Quantum Stochastic Walks

We just posted a paper in the arXiv.

Quantum stochastic walks: A generalization of classical random walks and quantum walks

We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions of a vertex as defined from its connectivity. We show how the family of possible QSW encompasses both the classical random walk (CRW) and the quantum walks (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on the line, its QW to CRW transition and transitions to genearlized QSWs that go beyond the CRW and QW. QSWs provide a new framework to the study of quantum walks with environmental effects as well as quantum algorithms.

I promise a simple explanation of Classical Random Walks soon!

What is Time-Dependent Density Functional Theory?

Time-Dependent Density Functional Theory looked like a mess when it was first explained to me. I probably made the face of a person who smells sushi for their first time, wondering if this is some sort of bad joke.

After all, quantum mechanics is supposed to be a theory of non-commuting observables that evolve in a linear fashion. TD-DFT is nothing like that, yet it claims to reproduce all the same effects. Fishy indeed.

TD-DFT first focuses on the density of the wave function, in particular, the position basis of it. This is relevant for chemical calculations where it is very important to know where are the electrons. Of course, the density is one of many observables that are relevant, but TD-DFT makes it stand out by letting this observable evolve by means of a functional of itself. In other words, you don’t fully evolve the wave function by means of an operator, but instead you have a very complicated, non-linear functional that takes as its input the density and lets it evolve. In practice, since the functional is non-linear, in practice, the evolution is done iteratively.

Runge and Gross proved that if you only cared about the evolution of one observable, the density, this procedure is equivalent to the full quantum mechanical evolution. In other words, you can map the evolution of a particular observable the wavefunction under Schrodinger’s equation into a functional of the same observable.

What you gain from this approach is a computational speedup. The prize paid is that writing the exact functional is actually a very hard problem, at least as hard as doing the full quantum mechanical evolution. However, in practice, approximated functional can be written down and used for real calculations that can predict properties for real materials. This technique is widely used, mostly as a black box toolkit used by many physical chemists around the world.

In our latest papers, we were able to show that this mapping can be also performed for open quantum systems instead of just Schrodinger’s equation. First, we developed the general theory of how the Runge-Gross theorem can be generalized, placing it in context of previous incomplete attempts. This paper was published in PCCP as a Hot Article. In it, we discuss how the theorem works even in the highly non-Markovian regime of an open quantum system.

In our second paper, we take this even further. The evolution of an observable of an open quantum system can be mapped to a functional for a close system. At first, this seemed counter-intuitive. After all, you cannot map the evolution of an open system into a closed system.

O te peinas o te haces rolos.

However, if you only care about one observable, and you are willing to use non-linear functionals, this can be done consistently, for just that observable. Since most of the code written for TD-DFT was for closed systems, our results shows that those techniques could be used to model open quantum systems. We feel that new chemical calculations with thermodynamic effects can now be explored with this theory.

Dr. Strangelove: It is not only possible, it is essential.

Our paper chosen as PCCP Hot Article!

Our paper

Time-dependent current-density functional theory for generalized open quantum system

Joel Yuen-Zhou, César Rodríguez-Rosario and Alán Aspuru-Guzik

In this article, we prove the one-to-one correspondence between vector potentials and particle and current densities in the context of master equations with arbitrary memory kernels, therefore extending time-dependent current-density functional theory (TD-CDFT) to the domain of generalized many-body open quantum systems (OQS). We also analyse the issue of A-representability for the Kohn–Sham (KS) scheme proposed by DAgosta and Di Ventra for Markovian OQS [Phys. Rev. Lett. 2007, 98, 226403] and discuss its domain of validity. We suggest ways to expand their scheme, but also propose a novel KS scheme where the auxiliary system is both closed and non-interacting. This scheme is tested numerically with a model system, and several considerations for the future development of functionals are indicated. Our results formalize the possibility of practising TD-CDFT in OQS, hence expanding the applicability of the theory to non-Hamiltonian evolutions.

TD-CDFT and Open Quantum Systems
TD-CDFT and Open Quantum Systems

has been published on PCCP and chosen as a Hot Article !

SQuInT, APS, Publications

I’m back.

After a long, tough, wintery and busy month, I’m back.

Progress report follows.

Alright, first, I went to SQuInT. The Southwestern Quantum Information and Technology conference isn’t true to its name. It was held in the Northwest, Seattle, where beautiful weather seemed to tunnel through the mountains’ potential just for us. The conference itself was very productive and I had the opportunity to see family, friends and collaborators.

In other news, we submitted a paper on Open Quantum Systems and Time Dependent Current Density Functional Theory titled Time-dependent current-density functional theory for generalized open quantum systems to the journal Physical Chemistry, Chemical Physics (PCCP). It has been accepted for publication and might appear in a special issue on Time Dependent Density Functional Theory.

We also submitted a related paper to another journal, paper titled Time-Dependent Density Functional Theory for Open Quantum Systems using Closed Systems. You can read it in the arXiv.

Finally, I went to the APS March Meeting, where 7,000 physicists took over the city of Pittsburgh, where I was able to find bars decorated with Roberto Clemente posters, where a restaurant served Carrucho (Conch). Sometimes I feel the APS March meeting is too big, too overwhelming, talks are too short, and there is too much going on simultaneously. But then I’m surprised by meeting people I hadn’t seen in almost 10 years now, and by how APS March meeting always lead to new collaborations.

Exciting times these are.

“I have wept three times in my life. Once when my first opera failed. Once again, the first time I heard Paganini play the violin. And once when a truffled turkey fell overboard at a boating picnic.”
-Gioachino Rossini