Began work on the tensor section

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Meeg Leeto 2023-07-13 18:03:52 +01:00
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3 changed files with 349 additions and 133 deletions

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@ -90,8 +90,6 @@
journal = {Nature}
}
@article{Terhal2018,
doi = {10.1038/s41567-018-0131-y},
url = {https://doi.org/10.1038/s41567-018-0131-y},
@ -361,3 +359,167 @@
title = {Quantum information with Rydberg atoms},
journal = {Reviews of Modern Physics}
}
@unpublished{Li2018,
author = {Riling Li and Bujiao Wu and Mingsheng Ying and Xiaoming Sun and Guangwen Yang},
note = {Unpublished},
title = {Quantum Supremacy Circuit Simulation on Sunway TaihuLight},
year = {2018},
archiveprefix = {arXiv},
primaryclass = {quant-ph},
eprint = {1804.04797}
}
@article{Vidal2003,
doi = {10.1103/physrevlett.91.147902},
url = {https://doi.org/10.1103/physrevlett.91.147902},
year = {2003},
month = oct,
publisher = {American Physical Society ({APS})},
volume = {91},
number = {14},
author = {Guifr{\'{e}} Vidal},
title = {Efficient Classical Simulation of Slightly Entangled Quantum Computations},
journal = {Physical Review Letters}
}
@article{Markov2008,
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month = jan,
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volume = {38},
number = {3},
pages = {963--981},
author = {Igor L. Markov and Yaoyun Shi},
title = {Simulating Quantum Computation by Contracting Tensor Networks},
journal = {{SIAM} Journal on Computing}
}
@unpublished{Pednault2020,
author = {Edwin Pednault and John A. Gunnels and Giacomo Nannicini and Lior Horesh and Thomas Magerlein and Edgar Solomonik and Erik W. Draeger and Eric T. Holland and Robert Wisnieff},
note = {Unpublished},
title = {Pareto-Efficient Quantum Circuit Simulation Using Tensor Contraction Deferral},
year = {2020},
archiveprefix = {arXiv},
primaryclass = {quant-ph},
eprint = {1710.05867v4}
}
@book{Joshi1995,
title = {Matrices and Tensors in Physics},
author = {Joshi, A.W.},
isbn = {9788122405637},
lccn = {94021798},
url = {https://books.google.co.ck/books?id=FesDylvUy00C},
year = {1995},
publisher = {Wiley}
}
@article{DeLathauwer2000,
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publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
volume = {21},
number = {4},
pages = {1253--1278},
author = {Lieven De Lathauwer and Bart De Moor and Joos Vandewalle},
title = {A Multilinear Singular Value Decomposition},
journal = {{SIAM} Journal on Matrix Analysis and Applications}
}
@incollection{Vasilescu2002,
doi = {10.1007/3-540-47969-4_30},
url = {https://doi.org/10.1007/3-540-47969-4_30},
year = {2002},
publisher = {Springer Berlin Heidelberg},
pages = {447--460},
author = {M. Alex O. Vasilescu and Demetri Terzopoulos},
title = {Multilinear Analysis of Image Ensembles: {TensorFaces}},
booktitle = {Computer Vision {\textemdash} {ECCV} 2002}
}
@article{Kolda2009,
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volume = {51},
number = {3},
pages = {455--500},
author = {Tamara G. Kolda and Brett W. Bader},
title = {Tensor Decompositions and Applications},
journal = {{SIAM} Review}
}
@article{Villalonga2019,
doi = {10.1038/s41534-019-0196-1},
url = {https://doi.org/10.1038/s41534-019-0196-1},
year = {2019},
month = oct,
publisher = {Springer Science and Business Media {LLC}},
volume = {5},
number = {1},
author = {Benjamin Villalonga and Sergio Boixo and Bron Nelson and Christopher Henze and Eleanor Rieffel and Rupak Biswas and Salvatore Mandr{\`{a}}},
title = {A flexible high-performance simulator for verifying and benchmarking quantum circuits implemented on real hardware},
journal = {npj Quantum Information}
}
@article{Guo2019,
doi = {10.1103/physrevlett.123.190501},
url = {https://doi.org/10.1103/physrevlett.123.190501},
year = {2019},
month = nov,
publisher = {American Physical Society ({APS})},
volume = {123},
number = {19},
author = {Chu Guo and Yong Liu and Min Xiong and Shichuan Xue and Xiang Fu and Anqi Huang and Xiaogang Qiang and Ping Xu and Junhua Liu and Shenggen Zheng and He-Liang Huang and Mingtang Deng and Dario Poletti and Wan-Su Bao and Junjie Wu},
title = {General-Purpose Quantum Circuit Simulator with Projected Entangled-Pair States and the Quantum Supremacy Frontier},
journal = {Physical Review Letters}
}
@article{Valiant2002,
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}
@article{Jozsa2008,
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}
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}

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@ -54,6 +54,7 @@
}
\ExplSyntaxOff
\def\viz.{\textit{viz}.}
\NewDocumentCommand{\Naturals}{}{\mathbb{N}}
\let\NN\Naturals
\NewDocumentCommand{\Integers}{}{\mathbb{Z}}
@ -404,7 +405,7 @@ transformations, and by extension quantum algorithms: quantum circuits.
\FootnoteLater{Rigorously, this depends on how the unitary is ``given''. Here,
consider that a unitary is given by specifying its action over each element
$\ket{j}$ of a set spanning the state space. By linearity (cf.\ the
$\ket{j}$ of a set spanning the state space. By linearity (\viz.\ the
\Postulate{evolution} postulate), this determines the action of the unitary
over any vector in the state space.}
@ -477,7 +478,7 @@ depth scale at most polynomially with input size.
associated to a qubit, i.e., $\Hilbert_2$. Multiple wires,
vertically aligned, represent the collective Hilbert space, given
by the tensor product of the individual $\Hilbert_2$ spaces
(cf.\ the \Postulate{composite systems} postulate).}
(\viz.\ the \Postulate{composite systems} postulate).}
\\[.8cm]
$\Qcircuit @C=1em { \lstick{{}_A} & \qw{/} & \qw }$ & Register &
\Description{Represents multiple wires, i.e., a subspace of dimension
@ -607,7 +608,7 @@ phase of $\alpha_j$\FootnoteNow. So, strong simulation may be taken to mean:
Note that, in all of the definitions above, we took the quantum circuits to be
described by a unitary operator, which may not be trivially true if measurements
are performed half-way in the circuit (cf.\ section~\ref{sec:density-matrix}),
are performed half-way in the circuit (\viz.\ section~\ref{sec:density-matrix}),
or if classical post-processing is employed. However, a well-known result,
which we review in appendix~\ref{appendix:delayed-measurement}, allows us to
defer all measurements to the end of the circuit, such that the whole of the
@ -617,7 +618,7 @@ computation is carried out unitarily.
Recall that Clifford gates are gates in the Clifford set, i.e., any quantum
circuit that can be written in terms of phase, Hadamard, and Controlled-Not
gates (cf.\ Table~\ref{table:basic-operations}). Then, the Gottesman-Knill
gates (\viz.\ Table~\ref{table:basic-operations}). Then, the Gottesman-Knill
theorem states the following:
\begin{theorem}[Gottesman-Knill \cite{Gottesman1999}]
@ -640,7 +641,11 @@ regime: not only by augmentation of the gate set -- the Clifford+$T$ gate set
is already universal -- but also by choice of the input state. Indeed, there
exist ``magic states'', such that a supply of these (pre-prepared) quantum
states and Clifford operations are enough to perform universal quantum
computation \cite{Bravyi2005}.
computation \cite{Bravyi2005}. Nonetheless, if Clifford gates dominate a
non-Clifford circuit, this structure may be exploited to speed-up computation,
by treating the simulation as a tensor network where the calculation of certain
tensors can be sped-up \cite{Bravyi2016} (\viz.\ section
\ref{sec:tensor-network-simulation}).
This move from Clifford-based computation to quantum universality entails a
``jump'', since Clifford circuits are not as powerful as classical circuits.
@ -648,6 +653,10 @@ In Ref.~\cite{VandenNest2010}, this computational gap is discussed and
eliminated, by giving a superclass of Clifford circuits, ``$HT$ circuits'', that
is equivalent to classical computation and can be weakly simulated.
Finally, note also that certain classes of efficiently simulatable circuits not
discussed here, such as \emph{matchgate} circuits, may relate non-trivially
with Clifford circuits \cite{Valiant2002,Jozsa2008,Brod2016}.
\subsubsection{Schr\"{o}dinger simulation}
\label{sec:schrodinger-simulation}
@ -719,6 +728,14 @@ paths; each path requires a linear amount of memory to compute, but there are
exponentially many paths to consider, which interfere among themselves. This is
illustrated in figure~\ref{fig:interference}.
A further advantage of this method is that some of the computational paths may
be simply ignored, at the expense of simulation fidelity. The computational
advantage of not having to compute these paths is sufficiently expressive to
allow for the simulation of real-world implementations thought to be impossible
to simulate (matching their fidelity) \cite{Markov2018}. These considerations
also apply to the \Schrodinger-Feynman method (section
\ref{sec:schrodinger-feynman-simulation}).
\begin{figure}
\hskip-.5cm\includegraphics{assets/interference.pdf}
\caption{The action of two Hadamard gates (eq.~\ref{eq:hadamard}) acting on
@ -733,6 +750,7 @@ illustrated in figure~\ref{fig:interference}.
\end{figure}
\subsubsection{Schr\"{o}dinger-Feynman simulation}
\label{sec:schrodinger-feynman-simulation}
It is possible to establish an intermediate scheme between \Schrodinger\
simulation (section \ref{sec:schrodinger-simulation}) and Feynman simulation
@ -757,12 +775,48 @@ $\BigO(2^{n-k} \log d)$ space \cite{Chen2017}.
This method is well suited for simulating circuits with a grid-like
connectivity graph between circuits, as is common in practical implementations
\cite{Arute2019,Kim2023,Saffman2010}, and so is used in multiple works pushing
the boundary of quantum advantage \cite{Chen2018,Smelyanskiy2016,Haner2017}.
the boundary of quantum advantage, allowing for simulation of circuits with
significantly more than $50$ qubits \cite{Chen2018,Li2018,Smelyanskiy2016,Haner2017}.
\subsubsection{Tensor network simulation}
\label{sec:tensor-network-simulation}
TODO
Closely related to the Feynman simulation technique, but generalizing the idea,
tensor-network based simulation regards quantum circuit simulation as a form of
tensor contraction \cite{Vidal2003,Markov2008,Pednault2020}.
\FootnoteLater{Terminology regarding tensors is not always consistent across
works. For example, the terms \emph{way}, \emph{order}, \emph{degree}, or
\emph{dimension} may be used instead of \emph{rank}
\cite{Joshi1995,DeLathauwer2000,Vasilescu2002,Kolda2009}.}
Recall that a tensor is an object generalizing matrices: a tensor has upper and
lower (contra- and co-variant) indices; its number of indices determines its
\emph{rank}, while the values that each index may take determine the
\emph{dimension} of the index\FootnoteNow. A given choice of indices yields a
\emph{component} of the tensor, e.g., $T^{ijk}_{lm}$ denotes the $(i,j,k;l,m)$th
component of the tensor $T$. A tensor $T$ is of type $(a,b)$ if it has $a$
contravariant indices and $b$ covariant indices. Two tensors may be
\emph{contracted} by summing over, respectively, a contravariant and
contravariant index of the same dimension over the product of the two tensors.
I.e., let $M$ and $N$ be two tensors of types $(a,b)$ and $(c,d)$, with
dimension $d$ on each index. The tensor whose components are given by
%
$$
\sum_{i_k=1}^d M^{i_1 \ldots i_{k-1} i_k i_{k+1} \ldots i_a}_{j_1 \ldots j_b}
N^{k_1 \ldots k_c}_{l_1 \ldots l_{k-1} i_k l_{k+1} \cdots l_d}
$$
%
is an $(a+c-1,b+d-1)$-type tensor.
Now, clearly, since a state vector is a vector (a rank-$1$ tensor), and each
operation in a quantum circuit may be represented by a matrix (a rank-$2$
tensor), it is possible to represent a quantum circuit acting on an input state
as a sequence of tensor products and contractions. The result will be a vector,
the components of which are the entries in the state vector resulting from the
action of the quantum circuit in the input state.
A citar \cite{Villalonga2019,Guo2019}
\appendix