Algebraic Codification and Discrete Isomorphic State Propagation for Exact Synthesis of NCV Quantum Circuits

Abstract

Exact synthesis of quantum circuits seeks depth-optimal gate layouts but is fundamentally bottlenecked by the continuous-variable scaling of unitary matrices or complex amplitudes. In this paper, we propose a mathematically novel approach for the exact synthesis of NCV (NOT, CNOT, Controlled-V , Controlled-V †) quantum circuits. We demonstrate that the quantum state evolution of basis vectors under NCV logic does not require continuous variable tracking; instead, it is strictly isomorphic to a discrete 4-valued cyclic group. By codifying quantum amplitudes and relative phases into a single scalar integer S ∈ {0, 1, 2, 3}, we bypass fractional and imaginary components entirely. We further give a binary algebraic decomposition of this scalar, reducing quantum gate interactions to exact bitwise XOR continuous relaxations. This formulation maps the exact synthesis problem into a highly constrained, pure Mixed-Integer Linear Programming (MILP) model. We mathematically prove the validity of this discrete encoding by demonstrating that it flawlessly preserves the NCV group structure, achieving computational efficiency by eliminating traditional large-M relaxations and dual-variable tracking