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Trinidad and Tobago Defence Force Promotes Youth Programs
Geraldine Cook/Diálogo May 07, 2019 Capacity Building 1 Comment
By Geraldine Cook/Diálogo
The Trinidad and Tobago Defence Force promotes youth programs to help young men and women at risk of falling prey to crime, drugs, and radicalization change their lives.
The Trinidad and Tobago Defence Force (TTDF) leads two youth programs as part of broader national effort to provide opportunity for underprivileged youths. The Military-Led Academic Training Programme (MiLAT), and the Civilian Conservation Corps (CCC), focus on driving misguided and underprivileged youths away from crime.
"In Trinidad and Tobago, the criminal landscape is fed by misguided youths often times with pent up, unexpanded energy that hasn't been positively channeled. It's critically important to provide avenues for the positive expression of the youth's energy, as it's much easier to prevent somebody from becoming a criminal than to change a criminal's mind," Retired TTDF Rear Admiral Hayden Pritchard, former chief of Defence Staff, told Diálogo. "It's important to find healthy outlets for young people, especially because of the competing interests of gangs, criminal networks, terrorist networks, and so on."
MiLAT and CCC represent the government of Trinidad and Tobago's efforts to expose young men and women to vocational and educational opportunities and behavioral changes they may not otherwise have. "The youth programs are like a second chance to help the youths make something of themselves before they get involved in criminal activities," said Trinidad and Tobago Reserve Major Cheryl Richardson, director of MiLAT and CCC. "It also helps them after they have been involved in criminal activity to get them back on their feet, to redirect their mind, fix their attitudes and behaviors. The students receive positive mentorship and reinforcement, academic instruction, certified training, career guidance, benefit from positive peer motivation, friendship, and brotherhood."
MiLAT Programme
Created in 2007, MiLAT is designed to provide academic alternatives for young men between 16 and 20 at risk of gang affiliation, drug trafficking, other criminal activities, and potential radicalization by external actors. MiLAT admits approximately 100 young men per year.
Kevin Narine, a 19-year-old MiLAT student, attributes the program to turning his life of street criminality into a success story through discipline, academics, and brotherhood. "I was a juvenile delinquent. [I was in] plenty of trouble in school," said Narine, who is six months shy of completing the two-year residential program. "MiLAT is life changing. I was at risk; it changed my life in a big way."
The students undergo a three-month induction training to adjust to military discipline and the academy's routine. Then they enroll in a combination of core subjects, such as Mathematics, English, Language Arts, and Life Skills, and optional subjects, including Information Technology, Social Studies, Human and Social Biology, Music, and Physical Education, among others.
As part of the certification program, MiLAT students participate in activities to learn about the environment, first aid, survival training, community service, and others. They prepare for the Caribbean Secondary Education Certificate and the Caribbean Certificate of Secondary Level of Competence, equivalent to a high-school degree.
"I understand the effectiveness of the youth programs," said TTDF Coast Guard Leading Writer Alexander Gershwin, a MiLAT instructor who was an at-risk teen before enrolling at CCC. "I offer [students] my own story. I give them my testimony, and it gives them hope. I become more relatable, and they come to me for advice, mentorship, and counseling."
Civilian Conversation Corps
CCC is a 1993 initiative designed to provide skills for young men and women between the ages of 16 and 25, with low levels of education and little or no work experience who are susceptible to engaging in criminal activity. Trinidad and Tobago's Ministry of Defense borrowed the CCC concept from a U.S. public relief program for unemployed, unmarried men during the Great Depression.
Matthew Taylor, a 21-year-old Trinidadian high-school dropout, dreams of becoming a graphic designer. When he heard about CCC, he jumped at the opportunity to change his life for the better. "Many youths have nothing to do and they have no motivation. CCC is a good program for the youth," Taylor told Diálogo. "We can learn and do something good with our lives," he added. Taylor is five months into CCC's six-month Computer Literacy course.
CCC counts seven educational centers in Trinidad and one in Tobago, and offers students more than 30 six-month courses, such as Building Maintenance, Child Care, Computer Literacy, Electrical Installation, Hairdressing, and Plumbing. The skills prepare them to join the work force through private partnerships with national organizations, to include the National Energy Skill Center and the Trinidad and Tobago Hospitality Institute. Since its creation, more than 35,000 students completed the program, earning a certification in their chosen field.
After graduation, CCC hires its top students as part of their leader and mentorship program. "CCC provided me with a stable environment to assist me in deciding what goals I wanted to set for my life and how to go about achieving them," said TTDF Corporal Adanna Hume-Borrell, a member of the TTDF Public Affairs Division, who specialized in photography at CCC and was later employed as a CCC assistant team commander. She joined TTDF in 2004.
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\section{Introduction}
\label{section:Introduction}

There is a natural relationship between quantum computation and evaluations of Tutte polynomials~\cite{aharonov2007polynomial, shepherd2010binary}. In particular, quantum probability amplitudes are proportional to evaluations of the Tutte polynomial of graphic matroids. In this paper we use this relationship to establish a classical heuristic algorithm for exactly computing quantum probability amplitudes. While this problem is known to be \mbox{\textsc{\#P}-hard} in general~\cite{fenner1999determining}, our algorithm focuses on exploiting structural properties of an instance to achieve an improved runtime over traditional methods. Previously it was known that this problem can be solved in time exponential in the treewidth of the underlying graph~\cite{markov2008simulating}.

The basis of our algorithm is a mapping between output probability amplitudes of quantum circuits and evaluations of the Tutte polynomial of graphic matroids~\cite{shepherd2009temporally, shepherd2010binary, bremner2010classical}. Our algorithm proceeds to evaluate the Tutte polynomial recursively using the deletion-contraction property. At each step in the recursion, our algorithm computes certain structural properties of the matroid in order to attempt to prune the computational tree. This approach to computing Tutte polynomials was first studied by Haggard, Pearce, and Royle~\cite{haggard2010computing}. Our algorithm can be seen as an adaption of their work to special points of the Tutte plane where we can exploit additional structural properties.

The performance of algorithms for computing Tutte polynomials based on the deletion-contraction property depends on the heuristic used to decide the ordering of the recursion~\cite{pearce2009edge, haggard2010computing, monagan2012new}. We consider several heuristics introduced by Pearce, Haggard, and Royle~\cite{pearce2009edge} and an additional heuristic, which is specific to our algorithm. We present some experimental results comparing the performance of these heuristics on two classes of random quantum circuits corresponding to dense and sparse instances.

The correspondence between output probability amplitudes of quantum circuits and evaluations of Tutte polynomials also allows us to obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants by a theorem of Pendavingh~\cite{pendavingh2014evaluation}. This gives rise to an alternative efficient classical algorithm for computing output probability amplitudes of Clifford circuits.

This paper is structured as follows. We introduce matroid theory in \mbox{Section~\ref{section:MatroidTheory}} and the Tutte polynomial in \mbox{Section~\ref{section:TuttePolynomial}}. In \mbox{Sections~\ref{section:PottsModelPartitionFunction}, \ref{section:InstantaneousQuantumPolynomialTime}, and \ref{section:QuantumComputationAndTheTuttePolynomial}}, we establish a mapping between output probability amplitudes of quantum circuits and evaluations of the Tutte polynomial of graphic matroids. This is achieved by introducing the Potts model partition function in \mbox{Section~\ref{section:PottsModelPartitionFunction}}, Instantaneous Quantum Polynomial-time circuits in \mbox{Section~\ref{section:InstantaneousQuantumPolynomialTime}}, and a class of universal quantum circuits in \mbox{Section~\ref{section:QuantumComputationAndTheTuttePolynomial}}. In \mbox{Section~\ref{section:EfficientClassicalSimulationCliffordCircuits}}, we use this mapping to obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants. We also obtain an efficient classical algorithm for computing the output probability amplitudes of Clifford circuits. We describe our algorithm in \mbox{Section~\ref{section:AlgorithmOverview}} and present some experimental results in \mbox{Section~\ref{section:ExperimentalResults}}. Finally, we conclude in \mbox{Section~\ref{section:ConclusionAndOutlook}}.

\section{Matroid Theory}
\label{section:MatroidTheory}

We shall now briefly introduce the theory of matroids. The interested reader is referred to the classic textbooks of Welsh~\cite{welsh1976matroid} and Oxley~\cite{oxley2006matroid} for a detailed treatment. Matroids were introduced by Whitney~\cite{whitney1935abstract} as a structure that generalises the notion of linear dependence. There are many equivalent ways to define a matroid. We shall define a matroid by the independence axioms.
\begin{definition}[Matroid]
    A matroid is a pair \mbox{$M=(\mathcal{S},\mathcal{I})$} consisting of a finite set $\mathcal{S}$, known as the \emph{ground set}, and a collection $\mathcal{I}$ of subsets of $\mathcal{S}$, known as the \emph{independent sets}, such that the following axioms are satisfied.
    \begin{enumerate}
        \item The empty set is a member of $\mathcal{I}$.
        \item Every subset of a member of $\mathcal{I}$ is a member of $\mathcal{I}$.
        \item If $A$ and $B$ are members of $\mathcal{I}$ and \mbox{$\abs{A}>\abs{B}$}, then there exists an \mbox{$x \in A {\setminus} B$} such that \mbox{$B\cup\{x\}$} is a member of $\mathcal{I}$.
    \end{enumerate}
\end{definition}

The rank of a subset $A$ of $\mathcal{S}$ is given by the \emph{rank function} \mbox{$r:2^{\mathcal{S}}\to\mathbb{N}$} of the matroid defined by \mbox{$r(A)\coloneqq\max\left(\abs{X} \mid X \subseteq A,X\in\mathcal{I}\right)$}. The \emph{rank} of a matroid $M$, denoted \mbox{$r(M)$}, is the rank of the set $S$.

The archetypal class of matroids are \emph{vector matroids}. A vector matroid \mbox{$M=(\mathcal{S},\mathcal{I})$} is a matroid whose ground set $\mathcal{S}$ is a subset of a vector space over a field $\mathbb{F}$ and whose independent sets $\mathcal{I}$ are the linearly independent subsets of $\mathcal{S}$. The rank of a subset of a vector matroid is the dimension of the subspace spanned by the corresponding vectors. We say that a matroid is \mbox{$\mathbb{F}$-\emph{representable}} if it is isomorphic to a vector matroid over the field $\mathbb{F}$. A matroid is a \emph{binary matroid} if it is \mbox{$\mathbb{F}_2$-representable} and is a \emph{ternary matroid} if it is \mbox{$\mathbb{F}_3$-representable}. A matroid that is representable over every field is called a \emph{regular matroid}.

Every finite graph \mbox{$G=(V,E)$} induces a matroid \mbox{$M(G)=(\mathcal{S},\mathcal{I})$} as follows. Let the ground set $\mathcal{S}$ be the set of edges $E$ and let the independent sets $\mathcal{I}$ be the subsets of $E$ that are a forest, i.e., they do not contain a simple cycle. It is easy to check that $M(G)$ satisfies the independence axioms. The rank of a subset $A$ of a cycle matroid is \mbox{$\abs{V}-\kappa(A)$}, where $\kappa(A)$ denotes the number of connected components of the subgraph with edge set $A$. The rank of the cycle matroid $M(G)$, denoted \mbox{$r(M(G))$} or simply $r(G)$, is the rank of the set $E$. The matroid $M(G)$ is called the \emph{cycle matroid} of $G$. We say that a matroid is \emph{graphic} if it is isomorphic to the cycle matroid of a graph.

Graphic matroids are regular. To see this consider assigning to the graph $G$ an arbitrary \emph{orientation} $D(G)$, that is, for each edge \mbox{$e=\{u,v\}$} in $G$, we choose one of $u$ and $v$ to be the \emph{positive end} and the other one to be the \emph{negative end}. Then construct the \emph{oriented incidence matrix} of $G$ with respect to the orientation $D(G)$.
\begin{definition}[Oriented incidence matrix]
    Let \mbox{$G=(V,E)$} be a graph and let $D(G)$ be an orientation of $G$. Then the oriented incidence matrix of $G$ with respect to $D(G)$ is the \mbox{$\abs{V}\times\abs{E}$} matrix \mbox{$A_{D(G)}=(a_{ve})_{\abs{V}\times\abs{E}}$} whose entries are
    \begin{equation}
        a_{ve} =
        \begin{dcases*}
            +1, & if $v$ is the positive end of $e$; \\
            -1, & if $v$ is the negative end of $e$; \\
            0, & otherwise.
        \end{dcases*} \notag
    \end{equation}
\end{definition}
The rows of the oriented incidence matrix $A_{D(G)}$ correspond to the vertices of $G$ and the columns correspond to the edges of $G$. Each column contains exactly one $+1$ and exactly one $-1$ representing the positive and negative ends of the corresponding edge. If the column space of $A_{D(G)}$ is the ground set of a vector matroid, then it is easy to see that a subset is independent if and only if it is a forest in $G$. Hence, the oriented incidence matrix provides a representation of a graphic matroid over every field.

A \emph{minor} of a matroid $M$ is a matroid that is obtained from $M$ by a sequence of \emph{deletion} and \emph{contraction} operations.
\begin{definition}[Deletion]
    Let \mbox{$M=(\mathcal{S},\mathcal{I})$} be a matroid and let $e$ be an element of the ground set. Then the deletion of $M$ with respect to $e$ is the matroid \mbox{$M{\setminus}\{e\}=(\mathcal{S}',\mathcal{I}')$} whose ground set is \mbox{$\mathcal{S}'=\mathcal{S}{\setminus}\{e\}$} and whose independent sets are \mbox{$\mathcal{I}'=\{I\subseteq\mathcal{S}{\setminus}\{e\} \mid I\in\mathcal{I}\}$}.
\end{definition}
The deletion of an element from the cycle matroid of a graph corresponds to removing an edge from the graph.
\begin{definition}[Contraction]
    Let \mbox{$M=(\mathcal{S},\mathcal{I})$} be a matroid and let $e$ be an element of the ground set. Then the contraction of $M$ with respect to $e$ is the matroid \mbox{$M/\{e\}=(\mathcal{S}',\mathcal{I}')$} whose ground set is \mbox{$\mathcal{S}'=\mathcal{S}{\setminus}\{e\}$} and whose independent sets are \mbox{$\mathcal{I}'=\{I\subseteq\mathcal{S}{\setminus}\{e\} \mid I\cup\{e\}\in\mathcal{I}\}$}.
\end{definition}
The contraction of an element from the cycle matroid of a graph corresponds to removing an edge from the graph and merging its two endpoints.

An element $e$ of a matroid is said to be a \emph{loop} if $\{e\}$ is not an independent set and said to be a \emph{coloop} if $e$ is contained in every maximally independent set. If an element $e$ of a matroid is either a loop or a coloop then the deletion and contraction of $e$ are equivalent.

\section{The Tutte Polynomial}
\label{section:TuttePolynomial}

We shall now briefly introduce the Tutte polynomial, which is a well-known invariant in matroid and graph theory.
\begin{definition}[Tutte polynomial of a matroid]
    Let \mbox{$M=(\mathcal{S},\mathcal{I})$} be a matroid with rank function \mbox{$r:2^{\mathcal{S}}\to\mathbb{N}$}. Then the Tutte polynomial of $M$ is the bivariate polynomial defined by
    \begin{equation}
        \mathrm{T}(M;x,y) \coloneqq \sum_{A \subseteq \mathcal{S}}(x-1)^{r(M)-r(A)}(y-1)^{\abs{A}-r(A)}. \notag
    \end{equation}
\end{definition}

The Tutte polynomial may also be defined recursively by the \emph{deletion-contraction property}.
\begin{definition}[Deletion-contraction property]
   Let \mbox{$M=(\mathcal{S},\mathcal{I})$} be a matroid. If $M$ is the empty matroid, i.e., \mbox{$\mathcal{S}=\varnothing$}, then
   \begin{equation}
        \mathrm{T}(M;x,y) = 1. \notag
    \end{equation}
    Otherwise, let $e$ be an element of the ground set. If $e$ is a loop, then
    \begin{equation}
        \mathrm{T}(M;x,y) = y\mathrm{T}(M{\setminus}\{e\};x,y). \notag
    \end{equation}
    If $e$ is a coloop, then
    \begin{equation}
        \mathrm{T}(M;x,y) = x\mathrm{T}(M/\{e\};x,y). \notag
    \end{equation}
    Finally, if $e$ is neither a loop nor a coloop, then
    \begin{equation}
        \mathrm{T}(M;x,y) = \mathrm{T}(M{\setminus}\{e\};x,y)+\mathrm{T}(M/\{e\};x,y). \notag
    \end{equation}
\end{definition}
The deletion-contraction property immediately gives an algorithm for recursively computing the Tutte polynomial. This algorithm is in general inefficient, but the performance may be improved by using isomorphism testing to reduce the number of recursive calls~\cite{sekine1995computing}. The performance of this algorithm depends on the heuristic used to choose elements of the ground set~\cite{pearce2009edge, haggard2010computing, monagan2012new}. Bj\"orklund et al.~\cite{bjorklund2008computing} showed that the Tutte polynomial can be computed in time exponential in the number of vertices.

The Tutte polynomial of a graph may be recovered by considering the Tutte polynomial of the cycle matroid of a graph and using the fact that the rank of a subset $A$ of a cycle matroid is \mbox{$\abs{V}-\kappa(A)$}, where $\kappa(A)$ denotes the number of connected components of the subgraph with edge set $A$.
\begin{definition}[Tutte Polynomial of a graph]
    Let \mbox{$G=(V,E)$} be a graph and let $\kappa(A)$ denote the number of connected components of the subgraph with edge set $A$. Then the Tutte polynomial of $G$ is a polynomial in $x$ and $y$, defined by
    \begin{equation}
        \mathrm{T}(G;x,y) \coloneqq \sum_{A \subseteq E}(x-1)^{\kappa(A)-\kappa(E)}(y-1)^{\kappa(A)+\abs{A}-\abs{V}}. \notag
    \end{equation}
\end{definition}

The Tutte polynomial is trivial to evaluate along the hyperbola \mbox{$(x-1)(y-1)=1$} for any matroid. In the case of graphic matroids, Jaeger, Vertigan, and Welsh~\cite{jaeger1990computational} showed that the Tutte polynomial is \mbox{\textsc{\#P}-hard} to evaluate, except along this hyperbola and when \mbox{$(x,y)$} equals one of nine special points.
\begin{theorem}[Jaeger, Vertigan, and Welsh~\cite{jaeger1990computational}]
    The problem of evaluating the Tutte polynomial of a graphic matroid at an algebraic point in the \mbox{$(x,y)$-plane} is \mbox{\emph{\#P-hard}} except when \mbox{$(x-1)(y-1)=1$} or when \mbox{$(x,y)$} equals one of \mbox{$(1,1)$}, \mbox{$(-1,-1)$}, \mbox{$(0,-1)$}, \mbox{$(-1,0)$}, \mbox{$(i,-i)$}, \mbox{$(-i,i)$}, \mbox{$(j,j^2)$}, \mbox{$(j,j^2)$}, or \mbox{$(j^2,j)$}, where \mbox{$j=\exp(2\pi i/3)$}. In each of these exceptional cases the evaluation can be done in polynomial time.
\end{theorem}
Vertigan~\cite{vertigan1998bicycle} extended this result to vector matroids.
\begin{theorem}[Vertigan~\cite{vertigan1998bicycle}]
    The problem of evaluating the Tutte polynomial of a vector matroid over a field $\mathbb{F}$ at an algebraic point in the \mbox{$(x,y)$-plane} is \mbox{\emph{\#P-hard}} except when \mbox{$(x-1)(y-1)=1$}, \mbox{$(x,y)$} equals \mbox{$(1,1)$}, or when
    \begin{enumerate}
        \item \mbox{$\abs{\mathbb{F}}=2$} and \mbox{$(x,y)$} equals one of \mbox{$(-1,-1)$}, \mbox{$(0,-1)$}, \mbox{$(-1,0)$}, \mbox{$(i,-i)$}, or \mbox{$(-i,i)$};
        \item \mbox{$\abs{\mathbb{F}}=3$} and \mbox{$(x,y)$} equals one of \mbox{$(j,j^2)$} or \mbox{$(j^2,j)$}, where \mbox{$j=\exp(2\pi i/3)$}; or
        \item \mbox{$\abs{\mathbb{F}}=4$} and \mbox{$(x,y)$} equals \mbox{$(-1,-1)$}.
    \end{enumerate}
    In each of these exceptional cases, except when \mbox{$(x,y)$} equals \mbox{$(1,1)$}, the evaluation can be done in polynomial time.
\end{theorem}
Snook~\cite{snook2012counting} showed that when \mbox{$(x,y)$} equals \mbox{$(1,1)$} and $\mathbb{F}$ is either a finite field of fixed characteristic or a fixed infinite field, then evaluating the Tutte polynomial is \mbox{\textsc{\#P}-hard}. It is an open problem to understand the complexity of evaluating the Tutte polynomial at \mbox{$(1,1)$} over any fixed field.

\section{The Potts Model Partition Function}
\label{section:PottsModelPartitionFunction}

The Potts model is a statistical physical model described by an integer \mbox{$q\in\mathbb{Z}^+$} and a graph \mbox{$G=(V,E)$}, with the vertices representing spins and the edges representing interactions between them. A set of edge weights \mbox{$\{\omega_e\}_{e \in E}$} characterise the interactions and a set of vertex weights \mbox{$\{\upsilon_v\}_{v \in V}$} characterise the external fields at each spin. A configuration of the model is an assignment $\sigma$ of each spin to one of $q$ possible states. The \emph{Potts model partition function} is defined as follows.
\begin{definition}[Potts model partition function]
    Let \mbox{$q\in\mathbb{Z}^+$} be an integer and let \mbox{$G=(V,E)$} be a graph with the weights \mbox{$\Omega=\{\omega_e\}_{e \in E}$} assigned to its edges and the weights \mbox{$\Upsilon=\{\upsilon_v\}_{v \in V}$} assigned to its vertices. Then the \mbox{$q$-state} Potts model partition function is defined by
    \begin{equation}
        \mathrm{Z}_\mathrm{Potts}(G;q,\Omega,\Upsilon) \coloneqq \sum_{\sigma\in\mathbb{Z}_q^V}w_G(\sigma), \notag
    \end{equation}
    where
    \begin{equation}
        w_G(\sigma) = \exp\left(\sum_{\{u,v\}\in E}\omega_{\{u,v\}}\delta(\sigma_u,\sigma_v)+\sum_{v\in V}\upsilon_v\delta(\sigma_v)\right). \notag
    \end{equation}
\end{definition}

The Potts model partition function with an external field is equivalent to the zero-field case on an augmented graph \mbox{$G'=(V',E')$}. To construct $G'$ from $G$, for each of the connected components \mbox{$\{C_i\}_{i=1}^{\kappa(E)}$} of $G$ add a new vertex $u_i$ and for every vertex \mbox{$v \in V(C_i)$} add an edge \mbox{$e_v=\{u_i,v\}$} with the weight $\upsilon_v$ assigned to it. Then we have the following proposition.
\begin{proposition}[{restate=[name=restatement]PottsModelPartitionFunctionExternalField}]
    \label{proposition:PottsModelPartitionFunctionExternalField}
    \begin{equation}
        \mathrm{Z}_\mathrm{Potts}(G;q,\Omega,\Upsilon) = q^{-\kappa(E)}\mathrm{Z}_\mathrm{Potts}(G';q,\Omega\cup\Upsilon,0). \notag
    \end{equation}
\end{proposition}
A similar proposition appears in Welsh's monograph~\cite{welsh1993complexity}; we prove \mbox{Proposition~\ref{proposition:PottsModelPartitionFunctionExternalField}} in \mbox{Appendix~\ref{section:PottsModelPartitionFunctionExternalField}}.

It will be convenient to consider the Potts model with weights that are all positive integer multiples of a complex number $\theta$. We shall implement this model on the augmented graph $G'$ with all weights equal to $\theta$ by replacing each edge with the appropriate number of parallel edges. Let us denote the partition function of this model by \mbox{$\mathrm{Z}_\mathrm{Potts}(G';q,\theta)$}. Then, we have the following proposition relating the partition function of this model to the Tutte polynomial of the augmented graph $G'$.
\begin{proposition}
    \label{proposition:PottsModelPartitionFunctionTuttePolynomialRelation}
    \begin{equation}
        \mathrm{Z}_\mathrm{Potts}(G';q,\theta) = q^{\kappa(E')}(e^\theta-1)^{r(G')}\mathrm{T}\left(G';x,y\right), \notag
    \end{equation}
    where \mbox{$x=\frac{e^\theta+q-1}{e^\theta-1}$} and \mbox{$y=e^\theta$}.
\end{proposition}
In particular, the \mbox{$q$-state} Potts model partition function is related to the Tutte polynomial along the hyperbola \mbox{$(x-1)(y-1)=q$}. For a proof of \mbox{Proposition~\ref{proposition:PottsModelPartitionFunctionTuttePolynomialRelation}}, we refer the reader to Welsh's monograph~\cite[Section 4.4]{welsh1993complexity}.

The \mbox{$2$-state} Potts model partition function specialises to the \emph{Ising model partition function}.
\begin{definition}[Ising model partition function]
    Let \mbox{$G=(V,E)$} be a graph with the weights \mbox{$\Omega=\{\omega_e\}_{e \in E}$} assigned to its edges and the weights \mbox{$\Upsilon=\{\upsilon_v\}_{v \in V}$} assigned to its vertices. Then the Ising model partition function is defined by
    \begin{equation}
        \mathrm{Z}_\mathrm{Ising}(G;\Omega,\Upsilon) \coloneqq \sum_{\sigma\in\{-1,+1\}^V}w_G(\sigma), \notag
    \end{equation}
    where
    \begin{equation}
        w_G(\sigma) = \exp\left(\sum_{\{u,v\}\in E}\omega_{\{u,v\}}\sigma_u\sigma_v+\sum_{v\in V}\upsilon_v\sigma_v\right). \notag
    \end{equation}
\end{definition}
\begin{proposition}
    \label{proposition:PottsIsingRelation}
    \begin{equation}
        \mathrm{Z}_\mathrm{Potts}(G;2,\Omega,\Upsilon) = w_G\mathrm{Z}_\mathrm{Ising}\left(G;\frac{\Omega}{2},\frac{\Upsilon}{2}\right), \notag
    \end{equation}
    where \mbox{$w_G=\exp\left(\frac{1}{2}\sum_{e\in E}\omega_e+\frac{1}{2}\sum_{v\in V}\upsilon_v\right)$}.
\end{proposition}
\begin{proof}
    The proof follows from some simple algebra.
\end{proof}

\section{Instantaneous Quantum Polynomial Time}
\label{section:InstantaneousQuantumPolynomialTime}

We shall now briefly introduce the class of commuting quantum circuits, known as \emph{Instantaneous Quantum Polynomial-time} (IQP) circuits~\cite{shepherd2009temporally}. These circuits exhibit many interesting mathematical properties. In particular, the output probability amplitudes of IQP circuits are proportional to evaluations of the Tutte polynomial of binary matroids~\cite{shepherd2010binary}. IQP circuits comprise only gates that are diagonal in the \mbox{Pauli-X} basis and are described by an \emph{X-program}.
\begin{definition}[X-program]
    An X-program is a pair \mbox{$(P,\theta)$}, where \mbox{$P=(p_{ij})_{m \times n}$} is a binary matrix and \mbox{$\theta\in[-\pi,\pi]$} is a real angle. The matrix $P$ is used to construct a Hamiltonian of $m$ commuting terms acting on $n$ qubits, where each term in the Hamiltonian is a product of Pauli-X operators,
    \begin{equation}
        \mathrm{H}_{(P,\theta)} \coloneqq -\theta\sum_{i=1}^m\bigotimes_{j=1}^nX_j^{p_{ij}}. \notag
    \end{equation}
    Thus, the columns of $P$ correspond to qubits and the rows of $P$ correspond to interactions in the Hamiltonian.
\end{definition}

An X-program induces a probability distribution $\mathcal{P}_{(P,\theta)}$ known as an \emph{IQP distribution}.
\begin{definition}[$\mathcal{P}_{(P,\theta)}$]
    For an X-program \mbox{$(P,\theta)$} with \mbox{$P=(p_{ij})_{m \times n}$}, we define $\mathcal{P}_{(P,\theta)}$ to be the probability distribution over binary strings \mbox{$x\in\{0,1\}^n$}, given by
    \begin{equation}
        \textbf{Pr}[x] \coloneqq \abs{\psi_{(P,\theta)}(x)}^2, \notag
    \end{equation}
    where
    \begin{equation}
        \psi_{(P,\theta)}(x) = \bra{x}\exp\left(-i\mathrm{H}_{(P,\theta)}\right)\ket{0^n}. \notag
    \end{equation}
\end{definition}

The principal probability amplitude \mbox{$\psi_{(P,\theta)}(0^n)$} of an IQP distribution is directly related to an evaluation of the Tutte polynomial of the binary matroid whose ground set is the row space of $P$.
\begin{proposition}
    \label{proposition:IQPTuttePolynomialRelation}
    Let \mbox{$(P,\theta)$} be an X-program with \mbox{$P=(p_{ij})_{m \times n}$}. Let \mbox{$M=(\mathcal{S},\mathcal{I})$} be the binary matroid whose ground set $\mathcal{S}$ is the row space of $P$, then,
    \begin{equation}
        \psi_{(P,\theta)}(0) = e^{i\theta(r(M)-m)}(i\sin(\theta))^{r(M)}\mathrm{T}\left(M;x,y\right), \notag
    \end{equation}
    where \mbox{$x=-i\cot(\theta)$} and \mbox{$y=e^{2i\theta}$}.
\end{proposition}
A similar result may be obtained for the other probability amplitudes. This can easily be seen when \mbox{$\theta=\frac{\pi}{2k}$} for \mbox{$k\in\mathbb{Z}^+$}, by firstly letting \mbox{$P\|^kx$} be the matrix obtained from $P$ by appending $k$ rows identical to $x$, and then observing that \mbox{$\psi_{(P,\theta)}(x)=-i\psi_{(P\|^kx,\theta)}(0^n)$}. For a proof of \mbox{Proposition~\ref{proposition:IQPTuttePolynomialRelation}} and a treatment of the general $\theta$ case, we refer the reader to Ref.~\cite[Section 3]{shepherd2010binary}.

We shall consider X-programs that are induced by a weighted graph.
\begin{definition}[Graph-induced X-program]
    For a graph \mbox{$G=(V,E)$} with the weights \mbox{$\left\{\omega_e\in[-\pi,\pi]\right\}_{e \in E}$} assigned to its edges and the weights \mbox{$\left\{\upsilon_v\in[-\pi,\pi]\right\}_{v \in V}$} assigned to its vertices, we define the X-program induced by $G$ to be an X-program $\mathcal{X}_G$ such that
    \begin{equation}
        \mathrm{H}_{\mathcal{X}_G} = -\sum_{\{u,v\} \in E}\omega_{\{u,v\}}X_uX_v-\sum_{v \in V}\upsilon_vX_v. \notag
    \end{equation}
\end{definition}
Any X-program can be efficiently represented by a graph-induced X-program~\cite{shepherd2009temporally}. The principal probability amplitude \mbox{$\psi_{\mathcal{X}_G}(0^n)$} of the IQP distribution generated by a graph-induced X-program is directly related to the Ising model partition function of the graph with imaginary weights.
\begin{proposition}[{restate=[name=restatement]IQPIsingModelPartitionFunctionRelation}]
    \label{proposition:IQPIsingModelPartitionFunctionRelation}
    Let \mbox{$G=(V,E)$} be a graph with the weights \mbox{$\Omega=\left\{\omega_e\in[-\pi,\pi]\right\}_{e \in E}$} assigned to its edges and the weights \mbox{$\Upsilon=\left\{\upsilon_v\in[-\pi,\pi]\right\}_{v \in V}$} assigned to its vertices, then,
    \begin{equation}
        \psi_{\mathcal{X}_G}\left(0^{\abs{V}}\right) = \frac{1}{2^{\abs{V}}}\mathrm{Z}_\mathrm{Ising}(G;i\Omega,i\Upsilon). \notag
    \end{equation}
\end{proposition}
\mbox{Proposition~\ref{proposition:IQPIsingModelPartitionFunctionRelation}} is well known~\cite{iblisdir2014low, fujii2017commuting}; we provide a proof in \mbox{Appendix~\ref{section:IQPIsingModelPartitionFunctionRelation}}. It will be convenient to consider graph-induced X-programs \mbox{$\mathcal{X}_{G(\theta)}$} with weights that are all positive integer multiples of a real angle $\theta$. As in \mbox{Section~\ref{section:TuttePolynomial}}, this model can be implemented on the augmented graph \mbox{$G'=(V',E')$} with all weights equal to $\theta$ by replacing each edge with the appropriate number of parallel edges. Let us denote the graph-induced X-program of this model by \mbox{$\mathcal{X}_{G'(\theta)}$}. Then, we have the following proposition.
\begin{proposition}[{restate=[name=restatement]IQPAugmentedGraphRelation}]
    \label{proposition:IQPAugmentedGraphRelation}
    \begin{equation}
        \psi_{\mathcal{X}_{G(\theta)}}\left(0^{\abs{V}}\right) = \psi_{\mathcal{X}_{G'(\theta)}}\left(0^{\abs{V'}}\right). \notag
    \end{equation}
\end{proposition}
We prove \mbox{Proposition~\ref{proposition:IQPAugmentedGraphRelation}} in \mbox{Appendix~\ref{section:IQPAugmentedGraphRelation}}. We also have the following proposition relating the principal probability amplitude to the Tutte polynomial of the augmented graph.
\begin{proposition}[{restate=[name=restatement]IQPAugmentedGraphTuttePolynomialRelation}]
    \label{proposition:IQPAugmentedGraphTuttePolynomialRelation}
    \begin{equation}
        \psi_{\mathcal{X}_{G'}(\theta)}\left(0^{\abs{V'}}\right) = e^{i\theta\left(r(G')-\abs{E'}\right)}(i\sin(\theta))^{r(G')}\mathrm{T}\left(G';x,y\right), \notag
    \end{equation}
    where \mbox{$x=-i\cot(\theta)$} and \mbox{$y=e^{2i\theta}$}. \notag
\end{proposition}
We prove \mbox{Proposition~\ref{proposition:IQPAugmentedGraphTuttePolynomialRelation}} in \mbox{Appendix~\ref{section:IQPAugmentedGraphTuttePolynomialRelation}}. Notice that if we let \mbox{$M=(\mathcal{S},\mathcal{I})$} be the binary matroid whose ground set $\mathcal{S}$ is the column space of the orientated incidence matrix $A_{D(G')}$ of $G'$ with an arbitrary orientation $D(G')$ assigned to it, then we can use \mbox{Proposition~\ref{proposition:IQPTuttePolynomialRelation}} to obtain \mbox{Proposition~\ref{proposition:IQPAugmentedGraphTuttePolynomialRelation}}.

\section{Quantum Computation and the Tutte Polynomial}
\label{section:QuantumComputationAndTheTuttePolynomial}

In this section we show that quantum probability amplitudes may be expressed in terms of the evaluation of a Tutte polynomial. We achieve this by showing that output probability amplitudes of a class of universal quantum circuits are proportional to the principal probability amplitude of some IQP circuit.

It will be convenient to define the following gate set.
\begin{definition}[$\mathcal{G}_\theta$]
    For a real angle \mbox{$\theta\in[-\pi,\pi]$}, we define $\mathcal{G}(\theta)$ to be the gate set
    \begin{equation}
        \mathcal{G}_\theta \coloneqq \{H, e^{i\theta X}, e^{i\theta XX}\}, \notag
    \end{equation}
    where $H$ denotes the Hadamard gate.
\end{definition}
It is easy to see that the gate set $\mathcal{G}_{\frac{\pi}{4}}$ generates the Clifford group and the gate set $\mathcal{G}_{\frac{\pi}{8}}$ is universal for quantum computation.

In the IQP model it is easy to implement the gates $e^{i\theta X}$ and $e^{i\theta XX}$. So in order to implement the entire gate set $\mathcal{G}_\theta$, it remains to show that we can implement the Hadamard gate. This can be achieved by the use of postselection when \mbox{$\theta=\frac{\pi}{4k}$} for \mbox{$k\in\mathbb{Z}^+$}~\cite{bremner2010classical}. To apply a Hadamard gate to the target state $\ket{\alpha}_t$ consider the following Hadamard gadget. Firstly introduce an ancilla qubit in the state $\ket{0}_a$ and apply the gate \mbox{$e^{\frac{i\pi}{4}(\mathbb{I}-X)_t(\mathbb{I}-X)_a}$} to \mbox{$\ket{\alpha}_t\ket{0}_a$}. Then measure qubit $t$ in the computational basis and postselect on an outcome of $0$. The output state of this gadget is then \mbox{$H\ket{\alpha}_a$}.

We shall consider quantum circuits that comprise gates from the set $\mathcal{G}_{\frac{\pi}{4k}}$ for an integer \mbox{$k\in\mathbb{Z}^+$}. Let $C_{k,n,m}$ denote such a circuit that acts on $n$ qubits and comprises $m$ Hadamard gates. Further let $\mathcal{X}_G(C_{k,n,m})$ denote the graph-induced X-program that implements the circuit $C_{k,n,m}$ by replacing each of the $m$ Hadamard gates with the Hadamard gadget. Then we have the following proposition.
\begin{proposition}
    \label{proposition:QuantumCircuitAmplitudeTuttePolynomialRelation}
    \begin{equation}
        \bra{0^n}C_{k,n,m}\ket{0^n} = \sqrt{2}^m\psi_{\mathcal{X}_G(C_{k,n,m})}\left(0^{n+m}\right). \notag
    \end{equation}
\end{proposition}
\begin{proof}
    The proof follows immediately from the application of the Hadamard gadgets.
\end{proof}

Any quantum amplitude may therefore be expressed as the evaluation of a Tutte polynomial by \mbox{Proposition~\ref{proposition:IQPAugmentedGraphRelation}}, \mbox{Proposition~\ref{proposition:IQPAugmentedGraphTuttePolynomialRelation}}, and \mbox{Proposition~\ref{proposition:QuantumCircuitAmplitudeTuttePolynomialRelation}}.

\section{Efficient Classical Simulation of Clifford Circuits}
\label{section:EfficientClassicalSimulationCliffordCircuits}

In this section we show how the correspondence between quantum computation and evaluations of the Tutte polynomial provides an explicit form for Clifford circuit amplitudes in terms of matroid invariants; namely, the \emph{bicycle dimension} and \emph{Brown's invariant}. This gives rise to an efficient classical algorithm for computing the output probability amplitudes of Clifford circuits. We note that it was first observed by Shepherd~\cite{shepherd2010binary} that to compute the probability amplitude of a Clifford circuit, it is sufficient to evaluate the Tutte polynomial of a binary matroid at the point $(x, y)$ equals $(-i, i)$, which can be efficiently computed by Vertigan's algorithm~\cite{vertigan1998bicycle}. We proceed with some definitions.

Let $V$ be a linear subspace of $\mathbb{F}_2^n$. The bicycle dimension and Brown's invariant are defined as follows.
\begin{definition}[Bicycle dimension]
    The bicycle dimension of $V$ is defined by
    \begin{equation}
        d(V) \coloneqq \dim(V \cap V^\perp). \notag
    \end{equation}
\end{definition}
\begin{definition}[Brown's invariant]
    If $\abs{\mathrm{supp}(x)} \equiv 0 \pmod{4}$ for all $x \in V \cap V^\perp$, then Brown's invariant $\sigma(V)$ is defined to be the smallest integer such that
    \begin{equation}
        \sum_{x \in V}i^{\abs{\mathrm{supp}(x)}} = \sqrt{2}^{d(V)+\dim(V)}e^{\frac{i\pi}{4}\sigma(V)}. \notag
    \end{equation}
\end{definition}

The following theorem of Pendavingh~\cite{pendavingh2014evaluation} provides an explicit form for the Tutte polynomial of a binary matroid at $(-i, i)$ in terms of the bicycle dimension and Brown's invariant.
\begin{theorem}[Pendavingh~\cite{pendavingh2014evaluation}]
    \label{theorem:TuttePolynomialExplicitForm}
    Let $V$ be a linear subspace of $\mathbb{F}_2^{\mathcal{S}}$ and let $M(V)$ be the corresponding binary matroid with ground set $\mathcal{S}$. If \mbox{$\abs{\mathrm{supp}(x)} \equiv 0 \pmod{4}$} for all \mbox{$x \in V \cap V^\perp$}, then,
    \begin{equation}
        \mathrm{T}(M(V);-i,i) = \sqrt{2}^{d(V)}e^{\frac{i\pi}{4}(2\abs{S}-3r(M)-\sigma(V))}. \notag
    \end{equation}
    Otherwise, $\mathrm{T}(M(V);-i,i)=0$. Further, $\mathrm{T}(M(V);-i,i)$ can be evaluated in polynomial time.
\end{theorem}
As an immediate consequence of \mbox{Theorem~\ref{theorem:TuttePolynomialExplicitForm}}, we obtain an explicit form for Clifford circuit amplitudes in terms the bicycle dimension and Brown's invariant of the corresponding matroid. Furthermore, we obtain an efficient classical algorithm for computing the output probability amplitudes of Clifford circuits. For similar results of this flavour see Refs.~\cite{guan2019stabilizer, gosset2020fast}.

\section{Algorithm Overview}
\label{section:AlgorithmOverview}

We shall now use the correspondence between quantum computation and evaluations of the Tutte polynomial to establish a heuristic algorithm for computing quantum probability amplitudes. To compute a probability amplitude, it is sufficient to compute the Tutte polynomial of a graphic matroid at \mbox{$x=-i\cot\left(\frac{\pi}{4k}\right)$} and \mbox{$y=e^{\frac{i\pi}{2k}}$} for an integer $k\geq2$~\cite{shepherd2009temporally, shepherd2010binary, bremner2010classical}. Our algorithm will use the deletion-contraction property to recursively compute the Tutte polynomial. At each step in the recursion, the algorithm will compute certain structural properties of the graph in order to attempt to prune the computational tree. Our algorithm can be seen an adaption of the work of Haggard, Pearce, and Royle~\cite{haggard2010computing} to special points of the Tutte plane.

We note that our approach differs from tensor network-based methods, which involve the contraction of a graph with tensors assigned to its vertices. These methods have been used to simulate quantum computations while exploiting structural properties of the graph~\cite{markov2008simulating, ogorman2019parameterization, gray2021hyper, huang2020classical, pan2021simulating}. However, our approach allows us to exploit an alternative class of structural properties. We proceed by describing the key aspects of our algorithm.

\subsection{Multigraph Deletion-Contraction Formula}
\label{section:MultigraphDeletionContractionFormula}

To improve the performance of our algorithm, we shall use the following deletion-contraction formula for multigraphs.
\begin{proposition}
    \label{proposition:MultigraphDeletionContractionFormula}
    Let \mbox{$G=(V,E)$} be a multigraph and let $e$ be a multiedge of $G$ with multiplicity $\abs{e}$. If $e$ is a loop, then
    \begin{equation}
        \mathrm{T}(G;x,y) = y^{\abs{e}}\mathrm{T}(G{\setminus}\{e\};x,y). \notag
    \end{equation}
    If $e$ is a coloop, then
    \begin{equation}
        \mathrm{T}(G;x,y) = \left(x+\sum_{i=1}^{\abs{e}-1}y^i\right)\mathrm{T}(G/\{e\};x,y). \notag
    \end{equation}
    Finally, if $e$ is neither a loop nor a coloop, then
    \begin{equation}
        \mathrm{T}(G;x,y) = \mathrm{T}(G{\setminus}\{e\};x,y)+\left(\sum_{i=0}^{\abs{e}-1}y^i\right)\mathrm{T}(G/\{e\};x,y). \notag
    \end{equation}
\end{proposition}
\mbox{Proposition~\ref{proposition:MultigraphDeletionContractionFormula}} can easily be proven from the deletion-contraction formula by induction; we omit the proof.

If $U$ is the underlying graph of $G$, then the number of recursive calls may be bounded by \mbox{$O\left(2^{\abs{E(U)}}\right)$}. Alternatively, we may bound the number of recursive calls in terms of the number of vertices plus the number of edges \mbox{$s=\abs{V(U)}+\abs{E(U)}$} in the underlying graph. The number of recursive calls $R_s$ is then bounded by \mbox{$R_s \leq R_{s-1}+R_{s-2}$}, which is precisely the Fibonacci recurrence. Hence the number of recursive calls is bounded by \mbox{$O\left(\phi^{\abs{V(U)}+\abs{E(U)}}\right)$}, where \mbox{$\phi=\frac{1+\sqrt{5}}{2}$} is the golden ratio~\cite{wilf2002algorithms}. A careful analysis shows that the number of recursive steps is bounded by \mbox{$O\left(\tau(U)\cdot\abs{E(U)}\right)$}, where $\tau(U)$ denotes the number of spanning trees in $U$~\cite{sekine1995computing}.

At each step in the recursion, we use the multigraph deletion-contraction formula to remove all multiedges that correspond to either a loop or a coloop in the underlying graph. This process contributes a multiplicative factor to the proceeding evaluation. Notice that when $G$ is a graph whose underlying graph is a looped forest, then every edge in the underlying graph is either a loop or a coloop. Hence, we obtain the following formula for the Tutte polynomial of $G$.
\begin{corollary}
    Let \mbox{$G=(V,E)$} be a multigraph whose underlying graph $U$ is a looped forest. Further, for each edge $e$ in $U$, let $\abs{e}$ denote its multiplicity in $G$. Then,
    \begin{equation}
        \mathrm{T}(G;x,y) = \prod_{\substack{e \in E(U) \\ \mathrm{loop}}}y^{\abs{e}}\prod_{\substack{e \in E(U) \\ \mathrm{coloop}}}\left(x+\sum_{i=1}^{\abs{e}-1}y^i\right). \notag
    \end{equation}
\end{corollary}
\begin{proof}
    The proof follows immediately from Proposition~\ref{proposition:MultigraphDeletionContractionFormula}.
\end{proof}

\subsection{Graph Simplification}
\label{section:GraphSimplification}

There are a number of techniques that we can use to simplify the graph at each step in the recursion. Firstly, we may remove any isolated vertices, since they do not contribute to the evaluation.

Secondly, when \mbox{$x=-i\cot\left(\frac{\pi}{4k}\right)$} and \mbox{$y=e^{\frac{i\pi}{2k}}$} for an integer \mbox{$k\in\mathbb{Z}^+$}, we may replace each multiedge with a multiedge of equal multiplicity modulo $4k$. To account for this, we multiply the proceeding evaluation by a efficiently computable factor. Specifically, we invoke the following proposition.
\begin{proposition}[{restate=[name=restatement]GraphEdgeSimplification}]
    \label{proposition:GraphEdgeSimplification}
    Fix \mbox{$k\in\mathbb{Z}^+$}. Let \mbox{$G=(V,E)$} be a multigraph and let \mbox{$G'=(V',E')$} be the graph formed from $G$ by taking the multiplicity of each multiedge in $G$ modulo $4k$. Then,
    \begin{equation}
        \mathrm{T}(G;x,y) = \left(ie^{\frac{i\pi}{4k}}\sin\left(\frac{\pi}{4k}\right)\right)^{\kappa(E)-\kappa(E')}\mathrm{T}(G';x,y), \notag
    \end{equation}
    where \mbox{$x=-i\cot\left(\frac{\pi}{4k}\right)$} and \mbox{$y=e^{\frac{i\pi}{2k}}$}. \notag
\end{proposition}
We prove \mbox{Proposition~\ref{proposition:GraphEdgeSimplification}} in \mbox{Appendix~\ref{section:GraphEdgeSimplification}}.

\subsection{Vertigan Graphs}
\label{section:VertiganGraphs}

The Tutte polynomial of a multigraph whose edge multiplicities are all integer multiples of an integer \mbox{$k\in\mathbb{Z}^+$} may be evaluated at the point \mbox{$x=-i\cot\left(\frac{\pi}{4k}\right)$} and \mbox{$y=e^{\frac{i\pi}{2k}}$} in polynomial time. This can be seen by the following proposition.
\begin{proposition}[{restate=[name=restatement]TuttePolynomialVertiganGraph}]
    \label{proposition:TuttePolynomialVertiganGraph}
    Fix \mbox{$k\in\mathbb{Z}^+$}. Let $G=(V,E)$ be a multigraph whose edge multiplicities are all integer multiples of $k$. Further let \mbox{$G'=(V',E')$} be the graph formed from $G$ by taking the multiplicity of each multiedge in $G$ divided by $k$. Then,
    \begin{equation}
        \mathrm{T}(G;x,y) = \left(\sqrt{2}e^{\frac{i\pi(1-k)}{4k}}\sin\left(\frac{\pi}{4k}\right)\right)^{-r(G)}\mathrm{T}(G';-i,i), \notag
    \end{equation}
    where \mbox{$x=-i\cot\left(\frac{\pi}{4k}\right)$} and \mbox{$y=e^{\frac{i\pi}{2k}}$}. \notag
\end{proposition}
We prove \mbox{Proposition~\ref{proposition:TuttePolynomialVertiganGraph}} in \mbox{Appendix~\ref{section:TuttePolynomialVertiganGraph}} and note that this is a special consequence of the \emph{$k$-thickening} approach of Jaeger, Vertigan, and Welsh~\cite{jaeger1990computational}. The Tutte polynomial may then be efficiently computed by Vertigan's algorithm~\cite{vertigan1998bicycle}; we call such a multigraph a \emph{Vertigan graph}. We may therefore prune the computational tree whenever the graph is a Vertigan graph with respect to $k$. Note that this corresponds to quantum circuits comprising gates from the Clifford group.

\subsection{Connected Components}
\label{section:ConnectedComponents}

The Tutte polynomial factorises over components.
\begin{proposition}
    \label{proposition:TuttePolynomialConnectedComponents}
    Let \mbox{$G=(V,E)$} be a graph with connected components $C=\{C_i\}_{i=1}^k$, then,
    \begin{equation}
        \mathrm{T}(G;x,y) = \prod_{i=1}^k\mathrm{T}(C_i;x,y). \notag
    \end{equation}
\end{proposition}
\mbox{Proposition~\ref{proposition:TuttePolynomialConnectedComponents}} can easily be proven from the deletion-contraction formula; we omit the proof. At each step in the deletion-contraction recursion, if the graph is disconnected, then we may use this property to prune the computational tree and hence improve performance.

\subsection{Biconnected Components}
\label{section:BiconnectedComponents}

An identical result holds for biconnected components.
\begin{proposition}[Tutte~\cite{tutte1954contribution}]
    \label{proposition:TuttePolynomialBiconnectedComponents}
    Let \mbox{$G=(V,E)$} be a graph with biconnected components $B=\{B_i\}_{i=1}^k$, then,
    \begin{equation}
        \mathrm{T}(G;x,y) = \prod_{i=1}^k\mathrm{T}(B_i;x,y). \notag
    \end{equation}
\end{proposition}
\mbox{Proposition~\ref{proposition:TuttePolynomialBiconnectedComponents}} can easily be proven from the deletion-contraction formula. For a proof, we refer the reader to Ref.~\cite[Section 3]{tutte1954contribution}.
Similarly to the connected component case, we may use this property to prune the computational tree and improve performance. Note that the biconnected components of a graph may be listed in time linear in the number of edges via depth-first search~\cite{hopcroft1973algorithm}.

\subsection{Multi-Cycles}
\label{section:MultiCycles}

The Tutte polynomial of a multigraph whose underlying graph is a cycle may be computed in polynomial time by invoking the following proposition.
\begin{proposition}[Haggard, Pearce, and Royle~\cite{haggard2010computing}]
    \label{proposition:TuttePolynomialMultiCycle}
    Let \mbox{$G=(V,E)$} be a multigraph whose underlying graph $U$ is an $n$-cycle with edges indexed by the positive integers. Further, for each edge $e$ in $U$, let $\abs{e}$ denote its multiplicity in $G$. Then,
    \begin{align}
        \mathrm{T}(G;x,y) &= \sum_{k=1}^{n-2}\left(\prod_{j=k+1}^ny_x\left(\abs{e_j}\right)\prod_{j=1}^{k-1}y_1\left(\abs{e_j}\right)\right) \notag \\
        &\quad+y_x\left(\abs{e_n}+\abs{e_{n-1}}\right)\prod_{j=1}^{n-2}y_1\left(\abs{e_j}\right), \notag
    \end{align}
    where \mbox{$y_x(j) \coloneqq x+\sum_{i=1}^{j-1}y^i$}.
\end{proposition}
\mbox{Proposition~\ref{proposition:TuttePolynomialMultiCycle}} can easily be proven from the deletion-contraction formula. For a proof, we refer the reader to Ref.~\cite[Theorem 4]{haggard2010computing}. We may use this proposition to prune the computational tree whenever the underlying graph is a cycle.

\subsection{Planar Graphs}
\label{section:PlanarGraphs}

The Tutte polynomial of a planar graph along the hyperbola \mbox{$(x-1)(y-1)=2$} may be evaluated in polynomial time via the Fisher-Kasteleyn-Temperley (FKT) algorithm~\cite{temperley1961dimer, kasteleyn1963dimer, kasteleyn1967graph}. We may therefore use this algorithm to prune the computational tree whenever the underlying graph is planar. Note that we may test whether a graph is planar in time linear in the number of vertices~\cite{hopcroft1974efficient}.

\subsection{Edge-Selection Heuristics}
\label{section:EdgeSelectionHeuristics}

The performance of our algorithm depends on the heuristic used to select edges. We shall consider six edge-selection heuristics: \emph{vertex order}, \emph{minimum degree}, \emph{maximum degree}, \emph{minimum degree sum}, \emph{maximum degree sum}, and \emph{non-Vertigan}. These edge-selection heuristics were first studied by Pearce, Haggard, and Royle~\cite{pearce2009edge}, with the exception of non-Vertigan, which is specific to our algorithm.

\emph{Vertex order}: The vertices of the graph are assigned an ordering. A multiedge is selected from those incident to the lowest vertex in the ordering and whose other endpoint is also the lowest vertex of any incident in the ordering. For contractions, the vertex inherits the lowest of the positions in the ordering.

\emph{Minimum degree}: A multiedge is selected from those incident to a vertex with minimal degree in the underlying graph.

\emph{Maximum degree}: A multiedge is selected from those incident to a vertex with maximal degree in the underlying graph.

\emph{Minimum degree sum}: A multiedge is selected from those whose sum of degrees of its endpoints is minimal in the underlying graph.

\emph{Maximum degree sum}: A multiedge is selected from those whose sum of degrees of its endpoints is maximal in the underlying graph.

\emph{Non-Vertigan}: A multiedge is selected from those whose multiplicity is not an integer multiple of $k$; we call such a multiedge \emph{non-Vertigan}. Using this edge-selection heuristic, the number of recursive calls may be bounded by \mbox{$O\left(2^{\nu(G)}\right)$}, where $\nu(G)$ denotes the number of non-Vertigan multiedges in $G$. This is due to the fact that both the deletion and contraction operation reduce the number of non-Vertigan multiedges by at least one. We note that this is similar to the \emph{Sum-over-Cliffords} approach studied in Refs.~\cite{bravyi2016trading, bravyi2016improved, bravyi2019simulation}.

\subsection{Other Methods}
\label{section:OtherMethods}

There are many other methods that may improve the performance of our algorithm, which we do not study. We shall proceed by discussing some of these.

\emph{Isomorphism testing}: During the computation the graphs encountered and the evaluation of their Tutte polynomial is stored. At each recursive step, we test whether the graph is isomorphic to one already encountered, and if so, we use the evaluation of the isomorphic graph instead. Haggard, Pearce, and Royle~\cite{haggard2010computing} showed that isomorphism testing can lead to an improvement in the performance of computing Tutte polynomials. Note that this may not be as effective when the input is a multigraph.

\emph{Almost planar}: At each step in the recursion, we may test whether the graph is close to being planar, and if so, select edges in such a way that the deletion and contraction operations give rise to a planar graph. For example, if the graph is \emph{apex}, that is, it can be made planar by the removal of a single vertex, then we may select a multiedge incident to such a vertex. Similarly, if the underlying graph is \emph{edge apex} or \emph{contraction apex}~\cite{lipton2017six}, then we may select a multiedge such that the deletion or the contraction operation gives rise to a planar graph.

\emph{$k$-connected components}: Similarly to the connected and biconnected component case, we may compute the Tutte polynomial in terms of its $k$-connected components~\cite{andrzejak1997splitting, bonin2004tutte}.

\section{Experimental Results}
\label{section:ExperimentalResults}

In this section we present some experimental results comparing the performance of the edge-selection heuristics described in \mbox{Section~\ref{section:EdgeSelectionHeuristics}} on two classes of random quantum circuits. Our experiments were performed using SageMath 9.0~\cite{sage2020sagemath}. The source code and experimental data are available at Ref.~\cite{mann2021data}.

The first class we consider corresponds to random instances of IQP circuits induced by dense graphs. Specifically, an instance is an IQP circuit induced by a complete graph with edge weights chosen uniformly at random from the set \mbox{$\{\frac{m\pi}{8} \mid m\in\mathbb{Z}/8\mathbb{Z}\}$}. This class of IQP circuits is precisely that studied in Ref.~\cite{bremner2016average}, where it is conjectured that approximating the corresponding amplitudes up to a multiplicative error is \textsc{\#P}-hard on average.

The second class we consider corresponds to random instances of IQP circuits induced by sparse graphs. Specifically, an instance is an IQP circuit induced by a random graph where each of the possible edges is included independently with probability $1/2$ and with edge weights chosen uniformly at random from the set \mbox{$\{\frac{m\pi}{8} \mid m\in\mathbb{Z}/8\mathbb{Z}\}$}.

We run our algorithm using each of the edge-selection heuristics to compute the principal probability amplitude of $64$ random instances of both the dense and sparse class on $12$ vertices. The performance of each edge-selection heuristic is measured by counting the number of leaves in the computational tree. Our experimental data is presented in \mbox{Appendix~\ref{section:TablesExperimentalData}}. We find that the non-Vertigan edge-selection heuristic performs particularly well for the dense class and the maximum degree sum edge-selection heuristic performs particularly well for the sparse class.

\section{Conclusion \& Outlook}
\label{section:ConclusionAndOutlook}

We established a classical heuristic algorithm for exactly computing quantum probability amplitudes. Our algorithm is based on mapping output probability amplitudes of quantum circuits to evaluations of the Tutte polynomial of graphic matroids. The algorithm evaluates the Tutte polynomial recursively using the deletion-contraction property while attempting to exploit structural properties of the matroid. We considered several edge-selection heuristics and presented experimental results comparing their performance on two classes of random quantum circuits. Further, we obtained an explicit form for Clifford circuit amplitudes in terms of matroid invariants and an alternative efficient classical algorithm for computing the output probability amplitudes of Clifford circuits.

\section*{Acknowledgements}

We thank Michael Bremner, Adrian Chapman, Iain Moffatt, Ashley Montanaro, Rudi Pendavingh, and Dan Shepherd for helpful discussions. This research was supported by the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme (QuantAlgo project), EPSRC grants EP/L021005/1, EP/R043957/1, and EP/T001062/1, and the ARC Centre of Excellence for Quantum Computation and Communication Technology (CQC2T), project number CE170100012. Data are available at the University of Bristol data repository, data.bris, at \url{https://doi.org/10.5523/bris.kbhgclva863q21tjkqpyr5uvq}.

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second and third year students.
have moved a mountain of clay and now they need to have a clay sabbatical.
winningest record of teachers in textiles, furniture, glass and ceramics.
Your #1 teacher would be Bruce Cochrane. It don't get better than him.
she leaves me she has got to take me with her.


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Suzanne Bonamici on Homeland Security
Sponsored defunding nuclear-armed aircraft and nuclear R&D or testing.
Bonamici co-sponsored SANE Act
Congressional Summary:Smarter Approach to Nuclear Expenditures Act: Prohibits using funds in 2014 or thereafter:
to arm a B-2 or B-52 aircraft with a nuclear weapon;
for the research, development, test, and evaluation (RDT&E) or procurement of a long-range penetrating bomber aircraft;
to make the F-35 Joint Strike Fighter aircraft capable of carrying nuclear weapons; or
for the B61 or W78 life extension program.
Requires that, beginning in FY2014, the Navy shall include no more than eight operational ballistic-missile submarines available for deployment.
Prohibits using DOD funds for FY2014 or thereafter to maintain more than 200 intercontinental ballistic missiles (ICBMs), and 250 submarine-launched ballistic missiles
Prohibits funding RDT&E or procurement of a new ICBM or for the medium extended air defense system.
Opponent's argument against bill: (Twitchy blog, "Who said what?", twitchy.com): North Korea has evidently been conducting nuclear device tests, and the timing couldn't be better, of course, as President Obama is expected to call for a reduction in nuclear arms tonight in his State of the Union address. Mass. Rep. Ed Markey is totall Source: H.R.1506 13-H1506 on Apr 11, 2013
End bulk data collection under USA PATRIOT Act.
Bonamici co-sponsored USA FREEDOM Act
Congressional summary:: Uniting and Strengthening America by Fulfilling Rights and Ending Eavesdropping, Dragnet-collection, and Online Monitoring Act or the USA FREEDOM Act:
Amends the Foreign Intelligence Surveillance Act (FISA) to require that the records sought pertain to an individual in contact with a foreign power.
Amends the USA PATRIOT Act to minimize the acquisition and retention of information and to prohibit its unauthorized dissemination.
Imposes additional requirements on the authorized use of pen registers and trap and trace devices (devices for recording incoming and outgoing telephone numbers).
Prohibits the searching of collections of communications of US persons.
Opponent's argument against (Electronic Frontier Foundation): The bill only addresses a small portion of the problems created by NSA spying. It does not touch problems like NSA programs to sabotage encryption standards; it does not effectively tackle the issue of collecting information on people outside of the US; and it doesn't address the authority that the government is supposedly using to tap the data links between service provider data centers, such as those owned by Google and Yahoo. The bill also does not address excessive secrecy; it won't deal with the major over-classification issues or the state secrets privilege.
Opponent's argument against (J. Kirk Wiebe, former NSA Senior Intelligence Analyst interview with TheRealNews.com): It's window dressing. Stopping bulk collection is a good step, but the only thing that's going to fix this is direct access into NSA's databases by an independent group of hackers, techie types, people like Snowden who know how to get into a network and look at things and verify that the data they're collecting and what they're doing with it complies with the Constitution. The NSA has essentially operated illegally--unconstitutionally--for 60% of its existence. Source: HR3361 & S1599 14-H3361 on Oct 29, 2013
Click here for definitions & background information on Homeland Security.
Click here for HouseMatch answers by Suzanne Bonamici.
Click here for a summary of Suzanne Bonamici's positions on all issues.
Click here for a summary of Suzanne Bonamici's positions on the AmericansElect.org quiz.
Click here for MyOcracy quiz answers by Suzanne Bonamici.
Click here for Huffington Post quiz answers by Suzanne Bonamici.
Click here for contact information for Suzanne Bonamici.
Click here for a Wikipedia profile of Suzanne Bonamici.
Click here for a Ballotpedia profile of Suzanne Bonamici.
Click here for other OR politicians.
Click here for OR primary archives.
Click here for OR secondary archives.
2012 Governor, House and Senate candidates on Homeland Security: Suzanne Bonamici on other issues:
OR Gubernatorial:
Kate Brown
OR Senatorial:
Jason Conger
Monica Wehby
OR politicians
OR Archives
PO Box 1632, Beaverton OR 97074
Page last updated: Mar 04, 2016

redpajama_set_name: RedPajamaCommonCrawl


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INCREDIBLES 2 Out on Bluray/DVD! Read this for some insider information and donuts. Yes, Incredible Donuts.
by Bert A. | Nov 5, 2018 | Disney
After 14 years INCREDIBLES 2 is finally open for everyone to see. If you've been following the blog you know that I had the chance to visit Pixar Animation Studios on an all expense paid press trip back in April to learn about the creation of INCREDIBLES 2. The trip was fascinating to say the least; there is so much creative energy within the four walls that make up Pixar. Now I'm not saying you have to read everything I've written but I am saying you'll appreciate the film more if you do read my coverage.
What can I expect to see in the INCREDIBLES 2 Blu-ray/DVD Bonus Features?
My kids and I have been having a blast watching all of the bonus features in the INCREDIBLES 2 Blu-ray/DVD bonus features! If you're a fan of the Parr family you have to add this to your personal movie collection at home.
"Incredibles 2" is packaged and released in several different ways, offering families the flexibility to watch the movie instantly and on a variety of devices of their choosing, including Digital 4K Ultra HD, HD and SD and physically as a 4K Ultra HD Combo Pack (4K Ultra HD, Blu-ray and Digital Copy), a Multi-Screen Edition (Blu-ray, DVD and Digital Copy) and a single DVD. Additionally, the Digital HD/SD, Blu-ray and DVD versions of the film have been modified in an effort to accommodate photosensitive viewers, however individual sensitivities may vary.
In "Incredibles 2," Helen (voice of Holly Hunter) is called on to lead a campaign to bring Supers back, while Bob (voice of Craig T. Nelson) navigates the day-to-day heroics of "normal" life at home with Violet (voice of Sarah Vowell), Dash (voice of Huck Milner) and baby Jack-Jack—whose super powers are about to be discovered. Their mission is derailed, however, when a new villain emerges with a brilliant and dangerous plot that threatens everything. But the Parrs don't shy away from a challenge, especially with Frozone (voice of Samuel L. Jackson) by their side. That's what makes this family so Incredible.
BONUS FEATURES (may vary by retailer)
Digital Exclusives:
The Coolest Guy in Show Business – In this partially illustrated documentary, Samuel L. Jackson reflects how his childhood and love of comics shaped his passion for film and imaginative storytelling.
2 SuperScene Breakdowns – Casual commentary-style pieces looking at specific scenes in the film (The Racoon Fight and Mrs. Incredible) through a particular creative focus like action choreography, set design or story.
Blu-ray & Digital:
All-New "Auntie Edna" Mini-Movie – When Bob Parr visits super-suit designer Edna Mode looking for help with his high-energy toddler Jack-Jack, Edna pulls an all-nighter designing a suit to harness the baby's seemingly limitless powers.
10 Deleted Scenes With Introductions – Suburban Escape, Kari Revisited, Return of the Supers, Chewed Out, Late Audition, Slow Day, Frozone and Honey, Restaurant Robbery, Fashion Show and Security Breakdown.
Super Stuff – From buildings and vehicles to costumes and props, every action movie requires a lot of really cool stuff. Meet the makers and learn what it takes to design and build such a uniquely incredible world.
Heroes & Villains – A collection of mini-docs about the backstory and major design ideas behind the "Incredibles 2" characters — featuring voice actors, director Brad Bird, and Pixar artists talking about the many elements that make these characters feel real.
Ralph Eggleston: Production Designer – This short piece explores the many ways a single production designer has influenced the look, feel and character of the Pixar universe, culminating in "Incredibles 2."
Strong Coffee: A Lesson in Animation with Brad Bird – Brad Bird's passion for animation dates back to his childhood and mentorship under Disney's Milt Kahl, and that enthusiasm and powerful insight emanates from every film he's made. Take a deep dive into Brad's early years at Disney Animation Studios and his time at Pixar.
Paths to Pixar: Everyday Heroes – At its heart, "Incredibles 2" is about family dynamics and the challenges of being a working parent. Meet the parents of Pixar as they discuss their personal connections to the film and their experience with stretching to balance work and family.
SuperBaby – A documentary/hip hop music video hybrid hosted by Frankie and Paige from Disney Channel's Bizaardvark. This piece explores how Jack-Jack came to life onscreen — from design to special effects to animation — all set to a hot beat.
Commentary – Get inside commentary from animators Alan Barillaro (supervising animator), Tony Fucile (supervising animator, story artist and character designer), Dave Mullins (supervising animator) and Bret Parker (animation second unit and crowds supervisor).
Theatrical Short: "Bao" – An aging Chinese mom suffering from empty nest syndrome gets another chance at motherhood when one of her dumplings springs to life as a lively, giggly dumpling boy.
Making "Bao" – Director Domee Shi shares her secret recipe for making an animated short — discussing how her rich cultural heritage, unique relationship with her mom, and her love of food all informed the making of the food-fantasy "Bao."
Outtakes & Stories – Raccoon Fight Story, Evelyn Animation Outtakes, Puppet Animator Interview, Outtakes Goofy Arms Story and SuperBaby Music Video.
Character Theme Songs, Vintage Toy Commercial TV Spots , Toolkit Montage and Global "Incredibles 2" Trailers
Theatrical Short: "Bao" & Commentary
INCREDIBLES 2 Insider Information: What you should know about INCREDIBLES 2
Where is INCREDIBLES 2 set? What Easter eggs should I look for in INCREDIBLES 2?
Why did it take so long to make INCREDIBLES 2? Director Brad Bird explains the creative process and what's changed in INCREDIBLES 2
Do you need a costume designer for an animated film? The costume design process of INCREDIBLES 2.
I promise you may be crying over a dumpling during Disney Pixar's Short Animated Film BAO
I toured Pixar Animation Studios and this is what it was like.
What is New for Pixar Fest at Disneyland
What is INCREDIBLES 2 about?
In "Incredibles 2," Helen (voice of Holly Hunter) is called on to lead a campaign to bring Supers back, while Bob (voice of Craig T. Nelson) navigates the day-to-day heroics of "normal" life at home with Violet (voice of Sarah Vowell), Dash (voice of Huck Milner) and baby Jack-Jack—whose super powers are about to be discovered. Their mission is derailed, however, when a new villain emerges with a brilliant and dangerous plot that threatens everything. But the Parrs don't shy away from a challenge, especially with Frozone (voice of Samuel L. Jackson) by their side. That's what makes this family so Incredible. Written and directed by Brad Bird ("Iron Giant," "The Incredibles," "Ratatouille") and produced by John Walker ("The Incredibles," "Tomorrowland") and Nicole Grindle ("Sanjay's Super Team" short, "Toy Story 3" associate producer), Disney•Pixar's "Incredibles 2" busts into theaters on June 15, 2018.
A BIT OF A STRETCH – Elastigirl is called on to lead a campaign to bring Supers back in "Incredibles 2"—a mission that involves a helicopter chase among other Incredible stunts. Featuring Holly Hunter as the voice of Helen aka Elastigirl, "Incredibles 2" opens in U.S. theaters on June 15, 2018. ©2018 Disney•Pixar. All Rights Reserved.
Follow INCREDIBLES 2 on all social media platforms.
#Incredibles2 #Incredibles2Event
Like INCREDIBLES 2 on Facebook: https://www.facebook.com/PixarTheIncredibles/
Follow Disney/Pixar on Twitter: https://twitter.com/disneypixar
Follow Disney/Pixar on Instagram: https://www.instagram.com/pixar/
Visit the official INCREDIBLES 2 website here: http://disney.com/incredibles2
INCREDIBLES 2 opens in theatres everywhere on June 15, 2018!
The Incredible Doughnut
This is just a fun way to show your INCREDIBLES 2 fanfare. These doughnuts are easy to make and perfect for an INCREDIBLES 2 themed birthday party! Just follow these simple steps and voila! You've got yourself the Incredible Doughnut.
First things first, after you've preheated the oven to 400 degrees F and prepared your doughnut pan, put all of the dry ingredients in one bowl and whisk to mix them well.
Then you'll want to blend the wet ingredients, making sure everything is combined all nice and smooth-like for that Incredible Doughnut experience. No lumps or bumps here!
Next, slowly add the wet ingredients to the dry ingredients and stir until well blended.
Then you'll divide the batter into three separate bowls. After you've done that use food coloring to make the INCREDIBLES colors – red, black and yellow. Once you get the desired color, add each color to the doughnut pan.
While those are baking, make the glaze, take the doughnuts out of the oven when they're done and let them completely cool. Then you'll glaze the doughnuts by dipping them into your glaze. Once you've glazed all of the doughnuts and let them completely cool, you can enjoy them because they are ready to eat!
Doughnut Ingredients:
1/4 Cup melted unsalted sweet cream butter
2 large egg (room temp)
Yellow, red, and black gel food coloring
Vanilla Glaze ingredients:
2 Cup powder sugar
Donut Directions:
Use a baking spray to prep your doughnut pan.
Set the pan to the side.
Whisk together the flour, baking powder, baking soda, cinnamon, salt and sugar in a large bowl.
Whisk together the whole milk, sour cream , and eggs into a separate bowl.
Whisk in the melted butter, vegetable oil, and vanilla
Gradually stir in the wet ingredients into the dry ingredients and stir just until combined
Divide the batter into three bowls.
Add several drops of Yellow food coloring in the first bowl.
Add several drops of Red food coloring in the first bowl.
Add several drops of Black food coloring in the first bowl.
Spoon equal amounts of the 3 batters into the doughnut pan.
Bake at 400 degrees for 9 minutes.
Allow doughnuts to cool a few minutes in the pan.
Transfer the donuts to a wire rack to completely cool.
Glaze Directions:
Whisk together the milk, powder sugar and vanilla extract until smooth.
Split the icing into 2 bowls.
Gradually pour the yellow icing into the white icing.
DO NOT STIR!!
Dip the doughnuts into the icing and turn the doughnut 90 degrees
Place the doughnut back onto the wire rack and repeat the previous steps for remaining doughnuts
Let dry for an hour before enjoying!
The Incredibles Doughnut
Course Snack
Keyword donuts, doughnuts
Doughnut Cooling Time 1 hour
2 large egg room temp
Yellow red, and black gel food coloring


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Q: Process more than 5000 rows from lookup activity in Azure Data factory I need to paas Id in another activity in data factory .Id is stored in blob storage in json format.
I am using Look-Up in my activity which will fetch data .But my pipeline gets failed when data is more than 5000.I need a solution for this.I didnt understand the existing solution in stack overflow.

A: Ah ok, well you cannot use OFFSET/LIMIT pagination sensibly in Cosmos and ADF cannot use continuation tokens.  Also you cannot LOOKUP >5000 results from blob or paginate the blob output.
If I had this problem I would try the following based on this idea Azure Data Factory DYNAMICALLY partition a csv/txt file based on rowcount.

*

*Use dataflow to get the data from cosmos and write to several json files using partitioning, each < 5000 rows (using the method described in the comment on the above link - using a surrogate and the MOD operator)

*ForLoop over those blobs

*Have a nested pipeline that does the lookup and calls the API, as you have now - now the lookup will only have max 5000 items

redpajama_set_name: RedPajamaStackExchange


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Het mineraal lopariet is een cerium-natrium-calcium-titanium-niobium-oxide met de chemische formule (Ce,Na,Ca)2(Ti,Nb)2O6.

Eigenschappen
Het opaak zwarte lopariet heeft een metallische tot diamantglans, een roodbruine streepkleur en een imperfecte splijting volgens het kristalvlak [100]. De gemiddelde dichtheid is 4,77 en de hardheid is 5,5. Het kristalstelsel is kubisch en het mineraal is zwak radioactief. De gamma ray waarde volgens het American Petroleum Institute is 24.983,65.

Naamgeving
De naam van het mineraal lopariet is afgeleid van het Russische woord voor Lappen, Lopar. Dit omdat het mineraal voornamelijk gevonden werd op het door Lappen bevolkte Kola schiereiland.

Voorkomen
Lopariet komt voor als primair mineraal in nefelien-syenieten. Het wordt ook gevormd in alkalische pegmatieten, waar het perovskiet vervangt. De typelocatie is gelegen in de Khibiny en Lovozero bergmassieven, schiereiland Kola, Noord-Rusland. Het wordt ook gevonden in Buer, Bjorkedalen, Langesundfjord, Telemarken, Noorwegen.

Zie ook
 Lijst van mineralen

Oxide (mineraal)
Ceriumhoudend mineraal
Natriumhoudend mineraal
Calciumhoudend mineraal
Niobiumhoudend mineraal
Titaniumhoudend mineraal
Radioactief mineraal

redpajama_set_name: RedPajamaWikipedia


MSC Foundation is currently the world's largest modern sports competition platform, it issues MSC tokens. Based on its P2P-based public blockchain MSCChain, the foundation has developed the MSC global new media copyright trading platform, the advertising platform and the MSC streaming media playing platform, realizing a point-to-point trading and dissemination of streaming media and benefiting content providers, advertisers, disseminators and individual viewers.
Yutaro Takeuchi specializes in C / C + +, Java, software development experience for 10 years, and good at developing sales management system. What's more, he also has project management experience for 5 years. He also is an expert of Salesforce.
Rein Kimura is a senior IoT product manager, leading a number of automated driving, road recognition, automotive intelligence systems, and health applications.


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May 21, 2020 | Part Of Treasury Secretary Mnuchin & Others Discuss Coronavirus Pandemic & Reopening Econcomy
Secretary Mnuchin on Coronavirus Pandemic and Reopening Economy
2020-05-21T20:00:28-04:00https://images.c-span.org/Files/7fe/20200521200141002_hd.jpgTreasury Secretary Steven Mnuchin participated in a virtual summit on the future of the American economy hosted by The Hill. Secretary Mnuchin said there was a "strong likelihood we will need another bill" regarding a fourth coronavirus related stimulus package. The secretary promised to review any loans from the CARES Act that were more than $2 million, but he declined to commit to testifying in front of the House coronavirus relief committee.
Treasury Secretary Steven Mnuchin participated in a virtual summit on the future of the American economy hosted by The Hill. Secretary Mnuchin said… read more
Treasury Secretary Steven Mnuchin participated in a virtual summit on the future of the American economy hosted by The Hill. Secretary Mnuchin said there was a "strong likelihood we will need another bill" regarding a fourth coronavirus related stimulus package. The secretary promised to review any loans from the CARES Act that were more than $2 million, but he declined to commit to testifying in front of the House coronavirus relief committee. close
Filter by Speaker All Speakers Bob Cusack Steven T. Mnuchin
Bob Cusack Editor-in-Chief Hill, The
Steven T. Mnuchin Secretary Department of the Treasury
[Hill, The][Hill, The]
Treasury Secretary Mnuchin & Others Discuss Coronavirus Pandemic & Reopening Econcomy
Senator Warner and Others on Coronavirus Pandemic and Reopening Economy
Senator Cardin on Workplace Safety and Coronavirus Pandemic
Representative Hurd on Coronavirus Pandemic Funding
Surgeon General Jerome Adams on Health Care and the Coronavirus Pandemic
Representative Rodney Davis on the Coronavirus Pandemic and the Economy
May 21, 2020 | 8:00pm EDT | C-SPAN 2
May 23, 2020 | 11:34pm EDT | C-SPAN 1
May 22, 2020 | 3:01am EDT | C-SPAN 2
May 22, 2020 | 6:38am EDT | C-SPAN RADIO
May 23, 2020 | 5:30pm EDT | C-SPAN RADIO
May 25, 2020 | 10:01am EDT | C-SPAN RADIO
May 25, 2020 | 11:00pm EDT | C-SPAN RADIO
See all on U.S. Economy Coronavirus Disease 2019
Senator Mark Warner (D-VA) participated in a virtual summit on the future of the American economy hosted by The Hill. Senator Warner…
Senator Ben Cardin (D-MD) participated in a virtual summit on the future of the American economy hosted by The Hill.…
Representative Will Hurd (R-TX) participated in a virtual summit on the future of the American economy hosted by The…
Surgeon General Dr. Jerome Adams participated in a virtual summit on the future of the American economy hosted by The Hill. Dr.…


Slope Unblocked rejoices the fast pace lovers for certainty! So, why not hurl yourself into this awesome addictive Slope and check out if you can conquer in such a long distance or not? Hurry up, guys! The ball on the top is waiting for you, and it needs your help to reach the end of the slope. In Cool Math Games Slope, you need to roll your ball down the paths without being fallen out of the platforms. Try to speed up your ball as fast as possible to get the high scores. Remember to dodge the disturbing walls or impediments on your way. Ok, let's roll in Slope Game at ABC Kids Games now!


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San Francisco – Oakland Bay Bridge
Free San Francisco – Oakland Bay Bridge Essays and Papers
The San Francisco-Oakland Bay Bridge
About eighty years ago, engineers were able to build a bridge on time and within budget with no problem; today, most of the projects take extra time and money to finish. What exactly happened between now and eighty years ago that caused such a drastic shift? It is clearly not the engineers nor the technology; in fact, those have only improved over time. Today, engineers are faced with much more than designing and building the infrastructure, as they were before. They compete with politicians for
Home Is Where the Kooks Are: San Francisco's Eccentrics Each community and each culture has its social norms. Whether vocalized or not, there is an unspoken set of rules that govern what is appropriate for those living amidst a certain culture. In Japan it is customary to remove your shoes before entering someone's home. In some Middle Eastern cultures, the women must cover their heads in public at all times. But in San Francisco it is customary to have no customs at all. You don't want to put on
Essay On Oakland
Apartment Living in Oakland, CA Overview There's a lot that could be said to convince you that renting apartments in Oakland would be one of the best decisions of your life. Maybe you'd be interested in the fact that Oakland is a smart city; it's one of the top cities when counting residents with bachelor's degrees. Maybe you'd be intrigued by the fact that Oakland is known for its ethnic diversity, as well as its sustainability practices. Many more facts could be spewed at you, and they'd all add
Little Brother Book Analysis
commonly known as Van, and Jose, live in San Francisco, California. These four students work to defend themselves against what they see as attacks by the government and schools on the Bill of Rights after a terrorist attack on the Oakland Bay Bridge in San Francisco, California. Little Brother portrays how surveillance has increased in San Francisco due to attacks and other illegal activities. As the story unfolds, surveillance increases in not only in San Francisco, but also throughout the entire world
The Forgotten City This paper will discuss the many unique aspects of Oakland, California. Divided into five major parts, Oakland is a very diverse city. The five major parts include: Downtown Oakland which is located in the heart of the city, East Oakland which consists of the majority of the city, West Oakland, North Oakland, and the Oakland hills where the terrain is quite different from other parts of the city. While many may perceive the population to mainly consist of African Americans and
Alcatraz Description
San Francisco is a very unique place.With a population of 725,000 its culture is unlike any other. The city has so many different experiences that you do not want to miss. If you ever plan to travel to San Francisco you should know about their attractions, transportation, and some of their interesting facts. San Francisco's attractions are unlike any other. One of the most popular attraction is Alcatraz. This is different from any other attraction because it is a former federal prison. Most people
Sales and Sponsorship for the Golden State Warriors
Hello Class, For this discussion, I have chosen the sports property of the Golden State Warriors. The Warriors play in the NBA within the Pacific Division and have a large local and regional fan base in the city of Oakland and state of California. Even though the Warriors have a large fan following, there are several ways of extending the Warriors brand reach beyond the basketball court with conventional sports minded consumers and non-sports fans. In order to accomplish this, the Warriors must
Process Essay: How To Successfully Escape Alcatraz
building via an unused utility corridor, and escaped Alcatraz Island aboard a raft.(wikipedia)In 1962, the prisoners escaped using a raft, and were never seen again. But if they left in like midnight I think they died in the freezing cold waters of San Francisco Bay. The Fake heads This prison break - dramatized by Clint Eastwood in the film Escape from Alcatraz - is one of the most daring and intricate escapes ever attempted. Three bank robbers spent months digging an escape tunnel
Racism In San Francisco
through your veins. I'm in San Francisco to execute some local triad boss, who got in over his head with the Boston mob. Its not a pretty job, but at least I can leave the frozen tundra of Massachusetts for the foggy grounds of the Zodiac Killer. I'm in San Francisco to end the rule of a Triad who had my boss's son beaten to a pulp after one too many hands of blackjack. …or at least, I should say, I'm on my way to San Francisco. It has been three hours since I left Oakland for "The City" and I haven't
The History of San Francisco Bay Area
The Bay Area is an interesting place with a unique geology and formation. The San Francisco Bay, also known as San Francisco Estuarine, consists of three bays, San Francisco Proper, San Pablo Bay, Carquinez Strait, and has a vast inland river delta. The river delta provides fresh water to the bays. The estuarine system provides food, safety, and shelter to many organisms and animals, including humans. (San Francisco Estuarine Wetlands) Along the bays lie major cities such as San Francisco, Oakland
The San Francisco Golden Gate Exhibition 1939-1940
The San Francisco Golden Gate Exhibition 1939-1940 In 1939, the city of San Francisco hosted an international exhibition, named the Golden Gate Fair in honor of the construction by the city of the world's two largest suspension bridges, the Golden Gate and San Francisco-Oakland, which spanned the San Francisco Bay. With all the skill that could be mustered by American engineers, an island was constructed amidst the Pacific waters, becoming the largest ever manmade island. Christened Treasure Island
San Francisco Descriptive Essay
"The City by the Bay," and "The Fog City" all describe the lovely San Francisco. San Francisco is a famous city located in Northern California and has many attractions that bring tourists from all around the world to this beautiful destination each year. San Francisco is known for the Golden Gate Bridge, Alcatraz Island, Pier 39, and much more! It is the fourth most populated city in California, with more than eight hundred thousand residents. Built on forty-three hills, San Francisco covers over forty-six
Jack London Square
Jack London Square The Golden Gate Bridge in San Francisco, the Gothic architecture of the University of Chicago, Mardi gras, the Great Wall of China--all are highly visible landmarks, characteristics, or events that are emblematic of a particular place. In a more subtle way, there are other "landmarks" that are less recognizable but nonetheless suggest a specific place. Perhaps it is the local mall, or spring tulips in your garden, or abandoned warehouses, or an annual Fourth of July parade
Description and Analysis of Monorails
for its entire 20-mile length. Note the blue line on the opposite bank, which indicates the path of a passenger monorail system between the City of Long Beach and Los Angeles Union Station, and extending along the river to Warner Center in the West San Fernando Valley, a continuous fifty miles from Long Beach
The 1906: The Cause Of The Earthquake Of 1906
At 5:12 a.m. on April 18, 1906, the people of San Francisco were abruptly awaken to one of the world's most expensive and impacting earthquakes of the 20th century. The magnitude of the earthquake is debatable, ranging from 7.7 to as high as 8.2. However, the most accepted measurement is 7.9. The main shock epicenter occurred offshore 3.2 km from the city, near Mussel Rock. It ruptured along the San Andreas Fault northward and southward for 476 km (296 miles). The quake was felt from Oregon to Los
As the cold waters rush into the San Francisco Bay, they crash up against an island standing in the strait. This rock is hidden by the fog and isolated by the chilling waters of the Pacific that flow in and out every day. It has a gloom that hangs about its rocky face most know it as Alcatraz but the men who experienced this island, referred to her as "The Rock". To the men confined there, it is not only the ultimate in isolation but the most ironic because they are there in the midst of the activity
History Of Epoxy Resins
polymer concrete was first used as a wearing course in the repair of popouts and spalled areas on the surfaces of various concrete bridge decks in California in 1957, on the San Francisco-Oakland Bay Bridge, and in industrial plants and warehouses. The epoxy polymer concrete consisted primarily of the epoxy resin system and clean, dry well-graded sand. By 1963, several bridges in various parts of the United
How To Solve The Foreclosure Crisis
How to solve the foreclosure crisis. Solving the foreclosure crisis will depend on the contribution, and reward, of many stakeholders. In one neighborhood alone, West Oakland in the San Francisco Bay Area, there were almost three hundred foreclosures and short sales this year alone. This has had many ill effects: displaced residents, blighted neighborhoods, torn social structures, increased crime and further erosion of home values, all contributing to yet more foreclosures in a downwardly spiraling
this to happen'. KT 39; Costs; economic and human. · Total economic damage of around $10 billion. · 42 died in Oakland, 60 miles from the epicentre. 400 injuries from the Cypress structure. · 59 water mains burst, but 100+ gas mains were ruptured. · 10 000 were left homeless, 62 died.· The Nimitz highway, and Cypress structure, and sections of the Bay Bridge collapsed, due to structural failure. They were subject to ten times the amount of force for which they were designed
Essay On The Black Panther Party
black beret. This image is formed through the thousands of posters and t-shirts once worn as a form of propaganda. The Black Panther Party may have been seen as a prominently male organization, but to everyone's surprise it was two-thirds female. In Oakland, California during October in 1966, Huey Newton and Bobby Seale founded the Black Panther Party for Self Defense. This revolutionary national organization aimed to challenge police brutality, poverty, and racial injustice. It devised a ten-point platform
Sensor node
Wireless sensor network
Model organism
Bedsore
John Charnley
Equilibrium constant
Bornean Orangutan
Biofilm
Geometric mean
Interquartile mean
Sex-determination system
Southern Ontario


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Retired Four-Star Air Force Gen. Demoted Two Ranks For "Inappropriate Sexual Acts"
February 02, 2017 Peter Reid
The Air Force has demoted retired Air Force Gen. Arthur Lichte to the rank of major General after finding that he engaged in "inappropriate sexual acts" on several occasions with a lower-ranking female officer before retiring in 2010, the service announced Wednesday.
He was demoted following a probe by the Air Force Office of Special Investigations after the female officer alleged last summer that she was assaulted by Lichte twice in 2007 while he was holding the rank of lieutenant general as the service's assistant vice chief of staff and Air Staff director at the Pentagon. The service found that he also engaged in an inappropriate sexual relationship again in 2009 with the same female officer while he was the four-star commander of U.S. Air Mobility Command at Scott Air Force Base in Illinois. The service did not substantiate the sexual assault allegations.
Lichte "did not commit a sexual assault and vehemently denies the unsworn allegations made against him," Lichte's attorney, Larry Youngner, said in a statement.
"General (Ret.) Arthur Lichte has continually asserted that he is deeply sorry for the pain he has caused his family, especially his strong and loving wife," the statement said. "He is regretful of the decisions he made that allowed him to find himself in this predicament. My client and his family ask for privacy to work through this difficult time. We will continue to appeal the Air Force's wrong decision concerning this unsworn accusation and the grade determination through the proper channels."
In December, then-Air Force Secretary Deborah James issued Lichte a letter of reprimand after reviewing the results of the investigation.
"You are hereby reprimanded!"James wrote in the letter of Dec. 6, 2016. "Your conduct is disgraceful and, but for the statute of limitations bar to prosecution, would be more appropriately addressed through the Uniform Code of Military Justice."
James slammed Lichte for putting the female officer "in a position in which she could have believed that she had no choice but to engage in these sex acts given your far superior grade, position, and significant ability to affect her career."
The heavily redacted 50 page report released by the Air Force said that the woman alleged that Lichte told the female officer, now a colonel, that if she told anyone what happened "he would deny it until the day he died."
The decision to reduce Lichte in rank was made by acting Air Force secretary, Lisa Disbrow, on Tuesday. On Monday, Secretary Jim Mattis withdrew a Defense Department certification that Lichte had served satisfactorily as a four-star general. He was reduced in rank to a two-star general because that was the last time he had performed satisfactorily. He also was docked roughly $5,000 a month in retirement pay.
"The Air Force takes all allegations of inappropriate conduct very seriously," Disbrow said in a statement Wednesday. "We expect our leaders to uphold the highest standards of behavior. These standards and rules underpin good order and discipline. Airmen at every level are held accountable."


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Climate Commission re-launches as Climate Council
September 24, 2013	Comments 0 Libby Dreyer
The former head of the Climate Commission, Professor Tim Flannery, and some of his colleagues, have re-launched the Commission today, as a community funded body.
The Commission was scrapped just one day after the new Coalition government was sworn in, after being established under the Gillard government as a body to inform the public about climate change.
Professor Flannery says after receiving enormous support from the public, he and other former commissioners have decided to re-launch as the Climate Council, and will volunteer their time to get it up and running.
He says it will be a big job to keep the body running without government funding, but he's confident they will receive enough donations from businesses and the general public, to keep it up and running.
Professor Flannery says just like the Commission, The Climate Council will also stay out of political matters, and will simply work to inform the Australian public abotu climate change.
Category National
Libby Dreyer	1729 posts
Libby has worked as a Senior Journalist at National Radio News since February 2011. She graduated with a Bachelor of Journalism from the University of Canberra in December 2010.
National	0 Comments
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Independents & Greens lash out at major parties over political funding
Independent MP Tony Windsor says he will make his voice heard next week, when parliament debates new legislation on political funding.
Economic growth declines but Hockey stands by forecast
Australia's economic growth has only risen 2 per cent in the June quarter, but Joe Hockey says it is in line with budget forecasts.


text:
Trick-Trick (* 28. Juni 1973 in Detroit; eigentlich Christian Mathis) ist ein US-amerikanischer Rapper und Mitglied der Hip-Hop-Gruppe Goon Sqwad.

Werdegang
Sein musikalisches Interesse wurde bereits früh durch seine Mutter geweckt, die Backgroundsängerin der Jazzlegende Alberta Adams war. Mathis wurde von Genres wie Gospel, Soul und Funk geprägt. Beeinflussende Künstler waren Phil Collins und George Clinton, sowie später Hip-Hop-Bands wie Public Enemy, N.W.A und Run DMC. Seinen Spitznamen Trick erhielt er bereits während seiner Jugend.

Die Karriere als Rapper begann 1992, als er von dem Plattenlabel Click Boom Records unter Vertrag genommen wurde. Unterbrochen wurde sie durch eine Haftstrafe. Nach einer Mordanklage wurde Mathis zunächst zu einer Gefängnisstrafe verurteilt, später jedoch nach dem Beweis seiner Unschuld freigelassen.

Nach seiner Entlassung nahm er 1995 als Mitglied der Band The Goon Squad den Titel Booty Bounce auf, der sich bei den lokalen Radiostationen seiner Heimatstadt Detroit zu einem Hit entwickelte. Weitere lokale Radiohits folgten und Trick-Trick etablierte sich als einer der angesehensten Rapper im Underground von Detroit.

Nach einem weiteren Gefängnisaufenthalt 2003/04 begann Trick-Trick ausgestattet mit einem Vertrag bei Motown an der Arbeit für sein Debütalbum. Für die Produktion von The People Vs. wurde er von befreundeten Rappern wie Eminem, Jazze Pha, Obie Trice, Miz Karona, Kon Artis, Proof und Diezel unterstützt. Ende Dezember 2005 kam das Album in den USA auf den Markt und erreichte auf Anhieb die Billboard-Charts. Die erste ausgekoppelte Single Welcome 2 Detroit stieg im März 2006 in die deutsche Hitparade ein.

Kritik
Trick-Trick erhielt Kritik von Seiten von Homosexuellenorganisationen, da sein Album The Villain Aufrufe zur Gewalt bis zum Mord von Homosexuellen enthält. Er ruft zum Beispiel dazu auf, die offen lesbischen US-Moderatorinnen Ellen DeGeneres und Rosie O'Donnell in die Luft zu sprengen. Die schweizerische Organisation Pink Cross unterband erfolgreich den Verkauf des Albums in den größten Plattenläden in der Schweiz.

Diskografie

Alben
 2006: The People vs.
 2008: The Villain

Singles
 2003: It's Goin' Down
 2005: Welcome 2 Detroit (feat. Eminem)
 2008: Let's Work
 2008: Let It Fly
 2008: Who Want It (feat. Eminem)
 2011: I Made It (feat. D12)
 2013: Smoke Weed Everyday
 2014: Twerk Dat Pop That! (feat. Eminem und Royce da 5′9″)
 2014: Detroit vs. Everybody (feat. Eminem, Royce da 5′9″, Big Sean, Danny Brown und Dej Loaf)

Einzelnachweise

Rapper
Pseudonym
Person (Detroit)
Musiker (Vereinigte Staaten)
US-Amerikaner
Geboren 1973
Mann


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