Deterministic Quantum Jump (DQJ) Method for Weakly Dissipative Systems

Abstract

Physical quantum systems are generically coupled to an environment, resulting in open system dynamics. A typical approach to simulating this dynamics is to propagate the density matrix of the system via the Lindblad master equation. This approach is numerically challenging due to the size of the density matrix, which has led to the development of quantum jump methods, which unravel the density matrix into an ensemble of state vectors. These methods utilize a stochastic sampling of the quantum jump times, which becomes inefficent for weakly dissipative dynamics, in which jumps are rare events. Here, we propose the deterministic quantum jump (DQJ) method, which we show to outperform standard quantum jump methods in the weakly dissipative regime, by removing the error of stochastic sampling. We describe the methodology at the single-jump and two-jump level, reconstructing the density matrix at the corresponding level. We demonstrate the performance of the method for two examples, the dissipative transverse-field Ising model, and the dissipative Kerr oscillator. Given that quantum technologies such as quantum computing have weakly dissipative quantum dynamics as their central focus, we propose this method to be utilized in that context, for exploring and understanding quantum technology platforms.

Figures (6)

• Figure 1: (a) Schematic generation of a single jump trajectory in the Deterministic Quantum Jump (DQJ) method. The initial state is propagated to the jump time $\tau$ with the effective Hamiltonian $H_{\text{eff}}$ (green). The jump time is an element of the deterministic and equally spaced jump time grid $\Sigma^1$ defined on the integration domain $t\in[0,T]$. The trajectory probability is recorded and a quantum jump performed (blue). After normalization, the state is propagated to the final time $T$. (b) DQJ calculates the density matrix evolution by accounting for all jump orders individually. Each jump trajectory with a jump from $\tau \in \Sigma^1$ is weighted by its trajectory probability in the interval $[0,T]$. (c) Regime of the DQJ and SQJ (red) method as a function of computational trajectories and target infidelity. DQJ outperforms (cyan) or is on par (striped) with the SQJ method (red) if the jump time grid spacing is smaller than the fastest density matrix entry evolution, $\Delta t< 1/f_{\max}$ (vertical dashed line) and higher jump orders are negligible (horizontal dashed line). The power-law exponent of the infidelity is $\alpha = 4/n$ for the $n$ jump DQJ method. SQJ scales much slower in the number of trajectories.
• Figure 2: Infidelity of the density matrix calculated with the DQJ (blue) and the modified SQJ (red) method with the true density matrix evolution. The infidelity is evaluated at time $T=1$ for a $5$ qubit TFIM system with $\gamma\, T \approx 0.03$. Numerically, we recover the asymptotic scaling of Eq. \ref{['equ:InfidScaling']}. The scaling functions (grey) serve as a guide to the eye. The minimal infidelity is set by the contribution of trajectories with more than two jumps.
• Figure 3: Top: Error of the $\langle\hat{X}\rangle$ evolution in the Kerr-oscillator. The DQJ and the SQJ method are compared to the full density matrix Lindbladian simulation. A total of $21$ trajectories were propagated for both methods. Bottom: Error scaling in the number of trajectories for the $X$-quadrature spectrum at frequency $\omega_0$. for the SQJ (red) and DQJ (blue) method compared to the true full density evolution. The dissipation strength is set to $\gamma\, T = 0.32$. The scaling functions (grey) in the number of trajectories for the SQJ is shown to guide the eye.
• Figure 4: (a) Schematic jump time integral of Eq. \ref{['equ:pn_rhon_integral']} at time $t = t_0$ and the sampled jump time trajectories (black dots) used in the DQJ approximation, Eq. \ref{['equ:pn_rhon_sum']}. (b) In SQJ, the integral is instead sampled with random points from the $\tilde{p}_\tau$ trajectory probability distribution. The jump time integral has a discontinuity (red) at the jump time $\tau = t_0$ for the time $t= t_0$.
• Figure 5: (a) Two-time jump time integral Eq. \ref{['equ:pn_rhon_integral2']}. DQJ integrates over the simplex spanned by $\tau_1 <\tau_2$ by sampling the jump times deterministically over the cartesian midpoint grid $\Sigma^2_\text{cartesian}$ and the barycenters $\Sigma^2_\text{bary}$ for $\tau_1 \lesssim \tau_2$. (b) SQJ again samples randomly from the trajectory probability distribution $\tilde{p}_2\left(\vec{\tau}\right)$.