# Meeting notes 11 07 18

**Paper-reading party**: Wed at 4pm in the theory lab.

*Greg*: About to go off and give a talk about CodeQuest. His talk has awesome animations.

*Tom*: Implementing optimizations over k-extendable states for general dimensions.

*Rowan*: Implementing iterative eigenvalue maximization. Greg suggests using ARPACK.

*Isaac*: Need to bound Markov chain mixing times. This paper was considered, but its setting is somewhat different.

*Paul*: Working on 2-D architectures for classical reversible arithmetic. One technique used is a 3:2 adder, which allows us to write a+b+c=u+v, where u,v each have one more bit than a,b,c do. Layouts could be

- linear nearest neighbor (LNN), e.g. on an ion trap.
- 2-D, like in a superconducting circuit

There is a diagram that I won't reproduce, but it gives a plausible way of implementing this in 2-D. Questions:

- Is this circuit optimal? Can computer search improve it?
- Are 7:3 or <math>2^n-1:n</math> adders ever better than iterating 3:2 adders?
- What role do long-range links play?

*Melanie*: Looking at quantum walks on welded trees, and considering their generalization to cone graphs.

*David*: Looking at lower bounds for the loops problem.

*Johnny*: Given 2- or 3-qubit interactions, can we simulate a many-body interaction? A *k*-local Hamiltonian is defined to be a Hamiltonian on *n* qubits that is a sum of terms, each acting on at most *k* qubits. One important problem is to determine the ground-state energy. Kempe-Kitaev-Regev proved that it is QMA-complete to determine whether this energy is <math>\leq a</math> or <math>\geq b</math>, where <math>b-a\geq 1/{\rm poly}(n)</math>.

**Crucial Lemma**: <math>H=H_1 + H_2</math> These act on a space <math>S\oplus S^\perp</math>. Let <math>\lambda(\cdot)</math> denote the smallest eigenvalue of a matrix. Assume that <math>\lambda(H_2|_{S^\perp}) = J > 2 \|H_1\|</math>. Then

<math>\lambda(H_1|_S) - \frac{\|H_1\|^2}{J - 2 \|H_1\|} \leq \lambda(H) \leq \lambda(H_1|_S) </math>

This is used as follows. Let <math>\tilde H = H+V</math>. Let <math>\lambda_1\leq \cdots \leq \lambda_d</math> be the eigenvalues of H and <math>|\psi_1\rangle, \ldots, |\psi_d\rangle</math>. Similarly, let <math>\tilde\lambda_1\leq \cdots \leq \tilde\lambda_d</math> be the eigenvalues of <math>\tilde H</math> and <math>|\tilde\psi_1\rangle, \ldots, |\tilde\psi_d\rangle</math>. Define the resolvent <math>\tilde G(z) = (zI-\tilde H)^{-1} = \sum_j (z-\tilde \lambda_j)^{-1} |\tilde\psi_j\rangle\langle\tilde\psi_j|</math>.

Pick a threshold <math>\lambda_*</math> and define <math>L_\pm</math> to be the subspaces corresponding to eigenvalues of H larger/smaller than <math>\lambda_*</math>. Define now <math>\Sigma_-(z) = zI_- - \tilde G_{--}^{-1}(z) </math>. Then for appropriate values of z, <math>\Sigma_-(z) \approx H_{\rm eff}</math>.

*Lukas*: Doing MUSIQC code with Georgia Tech. Issues of 'which encoded qubit goes on which wire.' Next up, diagonalizing the quantum transistor, and finding a good architecture for it.

*Kamil*: Superconducting Hamiltonians are like the Hamiltonian one gets for a system of coupled pendulums. Can we understand adiabatic evolution in the transitor in this model? Also, can we use this to simulate arbitrary single-particle potentials? (Isaac: this may be related to the idea of a 'universal' differential equation.)

In general, is looking at connections between classical and quantum dynamics. In some cases, this may give better classical simulations of quantum systems.

Also, has been looking at gaps and phase transitions, and the Ising model on a hypercube. The question is again how gaps protect quantum information. Wants a quantum automata that protects quantum information.