Distributed optimization of Lindblad equations for large-scale cavity QED systems

TL;DR

Results show that this framework significantly accelerates non-unitary evolution, providing a feasible solution for simulating large-scale open quantum systems where the number of dissipative channels $M$ is much larger than the Hamiltonian dimension $N$.

Abstract

This paper proposes a distributed computing framework for solving the Lindblad master equation in large-dimensional cavity QED systems. By leveraging the sparsity of the jump operator and combining this approach with the Cannon algorithm, the computational complexity of non-unitary terms is reduced from $O(MN^3)$ to $O(MN)$. For unitary terms, a combination of Taylor series approximation and the Cannon algorithm enables distributed matrix exponentiation, though scalability is limited by cross-processor communication. The proposed dynamic subspace construction method further reduces the Hamiltonian dimension: when $n_{\text{at}}=10$, the dimension is reduced to $5.63\%$ of the full Hamiltonian, with a memory footprint of only $0.32\%$. Results show that this framework significantly accelerates non-unitary evolution, providing a feasible solution for simulating large-scale open quantum systems where the number of dissipative channels $M$ is much larger than the Hamiltonian dimension $N$.

Figures (8)

• Figure 1: (online color) Illustration of Cannon's algorithm on a $4\times4$ processor grid. Panel (a): Initial distribution of blocks $A_{i,j}$ and $B_{i,j}$ on each processor $P_{i,j}$, followed by horizontal alignment (shifting $A_{i,j}$ left by $i$ steps). Panel (b): Vertical alignment: shifting $B_{i,j}$ up by $j$ steps. Panels (c)--(e): Three iterations of cyclic shifts and multiply--accumulate operations: after each multiplication step, $A$ blocks are shifted left cyclically and $B$ blocks are shifted up cyclically. Panel (f): Final block distribution after completing all iterations, yielding the product matrix $C=A\times B$. (Adapted from our previous work MiaoOzhigov2024.)
• Figure 2: (online color) Numerical update for the non-unitary term with dissipation channel $L_k = |j\rangle \langle i|$ and its reverse process. Panels (a.1)--(a.3) show the dissipative process, and panels (b.1)--(b.3) show its inverse process. All operations are applied to $\rho(t)$ to obtain $\rho(t+\Delta t)$. Panel (a.1): Population transfer from the diagonal element of state $|i\rangle$ to that of state $|j\rangle$, proportional to the original population of $|i\rangle$. Panel (a.2): Subtraction applied to all elements in the $i$-th row, each reduced by a factor proportional to itself. Panel (a.3): Subtraction applied to all elements in the $i$-th column, each reduced by a factor proportional to itself. Panel (b.1): Reverse population transfer from the diagonal element of state $|j\rangle$ to that of state $|i\rangle$, proportional to the original population of $|j\rangle$. Panel (b.2): Subtraction applied to all elements in the $j$-th row, each reduced by a factor proportional to itself. Panel (b.3): Subtraction applied to all elements in the $j$-th column, each reduced by a factor proportional to itself.
• Figure 3: (online color) The impact of different processor grids on dissipation channels. Panel (a) is a $3\times3$ processor grid, and panel (b) is a $2\times2$ processor grid.
• Figure 4: (online color) Tavis--Cummings model with $n_\text{at}$ two-level atoms. Panel (a) shows the excitation process, and panel (b) shows the de-excitation process. The initial state is shown in panel (c), where there exist $n_\text{at}$ excited atoms and no photons. In panel (d), an excited atom de-excites and becomes a ground-state atom, at which point a photon is released. In panel (e), when all atoms have transitioned to the ground state, there are $n_\text{at}$ photons in the optical cavity. As soon as a photon is released, it can escape into the external environment through a dissipation channel. (Adapted from our previous work MiaoOzhigov2024.)
• Figure 5: (online color) Dissipative dynamics of the Tavis--Cummings model with $n_\text{at}$ two-level atoms. Panels (a)--(f) correspond to atom numbers ranging from 5 to 10, respectively.