WORKLIST.md

Jun' 11

***Hypothesis :

*For ansatz of the form $H_{ev} = (1- \gamma) H_{prob} + \gamma H_{mix}$, tuning the $\gamma$ allows for the variation in the relative weight of the evolution between the two hamiltonians.

*We expect better results in sampling from a distribution generated by the Hamiltonian $H_{prob}$ by broadly two different approaches,

1. *Choosing the $H_{mix}$ specifically to respect the symmetries present in the $H_{prob}$
2. *And then fine tuning the relative weight $\gamma$ to enhance the sampling procedure

*Intuitively we expect that a specifically designed $H_{mix}$ would generate samples that are more representative of the typical set of the probability distribution corresponding to $H_{prob}$.

*However restricting the sample space considerably might substantially reduce the search space which we can get over by fine-tuning $\gamma$ over the range of the simulation. The paper suggests that the for perturbative values of $\gamma$ we should have the probability of transitions is given by, $\mathcal{P}(s \to s') : \propto ;: \frac{\gamma^2}{(E_{s}-E_{s'})^2} : \bra{s}H_{mix}\ket{s'} $ but similar analysis for different ranges of $\gamma$ is not known.

*We strongly believe that even for other ranges the transition probability depends significantly on the structure of $H_{mix}$, and thus suitably finding one would enhance the capability of your sampling scheme.

***Experiments :

1. **Confirming effect of $\gamma$ variation without enhanced (informed ?) $H_{mix}$ design.

   1. Take the usual $H^{v}{mix} = \Sigma{i=0}^{n} X_{i}$, (aka. vanilla mixer ) and obtain sampling results for a generic Random Ising Model hamiltonian.

   2. Average the effect over different instance of the randomly generated ising models.

   3. Run the simulations for different ranges of gamma, $\gamma_{perturbative} \approx {0.01}$, $\gamma_{low} \approx {0.1,0.2}$, $\gamma_{mid} \approx {0.5,0.6}$ , $\gamma_{high} \approx {0.8,0.9}$

      To observe :
      At high $\gamma$ the sampling algorithm will loose track of the information of the landscape it is supposed to sample. Aim to get an idea about the mid $\gamma$ range in which, we still manage to get good samples, or an upper bound such that an $\gamma > \gamma_{max}$ will fail to produce good samples.

2. **Confirming the effect of enhanced $H_{mix}$ design, for Random Ising Models, with $\gamma$ variation.

   1. Design $H_{mix}$ from the set of multi qubit interactions i.e ${ X_{i}X_{i+1}, X_{i}X_{i+1}X_{i+1}, X_{i}X_{j},}$. Restrict design to $X$ mixers only, for better interpretebilty.

   2. Check the sampling results for the choice of enhanced mixers against their pauli weight

   3. Rerun the simulations for different ranges of gamma, $\gamma_{perturbative} \approx {0.01}$, $\gamma_{low} \approx {0.1,0.2}$, $\gamma_{mid} \approx {0.5,0.6}$ , $\gamma_{high} \approx {0.8,0.9}$

      To observe : Using $H_{mix}$ of different pauli weight allows for transitions of higher hamming distance $d_{h}$, though this would allow for transversal of local minima this does not directly guarantee a better sampling of the the energy landscape because the algorithm would fail to make small transitions when required thus bypassing the ground states.

      We need to find a strategy of picking the right $H_{mix}$ (or a scheme of varying it over iterations ) combined with the information about the mid $\gamma$ range from the previous experiment this will let us balance between the different types of transitions to drive the sampling algorithm more accurately through the landscape.

3. **Confirming the effect of enhanced $H_{mix}$ design, for structured datasets, with $\gamma$ variation.

   1. Get an close approximation of the ideal Hamiltonian $H_{D}$ for the dataset $D$, by some means (?).
   2. Design $H_{mix}$ such that it enhances transitions $\ket{s} \to \ket{s'}$, such that ${\ket{s}, \ket{s'}}; \in D$ , over other transitions.
   3. Compare the sampling, by using $H_{D}$ as the problem hamiltonian and comparing the effect of using an enhanced $H_{mix}$ over using vanilla mixer $H^{v}_{mix}$ .

Meeting Aug'20

• run local sampling for multiple 3 secs
• sampling with interleaved mixers of different weights