Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

TL;DR

This paper introduces an efficient method for loading and decomposing the linearized representation of nonlinear systems on quantum computers, addressing post-linearization challenges with a novel Sigma basis approach.

Contribution

It proposes a new data access model and decomposition technique using Sigma basis to improve quantum simulation of nonlinear systems after linearization.

Findings

Sigma basis reduces decomposition terms exponentially

Efficient Hamiltonian loading up to truncation order N

Constructs circuits for each tensor component using unitary completion

Abstract

The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order $N$, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component $\mathcal{H}_{j}$ of the decomposition.