### Homework 2: Algorithmic Perspective

Although I've tried to make things correct/clear there are occassionally typos/confusing wording/etc. When there is a discrepancy between the problem I clearly mean to be asking and the one I've asked, please solve the former. The piazza website is a good place to help us clarify the problem statements.

When answering questions please write out in a complete and explanatory way what you're doing. Someone who doesn't know what's going on should be able to read it and understand (so in addition to equations, use words!).

#### Problem 1: Uniform RVB

In class we've seen that you can represent the uniform RVB state as a

• bosonic RVB wave-function
• fermionic RVB wave-function
• PEPS

Explicitly write out the uniform RVB states in these three different forms (you may choose which boundary conditions you want for each representation). For the bosonic and fermionic RVB wave-function you need to establish the values of $f_{ij}.$ (For the fermionic case, it is a nice exercise to figure out how to do this in general for planar states). For the PEPS, you need to explicitly write out a set of tensors whose contraction gives the amplitude for the uniform RVB state.

#### Problem 2: Write a variational Monte Carlo code

Follow the tutorial at [https://dl.dropboxusercontent.com/u/90045/site/Tutorials/VMCIntroTutorial/index.html] to write a variational Monte Carlo code translating appropriately from python to Julia where necessary.

#### Problem 3: Matrix Product States

This problem is designed to get you to think about conceptual questions with MPS.

• Our algorithms so far have required different matrices for each site even if we have a translationally invariant ground state.
• If I have a quantum state which is translationally invariant, can I have the same (pair of) matrices on each site in the MPS representation?
• If so, can I do it without the bond-dimension growing as the system size increases?
• (optional) If I have a quantum state which is not translationally invariant, can I have the same (pair of) matrices on each site in the MPS representation?
• Hint: Consider the quantum state $|000..\rangle + |111...\rangle$
• Consider the algorithm we discussed in class for converting from a state represented as a vector to a MPS. If there is a low entanglement rank over some cut, will this algorithm ensure that the size of the matrix in the MPS framework at that site is bounded by the entanglement rank (assuming I always truncate the matrices when my singular values are zero).

• Can you use the gauge freedom to make every site simultaneously left and right canonical?

• Write out an MPS for a one-dimensional Bosonic valence bond state that spans sites (1,6) (2,5) (3,4).