meeg_leeto
/
aced-review
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Finished(?) tensor section

This commit is contained in:
Meeg Leeto 2023-07-14 11:50:08 +01:00
parent 56be9d560a
commit 6b40dfb64b
3 changed files with 156 additions and 16 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

102
citations.bib
View File

@ -522,4 +522,106 @@
author = {Daniel J. Brod},
title = {Efficient classical simulation of matchgate circuits with generalized inputs and measurements},
journal = {Physical Review A}
}
@inproceedings{Lykov2022,
doi = {10.1109/qce53715.2022.00081},
url = {https://doi.org/10.1109/qce53715.2022.00081},
year = {2022},
month = sep,
publisher = {{IEEE}},
author = {Danylo Lykov and Roman Schutski and Alexey Galda and Valeri Vinokur and Yuri Alexeev},
title = {Tensor Network Quantum Simulator With Step-Dependent Parallelization},
booktitle = {2022 {IEEE} International Conference on Quantum Computing and Engineering ({QCE})}
}
@article{Gray2021,
doi = {10.22331/q-2021-03-15-410},
url = {https://doi.org/10.22331/q-2021-03-15-410},
year = {2021},
month = mar,
publisher = {Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
volume = {5},
pages = {410},
author = {Johnnie Gray and Stefanos Kourtis},
title = {Hyper-optimized tensor network contraction},
journal = {Quantum}
}
@article{Fried2018,
doi = {10.1371/journal.pone.0208510},
url = {https://doi.org/10.1371/journal.pone.0208510},
year = {2018},
month = dec,
publisher = {Public Library of Science ({PLoS})},
volume = {13},
number = {12},
pages = {e0208510},
author = {E. Schuyler Fried and Nicolas P. D. Sawaya and Yudong Cao and Ian D. Kivlichan and Jhonathan Romero and Al{\'{a}}n Aspuru-Guzik},
editor = {Itay Hen},
title = {{qTorch}: The quantum tensor contraction handler},
journal = {{PLOS} {ONE}}
}
@article{Schindler2020,
doi = {10.1088/2632-2153/ab94c5},
url = {https://doi.org/10.1088/2632-2153/ab94c5},
year = {2020},
month = jul,
publisher = {{IOP} Publishing},
volume = {1},
number = {3},
pages = {035001},
author = {Frank Schindler and Adam S Jermyn},
title = {Algorithms for tensor network contraction ordering},
journal = {Machine Learning: Science and Technology}
}
@inproceedings{Ibrahim2022,
author = {Ibrahim, Cameron and Lykov, Danylo and He, Zichang and Alexeev, Yuri and Safro, Ilya},
booktitle = {2022 IEEE High Performance Extreme Computing Conference (HPEC)},
title = {Constructing Optimal Contraction Trees for Tensor Network Quantum Circuit Simulation},
year = {2022},
volume = {},
number = {},
pages = {1-8},
doi = {10.1109/HPEC55821.2022.9926353}
}
@article{Pfeifer2014,
doi = {10.1103/physreve.90.033315},
url = {https://doi.org/10.1103/physreve.90.033315},
year = {2014},
month = sep,
publisher = {American Physical Society ({APS})},
volume = {90},
number = {3},
author = {Robert N. C. Pfeifer and Jutho Haegeman and Frank Verstraete},
title = {Faster identification of optimal contraction sequences for tensor networks},
journal = {Physical Review E}
}
@article{Liang2021,
author = {Liang, Ling and Xu, Jianyu and Deng, Lei and Yan, Mingyu and Hu, Xing and Zhang, Zheng and Li, Guoqi and Xie, Yuan},
journal = {IEEE Journal of Selected Topics in Signal Processing},
title = {Fast Search of the Optimal Contraction Sequence in Tensor Networks},
year = {2021},
volume = {15},
number = {3},
pages = {574-586},
doi = {10.1109/JSTSP.2021.3051231}
}
@article{Pan2022,
doi = {10.1103/physrevlett.128.030501},
url = {https://doi.org/10.1103/physrevlett.128.030501},
year = {2022},
month = jan,
publisher = {American Physical Society ({APS})},
volume = {128},
number = {3},
author = {Feng Pan and Pan Zhang},
title = {Simulation of Quantum Circuits Using the Big-Batch Tensor Network Method},
journal = {Physical Review Letters}
}

BIN
main.pdf
View File

Binary file not shown.

70
main.tex
View File

@ -592,13 +592,14 @@ notions more formally:
result from acting with the Hadamard gate is completely different.}
In practice, one may wish to simulate the circuit only up to some point, or to
inspect the intermediate state of a small-scale computation \cite{Bravyi2016}.
This motivates a different definition of strong simulation by some authors.
Namely, recalling Eq.~\eqref{eq:superposition}, note that knowledge of
$\alpha_j$ is enough to determine the probability of observing any measurement
in the computational basis. However, the converse is not true: because $\Pr[j]
= \abs{\alpha_j}^2$, knowledge of $\Pr[j]$ fails to inform about the complex
phase of $\alpha_j$\FootnoteNow. So, strong simulation may be taken to mean:
inspect the intermediate state of a small-scale computation
\cite{Bravyi2016,Haner2016}. This motivates a different definition of strong
simulation by some authors. Namely, recalling Eq.~\eqref{eq:superposition},
note that knowledge of $\alpha_j$ is enough to determine the probability of
observing any measurement in the computational basis. However, the converse is
not true: because $\Pr[j] = \abs{\alpha_j}^2$, knowledge of $\Pr[j]$ fails to
inform about the complex phase of $\alpha_j$\FootnoteNow. So, strong simulation
may be taken to mean:
\begin{definition}[Strong simulation, wave-function version \cite{Chen2017}]
\label{def:strong-simulation-wavefn}
@ -773,7 +774,7 @@ circuit of depth $d$, one requires $\BigO(n 2^{n-k} \cdot (2d)^{k+1})$ time, and
$\BigO(2^{n-k} \log d)$ 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
connectivity graph between qubits, as is common in practical implementations
\cite{Arute2019,Kim2023,Saffman2010}, and so is used in multiple works pushing
the boundary of quantum advantage, allowing for simulation of circuits with
significantly more than $50$ qubits \cite{Chen2018,Li2018,Smelyanskiy2016,Haner2017}.
@ -783,7 +784,7 @@ significantly more than $50$ qubits \cite{Chen2018,Li2018,Smelyanskiy2016,Haner2
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}.
tensor contraction \cite{Vidal2003,Markov2008,Pednault2020,Villalonga2019,Guo2019}.
\FootnoteLater{Terminology regarding tensors is not always consistent across
works. For example, the terms \emph{way}, \emph{order}, \emph{degree}, or
@ -809,14 +810,51 @@ $$
%
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.
Now, it is possible to represent a quantum state of $n$ qubits by an $n$-rank
tensor, where each index has dimension $2$:
%
\begin{equation}
\everymath={\displaystyle}
\begin{array}{r@{\hskip .5cm}c@{\hskip .5cm}l}
\psi^{i_1 i_2 \ldots i_n} & \leftrightarrow & \ket{\psi} = \sum_{i_1, \ldots, i_n = 0,1} \psi^{i_1 \ldots i_n} \ket{i_1}\cdots\ket{i_n}
\end{array}
\end{equation}
A citar \cite{Villalonga2019,Guo2019}
It follows that likewise any quantum operation involving $k$ qubits is given by
a $(k,k)$-type tensor, and the quantum state vector after the operation is given
by a tensor contraction. E.g.,
%
\begin{multline}
\ket{\phi} = G\ket{\psi} \quad\leftrightarrow\quad \\
\phi^{i'_a \ldots i'_k i_{k+1} \ldots i_n} = \sum_{\mathclap{i_a \ldots i_k = 0,1}} G^{i'_a \ldots i'_k}_{i_a \ldots i_k} \psi^{i_a \ldots i_k i_{k+1} \ldots i_n}
\end{multline}
%
where we have above, for simplicity of notation, the action of the gate $G$ to
be on the first $k$ qubits.
Therefore, it is possible to represent a quantum circuit acting on an input
state as a sequence of tensor products and contractions. The network of
contractions of the multiple tensors is designated by \emph{tensor network}.
The result of the contraction 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, making this a wave-function strong simulation method
(definition \ref{def:strong-simulation-wavefn}). The method can also be extended
to mixed states \cite{Markov2008}.
Depending on how the tensor network is contracted, the memory and time necessary
to respectively keep track of the tensor and contract it may vary dramatically
\cite{Lykov2022}. Thus, optimizing the procedure of finding the optimal
contraction ordering, known to be a computationally hard task in its generality,
is a central research topic
\cite{Pfeifer2014,Gray2021,Pednault2020,Fried2018,Schindler2020,Ibrahim2022,Liang2021}.
Nonetheless, one may upper bound the time complexity of the contraction as
$T^{\BigO(1)}\exp[\BigO(qD)]$, where $T$ is the total number of gates in the
circuit, $q$ is the maximum number of adjacent qubits involved in a single
operation, and $D$ is the depth of the circuit \cite{Markov2008}. Note how the
time complexity grows exponentially with the depth of the circuit; since current
real-world implementations are depth-limited by noise, this may not be an
impeditive factor when attempting to simulate ``quantum supremacy'' experiments
\cite{Gray2021,Pan2022}.
\appendix