Quantum Simulation Chapter

Signed-off-by: Robert C. Helling <helling@atdotde.de>

simulation.tex

+\section{The main idea: simulating a quantum system with another one} \label{sec:main_idea}
+
+Quantum simulation is the practice of simulating, or imitating, a quantum system $A$ with another quantum system $B$ that is somewhat similar to $A$. This idea was already proposed by R. Feynman in his famous 1981 paper~\cite{feynman1982simulating} that opened the way to quantum computers.
+
+As a rough example, suppose that you want to understand how the elastic head-on scattering between a proton and a neutron  (system $A$) works. You could either sit in the library and open particle physics books (but you don't feel like it), or build a particle collider (but it's expensive), or \emph{simulate} the pair of particles with, say, a pair of billiard balls (system $B$). In the latter option, you would involve a friend, go to a billiard saloon, let the billiard balls collide frontally and see what happens. This would give you extremely rough knowledge on what a proton-neutron collision looks like. This is an example of simulating a system $A$ with another system $B$. Since the behavior of billiard balls is best described with classical physics, we can call this a classical simulation of the quantum system $A$. Otherwise, if your money is not enough to build a nucleon collider but suffices for an atom collider, you can simulate the proton and the neutron with small atoms instead of billiard balls. This would be an example of a \emph{quantum simulation}, as you are simulating a quantum system with another quantum system.
+
+While keeping the same spirit as in the example above, let us consider an example of the kind of quantum simulations that experimental physicists actually do in the lab. Suppose we are interested in the dynamics of a system $A$ of electrons in a 2D crystal. As a first step towards analyzing this system, we assume that it is approximately described by the (unsolved) 2D Hubbard model, i.e., by the Hamiltonian
+\begin{equation} \label{eq:hubbard}
+	\hat{H}_{A} = - J \sum_{ \langle i,j \rangle , \sigma} \hat{c}^{\dagger}_{i,\sigma} \hat{c}_{j,\sigma} + U \sum_i \hat{n}_{i, \uparrow}\hat{n}_{i, \downarrow} \; .
+\end{equation}
+This Hamiltonian acts on the Fock space for electrons in a lattice, the indices $i$ and $j$ refer to the sites of the lattice, $\langle i,j \rangle$ refers to nearest neighbors, $\hat{c}^{(\dagger)}_{i,\sigma}$ is the annihilation (creation) operator for an electron at site $i$ and with spin $\sigma \in \{\uparrow, \downarrow \}$, $\hat{n}_{i, \sigma} = \hat{c}^{\dagger}_{i, \sigma} \hat{c}_{i, \sigma}$ is the number of electrons with spin $\sigma$ at site $i$, $J$ is the `hopping' parameter, and $U$ is the `on-site interaction' parameter. As a second step, one can engineer and build a 2D `simulating' system $B$ whose Hamiltonian $\hat{H}_{B}$ \emph{has the same form of} $\hat{H}_{A}$, or is as similar as possible to it. This simulating system should be easier to measure than the initial electron system, to get a practical advantage. As we will see, a possible choice is a simulating system of (cold) spin-1/2 \emph{atoms} in an `optical lattice'. The atoms of the simulating system $B$ then represent the electrons of the original system $A$. By appropriately tuning the experimental set-up (e.g., the lattice spacing), one can explore different values for the parameters $J$ and $U$ appearing in $\hat{H}_{A}$. One can finally run experiments, e.g., state preparation and time evolution, of the simulating system, and take measurements on it, e.g., snapshots of the positions of the atoms. One thus gains experimental information on the system $B$ of atoms in an optical lattice, hence information on $\hat{H}_{B}$, hence information on $\hat{H}_{A}$, hence information on the system $A$ of electrons in a crystal.
+
+\section{Analog and digital quantum simulation}
+
+The simulations discussed so far are called `analog' simulations. This refers to the fact that one simulates system $A$ with an \emph{analogous} system $B$. More broadly, in science and technology the word analog refers to the act of reproducing some event (for example, the flow of time; or the performance of a singer) in a direct way with a device (such as an analog clock/hourglass; or a tape of an audiocassette/a vinyl disk + another device reading it), also named analog. This is opposed to `digital' simulations, where the event gets described in terms of numbers (hence the word `digital') corresponding to a discrete bunch of sampled instants, and reproduced with a digital device. For example, this is the case for a digital clock, or for a compact disk containing micrometer-long `pits and lands' encoding bits, which in turn encode the sound originally produced by the singer.
+
+In the study of quantum systems, not only analog, but also digital simulations are object of research interest. For example, one can simulate on a classical computer (CC) the proton-neutron collision mentioned above by numerically solving the time-dependent Schr\"odinger equation for the two-particle wavefunction $\psi_t(x_p, x_n)$, and displaying the time evolution of the system with a video of the time-dependent graph of $\psi_t$ (or with a video of two little balls following the trajectory given by the expectation values $x_p(t) = \langle \psi_t| \hat{X_p} | \psi_t \rangle$ and $x_n(t) = \langle \psi_t| \hat{X_n} | \psi_t \rangle$, where $\hat{X}_{p/n}$ is the position operator of the proton/neutron). What's more recent and interesting to us, there are also digital simulations on \emph{quantum} computers (QC) made up of qubits. For example, consider an arbitrary quantum system $A$. Again, suppose that you are interested in the time evolution of $A$. Let $\mathcal{H}_A$ be the Hilbert space for $A$, and let $\hat{U}_A: \mathcal{H}_A \rightarrow \mathcal{H}_A$ be the unitary time evolution operator relative to a fixed time interval. One may be interested in studying the time-evolved version $\hat{U}_A| \psi_A \rangle$ of arbitrary initial state vectors $| \psi_A \rangle \in \mathcal{H}_A$. However, it may be experimentally unfeasible to manipulate/measure/prepare arbitrary states of the system $A$. Therefore, one may instead do that on a better controllable quantum computer, denoted with $B$. This can be done as follows:
+\begin{enumerate}
+	\item establish a unitary mapping from (a relevant subspace of) $\mathcal{H}_A$ to (a subspace of) the Hilbert space $\mathcal{H}_B = \mathbb{C}^{2^n}$ of the quantum computer (with $n$ the number of qubits);
+	\item find a suitable representation of the unitary operator $\hat{U}_B$ on $\H_B$ that corresponds to $\hat{U}_A$;
+	\item break down $\hat{U}_B$ as a sequence, or circuit, of one- or two-qubit gates;
+	\item prepare the quantum computer in the desired initial state $\psi_B$;
+	\item implement the circuit of gates;
+	\item measure the final state $\hat{U}_B | \psi_B \rangle$ (which implies that several rounds of this whole scheme are in order).
+\end{enumerate}
+When quantum computers are used, or even specifically designed, for such digital quantum simulations, they are also referred to as `digital quantum simulators'. The above procedure is after all similar in spirit to the idea of simulating on a classical computer the proton-neutron collision, explained above, except the new procedure is quantum. It is instead conceptually different from the idea of simulating the proton-neutron scattering with a scattering experiment involving, say, two atoms---although similar in that both procedures are quantum. We summarize the various possibilities for simulating the proton-neutron scattering in the following table:
+
+\begin{table}[h]
+	\centering
+	\begin{tabular}{|c|c|c|}
+		\hline
+		& \textbf{Analog} & \textbf{Digital} \\  
+		\hline
+		\textbf{Classical} & Collide billiard balls & Solve time-dependent Schr\"odinger equation on a CC  \\  
+		\hline
+		\textbf{Quantum} & Collide small atoms & Implement time evolution operator on a QC \\  
+		\hline
+	\end{tabular}
+	\caption{Different types of simulation of proton-neutron scattering.}
+\end{table}
+
+\section{Purposes}
+
+For the rest of this chapter, we will focus on \emph{analog} quantum simulations, since digital quantum simulations are just a particular type of quantum computations (see the other chapters). We now list three main applications of analog quantum simulators. Notice the partial overlap with some topics of quantum computing.
+\begin{itemize}
+	\item \textbf{Time evolution}. Consider a quantum system $A$ simulating another quantum system $B$ (for example, spin-1/2 atoms simulating electrons in a crystal, see Sec. \ref{sec:main_idea}). Then, by letting the system $B$ evolve in time and measuring it, one gains insights into the time evolution of the system $A$.
+	\item \textbf{Implementation of unitaries (beyond the time evolution operator).} As a generalization of the previous application, one can implement unitaries $\hat{U}$ on the simulating system $B$ beyond simple time evolution operators of the form $\hat{U} = \exp(- \textup{i} t \hat{H_B})$. For example, $\hat{U}$ could be a unitary on the Hilbert space of $B$ that mathematically resembles the time evolution of another system, which however need not be in any way similar to the system $B$. Such simulations are done with an approach between analog and digital. For example, one could decompose the unitary $\hat{U}$ as a product of simpler unitaries, $\hat{U} = \hat{U_N} \dots \hat{U_1}$, and implement each unitary on the device $B$, one after another, by changing the tunable parameters (intensity of lasers, lattice spacing, depth of the potential minima making up the lattice sites, ...) of $B$. That is, one implements $\hat{U}_1 = \exp(\textup{i} t_1 \hat{H}_B(\lambda_1))$ as the time evolution operator of the system $B$ relative to specific set-up parameters $\lambda_1$ and a suitable time $t_1$, then changes the parameters to $\lambda_2$, ...
+	\item \textbf{Ground state preparation via adiabatic evolution.} The simulating system $B$ typically features some tunable parameters. One can then start with a choice of the parameters (e.g., very large lattice spacing) for which the ground state of $B$ is easy to prepare, and then slowly change the parameters (e.g., decrease the lattice spacing) until they reach the values of interest. By the adiabatic theorem~\cite{sakurai}, if at time $0$ the system is in the ground state of the initial Hamiltonian $\hat{H}(0)$, the system will be at each time $t$ approximately in the ground state of $\hat{H}(t)$. Eventually, one ends up with the ground state of the Hamiltonian $\hat{H}_{t_{final}} = \hat{H}_B$ resembling the original system Hamiltonian $\hat{H}_A$. Measuring this state of $B$ thus unveils information on the ground state of the original system. Note that the parameter tuning is usually impossible in the original system $A$, where parameters are just fixed by nature.
+	\item \textbf{Quantum transport}. Quantum simulators allow us to have a microscopical look at thermodynamically interesting systems, such as a two-compartment box where one compartment is empty and the other one contains particles. At time zero, the wall between the compartments is removed, and the redistribution of the particles throughout the box is kept track of with snapshots that resolve the individual particles.
+\end{itemize}
+
+\section{(Dis)advantages of the analog quantum simulation method}
+
+Similar to the expected advantages of more general quantum hardware, the main strong point of the method of analog quantum simulation is that the size of a quantum simulator $B$ needed to study a given system $A$ scales linearly with the size of $A$. For example, to study the 2D Hubbard model on a square lattice with $L \times L$ sites, one needs to build a similar square lattice with the same number of sites $L \times L$. This is in stark contrast with the case of computations on a classical computer, where the resources (memory and time) needed to study a quantum system typically scale linearly with the dimension of the Hilbert space of the system under investigation, that is, exponentially in the size of that system. Another advantage is that analog quantum simulation bypasses the idea of building a universal quantum computer, i.e., each analog quantum simulator is specifically built for a restricted class of target Hamiltonians (e.g., Hamiltonians for electrons in a lattice), allowing for a more flexible and system-specific design. In particular, overheads are prevented that instead plague the construction and use of a universal quantum computer. As examples of such overheads, consider the ancillary qubits or non-local gates that incur when one `maps' an arbitrary system $A$, e.g., a system of $n$ fermionic modes, to a quantum computer which does not fully resemble $A$, e.g., an $n$-qubit computer---if one simulates electrons with fermionic atoms, then the fermionic canonical anticommutation relations of operators are automatically taken into account, while when mapping fermionic modes to qubits additional care and resources are needed (see the Jordan-Wigner mapping in Ref.~\cite{lieb1961JordanWigner}).
+
+Non-universality is also a disadvantage, as one needs to build one different device for each class of models. Also, for a given initial system $A$ there may simply exist no similar and easily-controllable simulating system $B$ that does the job. What's more, methods such as Hamiltonian learning need to be used in order to ensure, e.g., that the Hamiltonian $\hat{H}_A$ of the original system and that of the simulating system, $\hat{H}_B$, are actually similar, see Ref.~\cite{DaleyBloch2022}. In the same reference it is also argued that calibration accuracy represents the bottleneck on the precision of analog quantum simulations. Noise and decoherence are also there (but that is in common with all quantum devices).
+
+\section{Platforms}
+Analog quantum simulators come in different types, called `platforms'. Many of these are also used for quantum computers. Here are the main ones:
+\begin{itemize}
+	\item Cold atoms in optical lattices (see Sec.~\ref{sec:cold_atoms});
+	\item Electrostatically trrapped ions, with qubits made up of couples of electronics states;
+	\item Atom arrays with Rydberg interactions involving highly-excited electrons (see Ref.~\cite{Scholl2021});
+	\item Superconducting circuits.
+\end{itemize}
+
+\subsection{Ultracold atoms in optical lattices} \label{sec:cold_atoms}
+
+We finally have a closer look at the platform of (neutral) ultracold atoms in optical lattices. The idea goes back to 1998~\cite{cirac1998coldatoms}, while the first experiments took place in 2002~\cite{bloch2002, bloch2012, Choi_2016}. A 1D version of this set-up is represented in Fig.~\ref{fig:setup}. Laser beams are used to build a 2D standing wave that is able to trap atoms through the so-called optical dipole force, hence the name optical lattice. Each minimum of the corresponding potential felt by the atoms can host a number of atoms that depend on the type of atom (fermion/boson, and spin degree of freedom; a typical example is bosonic rubidium atoms). The third spatial dimension is used for taking snapshots of the atomic positions. In the case of spin-1/2 fermionic atoms, the actual Hamiltonian for such a system is very similar to the Hubbard Hamiltonian of Eq.~\eqref{eq:hubbard}. Indeed, several experiments with cold atoms in an optical lattice were designed to get insights into the Hubbard model for electrons in a crystal. The hopping, or tunneling, parameter $J$ can be tuned by tuning the lattice spacing and/or the amplitude of the potential, while the on-site interaction parameter $U$ depends on the way the atoms interact. As opposed to the electron-electron repulsion, here the major interactions are collisional. These are short-range/on-site interactions, in agreement with the Hubbard Hamiltonian, see Eq.~\eqref{eq:hubbard}.
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.7\textwidth]{graphics/atoms_lattice_setup.png}
+	\caption{The set-up for experiments with cold atoms in an optical lattice.}
+	\label{fig:setup}
+\end{figure}
+
+A typical experiment is the following, which studies the time evolution:
+\begin{enumerate}
+	\item `load' the optical lattice with atoms in a given initial state-vector $| \psi_0 \rangle$;
+	\item let the system evolve for a suitable time;
+	\item take measurements (mainly snapshots of the atoms' positions, possibly spin-resolved);
+	\item reconstruct the final state-vector.
+\end{enumerate}
+
+Figures~\ref{fig:bloch_setup},~\ref{fig:bloch_snapshots} and~\ref{fig:bloch_spin} are taken from Ref.~\cite{bloch2012}. Finally, we include Fig.~\ref{fig:barredo_eiffel} from Ref.~\cite{Barredo2018}, which shows that control of atoms in 3D structures is also possible.
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.7\textwidth]{graphics/bloch_setup.png}
+	\caption{More on the set-up for experiments with cold atoms in an optical lattice (figure reproduced from Ref.~\cite{bloch2012}).}
+	\label{fig:bloch_setup}
+\end{figure}
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.7\textwidth]{graphics/bloch_snapshots.png}
+	\caption{Snapshots of atoms and the corresponding reconstructed positions (figure reproduced from Ref.~\cite{bloch2012}).}
+	\label{fig:bloch_snapshots}
+\end{figure}
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.7\textwidth]{graphics/bloch_spin.png}
+	\caption{Spin-resolved preparation of states and snapshots are also feasible (figure reproduced from Ref.~\cite{bloch2012}).}
+	\label{fig:bloch_spin}
+\end{figure}
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.7\textwidth]{graphics/barredo_eiffel.png}
+	\caption{3D structures of atoms (figure reproduced from Ref.~\cite{Barredo2018}).}
+	\label{fig:barredo_eiffel}
+\end{figure}
\ No newline at end of file

tqc.bib

@@ -5,4 +5,117 @@
   volume={21},
   number={6/7},
   year={1982}
-}
\ No newline at end of file
+}
+
+@book{Sakurai,
+	place={Cambridge},
+	edition={3},
+	title={Modern Quantum Mechanics},
+	publisher={Cambridge University Press},
+	author={Sakurai, J. J. and Napolitano, Jim},
+	year={2020}}
+
+@article{Scholl2021,
+author = {Scholl, Pascal and Schuler, Michael and Williams, Hannah and Eberharter, Alexander and Barredo, Daniel and Schymik, Kai-Niklas and Lienhard, Vincent and Henry, Louis-Paul and Lang, Thomas and Lahaye, Thierry and Läuchli, Andreas and Browaeys, Antoine},
+year = 2021,
+month = 07,
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+volume = 595,
+journal = {Nature},
+doi = {10.1038/s41586-021-03585-1}
+}
+@article{Choi_2016,
+author = {Jae-yoon Choi  and Sebastian Hild  and Johannes Zeiher  and Peter Schauß  and Antonio Rubio-Abadal  and Tarik Yefsah  and Vedika Khemani  and David A. Huse  and Immanuel Bloch  and Christian Gross },
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+journal = {Science},
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+pages = {1547-1552},
+year = {2016},
+doi = {10.1126/science.aaf8834},
+URL = {https://www.science.org/doi/abs/10.1126/science.aaf8834},
+eprint = {https://www.science.org/doi/pdf/10.1126/science.aaf8834}
+}
+
+@article{bloch2012,
+	risfield_0_da = {2012/04/01},
+	author = {Bloch, Immanuel and Dalibard, Jean and Nascimbène, Sylvain},
+	doi = {10.1038/nphys2259},
+	issn = {1745-2481},
+	journal = {Nature Physics},
+	number = {4},
+	pages = {267–276},
+	title = {Quantum simulations with ultracold quantum gases},
+	volume = {8},
+	year = {2012}
+}
+
+@article{DaleyBloch2022,
+	risfield_0_da = {2022/07/01},
+	author = {Daley, Andrew J. and Bloch, Immanuel and Kokail, Christian and Flannigan, Stuart and Pearson, Natalie and Troyer, Matthias and Zoller, Peter},
+	doi = {10.1038/s41586-022-04940-6},
+	issn = {1476-4687},
+	journal = {Nature},
+	number = {7920},
+	pages = {667–676},
+	title = {Practical quantum advantage in quantum simulation},
+	volume = {607},
+	year = {2022}
+}
+
+@article{lieb1961JordanWigner,
+	title = {Two soluble models of an antiferromagnetic chain},
+	journal = {Annals of Physics},
+	volume = {16},
+	number = {3},
+	pages = {407-466},
+	year = {1961},
+	issn = {0003-4916},
+	doi = {https://doi.org/10.1016/0003-4916(61)90115-4},
+	url = {https://www.sciencedirect.com/science/article/pii/0003491661901154},
+	author = {Elliott Lieb and Theodore Schultz and Daniel Mattis}
+}
+
+@article{cirac1998coldatoms,
+	title = {Cold Bosonic Atoms in Optical Lattices},
+	author = {Jaksch, D. and Bruder, C. and Cirac, J. I. and Gardiner, C. W. and Zoller, P.},
+	journal = {Phys. Rev. Lett.},
+	volume = {81},
+	issue = {15},
+	pages = {3108--3111},
+	numpages = {0},
+	year = {1998},
+	month = {Oct},
+	publisher = {American Physical Society},
+	doi = {10.1103/PhysRevLett.81.3108},
+	url = {https://link.aps.org/doi/10.1103/PhysRevLett.81.3108}
+}
+
+
+@article{Barredo2018,
+	author    = {Daniel Barredo and Vincent Lienhard and Sylvain de Léséleuc and Thierry Lahaye and Antoine Browaeys},
+	title     = {Synthetic three-dimensional atomic structures assembled atom by atom},
+	journal   = {Nature},
+	year      = {2018},
+	volume    = {561},
+	number    = {7721},
+	pages     = {79--82},
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+	url       = {https://doi.org/10.1038/s41586-018-0450-2},
+	issn      = {1476-4687}
+}
+
+@article{bloch2002,
+	author    = {Markus Greiner and Olaf Mandel and Tilman Esslinger and Theodor W. Hänsch and Immanuel Bloch},
+	title     = {Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms},
+	journal   = {Nature},
+	year      = {2002},
+	volume    = {415},
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+	url       = {https://doi.org/10.1038/415039a},
+	issn      = {1476-4687}
+}
+

tqc.tex

@@ -205,6 +205,10 @@ jada jada jada
 \chapter{Algebraic Quantum Theory\\\large Robert C. Helling}
 \renewcommand{\authormark}{Robert C. Helling: Algebraic Quantum Theory}
 \input AQM
+\chapter{Quantum simulations\\\large Damiano Aliverti-Piuri}
+\renewcommand{\authormark}{Damiano Aliverti-Piuri: Quantum Simulation}
+\input simulation
+
 
 \newpage
 \chapter*{Bibliography}
@@ -215,4 +219,4 @@ jada jada jada
 \bibliographystyle{is-plain}
 
 
-\end{document}
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+\end{document}