Tensor Network States Via Time-Dependent Variational Monte Carlo Enables Real-Time Two-Dimensional Dynamics

Simulating how systems evolve over time in two dimensions presents a significant challenge for physicists, particularly when dealing with complex quantum interactions. Yantao Wu, along with colleagues, now addresses this problem by developing a new computational framework, time-dependent variational Monte Carlo for projected entangled pair states, that reliably tracks these dynamics. The team achieves this breakthrough by eliminating mathematical ambiguities within the simulation and leveraging the inherent structure of quantum systems to create a stable and efficient method. This allows them to model a range of physical phenomena, including the behaviour of electrons in exotic materials and the flow of energy in quantum systems, with unprecedented accuracy and opens new avenues for exploring complex quantum behaviour using classical computers and complementing emerging quantum simulators.

The team achieves this breakthrough by eliminating mathematical ambiguities within the simulation and leveraging the inherent structure of quantum systems to create a stable and efficient method. This allows them to model a range of physical phenomena, including the behaviour of electrons in exotic materials and the flow of energy in quantum systems, with unprecedented accuracy and opens new avenues for exploring complex quantum behaviour using classical computers and complementing emerging quantum simulators.

Memory Optimisation via Small-o Trick

This research details numerical simulations of various quantum many-body systems using Tensor Network States, specifically Projected Entangled Pair States and Time-Dependent Variational Monte Carlo. A key focus is on memory optimisation techniques, particularly a method called the small-o trick, to enable simulations with larger bond dimensions and system sizes. The small-o trick addresses the problem of large memory requirements in tVMC simulations by exploiting the structure of PEPS, focusing on storing only the non-zero elements of the wave function’s matrix representation using a selection tensor. This significantly reduces memory usage, allowing for simulations with larger bond dimensions and system sizes, demonstrated through simulations of Chern insulators, Bosonic Hofstadter models, Z2 gauge theories, and superfluid bosons.

Stable Real-Time Simulation of Quantum Dynamics

Scientists have achieved a breakthrough in simulating two-dimensional quantum dynamics using projected entangled pair states and time-dependent variational Monte Carlo. This work overcomes long-standing challenges in accurately modeling the real-time evolution of quantum systems in two dimensions, previously limited by computational cost and instability. The team developed a stable and efficient tVMC framework for PEPS, enabling simulations in regimes previously inaccessible to classical methods. A key achievement lies in the analytical removal of gauge redundancies within the PEPS structure and exploiting the inherent locality of tensor networks, resulting in a numerically well-conditioned stochastic reconfiguration equation solved using efficient Cholesky decomposition. Experiments demonstrate the method’s versatility through simulations of chiral edge propagation in a Chern insulator, fractionalized charge transport in a fractional Chern insulator, vison confinement dynamics in a Z2 lattice gauge theory, and superfluidity in interacting bosons.

All simulations were performed on lattices of 12×12 or 13×13, with evolution times ranging from 10 to 12, utilizing modest computational resources, requiring only 1 to 5 days on a single GPU card. In cases where exact benchmarks existed, the PEPS-tVMC method matched free-fermion dynamics with high accuracy up to 12, with the SR residual remaining remarkably low, reaching values between 10⁻⁹ and 10⁻²⁵. These results establish PEPS-tVMC as a practical and versatile tool for studying real-time quantum dynamics in two dimensions, significantly extending the reach of classical tensor-network simulations for investigating elementary excitations and providing a valuable computational counterpart to emerging quantum simulators. The team visualized the unidirectional motion of chiral edge modes, tracked the flow of fractional charge, observed vison confinement, and identified signatures consistent with a critical velocity in superfluid systems, demonstrating the method’s broad applicability.

Stable Real-Time Quantum Dynamics Simulations Achieved

Scientists have developed a new computational framework that enables reliable simulation of real-time dynamics in two-dimensional quantum systems, overcoming a longstanding challenge in the field. The team successfully combined projected entangled pair states, a method for representing quantum states, with time-dependent variational Monte Carlo, a technique for evolving these states over time. A key achievement was the analytical removal of gauge redundancies within the PEPS framework, which previously caused numerical instability and limited the duration of simulations. This advancement allows for stable and efficient calculations of low-energy dynamics, demonstrated through simulations of chiral edge propagation, fractionalized charge transport, vison confinement, and superfluidity.

The method accurately reproduces known results where benchmarks exist and establishes PEPS-tVMC as a versatile tool for studying elementary excitations and transport phenomena in two-dimensional quantum systems with modest computational resources. The researchers note that the method’s accuracy relies on the system remaining within a regime compatible with the PEPS representation. Future research directions include extending the method to finite temperatures and open quantum systems, as well as further investigating the relationship between PEPS bond dimension and dynamical fidelity. The team also highlights the potential for PEPS-tVMC to serve as a valuable benchmark for validating emerging quantum simulation platforms and interpreting data from real-time experiments probing two-dimensional dynamics. Overall, this work demonstrates that PEPS, previously considered difficult for time evolution, can accurately and stably simulate large-scale two-dimensional quantum dynamics at low energies.