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Chinese Control Conference (CCC)

Accelerated Simulation of Non-Markovian Quantum Systems Using Quantum Trajectory

Yanfang Wang, Bo Yue, Shibei Xue

Published Jul 24, 2023• 0 citations
Publisher

Abstract

Quantum trajectory can be viewed as a numerical simulation method for wavefunctions which consume smaller memories compared to density matrices in master equations. In this paper, we adopt quantum trajectory to simulate the evolution of a non-Markovian quantum system modeled by an augmented system and further use parallel computing to improve efficiency. Compared to the Lindblad master equation solver in QuTiP, it can run more efficiently using less memory. In the example of a 8-spin-chain non-Markovian system, it only takes about 0.57% of the memory footprint and 4.12% of the runtime of the solver for the Lindblad master equation.

DOI
10.23919/ccc58697.2023.10240942

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Quantum trajectory can be viewed as a numerical simulation method for wavefunctions which consume smaller memories compared to density matrices in master equations. In this paper, we adopt quantum trajectory to simulate the evolution of a non-Markovian quantum system modeled by an augmented system and further use parallel computing to improve efficiency. Compared to the Lindblad master equation solver in QuTiP, it can run more efficiently using less memory. In the example of a 8-spin-chain non-Markovian system, it only takes about 0.57% of the memory footprint and 4.12% of the runtime of the solver for the Lindblad master equation.
Simulation of open quantum systems is undoubtedly critical to design of quantum devices and applications of quantum technologies since dynamics of open quantum systems cannot be solved analytically in general. However, the dimension of open quantum systems grows exponentially as the number of their subsystems increases or as involving complicated environments, which reduces the computational efficiency of the simulation. In this paper, we boost numerical simulation of master equations of open quantum systems from several aspects with which the total time and the occupied memory in the simulation are significantly reduced without losing precision. Our method can be applied to the simulation of non-Markovian quantum systems described by an augmented system model as well. We conduct numerical experiments on simulation of a non-Markovian spin chain. Compared with the algorithm in a well-known toolbox, QuTiP, our method can take 26.3% time and less memory under fair conditions.
We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension NNN instead of a complex density matrix of dimension N2{N}^{2}N2, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor NNN advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor N2{N}^{2}N2 is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.
For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.
The Monte Carlo wave function method or the quantum-trajectory-jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. We extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Markovian jump approach and the computational efficiency of this method is also discussed.
Open quantum systems interacting with structured environments is important and manifests non-Markovian behavior, which was conventionally studied using a quantum trajectory stochastic method. In this paper, by dividing the effects of the environment into two parts, we propose a deterministic method without using a quantum trajectory. This method is more efficient and accurate than the stochastic method in most Markovian and non-Markovian cases. We also extend this method to the generalized Lindblad master equation.
Open quantum system interacting with structured environment is important and manifests non- Markovian behavior, which was conventionally studied using quantum trajectory stochastic method. In this paper, by dividing the effects of the environment into two parts, we propose a deterministic method without using quantum trajectory. This method is more efficient and accurate than stochastic method in most Markovian and non-Markovian cases. We also extend this method to the generalized Lindblad master equation.