Quantum simulation from the bottom up: the case of rebits

TL;DR

This paper demonstrates how a real-amplitude quantum computer can simulate nonlinear quantum evolutions, expanding the quantum computing toolbox and offering new insights into bottom-up simulation of non-physical operators.

Contribution

It introduces a method to simulate all real-unitary evolutions on qubits using an extended unitary, real-amplitude quantum computer, and explores the classical simulability of a subgroup called R-Cliffords.

Findings

Real-amplitude quantum computers can simulate nonlinear quantum evolutions.

R-Clifford subgroup can be efficiently classically simulated.

The approach broadens the scope of quantum simulation and computation.

Abstract

Typically, quantum mechanics is thought of as a linear theory with unitary evolution governed by the Schr\"odinger equation. While this is technically true and useful for a physicist, with regards to computation it is an unfortunately narrow point of view. Just as a classical computer can simulate highly nonlinear functions of classical states, so too can the more general quantum computer simulate nonlinear evolutions of quantum states. We detail one particular simulation of nonlinearity on a quantum computer, showing how the entire class of $\mathbb{R}$-unitary evolutions (on $n$ qubits) can be simulated using a unitary, real-amplitude quantum computer (consisting of $n+1$ qubits in total). These operators can be represented as the sum of a linear and antilinear operator, and add an intriguing new set of nonlinear quantum gates to the toolbox of the quantum algorithm designer. Furthermore, a subgroup of these nonlinear evolutions, called the $\mathbb{R}$-Cliffords, can be efficiently classically 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 to simulate non-physical ones that we do not is what we call bottom-up simulation, and we give some examples of its broader implications.