Bravyi-Kitaev encoding for fermion-to-qubit mappings.

 The BK transform uses a Fenwick tree structure to achieve O(log n) Pauli
 weight per ladder operator, compared to O(n) for the Jordan-Wigner transform.

 This module delegates to MajoranaEncoding with Fenwick-tree index sets.

 Reference:
   Seeley, Richard, Love — "The Bravyi-Kitaev transformation for quantum
   computation of electronic structure" (arXiv:1208.5986)

 A purely functional Fenwick tree (binary indexed tree) parameterized by
 an associative combining operation.

 The tree stores partial aggregates over an array of n values, supporting:
   - point update   : O(log n)
   - prefix query   : O(log n)
   - range query    : via prefix difference when the combine has an inverse

 The index sets used by the Bravyi-Kitaev encoding (update, parity,
 occupation, remainder) all fall out naturally from the Fenwick tree
 structure, making this a good foundation for the BK transform.

 Generic fermion-to-qubit encoding via Majorana decomposition.

 Every encoding in this family maps a single fermionic ladder operator
 to a pair of Majorana operators (c, d) built from three index sets:

   U(j, n)  — update set:     qubits that flip when occupation of mode j changes
   P(j)     — parity set:     qubits encoding the parity n₀ ⊕ … ⊕ n_{j−1}
   Occ(j)   — occupation set: qubits encoding whether mode j is occupied

 The Majorana operators are:
   c_j = X_{U(j)∪{j}} · Z_{P(j)}
   d_j = Y_j · X_{U(j)} · Z_{(P(j)⊕Occ(j))∖{j}}

 And the ladder operators follow:
   a†_j = ½(c_j − i·d_j)     a_j = ½(c_j + i·d_j)

 Different choices of index-set functions yield different encodings:
   Jordan-Wigner :  U = ∅,              P = {0…j−1},     Occ = {j}
   Parity        :  U = {j+1…n−1},      P = {j−1}?,      Occ = {j−1,j}?
   Bravyi-Kitaev :  U, P, Occ from Fenwick tree structure

 General tree-based fermion-to-qubit encodings.

 This module implements two approaches:

 (1) Index-set approach (Havlíček et al. arXiv:1701.07072):
     Works correctly ONLY for Fenwick trees (where all ancestors > j).
     Uses Update/Parity/Occupation sets with the generic MajoranaEncoding.

 (2) Path-based ternary-tree approach (Bonsai: arXiv:2212.09731,
     Jiang et al.: arXiv:1910.10746):
     Works for ANY ternary tree. Constructs Majorana strings directly
     from root-to-leg paths. Each node has 3 descending links labeled
     X, Y, Z.  Each leg yields a Pauli string by collecting the labels
     along its root-to-leg path. Paired legs give the two Majoranas
     for each fermionic mode.

 Different tree shapes yield different encodings:
   - Chain (linear tree):   recovers Jordan-Wigner
   - Fenwick tree:          recovers Bravyi-Kitaev
   - Balanced ternary tree: achieves O(log₃ n) Pauli weight (optimal)