Posts

The transmon qubit, first proposed in 2007 by Koch et al. 1, has become a top contender for realizing large-scale NISQ devices. The transmon is based on an anharmonic oscillator realized by a Josephson junction shunted to a capacitor. Anharmonicity — i.e. unevenly separated energy levels, c.f. the harmonic oscillator — is the key ingredient enabling a stable two-level subsystem where the qubit state will live. The circuit diagram (taken from 1), shown below,...

As this planet completes yet another revolution around a pocket of spacetime curvature abounding with scintillating, mingling ions — and while the sparks seem to be flying in all of the wrong ways here on Earth — I wanted to share some thoughts on a few of the books I read this year. I will in no way do these works justice, but hopefully, I can convey some of the insights, joys, laughs, and tears that they brought me....

In the realm of mathematical finance there are two relatively disjoint worlds: the “buy-side” and the “sell-side”. Buyers are analyzing the available financial products (e.g. stocks, bonds, derivatives, etc.) and are trying to predict, and profit off of, the future prices of these assets—i.e. they are trying to estimate the real-world probability distribution, commonly denoted $P$, of the future stochastic price movements. On the other side, sellers (aka market-makers) are trying to set a fair price for more illiquid assets, like derivatives, using the prices of more liquid assets, like stocks, which are priced by supply and demand....

In 2014, Ian Goodfellow, then at OpenAI, published his seminal paper, titled “Generative Adversarial Networks”1, detailing how competition between generator and discriminator functions, approximated by neural networks, can train the generator to produce realistic images. In this article I will be discussing the theory behind this idea, my own implementation in Julia (mirroring the network structure in Goodfellow’s original paper), and show some of the images I was able to get out....

So you want inline interactive plots in your blog posts? But you don’t want to learn javascript and there wasn’t an obvious way to embed matplotlib? If your comfortable with (or willing to learn) Julia, there is a simple and elegant solution! Makie.jl is a julia plotting package with a lot of functionality, including a backend system consisting of WGLMakie and JSServe which provides a way to embed makie plots—including 3d visualizations—right into html and inserted into markdown jekyll blog posts....

Courses and textbooks, at least in undergrad, often gloss over the details of the Legendre transformation, which converts convex functions of one variable into another convex function of the “conjugate” variable. Used ubiquitously in physics, from thermodynamics to quantum field theory, this mathematical method plays a central role in connecting some of the most fundamental concepts, and yet, at least to me for a while, was a mysterious black-box procedure. I am now going to attempt to illuminate this technique....

We saw in a previous post that for a non-interacting theory (i.e. $V(\varphi) = 0$) that the generating functional can be written as $$ Z[J] = e^{\frac{1}{2} J \cdot K^{-1} \cdot J}. $$ We hinted that it is not always the case that $K$ can be naively inverted. The issue arises when we consider the Maxwell action for a $U(1)$ gauge potential $A_\mu$: $$ S(A) = \int d^4 x \left[ \frac{1}{2} A_\mu \left( \partial^2 g^{\mu \nu} - \partial^\mu \partial^\nu \right) A_\nu + A_\mu J^\mu \right]....

Quantum field theory is the study of various types of fields, the interactions between these fields, and the correlation functions describing the dynamics of the entire system. Fields are really just functions that label each spacetime coordinate with some type of mathematical object. We can talk about spinor fields, scalar fields, and even gauge fields, which can be described by vectors or tensors that transform in certain ways under an associated gauge group....

The development of mathematical finance, much like the processes it aims to study, has had a particularly jumpy history. A major leap came in 1973, when the Black-Scholes option pricing model was published and mathematically understood. The essential idea is to model the underlying asset $S_t$ of an option as a geometric brownian motion, with a stochastic differential equation (SDE), given by $$ dS_t = \mu S_t \ dt + \sigma S_t \ dW_t, $$...

Pseudovectors, or axial vectors as they are sometimes referred to as, are commonly encountered, but mysterious mathematical objects. In physics they arise in many different areas, particularly when cross products are involved - e.g., magnetic fields, angular momentum, and torque. The goal of this post is to address two seemingly different definitions of a pseudovector, in 3 dimensions, and in the process unravel this mystery. Definition 1 A pseudovector is a tensor on $\mathbb{R}^3$ that transforms like a vector under proper rotations, but picks up a sign under an improper rotation, like a reflection....

Just about a year ago today, I began working on implementing an algorithm my undergrad research advisor had devised to speed up the Metropolis algorithm, in the regime where the acceptance probability is very low, which is the case in lattice simulations of quantum gravity. Quantum Gravity Physics has experienced its most rapid advancement when theories are unified: electromagnetism $\leftarrow$ electricity + magnetism + light general relativity $\leftarrow$ special relativity + curved space-time (gravity) quantum field theory $\leftarrow$ quantum mechanics + special relativity + electromagnetism + matter + nuclear forces And hopefully soon…...

Introductions are tricky. Where are you supposed to begin? I think I’ve had a pretty interesting life so far, of course, I am biased, but I’m not going to go deep into any of that, right now. As an undergraduate I had the opportunity to work on many interesting projects, most of which were computational, and can be found on my github with poor to nonexistent documentation. I thought about cleaning up my past work and presenting it here, but while that might be interesting to readers (if there are any), it sounds boring to me....