From e3ab46182225d7c88dd9bb8961fa0e2277d81020 Mon Sep 17 00:00:00 2001 From: Meeg Leeto Date: Mon, 17 Jul 2023 11:08:46 +0100 Subject: [PATCH] Added noise section --- citations.bib | 61 +++++++++++++++++++++++++++++++++ main.pdf | Bin 494529 -> 23926 bytes main.tex | 91 +++++++++++++++++++++++++++++++++++++++----------- 3 files changed, 132 insertions(+), 20 deletions(-) diff --git a/citations.bib b/citations.bib index d7b9345..c72a92c 100644 --- a/citations.bib +++ b/citations.bib @@ -483,6 +483,21 @@ journal = {Physical Review Letters} } +@article{McCaskey2018, + doi = {10.1371/journal.pone.0206704}, + url = {https://doi.org/10.1371/journal.pone.0206704}, + year = {2018}, + month = dec, + publisher = {Public Library of Science ({PLoS})}, + volume = {13}, + number = {12}, + pages = {e0206704}, + author = {Alexander McCaskey and Eugene Dumitrescu and Mengsu Chen and Dmitry Lyakh and Travis Humble}, + editor = {Nicholas Chancellor}, + title = {Validating quantum-classical programming models with tensor network simulations}, + journal = {{PLOS} {ONE}} +} + @article{Valiant2002, doi = {10.1137/s0097539700377025}, url = {https://doi.org/10.1137/s0097539700377025}, @@ -634,4 +649,50 @@ archiveprefix = {arXiv}, primaryclass = {quant-ph}, eprint = {2306.14887} +} + +@article{Preskill2018, + doi = {10.22331/q-2018-08-06-79}, + url = {https://doi.org/10.22331/q-2018-08-06-79}, + year = {2018}, + month = aug, + publisher = {Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften}, + volume = {2}, + pages = {79}, + author = {John Preskill}, + title = {Quantum Computing in the {NISQ} era and beyond}, + journal = {Quantum} +} + +@book{Breuer2002, + title = {The Theory of Open Quantum Systems}, + author = {Breuer, H.P. and Petruccione, F.}, + isbn = {9780198520634}, + lccn = {2002075713}, + url = {https://books.google.pt/books?id=0Yx5VzaMYm8C}, + year = {2002}, + publisher = {Oxford University Press} +} + +@article{Bassi2008, + doi = {10.1103/physreva.77.032323}, + url = {https://doi.org/10.1103/physreva.77.032323}, + year = {2008}, + month = mar, + publisher = {American Physical Society ({APS})}, + volume = {77}, + number = {3}, + author = {Angelo Bassi and Dirk-Andr{\'{e}} Deckert}, + title = {Noise gates for decoherent quantum circuits}, + journal = {Physical Review A} +} + +@software{qsim2020, + author = {Quantum AI team and collaborators}, + title = {qsim}, + month = Sep, + year = 2020, + publisher = {Zenodo}, + doi = {10.5281/zenodo.4023103}, + url = {https://doi.org/10.5281/zenodo.4023103} } \ No newline at end of file diff --git a/main.pdf b/main.pdf index 200c8d192ac5ad5e5c6685695b6136667d92e2fa..02a05037e93904994ab08ab42a0bba55963ca2fd 100644 GIT binary patch delta 6389 zcmZ`-cRbbq_wQbNZ&`8exWYBBnQWqrxOVo=-sF<(vfgp6tP~<#dt_(tRCW=vxhSjR zN51vx^Zh=4|D5$YuXE1poag&K9{04NnD-k%2{DN+f`9-XGvJL2t&{;>C8W7!A}A>6 zdQKe9iRU@70o-vFz+y0fm53GCCSnK532*=j98r#6r@x$S&;7j4iSIeFI42(G#0HQf 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In this section, we review some key theoretical results regarding classical simulation of quantum computations, as well as some general techniques. -\subsection{Strong simulation \texorpdfstring{\textit{vs.\!}}{vs.} weak simulation} +\subsection{Strong simulation \texorpdfstring{\textit{vs.}}{vs.}~weak simulation} As defined in section \ref{sec:quantum-computation-fundamentals}, the final output of a quantum computation results from a measurement. Due to the nature of @@ -693,10 +693,10 @@ Ref.~\cite{Haner2017}, this is used to ensure that compute resources are maximally utilized via parallelization. Following a different strategy, in Ref.~\cite{Haner2016}, advance knowledge of the action of common blocks of operations in quantum computing is used to speed up over the simulation of each -gate individually. Gate clustering, different encoding techniques and -cache-related considerations, as well as employment of compiler intrinsics, -also allow for speed improvements -\cite{Fatima2021,Haner2016,Haner2017,Markov2018}. +gate individually. Gate fusion, different encoding techniques and cache-related +considerations, as well as employment of compiler intrinsics, also allow for +speed improvements +\cite{Fatima2021,Smelyanskiy2016,Haner2016,Haner2017,Markov2018}. Otherwise, the problem may be regarded as a classical large-sparse-matrix and vector product, a well-researched problem (see, e.g., @@ -786,14 +786,14 @@ This method is well suited for simulating circuits with a grid-like 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}. +significantly more than $50$ qubits \cite{Chen2018,Li2018,Haner2017}. \subsection{Tensor network simulation} \label{sec:tensor-network-simulation} 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,Villalonga2019,Guo2019}. +tensor contraction \cite{Vidal2003,Markov2008,McCaskey2018,Pednault2020,Villalonga2019,Guo2019}. \FootnoteLater{Terminology regarding tensors is not always consistent across works. For example, the terms \emph{way}, \emph{order}, \emph{degree}, or @@ -822,12 +822,10 @@ is an $(a+c-1,b+d-1)$-type tensor. 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} +\begin{multline} + \psi^{i_1 i_2 \ldots i_n} \quad \leftrightarrow \quad \\ + \ket{\psi} = \sum_{i_1, \ldots, i_n = 0,1} \psi^{i_1 \ldots i_n} \ket{i_1}\cdots\ket{i_n} +\end{multline} 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 @@ -838,7 +836,7 @@ by a tensor contraction. E.g., \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 +where we have, for simplicity of notation, taken 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 @@ -860,11 +858,64 @@ 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, and in conjunction with -other characteristics, like qubit connectivity, this may not be an impeditive -factor when attempting to simulate ``quantum supremacy'' experiments -\cite{Gray2021,Pan2022,Tindall2023}. +time complexity grows exponentially with the depth of the circuit; this +motivated the increase of depth in ``quantum supremacy'' experiments, though +other characteristics, like qubit connectivity or better contraction orderings, +may still allow for tensor-based simulation \cite{Gray2021,Pan2022,Tindall2023}. + +\subsection{Noise simulation} + +In the previous subsections, we have considered simulation of ``perfect'' +quantum circuits, i.e., circuits that do not simulate the presence of noise +(even though they may consider the presence of noise to speed-up the simulation, +such as in \cite{Markov2018}). However, the presence of noise is practically +unavoidable in current experimental settings \cite{Preskill2018}. Therefore, it +may be useful to simulate a noisy quantum circuit. + +\FootnoteLater{For the reader unfamiliar with quantum channels, they may regard + them as a generalization of measurements in the mixed state formulation.} + +Typically, the density matrix formalism (\viz.\ section +\ref{sec:density-matrix}) is used to describe the quantum state resulting from +a quantum circuit affected by noise, by modelling the effects of noise as +\emph{quantum channels}, i.e., completely positive, convex-linear, +non-trace-increasing maps on density matrices\FootnoteNow \cite{Nielsen2012}. +These channels, in turn, reflect physical equational models of noise +\cite{Breuer2002}. + +While, for example, tensor-based simulation (section +\ref{sec:tensor-network-simulation}) naturally extends to density matrix +simulation, it may seem that the overhead of maintaining a density matrix while +employing \Schrodinger\ or Feynman simulation would be impeditive. However, it +turns out that the effect of the usual noise channels can be rephrased as the +statistical average of random, non-unitary gates in a quantum circuit +\cite{Bassi2008}. Therefore, with an overhead due to repeating the simulation +multiple times to collect statistics, it is possible to extend also the +pure-state based methods to simulate noise. + +\section{Simulation software} + +In this section we examine some of the publicly available software for +simulating quantum circuits. It is important to note that we do not claim to be +exhaustive in the number of simulators considered, nor on the benchmarking the +considered solutions. The large and growing number of quantum circuit +simulators available, and the different possible goals for such software (e.g., +small-scale \textit{vs.}~supremacy-scale simulation), would make it impossible +for a complete and even comparison. Thus, we focused on offline, small-scale +simulators, such as those that a researcher might use to validate their +algorithms in toy-settings, and that are recognized in the industry as common. + +\subsection{Methodology} + +\FootnoteLater{As of July $16$, $2023$, the Github repositories for the Cirq, + the Microsoft Quantum, and the Qiskit simulators have, respectively, + $3,\!823$, $3,\!737$, and $3,\!678$, ``stars''.} + +We began with the $3$ most popular (at the time of writing) quantum circuit +simulators under the Github ``quantum-computing'' tag\FootnoteNow. Then, we +considered every quantum circuit simulator for which Qiskit provided a backend +interface. For providers of quantum hardware interfacing with Qiskit, we +considered the provider's simulation solution, if it existed. \appendix