Quantum Simulation of Open Quantum Dynamics via Non-Markovian Quantum State Diffusion

TL;DR

A hybrid quantum-classical algorithm designed for simulating dissipative dynamics in system with non-Markovian environment, leading to a substantial reduction in qubit requirements is introduced.

Abstract

Quantum simulation of non-Markovian open quantum dynamics is essential but challenging for standard quantum computers due to their non-Hermitian nature, leading to non-unitary evolution, and the limitations of available quantum resources. Here we introduce a hybrid quantum-classical algorithm designed for simulating dissipative dynamics in system with non-Markovian environment. Our approach includes formulating a non-Markovian Stochastic Schrödinger equation with complex frequency modes (cNMSSE) where the non-Markovianity is characterized by the mode excitation. Following this, we utilize variational quantum simulation to capture the non-unitary evolution within the cNMSSE framework, leading to a substantial reduction in qubit requirements. To demonstrate our approach, we investigated the spin-boson model and dynamic quantum phase transitions (DQPT) within transverse field Ising model (TFIM). Significantly, our findings reveal the enhanced DQPT in TFIM due to non-Markovian behavior.

Figures (4)

• Figure 1: Our strategy to simulate the non-Markovian open quantum dynamics. We consider a quantum system interacting with a thermal environment at temperature $T$ consisting of bosonic modes which are linearly coupled to the quantum system. After tracing out the environmental degrees of freedom, this many-body problem is then treated using the non-Markovian stochastic Schrödinger equation with complex frequecy modes(cNMSSE). In cNMSSE, only a few modes holds significance, allowing for efficient solutions through variational quantum algorithms. This approach enables the exploration of properties like dynamical quantum phase transitions (DQPT) in the transverse field Ising model with alleviated quantum resources.
• Figure 2: (a) Population dynamics of the spin-boson model with parameters $\epsilon=1.0,~\Delta=1.0,~\eta=0.5$, at a characteristic frequency $\gamma=0.25$ and temperature $\beta=0.5$, by averaging over $10^4$ (red, dashed) trajectories. (b) Stochastic noise and population dynamics are shown for two independent trajtories (red, dashed). (c) The quantum circuit diagram employed for this model. The numerical exact results (blue, solid) are obtained using matrix product state method.
• Figure 3: (a) Stochastic noise and rate function $\Lambda(t)$ for one trajectory of different $\gamma$ (which is set to $50$, $5$, $0.5$ from left to right), other parameters of TFIM: $J=2$, $B=1/0.42$, and $\Gamma=1$(red, dashed). (b) The quantum circuit diagram employed for this model. The numerical exact results (blue, solid) are obtained using matrix product state method.
• Figure 4: rate function $\Lambda(t)$ of transverse field Ising model with parameters $J=2$, $B=1/0.42$ and $\Gamma=1$. In the Markovian range($\gamma=50$), the result of quantum simulation(solid line) coincides with that of Lindblad master equation calculated by QuTiP(crosses). When decreasing $\gamma$, the environment transitions to the non-Markovian regime, resulting in more occurrences of DQPT.