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

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

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

> 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