Geometric Optimization Techniques in Quantum Research

Gradient Flows for Optimisation and Quantum Control: Foundations and Applications https://lnkd.in/exxgq-P5 For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications to quantum control: we extend former results on unitary groups to closed subgroups with tensorproduct structure, where the finest product partitioning consists of purely local unitary operations. Moreover, the framework is kept sufficiently general for setting up gradient flows on (sub-)manifolds, Lie (sub-)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This part of the paper is meant to serve as foundation for some recent and new achievements, and as setting for further research. Exploiting the differential geometry of quantum dynamics under different scenarios helps to provide highly useful algorithms: (a) On an abstract level, gradient flows may establish the exact upper bounds of pertinent quality functions, i.e. upper bounds reachable within the underlying manifold of the state space dynamics; (b) in a second stage referring to a concrete experimental setting, gradient flows on the space of piecewise constant control amplitudes in R m may be set up to yield (approximations to) optimal control for quantum devices under realistic conditions. Illustrative examples and new applications are given, such as figures of merit on the subgroup of local unitary action on n qubits relating to distance measures of pure-state entanglement. We establish the correspondence to best rank-1 approximations of higher order tensors and show applications from quantum information, where our gradient flows on the subgroup of local unitary operations provide a numerically stable alternative to tensor-svd techniques.

Any new approach to having a more efficient quantum encoding method in QML? Here's an interesting and novel perspective. A new study titled "A Qubit-Efficient Hybrid Quantum Encoding Mechanism for Quantum Machine Learning" introduces an interesting approach to address a significant barrier in Quantum Machine Learning (QML): efficiently embedding high-dimensional datasets onto noisy, low-qubit quantum systems. The research proposes Quantum Principal Geodesic Analysis (qPGA), a non-invertible method for dimensionality reduction and qubit-efficient encoding. Unlike existing quantum autoencoders, which can be constrained by current hardware and may be vulnerable to reconstruction attacks, qPGA offers a robust alternative. Key outcomes of this study include: * Qubit-efficient encoding: qPGA leverages Riemannian geometry to project data onto the unit Hilbert sphere (UHS), generating outputs inherently suitable for quantum amplitude encoding. This technique significantly reduces qubit requirements for amplitude encoding, allowing high-dimensional data to be mapped onto small-qubit systems. * Preservation of data structure: The method preserves the neighborhood structure of high-dimensional datasets within a compact latent space. Empirical results on MNIST, Fashion-MNIST, and CIFAR-10 datasets show that qPGA preserves local structure more effectively than both quantum and hybrid autoencoders. * Enhanced resistance to reconstruction attacks: Due to its non-invertible nature and lossy compression, qPGA enhances resistance to reconstruction attacks, offering better defense against data privacy leakage compared to quantum-dependent encoders like Quantum Autoencoders (QE) and Hybrid Quantum Autoencoders (HQE). * Noise-resilient and scalable: Initial tests on real hardware and noisy simulators confirm qPGA's potential for noise-resilient performance, offering a scalable solution for advancing QML applications. The study also provides theoretical bounds quantifying qubit requirements for effective encoding onto noisy systems. Here more details: https://lnkd.in/dSz_xM2q #qml #machinelearning #datascience #ml #quantum

Conventional wisdom suggests that the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing differ fundamentally, especially when QAOA uses angles that do not vanish with problem size. In our new work (led by Sami Boulebnane), we rigorously show that, for certain “constant‐angle” schedules, QAOA does replicate linear‐time quantum annealing behavior—precisely in the regime where QAOA achieves its best performance. • We prove this equivalence for the Sherrington‐Kirkpatrick (SK) model, a classical spin‐glass benchmark, and provide evidence that the same reasoning may extend to other constrained optimization problems. • Because QAOA can approximate annealing without tiny Trotter steps, we reduce the required circuit depth by a factor linear in the number of variables. • Our analysis employs a novel series expansion for QAOA observables at arbitrary depth, providing new tools to study how QAOA scales. Special thanks to the great team: Sami Boulebnane, James Sud, and Marco Pistoia. Link: https://lnkd.in/eBfCrffj

Quantum state tomography, the process of reconstructing an unknown quantum state, traditionally suffers from computational demands that grow exponentially with system size, a significant barrier to progress in quantum technologies. S. M. Yousuf Iqbal Tomal and Abdullah Al Shafin, both from BRAC University, now present a new approach, geometric latent space tomography, which overcomes this limitation while crucially preserving the underlying geometric structure of quantum states. Their method combines classical neural networks with quantum circuit decoders, trained to ensure that distances within the network’s ‘latent space’ accurately reflect the true distances between quantum states, measured by the Bures distance. This innovative technique achieves high-fidelity reconstruction of quantum states and reveals an intrinsic, lower-dimensional structure within the complex space of quantum possibilities, offering substantial computational advantages and enabling direct state discrimination and improved error mitigation for quantum devices. https://lnkd.in/eSpH3YhD

Fresh off the press in APL Quantum: Reimagining Magnetic Material Design with Quantum-Inspired Logic. 🔬 In my previous posts, I talked about the "#Quantum On-Ramp" and our new #MATLAB integration. Today, I want to share the rigorous scientific validation behind it. Our latest research, "Design of magnetic lattices with a quantum-inspired evolutionary optimization algorithm," has just been published in APL Quantum journal. This work was a fantastic collaboration between Prof. Pınar Acar and her talented students (Zekeriya Ender Eger and Waris Khan) at Virginia Tech, alongside the dedicated team at BQP, led by Rut Lineswala, with Dr. Eswara Sai Kumar Kandula, Udbhav Sharma, & Priyo. 🤝 The Challenge: Designing magnetic #materials at the lattice level is a massive computational bottleneck. Using the Ising model to find the #magnetic equilibrium under uncertainty (like fluctuating temperature and external fields) results in a high-dimensional, non-convex optimization landscape with countless local minima. The Breakthrough: We utilized the Quantum-Inspired Evolutionary Optimization (QIEO) algorithm, one of our core capabilities in QuantumNOW™ solver, to tackle this. Key Findings: 1. Large-Scale Success: While traditional methods like Genetic Algorithms (GA) and Simulated Annealing (SA) struggle as dimensionality grows, our QIEO solver successfully optimized 50x50 lattices systems where conventional classical methods often become computationally impractical. We reported 20X+ speedup vs SA in that case. 2. Handling Uncertainty: The algorithm remained robust even when accounting for stochastic variables in temperature and magnetic fields, proving that quantum logic can solve real-world "noise" problems today. 3. HPC Efficiency: This entire study was conducted on existing high-performance computing (#HPC) infrastructure, creating the foundation for hybrid QC + HPC environments. Why this matters for your R&D team? The exact algorithms validated in this paper are available right now in the #BQPhy Toolbox for MATLAB. Whether you are working on magnetic materials, satellite trajectories, or complex design optimization, you can leverage this "Quantum On-Ramp" today. You can read the full paper here:

Parity Quantum Optimization: Encoding Constraints “Constraints to optimization problems are crucial for many problems that are encountered in science, technology, and industry, ranging from scheduling problems to quantum chemistry. Quantum computing as a new paradigm of computing, which aims, among other things, at enhancing optimization algorithms by making use of quantum phenomena, may improve upon existing algorithms to solve these kinds of problems. However, quantum computers are limited in coherence, control, and connectivity which makes encoding of optimization problems one of the current grand challenges in the field. Constraints are an additional complication to the encoding challenge and they are typically encoded via large energy penalties given as quadratic terms leading to fully connected interactions. “ “To encode constraints we introduce a combination of exchange interactions and spin-flip terms in combination with the parity encoding. The parity trans-formation encodes optimization problems in a lattice gauge model with local 3-body and 4-body interac-tions on a square lattice. We introduce exchange terms that only act on qubits that are part of the constraints and spin-flip terms that act on the rest of the qubits. Using a compiler , qubits can be arranged on the square lattice with flexibility.”   By Maike Drieb-Schön , Kilian Ender , Younes Javanmard, and Wolfgang Lechner   ParityQC Universität Innsbruck Link https://lnkd.in/dJZknhiN

💡 𝗪𝗮𝗻𝘁 𝘁𝗼 𝘀𝗲𝗲 𝗵𝗼𝘄 𝗴𝗲𝗼𝗺𝗲𝘁𝗿𝘆 𝗰𝗮𝗻 𝗴𝘂𝗶𝗱𝗲 𝗹𝗲𝗮𝗿𝗻𝗶𝗻𝗴? I’ve been working on a geometry-aware dynamic approach for quantum ML, where the search space is constrained to a structured manifold. 📌 But the key idea is this: The model adapts itself based on what it diagnoses during training. Using signals like: • Shannon entropy • KL-divergence (diversity) • Reward dynamics & gradients It can detect issues and react. Example from a recent run: → 𝗗𝗶𝗮𝗴𝗻𝗼𝘀𝗶𝘀: 𝗘𝗮𝗿𝗹𝘆 𝗟𝗼𝘄 𝗘𝘅𝗽𝗿𝗲𝘀𝘀𝗶𝘃𝗶𝘁𝘆 → 𝗔𝗰𝘁𝗶𝗼𝗻: 𝗶𝗻𝗰𝗿𝗲𝗮𝘀𝗲𝗱 𝗰𝗶𝗿𝗰𝘂𝗶𝘁 𝗱𝗲𝗽𝘁𝗵 (𝟭 → 𝟮 𝗹𝗮𝘆𝗲𝗿𝘀, 𝗽𝗿𝗲𝘀𝗲𝗿𝘃𝗶𝗻𝗴 𝗹𝗲𝗮𝗿𝗻𝗲𝗱 𝘄𝗲𝗶𝗴𝗵𝘁𝘀) → 𝗢𝘂𝘁𝗰𝗼𝗺𝗲: 𝘁𝗿𝗮𝗻𝘀𝗶𝘁𝗶𝗼𝗻 𝘁𝗼 𝗛𝗲𝗮𝗹𝘁𝗵𝘆 𝗘𝘅𝗽𝗹𝗼𝗿𝗮𝘁𝗶𝗼𝗻 + 𝗶𝗺𝗽𝗿𝗼𝘃𝗲𝗱 𝗿𝗲𝘄𝗮𝗿𝗱𝘀 So instead of blind optimization, the system: → 𝗜𝗻𝘁𝗲𝗿𝗽𝗿𝗲𝘁𝘀 𝗶𝘁𝘀 𝗹𝗲𝗮𝗿𝗻𝗶𝗻𝗴 𝘀𝘁𝗮𝘁𝗲 → 𝗔𝗱𝗷𝘂𝘀𝘁𝘀 𝗮𝗿𝗰𝗵𝗶𝘁𝗲𝗰𝘁𝘂𝗿𝗲 & 𝗵𝘆𝗽𝗲𝗿𝗽𝗮𝗿𝗮𝗺𝗲𝘁𝗲𝗿𝘀 𝗱𝘆𝗻𝗮𝗺𝗶𝗰𝗮𝗹𝗹𝘆 Still early, but already showing promising behavior. #QuantumAI #MachineLearning #QuantumComputing #Optimization

> sharing resource < Impressive work this morning: "Quantum Geometric Machine Learning" a thesis by Elija Perrier under the supervision of Chris Ferrie and Dacheng Tao Introduction: The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent combination of such geometric and symmetry-based methods with quantum machine learning (QML) has provided a rich opportunity to contribute to solving a number of persistent challenges in fields such as QML parametrisation, quantum control, quantum unitary synthesis and quantum proof generation. In this thesis, we combine state of-the-art machine learning methods with techniques from differential geometry and topology to address these challenges. We present a large-scale simulated dataset of open quantum systems to facilitate the development of quantum machine learning as a field. We demonstrate the use of deep learning greybox machine learning techniques for estimating approximate time-optimal unitary sequences as geodesics on subRiemannian symmetric space manifolds. Finally, we present novel techniques utilising Cartan decompositions and variational methods for analytically solving quantum control problems for certain classes of Riemannian symmetric space. [...] Link: https://lnkd.in/e4FjBRhz #quantummachinelearning #quantumcomputing #geometricalmachinelearning #research #thesis

⚛️ Time-Aware Qubit Assignment and Circuit Optimization for Distributed Quantum Computing 📑 Abstract—The emerging paradigm of distributed quantum computing promises a potential solution to scaling quantum computing to currently unfeasible dimensions. While this approach itself is still in its infancy, and many obstacles must still be overcome before its physical implementation, challenges from the software and algorithmic side must also be identified and addressed. For instance, this paradigm shift requires a new form of compiler that considers the network constraints in general as well as phenomena arising due to the nature of quantum communication. In distributed quantum computing, large circuits are divided into smaller subcircuits such that they can be executed individually and simultaneously on multiple QPUs that are connected through quantum channels. As quantum communication, for example, in the form of teleportation, is expensive, it must be used sparingly. We address the problem of assigning qubits to QPUs to minimize communication costs in two different ways. First by applying time-aware algorithms that take into account the changing connectivity of a given circuit as well as the underlying network topology. We define the optimization problem, use simulated annealing and an evolutionary algorithm and compare the results to graph partitioning and sequential qubit assignment baselines. In another approach, we propose an evolutionary-based quantum circuit optimization algorithm that adjusts the circuit itself rather than the schedule to reduce the overall communication cost. We evaluate the techniques against random circuits and different network topologies. Both evolutionary algorithms outperform the baseline in terms of communication cost reduction. We give an outlook on how the approaches can be integrated into a compilation framework for distributed quantum computing. ℹ️ Sunkel et al - Institute for Informatics, LMU Munich, Munich, Germany - 2025

Our paper "Equivariant Quantum Approximate Optimization Algorithm" is finally out in IEEE Transactions on Quantum Engineering! Many combinatorial optimization problems have natural symmetries, but standard QAOA mixer doesn't take advantage of them. We introduce new symmetry aware mixers that are tailored to the structure of the problem and can be efficiently implemented as quantum circuits. Through numerical experiments on the coloring and partitioning problems on graphs, we demonstrate that these mixers outperform the standard approach. We also explain why warm start QAOA often fails to improve results, identifying a fundamental limitation that prevents guaranteed convergence. Congratulations and huge thanks to our co-authors Boris Tsvelikhovskiy and Yuri Alexeev! Check our paper at https://lnkd.in/e3xAAPNW #QuantumComputing #QAOA #QuantumOptimization #IEEETQE #QuantumEngineering #HybridAlgorithms #Research #VariationalQuantumAlgorithms