Matrix product states (MPS) provide a memory efficient way to store high dimensional many body quantum systems [1]. Each degree of freedom is assigned a tensor and connecting each are bond matrices which capture correlations between different degrees of freedom as depicted in Fig 1. (a). Next efficient time evolution is then achieved by time evolving block decimation (TEBD) [2]. The essence of TEBD is that given a local Hamiltonian and a small time step the resulting unitary transformation can be decomposed into a collection of commuting two-site and single site gates acting on our MPS as shown in Fig 1. (b). With this formalism we care able to describe 1D waveguide systems with dispersion profiles up to arbitrary order and evolve states under a Hamiltonian with a$\chi^{(3)}$non linearity [3]. Linear loss and two photon absorption can be included in the model through the materials imaginary dispersion data. Arbitrary spatial, temporal and spectral pump profiles may also be included in the driving of parametric processes. Given an initial quantum field and classical drives we can simulate the temporal dynamics of the quantum field. From this we extract the familiar measurable quantities including spectral content, temporal profile, correlators or any well defined observable of the field. In Fig 2. (c) We show the spectrum of a weak coherent state at$2050\text{nm}\ (\vert\alpha\vert =1)\ (\text{red})$, which undergoes stimulated 4 wave mixing via a classical pump at 2070nm (purple). After the interaction the resulting spectra (blue) contains stimulated photon generation at the phase matched idler wavelength of 2090nm and parametric amplification of the stimulating field. Fig 2. (d) shows an example of the field expectation value and variance for the underlying quantum state before (red) and after (blue) the interaction.
Matrix product states (MPS) provide a memory efficient way to store high dimensional many body quantum systems [1]. Each degree of freedom is assigned a tensor and connecting each are bond matrices which capture correlations between different degrees of freedom as depicted in Fig 1. (a). Next efficient time evolution is then achieved by time evolving block decimation (TEBD) [2]. The essence of TEBD is that given a local Hamiltonian and a small time step the resulting unitary transformation can be decomposed into a collection of commuting two-site and single site gates acting on our MPS as shown in Fig 1. (b). With this formalism we care able to describe 1D waveguide systems with dispersion profiles up to arbitrary order and evolve states under a Hamiltonian with aχ(3)non linearity [3]. Linear loss and two photon absorption can be included in the model through the materials imaginary dispersion data. Arbitrary spatial, temporal and spectral pump profiles may also be included in the driving of parametric processes. Given an initial quantum field and classical drives we can simulate the temporal dynamics of the quantum field. From this we extract the familiar measurable quantities including spectral content, temporal profile, correlators or any well defined observable of the field. In Fig 2. (c) We show the spectrum of a weak coherent state at2050nm(∣α∣=1)(red), which undergoes stimulated 4 wave mixing via a classical pump at 2070nm (purple). After the interaction the resulting spectra (blue) contains stimulated photon generation at the phase matched idler wavelength of 2090nm and parametric amplification of the stimulating field. Fig 2. (d) shows an example of the field expectation value and variance for the underlying quantum state before (red) and after (blue) the interaction.
Quantum memories are essential components in applications ranging from quantum computing to quantum communication networks. However, their practical utility is constrained by short coherence times, motivating the search for new physical systems that can inherently protect stored information. Discrete time crystals (DTCs)—periodically driven many-body systems exhibiting stable subharmonic oscillations that break discrete time-translation symmetry—offer a promising approach, as they are theoretically able to shield encoded information from local perturbations, making them compelling candidates for nextgeneration, passively protected quantum memories. In this work, we employ a tensor-network framework that models a quantum memory as a DTC. We employ the timeevolving block-decimation (TEBD) algorithm to perform both real- and imaginary-time evolution of a matrix-product-state (MPS) representation, thereby efficiently capturing the large many-body Hilbert space while tracking entanglement growth, sub-harmonic spectral responses, and memory-fidelity metrics over experimentally relevant timescales. By sweeping the drive strength, interaction range, and disorder, we map the phase diagram, pinpoint regimes that sustain time-crystalline order, and set the stage to model their coherence lifetimes.
Understanding the quantum evolution of light in nonlinear media is central to the development of next-generation quantum technologies. Yet, modeling these processes remains computationally demanding, as the required resources grow rapidly with photon number and phase-space resolution. Here, we introduce a tensor-network approach that efficiently captures the dynamics of nonlinear optical systems in a continuous-variable representation. Using the matrix product state (MPS) formalism, both quantum states and operators are encoded in a highly compressed form, enabling direct numerical integration of the Schrödinger equation. We demonstrate the method by simulating degenerate spontaneous parametric down-conversion (SPDC) and show that it accurately reproduces established theoretical benchmarks - energy conservation, pump depletion, and quadrature squeezing - even in regimes where conventional Fock-basis simulations become infeasible. For high-intensity pump fields (α=100), the MPS representation achieves compression ratios below 3×10−4 while preserving physical fidelity. This framework opens a scalable route to modeling multimode quantum light and nonlinear optical phenomena beyond the reach of traditional methods.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
We introduce the concept of concatenated tensor networks to efficiently describe quantum states. We show that the corresponding concatenated tensor network states can efficiently describe time evolution and possess arbitrary block-wise entanglement and long-ranged correlations. We illustrate the approach for the enhancement of matrix product states, i.e. one-dimensional (1D) tensor networks, where we replace each of the matrices of the original matrix product state with another 1D tensor network. This procedure yields a 2D tensor network, which includes—already for tensor dimension 2—all states that can be prepared by circuits of polynomially many (possibly non-unitary) two-qubit quantum operations, as well as states resulting from time evolution with respect to Hamiltonians with short-ranged interactions. We investigate the possibility of efficiently extracting information from these states, which serves as the basic step in a variational optimization procedure. To this aim, we utilize the known exact and approximate methods for 2D tensor networks and demonstrate some improvements thereof, which are also applicable e.g. in the context of 2D projected entangled pair states. We generalize the approach to higher dimensional and tree tensor networks.
Ultrashort pulses propagating in nonlinear nanophotonic waveguides can simultaneously leverage both temporal and spatial field confinement, promising a route towards single-photon nonlinearities in an all-photonic platform. In this multimode quantum regime, however, faithful numerical simulations of pulse dynamics naïvely require a representation of the state in an exponentially large Hilbert space. Here, we employ a time-domain, matrix product state (MPS) representation to enable efficient simulations by exploiting the entanglement structure of the system. To extract physical insight from these simulations, we develop an algorithm to unravel the MPS quantum state into constituent temporal supermodes, enabling, e.g., access to the phase-space portraits of arbitrary pulse waveforms. As a demonstration, we perform exact numerical simulations of a Kerr soliton in the quantum regime. We observe the development of non-classical Wigner-function negativity in the solitonic mode as well as quantum corrections to the semiclassical dynamics of the pulse. A similar analysis of χ ( 2 ) simultons reveals a unique entanglement structure between the fundamental and second harmonics. Our approach is also readily compatible with quantum trajectory theory, allowing full quantum treatment of propagation loss and decoherence. We expect this work to establish the MPS technique as part of a unified engineering framework for the emerging field of broadband quantum photonics.
[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.