Quantum algorithm for the collision-coalescence of cloud droplets

TL;DR

This study explores the use of quantum computing for calculating the collision-coalescence process of cloud droplets, which dominates the size growth of liquid particles in the cloud microphysics.

Abstract

Quantum computing is gaining attention as a new approach for solving complex problems in many scientific fields. In atmospheric and oceanic sciences, it may help reduce computational costs of simulating large and nonlinear systems. However, research into the use of quantum computers in this area is still in its earlier stage, and suitable applications have not been established yet. This study explores the use of quantum computing for calculating the collision-coalescence process of cloud droplets, which dominates the size growth of liquid particles in the cloud microphysics. Inspired by the quantum algorithms developed in the field of financial engineering, we propose a new algorithm based on a master equation that describes the time evolution of the droplet mass distribution. Our algorithm uses the quantum amplitudes to encode the probability distribution of droplet mass and calculates the expected number of droplets via the quantum amplitude estimation. Our resource analysis shows that the number of T gates scales as $O(N^2)$, where $N$ is the number of bins of the mass distributions. This is an essential improvement over the classical methods that scale only exponentially with $N$. This efficiency improvement is achieved by using quantum arithmetic in the superposition and by encoding the transition histories instead of the full distributions at each time step. Our results suggest that the collision-coalescence process is one of the promising targets of quantum computing in the field of atmospheric science.

Figures (12)

• Figure 1: The schematic diagram of the probability division process over multiple steps. The numbers in parentheses represent the mass distribution $\bar{\bm{n}}$. For convenience, the states $(N,0,\ldots,0)$ are denoted as $\bar{\bm{n}}_0$, and $(N-2,1,\ldots,0)$ as $\bar{\bm{n}}_1$. The arrows indicate the probability flow. The transition probability from the state $\bar{\bm{n}}_i$ by the transition labeled $h$ is denoted as $r_h(\bar{\bm{n}}_i)$ above the arrows. The probability of no transition is represented by $r_0$. The transition probabilities associated with the transitions with labels $1$ and $2$ are denoted as $r_1$ and $r_2$, respectively.
• Figure 2: The schematic diagram of the probability division process for a single time step. The probability $P(\bar{\bm{n}})$ is first divided into $s_H P(\bar{\bm{n}})$ and $r_H P(\bar{\bm{n}})$, where $r_H$ is the transition probability associated with the transition $H$, and $s_H$ is the remaining part of this probability division ($s_H=1-r_H$). Similarly, the probability division following the transition $(H-1)$ is conducted, and $s_H P(\bar{\bm{n}})$ is divided into $s_{H-1} P(\bar{\bm{n}})$ and $r_{H-1} P(\bar{\bm{n}})$. After performing the probability division for every possible pair of bins, the probabilities at the next time level are obtained (indicated by the blue dots).
• Figure 3: The quantum circuit for a single time step. The left-hand-side of the equation is the input and output states of the time step, while the right-hand-side illustrates the detailed operations within the time step. These segments enclosed by blue dashed lines on the right-hand-side include sequential applications of probability division for all possible bin pairs (omitted for simplicity, represented by dots in the figure). The left one calculates the modified transition probability and applies the probability division to the quantum phase. The central one uncomputes the auxiliary qubits $\mathcal{A}$ to $\ket{1.0}_\mathcal{A}$. The right one updates the mass distribution $\ket{\bar{\bm{n}}}_{\mathcal{N}}$ to the next time step based on the transition label $\ket{h}_{\mathcal{H}_m}$.
• Figure 4: The quantum circuit for calculating the time evolution of the probability distribution over all time steps. The circuit begins with the initialization of the droplet mass distribution and the auxiliary qubits $\mathcal{A}$ in the state $\ket{1.0}_\mathcal{A}$. Subsequently, the quantum gates $U_{\Delta t}$ are sequentially applied $M$ times. The transition history qubits $\mathcal{H}_m$ is used for the $m$-th time step.
• Figure 5: The quantum circuit for calculating the expected value of the number of droplets in the $i$-th bin. The expected value is encoded into the quantum amplitude of the auxiliary qubit $\mathcal{D}$.