On the power quantum computation over real Hilbert spaces

TL;DR

This paper investigates the computational power of quantum complexity classes when restricted to real Hilbert spaces, showing that many classes retain their power under this restriction, similar to the complex case.

Contribution

It proves that several quantum complexity classes over complex spaces have equivalent power when restricted to real Hilbert spaces.

Findings

BQP over reals equals BQP over complex numbers

QMA(k), QIP(k), QMIP, and QSZK retain their power over reals

Complex-to-real transformation preserves computational class power

Abstract

We consider the power of various quantum complexity classes with the restriction that states and operators are defined over a real, rather than complex, Hilbert space. It is well know that a quantum circuit over the complex numbers can be transformed into a quantum circuit over the real numbers with the addition of a single qubit. This implies that BQP retains its power when restricted to using states and operations over the reals. We show that the same is true for QMA(k), QIP(k), QMIP, and QSZK.