Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is 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 is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SUloc(2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SUloc(2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.
Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is 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 is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SUloc(2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SUloc(2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.
Many theoretical problems in quantum technology can be formulated and addressed as constrained optimization problems. The most common quantum mechanical constraints such as, e.g., orthogonality of isometric and unitary matrices, CPTP property of quantum channels, and conditions on density matrices, can be seen as quotient or embedded Riemannian manifolds. This allows to use Riemannian optimization techniques for solving quantum-mechanical constrained optimization problems. In the present work, we introduce QGOpt, the library for constrained optimization in quantum technology. QGOpt relies on the underlying Riemannian structure of quantum-mechanical constraints and permits application of standard gradient based optimization methods while preserving quantum mechanical constraints. Moreover, QGOpt is written on top of TensorFlow, which enables automatic differentiation to calculate necessary gradients for optimization. We show two application examples: quantum gate decomposition and quantum tomography.
This article shows how C-numerical-range related new strucures may arise from practical problems in quantum control – and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function . In quantum control of n qubits one may be interested (i) in having U∈SU(2 n ) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e., to the n-fold tensor product . Interestingly, the latter then leads to a novel entity, the local C-numerical range W loc(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying article on Relative C-Numerical Ranges for Application in Quantum Control and Quantum Information [Dirr, G., Helmke, U., Kleinsteuber, M. and Schulte-Herbrüggen, T., 2008 Dirr, G, Helmke, U, Kleinsteuber, M and Schulte-Herbrüggen, T. 2008. Relative C-numerical ranges for application in quantum control and quantum information. Linear and Multilinear Algebra, 56: 27–51. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], Linear and Multilinear Algebra, 56, 27–51]. We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. (3) We conclude by connecting the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient-flow algorithms.
Several tensor networks are built of isometric tensors, i.e. tensors satisfying W\dagger W = \mathbb{1} W † W = 1 . Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.
We establish a rigorous existence theory for the quantum splines introduced by Brody, Holm, and Meier in Physical Review Letters (2012). These curves arise as solutions of a variational problem on the unitary group describing optimally controlled quantum evolutions. By formulating the problem within a geometric gradient flow framework for Riemannian spline interpolation, we construct a well-posed fourth order evolution whose asymptotic limits realize the desired quantum splines. The analysis requires adapting the variational structure to boundary conditions dictated by the physical model, which are not directly amenable to the setting in our recently developed framework for gradient flows of Riemannian spline interpolation. We show that, despite these difficulties, the modified system admits a rigorous analytical treatment, yielding both existence and a constructive procedure for generating quantum splines. Our results provide a mathematical foundation for the variational description of smooth quantum control trajectories and clarify the analytical structure underlying their formation.
Accurate derivatives are important for locally traversing and converging efficiently in quantum optimization landscapes. By deriving analytically exact control derivatives (gradient and Hessian) for unitary control tasks, we show here that the computational feasibility of meeting this accuracy requirement depends on the choice of propagation scheme and problem representation. Even when exact propagation is sufficiently cheap it is, perhaps surprisingly, much more efficient to optimize the (appropriately) approximate propagators: Approximations in the dynamics are traded off for significant complexity reductions in the exact derivative calculations. Importantly, past the initial analytical considerations, only standard numerical techniques are explicitly required with straightforward application to realistic systems. These results are numerically verified for two concrete problems of increasing Hilbert space dimensionality. The best schemes obtain unit fidelity to machine precision, whereas the results for other schemes are separated consistently by orders of magnitude in computation time and in worst-case ten orders of magnitude in achievable fidelity. Since these gaps continually increase with system size and complexity, this methodology allows numerically efficient optimization of very high-dimensional dynamics, e.g., in many-body contexts, operating in the high-fidelity regime.
Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science and becomes especially challenging for high-dimensional complex structure problems. One specific issue is that of quantum process tomography, in which the goal is to retrieve the underlying quantum process based on a given set of measurement data. In this paper, we introduce a modified version of stochastic gradient descent on a Riemannian manifold that integrates recent advancements in numerical methods for Riemannian optimization. This approach inherently supports the physically driven constraints of a quantum process, takes advantage of state-of-the-art large-scale stochastic objective optimization, and has superior performance to traditional approaches such as maximum likelihood estimation and projected least squares. The data-driven approach enables accurate, order-of-magnitude faster results, and works with incomplete data. We demonstrate our approach on simulations of quantum processes and in hardware by characterizing an engineered process on quantum computers.