Explicit Quantum Circuit for Simulating the Advection-Diffusion-Reaction Dynamics

TL;DR

This paper evaluates the convergence of Carleman linearization for ADR equations, analyzes quantum implementation challenges, and proposes a block-encoding approach to reduce complexity, highlighting the current hardware limitations.

Contribution

It introduces a quantum algorithm framework for ADR equations using Carleman linearization and block-encoding, addressing complexity issues and implementation feasibility.

Findings

Five Carleman iterates suffice for broad parameter ranges

Carleman ADR matrix requires exponential Pauli gates, hindering current hardware implementation

Block-encoding reduces complexity from exponential to polynomial

Abstract

We assess the convergence of the Carleman linearization of advection-diffusion-reaction (ADR) equations with a logistic nonlinearity. It is shown that five Carleman iterates provide a satisfactory approximation of the original ADR across a broad range of parameters and strength of nonlinearity. To assess the feasibility of a quantum algorithm based on this linearization, we analyze the projection of the Carleman ADR matrix onto the tensor Pauli basis. It is found that the Carleman ADR matrix requires an exponential number of Pauli gates as a function of the number of qubits. This prevents the practical implementation of the Carleman approach to the quantum simulation of ADR problems on current hardware. We propose to address this limitation by resorting to block-encoding techniques for sparse matrix employing oracles. Such quantum ADR oracles are presented in explicit form and shown to turn the exponential complexity into a polynomial one. However, due to the low probability of successfully implementing the nonunitary Carleman operator, further research is needed to implement the multi-timestep version of the present circuit.