Computational Methods for Quantum Systems

⚛️ Who can compete with quantum computers? Lecture notes on quantum inspired tensor networks computational techniques 📜 This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algo- rithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the “quantics” representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross–Pitaevskii) within the “quantics” representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements. ℹ️ Waintal et al - 2026

Quantum Linear System Solvers: A Survey of Algorithms and Applications https://lnkd.in/e8fsHKgP Solving linear systems of equations plays a fundamental role in numerous computational problems from different fields of science. The widespread use of numerical methods to solve these systems motivates investigating the feasibility of solving linear systems problems using quantum computers. In this work, we provide a survey of the main advances in quantum linear systems algorithms, together with some applications. We summarize and analyze the main ideas behind some of the algorithms for the quantum linear systems problem in the literature. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. We note its limitations and reliance on computationally expensive quantum methods, then highlight subsequent research efforts which aimed to address these limitations and optimize runtime efficiency and precision via various paradigms. We focus in particular on the post-HHL enhancements which have paved the way towards optimal lower bounds with respect to error tolerance and condition number. By doing so, we propose a taxonomy that categorizes these studies. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, we explore the foundational work that have inspired and informed their development, as well as subsequent refinements. Finally, we discuss the potential applications of these algorithms in differential equations, quantum machine learning, and many-body physics.

The paper below introduces a novel computational framework for understanding&modeling the interaction of molecules through the concept of “molecular holograms” I.e., spatiotemporal representations that encode the quantum&chemical properties of molecules (e.g., electronic distributions and reactive behaviors). The computational approach combines quantum mechanics, ML, & holographic imaging techniques to build a predictive&interpretable model of molecular systems. Molecular holography refers to a high-dimensional representation of molecules that encapsulates their spatial/temporal properties including electronic distributions, spin states, and other quantum mechanical descriptors. Spatiotemporal modeling involves tracking the dynamic behavior of molecules in space&time by integrating quantum mechanical simulations w/data-driven models that account for complex temporal dependencies, such as reaction kinetics. Methods: Time-dependent Density Functional Theory was used to simulate the electronic structure of molecules while molecular dynamics simulations provided insight into temporal evolution. The molecular holograms are generated by encoding wavefunction data into a multidimensional space using Fourier transforms integrating position, momentum, &electronic density. DL models (e.g., graph neural networks, recurrent networks) are trained on holographic data to learn patterns and predict outcomes like reactivity&stability. The molecular holograms enable precise predictions of reaction pathways, transition states, and activation energies. The method facilitates the design of molecules with desired properties by analyzing holograms for stability, reactivity, &functionality. The framework can identify molecular interactions in biological environments, aiding in drug-target binding predictions. The authors demonstrate the effectiveness of the method by applying it to a diverse dataset of molecular systems, including organic reactions, enzyme dynamics, & nanomaterial design. Comparative analysis shows that holographic models outperform traditional descriptors (e.g., molecular fingerprints) in terms of predictive accuracy&interpretability. This framework was able to predict complex non-linear phenomena (e.g., electron delocalization&excited-state dynamics). Molecular holograms provide a visually interpretable & mathematically rigorous framework; The integration of ML accelerates computations without compromising accuracy; The framework is applicable across a wide range of molecular systems. Unfortunately the computational cost remains high for large-scale systems & holographic encoding is sensitive to noise in input data, which may limit accuracy for certain classes of molecules. Future steps: Develop noise-robust holographic encoding algos; Scale up the approach for macromolecular systems (e.g., proteins, polymers); Extend the temporal resolution for ultra-fast processes (e.g., femtosecond reactions).

I’m excited to share this new work from our IBM Quantum team in collaboration with Oak Ridge National Laboratory. This is a major demonstration of what we mean by realizing useful Quantum-centric supercomputing. Building on the chemistry work developed with RIKEN (https://lnkd.in/eK8jW-Wp) last year, and the previous Krylov demonstration with University of Tokyo (https://lnkd.in/eae_8zGc), the IBM Quantum and ORNL teams developed a quantum algorithm for ground states with convergence guarantees similar to phase estimation, while retaining the error mitigation aspect of sample-based methods. Putting together sample-based approaches and Krylov methods, we call this sample-based Krylov quantum diagonalization (SKQD). The algorithm can be used to compute ground state energies of quantum systems for many lattice models relevant in materials science and high-energy physics. SKQD is demonstrated experimentally on 85 qubits and 6,000 two-qubit gates on IBM quantum processors, against the ground state of the Anderson impurity model, obtaining high accuracies for problem sizes beyond the reach of exact diagonalization. This marks one of the largest implementations of quantum diagonalization to date, and points at how quantum computing, combined with classical computation in quantum-centric supercomputing environments, will enable us to push beyond classical methods for interesting applications. These new results also show again how algorithmic discovery is essential, especially for quantum-centric supercomputing architectures. Classical algorithms for materials science have made an impressive progress in the last decades. However, by thinking of quantum-classical workflows where quantum can deliver a value that cannot be matched by classical, we will move closer to demonstrating quantum advantage. Congratulations again to the team on this achievement. Check out the paper here: https://lnkd.in/epwCrG5R.

The Schrödinger Equation Gets Practical: Quantum Algorithm Speeds Up Real-World Simulations Quantum computing has taken a major leap forward with a new algorithm designed to simulate coupled harmonic oscillators, systems that model everything from molecular vibrations to bridges and neural networks. By reformulating the dynamics of these oscillators into the Schrödinger equation and applying Hamiltonian simulation methods, researchers have shown that complex physical systems can be simulated exponentially faster on a quantum computer than with traditional algorithms. This breakthrough demonstrates not only a practical use of the Schrödinger equation but also the deep connection between quantum dynamics and classical mechanics. The study introduces two powerful quantum algorithms that reduce the required resources to only about log(N) qubits for N oscillators, compared to the massive computational demands of classical methods. This exponential speedup could transform fields such as engineering, chemistry, neuroscience, and material science, where coupled oscillators serve as the backbone of real-world modeling. By bridging theory and application, this research underscores how quantum computing is redefining problem-solving in physics and beyond. With proven exponential advantages and the ability to simulate systems once thought computationally impossible, this quantum algorithm marks a milestone in quantum simulation, Hamiltonian dynamics, and real-world physics applications. The findings point toward a future where quantum computers can accelerate scientific discovery, optimize engineering designs, and even open new frontiers in AI and computational neuroscience. #QuantumComputing #SchrodingerEquation #HamiltonianSimulation #QuantumAlgorithm #CoupledOscillators #QuantumPhysics #ComputationalScience #Neuroscience #Chemistry #Engineering

Quantum computing is pushing the boundaries of chemical simulations to unprecedented accuracy! In a groundbreaking study recently published in The Journal of Chemical Theory and Computation, researchers from IBM Quantum® and Lockheed Martin demonstrated a significant milestone in quantum chemistry, the application of sample-based quantum diagonalization (SQD) techniques to accurately model "open-shell" molecules. Why is this critical? Open-shell molecules like CH₂ (methylene) have unpaired electrons, resulting in complex electronic structures that classical computational methods struggle to simulate accurately. Methylene is particularly intriguing because its high reactivity and magnetic properties significantly influence combustion processes, atmospheric chemistry, and even interstellar phenomena. By harnessing quantum computing, researchers successfully calculated CH₂’s singlet-triplet energy gap—a notoriously difficult challenge for classical approaches. This advancement paves the way for accurately predicting chemical reactivity and designing novel materials crucial for aerospace, catalysis, and sensor technologies. Quantum computing is becoming a transformative tool in real-world chemical research. Explore the full details of this landmark study below #QuantumComputing #QuantumChemistry #IBMQuantum #LockheedMartin #OpenShellMolecules #AerospaceInnovation #MaterialsScience #ChemicalSimulation

We’re excited to share our latest work, “Molecular Quantum Computations on a Protein”! In this paper, we present a fragment-based, quantum-centric workflow that harnesses quantum hardware and classical computing to compute the electronic structure for a 20 amino acid protein (Trp-cage) — demonstrating scalable quantum/classical CI simulations for large biomolecular systems. This approach represents a promising route towards bringing quantum computing to complex biological molecules, with implications for materials science, drug design, and beyond.  You can find the paper here:https://lnkd.in/gsxfmjU9 #IClevelandClinic #IBMQuantum #QuantumComputing #QuantumChemistry