Heuristic Simulates 2D Transverse-Field Ising Models, Enabling Study of Out-of-Equilibrium Dynamics

Understanding the behaviour of complex materials requires simulating how they evolve over time, but accurately modelling these ‘out-of-equilibrium dynamics’ presents a significant computational challenge, particularly for systems exhibiting strong entanglement. Salvatore Mandrà, Nikita Astrakhantsev, Sergei Isakov, and colleagues at Google Quantum AI have developed a new method to accelerate these simulations, offering a way to overcome the exponential resources typically required. Their approach focuses on efficiently calculating key properties of the system, such as energy and correlations, by intelligently rescaling results from a simplified simulation, and crucially, this technique works across a wider range of energy levels than previously possible. By applying this heuristic to the two-dimensional Transverse-Field Ising Model, the team demonstrates a substantial speed-up in calculations performed on standard hardware, paving the way for more detailed investigations into the behaviour of complex quantum materials.

Dynamic Decimation Controls Entanglement Growth

Out-of-equilibrium dynamics in many-body quantum systems are characterised by highly entangled wave functions, posing a significant challenge for numerical simulations, particularly when studying thermalisation or pre-thermalisation. This work introduces a new approach for simulating the dynamics of two-dimensional transverse-field Ising models using Matrix Product States (MPS), leveraging a dynamically adjusted decimation strategy to control entanglement growth and enable simulations over longer timescales than previously possible. Specifically, the researchers implemented a novel decimation scheme that prioritises retaining the most important singular values during time evolution, minimising truncation error and maintaining accuracy. This allows investigation of dynamical properties, such as the emergence of quasi-particle excitations and the approach to thermal equilibrium, in systems with up to 16×16 lattice sites and evolution times exceeding 100 ħ⁻¹ units. The TFIM is a standard tool in condensed matter physics for studying phase transitions and quantum magnetism. The researchers also investigate periodically driven systems, known as Floquet systems, and the phenomenon of prethermalisation, where a system appears to reach a thermal-like state before reaching true thermal equilibrium. A major challenge in digital quantum simulation is the accumulation of errors due to the Trotter decomposition, which approximates the time evolution operator.

The team’s novel contribution is a rescaling heuristic for Matrix Product States (MPS), a powerful method for simulating one-dimensional and quasi-one-dimensional quantum systems. The rescaling heuristic aims to reduce errors and improve the efficiency of MPS calculations by adjusting the parameters of the MPS to better represent the quantum state. They built their own MPS simulator to implement and test this new technique. To verify their results, the researchers employed multiple methods, including exact diagonalisation, which solves for the exact eigenstates of small systems, and other tensor network simulations.

They also used sparse Pauli dynamics and influence functional belief propagation to simulate quantum dynamics in two-dimensional lattices, carefully comparing their MPS results with exact diagonalisation results for systems up to 6×8 to assess the accuracy of the rescaling heuristic. The results demonstrate that the rescaling heuristic improves the accuracy of MPS simulations, allowing them to study larger systems and longer timescales. The researchers discovered that rescaling MPS results with a factor dependent only on the fidelity of the wave function allows for accurate estimation of physical observables using significantly smaller bond dimensions than previously required, enabling simulations of larger systems and longer timescales. The team successfully applied this rescaling technique to simulate the TFIM, comparing results to exact simulations on smaller grids and, where available, recent experimental data. They found that rescaling effectively converges to the expected thermal values for single-body observables, and with a refined scaling factor, also for two-body observables like the Ising order parameter. While the simulations show quantitative disagreement with some experimental data, the authors acknowledge that their extrapolation relies on exact results limited to smaller systems, potentially impacting the reliability of predictions for larger grids. Future work could focus on extending the extrapolation methods or applying this rescaling heuristic to other models beyond the TFIM.