Double-bracket quantum algorithms find a clear place on the quantum algorithms landscape

As a newbie in quantum algorithms, when Marek first told me about double-bracket compilations, a question that never went away was how did it relate to the enormous field of existing techniques out there. This year, looking back, I feel very pleased that we’ve understood this question with a much greater clarity.

It started with the discovery (which our team would like to consistently credit to David Gosset for pointing us to) that double-bracket flow equations (which are essentially non-linear Schrodinger equations) are solved by imaginary-time evolution. This clarifies a central role of DBQAs in quantum simulation: it gives ITE a clear implementation recipe, that comes with theoretical performance guarantees. Our article with collaborators from EPFL and Fraunhofer Berlin studies this connection comprehensively, establishing both theoretical and numerical findings on how well DBQAs work for cooling! In particular, we also spelt out the fact that DBFs are gradient descents on a Riemannian manifold, something that was already observed by Brockett in his seminal results from 1991.

After this, the insights kept increasing: after a series of intense interactive sessions, Marek, Yudai and Jeongrak step-by-step established that one can implement arbitrary polynomials quite well, and do quantum signal processing with DBQAs. The connection to Grover’s algorithm also became increasingly clear in the process!

On the overall, this was a satisfying learning journey for all of us at the in.Q algorithms subgroup. I hope that the community of quantum algorithms will benefit from the DBQA approach. In particular, using the analytics captured by a non-linear evolution, we have a better handle on designing quantum algorithms that would avoid problems such as overfitting and post-selection.