Finished(?) tensor section

citations.bib

@@ -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}
 }

main.tex

@@ -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}.
+inspect the intermediate state of a small-scale computation
-This motivates a different definition of strong simulation by some authors.
+\cite{Bravyi2016,Haner2016}. This motivates a different definition of strong
-Namely, recalling Eq.~\eqref{eq:superposition}, note that knowledge of
+simulation by some authors. Namely, recalling Eq.~\eqref{eq:superposition},
-$\alpha_j$ is enough to determine the probability of observing any measurement
+note that knowledge of $\alpha_j$ is enough to determine the probability of
-in the computational basis. However, the converse is not true: because $\Pr[j]
+observing any measurement in the computational basis. However, the converse is
-    = \abs{\alpha_j}^2$, knowledge of $\Pr[j]$ fails to inform about the complex
+not true: because $\Pr[j] = \abs{\alpha_j}^2$, knowledge of $\Pr[j]$ fails to
-phase of $\alpha_j$\FootnoteNow. So, strong simulation may be taken to mean:
+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
+Now, it is possible to represent a quantum state of $n$ qubits by an $n$-rank
-operation in a quantum circuit may be represented by a matrix (a rank-$2$
+tensor, where each index has dimension $2$:
-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,
+\begin{equation}
-the components of which are the entries in the state vector resulting from the
+    \everymath={\displaystyle}
-action of the quantum circuit in the input state.
+    \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