A bottleneck and largest-weight greedy Birkhoff-von Neumann decomposition reduces LCU permutation terms from O(N²) to O(N log(1/ε)) or ~2N, halving ancilla qubits while setting normalization constant α=1.
Explicit quantum circuits for block encodings of certain sparse matrices,
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Lowering LCU Circuit Width through Maximum-Weight Birkhoff-von Neumann Decomposition
A bottleneck and largest-weight greedy Birkhoff-von Neumann decomposition reduces LCU permutation terms from O(N²) to O(N log(1/ε)) or ~2N, halving ancilla qubits while setting normalization constant α=1.