Author Archives: wdengquantum.me

Problem: What does it mean for a topological space \(X\) to be locally homeomorphic to a topological space \(Y\)? Hence, what does it mean for a topological space \(X\) to be locally Euclidean? Solution: \(X\) is said to be locally … Continue reading →

Problem: Explain how a hard-margin support vector machine would perform binary classification. Solution: Conceptually, it’s simple. Given a training set of \(N\) feature vectors \(\mathbf x_1,…,\mathbf x_N\in\mathbf R^n\) each associated with some binary target label \(y_1,…,y_N\in\{-1,1\}\) (notice the \(2\) binary … Continue reading →

Problem: In numerical computation, what are the \(2\) main kinds of rounding error? Solution: Overflow error (\(N\approx\infty\)) but perhaps even more dangerous is underflow error (\(\varepsilon\approx 0\)) which are in some sense inverses of each other: \[0=\frac{1}{\infty}\] Problem: Explain how … Continue reading →

Problem: Draw a schematic of a binary symmetric channel (BSC) with bit flip probability \(p_f\). Solution: Classically, one has: On the other hand, taking a more quantum perspective, in the computational basis \((|0\rangle,|1\rangle)\) one might define a binary symmetric channel … Continue reading →

Problem: State the form of the singular value decomposition (SVD) of an arbitrary linear operator \(X:\mathbf C^m\to\mathbf C^n\). Solution: The SVD of \(X\) is given by: \[X=U_2\Sigma U^{\dagger}_1\] where \(U_1\in U(m)\) and \(U_2\in U(n)\) are unitary operators and \(\Sigma:\mathbf C^m\to\mathbf … Continue reading →

Problem: The Hamiltonian of an atom may be written in the perturbative form: \[H=H_0+V\] what are \(H_0\) and \(V\)? Solution: If the atom has \(N\) electrons and atomic number \(Z\) (if neutral then \(Z=N\) but here one can also allow … Continue reading →

Problem: Given a function \(f(x,p)\) defined on a single-particle classical phase space \((x,p)\) in \(1\) dimension, define the Weyl transform \(\hat f\) of \(f(x,p)\). Solution: \[\hat f=\int\frac{dx’dp’}{h}e^{i(p’X+x’P)/\hbar}\tilde f(x’, p’)\] where the phase-space Fourier transform \(\tilde f(x’,p’)\) of \(f(x,p)\) is given … Continue reading →

Problem: At a high level, what is the goal of classical hydrodynamics? Solution: The program of classical hydrodynamics seeks to bridge the physics of a many-body system at different length scales. The idea is to start from microscopics (i.e. Newton’s … Continue reading →

Problem: Prove the Sokhotski-Plemelj theorem on the real line \(\textbf R\): \[\frac{1}{\omega-\omega’\pm i0^+}=\mathcal P\frac{1}{\omega-\omega’}\mp i\pi\delta(\omega-\omega’)\] Solution: (in this solution, the function \(\chi(\omega)\) is assumed to be analytic on the integration interval in \(\textbf R\) to be able to apply the … Continue reading →

Problem: Distinguish between a supervised data set and unsupervised data set. Solution: Supervised data is of the form \(\{(\textbf x_1,y_1),…,(\textbf x_N,y_N)\}\). Unsupervised data consists of stripping away the target labels \(y_1,…,y_N\), leaving behind just the feature vectors \(\{\textbf x_1,…,\textbf x_N\}\). … Continue reading →