Work on the Feynman simulation section
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@ -71,7 +71,7 @@
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doi = {10.48550/ARXIV.QUANT-PH/9511026},
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archiveprefix = {arXiv},
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primaryclass = {quant-ph},
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note = {\url{https://arxiv.org/abs/quant-ph/9511026}},
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note = {Unpublished},
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keywords = {Quantum Physics (quant-ph), FOS: Physical sciences, FOS: Physical sciences},
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copyright = {Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004}
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}
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@ -119,7 +119,7 @@
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eprint = {1807.10749},
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archiveprefix = {arXiv},
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primaryclass = {quant-ph},
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note = {\url{https://arxiv.org/abs/1807.10749}}
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note = {Unpublished}
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}
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@article{Bravyi2016,
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@ -217,7 +217,7 @@
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eprint = {1601.07195},
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archiveprefix = {arXiv},
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primaryclass = {quant-ph},
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note = {\url{https://arxiv.org/abs/1601.07195}},
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note = {Unpublished},
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keywords = {Quantum Physics (quant-ph), Distributed, Parallel, and Cluster Computing (cs.DC), FOS: Physical sciences, FOS: Physical sciences, FOS: Computer and information sciences, FOS: Computer and information sciences}
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}
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@ -248,12 +248,47 @@
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}
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@inproceedings{Haner2016,
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doi = {10.1109/sc.2016.73},
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url = {https://doi.org/10.1109/sc.2016.73},
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year = {2016},
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month = nov,
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doi = {10.1109/sc.2016.73},
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url = {https://doi.org/10.1109/sc.2016.73},
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year = {2016},
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month = nov,
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publisher = {{IEEE}},
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author = {Thomas Haner and Damian S. Steiger and Mikhail Smelyanskiy and Matthias Troyer},
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title = {High Performance Emulation of Quantum Circuits},
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author = {Thomas Haner and Damian S. Steiger and Mikhail Smelyanskiy and Matthias Troyer},
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title = {High Performance Emulation of Quantum Circuits},
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booktitle = {{SC}16: International Conference for High Performance Computing, Networking, Storage and Analysis}
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}
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@article{Xiao2023,
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doi = {10.1145/3604606},
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url = {https://doi.org/10.1145/3604606},
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year = {2023},
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month = jun,
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publisher = {Association for Computing Machinery ({ACM})},
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author = {Guoqing Xiao and Chuanghui Yin and Tao Zhou and Xueqi Li and Yuedan Chen and Kenli Li},
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title = {A Survey of Accelerating Parallel Sparse Linear Algebra},
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journal = {{ACM} Computing Surveys}
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}
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@article{Bernstein1997,
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doi = {10.1137/s0097539796300921},
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url = {https://doi.org/10.1137/s0097539796300921},
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year = {1997},
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month = oct,
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publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
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volume = {26},
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number = {5},
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pages = {1411--1473},
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author = {Ethan Bernstein and Umesh Vazirani},
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title = {Quantum Complexity Theory},
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journal = {{SIAM} Journal on Computing}
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}
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@unpublished{Boixo2017,
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author = {Sergio Boixo and Sergei V. Isakov and Vadim N. Smelyanskiy and Hartmut Neven},
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note = {Unpublished},
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title = {Simulation of low-depth quantum circuits as complex undirected graphical models},
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year = {2017},
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archiveprefix = {arXiv},
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primaryclass = {quant-ph},
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eprint = {1712.05384}
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}
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70
main.tex
70
main.tex
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@ -17,6 +17,7 @@
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\usepackage{qcircuit}
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\usepackage{calc}
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\usepackage{marginnote}
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\usepackage{mathtools}
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\newtheorem{theorem}{Theorem}
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\newtheorem{lemma}{Lemma}
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@ -254,6 +255,7 @@ correctly described by a superposition. To see this, suppose a single qubit,
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and a Hadamard gate:
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%
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\begin{equation}
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\label{eq:hadamard}
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\def\arraycolsep{8pt}
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\begin{array}{cc}
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\def\arraycolsep{4pt}
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@ -391,8 +393,9 @@ $\rho$, then subsystem $A$ is described by density operator
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\rho_A = \Tr_B(\rho) = \sum_j (I_A \otimes \bra{j}_B) \rho (I_A \otimes \ket{j}_B)
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\end{equation}
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%
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where $\Tr_B$ is the newly defined \emph{partial trace} operation, and we take
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$\{\ket{j}\}_j$ to be a basis over the state space of $B$.
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where $\Tr_B$ is the newly defined \emph{partial trace} operation, $I_A$ is the
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identity in the state space of $A$, and we take $\{\ket{j}\}_j$ to be a basis
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over the state space of $B$.
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\subsubsection{Quantum circuits}
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@ -643,7 +646,7 @@ In Ref.~\cite{VandenNest2010}, this computational gap is discussed and
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eliminated, by giving a superclass of Clifford circuits, ``$HT$ circuits'', that
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is equivalent to classical computation and can be weakly simulated.
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\subsubsection{Schr\"{o}dinger Simulation}
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\subsubsection{Schr\"{o}dinger simulation}
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\Schrodinger\ simulation refers to the straightforward approach of maintaining
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the global state-vector, updating it as new unitary operations are encountered
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@ -652,13 +655,72 @@ version of strong simulation (definition \ref{def:strong-simulation-wavefn}).
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This method also requires, by definition, that $2^n$ complex values are
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maintained for a state of $n$ qubits, such that it cannot physically scale
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beyond a certain number of qubits (about $45$-$50$ qubits, corresponding to a
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petabyte or more of memory, if each amplitude is represented within $8$ bytes).
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petabyte or more of memory, if each amplitude is represented within $8$ bytes;
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barring adaptative models admitting error, such as in Ref.~\cite{DeRaedt2019}).
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Despite this constraint, a significant amount of research has been devoted to
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\Schrodinger\ simulation in the memory-tractable regime ($n \lesssim 50$),
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specifically in pushing this limit and speeding up the running time
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\cite{Smelyanskiy2016,DeRaedt2019,Jones2019,Haner2017,Haner2016,Niwa2002,DeRaedt2006}.
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A key observation is that, if each gate is considered at a time, the
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matrix-vector products being calculated are of very sparse and strongly
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structured matrices. Namely, the gates are \emph{local}, in the sense that they
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involve non-trivially at most a small number $k$ of qubits. For example, in
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Ref.~\cite{Haner2017}, this is used both to ensure that compute resources are
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maximally utilized, and to distribute the computation to some extent. Following
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a different strategy, in Ref.~\cite{Haner2016}, advance knowledge of the action
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of common blocks of operations in quantum computing is used to speed up over the
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simulation of each gate individually.
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Otherwise, the problem may be regarded as a classical
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large-sparse-matrix and vector product, a well-researched problem (see, e.g.,
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Ref.~\cite{Xiao2023}).
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\subsubsection{Feynman simulation}
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\label{sec:feynman-simulation}
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The Feynman simulation method \cite{Boixo2017,Bernstein1997} trades the $2^n$
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memory requirement by an exponential time computation, but in linear space.
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Being also a form of the wave-function version of strong simulation (definition
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\ref{def:strong-simulation-wavefn}), the Feynman simulation method follows from
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noticing the following:
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%
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\begin{equation}
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\def\ProdStrut{\rule[-0.3\baselineskip]{0pt}{7pt}}
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\everymath={\displaystyle}
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\begin{array}{l}
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\bra{x_b} U_L U_{L-1} U_{L-2} \cdots U_2 U_1 \ket{0_b} = {} \\[1em]
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{} = \bra{x_b} U_L (\Sigma_{j=0}^{2^n-1} \dyad{j_b}) U_{L-1} (\Sigma_{j'=0}^{2^n-1} \dyad{j'_b}) \\[.5em]
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\hfill U_{L-2} \cdots U_2 (\Sigma_{j''=0}^{2^n-1} \dyad{j''_b}) U_1 \ket{0_b} = {} \\[1em]
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\hfill {} = \sum_{\mathclap{\{b_{(t)}\} \in \{0,1,\ldots,2^n-1\}^{L-1}}} \mkern 65mu \prod_{\ProdStrut t=0}^{L-1} \bra{b_{(t+1)}} U_{t} \ket{b_{(t)}}
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\end{array}
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\end{equation}
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%
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since $\{\ket{j_b}\}_{j=0,\ldots,2^n-1}$ form an orthonormal basis of the state
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vector space, and letting $\ket*{b_{(L)}} \equiv \ket{x_b}$. Now, take each of
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the $U_j$ to be the (local, sparse) unitary corresponding to a quantum gate in
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a quantum circuit, such that $L$ is the depth of the quantum circuit. This
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approach may be interpreted as a discrete version of the Feynman path integral
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formulation, where, simply put, every possible ``computation path'' is
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considered separately, in order to determine the resulting constructive or
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destructive interference between the paths; each path requires a linear amount
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of memory to compute, but there are exponentially many paths to consider. This
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is illustrated in figure~\ref{fig:interference}.
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\begin{figure}
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\hskip-.5cm\includegraphics[width=0.75\linewidth]{assets/interference.pdf}
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\caption{The action of two Hadamard gates (eq.~\ref{eq:hadamard}) acting on
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a single qubit initialized to $\ket{0}$, as a trivial example of
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interference. Each node's tone reflects the absolute value of the
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associated amplitude: darker corresponds to greater amplitude. The node
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with a negative amplitude is marked with a $(-)$. In a Feynman path
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integral interpretation (see section \ref{sec:feynman-simulation}),
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each of the drawn paths is considered separately, and then summed,
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resulting in the destructive interference of the $\ket{1}$ state.}
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\label{fig:interference}
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\end{figure}
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\appendix
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\section{Delayed measurement}
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