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PAMM

Conjugate gradient algorithms for best rank‐1 approximation of tensors

O. Curtef, Gunther Dirr, Uwe Helmke B

Published Dec 1, 2007• 6 citations
Publisher

Abstract

Abstract Motivated by considerations of pure state entanglement in quantum information, we consider the problem of finding the best rank‐1 approximation to an arbitrary r ‐th order tensor. Reformulating the problem as an optimization problem on the Lie group SU (n1) ⊗ … ⊗ SU (nr) of so‐called local unitary transformations and exploiting its intrinsic geometry yields a new approach, which finally leads to Riemannian variant of the conjugate gradient algorithm. Numerical simulations support that our method offers an alternative to the higher‐order power method for computing the best rank‐1 approximation to a tensor. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pamm.200700706

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Abstract. In this paper, we propose a preconditioned Riemannian conjugate gradient method for computing ground states of spin-1 Bose-Einstein condensates, which can be reformulated as an optimization problem with the mass and magnetization constraints. The energy functional and constraints can be discretized using Fourier pseudospectral schemes, thereby transforming the problem into an optimization task on a manifold. We derive vector transports by differentiating three existing retractions and propose an initial step size selection strategy based on the second-order approximation of the energy function for the proposed Riemannian optimization algorithm. In addition, a preconditioner is derived and used. To further accelerate the convergence, we combine the proposed algorithm with a multigrid method. Numerical experiments demonstrate the efficiency and accuracy of the proposed method while confirming the usefulness of our preconditioning technique and the effectiveness of the proposed strategy for initial step size selection.
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.
Abstract In this paper we study the relationship between factorization problems on SU(2n) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract When dealing with quantum many‐body systems one is faced with problems growing exponentially in the number of particles to be considered. To overcome this curse of dimensionality one has to consider representation formats which scale only polynomially. Physicists developed concepts like matrix product states (MPS) to represent states of interest and formulated algorithms such as the density matrix renormalization group (DMRG) to find such states. We consider the standard Lanczos algorithm and formulate it for vectors given in the MPS format. It turns out that a restarted version which includes a projection onto the MPS manifold gives the same approximation quality as the well‐established DMRG method. Moreover, this variant is more flexible and provides more information about the spectrum. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)