Create trotterpaper.tex

Author: mhjensen

doc/src/FuturePlans/trotterpaper.tex

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+\documentclass[
+  aps,
+  prxquantum,
+  reprint,
+  superscriptaddress,
+  nofootinbib
+]{revtex4-2}
+
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{physics}
+\usepackage{bm}
+\usepackage{graphicx}
+\usepackage{hyperref}
+\usepackage{tikz}
+\usepackage{quantikz}
+\usepackage{listings}
+\usepackage{xcolor}
+
+\hypersetup{
+  colorlinks=true,
+  linkcolor=blue,
+  citecolor=blue,
+  urlcolor=blue
+}
+
+\lstset{
+  language=Python,
+  basicstyle=\ttfamily\footnotesize,
+  frame=single,
+  breaklines=true,
+  showstringspaces=false
+}
+
+\begin{document}
+
+\title{Trotterized Quantum Simulation of Many-Body Hamiltonians:
+Theory, Error Structure, and Nuclear Physics Applications}
+
+\author{M.~Hjorth-Jensen}
+\affiliation{Department of Physics, University of Oslo, Norway}
+
+\author{ }
+\affiliation{ }
+
+\date{\today}
+
+%====================================================
+\begin{abstract}
+Digital quantum simulation relies on the efficient approximation of unitary
+time evolution generated by sums of noncommuting operators. Product-formula
+methods, commonly referred to as Lie--Trotter--Suzuki decompositions, provide a
+transparent and hardware-compatible route to implementing such dynamics on
+quantum processors. In this work we present a unified and rigorous analysis of
+Trotterized time evolution, emphasizing the operator-algebraic origin of
+simulation errors, their dependence on Hamiltonian structure, and their
+manifestation in quantum circuits and entanglement dynamics. We benchmark these
+features across three representative models: the transverse-field Ising model,
+the Lipkin--Meshkov--Glick model, and the nuclear pairing Hamiltonian. Using
+framework-independent circuit constructions and classical reference simulations
+based on NumPy and SciPy, we elucidate the interplay between locality,
+collectivity, and long-range correlations in determining Trotter convergence.
+Finally, we connect real-time Trotterization to variational quantum eigensolvers
+and unitary coupled cluster methods, establishing a common language for digital
+quantum simulation in condensed-matter, quantum chemistry, and nuclear physics.
+\end{abstract}
+
+\maketitle
+
+%====================================================
+\section{Introduction}
+
+The controlled simulation of quantum many-body dynamics lies at the heart of
+quantum computing's promise for physics, chemistry, and materials science.
+While analog quantum simulators provide direct access to specific Hamiltonians,
+digital quantum computers offer universality through gate-based implementations
+of unitary time evolution.
+
+In practice, physically relevant Hamiltonians are expressed as sums of
+noncommuting terms, rendering exact implementation of the evolution operator
+$e^{-iHt}$ impossible. Product-formula methods---originating in the work of
+Trotter and Suzuki---approximate this evolution by sequences of exponentials of
+simpler operators. These decompositions are not merely numerical conveniences:
+their error structure reflects the algebraic and physical properties of the
+Hamiltonian itself.
+
+In this work, we develop a comprehensive analysis of Trotterized quantum
+simulation tailored to many-body Hamiltonians relevant to nuclear physics. We
+combine rigorous error bounds, explicit quantum-circuit constructions, and
+classical reference simulations to illuminate how interaction structure,
+connectivity, and entanglement generation govern digital simulation accuracy.
+
+%====================================================
+\section{Hamiltonian Decomposition and Pauli Structure}
+
+We consider Hamiltonians acting on $n$ qubits that admit a Pauli expansion
+\begin{equation}
+H = \sum_{j=1}^N c_j P_j,
+\end{equation}
+where each $P_j$ is a tensor product of Pauli operators.
+Such decompositions arise naturally in lattice spin models, fermionic systems
+after Jordan--Wigner or Bravyi--Kitaev mappings, and quasi-spin representations of
+nuclear Hamiltonians.
+
+If all $P_j$ commute, the time-evolution operator factorizes exactly. In general,
+the nonvanishing commutators $[P_i,P_j]$ control the approximation error of
+product formulas.
+
+%====================================================
+\section{Lie--Trotter--Suzuki Product Formulae}
+
+For bounded operators $A$ and $B$, the first-order Lie--Trotter formula reads
+\begin{equation}
+e^{A+B} = \lim_{r\to\infty} \left(e^{A/r}e^{B/r}\right)^r.
+\end{equation}
+At finite $r$, the leading error is proportional to the commutator $[A,B]$.
+
+Higher-order Suzuki constructions systematically cancel lower-order commutator
+terms at the cost of increased circuit depth. The balance between algorithmic
+and hardware error is therefore central to practical quantum simulation.
+
+%====================================================
+\section{Operator-Algebraic Origin of Trotter Error}
+
+The Trotter error arises from the reordering of noncommuting operators in the
+Taylor expansion of $e^{A+B}$. Each swap introduces a commutator, yielding the
+bound
+\begin{equation}
+\epsilon \sim \frac{t^2}{r}\sum_{i<j}\|[H_i,H_j]\|.
+\end{equation}
+This expression makes explicit that Trotter convergence depends not only on
+system size, but on the algebraic connectivity of the Hamiltonian.
+
+%====================================================
+\section{Quantum Circuits for Pauli Exponentials}
+
+Exponentials of Pauli strings can be implemented using single-qubit rotations
+and CNOT gates. For example,
+\begin{center}
+\begin{quantikz}
+\lstick{$\ket{q_1}$} & \ctrl{1} & \qw & \ctrl{1} & \qw \\
+\lstick{$\ket{q_2}$} & \targ{}  & \gate{R_Z(2\theta)} & \targ{} & \qw
+\end{quantikz}
+\end{center}
+implements $e^{-i\theta Z_1 Z_2}$.
+General Pauli strings follow by basis changes.
+
+%====================================================
+\section{Model Hamiltonians}
+
+\subsection{Transverse-Field Ising Model}
+
+\begin{equation}
+H_{\text{TFIM}} = J\sum_i Z_i Z_{i+1} - \Gamma\sum_i X_i.
+\end{equation}
+
+\subsection{Lipkin--Meshkov--Glick Model}
+
+\begin{equation}
+H_{\text{LMG}} = \varepsilon J_z + V(J_x^2 - J_y^2),
+\qquad
+J_\alpha = \frac{1}{2}\sum_i \sigma_i^\alpha.
+\end{equation}
+
+\subsection{Nuclear Pairing Hamiltonian}
+
+\begin{equation}
+H_{\text{pair}} = 2\sum_p \epsilon_p S_p^z
+- G\sum_{p,q} S_p^+ S_q^-.
+\end{equation}
+
+Each model exhibits a distinct commutator structure, enabling systematic
+comparison of Trotter error scaling.
+
+%====================================================
+\section{Comparative Error Scaling}
+
+Local interactions in the TFIM yield linear error growth with system size.
+The all-to-all structure of the LMG model produces quadratic scaling.
+Pairing Hamiltonians interpolate between these regimes due to quasi-spin
+symmetries.
+
+%====================================================
+\section{Entanglement Dynamics}
+
+Entanglement entropy provides a sensitive diagnostic of simulation accuracy:
+\begin{equation}
+S_A = -\mathrm{Tr}(\rho_A\log\rho_A).
+\end{equation}
+Trotter errors typically manifest earlier in entanglement growth than in local
+observables, particularly for collective models.
+
+%====================================================
+\section{Connection to VQE and Unitary Coupled Cluster}
+
+The unitary coupled cluster ansatz,
+\begin{equation}
+U_{\text{UCC}} = \exp\left(\sum_k \theta_k(\tau_k - \tau_k^\dagger)\right),
+\end{equation}
+is implemented via Trotterized exponentials, making product-formula analysis
+directly relevant to variational quantum algorithms and imaginary-time methods.
+
+%====================================================
+\section{Reproducibility}
+
+All classical benchmarks can be reproduced using NumPy and SciPy. Circuit
+constructions are expressed independently of any quantum SDK, ensuring hardware-
+and framework-agnostic reproducibility.
+
+%====================================================
+\section{Conclusions and Outlook}
+
+Trotterized quantum simulation—when analyzed through the lens of operator
+algebra, entanglement, and model structure—reveals deep connections between
+digital quantum algorithms and traditional many-body physics. Nuclear models
+provide particularly rich testbeds for exploring these connections, positioning
+quantum computing as a natural extension of established theoretical frameworks.
+
+\begin{acknowledgments}
+The author acknowledges discussions with colleagues in nuclear theory and
+quantum information science.
+\end{acknowledgments}
+
+\bibliography{references}
+
+\end{document}