Quantum Computers Tackle Non-Markovian Dynamics, Enhancing Simulation of Physical Processes

Non-Markovian quantum dynamics, the behavior of quantum systems that do not follow the Markov property, is a significant area of study in quantum physics. It is relevant to condensed phase chemistry, many-body physics, quantum biology, and quantum error correction. Quantum computers can potentially offer an advantage in simulating quantum dynamics. The authors of the article present a non-Markovian quantum algorithm with a NISQ (Noisy Intermediate-Scale Quantum) friendly focus: the time-dependent variational algorithm (TDVA). They use the ensemble averaged classical path (EACP) to capture the non-Markovian dynamics in a finite temperature bath. The algorithm was validated on the simulator and demonstrated on the IBM quantum device.

What is Non-Markovian Quantum Dynamics and Why is it Important?

Non-Markovian quantum dynamics refers to the behavior of quantum systems that do not follow the Markov property, which states that the future state of a system depends only on its current state and not on its past states. This is a significant area of study in quantum physics, as many physical and chemical processes in the condensed phase environment exhibit non-Markovian quantum dynamics. These processes are challenging to simulate on classical computers due to the exponential growth of resource requirements with respect to system size and the degree of non-Markovianity.

The study of non-Markovian quantum dynamics is directly relevant to condensed phase chemistry, many-body physics, quantum biology, and quantum error correction. Recent advances have uncovered many interesting phenomena in open quantum systems such as non-equilibrium phase transitions, entangled state preparation through reservoir engineering, and information backflow. The spin-boson model and its multistate extension are commonly used frameworks for studying quantum dynamics in condensed phase chemical environments, which often exhibit non-Markovian behavior.

How Can Quantum Computers Help?

Quantum computers can potentially offer an advantage in simulating quantum dynamics, as first conjectured by Feynman and demonstrated by Lloyd. A wealth of literature exists for Hamiltonian simulation algorithms, with Low and Chuang having realized the optimal query complexity. For open quantum systems, much work has been focused on Markovian dynamics, but the development of quantum algorithms for non-Markovian time evolution is still in its infancy.

In this context, the authors of the article present a non-Markovian quantum algorithm with a NISQ (Noisy Intermediate-Scale Quantum) friendly focus: the time-dependent variational algorithm (TDVA). They work with the spin-boson model and use the ensemble averaged classical path (EACP) to capture the non-Markovian dynamics in a finite temperature bath.

What is the Ensemble Averaged Classical Path (EACP) Approximation?

The EACP approximation is a method used to capture the non-Markovian dynamics of a quantum system. The Hamiltonian for a quantum system linearly coupled to its harmonic bath can be written in a specific form, where the system and bath coordinates and the system-bath coupling strength are denoted by specific variables. The bath’s influence on the system can be seen as having a time-dependent driving force.

The nonlocal memory kernel, termed back reaction, is partially responsible for the non-Markovian dynamics. The other important contributor is from the integration of the phase space variables from the bath. The effects of these two contributions to the non-Markovianity are delineated by Makri using path integral formulation.

How Does the Algorithm Work?

In the absence of the back reaction, the reduced density matrix (RDM) expresses in a specific form, where the influence of the bath on the system’s dynamics is described by a specific equation. This equation is time local, meaning that the propagation of the RDM can be done iteratively with time.

When integrating over the Wigner distribution, the equation turns into another form, immediately manifesting the non-Markovian effect in the double time integration. Therefore, tracing out the bath degrees of freedom introduces non-Markovianity. On the other hand, the authors can employ the reverse by introducing additional degrees of freedom to remove the non-Markovian effect.

When the back reaction is included, an additional degree of freedom is introduced. This framework is naturally adapted to any anharmonic bath with non-linear coupling to the system and is also well suited for simulating spin chain dynamics in a dissipative environment. The authors validated the algorithm on the simulator and demonstrated its performance on the IBM quantum device.