Preliminary description of DBR duration scheduling

examples/dbi/README.md

@@ -36,14 +36,18 @@ For theoretical considerations the canonical bracket is useful.
 For this we need the notation of the dephasing channel $\Delta(H)$ which is equivalent to `np.diag(h)`.
  $M = [\Delta(H),\sigma(H)]= [H,\sigma(H)]= [\Delta(H),H]$
  The canonical bracket appears on its own in the monotonicity relation above and gives an unconditional reduction of the magnitude of the off-diagonal terms
- $$||\sigma(H_0(s))||^2- ||\sigma (H_0 )||^2= -2s \|M\|^2+O(s^2)$$
+ $$||\sigma(H_0(s))||^2- ||\sigma (H_0 )||^2= -2s ||M||^2+O(s^2)$$
 - the multi qubit Pauli Z generator with $Z(\mu) = (Z_1)^{\mu_1}\ldots (Z_L)^{\mu_L}$ where we optimize over all binary strings $\mu\in \{0,1\}^L$
 - the magnetic field $D = \sum_i B_i Z_i$
 - the two qubit Ising model $D  = \sum_i B_i Z_i + \sum_{i,j} J_{i,j} Z_i Z_j$, please follow the tutorial by Matteo and use the QIBO ising model for that with $h=0$
 
 
 ### How to choose s?
 
+The theory above shows that in generic cases the DBR will have a linear diagonalization effect (as quantified by $||\sigma(H_0(s))||$).
+This can be further expanded with Taylor expansion and the Qibo implementation comes with methods for fitting the first local minimum.
+Additionally a grid search for the optimal step is provided for an exhaustive evaluation and hyperopt can be used for a more efficient 'unstructured' optimization; additionally simulated annealing is provided which sometimes outperforms hyperopt (and grid search), see example notebooks.
+The latter methods may output DBR durations $s_k$ which correspond to secondary local minima.