# Meeting notes 11 08 22

Aram: Working on showing that random quantum circuits are poly(n)-designs, with Fernando.

Johnny: Why does making a projective measurement and forgetting the outcome increase entropy? One explanation is that the measurement can be modeled as applying a random phase, and then we can appeal to the concavity of the entropy function.

Tom: Going to rewrite code in C. But maybe it should be profiled first.

Isaac: Rowan isn't here, but he's working on exactly diagonalizing Hamiltonians, which is working up to n=20 or so. Now is looking at quantum Monte Carlo, which gives rise to the 2-D classical Ising model. On the theory side, we need to sample from closed paths on a graph. Also, what about turning local Hamiltonians into frustration-free Hamiltonians, like Steve talked about? This is done as follows. Suppose that $H=\sum_i H_i$, with ground state energy 0, gap $\Delta$ and the diagonals of local terms adjusted so that their expectation w.r.t. the ground state is 0 (although they are not necessarily p.s.d.). Now define $\tilde H_i = \int {\rm d}t e^{iHt} H_i e^{-iHt} w(t)$, where w(t) has Fourier transform satisfying $\hat w(\epsilon) = \left\{\begin{matrix}1 & \text{for }|\epsilon|<\Delta \\ 0 & \text{otherwise}\end{matrix}\right.$

If the original Hamiltonian is stoquastic, then is the modified Hamiltonian as well? Note: this technique is known as apodization.

David: Looking at Euclidean domains, which are integral domains with a division algorithm, which in turn implies a pre-order on the elements of the ring.

Previously, had looked at the condition that R/<x> is a PID. However (Isaac) this implies that R is a field.

Also, what if we assume that x is an element of R that can be written as the product of two primes, p and q. Then $R/\langle x \rangle \cong \left\langle\left\langle p + \langle x\rangle \right\rangle\right\rangle \oplus \left\langle\left\langle q + \langle x\rangle \right\rangle\right\rangle$, and $R/\langle q \rangle$ is a field. Unfortuantely it's hard to access this without knowing p,q.

Kamil: Searching for oracle speedups. There are $2^{2^n}$ oracles that take n bits of inputs, and the number of subsets of this is $2^{2^{2^n}}$. Wow. For each one, we want to find the optimal quantum algorithm, and the optimal classical algorithm, for the problem.

The format of group meeting is also up in the air. One idea is to have shorter meetings 1/2 or 3/4 of the time, with occasional longer meetings. Another is to have group meeting from 5-7 with a pizza break in the middle. Another idea is to have a problems wiki.