Distributed Phase Tags: A Locality-Preserving Realification Compiler for Misty-State Quantum Programs

Abstract

Complex amplitudes can be simulated over real Hilbert spaces by adjoining a two-dimensional real/imaginary degree of freedom. In a multipartite program, however, one globally shared phase tag is not a locality-preserving program transformation: every party would have to act on the same auxiliary system. We recast the distributed real simulation of multipartite quantum mechanics as a syntax-directed compiler into a real misty-state intermediate language. For $n$ parties we give explicit phase-reference codewords $|0\rangle_{P_n}$ and $|1\rangle_{P_n}$ on $n$ local tags and prove that the same real operator $J=XZ$ on any single tag realizes the logical action of multiplication by $i$. If a complex party-local gate is $A=A_R+iA_I$, its real translation is \[ \widehat A_j=A_R\otimes I+A_I\otimes J_j, \] which acts only on party $j$'s data and local phase tag. We prove an intertwining theorem for arbitrary finite party-local pure-state circuits, exact preservation of computational-basis measurement probabilities, preservation of the run-time interaction graph, and one-rebit-per-party overhead. For the canonical three-party GHZ program, the translation lies entirely inside the ordinary black-and-white misty gate fragment: controlled-$J$ replaces the complex phase operation, while the invariant $J_jJ_k|0\rangle_P=-|0\rangle_P$ transfers the required branch sign. This yields a short correctness proof for all four promised GHZ inputs and explains the earlier 64-path cancellation calculation at a single logical phase-code boundary. We distinguish this program realification problem from the weaker task of finding any real strategy with the same correlations, and state precisely which parts are known real-Hilbert-space simulation and which parts belong to the misty-language compilation layer developed here.