Quantum simulation from the bottom up: the case of rebits

Typically, quantum mechanics is thought of as a linear theory with unitary evolution governedby the Schrödinger equation. While this is technically true and useful for a physicist, with regardsto computation it is an unfortunately narrow point of view. Just as a classical computer can simu-late highly nonlinear functions of classical states, so too can the more general quantum computersimulate nonlinear evolutions of quantum states. We detail one particular simulation of nonlinearityon a quantum computer, showing how the entire class of R-unitary evolutions (on n qubits) can besimulated using a unitary, real-amplitude quantum computer (consisting of n+ 1 qubits in total).These operators can be represented as the sum of a unitary and antiunitary operator, and add anintriguing new set of nonlinear quantum gates to the toolbox of the quantum algorithm designer.Furthermore, a subgroup of these nonlinear evolutions, called the R-Cliffords, can be efficientlyclassically simulated, by making use of the fact that Clifford operators can simulate non-Clifford(in fact, non-linear) operators. This perspective of using the physical operators that we have tosimulate non-physical ones that we do not is what we call bottom-up simulation, and we give someexamples of its broader implications.