Efficient Gaussian Simulations of Fermionic Open Quantum Systems

Abstract

We review existing classical simulation methods for performing fermionic Gaussian operations and develop new methods to address the gap by adhering to the fundamental theoretical framework established by Bravyi [Quantum Info. Comput. 5, 216 (2005)] for the most general fermionic Gaussian processes. Throughout this attempt, the focus remains on the unified approach that can be applied to generic fermionic Gaussian operations. This is beneficial since the selection of simulation methods has often been based on an ad hoc choice, heavily influenced by the specific model and circumstances, rather than on a systematic approach.

Figures (8)

• Figure 1: A typical example of fermionic random circuit with alternating layers of unitary and measurement layers. Each unitary layer (labeled $U$) describes fermionic gate of the Gaussian form $\hat{U}:=\exp\left(\sum_{ij}\hat{c}_i H_{ij}\hat{c}_j\right)$, where $\hat{c}_i$ are Majorana fermion operators and $H$ is a real anti-symmetric matrix. Each measurement layer consists of projective measurements (depicted as measurement devices) or dissipation processes (labeled $\mathcal{D}$ in the picture) of randomly selected modes.
• Figure 2: Comparison of the logarithmic negativity $\operatorname{LN}(L,L/2)$ from direct numerical solution of the Lindblad equation (blue solid line) and Wick simulation (orange dots) in the dissipative (a) and projective (b) Hatano-Nelson model for $L=6$ and $\gamma/J=0.5$. For the simulation, the step size in time was $dt=0.01/\gamma$ and the sample size was $N=500$.
• Figure 3: Comparison of the mixed-state logarithmic negativity (LN) from the direct numerical solution of the Lindblad equation (blue solid line) and Monte Carlo simulation (orange dots) for the problem of continuous monitoring in the Kitaev chain of Majorana fermions. In both cases, the chemical potential is $\mu/\Delta=0.2$, the hopping amplitude is $J/\Delta=1$, the monitoring rate is $\gamma/\Delta=0.1$ and the system size is $L=6$. The simulation for $N=500$ quantum trajectories were performed with step size of time $dt=0.001/\gamma$, whereas the values of logarithmic negativity are plotted in the step size $dt'=0.02/\gamma$.
• Figure 4: The average entanglement entropy $\langle{\operatorname{EE}(L,L/2)}\rangle$ from the simulation of random circuits with the unitary layer governed by the Kitaev chain of Majorana fermions as a function of time (a, c) and size (b, d). In panels (a) and (b), the interaction time $\tau$ has been selected randomly from an exponential distribution over interval $[0,2\pi L/E_\mathrm{max}]$, where $E_\mathrm{max}\sim{J}$ refers to the largest energy scale of the system. In panels (c) and (d), the interaction time $\tau$ has been selected randomly from an exponential distribution $P(\tau)\propto\exp(-L\gamma\tau)$. The plot legends refer to system size $L$ in panels (a, c); to measurement probability $p$ in panel (b); and to $\gamma$ in panel (d). Other parameter values are $J/\Delta=1$ and $\mu/\Delta=0.2$.
• Figure 5: The average entanglement entropy $\langle{\operatorname{EE}(L,L/2)}\rangle$ for the continuous monitoring of the Kitaev chain of Majorana fermions, calculated from the simulation based on the Lindblad equation, as a function of time (a) and size (b). The plot legends in panels (a) and (b) refer to the system size $L$ and monitoring rate $\gamma$, respectively. Other parameter values are $J/\Delta=1$ and $\mu/\Delta=0.2$.