Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies

Abstract

Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on manifolds of unitary and isometric matrices as well as the cone of positive-definite matrices. Combining this approach with the up-to-date computational methods of automatic differentiation, we demonstrate the efficacy of the Riemannian optimization in the study of the low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of a tensor network in the form of the multiscale entanglement-renormalization ansatz, preparation of arbitrary states (including highly entangled ones) in the circuit implementation of quantum computation, decomposition of quantum gates, and tomography of quantum states. Universality of the developed approach together with the provided open source software enable one to apply the Riemannian optimization to complex quantum architectures well beyond the listed problems, for instance, to the optimal control of noisy quantum systems.

arXiv comment27 pages, 15 figures, a note on relevant research is added

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• Tensor networks for quantum machine learning
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• Optimizing quantum circuits with Riemannian gradient flow
• Tree tensor network classifiers for machine learning: from quantum-inspired to quantum-assisted
• Gradient-descent quantum process tomography by learning Kraus operators
• Compressive gate set tomography
• Tensor networks for quantum computing
• Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus
• Quantum-classical eigensolver using multiscale entanglement renormalization
• Multipartite correlations in quantum collision models
• Robustly learning the Hamiltonian dynamics of a superconducting quantum processor
• Efficient MPS representations and quantum circuits from the Fourier modes of classical image data
• A tensor network discriminator architecture for classification of quantum data on quantum computers
• The Comparison of Riemannian Geometric Matrix-CFAR Signal Detectors
• NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization
• Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states
• Finite-depth scaling of infinite quantum circuits for quantum critical points
• Gradient-descent methods for fast quantum state tomography
• Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems
• Exploring postselection-induced quantum phenomena with time-bidirectional state formalism
• Controlling quantum many-body systems using reduced-order modelling
• Learning topological states from randomized measurements using variational tensor network tomography
• Deep Circuit Compression for Quantum Dynamics via Tensor Networks
• Quantum channels, complex Stiefel manifolds, and optimization
• Automatic Structural Search of Tensor Network States including Entanglement Renormalization
• Simulating quantum circuits using the multi-scale entanglement renormalization ansatz
• Tensor-Programmable Quantum Circuits for Solving Differential Equations
• Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo
• Diagonal Isometric Form for Tensor Network States in Two Dimensions
• Unsupervised Learning of Effective Quantum Impurity Models
• Benchmarking Single-Qubit Gates on a Neutral Atom Quantum Processor