Difference between revisions of "Meeting notes 11 08 22"

Revision as of 01:40, 23 August 2011

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 <math>H=\sum_i H_i</math>, with ground state energy 0, gap <math>\Delta</math> 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 <math>\tilde H_i = \int {\rm d}t e^{iHt} H_i e^{-iHt} w(t) </math>, where w(t) has Fourier transform satisfying <math>\hat w(\epsilon) = \left\{\begin{matrix}1 & \text{for }|\epsilon|<\Delta \\ 0 & \text{otherwise}\end{matrix}\right.</math>

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 <math>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 </math>, and <math>R/\langle q \rangle</math> is a field. Unfortuantely it's hard to access this without knowing p,q.

Kamil: Searching for oracle speedups. There are <math>2^{2^n}</math> oracles that take n bits of inputs, and the number of subsets of this is <math>2^{2^{2^n}}</math>. 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.