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 above $3\cdot 10^3$ 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.
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 above 3⋅103 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.
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.
In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.
Quantum optics has been, from its beginning, a driving force both for the exploration of fundamental limits of the quantum world and for conceiving seminal ideas and applications of the so-called quantum technologies. The last 20 years have seen a rapid development of ideas and proof-of-principle experiments involving the fields of quantum communication, quantum computation, quantum metrology and quantum simulation. The so called continuous variable (CV) approach to quantum optics, which uses many photon in collective states, and continuous observables like the quadratures of the electric field to encode information, has many interesting properties, especially from a communications perspective. It is inherently broadband and compatible with standard telecom infrastructures. Moreover, in CV, entanglement, one of the fundamental resource of quantum optics, can be generated deterministically. One of the main challenges for all quantum information technologies is scalability, being able to generate and manipulate a large number of quantum resources to achieve practical tasks efficiently. One approach to solve the issue of scalability is to use highly multimode quantum states. The quantum description of the electromagnetic field associate each photon (particle of light) with a mode (way for light to propagate). In a mutlimodal approach, we look at the quantum state bases and optical modes bases conjointly and tailor quantum fields not only in given modes, but also optimize the spatio-temporal shapes of the modes in which the state is defined. This opens wide perspectives for treating complex quantum states. In particular, using ultra-fast pulses of light which contain many temporal/spectral modes, we are able to generate large and complex entangled states of light using simple resources. In this thesis, we used an optical parametric oscillator pumped synchronously (SPOPO) with a frequency comb to generate multimode squeezed vacuum states which can be used to form cluster states: a large collection of modes (the nodes) entangled to each other in a certain way. These state are the basic resource for Measurement Based Quantum Computation (MBQC). This set-up has the advantage to generate entanglement in many mode with a single device. It is also highly tunable. Indeed, by tuning the spectrum of the OPO pump with a pulse shaper, one can tailor the properties of the generated quantum state. In this work, we focus on the optimization of the pump spectral shape to generate specific states. Using simulations based on Machine Learning Algorithms (MLA), we find optimal pump profile for typical target states. We then implement those shapes on the experimental set-up and measure the resulting states using multipixel homodyne detection. We also study intra-cavity dispersion effects. Dispersion inside the SPOPO cavity is indeed one of the main factor that limits the number of entangled modes in the generated quantum states. A systematic study of dispersion effects is therefore necessary to model the SPOPO output accurately. This works paves the way toward a fully tunable device that can be optimized in real time to generate specific quantum resources tailored for specific tasks.
This paper explores the application of tensor network methods for simulating molecular dynamics on quantum computers. Traditional molecular dynamics simulations are computationally expensive, particularly for complex systems exhibiting strong quantum correlations. Tensor networks provide a powerful framework for representing and manipulating high-dimensional quantum states, potentially offering a pathway to efficiently simulate molecular dynamics by directly leveraging quantum computational resources. This work details the theoretical framework, outlining how tensor network techniques, specifically Matrix Product States (MPS) and Tree Tensor Networks (TTN), can be adapted for this purpose. We present a simplified model of a diatomic molecule to illustrate the core concept: the construction of a tensor network representation of the molecular wavefunction, followed by the application of time evolution operators to simulate dynamics. The focus is on demonstrating the potential of this approach rather than a fully optimized simulation, highlighting key challenges and future research directions. The efficiency gains compared to classical methods are discussed, acknowledging the significant technological hurdles involved in realizing this approach on near-term quantum devices. We investigate the scalability of the method, considering the limitations imposed by the bond dimension of the tensor network. The ultimate goal is to demonstrate how tensor networks can provide a viable strategy for tackling computationally intractable molecular dynamics problems, paving the way for a new era of quantum-enhanced molecular simulations.
Tensor network formalisms have emerged as powerful tools for simulating quantum state evolution. While widely applied in the study of optical quantum circuits, such as boson sampling, existing tensor network approaches fail to address the complexity mismatch between tensor contractions and the calculation of photon-counting probability amplitudes. Here, we present an alternative tensor network framework that exploits the input-output relations of quantum optical circuits encoded in the unitary interferometer matrix. Our approach bridges the complexity gap by enabling the computation of the permanent—central to boson sampling—with the same computational complexity as the best known classical algorithm based on a graphical representation of the operator-basis matrix product states that we introduce. Furthermore, we exploit the flexibility of tensor networks to extend our formalism to incorporate partial distinguishability and photon loss, two key imperfections in practical interferometry experiments. This work offers a significant step forward in the simulation of large-scale quantum optical systems and the understanding of their computational complexity.
Modern approaches to generative modeling of continuous data using tensor networks incorporate compression layers to capture the most meaningful features of high-dimensional inputs. These methods, however, rely on traditional Matrix Product States (MPS) architectures. Here, we demonstrate that beyond a certain threshold in data and bond dimensions, a comb-shaped tensor network architecture can yield more efficient contractions than a standard MPS. This finding suggests that for continuous and high-dimensional data distributions, transitioning from MPS to a comb tensor network representation can substantially reduce computational overhead while maintaining accuracy.