Adding Support for Operator Functions

[Gavin-Rockwood]

I am wondering if there is any plans add support for functions of operators, particularly exponential and trig. I tried doing this myself but I don't have a good enough understanding of how Symbolics.jl works! The reason I ask this is because I work a lot with superconducting circuits and am looking for a way to represent them abstractly in symbolic Hamiltonians that can be easily manipulated before I turn them into truncated matrices for numerical calculations. The kind of work flow I imagine is something starting with a transmon+resonator Hamiltonian

$H =4E_C \hat{n}^2-E_J\cos\hat\phi + \omega \hat{a}^\dagger\hat{a}+g\hat{n}(\hat{a}^\dagger+\hat{a})$

and then being able to do stuff like rewriting $\hat{n}$, $\hat\phi$ in terms of bosonic operators $\hat{b}, \hat{b}^\dagger$, taylor expanding $\cos\phi$ and even performing conjugations $e^{-\epsilon \hat{O}}He^{\epsilon \hat{O}}$ which can be expanded to some order in $\epsilon$ using Baker-Campbell-Hausdorff (such as doing a Schrifer-Wolff transformation).

[ChristophHotter]

Hi @Gavin-Rockwood

We do not really have plans for this at the moment, but I think this could be really useful. So if you would like to implement this I would be happy to help.

Defining general rules for exp(c*op) would already cover a lot. To do this, I think, one needs to define exp(x::QNumber) and then implement a rule exp(c*op1)*op2 -> ... which depends on the operators op1 and op2. You can look into fock.jl for the implementation of rewriting rules.

@david-pl I remember that you did something like this in the past. Do you have some comments?

[david-pl]

I looked into this briefly, but only in the context of cumulant expansion. Since the function you're referring here are infinite power series of operators, it's not easy to generalize this. I did come up with some specific solutions, such as a cumulant expansion of $\langle\cos(x)\rangle$, but in the end nothing really came out of it.

Now that we have this package separated from QuantumCumulants, it makes more sense to have these functions in place. Even though we won't be able to support them directly in QuantumCumulants.

[Gavin-Rockwood]

Hey! I am just getting back to this project for this I remembered I never responded. I think before I go in and start implementing some of this here, I might give some motivation as to why I'm interested along with some of the other symbolic stuff I want to implement.
A lot of my work is in superconducting circuits and I am working on a package for related calculations. I am hoping to have an analytic component, where circuits can be defined in terms of charge + phase operators (or X, P ops) and then relevant symbolic calculations can be performed on these Hamiltonian. Part of that includes approximate SW transformations, RWA approximations and some other unitary operations. I was thinking about using SQA as the fundamentals for that but I am not sure about how easy it would be to add the the required operators and functionality. Ultimately the idea here would be to have a symbolic component that can be seamlessly integrated into something like QuantumToolbox (or maybe iTensor) for numerical calculations.

I would be curious to know if expanding some of the functionality of SQA in this direction to do more general quantum algebra might be of interest.