Lowering LCU Circuit Width through Maximum-Weight Birkhoff-von Neumann Decomposition
Year: 2026
Type: article
Abstract: Any square matrix can be transformed into a doubly stochastic matrix via Sinkhorn scaling with diagonal matrices or simply completing to a larger dimensional matrix. Standard Birkhoff-von~Neumann and Pauli decompositions represent such matrices as linear combinations of $O(N^2)$ permutation or Pauli terms, leading to a large ancilla overhead in a quantum Linear Combination of Unitaries (LCU) implementation. We prove that a bottleneck variant of Birkhoff's algorithm reduces the number of permutations to $O(N\log(1/\varepsilon))$, where $\varepsilon$ is the $\ell_1$-norm approximation error of the reconstructed matrix, and demonstrate empirically that a largest‑weight greedy variant requires only $\approx 2N$ terms for dense matrices (the exact average observed is $\approx 2.4N$). The quadratic reduction in term count directly shrinks the ancilla register from $2\log_2 N$ to $\log_2 N$ qubits, shortens the SELECT circuit, and is especially valuable in fixed‑Hadamard LCU architectures whose success probability scales with $1/K$. The approach enables compact quantum implementations of dense operators appearing in optimal transport, non‑Hermitian simulation, and other settings amenable to Sinkhorn preconditioning. Furthermore, because the decomposition is a convex combination, the LCU normalization constant is exactly $\alpha = 1$, and the uniform superposition is an eigenvector of the target matrix with eigenvalue~1. This structure can be exploited to achieve high success probability without amplitude amplification in many practical scenarios, including quantum walks and Markov chain simulations.
Author: DASKIN, Ammar
Institution: Istanbul Medeniyet University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161848596