For reference, I think the story in Hunter and Nachtergaele’s “Applied Analysis”, , chapters 6-7 is really nice. But it’s too technical for my audience.

I feel like this is such a natural thing, and yet every reference I find either doesn’t give a coherent story for the Hilbert space picture or is too mathematically advanced for my students.

Join millions who have switched to Grok.

I have a reference request. I want a physics undergrad friendly book, freely available through a university library, which develops Fourier series/transforms from the perspective of L2 as a Hilbert space.

RT @MartinLaroo: How fast can quantum circuits compile group designs? Recent work showed that designs over the n-qu….

Exciting stuff!.

Time for @JointMath! Alongside Carlos Ortiz Marrero and Jason Saied, I’m organizing the sessions on variational methods in quantum computing on Thursday, and presenting in the 2nd session on topological insulators on Friday. Lots of great talks lined up, hope to see you there!.

RT @MvsCerezo: 🚨Hey Quantum Community!. LANL is looking for a postdoc to work on Quantum Error Correction!!. If you are interested or know….

RT @Isaac__kim: I have an opening for a Postdoc position this year. ( If you can spread the word, that will help me….

RT @MartinLaroo: Happy to share that our proof of the barren plateau - Lie algebra conjecture is now published on Nature Communications! h….

RT @jenseisert: Full classification of Pauli Lie algebras. We provide a comprehensive classification of Lie algebr….

RT @MvsCerezo: It's so nice to see this Quantum Computing Summer School published in PRX Quantum. Here we present tools to build equivari….

As for what's next: I'm excited to announce that I will be joining @UCBerkeley's math department as a Morrey Jr. Assistant Professor. Looking forward to continuing to explore the wonderful world of quantum!.

(14/14) Chapter 7 and 8 contain the meat of new results. We will be writing these up soon in a separate, much briefer submission--stay tuned!.

(13/14) Chapter 5 consolidates 3 main viewpoints of the AKLT chain: as a valence bond state, a finitely correlated state, and a matrix product state. AFAIK, there is no other resource which works through all 3 in the same place.

(12/14) Chapter 4 carefully works through the necessary theory of projective representations of compact Lie groups, so we can talk meaningfully about Borel group cohomology H^2(G,U(1)).

(11/14) Chapter 3 also revisits classic examples from spin systems and condensed matter with a rep theory bent. Again, common knowledge to the masters, but very difficult to find.

(10/14) Chapter 3 is a crash course in representation theory for a quantum context. I gave the most attention to a careful working of Spin and SU(2)--if you want to understand spin the way a mathematician does, I wrote this for you!.

(9/14) There are many gaps in the quantum spin systems literature, knowledge common to the masters but that which required a ton of work for me as a grad student to consolidate. This is the meat of Chapters 3-5.

(8/14) There's a lot more to say about new results in Chapters 7 and 8, including an extension of Ogata's definition of an SPT phase to compact Lie groups. But the other reason this thesis took so long was the background.

(7/14) We also prove that these ground states have a 2-local parent Hamiltonian. This is rather surprising: by increasing local dimension n, the ground states have unbounded correlation and injectivity length. But still, always a 2-local parent Hamiltonian.