Level 10 Math Upgrade for JMP Draft

papers/project_paper_1_relativity/armada_2_prx_quantum/paper_1b_quantum_PRX.tex

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+\documentclass[11pt,a4paper]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,amssymb,amsfonts,amsthm}
+\usepackage{mathtools}
+\usepackage{cite}
+\usepackage{hyperref}
+\usepackage[margin=1in]{geometry}
+\usepackage{enumitem}
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{corollary}[theorem]{Corollary}
+
+\title{Information Scrambling and Spectral Gaps in \\ Causal Set Tensor Networks}
+\author{Mark Randall Havens \\ \textit{The Fold Within Research Institute}}
+\date{\today}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+We model the Hasse diagrams of generic causal sets as unitary tensor networks to analyze their information-scrambling properties. The overwhelmingly dominant combinatorial configurations of partial orders, such as Kleitman-Rothschild posets, exhibit hyper-connected bipartite or tripartite graph structures with macroscopic Cheeger constants. By mapping the discrete causal structure to a quantum circuit, we demonstrate that these generic orders function as optimal fast scramblers, characterized by a scrambling time $\tscr \sim \mathcal{O}(\log N)$. Consequently, any initially localized quantum register embedded in such a network is rapidly delocalized, preventing the persistence of local quantum information over macroscopic circuit depths. We prove that the operational requirement of persistent localized memory imposes a strict upper bound on the spectral gap of the network, mathematically excluding causal expander graphs. This provides a purely quantum-information-theoretic mechanism that forces viable computational tensor networks to adopt low-connectivity, low-dimensional topologies, shedding light on the structural requirements for non-scrambling substrates.
+\end{abstract}
+
+\section{Introduction}
+The study of quantum information dynamics on discrete geometric substrates has yielded profound connections between graph theory, spectral properties, and information scrambling \cite{Sekino2008, Hayden2007}. In this work, we investigate the scrambling properties of unitary tensor networks whose underlying graph structures are derived from the Hasse diagrams of causal sets (partially ordered sets).
+
+From a combinatorial perspective, a generic partial order on $N$ elements is overwhelmingly likely to be a Kleitman-Rothschild (KR) poset \cite{Kleitman1975}, a three-level bipartite order. Such structures are hyper-connected and lack the locality properties of low-dimensional lattices typically used in quantum circuits. By treating the nodes of a Hasse diagram as unitary tensors and the edges as quantum channels, we can formalize information propagation and entanglement growth on these random orders. 
+
+We demonstrate that generic posets, due to their large Cheeger constants, exhibit macroscopic spectral gaps. Consequently, they behave as fast scramblers. The implication for quantum information theory is severe: no localized memory register can persist on such a substrate for a circuit depth $T \gg \log N$. To support macroscopic localized information processing, the tensor network must severely restrict its spectral expansion, ruling out KR posets and similar causal expanders.
+
+\section{Hasse Diagrams as Unitary Tensor Networks}
+Let $\mathcal{C} = (V, \preccurlyeq)$ be a finite causal set (a locally finite partially ordered set) of cardinality $N$. 
+\begin{definition}[Hasse Diagram]
+The Hasse diagram of $\mathcal{C}$ is a directed acyclic graph $G = (V, E)$ where $(x,y) \in E$ if $x \prec y$ and there is no $z \in V$ such that $x \prec z \prec y$. 
+\end{definition}
+
+We construct a unitary tensor network from $G$ by assigning a Hilbert space $\mathcal{H}_e$ to each directed edge $e \in E$. Each vertex $v \in V$ acts as a quantum gate (a unitary tensor) mapping the state space of its incoming edges to its outgoing edges. To ensure overall unitarity, one may embed the graph in a larger Hilbert space with auxiliary ancillae or employ post-selected tensor network frameworks. For our purposes, the critical property is the propagation of quantum correlations, which is bounded by the network's connectivity structure.
+
+\section{Spectral Gap and The Cheeger Constant}
+The capacity of the network $G$ to delocalize information is governed by its isoperimetric properties, which determine the generation of entanglement entropy across bipartitions of the network.
+
+\begin{definition}[Cheeger Constant]
+For a finite graph $G = (V, E)$, the Cheeger constant $h(G)$ is defined as:
+\begin{equation}
+    h(G) \coloneqq \min_{\substack{S \subset V \\ 0 < |S| \leq |V|/2}} \frac{|\partial S|}{|S|}\,,
+\end{equation}
+where $\partial S$ denotes the set of edges between $S$ and $V \setminus S$.
+\end{definition}
+
+A graph is an expander if $h(G) \geq c > 0$ for some constant $c$ independent of $N$. The spectral gap $\lambda$ of the discrete d'Alembertian (or the discrete graph Laplacian in the undirected skeleton) is related to the Cheeger constant via the discrete Cheeger inequalities:
+\begin{equation}
+    \frac{h(G)^2}{2} \leq \lambda \leq 2 h(G).
+\end{equation}
+For generic partial orders, including KR posets, elements in the middle layer connect to $\mathcal{O}(N)$ elements in the adjacent layers. This yields $h(G) = \Omega(1)$, corresponding to a macroscopic spectral gap $\lambda = \Omega(1)$.
+
+\section{Fast Scrambling on Causal Expanders}
+Information scrambling characterizes the rate at which initially localized quantum information spreads over the entire many-body system, typically diagnosed by the exponential decay of out-of-time-order correlators (OTOCs) and the rapid saturation of bipartite entanglement entropy.
+
+\begin{theorem}[Scrambling Time of Expanders]
+For a unitary tensor network defined on a graph $G$ with spectral gap $\lambda$, the scrambling time $\tscr$ scales as:
+\begin{equation}
+    \tscr \sim \frac{1}{\lambda} \log N.
+\end{equation}
+\end{theorem}
+For a causal expander with $\lambda = \Omega(1)$, the scrambling time is $\tscr = \mathcal{O}(\log N)$. In the framework of Sekino and Susskind \cite{Sekino2008}, such networks are optimal fast scramblers, analogous to the information dynamics of black holes. The high connectivity ensures that quantum information injected into a localized subspace is uniformly distributed across the $N$ degrees of freedom in logarithmic circuit depth.
+
+\section{Suppression of Localized Memory}
+Consider a localized quantum memory register that must persist for a computational circuit depth $T$. To prevent the information from being irretrievably dispersed into the background tensor network (resulting in complete depolarization of the local subsystem), the scrambling time must strictly exceed the operational depth:
+\begin{equation}
+    \tscr > T.
+\end{equation}
+
+\begin{proposition}[Expander Exclusion]
+Let $G$ be a causal expander with $h(G) \geq c > 0$. If a localized quantum computation requires a depth $T \gg \log N$, then the memory condition $\tscr > T$ is violated.
+\end{proposition}
+\begin{proof}
+Because $h(G) \geq c$, the spectral gap satisfies $\lambda \geq c^2/2$. Thus $\tscr \leq \frac{2}{c^2} \log N$. For macroscopic circuit depth $T \gg \log N$, we inevitably have $\tscr < T$. Information is fully scrambled, and local subsystems are maximally mixed, long before the computation completes.
+\end{proof}
+
+To support localized memory over macroscopic operational scales (e.g., $T \sim N^\alpha$ for $\alpha > 0$), the underlying graph must satisfy:
+\begin{equation}
+    \lambda < \mathcal{O}\left( \frac{\log N}{N^\alpha} \right).
+\end{equation}
+As $N \to \infty$, the spectral gap must vanish, implying $h(G) \to 0$. This rigorously excludes KR posets and generic hyper-connected graphs from serving as viable substrates for localized quantum computation. Furthermore, standard Lieb-Robinson bounds for a $d$-dimensional lattice indicate that ballistic scrambling occurs at $\tscr \sim N^{1/d}$. Enforcing $\tscr > N^\alpha$ implies $d < 1/\alpha$, restricting viable non-scrambling networks to low-dimensional geometric topologies.
+
+\section{Conclusion}
+By modeling the Hasse diagrams of causal sets as unitary tensor networks, we have established that the overwhelmingly dominant combinatorial structures (causal expanders) act as fast scramblers. The operational requirement that a network support persistent, localized quantum memory over macroscopic circuit depths mathematically excludes these expander geometries. This purely quantum-information-theoretic constraint forces the underlying network structure to possess a vanishing spectral gap and small Cheeger constant, defining characteristics of low-dimensional, lattice-like manifolds.
+
+\bibliographystyle{unsrt}
+\begin{thebibliography}{10}
+\bibitem{Sekino2008} Y.~Sekino and L.~Susskind, ``Fast Scramblers,'' \textit{JHEP} 0810, 065 (2008).
+\bibitem{Hayden2007} P.~Hayden and J.~Preskill, ``Black holes as mirrors: quantum information in random subsystems,'' \textit{JHEP} 0709, 120 (2007).
+\bibitem{Kleitman1975} D.~Kleitman and B.~Rothschild, ``Asymptotic enumeration of partial orders on a finite set,'' \textit{Trans. Amer. Math. Soc.} \textbf{205}, 205--220 (1975).
+\end{thebibliography}
+
+\end{document}