Author Archives: wdengquantum.me

Problem: Why is it important to learn how to learn? Solution: In a post-AGI world where each day feels like swallowing a fire hose of new (mis!)information (especially as a researcher), the most important arrow one can have in the … Continue reading →

Problem: What is an estimator \(\hat{\theta}\) for some parameter \(\theta\) of a random variable? Distinguish between point estimators and interval estimators. Solution: In the broadest sense, an estimator is any function of \(N\) i.i.d. draws of the random variable \(\hat{\theta}(X_1,…,X_N)\). … Continue reading →

Just as the fundamental theorem of single-variable calculus \(\int_{x_1}^{x_2}f'(x)dx=f(x_2)-f(x_1)\) is the key insight on which the entire subject of single-variable calculus rests, there is an analogous sense in which one can consider a fundamental theorem of classical computing to be … Continue reading →

Problem: Given a linear operator \(H\) on some vector space, define the resolvent operator \(G_H(E)\) associated to \(H\). Solution: The resolvent \(G_H(E)\) of \(H\) is the operator-valued Mobius transformation of a complex variable \(E\in\textbf C\) defined by the inverse: \[G_H(E):=\frac{1}{E1-H}\] … Continue reading →

Although Feynman diagrams are often first encountered in statistical/quantum field theory contexts where they are employed in perturbative calculations of partition/correlation functions based on Wick’s theorem, there is a lot of “fluff” in these cases that obscures their underlying simplicity. … Continue reading →

Problem #\(1\): What is the Landau-Ginzburg free energy functional \(F[m]\) for the Ising model? Solution #\(1\): It is defined implicitly through: \[e^{-\beta F[m]}=\sum_{\{\sigma_i\}\to_{\text{c.g.}}m(\textbf x)}e^{-\beta E_{\{\sigma_i\}}}\] where \(E_{\sigma_i}=-E_{\text{ext}}\sum_{i=1}^N\sigma_i-E_{\text{int}}\sum_{\langle i,j\rangle}\sigma_i\sigma_j\) is the energy of a given spin microstate \(\{\sigma_i\}\) and “\(\text{c.g.}\)” is … Continue reading →

Problem #\(1\): Define the Poincaré group. Solution #\(1\): In words, the Poincaré group is the isometry group of Minkowski spacetime \(\textbf R^{1,3}\). Mathematically, it is the semidirect product \(\textbf R^{1,3}⋊O(1,3)\) of the normal subgroup \(\textbf R^{1,3}\) of spacetime translations with … Continue reading →

Problem #\(1\): Write down a general \(\phi\)-dependent perturbation to the Klein-Gordon Lagrangian density \(\mathcal L\) for a real scalar field \(\phi\), and explain why in practice only the first \(2\) terms of such a perturbation need to be considered. Solution … Continue reading →

The purpose of this post is to explain what the toric code is, and its potential use as a fault-tolerant quantum error correcting stabilizer surface code for topological quantum computing. To begin, consider an \(N\times N\) square lattice \(\Lambda\) with … Continue reading →

The purpose of this post is to explain why, experimentally, one only observes \(4\) electric dipole transitions in the IR spectrum of buckminsterfullerene, also known as \(\text C_{60}\) or informally as the buckyball: Buckyball Basics The simplest conceptual way to … Continue reading →