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‎doc/src/week11/Latexfiles/analogdigital.tex‎

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\documentclass{beamer}
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\usepackage{amsmath,amsfonts,amssymb,bm}
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\usepackage{quantikz}
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\usepackage{graphicx}
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\usetheme{Madrid}
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\title{Analog and Digital Quantum Computing}
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\subtitle{From Hamiltonian Dynamics to Variational Algorithms}
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\author{Morten Hjorth-Jensen}
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\date{Spring 2026}
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\begin{document}
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\frame{\titlepage}
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%================================================
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\section{Motivation}
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%================================================
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\begin{frame}{Quantum Computing as Physics}
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\[
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i\hbar \frac{d}{dt} |\psi\rangle = H |\psi\rangle
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\]
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\[
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|\psi(t)\rangle = e^{-iHt}|\psi(0)\rangle
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\]
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\begin{itemize}
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\item Computation = control of quantum dynamics
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Two Paradigms}
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\begin{itemize}
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\item Digital quantum computing
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\item Analog quantum computing
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\end{itemize}
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\begin{block}{Key distinction}
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Discrete gate decomposition vs continuous evolution
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\end{block}
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\end{frame}
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%================================================
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\section{Digital Quantum Computing}
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%================================================
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\begin{frame}{Gate-Based Model}
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\[
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U = \prod_i U_i
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\]
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\begin{itemize}
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\item Universal computation
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Quantum Circuit Example}
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\[
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\begin{quantikz}
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\lstick{|0\rangle} & \gate{H} & \ctrl{1} & \meter{} \\
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\lstick{|0\rangle} & \qw & \targ{} & \meter{}
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\end{quantikz}
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Trotterization}
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\[
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e^{-i(H_1+H_2)t} \approx
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\left(e^{-iH_1\Delta t}e^{-iH_2\Delta t}\right)^n
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Example: QAOA}
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\[
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|\psi\rangle =
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\prod_k e^{-i\beta_k H_M} e^{-i\gamma_k H_C}|+\rangle
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Example: HHL}
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\begin{itemize}
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\item Spectral decomposition
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\item Implements $A^{-1}$
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Advantages}
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\begin{itemize}
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\item Universal
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\item Programmable
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\item Error correction possible
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Limitations}
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\begin{itemize}
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\item Deep circuits
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\item Noise
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\item Trotter errors
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\end{itemize}
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\end{frame}
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%================================================
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\section{Analog Quantum Computing}
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%================================================
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\begin{frame}{Analog Model}
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\[
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|\psi(t)\rangle = e^{-iH_{\text{sim}}t}|\psi(0)\rangle
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Simulation Principle}
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\[
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H_{\text{sim}} \approx H_{\text{target}}
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Physical Platforms}
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\begin{itemize}
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\item Cold atoms
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\item Trapped ions
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\item Rydberg atoms
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Quantum Annealing}
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\[
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H(t) = (1-s)H_M + sH_C
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Advantages}
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\begin{itemize}
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\item Natural dynamics
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\item Efficient for many-body systems
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Limitations}
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\begin{itemize}
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\item Not universal
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\item Limited control
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\end{itemize}
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\end{frame}
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%================================================
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\section{Analog Simulation of Many-Body Models}
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%================================================
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\begin{frame}{Ising Model}
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\[
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H = -J \sum_{ij} Z_i Z_j - h \sum_i X_i
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\]
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\begin{itemize}
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\item Implemented in trapped ions, Rydberg systems
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Hubbard Model}
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\[
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H = -t \sum_{\langle ij\rangle} c_i^\dagger c_j
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+ U \sum_i n_{i\uparrow} n_{i\downarrow}
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\]
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\begin{itemize}
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\item Realized in optical lattices
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Why Analog is Powerful}
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\begin{itemize}
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\item Direct access to many-body dynamics
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\item No exponential classical cost
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\end{itemize}
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\end{frame}
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%================================================
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\section{Connections to Linear Response and TDHF}
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%================================================
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\begin{frame}{Linear Response Equation}
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\[
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(\omega I - M)x = b
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{TDHF Equation}
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\[
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i \frac{d\rho}{dt} = [h[\rho],\rho]
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Linearized TDHF}
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\[
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(\omega I - \mathcal{L})\delta\rho = s
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Connection to HHL}
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\begin{itemize}
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\item Same mathematical structure:
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\[
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A x = b
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\]
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\item Inverse operator = response function
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\end{itemize}
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\end{frame}
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%================================================
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\section{Variational Algorithms}
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%================================================
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\begin{frame}{Variational Principle}
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\[
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E = \langle \psi(\theta)|H|\psi(\theta)\rangle
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{ADAPT-VQE}
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\begin{itemize}
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\item Adaptive ansatz construction
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\item Gradient selection:
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\[
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\langle [H,A_k] \rangle
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\]
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Connection to Coupled Cluster}
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\begin{itemize}
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\item BCH expansion
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\item Many-body correlations
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Variational Dynamics}
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\begin{itemize}
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\item McLachlan principle
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\[
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\delta \| (i\partial_t - H)|\psi\rangle \| = 0
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\]
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\end{itemize}
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\end{frame}
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%================================================
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\section{Quantum Control Theory}
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%================================================
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\begin{frame}{Control Hamiltonian}
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\[
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H(t) = \sum_k u_k(t) H_k
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\]
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Optimal Control}
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\begin{itemize}
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\item Find $u_k(t)$ to achieve target state
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Digital vs Analog Control}
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\begin{itemize}
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\item Digital: discrete pulses
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\item Analog: continuous control
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{QAOA as Control}
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\begin{itemize}
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\item Piecewise constant control protocol
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\end{itemize}
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\end{frame}
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%================================================
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\section{Unified View}
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%================================================
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\begin{frame}{Unified Framework}
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\begin{itemize}
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\item Analog: continuous evolution
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\item Digital: discretized evolution
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\item Variational: optimized evolution
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Core Mathematical Object}
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\[
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e^{-iHt}
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\]
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\begin{itemize}
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\item Everything reduces to controlling this operator
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Conceptual Chain}
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\begin{itemize}
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\item Analog simulation → real dynamics
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\item Digital circuits → approximated dynamics
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\item Variational methods → optimized dynamics
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\end{itemize}
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\end{frame}
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%================================================
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\section{Outlook}
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%================================================
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\begin{frame}{Future Directions}
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\begin{itemize}
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\item Hybrid analog-digital systems
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\item Quantum simulation of materials
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\item Quantum-enhanced many-body theory
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\end{itemize}
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\end{frame}
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%------------------------------------------------
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\begin{frame}{Summary}
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\begin{itemize}
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\item Digital: universal and programmable
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\item Analog: efficient and physics-driven
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\item Variational: bridge between both
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\end{itemize}
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\end{frame}
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\end{document}

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