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\begin{document}
\title{Notes - Other stuff}
\author[1]{Luca Innocenti}%
\affil[1]{Affiliation not available}%
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\date{\today}
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\section{Other things}
{\label{643089}}
\cite{Woods_2012}:~\emph{Optical computing: Photonic neural networks}.
\cite{Wood_2011}:~\emph{Tensor networks and graphical calculus for
open quantum systems}.
\section{Entanglement}
{\label{643089}}
\cite{Ndagano_2017}:~\emph{A witness to quantify high-dimensional
entanglement}.
\section{Quantum supremacy}\label{quantum-supremacy}
Realising a quantum device that computationally outperforms
state-of-the-art classical supercomputers for a certain task that is
proably intractable classicaly has become a key milestone in the field
of quantum simulation and computing. This goal is often referred to as
\emph{quantum supremacy} \cite{preskill2012quantum}. Such \emph{quantum
computational supremacy} is not meant in the sense of the simulation of
a given system being intractable with classical supercomputers, using
the best known algorithms to date. Instead, such \emph{quantum
supremacy} is usually meant to refer to schemes for which the quantum
computational supremacy can be related to a notion of computational
complexity. For such a quantum supremacy scheme to be physically
realisable in principle, in the absence of quantum error correction, it
is crucial to assess that the hardness of the task is robust under
physically realistic errors.
Among the proposals for quantum supremacy architectures that are robust
against physical errors are \cite{Aaronson_2011,Bremner_2016,Bremner_2017,boixo2016characterizing,Gao_2017,bermejovega2017architectures,Morimae_2017}. Only some of these are
physically realistic with present-day technology, being them feasible
with experimental platforms such as linear optics \cite{Aaronson_2011},
superconductive qubits \cite{Bremner_2017, boixo2016characterizing}, ion traps or cold atoms in
optical lattices \cite{bermejovega2017architectures}. The computational task that is
solved in all of these proposals is a \emph{sampling} task, in
particular, the task of sampling from the output probability distriution
of a certain random time-evolution.
\subsection{Stockmeyer's algorithm and anti-concentration
theorems}\label{stockmeyers-algorithm-and-anti-concentration-theorems}
A central ingredient of all existing quantum supremacy proofs is
Stockmeyer's algorithm, that implies a collapse of the polynomial
hierarchy if sampling from the output distribution of the respective
circuits is \#P-hard on average. In order for this hardness argument to
be valid, one crucially requires so-called \emph{anti-concentration
bounds} for the output probability distribution of the respective random
circuits \cite{Lund_2017}.
See \cite{hangleiter2017anticoncentration} for more stuff and results regarding
anti-concentration theorems.
\section{Separability problem}\label{separability-problem}
While the separability problem for pure states is solved entirely via
Schmidt decomposition of state vectors (!), the problem of how to
characterize the set \(\mathcal S\) of separable \emph{mixed}
states, and to decide whether a given mixed state is separable or
entangled, is known to be a very difficult problem in general. In fact,
it has been shown to be NP-hard to solve \cite{gharibian2008strong}.
\href{https://arxiv.org/pdf/1709.00214.pdf}{Santagati et al. (Sept
2017)}: \emph{Integrated two-qubit entangling quantum logic on silicon
photonic processor}. They implement 2 qubits with path encoding of two
photons.
\href{http://arxiv.org/pdf/1708.09784}{Benedetti, Realpe-Gomez,
Perdomo-Ortiz (Aug 2017)}: \emph{Quantum-assisted Helmholtz machines: A
quantum-classical deep learning framework for industrial datasets in
near-term devices}. They introduce the ``quantum-assisted Helmholtz
machine'', a hybrid quantum-classical framework to tackle
high-dimensional real-world machine learning datasets on continuous
variables. The idea is use deep learning to extract a low-dimensional
binary representation of data, suitable for small quantum processors to
assist in the training. They show the concept using the 1644 qubits
D-Wave 2000Q machine training on subsamples MNIST dataset.
\section{Symmetrizing a tensor
product}\label{symmetrizing-a-tensor-product}
Consider some \(2\times2\) unitary matrix \( U \). The
tensor product of \( U \) with itself is given by:
\[
U \otimes U = \begin{pmatrix}
u_{11} u_{11} & u_{11} u_{12} & u_{12} u_{11} & u_{12} u_{12} \\
u_{11} u_{21} & u_{11} u_{22} & u_{12} u_{21} & u_{12} u_{22} \\
u_{21} u_{11} & u_{21} u_{12} & u_{22} u_{11} & u_{22} u_{12} \\
u_{21} u_{21} & u_{21} u_{22} & u_{22} u_{21} & u_{22} u_{22}
\end{pmatrix}.
\] If \( U \) described the evolution of a
single particle over two modes, then \( U \otimes U \) can be thought
of as describing the evolution of two distinguishable particles through
the same evolution. The second and third rows/columns correspond to the
two possible states with one particle in one mode and the other in the
other mode.
What happens however when the particles are \emph{in}distinguishable? It
is possible to readily compute the matrix giving the corresponding
scattering amplitudes between states of indistinguishable particles by
simply symmetrizing (or antisymmetrizing for fermions) the appropriate
rows/columns of \( U \otimes U \). In this case this means to sum
second and third rows columns, adding appropriate normalization factors,
obtaining: \[
\mathcal S (U \otimes U) = \begin{pmatrix}
u_{11} u_{11} & \sqrt2 u_{11} u_{12} & u_{12} u_{12} \\
\sqrt{2} u_{11} u_{21} & u_{11} u_{22} + u_{12} u_{21} & \sqrt{2} u_{12} u_{22} \\
u_{21} u_{21} & \sqrt{2} u_{21} u_{22} & u_{22} u_{22}
\end{pmatrix}.
\] Note that this is the same exact result
that would be obtained by computing the permanents of the
\( 2\times 2 \) submatrices of \( U \), according to the
rule (see for example \cite{Aaronson_2011} for a complete exposition of
this result): \[
\mathcal A(\mathbf{r})
\]
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