Add several new expressions to planning doc

docs/planning.md

@@ -5,18 +5,26 @@
 
 - Pauli Matrix $\sigma_i \in \{ I,X,Y,Z \}$
 - Pauli String $\mathcal{\hat{P}} = \bigotimes_i \sigma_i$
+- State vector ${\ket{\psi}}$ and a set of ${n}$ state vectors $\ket{\psi_t}$ represented as columns in matrix
 - Sum of weighted Pauli strings (currently called `PauliOp`) $A_k = \sum_i h_i \mathcal{\hat{P_i}}$
 - Sum of summed weighted Pauli strings (currently called `SummedPauliOp`) $B = \sum_k \sum_i h_{ik} \mathcal{\hat{P_i}}$
 
+
 # List of Operations
 
 Here's a terse list of the type of operations we want to support in `fast_pauli` (this list will grow over time):
 
-1. Pauli String to sparse matrix (Pauli Composer)
-2. $\mathcal{\hat{P}} \ket{\psi}$
-3. $\mathcal{\hat{P}} \ket{\psi_t}$
-4. $\big( \sum_i h_i \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
-5. $\big(\sum_k \sum_i h_{ik} \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
-6. $\big(\sum_k x_{tk} \sum_i h_{ik} \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
-7. $\bigg(\sum_k \big( \sum_i h_{ik} \mathcal{\hat{P}}_i \big)^2 \bigg) \ket{\psi_t}$
-8. Calculate $\bra{\psi_t} \{ \mathcal{\hat{P_i}}, \hat{A_k} \} \ket{\psi}$ and $\bra{\psi} \mathcal{\hat{P_i}} \ket{\psi}$
+1.  Pauli String to sparse matrix (Pauli Composer)
+2.  $\mathcal{\hat{P}} \ket{\psi}$
+3.  $\mathcal{\hat{P}} \ket{\psi_t}$
+4.  $\bra{\psi_t} \mathcal{\hat{P_i}} \ket{\psi_t}$
+5.  $\bra{\psi_t} \mathcal{x_{ti}\hat{P_i}} \ket{\psi_t}$
+6.  $\big( \sum_i h_i \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
+7.  $\big(\sum_k \sum_i h_{ik} \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
+8.  $\big(\sum_k x_{tk} \sum_i h_{ik} \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
+9.  $\big(\sum_k ( \sum_i h_{ik} \mathcal{\hat{P}}_i )^2 \big) \ket{\psi_t}$
+10. $\bra{\psi_t} \{ \mathcal{\hat{P_i}}, \hat{A_k} \} \ket{\psi_t}$
+11. $\bra{\psi_t} ( \sum_i h_{ik} \mathcal{\hat{P}}_i ) \ket{\psi_t}$
+12. $\bra{\psi_t} \big(\sum_k x_{tk} \sum_i h_{ik} \mathcal{\hat{P}}_i \big) \ket{\psi_t}$
+13. $\bra{\psi_t} ( \sum_i h_{ik} \mathcal{\hat{P}}_i )^2 \ket{\psi_t}$
+14. $\bra{\psi_t} \big(\sum_k ( \sum_i h_{ik} \mathcal{\hat{P}}_i )^2 \big) \ket{\psi_t}$