Quantum dynamics describe how objects obeying the laws of quantum mechanics interact with their surroundings, ultimately enabling predictions about how matter behaves. Accurate quantum dynamics simulations inform the development of new materials, solar cells, batteries, sensors, and many other cutting-edge technologies. They’re also a critical tool in designing and building useful quantum computers, including the design of novel types of qubits, improving gate fidelities, and performing device calibration.

In practice, simulating quantum systems is extremely challenging. The standard steps of a dynamics simulation include preparing a quantum state, evolving it in time, and then measuring some property of the system such as the system’s average energy or transition probabilities between its energy levels. In practice, this means solving differential equations governed by the Schrodinger Equation or the Lindblad Master Equation. Many-body quantum systems are represented by exponentially large Hilbert Spaces which make exact solutions intractable for conventional simulation methods.

To overcome this, clever approximations and numerical methods are used instead. The challenge is finding approximate methods that are computationally efficient while retaining a high degree of accuracy. Techniques like tensor networks can efficiently compute the dynamics of large-scale quantum systems, but struggle with highly entangled systems. New tools are needed to extend the reach of simulation techniques and explore more interesting and relevant systems.

Researchers Jens Eisert and Steven Thomson from the Free University of Berlin used NVIDIA GPUs to develop and test a powerful new method for simulating quantum dynamics. Their article Unravelling Quantum Dynamics Using Flow Equations, recently featured in the journal Nature Physics, provides a powerful new GPU-accelerated method to simulate these systems.

## Streamlined dynamics simulations

Jens and Steven tackled the challenge of simulating quantum systems using the method of *flow equations*. Instead of taking a single quantum state and evolving it in time, the flow equations method diagonalizes the Hamiltonian matrix * *describing the quantum system. This is accomplished by applying a large number of infinitesimally small unitary transformations (, where is a unitary matrix) to the initial .

The full unitary transform is a time-ordered integral over the dummy flow time variable . A time-ordered integral ensures that each step corresponds to the evolution of the Hamiltonian chronologically as goes from 0 to infinity. It turns out that this numerical task can be efficiently parallelized using GPUs, offering a tractable approach to simulating a system’s dynamics.

The primary advantage of flow equations is that the simulation is not limited by the degree of entanglement, but by the desired accuracy of the numerical procedure. This means that the error is a mathematical truncation that tends to be far less restrictive than the so-called ‘entanglement barrier’, and can be systematically improved when higher accuracy is required.

The second advantage is that a two or three dimensional system can easily be “unfolded” into a one-dimensional representation and solved with flow equations (Figure 1). The ability to simulate multidimensional systems is crucial for real-world quantum applications, which generally require consideration of more than one dimension.

Unfortunately, flow equations are not a panacea for simulating quantum dynamics. They struggle to converge when the initial Hamiltonian has multiple states with nearly identical energies, a common occurrence for some of the most interesting cases. This led Jens and Steven to propose the innovative idea of using so-called *scrambling transforms*. Using these to ‘scramble’ the initial Hamiltonian with an additional transformation helps remove degeneracies which would otherwise impede the diagonalization procedure (Figure 2).

## Large-scale GPU-enabled dynamics simulations

Studies using the flow equation technique have been largely analytical, leveraging pen and paper to find clever ways of avoiding unwieldy calculations. In 2023, Steven and his colleague Marco Schirò published foundational work for turning this promising technique into a powerful and more reliable numerical method, which can leverage the strengths of NVIDIA GPUs. For details, see Local integrals of Motion in Quasiperiodic Many-Body Localized Systems.

The method is well suited for parallelisation, as the many underlying matrix and tensor multiplications can be efficiently split into many smaller operations. A single NVIDIA GPU (such as the NVIDIA RTX A5000 used by Steven) runs operations on tens of thousands of cores, providing a huge speedup compared to even the best multicore CPUs.

The gap between CPU and GPU calculations grows quickly, even when only considering relatively small systems and modest GPU resources (Figure 3). Performing 24 particle simulations, which required over 2 hours to run on CPU, could be completed in under 15 minutes on a single NVIDIA GTX 1660Ti GPU. Even higher speedups are expected using more powerful data-center grade GPUs like the NVIDIA H100 Tensor Core, which alleviates the memory bottleneck.

The speedup provided by GPUs enables the flow-equation technique to be employed for larger scale 2D systems, unlocking a new frontier for numerical simulations of quantum matter.

According to Steven Thomson,* *“GPUs were absolutely essential to the success of this work, and our numerical technique was developed specifically to make use of their strengths. Without them, our simulations would have taken tens or hundreds of times longer to run. This would have not only taken unreasonably long, but would also have come with a huge environmental cost due to the energy required to run our simulations for such a long time.”

## A new dimension for quantum dynamics

Future work will explore flow equation simulations of larger 2D and 3D systems, leveraging multi-node GPU systems to further push the boundaries of quantum dynamics simulations. By building on the foundation laid by Jens and Steven, researchers will be able to simulate a wider variety of quantum systems than ever before, complementing the strengths and weaknesses of existing methods such as tensor networks.

## Get started accelerating your research

This groundbreaking work was possible in part thanks to the NVIDIA Academic Grant Program, which grants researchers free access to NVIDIA compute resources to further their work. Researchers focused on generative AI and large language models (LLMs), simulation and modeling (including quantum computing), data science, graphics and vision, and edge AI are encouraged to apply.

To learn more about NVIDIA initiatives related to quantum computing and simulation including tools like CUDA-Q for developing large-scale quantum applications, visit NVIDIA Quantum.