Gate Optimization in DiagCoulombEvolutionJW

[jamie-k-kim]

Changed the gate decomposition in the number operator representation (_diag_coulomb_evo_num_rep_jw). Right now it yields PhaseGates and CPhaseGates for every diagonal Coulomb term. This new approach accumulates all single-qubit Z-rotations into a coefficient array (z_coeffs), applies them all at the start using a single PhaseGate per qubit, and then uses RZZGates for the two-qubit interaction terms.

By the Jordan-Wigner mapping for number operators, $n_p = \frac{I-Z_p}{2}$ and the interaction term $n_p n_q$ expands to $\frac{I - Z_p - Z_q + Z_p Z_q}{4}$. So for single-qubit Z terms, each term $Z_{pq} n_p n_q$ contributes a factor of $\frac{t Z_{pq}}{4}$ to the $Z_p$ and $Z_q$ phases. These are accumulated via coef = time * this_mat[i, j] / 4.0. And for two-qubit ZZ terms, each term contributes $\exp\left(-i \frac{t Z_{pq}}{4} Z_p Z_q\right)$, which is equivalent to RZZGate(0.5 * time * Z_pq).

For a 50-orbital system with 100 trials:

	Avg. CX count	Avg. RZ count	Avg. depth
Current	4883.04	4883.04	1541.07
New	4842.04	4842.04	1414.03

Transpilation speedup: 24.9% (MacBook Pro M1)

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