Phase-Encoding and Sequence-Algebraic Algorithms for Matrix Permanents:Exact Exponent Improvements and Structured Approximations

摘要: The permanent of an N*N matrix is a central but computationally hard quantity. Exact evaluation is #P-complete and requires O(N 2^N) operations by Ryser or Glynn algorithms. We introduce a unified phase-encoding and sequence-algebraic framework that reformulates the permanent’s combinatorial structure via phase-modulated sequence aggregates inspired by coding theory. This yields (i) an exact deterministic algorithm with complexity O(2^{N-4} poly(N)), improving the exponential factor, and (ii) a structured polynomial-time approximation scheme for nonnegative matrices based on phase-encoded decompositions and ratio-calibrated estimation. Numerical results up to N=20 validate both accuracy and efficiency. The framework also suggests structural connections relevant to boson-sampling and quantum optical inference.