\begin{algorithm}
\caption{MPS Encoding Procedure}
\hspace*{\algorithmicindent} 
\begin{algorithmic}[1]
\Require{ A degree-$p$ piece-wise function  $f_\ell(x)  = \sum_{j=0}^p a_{j}^{(\ell)} x^j $. System size ${N}$. Domain [a,b]. Support bit $k$.} 
\Ensure{A $\chi \le 2^k(p+1)$ MPS, $\bf{M}_T$ which encodes $f_\ell(x)$}
\Statex
\For{$\ell \gets 1$ to $2^k$}   
    \State {Encode $f_\ell(x)$ into ${\bf M}_\ell$ on domain [a,b]}
    \State {Zero out  ${\bf M}_\ell$ outside domain $D_\ell$}
    \EndFor
\State \Return{ ${\bf M}_T \leftarrow  \sum_{\ell=0}^{2^k} {\bf M}_\ell$}

\end{algorithmic}

\label{alg:mps_enc}
\end{algorithm}

\begin{algorithm}
\caption{MPS Encoding Procedure}
\hspace*{\algorithmicindent}
\begin{algorithmic}
[1]
\Require{ A degree-$p$ piece-wise function $f_\ell(x) = \sum_{j=0}^p a_{j}^{(\ell)} x^j $. System size ${N}$. Domain [a,b]. Support bit $k$.}
\Ensure{A $\chi \le 2^k(p+1)$ MPS, $\bf{M}_T$ which encodes $f_\ell(x)$}
\Statex
\For{$\ell \gets 1$ to $2^k$}
\State {Encode $f_\ell(x)$ into ${\bf M}_\ell$ on domain [a,b]}
\State {Zero out ${\bf M}_\ell$ outside domain $D_\ell$}
\EndFor
\State \Return{ ${\bf M}_T \leftarrow \sum_{\ell=0}^{2^k} {\bf M}_\ell$}
\end{algorithmic}
\end{algorithm}
{
    "title": "quantum state preparation of normal distributions using matrix product   states",
    "id": "2303.01562",
    "abstract": "state preparation is a necessary component of many quantum algorithms. in this work, we combine a method for efficiently representing smooth differentiable probability distributions using matrix product states with recently discovered techniques for initializing quantum states to approximate matrix product states. using this, we generate quantum states encoding a class of normal probability distributions in a trapped ion quantum computer for up to 20 qubits. we provide an in depth analysis of the different sources of error which contribute to the overall fidelity of this state preparation procedure. our work provides a study in quantum hardware for scalable distribution loading, which is the basis of a wide range of algorithms that provide quantum advantage.",
    "categories": "quant-ph",
    "doi": "10.1038/s41534-024-00805-0",
    "created": "2023-03-02",
    "updated": "2024-02-16",
    "authors": [
        "jason iaconis",
        "sonika johri",
        "elton yechao zhu"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2303.01562"
}
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