Alright, so this paper was referenced in Steffen’s 2012 work, as one of the primary explanations for decoherence error. This paper is quite experimental, but let’s try to understand the basic principles in here.
The primary takeaway is to show that the primary source of error in a superconducting quantum bit is found from the dielectric loss within the insulating materials. Martinis suggests that the reason for this decoherence can be described by examining the two-level states (TLS) of the dielectric, and using that to model the loss.
First, recall that a Josephson junction is made up of a SIS layer, or Superconducting-Insulator-Superconductor material. In addition, normal capacitors are also made of a Capacitor-Insulator-Capacitor type. The paper mentions that there is decoherence loss in both the “bulk insulating materials” and in the “tunnel barrier” (of the JJ), but that the decoherence manifests differently in each of the cases.
The first takeaway is that the dielectric loss at low temperatures and low drive amplitudes behave differently than at high temperatures. That is, the loss is no longer linear at lower temperatures, and cannot be modeled based on previous data at high temperatures. In addition, it appears that the dielectric loss is especially bad when using non-crystalline substrates for the insulator. The experimental results show that even though the dielectric loss is low at high temperatures, there is a rapid ramp-up of loss as the drive amplitude decreases. Therefore, it is no longer reasonable to use high temperature approximations for the work done in superconducting circuits. This seems to be particularly surprising, as one might more typically expect there to be less loss as the temperature approaches zero.
The reason for this difference, seems especially interesting. According to the article, conventional resistors are often modeled as a bosonic bath, which is a model created by Feynman and Vernon in 1963. The “bosonic” part means, I believe, that the particles in the environment (bath) behave with Bose-Einstein statistics, and quantum dissipation loss is modeled by considering how the quantum particle is coupled to that bath. A related model, called the spin-boson model, probes this in more depth for quantum systems that have two levels. However, the Martinis paper suggests that the treating the bath as bosons is inaccurate, and it would be better to consider it as a fermionic bath instead.
The paper suggests that one way to prevent such errors from accumulating is by making the insulators very very small. At the low volume limit, the assumption that small defects in the dielectric becomes false, as the defects are then better modeled as a discrete number of defects. This qualitatively changes the model, limiting the maximum amount of decoherence that is present.
Next, the paper tries to explain why the earlier models were incorrect, given the new data. They argue that the two-level state fluctuations are a result of charge fluctuations, rather than fluctuations of the critical current as was previously believed. This was verified by examining the distribution of the splitting sizes using the standard TLS tunneling model.
After this part, I get more and more lost in the rest of the paper! but one thing that I did find interesting was an application of Fermis’s golden rule to calculate the decay rate of the qubit at large-area junctions. It’s always quite cool to see equations that I’ve learned show up in modern scientific papers :)
The conclusion of the paper is that the dielectric is always going to be lossy, but if you make the insulator small enough, it won’t be lossy enough to make a huge difference. Instead, the loss will plateau at some point, and therefore be manageable.