Abstract The study of many-body quantum dynamics in strongly correlated systems is extremely challenging. To date, few numerical methods exist that are capable of simulating the non-equilibrium dynamics of two-dimensional quantum systems, which is partly due to complexity theoretic obstructions. In this work, we present a technique able to overcome this obstacle, by combining continuous unitary flow techniques with the newly developed method of scrambling transforms. We overcome the assumption that approximately diagonalizing the Hamiltonian cannot lead to reliable predictions for relatively long times. Rather, we show that the method achieves good accuracy in both localized and delocalized phases and makes reliable predictions for a number of quantities including infinite-temperature autocorrelation functions. We complement our findings with rigorous incremental bounds on the truncation error. Our approach shows that, in practice, the exploration of intermediate-scale time evolution may be more feasible than is commonly assumed, challenging near-term quantum simulators.
Abstract The study of many-body quantum dynamics in strongly correlated systems is extremely challenging. To date, few numerical methods exist that are capable of simulating the non-equilibrium dynamics of two-dimensional quantum systems, which is partly due to complexity theoretic obstructions. In this work, we present a technique able to overcome this obstacle, by combining continuous unitary flow techniques with the newly developed method of scrambling transforms. We overcome the assumption that approximately diagonalizing the Hamiltonian cannot lead to reliable predictions for relatively long times. Rather, we show that the method achieves good accuracy in both localized and delocalized phases and makes reliable predictions for a number of quantities including infinite-temperature autocorrelation functions. We complement our findings with rigorous incremental bounds on the truncation error. Our approach shows that, in practice, the exploration of intermediate-scale time evolution may be more feasible than is commonly assumed, challenging near-term quantum simulators.
Simulating quantum many-body systems is essential for understanding complex phenomena in condensed matter, chemistry, and materials science. Quantum computing offers a way to overcome the limitations of classical methods by operating directly in exponentially large Hilbert spaces. In particular, classical methods often fail to capture the dynamics of strongly correlated systems, which makes quantum algorithms a promising alternative for exploring these challenging regimes. This thesis uses quantum dynamics simulations to probe the behaviour of many-body systems. We first introduce a spectroscopy-based protocol to extract excitation spectra from the time evolution of observables and analyse its performance on near-term quantum devices. We show that the protocol requires only short evolution times and is resilient to errors, facilitating its implementation on real hardware. Building on this, we simulate fermionic excitation spectra using current quantum devices, introducing resource optimisations - including an approximate free-fermion initial state, a unitary quench without ancillae, and reduced Trotter steps - that significantly lower circuit depth. These improvements enable simulations at system sizes competitive with the state of the art, yielding results that match classical benchmarks. While most studies focus on unitary dynamics of isolated systems, open quantum dynamics is increasingly relevant for describing non-equilibrium processes across biology, materials science, and quantum control. We extend our scope to systems coupled to an environment and governed by Lindbladian evolution. By studying simulation complexity through physically meaningful quantities - such as correlation length and mixing time - we explore whether open-system dynamics can be as tractable as their unitary counterparts. These findings provide insights into the computational complexity landscape of open-system simulation and highlight regimes where quantum methods offer a clear advantage.
The study of many-body quantum physics has been recently challenged by the appearance of an increasing number of experimental platforms where genuine quantum phenomena take place notwithstanding the presence of an environment and of dissipation. Exciton-polaritons, lossy atomic and molecular gases, cavity QED arrays, arrays of trapped ions and Rydberg atoms, are only few prominent examples of a long list. The present work is ideally divided into two parts, the former is devoted to methodological results; the latter to the theoretical characterization of experimental data. Whereas the notion of equilibrium has been a fruitful guide to the development of standard many-body physics, these setups are inherently out of equilibrium and their modelisation requires the introduction of new theoretical tools. Several methods for addressing this dissipative out-of-equilibrium dynamics have been proposed, based for instance on quantum trajectories, truncated Wigner expansions, tensor networks, machine learning, but the solution of many-body physics for open quantum systems remains a formidable task. Clearly, techniques developed in the framework of Hamiltonian closed systems are a continuous source of inspiration for novel developments, and in this work we present the generalization of one such technique, the so-called flow equations, to the dissipative framework where the generator of the dynamics is a linear non-Hermitian operator; notable examples include non-Hermitian Hamiltonians and Lindblad master equations. In the second part of this work we focus on two-body lossy dynamics in strongly-correlated cold atomic gases. In particular, we derive an effective description for the long-time dynamics of a spin ½ fermionic system by extending a pre-existing theory for the bosonic case.
Reliable numerical computation of quantum dynamics is a fundamental challenge when the long-ranged quantum entanglement plays essential roles as in the cases governed by quantum criticality in strongly correlated systems. Here we apply a method that utilizes reliable short-time data of physical quantities to accurately forecast long-time behavior of the strongly entangled systems. We straightforwardly employ the simple dynamic mode decomposition (DMD), which is commonly used in fluid dynamics. Despite the simplicity of the method, the effectiveness and applicability of the DMD in quantum many-body systems such as the Ising model in the transverse field at the critical point are demonstrated, even when the time evolution at long time exhibits complicated features such as a volume-law entanglement entropy and consequential power-law decays of correlations characteristic of systems with long-ranged quantum entanglements unlike fluid dynamics. The present method, though simple, enables accurate forecasts amazingly at time as long as nearly an order of magnitude longer than that of the short-time training data. Effects of noise on the accuracy of the forecast are also investigated, because they are important especially when dealing with the experimental data. We find that a few percentages of noise do not affect the prediction accuracy destructively.
We present a method for simulating operator dynamics in out-of-equilibrium quantum systems. Due to the rapid growth of complexity in these systems, this is typically speaking an intractable task. However, exceptional progress has been made in recent years to sidestep this barrier, with the introduction of a number of methods that make use of a truncation of the simulation to low-weight (the number of non-trivial terms in a Pauli string basis expansion) observables, which turns out to be a good approximation for many dynamical quantities of interest, e.g., two-point infinite-temperature correlation functions between local operators. In this work, we extend this idea to a leaner truncation protocol, truncating operators based on their diameter - that is, the size of the region on the lattice on which they are non-trivially supported. Using existing analysis for generic circuits we argue that this kind of truncation protocol is physically well-motivated, and show via extensive numerical simulations for a number of systems of interest (here, the kicked Ising model and the Heisenberg XXZ model) that it is effective, and allows us to efficiently and accurately extract local correlation functions and transport properties.
A powerful perspective in understanding nonequilibrium quantum dynamics is through the time evolution of its entanglement content. Yet apart from a few guiding principles for the entanglement entropy, to date, much less is known about the refined characteristics of entanglement propagation. Here, we unveil signatures of the entanglement evolving and information propagating out of equilibrium, from the view of the entanglement Hamiltonian. We investigate quantum quench dynamics of prototypical Bose-Hubbard model using state-of-the-art numerical technique combined with conformal field theory. Before reaching equilibrium, it is found that a current operator emerges in the entanglement Hamiltonian, implying that entanglement spreading is carried by particle flow. In the long-time limit the subsystem enters a steady phase, evidenced by the dynamic convergence of the entanglement Hamiltonian to the expectation of a thermal ensemble. Importantly, the entanglement temperature in steady state is spatially independent, which provides an intuitive trait of equilibrium. These findings not only provide crucial information on how equilibrium statistical mechanics emerges in many-body dynamics, but also add a tool to exploring quantum dynamics from the perspective of the entanglement Hamiltonian.
The jump unraveling of a quantum master equation decomposes the dynamics of an open quantum system into abrupt jumps, interspersed by periods of coherent dynamics when no jumps occur. Such open quantum systems are ubiquitous in quantum optics and mesoscopic physics, hence the need for efficient techniques for their stochastic simulation. Numerical simulation techniques fall into two main categories. The first splits the evolution into small time steps and determines stochastically for each step if a jump occurs or not. The second, known as Monte Carlo wave-function simulation, is based on the reduction of the norm of an initially pure state in the conditional no-jump evolution. It exploits the fact that the purity of the state is preserved by the finest unraveling of the master equation. In this paper, we present an alternative method for the simulation of the quantum jump unraveling, inspired by the classical Gillespie algorithm. The method is particularly well suited for situations in which a large number of trajectories is required for relatively small systems. It allows for non-purity-preserving dynamics, such as the ones generated by partial monitoring and channel merging. We describe the algorithm in detail and discuss relevant limiting cases. To illustrate it, we include four example applications of increasing physical complexity and discuss the performance of the algorithm across regimes of interest for open quantum systems simulation. Publicly available implementations of our code are provided in julia and mathematica.