposition is multiplicity-free, i.e., distinct $S_{n-1}$-irrep $S^{\lambda\rho}$ appear only once in the decomposition. The so-called Bratteli diagrams in Fig.\ref{Bratteli} show how how different irreps are decomposed. Continuing the decomposition process for $S_{n-2},...,S_1$, the original space $S^\lambda$ will be written as a direct sum of 1-dimensional subspaces ( $S_1$-irreps are 1-dimensional):
\begin{align*}
	S^\lambda & = \bigoplus_{S_{n-1},\rho} S^{\lambda\rho} = \cdots = \bigoplus_{S_{n-1},\rho} \cdots \bigoplus_{S_1,\tau} S^{\lambda\rho\sigma \cdots \tau}.
\end{align*}
Each 1-dimensional subspace $S^{\lambda\rho\sigma \cdots \tau}$ can be represented by a nonzero vector in it. Normalizing them, we obtain an orthonormal basis $\{\ket{v_T}\}$ of $S^\lambda$ called the \textit{Gelfand-Tsetlin basis (GZ)} or \textit{Young-Yamanouchi basis}. Indices $\lambda, \rho, \sigma,...,\tau$ form a path in the Bratteli diagram (see Fig.\ref{Bratteli}) and can be used to define a standard Young tableau $T$ (Fig.\ref{Path}). Young basis vectors are in one-to-one correspondence with standard Young tableaux \cite{Sagan01}. The branching rule is also discussed in $\operatorname{SU}(d)$ representation theory and some authors refer to the $\operatorname{SU}(d)$-irrep basis as the GZ-basis if it is constructed in a similar manner.
\begin{figure}[H]
	\centering
	\includegraphics[width=3.5in]{Figures/Path.pdf}
	\caption{The Standard Young tableau defined by the path in Fig.\ref{Bratteli}}
	\label{Path}
\end{figure}

Let us introduce the central concept used in our work:

\begin{definition*}
	For $1 < k \leq n$, the \textit{Young-Jucys-Murphy element}, or YJM-element for short, is defined as a sum of transpositions $X_k = (1,k) + (2,k) + \cdots + (k-1,k) \in \mathbb{C}[S_n]$. We set $X_1 = 0$ as a convention.
\end{definition*}

As the name indicates, this concept was developed by Young \cite{Young1977}, Jucys \cite{Jucys1974} and Murphy \cite{Murphy1981}.  Okounkov and Vershik showed that  YJM-elements generate the \textit{Gelfand-Tsetlin subalgebra} $\operatorname{GZ}_n \subset \mathbb{C}[S_n]$~\cite{Okounkov1996}, which is a maximal commutative subalgebra consisting of all centers $Z[S_k]$ of $\mathbb{C}[S_k]$ for $k=1,...,n$. Another striking fact is that all YJM-elements are strictly diagonal (indeed, representation of $\operatorname{GZ}_n$ consists of all diagonal matrices) in a Young basis $\{\ket{v_T}\}$ whose eigenvalues can be read out directly from the standard Young tableaux $T$. To be precise, let $\lambda$ be a Young diagram. Since this is a 2-dimensional diagram, we can naturally assign integer coordinates to its boxes. The \textit{content} of each of its boxes is determined by the $x$-coordinate minus $y$-coordinate. Suppose $T$ is a standard Young tableau of $\lambda$. Arranging all contents with respect to $T$, we obtain the \textit{content vector} $\alpha_T$. For instance, the Young diagram $\lambda = (4,2)$ has contents $0,1,2,-1$. The specific standard tableau $T$ in Fig.\ref{Path} has content vector $\alpha_T = (0,-1,1,2,0,3)$. Let $\ket{v_T}$ denote a Young basis vector corresponding to $T$. Measuring by YJM-elements, we have $X_1 \ket{v_T} = 0, X_2 \ket{v_T} = -\ket{v_T}, X_3 \ket{v_T} = \ket{v_T}$ and so forth. Each Young basis vector is determined uniquely by its content vector (see \cite{Okounkov1996} and \cite{Tolli2009} for more details).

\subsection{Correspondence between Spin Labels and Content Vectors}\label{s2.B}

Working from the perspective of Schur-Weyl duality requires, at least theoretically, using the \emph{Schur basis} rather than the computational basis. As we introduced above, it can be the Young basis constructed by the branching rule of $S_n$. However, a more conventional way is conducting sequential coupling and Clebsch-Gordan decompositions of $\operatorname{SU}(d)$ \cite{harrow2005PhD, pqc, sergii1}. We now turn to the question whether this $\operatorname{SU}(d)$ irrep basis, labeled by $d-1$ Casimir operators \cite{Biedenharn1}, is the same as the Young basis of $S_n$ labeled by $n$ YJM-elements. This was conjectured to be true in \cite{harrow2005PhD} and surfaced again in \cite{krovi} when authors introduced an efficient Quantum Schur Transform (QST). An affirmative answer to this conjecture is crucial in this work for two reasons: $(a)$ The Young basis is algebraic. Thus, the gate action drawing from the group algebra $\mathbb{C}[S_n]$ is basis-independent. In particular, it can be implemented directly in the computational basis without computing the Fourier coefficients -- this is a key observation that underpins the super-exponential quantum speed-up. $(b)$ This identification allows us to efficiently generate quantum states required for optimization and learning tasks. As we will show in Section \ref{s3.B} in the qubit case, this initialization will be done using circuits of constant depth.

For two-row Young diagrams, this conjecture was shown to be correct in \cite{Pauncz2018}, where the author studies the question by $\frac{1}{2}$-spin eigenfunctions instead of YJM-elements. Our first mathematical result shows that the general case for $\operatorname{SU}(d)-S_n$ duality still holds and can be proven in a surprisingly easy way using YJM-elements and the Okounkov-Vershik approach. The proof is presented in the SM.

\begin{lemma}\label{J_k^2-X_i}
	Under $\operatorname{SU}(d)-S_n$ duality, sequentially coupled Casimir operators commute with YJM-elements.
\end{lemma}

To illustrate this result, consider the \textit{sequentially coupled total spin basis} $\ket{j_1, j_2,..., j_{n-1}, j_n, m}$ of $\operatorname{SU}(2)$. The \textit{spin component} $m$ and \textit{spin labels} $j_k$ are determined by spin operator $S_z^n = \sum_{i = 1}^n S^z_i$ and sequential coupled Casimir operators $J_k^2 = (S_x^k)^2 + (S_y^k)^2 + (S_z^k)^2$ respectively. Since they commute with YJM-elements, $J_k^2 X_i \ket{j_1, j_2,..., j_{n-1}, j_n, m } = X_i J_k^2 \ket{j_1, j_2,..., j_{n-1}, j_n, m}$. We thus arrive at the following result:

\begin{theorem}\label{Spin-Content}
	YJM-elements are strictly diagonal under the $\operatorname{SU}(d)$ irrep basis. Conversely, sequentially coupled Casimir operators are strictly diagonal under the Young basis. Hence the $\operatorname{SU}(d)$ irrep basis and the Young basis are equivalent.
\end{theorem}

We henceforth denote both bases with unifying notation $\ket{v_T^\mu}$, where the subscript $T$ denotes a standard Young tableau, or equivalently the corresponding GZ-path from the Bratteli diagram, and the superscript $\mu$, as explained in Section \ref{s2.A}, comes with the permutation module $M^\mu$. An example in the next paragraph shows that basis vectors with the same subscript $T$ but different superscripts $\mu$ come from multiple copies of an $S_n$-irrep. We do not need to distinguish these copies mostly except in Section \ref{s3.B}, so the superscript $\mu$ will often be omitted. A detailed study on labeling multiple $SU(d)$ states can be found in \cite{Biedenharn2}.

At the end of this section, we illustrate diagrammatically the correspondence between spin labels and content vectors as a corollary of the preceding theorem plus Schur-Weyl duality. For brevity, consider $\ket{v_T}$ from Fig.\ref{Path}. Its content vector $\alpha_T$ equals $(0,-1,1,2,0,3)$. The first spin-label is definitely $j_1 = \frac{1}{2}$.
Because $\ket{v_T}$ is both a Young basis element and a total spin state, the next spin label should also be recorded according to the path in Fig.\ref{Path}. That is, $j_2 = j_1 + \frac{1}{2}$ if the box containing $2$ is put in the first row of the Young diagram. Otherwise, we subtract $\frac{1}{2}$. Consequently, we obtain spin labels $(\frac{1}{2}, 0, \frac{1}{2}, 1, \frac{1}{2}, 1)$. Since the total spin (last spin label) of $\ket{v_T^\mu}$ is $1$, there are three possible choices of $z$-spin components $j = 1,0,-1$ and they can be found in three different permutation modules (Fig.\ref{SchurWeyl} $(c)$).
The largest one takes $\ket{v_T^0}$ with $\mu = (3,3)$. Note that the pattern of constructing $\operatorname{SU}(2)$ spin labels is simply the familiar branching rule seen in $\operatorname{SU}(2)$-irreps \cite{sergii2}. This pattern still holds for spin label-content vector correspondence in $\operatorname{SU}(d)$ case. The refined Schur-Weyl duality by incorporating the equivalence between SU($d$) irrep bases and the Young bases is at the heart of the quantum Schur transform. Furthermore, in the following sections, we show how the refined Schur-Weyl duality leads to a quantum super-exponential speed-up in performing $S_n$ equivariant quantum convolutions which can be applied to a wide array of quantum machine learning and optimization tasks.

\section{Quantum Circuits as Natural Fourier Space}  \label{s3}

Following \cite{kondor1,kondor2}, given a compact group $G$, it is tempting to implement $G$-equivariant convolutions in the quantum circuits, where the equivariance property is given in the corresponding group algebra $L^2(G, \ast)$ equipped with convolution operator $\ast$. For $f_1, f_2 \in \mathbb{C}[S_n]$:
\begin{align}
	(f_1 \ast f_2)(\sigma) \equiv \sum_{\tau \in S_n} f_1(\tau  )f_2(\tau^{-1}\sigma),
\end{align}
where the equivariance is defined with respect to the left action $L_\eta$ on $\mathbb{C}[S_n]$:
\begin{align*}
	L_\eta (f_1 \ast f_2)(\sigma) = (L_\eta f_1 \ast f_2)(\sigma) = (f_1 \ast f_2)(\eta^{-1}\sigma).
\end{align*}
On the other hand, just like the case of $S_n$, any representation ($\pi_G, V$) can be extended to the group algebra by defining $\widetilde{\pi}_G(f)  = \sum_{g \in G} f(g) \pi(g)$, which is a homomorphism with respect to the convolution: $\widetilde{\pi}_G(f_1 \ast f_2) = \widetilde{\pi}_G(f_1) \widetilde{\pi}_G(f_2)$ \cite{Goodman2009} (RHS is a matrix product). Therefore, the so-called equivariance property is just the associativity of $\widetilde{\pi}_G$ on the representation space or Fourier space in the sense that $\widetilde{\pi}_G(f)$ encodes Fourier coefficients of $f$ \cite{kondor1,kondor2}. In what follows, this perspective will be applied to $S_n$ representation in the context of a recently studied computational model.

Permutational Quantum Computing (PQC) \cite{pqc} takes inspiration from Topological Quantum Computing (TQC) and Penrose spin network model~\cite{penrose1971angular}. In particular, PQC can be seen to mimic some features of TQC, where total spin bases play the role of anyons and computation proceeds by qubit interchange and spin measurement. Suppose we prepare an appropriate Young basis element $\ket{v_T} = U_{\operatorname{Sch}} \ket{x}$ for some $x \in \{0, 1\}^n$, where $U_{\operatorname{Sch}}$ is the quantum Schur transform (QST) in qudits and $\mu$ is the corresponding SU($d$) label (Dykin integer), such that we can measure the matrix elements: $\bra{v_{T'}} \pi(\sigma) \ket{v_T}$. Using $\mathcal{O}(n^2)$ SWAP gates, Jordan argues that the measurements form the permutational quantum polynomial (PQP) class \cite{pqc}. Capable of computing the matrix elements of the $S_n$-irreps in polynomial time was conjectured to capture the non-classical aspects of quantum computing. However, a classical sampling method was later found in \cite{sergii1} for qubits. In this work, we extend the class of PQP (for which we call PQP+) in the hope to capture some of the non-classical aspects by extending $\pi$ as the algebra homomorphism $\widetilde{\pi}(f)$ for some $f = \sum c_i \sigma_i \in \mathbb{C}[S_n]$. To be specific, consider a class of problems that can be solved by measuring
\begin{align*}
	\bra{v_{T'}}\widetilde{\pi}(f) \ket{v_T} \vcentcolon = \sum c_i \bra{v_{T'}} \pi(\sigma_i) \ket{v_T}.
\end{align*}
in polynomial time via Fourier space convolution. In particular, we define PQP+ by the set of polynomially-bounded quantum algorithms able to approximate the matrix elements of the so-called Hamiltonian simulation in qudits:
\begin{align} \label{autocorr}
	\begin{aligned}
		\exp(-i \widetilde{\pi}(f)) & = \sum_{n = 0} \frac{(-i)^n}{n!} (\widetilde{\pi}(f))^n \\
		& = \sum_{n = 0} \frac{(-i)^n}{n!} \widetilde{\pi}(\underbrace{f \ast \cdots \ast f}_{n \operatorname{ many}} ).
	\end{aligned}
\end{align}
Note that the Wedderburn theorem implies that Equation \eqref{autocorr} encompasses all \emph{unitary} $S_n$-Fourier coefficients. If $f$ is supported on $k$-qubits (acting on arbitrary k qubits), so is its auto-correlation $f \ast \cdots \ast f $, and we can efficiently simulate it using standard Hamiltonian simulation methods. The corresponding Hamiltonian evolution need not be $k$-local, and there is an efficient quantum Schur transform proposed with gate complexity $\operatorname{Poly}(n, \log d,\log(1/\epsilon))$ \cite{harrow2005PhD,Harrow06,krovi}. Thus, it follows that the $\mathbb{C}[S_n]$ generalization to PQC is able to achieve super-exponential speed-up computing the matrix element of $\mathbb{C}[S_n]$ Fourier coefficients of the following form, where no polynomial-time classical algorithm is found.

\begin{theorem} \label{thmpqcd}
	Consider a $k$-local $\mathbb{C}[S_n]$ Hamiltonian $H = \widetilde{\pi}(f)$ with Young basis elements $\ket{v_T}, \ket{v_{T'}}$ prepared using the efficient high dimensional QST. The quantum circuit of qudits is able to simulate the matrix element:
	\begin{align*}
		\bra{v_{T'}}\exp(-itH) \ket{v_T} =\delta_{T', T} \bra{v_{T'}}\exp(-itH) \ket{v_T},
	\end{align*}
	in $\mathcal{O}\left( tCk^3 n^k \frac{\log (tC kn^k/\epsilon)}{\log \log (tC kn^k/\epsilon)}\right)$ SWAP gates with the constant $C = \max_i \vert c_i \vert$.
\end{theorem}
\begin{proof}
	The proof is based on the method of Linear Combination Unitaries (LCU) from \cite{Childs_2012}. First, we determine the operator norm of $H$. A standard result for the symmetric group $S_n$ is that the number of permutations supported exactly on $l$ elements, or equivalently with $n-l$ elements being fixed is calculated by the \textit{derangement}:
	\begin{align*}
		D_l = \frac{n!}{(n-l)!} (\frac{1}{2!} - \frac{1}{3!} + \cdots + (-1)^l \frac{1}{l!} ).
	\end{align*}
	When $H$ is $k$-local, then it contains at most $D_2 + \cdots + D_k$ different permutations from $S_n$. Since $n!/ (12 (n-l)!)  \leq D_l \leq n!/ (4 (n-l)!)$, the number is of order $\mathcal{O}(k n^k)$. With $C = \max_i \vert c_i \vert$, $\Vert H \Vert = \Vert \sum c_i \sigma_i \Vert \leq \mathcal{O}(C k n^k)$.

	Consider the Hamiltonian simulation via the product formula:
	\begin{align*}
		[\exp(-i \Delta t H)]^M = \exp(-itH),
	\end{align*}
	where we set $M = t C k n^k$ so that $\Delta t \Vert H \Vert = t \Vert H \Vert/M = \mathcal{O}(1)$. In order to bound the total error by $\epsilon$, each product term should be simulated within error $\tilde{\epsilon} = \epsilon/M$. Then following \cite{Childs_2012} we take the truncated Taylor series of Equation \eqref{autocorr} to order $K$ such that
	\begin{align*}
		& \Vert \exp(-i \Delta t H) - \sum_{k = 0}^{K-1} \frac{(-i\Delta t H)^k}{k!} \Vert = \frac{(\Delta t \Vert H \Vert )^K}{K!} \leq \tilde{\epsilon}.
	\end{align*}
	Using $(k/e)^k \leq k!$, the above inequality holds when $K \log (K/\Delta t \Vert H \Vert e) \geq \log (1/\tilde{\epsilon})$. Since $\Delta t \Vert H \Vert = \mathcal{O}(1)$, we can further relax the condition to require $K \log K \geq \log (1/\tilde{\epsilon})$. Let $K' \vcentcolon \geq \frac{\log 1/\tilde{\epsilon}}{\log \log 1/\tilde{\epsilon}}$. Then
	\begin{align*}
		K' \log K' \geq \log \frac{1}{\tilde{\epsilon}} - \log \frac{1}{\tilde{\epsilon}} \frac{\log \log 1/\tilde{\epsilon}}{\log \log \log 1/\tilde{\epsilon}} \approx  \log \frac{1}{\tilde{\epsilon}}
	\end{align*}
	for small $\tilde{\epsilon}$. Setting $K = K'$ this bound also recovers the result from \cite{Berry_2015}.

The Taylor expansion gives that:
	\begin{align*}
		&\exp(-i \Delta t H) = \sum_{m=0} \sum^N_{i_1,..., i_m} \frac{(-i\Delta t)^m}{m!}c_{i_1}...c_{i_m} \sigma_{i_1} \cdots \sigma_{i_m} \\
		& = \sum^K_{m=0} \sum^N_{i_1,..., i_m} \widetilde{c}_{{i_1}...{i_m}} \widetilde{\sigma}_{i_1,... i_m} + \mathcal{O}(\tilde{\epsilon}),
	\end{align*}
	where $N = \mathcal{O}(kn^k)$ is the total number of k-local permutations and $c_i$ is the corresponding coefficients with $\|c_i \| \leq Ck n^k$ To apply the LCU, we also set:
	\begin{align*}
		\widetilde{c}_{{i_1}...{i_n}} &= \frac{(\Delta t)^n}{n!} c_{i_1}...c_{i_n}, \quad \widetilde{\sigma}_{i_1,..., i_n} &= (-i)^n \sigma_{i_1}\cdots \sigma_{i_n}.
	\end{align*}

	We need $\mathcal{O}(\log NK) $ many selection ancilla qubits to implement $\exp(-i\Delta t H)$ using:
		$\mathcal{O}(\exp(\Delta t \|c_i\|_1) \approx \mathcal{O}(1)$ applications of oblivious amplitude amplification.
	Since any $k$-local permutation $\sigma_i$ can be written in $\mathcal{O}(k^2)$ SWAPs, the total circuit complexity is:
	\begin{align*}
		\mathcal{O}\left(tCk^3 n^k \frac{\log (tC kn^k/\epsilon)}{\log \log (tC kn^k/\epsilon)}\right).
	\end{align*}
\end{proof}

Note that in our work we assume that qudit SWAP gates are easy to implement with constant overhead. We can thus disregard the polynomially-scaling constant on $k$ in the approximation and write the total circuit complexity $\mathcal{O}( t n^k \log (tn^k/\epsilon)/{\log \log (tn^k/\epsilon)}$.
 Theorem \ref{thmpqcd} establishes a generic quantum super-exponential speed-up. We also present a practically relevant quantum super-exponential speed-up for qubits using the following beautiful identity:
\begin{align} \label{trans}
	\pi((i \: j)) = 2\hat{\boldsymbol{S}}_{i} \cdot \hat{\boldsymbol{S}}_{j} + \frac{1}{2}I,
\end{align}
where $\hat{\boldsymbol{S}}_{i}$ is further expanded as the half of standard Pauli operators $\{X, Y, Z\}$. For example, consider $f = (1 \: 2) + (2 \: 3) + (3 \: 4) + (4 \: 1)$. Therefore under $\widetilde{\pi}$,
\begin{align*}
	H_P \equiv \widetilde{\pi}(f) = 2(\hat{\boldsymbol{S}}_{1} \cdot \hat{\boldsymbol{S}}_{2} + \hat{\boldsymbol{S}}_{2} \cdot \hat{\boldsymbol{S}}_{3} + \hat{\boldsymbol{S}}_{3} \cdot \hat{\boldsymbol{S}}_{4} + \hat{\boldsymbol{S}}_{4} \cdot \hat{\boldsymbol{S}}_{1} + I),
\end{align*}
where $H_P$ is simply the 1D Heisenberg chain with a periodic boundary condition of 4 spins. Equation \eqref{trans} was first discovered by Heisenberg himself \cite{Heisenberg1928, Klein_1992} (an elementary proof can be found in SM) and more recently noted by \cite{Seki_2020} in analyzing the ground state property of 1-D Heisenberg chain using symmetry adapted VQE. This identity also supports our numerical simulation on 2-D Heisenberg model in Section  \ref{s6}.

\begin{theorem}\label{thmpqc}
	Suppose $f \in \mathbb{C}[S_n]$ is $k$-local. Consider $H = \widetilde{\pi}(f)$ together with states $\ket{v_T}, \ket{v_{T'}}$. The quantum circuit can simulate the matrix elements:
	\begin{align*}
		\bra{v_{T'}}\exp(-itH) \ket{v_T} =\delta_{T', T} \bra{v_{T'}}\exp(-itH) \ket{v_T},
	\end{align*}
	using $\mathcal{O}\left( t L(k)n^{k} \frac{\log(tL(k)n^k/\epsilon)}{\log\log(tL(k)n^k/\epsilon)}\right)$  Pauli gates, with the constant $L(k)  = \max_i \vert c_i \vert 2^{-1+k}$.
	\begin{proof}
		Since $f = \sum c_i \sigma_i$ is $k$-local, each $\sigma_i$ acts on at most $k$ qubits and can be expanded by at most $k-1$ adjoint transpositions $\prod_{j = 1}^{k-1} \tau_{i_j}$. By Equation \eqref{trans}, it equals
		\begin{align} \label{ham_Pauli}
			\begin{aligned}
			& \prod_{j = 0}^{k-1} (2\hat{\boldsymbol{S}}_{i_j} \cdot \hat{\boldsymbol{S}}_{i_{j+1}} + \frac{1}{2} I) \\
			= & \sum_{j = 0}^{k-1} \binom{k-1}{j} 2^{k-1-2j} \underbrace{( \hat{\boldsymbol{S}}_{i_1} \cdot \hat{\boldsymbol{S}}_{i_2}) \circ \cdots \circ ( \hat{\boldsymbol{S}}_{i_{k-1}} \cdot \hat{\boldsymbol{S}}_{i_k}  ) }_{j \operatorname{ \text{many terms replaced by}} I}.
	     \end{aligned}
		\end{align}
		The operator norm of $\hat{\boldsymbol{S}}_{i_j} \cdot \hat{\boldsymbol{S}}_{i_{j+1}}$ can be seen directly from Equation \eqref{trans} as $\frac{3}{4}$. Therefore,
		\begin{align*}
			\Vert \sigma_i \Vert \leq  \sum_{j = 0}^{k-1} 2^{k-1-2j} (\frac{3}{4})^{k-1-j} \binom{k-1}{j} = 2^{k - 1},
		\end{align*}
		where the last step uses binomial theorem. By Theorem \ref{thmpqcd}, $H$ contains at most $\mathcal{O}(kn^k) \sim \mathcal{O}(n^k)$ many $k$-local permutations and hence written explicitly in $k$-local Pauli gates:
		\begin{align} \label{ham_norm}
			\begin{aligned}
				\Vert H \Vert \leq \sum_i^{\mathcal{O}(n^k)} L(k) = n^k L(k),
			\end{aligned}
		\end{align}
		where $L(k) =\max ({c_i})2^{-1+k}$. The operator norm of $H$ is thus bounded by $n^kL(k)$. The larger bound here reflects the fact that we are using $k$-local Pauli gates, whose coefficient 1-norm is also bounded by the above expression. Suppose the complete expansion of $H$ by Pauli gates is $\sum_i \beta_i H_i$ with $H_i$ being products of $k$-local Pauli gates like in Equation \eqref{ham_Pauli}. Then the Taylor expansion gives that:
	\begin{align*}
		&\exp(-i \Delta t H) = \sum_{n=0} \sum^N_{i_1,..., i_n} \frac{(-i\Delta t)^n}{n!}\beta_{i_1}...\beta_{i_n} H_{i_1}\cdots H_{i_n} \\
		& = \sum^K_{n=0} \sum^N_{i_1,..., i_n} \widetilde{\beta}_{{i_1}...{i_n}} \widetilde{H}_{i_1,..., i_n} + \mathcal{O}(\tilde{\epsilon}),
	\end{align*}
	where $N = \mathcal{O}(k^2 n^k 4^k)$ is the total count of Pauli operators and $\beta_{i}$ is the corresponding coefficients with $\|\beta_i \| \leq L(k) n^k$ by Equation \eqref{ham_norm}.

		Applying the technique of LCU as in Theorem \ref{thmpqcd}, we need $\mathcal{O}(\log NK)$ many selection ancilla qubits to implement $\exp(-i\Delta t H)$ using:
		$\mathcal{O}(\exp(\Delta t \|\beta_i\|_1) \approx \mathcal{O}(1)$ applications of oblivious amplitude amplification. For each selection, we need at most $\mathcal{O}(K)$ times of implementation of $k$-local Pauli operators. Therefore, the total circuit complexity is simply:
		\begin{align*}
			\mathcal{O}\left(M \frac{\log(M/\epsilon)}{\log \log(M/\epsilon)}\right) = \mathcal{O}\left(tL(k)n^k \frac{\log(tL(k)n^k/\epsilon)}{\log \log(tL(k)n^k/\epsilon)}\right).
		\end{align*}
	    This finishes the proof.
	\end{proof}
\end{theorem}

A more involved derivation using Pauli gates in Theorem \ref{thmpqc} in the form of unitary coupled clusters suggests a potentially hardware-friendly implementation on quantum computers \cite{Anand_2021}. However, the bounds become loose as $L(k)$ grows exponentially with $k$. However, we show that, for any practical application of quantum machine learning and optimization tasks, $k$ can be taken small, for instance $4$-local in the case of solving the Heisenberg model in Section \ref{s4} and \ref{s6}.

\begin{figure}[H]
	\centering
	\includegraphics[width=2in]{Figures/PQP.pdf}
	\caption{The relationship between complexity classes. It is not known whether PQP+ strictly contains PQP.}
	\label{PQP}
\end{figure}

Our results strongly suggest that the PQP+ class is able to capture some non-classical aspects of quantum computation in the qudit case. The above form of $\mathbb{C}[S_n]$ Hamiltonian simulation by Equation \eqref{trans} encompasses a large class of physically relevant problems, such as the Heisenberg spin model. Naturally, multiplication of $\mathbb{C}[S_n]$ Hamiltonian operators correspond to $\mathbb{C}[S_n]$ convolution, which leads to further useful applications of machine learning and optimization tasks.
In what follows, we focus on utilizing the quantum super-exponential speed-up to address quantum machine learning tasks by designing alternating variational $S_n$ equivariant convolution ans{\"a}tze. From this point of view, PQP+ may be interpreted as the natural complexity class for \emph{quantum $S_n$-Fourier space activation} for quantum machine learning and optimization tasks.

\subsection{$S_n$ Equivariant Convolutional Quantum Alternating Ans{\"a}tze}

Consider the YJM-elements $\{ X_1, ..., X_n \}$ introduced in Section \ref{s2.A} which generate the maximally commuting subalgebra $\operatorname{GZ}_n$. These elements take the Young basis, which is equivalent to any sequential coupled $\operatorname{SU}(d)$-irrep basis by Theorem \ref{Spin-Content}, as their common eigenbases. Eigenvalues can be read off directly by the associated content vectors \cite{Okounkov1996}. The use of YJM-elements allows us to design the following \textit{mixing Hamiltonian}:
\begin{align}
	H_M = \sum_{i_1,..., i_N} \beta_{i_1... i_N}X_{i_1} \cdots X_{i_N}.
\end{align}
The YJM-Hamiltonian is still strictly diagonal in the Young basis. Suppose we can initialize any Young basis element $\ket{\Psi_{\operatorname{init}}}$ from $S_n$-irreps via QST.  The following ansatz is developed in the form similar to that of QAOA (Pauli $X$ is diagonal wrt. $\ket{+}^{\otimes n}$) \cite{Farhi_2014},
\begin{align*}
	\cdots \exp (-i H_M ) \exp (-i \gamma H_P) \exp  (-i H_M )  \exp(-i \gamma^{\prime} H_P) \cdots.
\end{align*}
To be specific, we have defined a family of mixing operators $H_M$ parameterized by $\beta_{i_1... i_N}$ that is diagonal in the Young basis, which naturally preserves each $S_n$-irrep determined by the initialized states. Differing from the original purpose of using QAOA to solve constraint satisfaction problem, our $S_n$ Equivariant Convolutional Quantum Alternating Ans{\"a}tze ($S_n$-CQA) are applied to solve problems with global $\operatorname{SU}(d)$ symmetry or explicitly exhibits permutation equivariance, as long as the problem Hamiltonian can be efficiently simulated in quantum circuits. Indeed, most physical examples involve only 2 or 3 local spin-interactions such as the Heisenberg model studied in Section \ref{s6}, thus by Theorem \ref{thmpqc} can be efficiently simulated.

The $k$-dependent constant in Theorems \ref{thmpqcd} and  \ref{thmpqc} has, unfortunately, exponential scaling. So in practice, we would like to have perhaps at most 3 or 4-local terms for Hamiltonian simulation in ans{\"a}tze. We focus on the following mixing Hamiltonian consisted of only first and second order products of YJM-elements:
\begin{align} \label{mixer2}
	H_M = \sum_{k \leq l} \beta_{kl} X_k X_l,
\end{align}
whose evolution can be efficiently simulated in $\mathcal{O}(n^4 \log(n^4/\epsilon)/\log\log(n^4/\epsilon))$. In the next section, we prove that with a mixture as in the Equation \eqref{mixer2}, the $S_n$-CQA ans{\"a}tze are \textit{dense}: they are able to approximate any unitary from every $S_n$-irrep block (Fig.\ref{SchurWeyl} $(b)$). This can be seen as a restricted version of universal quantum computation to $S_n$-irreps. Since the 4-local $S_n$-CQA is an all-you-need approximation algorithm within PQP+, it is strongly suggestive that the new PQP+ class contains circuits that can approximate matrix elements of all the $S_n$ Fourier coefficients, for a polynomial length of alternating layer $p$. Moreover, the 4-local $S_n$-CQA is also the universal tool to approximate the solution of the problem with global $\operatorname{SU}(d)$ symmetry, such as the Heisenberg models, due to its nature as variational ans{\"a}tze.

\subsection{State Preparation for $S_n$-CQA Ans{\"a}tze}\label{s3.B}

To investigate evaluation of the matrix elements of $S_n$ Fourier coefficients, we were confined to the Young basis, which requires the implementation of QST \cite{harrow2005PhD,Harrow06,krovi}. However, for a wide variety of quantum machine learning and optimization tasks, such as  determining the ground state sign structure of frustrated magnets, it is often advantageous to relax the constraints and ask how easy it is to initialize the states that live in any given $S_n$-irrep. An algorithm to initialize a state in $(\frac{n}{2}, \frac{n}{2})$-irrep is given in~\cite{Seki_2020}. We generalize this result to an arbitrary $S_n$-irrep even in $\operatorname{SU}(d)-S_n$ duality. The key is to utilize different permutation modules and multiplicities of $S_n$-irreps as in Fig.\ref{SchurWeyl} $(c)$. The construct the algorithm inductively: first consider the $\operatorname{SU}(2)-S_n$ duality in which a $(\lambda_1, \lambda_2)$-$S_n$-irrep is dual to $\frac{1}{2}(\lambda_1 - \lambda_2)$-$\operatorname{SU}(2)$-irrep. Let
\begin{align}\label{psi-init}
	\ket{\Psi_{\operatorname{init}}} = \ket{\underbrace{0\cdots0}_{k \operatorname{many}}} \otimes \underbrace{ \ket{s} \otimes \cdots \otimes \ket{s}}_{\frac{n-k}{2} \operatorname{many}},
\end{align}
where $\ket{s} = \frac{1}{\sqrt{2}}(\ket{01} - \ket{10})$ is one of the Bell states and we assume $n-k$ is even. Then we have

\begin{lemma}
	Let $\mu = \lambda = (\frac{n+k}{2}, \frac{n-k}{2})$. The initialized state $\ket{\Psi_{\operatorname{init}}}$ is contained in $S^\lambda$ and belongs to the permutation module $M^\mu$.
\begin{proof}
	Acting by the spin operator $S^z = \sum_i S^z_i$, it is easy to check that the spin component of $\ket{\Psi_{\operatorname{init}}}$ equals $j = k/2$ hence it belongs to $M^\mu$. By Theorem \ref{Spin-Content}, it can be expanded by total spin basis/Young basis elements from $M^\mu$. Thus $\ket{\Psi_{\operatorname{init}}} = \sum_T \alpha_T \ket{v^{j}_T}$. Since $J_+ \ket{\Psi_{\operatorname{init}}} = 0$, the above summation only takes elements with total spin equal to $j$ and all these $\ket{v^{j}_T}$ with nonzero coefficients are highest weight vectors relative to $\operatorname{SU}(2)$. Therefore the result follows by Schur-Weyl duality.
\end{proof}
\end{lemma}

We illustrate how to expand $\ket{\Psi_{\operatorname{init}}}$ as a linear combination of Young basis elements by providing several examples: $(a)$ let $\ket{\Psi_{\operatorname{init}}} = \ket{s}^{\otimes \frac{n}{2}}$. This is the state used in \cite{Seki_2020}. One can check by YJM-elements that it is exactly a Young basis element. $(b)$ for a more involved case, consider the $(4,2)$-irrep of $S_6$ and write:
\begin{align*}
	& \ket{\Psi_{\operatorname{init}}} = \ket{00} \otimes \ket{s} \otimes \ket{s} \\
	= & \frac{2}{3}\ket{v_{T_1}} - \frac{\sqrt{2}}{3}\ket{v_{T_2}} - \frac{\sqrt{2}}{3} \ket{v_{T_3}}  + \frac{1}{3} \ket{v_{T_4}},
\end{align*}
where $\ket{v_{T_i}}$ are Young basis elements with paths $T_i$ determined by the branching rule in Figure~\ref{Initialization}.

\begin{figure}[H]
	\centering
	\includegraphics[width=3.5in]{Figures/Initialization.pdf}
	\caption{Decomposing the initial state by content vectors/spin labels.}
	\label{Initialization}
\end{figure}

Let $\ket{v_T}$ be a Young basis element of $S_{n-2}$. To be consistent with the above lemma, $\ket{v_T}$ is assumed to be the highest weight vector of total spin $j$. Tensoring with the singlet $\ket{s}$ and decomposing via Clebsch-Gordon rule, $\ket{v_T} \otimes \ket{s}$ is spanned by two highest weight vectors of total spin $j \pm \frac{1}{2}$. Then the spin label-content vector duality in Section \ref{s2.B} yields the above branches. Moreover, by reordering the tensor products of $\ket{0\cdots0}$ and $\ket{s}$ in Eq.\eqref{psi-init} yields different types of linear combinations of Young Bases. By checking the Clebsch-Gordon rule and Bratteli diagram, it is easy to see that each re-ordering yields a different expansion of Young basis elements. Fig.\ref{Initialization2} illustrates two more cases of $\ket{s} \otimes \ket{0} \otimes \ket{s} \otimes \ket{0}$ and $\ket{s} \otimes \ket{00} \otimes \ket{s}$.

\begin{figure}[H]
	\centering
	\includegraphics[width=3in]{Figures/Initialization2.pdf}
	\caption{Reordering tensor products yields different Young basis expansions.}
	\label{Initialization2}
\end{figure}

This method can be generalized to $\operatorname{SU}(d)-S_n$ duality. For instance, when $d = 3$ to initialize states for three-row Young diagrams, let us consider the upper, down, strange states $u,d,s$ of $SU(3)$. Let
\begin{align*}
	\ket{\Psi_0} = \ket{\underbrace{u \cdots u}_{k \operatorname{many}}} \otimes \underbrace{ (ud - du) \otimes \cdots \otimes (ud - du) }_{\frac{n-k}{2} \operatorname{many}}.
\end{align*}
This state lies in the $(n+k, n-k, 0)$-irrep. Tensoring with $SU(3)$-singlet $\ket{s} = \frac{1}{\sqrt{6}} (uds - usd + dsu - dus + sud - sdu)$, $\ket{\Psi_{\operatorname{init}}} = \ket{\Psi_0} \otimes \ket{s}^{\otimes l}$ is a state from the $(n+k+l, n-k+l, l)$-irrep. Its expansion can still be tracked by the branching rule as in Fig. \ref{Initialization} \& \ref{Initialization2}.

\begin{figure*}[ht]
	\centering
	\includegraphics[width=0.9\textwidth]{Figures/Circuit.pdf}
	\caption{$S_6$-CQA circuit with state initialization in Fig.\ref{Initialization}.}
	\label{Circuit}
\end{figure*}

\section{$S_n$-CQA Ans{\"a}tze is Dense in $S_n$ Irreps} \label{s4}

In this section, we show that the $S_n$-CQA ans{\"a}tze are dense in every $S_n$-irrep. This restricts universal quantum computation on $\operatorname{U}(d^n)$ to $S_n$-irrep blocks (Fig.\ref{SchurWeyl} $(b)$) because our ans{\"a}tze preserve $\operatorname{SU}(d)$ symmetry. This is of interest for three reasons: $(a)$ the density result indicates that $S_n$-CQA ans{\"a}tze is an universal approximator in PQP+ and it is the theoretical guarantee of our numerical simulations. $(b)$ Similarly to all above theorems, the result is still valid for qudits under $\operatorname{SU}(d)-S_n$ duality. We exhibit again the advantage of working with the symmetric group $S_n$ as there is no need to deal with complicated $\operatorname{SU}(d)$-generators in the proof. $(c)$ When changed from Young basis to computational basis, our results form a new proof to the universality of a broad class of QAOA ans{\"a}tze.

Mathematically, we aim to show that the subgroup generated by $S_n$-CQA ans{\"a}tze is dense in $\operatorname{U}(\dim S^{\lambda})$. However, arguing directly on the level of the Lie group is complicated. Instead, we prove that the generated Lie algebra is isomorphic with $\mathfrak{u}(\dim S^{\lambda})$. Then combining with some well-known results from the Lie group, we complete the proof. Let us first consider a general lemma where we recall some facts from Cartan subalgebra and root system. The key observation we will rely on is the fact that one can generate a semi-simple Lie algebra only by two elements \cite{k51}. Our case is somewhat different: we will first work on the complex general linear algebra $\mathfrak{gl}(d,\mathbb{C})$ which is not semi-simple, but it is easy to find its Cartan subalgebra -- the collection $\mathfrak{d}(d)$ of all diagonal matrices. Let $A$ be a matrix with nonzero off-diagonal elements $c_{ij}$. It can be thought of as a perturbation from $\mathfrak{d}(d)$. We want to know how large the subalgebra generated by $A$ and $\mathfrak{d}(d)$ would be. More precisely:

\begin{lemma}\label{gl}
	Let $E_{ij} \in \mathfrak{gl}(d,\mathbb{C})$ be the matrix unit with entry $1$ at $(i,j)$ and $0$ elsewhere. Given any matrix $A$, let $\mathcal{I} \subset \{1,...,d\} \times \{1,...,d\}$ be the index set of $A$ assigned with nonzero off-diagonal entries $c_{kl}$. Then the Lie subalgebra generated by $\mathfrak{d}(d)$ and $A$ contains
	\begin{align*}
		\mathfrak{d}(d) \bigoplus_{(i,j) \in \mathcal{I}} R_{ij},
	\end{align*}
	where $R_{ij}$ is the 1-dimensional root space spanned by $E_{ij}$.
\end{lemma}

Proofs in this subsection are rather technical and we relegate them to the SM. Intuitively speaking, the $\operatorname{GZ}_n$ defined in Section \ref{s2.A} corresponds precisely to the Cartan subalgebra of $\mathfrak{gl}(\dim S^\lambda,\mathbb{C})$. We also want to emphasize that to compute the root $a_{kl}$ of $E_{kl}$ under $\operatorname{GZ}_n$,  YJM-elements are not enough. We need a complete span set consisting of all high-order products $X_{i_1} \cdots X_{i_N}$ and that pose the problem for a practical ansatz design, which requires $k$-local $\mathbb{C}[S_n]$ Hamiltonian in order to be efficiently simulated by quantum circuits (Theorem \ref{thmpqcd}). By basic Lie algebra theory, a root $a_{kl}$ is computed by Lie brackets $[X_{i_1} \cdots X_{i_N}, E_{kl}]$. Since $\ket{v_T} = \alpha_T(i) \ket{v_T}$ for $i = 1,...,n$ with $\alpha_T$ as the content vector, $a_{kl}(i) = [X_i, E_{kl}] =  \alpha_{T_k}(i) - \alpha_{T_l}(i)$. Components of $a_{kl}$ under $X_{i_1} \cdots X_{i_N}$ are obtained by taking products of content vectors.

The problem Hamiltonians $H_P$ of interest are complicated in general and hard to diagonalize classically. In our discourse, we require $H_p$ to be \emph{path-connected}. A Hamiltonian is of this kind if its associated index graph $\mathcal{G}_{H_P}$ is connected. For example, the Pauli $X$ and $Y$ are path-connected while $Z$ is not. This condition will be discussed further after Theorem \ref{ex2} as well as in Section \ref{s6}. Path-connectedness is also seen in the famous Perron-Frobenius theorem and applied to graph theory.

\begin{lemma}\label{u}
	Let $H_P$ be a path-connected Hamiltonian. Then the generated Lie algebra $\langle \mathfrak{d}(d), H_P \rangle = \mathfrak{gl}(d,\mathbb{C})$. Consider $\mathfrak{d}_\mathbb{R}(d)$ consisting of all real-valued diagonal matrices. Generated over $\mathbb{R}$, $\langle i\mathfrak{d}_\mathbb{R}(d), iH_P \rangle_\mathbb{R} = \mathfrak{u}(d)$.
\end{lemma}

With respect to the Young basis, YJM-elements with their high-order products have real diagonal entries. Thus, they generate all real diagonal matrices in $\mathfrak{d}(\dim S^\lambda)$. Since a learning ansatz only has real parameters, the following theorem follows.

\begin{theorem}\label{ex1}
	Let $H_P$ be any path-connected Hamiltonian and let $X_{\operatorname{GZ}} \in \operatorname{GZ}_n$ denote a real linear combination of elements from $\{X_i, X_i X_j, X_i X_j X_k,...\}$. Given any $S^\lambda$, the subgroup $H \vcentcolon = \langle \exp(i\beta X_{\operatorname{GZ}} ), \exp(i\gamma H_P) \rangle$ is dense in $\operatorname{U}(\dim S^\lambda)$. Thus the ans{\"a}tze constructed by alternating exponentials can approximate any eigenstate of $H_P$.
\end{theorem}

As we mentioned, this ansatz is impractical because implementing a general element $X_{\operatorname{GZ}}$ would involve non-local high-order products of YJM-elements. These high-order products are used to define the root $a_{kl}$ with the property that $a_{k_s l_s} - a_{k_t l_t} \neq 0$ for any $s \neq t$. This is the crucial step to establish Lemma \ref{gl}. Fortunately, we can show that the inequality still holds when restricted to partially-defined roots by first- and second-order products of YJM-elements. Therefore, designing $H_M$ up to second-order products Eq.\eqref{mixer2} is quite enough to generate $\bigoplus_{(i,j) \in \mathcal{I}} R_{ij}$. However, only taking YJM-elements cannot guarantee the inequality as shown in Fig.\ref{Roots}.

\begin{figure}[H]
	\centering
	\includegraphics[width=3.0in]{Figures/Roots.pdf}
	\caption{Four standard tableaux/Young basis elements/content vectors of $(4,2)$-irrep of $S_6$. Differences between the first two and the last two content vectors are the same.} \label{Roots}
\end{figure}

\begin{definition}
	For any standard Young tableau $T$ of shape $(\lambda_1,...,\lambda_k)$, let $Y_i \vcentcolon = X_i + k I$. The \textit{shifted content vector} $\beta_T$ measured by $Y_i$ only contains positive integers. Using shifted content vector facilitates the following proof. Measuring via second-order product $Y_i Y_j$, we obtain an $(n \times n)$-component vector $\beta^{(2)}_T(r,s)$. It can also be written as a real symmetric matrix $(\beta_T)^T \beta_T$ and hence called \textit{tensor product content vector}.
\end{definition}

\begin{lemma}\label{dif}
	Different standard Young tableaux/Young basis elements have different tensor product content vectors. Moreover, let $T_i, T_j, T_k, T_l$ be four standard Young tableaux of size $n$. We require that $T_i \neq T_j$ and $T_k \neq T_l$. If all underlying Young diagrams are the same, we further require that $T_i \neq T_k$ or $T_j \neq T_l$ (this requirement comes from the computation of roots above). Then
	\begin{align*}
		\beta^{(2)}_i - \beta^{(2)}_j \neq \beta^{(2)}_k - \beta^{(2)}_l
	\end{align*}
	That is, when extended to tensor product content vectors, the differences of unequal standard Young tableaux ae still different.
\end{lemma}

Finally, we can conclude with:

\begin{theorem}\label{ex2}
	The subgroup generated by $Y_k Y_l$ with a path-connected Hamiltonian $H_P$ is still dense. Since $Y_k Y_l = (X_k + kI) (X_l + kI) = X_k X_l + k(X_k + X_l) + k^2 I$ and since $\exp(i\theta I)$ is simply a phase term, a $S_n$-CQA ansatz is written as
	\begin{align} \label{CQA}
	\begin{aligned}
		\cdots & \exp(-i \sum_{k,l} \beta_{kl} X_k X_l ) \exp(-i\gamma H_P)  \\
		& \exp(-i \sum_{k,l} \beta'_{kl} X_k X_l) \exp(-i\gamma' H_P) \cdots,
	\end{aligned}
    \end{align}
	where we redefine $X_1$ as $I$ with which any first-order YJM-element $X_i$ can be written as $X_i X_1$.
\end{theorem}

We present all the proofs for Young diagrams with arbitrary $\operatorname{SU}(d)-S_n$ duality in the SM. Working from the perspective of $S_n$ also frees us from computing complicated $\operatorname{SU}(d)$ generators.

Consider now the case when $H_P$ is not path-connected. That is, $H_P$ is block diagonal (after a possible re-cording of basis elements) in $S^\lambda$. It is straightforward to check that Theorem \ref{ex2} still holds within each sub-block of $H_P$. Suppose our task is to find the eigenvector $v_0$ of $H_P$ with the lowest eigenvalue of $H_P$ in $S^\lambda$. There is generally no prior knowledge about which sub-block $v_0$ is living. The brute-force way to find the minimum is by taking a collection of initial states from each of these sub-blocks and applying the theorem repeatedly. One way to do this is by implementing the efficient QST which gives us access to all Young basis elements. The state initialization proposed in Section \ref{s3.B} with constant-depth may take a hit (forcing the depth of the circuit to increase) if the problem Hamiltonian is not path-connected. (see Section \ref{s6}).

\subsection*{Universality of QAQA}

The first proof of the universality of the QAOA ans{\"a}tze was given in \cite{Lloyd_2018}, where the authors considered problem Hamiltonian of the first-order and second-order nearest-neighbor interaction. \cite{Morales_2020} subsequently generalized the result to broader families of ans{\"a}tze defined by sets of graphs and hyper-graphs. We now describe a new proof based on the techniques developed in this paper that covers novel, broader family of QAOA ans{\"a}tze. More precisely, we change to the computational basis $\{\ket{e_i}\}_{i=1}^{2^n}$ in which all tensor products $\tilde{Z}_{r_1...r_s} \vcentcolon = Z_{r_1} \otimes \cdots \otimes Z_{r_s}$ of Pauli basis can span any diagonal matrix. In the language of Lie algebra, $Z_i$ generates the Cartan subalgebra $\mathfrak{d}(2^n)$ of $\mathfrak{gl}(2^n,\mathbb{C})$. Roots can be computed by diagonal elements of $Z_i$ (cf. Lemma \ref{gl}) which are just spin components of computational basis. We denote by $H_Z$ the Hamiltonian composed by $\tilde{Z}_{r_1...r_s}$ and let $H_X$ be the uniform summation of Pauli $X$ operators (we do not write its explicit form to avoid any confusion with the notation of YJM-elements). The Hamiltonian $H_X$ is path-connected under $\{\ket{e_i}\}$. Then one can argue similarly as in Theorem \ref{ex1} that the alternating ans{\"a}tze generated by $H_Z$ and $H_X$ are universal. One may also expect $H_Z$ being composed by local operators. This holds true, but unlike $H_Z$ in \cite{Lloyd_2018} which contains only nearest neighbor terms $Z_j Z_{j+1}$, it is necessary to take all $\mathcal{O}(n^2)$ terms of second-order product $Z_k Z_l$ in our proof. The resulting Hamiltonian $H_Z$ is still simple though and the proof works for both odd and even number of qubits  \cite{Morales_2020}. Moreover, replacing $H_X$ by any other path-connected Hamiltonian, e.g., antiferromagnetic Heisenberg Hamiltonian \cite{Tasaki2020}, still guarantees the universality, and this fact enables one to experiment with a wide range of mixing Hamiltonians. In summary,

\begin{theorem}\label{QAOA-Univ}
	Let $H_X$ be the uniform summation of Pauli $X$ operators or any other path-connected Hamiltonian on computational basis, the QAOA-ansatz generated by $H_X, H_Z = \sum \beta_{kl} Z_k Z_l$ is dense in $\text{U}(2^n)$, i.e., it is universal.
\end{theorem}

\section{Relation to Adiabatic Quantum Computing} \label{s5}

Similar to the limiting $p \rightarrow \infty $ behavior of QAOA, our ansatz corresponds to the adiabatic quantum evolution for each $S_n$ irreducible representation subspace. The standard interpolating Hamiltonian is given by:
\begin{align}
\begin{aligned}
    \widehat{H}(s) &= s H_p + \sum_{k \leq \ell}(1-s) \beta_{k \ell}(s) X_k X_{\ell}
    \\&= s H_p + (1-s) H_M(s),
\end{aligned}
\end{align}
where $s$ ranging from $0$ to $1$ is the time-parameterized path with $s(0) = 0$ and $s(T) = 1$. In contrast with the standard interpolating Hamiltonian, we set the coupling strength parameters $\{\beta_{k \ell}\}$ as path dependent I.I.D. random variables drawn from some possibly unknown distribution $\mathcal{D}$. Using the Central Limit Theorem, it suffices to draw $\{\beta_{k \ell}\}$ from normal distributions (when $n$ is large), say $\mathcal{N}(0, \sigma)$. We shall work in SU(2)-$S_n$ duality but generalization to qudits is straightforward.

Choose an irrep subspace $S^\lambda$ with the instantaneous ground and first excited states $\ket{v_0(s)}, \ket{v_1(s)}$ and the spectral gap $\Delta = \operatorname{min}_{s \in [0,1]} E_1(s) - E_0(s)$. The adiabatic theorem implies that $\ket{v_0(s =T)}$ is $L^2$-$\epsilon$ close to the ground state of $H_p$ if:
\begin{align}
&T \gg \frac{|\bra{v_1(s)}\partial_s \widehat{H}(s)\ket{v_0(s)} |}{\Delta^2},
\end{align}
where:
\begin{align*}
     \partial_s \widehat{H}= {{H}_{p}}-\sum\limits_{k\le \ell }{{{\beta }_{k,\ell }}}(s){{X}_{k}}{{X}_{\ell }}-s\sum\limits_{k\le \ell }{{{\partial }_{s}}}{{\beta }_{k,\ell }}(s){{X}_{k}}{{X}_{\ell }}.
\end{align*}

The quantum evolution in the Heisenberg picture is:
\begin{align*}
    U(T) = \mathcal{T}\operatorname{exp}(-i \int^T_0 dt \widehat{H}(s(t))).
\end{align*}
Taken $T = N \Delta t \equiv \Delta t_N$ where $s(\Delta t_j)$ is constant at this small time increments, the standard Trotterization technique implies that:
\begin{align}
\begin{aligned}
    U(T) \approx \prod^N_{j=1}&\operatorname{exp}(-i\Delta t_j s(\Delta t_j)) \Pi^n_{k \leq \ell} \\
    &\operatorname{exp}(-i \Delta t_j (1 -s(\Delta t_j) )\beta_{k, \ell}(\Delta t_j) X_k X_{\ell}).
\end{aligned}
\end{align}
Setting $\gamma_j = \Delta t_j s(\Delta t_j)$ and $(\widetilde{\beta}_{_{k, \ell}})_j = \Delta t_j (1 -s(\Delta t_j) )\beta_{k, \ell}(\Delta t_j)$, we recover the $S_n$-CQA ans{\"a}tze at $p = N$. The variational parameters $\{(\widetilde{\beta}_{_{k, \ell}})_j\}_{k, \ell}$ for the classical optimization corresponds to the randomized coupling strength parameters for the initial Hamiltonian strictly diagonal in the total spin basis elements.

The adiabatic evolution time $T$ may be exponential if the spectral gap $\Delta $ is exponentially small as the number of spins grows. There are many works involving techniques to amplify the spectral gap, such as modifying the initial and final Hamiltonian \cite{Farhi2011, Altshuler2012}, quantum adiabatic brachistochrone \cite{Rezakhani_2009}, and adding non-stoquastic Hamiltonian \cite{Zeng2016, Seoane_2012}, to name a few. We refer interested readers to \cite{Lidar_2018} for further discussion. Therefore, one may ask whether the path-dependent randomized coupling coefficients would lead to amplification of the spectral gap so that the final evolution time $T$ can be polynomial bound. In particular, one might need to consider the dynamics of the random path-dependent coupling $\beta_{k, \ell}(s)$, such as the Langevin or diffusion processes, supplemented by systems of stochastic differential equations.

The answer to this question will shed light on the existence of a quantum advantage in finding the exact eigenstate of the problem Hamiltonian. Furthermore, by relating it to the adiabatic evolution, similarly to QAOA, this would imply that the $S_n$-CQA ans{\"a}tze enjoys a guaranteed performance improvement as the number of alternating layers $p$ increases. We leave this as an open question. The success in approximating the sign structure of the ground state in numerical simulation from the $S_n$-CQA ans{\"a}tze may serve as heuristic evidence for such a potential quantum advantage, in addition to the super-exponential quantum speed-up in performing Fourier space convolution in quantum circuits.

\section{$\mathbb{C}[S_n]$ Symmetries of $J_1$-$J_2$ Heisenberg Hamiltonian } \label{s6}

The spin-$1/2$ $J_1$-$J_2$ Heisenberg model is defined by the Hamiltonian:
\begin{align} \label{S_ham}
	\hat{H}_p=J_{1} \sum_{\langle i j\rangle} \hat{\boldsymbol{S}}_{i} \cdot \hat{\boldsymbol{S}}_{j}+J_{2} \sum_{\langle \langle i j\rangle \rangle } \hat{\boldsymbol{S}}_{i} \cdot \hat{\boldsymbol{S}}_{j},
\end{align}
where $\hat{\boldsymbol{S}}_{i} = (\hat{S}_{i}^{x}, \hat{S}_{i}^{y}, \hat{S}_{i}^{z})$ represents the spin operators at each site of a lattice. The symbols $\langle\cdots\rangle$ and $\langle\langle\cdots\rangle\rangle$ indicate pairs of nearest and next-nearest neighbor sites, respectively. The $J_1-J_2$ model has been the subject of intense research over its speculated novel spin-liquid phases at frustrated region \cite{Balents_2010}. The unfrustrated regime $(J_2 = 0 \operatorname{ or } J_1 = 0)$ for the anti-ferromagnetic Heisenberg model is characterized by the bipartite lattices, for which the sign structures of the respective ground states are analytically given by the  \textit{Marshall-Lieb-Mattis theorem} \cite{Lieb62}. The system is known to be highly frustrated when $J_1$ and $J_2$ are comparable $J_2/J_1 \approx 0.5$ \cite{Bukov_2021} and near the region of two phase transitions from Neel ordering to the quantum paramagnetic phase  and from quantum paramagnetic to colinear phase,  where no exact solution is known. Moreover, little is known about the intermediate quantum paramagnetic phase -- recent evidence of deconfined quantum criticality \cite{Nahum_2015,Wen_2016} sparked further interest in studying these regimes. Gaining  physical insights in the intermediate quantum paramagnetic phase requires solving the problem of the ground state sign structure the system approaches the phase transition. Recently, there were a number of numerical attempts to address the existence of the $\operatorname{U}(1)$ gapless spin liquid phase, using recently the tensor networks \cite{Liu_2021}, restricted Boltzmann Machine (RBM) \cite{Nomura_2021}, convolutional neural network (CNN) \cite{Choo_2019,Liang_2018,Castelnovo_2020}, and graphical neural network (GNN) \cite{Kochkov_2021} -- all yielding partial progress.

\subsection*{Global $\operatorname{SU}(2)$ Symmetry and Challenges in NQS Ans\"{a}tze}

Taking advantage of the global $\operatorname{SU}(2)$ symmetry, we address this problem in a different way: we recast the Hamiltonian in Equation \eqref{S_ham} using Equation \eqref{trans}. An immediate consequence is that the resulting Heisenberg Hamiltonian can be expressed in the Young basis where every $S_n$-irrep is indexed by the total spin-label $j$. Mapping to this basis can be done using the constant-depth circuit state initialization in Section III.C. Using our $\mathbb{C}[S_n]$ variational ansatz leads to a more efficient algorithm by polynomially reducing the space. In the NISQ application, especially between 10 to 50 qubits, we have much better scaling see Fig.\ref{fig_scale}.

\begin{figure} [!ht]
\includegraphics[width = 0.40\textwidth]{Figures/dim_plot1.png}
\centering
\caption{The scaling properties of small total spin irreps dimension respectively. The graph shows the scale: $2^n/ \operatorname{Dim}(S^{\lambda})$ with the partitions $\lambda_1 = (n/2, n/2) $ (red), $\lambda_2 = (n/2 +1, n/2-1)$ (grey), $\lambda_3 = (n/2 + 2, n/2 -2)$ (orange) The orange dashed line is $\exp(-n/8)$ the exponential decay. since the plot starts at $n=8$.} \label{fig_scale}
\end{figure}

Numerous efforts in applying NQS variational architecture to represent the complicated sign structure in the frustrated regime essentially use the energy as the only criterion for assessing its accuracy. This would result in the optimized low-energy variational states in frustrated regime still obeying the Marshall sign rules even though the true ground state is likely to deviate from it significantly \cite{Castelnovo_2020}, or breaks the $\operatorname{SU}(2)$ symmetry \cite{Choo_2019}.  The preservation of spatial symmetry has been the core topic of discussion in the literature, with proposed $\mathcal{C}_4$ equivariant CNN. However, on the 2D model Heisenberg model, the spatial symmetry consideration can only reduce the search space redundancy by a constant factor, thus scaling very poorly at even intermediate $n$. By reinforcing $\operatorname{SU}(2)$ symmetry, we achieve a polynomial reduction of Hilbert space and ensure the result to be physically reasonable, hence offering a second criterion to assess the variational ans\"{a}tze.

\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Lattice.pdf}
\caption{Rectangular and Kagome lattices.}
\label{Lattice}
\end{figure*}

The number of qubits scaling linearly with the number of qubits naturally circumvent the issue of having generalization property, a crucial property for the NQS ans{\"a}tze to function \cite{Westerhout_2020}. In fact, in Section \ref{s3}, we showed making use of the representation theory of the symmetric group this leads to the super-exponential quantum speed-up. To this end, it is unlikely that any classically trained ans{\"a}tze are capable to reinforce the global SU(2) symmetry of the system.

\subsection{Numerical Simulation}
We provide numerical simulations to showcase  the effectiveness of the $S_n$-CQA ans{\"a}tze, using JAX automatic differentiation framework \cite{JAX}.  The implementation of the $S_n$-CQA ans{\"a}tze utilizes the classical Fourier space activation by working in the $S_n$ irreducible representation subspace where the ground state energy lies.  This would impact the stability of the numerical simulations, which imply the best-suited models are with 8-16 spins. This bottleneck in computational resource, as shown in Section \ref{s3}, presents no issue for a potential larger-scale implementation on quantum computers. The benchmarked examples with RBM and Group-equivariant Convolutional Neural Network (GCNN) \cite{Roth_2021} are drawn from NetKet \cite{NetKet} tutorial \url{https://www.netket.org/tutorials.html}, which form the baseline comparison. Note that these benchmark algorithms we implemented do not preserve space continuous symmetry such as SU(2) or U(1). For numerical simulation of $S_n$-CQA, we perform random initialization of the parameters. We found that the random initialization already returns the energy which is within roughly $10^{-2}$ precision within ED ground state energy and non-oscillating descents around the ED ground state energy comparing with that of GCNN and RBM. This is likely due to the fact that we used the $S_n$-Fourier space activation with real-valued trial wavefunctions with explicit SU(2) symmetry. We record the optimized energy for the $S_n$-CQA ans{\"a}tze every five iterations, and we set the number of alternating layers $p=4$ for the $3 \times 4$ lattice and $p=6$ for the 12-spin Kagome lattice. In the implementation, we shift the Hamiltonian to $\widetilde{H}(\lambda) = H(\lambda)+ m \operatorname{1}_{d_{\lambda}} $ to ensure $\widetilde{H}(\lambda )$ is positive semi-definiteness in the $S_n$-irrep specificity by the partition $\lambda = (\lambda_1, \lambda_2)$ with the total spin label $j = (\lambda_1 -\lambda_2)/2$ (Section \ref{s2.A}), where $m$ is the total number of transpositions. We only take the real (normalized) part of the wavefunction $\operatorname{Re}(\psi) = \psi + \psi^*$. This can be seen as a post-processing step for the realization of $S_n$-CQA on a quantum computer. We use the Nesterov-accelerated Adam \cite{Dozat_2016} for the $S_n$-CQA optimization with hyper-parameters: $\operatorname{betas} = [0.99, 0.999]$. We also utilize NetKet's ED (Exact Diagonalization) result for the comparison with the exact ground state energy. The ground state is additionally calculated in the Young (Schur) basis by diagonalizing the Heisenberg Hamiltonian $H(\lambda)$ in the irrep $\lambda$ where the ground state lives. It is worth mentioning that our optimized ground state is strictly real-valued and has explicitly SU(2) symmetry, offering the missing yet essential physical interpretation. We provide code and Jupiter notebook in open-access on Github in python. The numerical simulations are run in CPU platform with the 9th Gen 1.4 GHz Intel Core i5 processors.

\subsubsection*{$3 \times 4$ Rectangular Lattice }
In frustrated region of $J_2/J_1 = 0.5, J_2/J_1=0.8$, we found that the ground state lives entirely in the total spin $0$ irrep, corresponding to the partition $\lambda = (4,4)$. In the case of $J_2 = 0.5$, we report that the $S_n$-CQA ans{\"a}tze are able to smoothly converge to the ground state, resulting in the overlap between the ED ground state $0.9973676$, with the final optimized ground state energy: $-5.339714$, roughly within $10^{-2}$ precision of the ED ground state energy. For $J_2 =0.8$, the $S_n$-CQA returns the optimized energy $-5.726855$, within $10^{-2}$ precision of the ED ground state energy. As a result, the overlap between the ED ground state is $0.9996319$. We notice that the  $S_n$-CQA seem always to converge to the ground state with reasonable good accuracy without the issue of trapping in local minima, regardless of initialization (random initialization from Gaussian is used). The Learning rate used here is $0.002$ and $0.001$ for the $S_n$-CQA respectively. For the GCNN layers in both $J_2$ values we set the feature dimensions of hidden layers $(8,8,8,8)$ and $1024$ samples with the learning rate set for $0.02$. For the RBM model, we fix the learning rate $0.02$ with $1024$ samples.

\begin{figure*}[ht]
\centering
\includegraphics[width=0.45\textwidth]{Figures/6square_J05.png}
\includegraphics[width=0.45\textwidth]{Figures/6square_J08.png}
\includegraphics[width=0.45\textwidth]{Figures/Kagome_J05.png}
\includegraphics[width=0.45\textwidth]{Figures/Kagome_J08.png}
\caption{For the $3 \times 4$ lattice, in either case, the $S_n$-CQA ans{\"a}tze are able to converge to the ground state with more than $0.99$ overlap with the ED ground state. For the Kagome lattice, at both frustration level $(a)$ (down, left) and $(b)$ (down, right), we see that the $S_n$-CQA ans{\"a}tze are able to approach the ground state energy. In particular, the overlap with the ED ground state in $(a)$ is $0.98525$ and in $(b)$ is $0.9967997$. The above figures indicate that the absence of oscillating behavior in gradient descent and that the $S_n$-CQA returns real-valued expectation value strictly above and approaching the exact ground state energy. The numerical results are subject to room for further development, for instance-- with better gradient descent algorithms such as to utilize the Hessian--since we have only 200-500 learnable parameters to optimize.   Therefore, we expect the performance and convergence rate of $S_n$-CQA ans{\"a}tze to further increase with perhaps more refined tuning.}
\label{simulation}
\end{figure*}

\subsubsection*{12-Spin Kagome Lattice}
We found by comparing with ED result that the ground state of 12-spin Kagome lattice lives in the total spin $2$ irrep, corresponding to partition $[8, 4]$ in $J_2 =0$, which suggests it to be $5$-fold degenerate. For the both frustration level $J_2/J_1 = 0.5$ and $J_2/J_1 = 0.8$, the ground state lives in total spin 0 irrep, which appear to be non-degenerate.
We aim to learn the ground state for the $12$-spin Kagome lattice at $J_2/J_1 = 0.5$ and $J_2/J_1 = 0.8$. In the case of $J_2/J_1 = 0.5$, the optimized ground state energy by the $S_n$-CQA ans{\"a}tze at the end of iteration returns $-4.12269$, falling roughly within $10^{-2}$ precision with the ED result. We expect further modifications (for instance, longer iterations, better-suited gradient descent algorithms, increasing $p$, etc.) would improve this result to approaching the exact ground state even closer. The overlap between the ED ground state is $0.98525$. In the case $J_2/J_1 = 0.8$, we have the final optimized energy $4.881405$, within roughly $10^{-2}$ precision from the ED ground state energy. As a result, the overlap is $0.9967997$. The performance in terms of both optimized energy and the overlap suffers slightly drawback compared to the learning in the $3 \times 4$ rectangular lattice. This drop in performance is likely due to the existence of the strong geometric frustration in the antiferromagnetic Heisenberg Kagome lattices. The learning rate is set for $0.001$ and $0.002$ for the $J_2/J_1 = 0.5$ and $0.8$ respectively. We set the GCNN in both frustration points of feature dims $(8,8,8,8)$ with $1024$ samples. The learning rate in both frustrations is set to be $0.02$. The RBM implementation uses $1024$ samples with a learning rate $0.02$ for both cases.

\section*{Discussion}

In this paper, we introduce a framework to design non-Abelian group-equivariant quantum variational ans{\"a}tze. With the help of the refined Schur-Weyl duality we demonstrated that quantum circuits present a \emph{natural} and theoretically sound option compared to their classical counterparts when performing Fourier space activation. In particular, the quantum Fourier space activation (within PQP+) enjoys a super-exponential quantum speed-up compared with the best-known result in classical FFT over the symmetric group. The density result of the $S_n$-CQA ana{\"a}tze indicates a wide array of other practical problems, in addition to finding the ground state of frustrated magnets. Typical problems would explicitly encode permutation-equivariant structure or exhibit global SU($d$) symmetry. Furthermore, not only do we show that the $S_n$-CQA ans{\"a}tze are dense in every $S_n$ irrep, but the proof techniques can be used to show the universality of QAOA. We illustrate the remarkable efficacy of our approach by finding the ground state of the Heisenberg antiferromagnet $J_1$-$J_2$ spins in a $3 \times 4$ rectangular lattice and 12-spin Kagome lattice in highly frustrated regimes near the speculated phase transition boundaries. We provided strong numerical evidence that our $S_n$-CQA can approximate the ground state with high degree of precision, and strictly respecting SU($2$) symmetry. This opens up new avenues for using representation theory and quantum computing in solving quantum many-body problems.

\subsubsection*{Open Problems}

We conclude with several interesting open problems: $(a)$ In our work we intruduced PQP+ computational model. We would like to find out the extent of its computational power. In particular, it is interesting to investigate whether quantum circuits can (in polynomial time) approximate matrix elements of any $S_n$ Fourier coefficients. A natural starting place is perhaps based on the density result of $S_n$-CQA ans{\"a}tze in each $S_n$ irrep by asking if a polynomial bounded number if alternating layers $p$ are able to approximate any matrix element of $S_n$ Fourier coefficients. A detailed study of this question will shed some light on the nature and scope of the prospective quantum advantage. $(b)$ It would be important to investigate whether the path-dependent coupling strength parameters $\beta_{kl}$ lead to potential amplitude amplification of the spectral gap in the adiabatic path. In particular, one might need to address the physical dynamics of the random path-dependent coupling strengths. $(c)$ More generally, the quantum speed-up we demonstrated here is inherently connected to the PQP+. Are there other quantum speed-ups within this framework? In particular, $(b)$ suggests a possible route related to quantum annealing. Another possible route may have to do with measurement-based quantum advantage. For instance, see \cite{qml_pres1}. Therefore, one might want to ask if our $S_n$-CQA ans{\"a}tze have other sources of quantum exponential speed-up.  $(d)$ Another open direction would be to benchmark the performance of the $S_n$-CQA ans{\"a}tze in various Heisenberg models and to implement the $S_n$-CQA ans{\"a}tze on a quantum device.

\section*{Code Availability}
The codes for the numerical simulation can be found at \url{https://github.com/hanzheng98/Sn-CQA}. The C++ implementation of $S_n$ operations can be found at \url{https://github.com/risi-kondor/Snob2}. Data availability is upon request by emailing  \url{hanz98@uchicago.edu}.

\section*{Acknowledgement}
We thank Liang Jiang, Antonio Mezzacapo, Kristan Temme, Miles Stoudenmire, Wenda Zhou, Alexander Bogatskiy, Hy Truong Son, Horace Pan, Jiequn Han, and Erik H. Thiede for useful discussions. We thank Chih-chan Tien, Christopher Roth, and Giuseppe Carleo for helping in numerical simulation debugging. JL is supported in part by International Business Machines (IBM) Quantum through the Chicago Quantum Exchange, and the Pritzker School of Molecular Engineering at the University of Chicago through AFOSR MURI (FA9550-21-1-0209). SS acknowledges support from the Royal Society University Research Fellowship.

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv