transmon circuit diagram

qubits embedded in a multilevel system

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,...

January 31, 2024 · 8 min

Legendre transformations

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....

December 9, 2021 · 4 min

ghosts, gauges, and generating functionals

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]....

October 31, 2021 · 5 min

the central identity of quantum field theory

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....

September 10, 2021 · 7 min


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....

June 13, 2021 · 9 min

the Freeman method

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…...

June 9, 2021 · 3 min