# Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Expansion over the Basis of $SU\left(N\right)$ Group Generators

## 3. Models

## 4. Algorithm

#### 4.1. Initialization (Sequential; Performed on Every Node of a Cluster)

#### 4.2. Data Preparation and Its Parallelized via Message Passing Interface (MPI)

#### 4.2.1. Computation of Expansion Coefficients ${h}_{j}$ and ${l}_{j}$ for H and L

#### 4.2.2. Computation of Coefficients ${f}_{mns}$, ${d}_{mns}$, ${z}_{mns}$ by Formulas (11) and (12)

#### 4.2.3. Computation of Coefficients ${q}_{sm}$ by Formula (13)

#### 4.2.4. Computation of Coefficients ${k}_{s}$ by Formula (14)

#### 4.2.5. Computation of Coefficients ${r}_{sm}$ by Formula (15)

#### 4.2.6. Computation of the Initial $v\left(0\right)$

#### 4.3. ODE Integration (Paralleled via MPI + OpenMP + SIMD)

## 5. Algorithm Performance

#### 5.1. Data Preparation Step

#### 5.2. The ODE Integration Step

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The parallel data preparation pipeline for the dimer model. On Panel A, we sketch distribution of nonzero elements of matrices S, J, and D, forming basis $\left\{F\right\}$, Equations (3)–(5), respectively. Panel B depicts locations of nonzero elements in tensors f and d, Equation (11), which are not stored in memory but computed on the fly, during the Data Preparation step. Panels C, D, and E show how matrices Q, Equation (13), K, Equation (14), and R, Equation (15), are computed in parallel by two MPI processes (steps 2.3–2.5 in Table 1).

**Figure 3.**Memory consumption per one node (

**left**) and computation time (

**right**) of the Data Preparation step for the dense model.

**Figure 4.**Computation time of ODE Integration step for the sparse (

**left**) and dense (

**right**) models. Numerical integration was performed for one period of modulation T, with 20,000 steps per period.

**Figure 5.**Histogram of the complex eigenvalues, ${\lambda}_{i}$, $i=2,3,\dots .,{N}^{2}$, of a single realization of a random Lindbladian (see Section 3) for $N=200$. The shown area was resolved with a grid of $100\times 100$ cells; the number of eigenvalues in every cell was normalized by the cell area. Altogether, ${N}^{2}-1=39999$ eigenvalues are presented (except ${\lambda}_{1}=1$). The bright thick line corresponds to the universal spectral boundary derived analytically in [17].

Step | Substep (the Sequential Algorithm) | Parallelization |
---|---|---|

1. Initialization | 1.1. Read the initial data from configuration files. 1.2. Allocate and initialize memory. | Sequential step, all the operations are performed on every node of a cluster. |

Data Preparation cycle: | This step is parallelized, computation time and memory costs are distributed among cluster nodes via Message Passing Interface (MPI). | |

2.1. Compute the coefficients ${h}_{i}$, ${l}_{i}$ of the expansion of the matrices H and L in the basis $\left\{{F}_{i}\right\}$. | Step 2.1. (Figure 1, Panel A) is not resource demanding and, therefore, is performed on each cluster node independently. | |

2. Data Preparation | 2.2. Compute the coefficients ${f}_{ijk}$, ${d}_{ijk}$, ${z}_{ijk}$ by formulas (11,12). | Step 2.2. (Figure 1, Panel B) is memory demanding in a straightforward implementation. Unlike the sequential algorithm, we compute only nonzero elements of the tensors when they are needed. |

2.3. Compute the coefficients ${q}_{sm}$ by formula (13). 2.4. Compute the coefficients ${k}_{s}$ by formula (14). 2.5. Compute the coefficients ${r}_{sm}$ by formula (15). | Parallel steps 2.3, 2.4, and 2.5 are sketched in Figure 1 (Panels C, D, and E, respectively). These steps are time and memory consuming and are executed in parallel on cluster nodes. | |

2.6. Compute the initial value $v\left(0\right)$. | Step 2.6 is not resource demanding and is realized on each cluster node independently. | |

3. ODE Integration | 3.1. Integrate the ODE (10), over time to $t=T$ by means of the Runge–Kutta method. 3.2. Compute $\rho \left(T\right)$ by formula (7). | This step is parallelized via MPI (among cluster nodes), OpenMP (among CPU cores on every node), and SIMD instructions (on every CPU core). |

4. Finalization | 4.1. Save the results. 4.2. Release memory. | Here we gather results from computational nodes, save the results, and finalize MPI. |

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**MDPI and ACS Style**

Meyerov, I.; Kozinov, E.; Liniov, A.; Volokitin, V.; Yusipov, I.; Ivanchenko, M.; Denisov, S.
Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm. *Entropy* **2020**, *22*, 1133.
https://doi.org/10.3390/e22101133

**AMA Style**

Meyerov I, Kozinov E, Liniov A, Volokitin V, Yusipov I, Ivanchenko M, Denisov S.
Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm. *Entropy*. 2020; 22(10):1133.
https://doi.org/10.3390/e22101133

**Chicago/Turabian Style**

Meyerov, Iosif, Evgeny Kozinov, Alexey Liniov, Valentin Volokitin, Igor Yusipov, Mikhail Ivanchenko, and Sergey Denisov.
2020. "Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm" *Entropy* 22, no. 10: 1133.
https://doi.org/10.3390/e22101133