Through a new survey, researchers show how
mathematical representations named 'tensor trains' can help to capture and
simulate the dynamics of evolving quantum systems across a range of different
scenarios.

Many quantum systems are heavily influenced by their surrounding
environments, making them incredibly challenging to describe theoretically. To
capture the dynamics and evolution of these systems,
researchers often use mathematical representations named tensor trains.

Through new research published in *The European Physical Journal Special Topics*,
a team of researchers from four different institutions in France show how
tensor trains can be implemented to describe and simulate quantum systems.

The team's findings could lead to helpful new
insights in many scenarios where evolving quantum systems interact with their
environments. Among the most important of these is quantum computing—where
reducing the interaction between qubits and their surroundings can help them
retain their quantum information for
longer.

Systems containing many interacting quantum particles can be
governed by many different variables, making it incredibly challenging for
researchers to capture their dynamics and evolution. In many cases, however,
their overall properties may only depend on a limited number of variables—so
the system as a whole can be described as a subsystem embedded in its
environment.

Today, researchers often represent and
simulate these embedded subsystems using mathematical structures named tensors:
with higher-order tensors describing systems governed by higher numbers of
variables. To simplify their descriptions further, tensor trains can be used to
break down higher-order tensors into sums of lower-order tensors.

To do this, the team turned to a theoretical framework named 'hierarchical
equations of motion' (HEOM). In their study, they demonstrated how the tensor
train format can be implemented using HEOM, with an aim to describe the
evolution and dynamics of quantum systems embedded in their environments.

The team now hopes their survey will become a valuable resource for researchers in future studies, helping them to explore a wide array of evolving quantum systems using accurate simulations.

**Source:** phys.org

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