The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is an important task in the simulation of quantum many-body systems. As the dimension of the underlying vector space grows exponentially in the number of particles, one has to consider appropriate subsets promising both convenient approximation properties and efficient computations. The variational ansatz for this numerical approach leads to the minimization of the Rayleigh quotient. The Alternating Least Squares technique is then applied to break down the eigenvector computation to problems of appropriate size, which can be solved by classical methods. Efficient computations require fast computation of the matrix-vector product and of the inner product of two decomposed vectors. To this end, both appropriate representations of vectors and efficient contraction schemes are needed.Here approaches from many-body quantum physics for onedimensional and two-dimensional systems (Matrix Product States and Projected Entangled Pair States) are treated mathematically in terms of tensors. We give the definition of these concepts, bring some results concerning uniqueness and numerical stability and show how computations can be executed efficiently within these concepts. Based on this overview we present some modifications and generalizations of these concepts and show that they still allow efficient computations such as applicable contraction schemes. In this context we consider the minimization of the Rayleigh quotient in terms of the parafac (CP) formalism, where we also allow different tensor partitions. This approach makes use of efficient contraction T. Huckle et al. / Linear Algebra and its Applications 438 (2013) 750-781 751 schemes for the calculation of inner products in a way that can easily be extended to the mps format but also to higher dimensional problems.
We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (mps) for the desired eigenvectors.This paper summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of Matrix Product States (Tensor Trains or Tensor Chains) and their canonical normal forms we show how symmetry properties of the vector translate into relations between the mps matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context we analyze different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This paper provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.
Quantum control plays a key role in quantum technology, in particular for steering quantum systems. As problem size grows exponentially with the system size, it is necessary to deal with fast numerical algorithms and implementations. We improved an existing code for quantum control concerning two linear algebra tasks: The computation of the matrix exponential and efficient parallelisation of prefix matrix multiplication.For the matrix exponential we compare three methods: the eigendecomposition method, the Padé method and a polynomial expansion based on Chebyshev polynomials. We show that the Chebyshev method outperforms the other methods both in terms of computation time and accuracy. For the prefix problem we compare the tree-based parallel prefix scheme, which is based on a recursive approach, with a sequential multiplication scheme where only the individual matrix multiplications are parallelised. We show that this fine-grain approach outperforms the parallel prefix scheme by a factor of 2-3, depending on parallel hardware and problem size, and also leads to lesser memory requirements.Overall, the improved linear algebra implementations not only led to a considerable runtime reduction, but also allowed us to tackle problems of larger size on the same parallel compute cluster.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.