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Register a new userVariational optimization algorithms for uniform matrix product states
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Added by
Valentin Zauner-Stauber
Publication date
2017
fields
Physics
and research's language is
English
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We combine the Density Matrix Renormalization Group (DMRG) with Matrix Product State tangent space concepts to construct a variational algorithm for finding ground states of one dimensional quantum lattices in the thermodynamic limit. A careful comparison of this variational uniform Matrix Product State algorithm (VUMPS) with infinite Density Matrix Renormalization Group (IDMRG) and with infinite Time Evolving Block Decimation (ITEBD) reveals substantial gains in convergence speed and precision. We also demonstrate that VUMPS works very efficiently for Hamiltonians with long range interactions and also for the simulation of two dimensional models on infinite cylinders. The new algorithm can be conveniently implemented as an extension of an already existing DMRG implementation.
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Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials, and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.
A variational ansatz for momentum eigenstates of translation invariant quantum spin chains is formulated. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Unlike previous approaches, the ansatz includes topologically non-trivial states (kinks, domain walls) for systems with symmetry breaking. The method is benchmarked using the spin-1/2 XXZ antiferromagnet and the spin-1 Heisenberg antiferromagnet and we obtain surprisingly accurate results.
Any quantum process is represented by a sequence of quantum channels. We consider ergodic processes, obtained by sampling channel valued random variables along the trajectories of an ergodic dynamical system. Examples of such processes include the effect of repeated application of a fixed quantum channel perturbed by arbitrary correlated noise, or a sequence of channels drawn independently and identically from an ensemble. Under natural irreducibility conditions, we obtain a theorem showing that the state of a system evolving by such a process converges exponentially fast to an ergodic sequence of states depending on the process, but independent of the initial state of the system. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk. Further applications and physical implications of our results are discussed in the companion paper [11].
For the past twenty years, Matrix Product States (MPS) have been widely used in solid state physics to approximate the ground state of one-dimensional spin chains. In this paper, we study homogeneous MPS (hMPS), or MPS constructed via site-independent tensors and a boundary condition. Exploiting a connection with the theory of matrix algebras, we derive two structural properties shared by all hMPS, namely: a) there exist local operators which annihilate all hMPS of a given bond dimension; and b) there exist local operators which, when applied over any hMPS of a given bond dimension, decouple (cut) the particles where they act from the spin chain while at the same time join (glue) the two loose ends back again into a hMPS. Armed with these tools, we show how to systematically derive `bond dimension witnesses, or 2-local operators whose expectation value allows us to lower bound the bond dimension of the underlying hMPS. We extend some of these results to the ansatz of Projected Entangled Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS to: a) derive some theoretical limitations on the use of hMPS and hPEPS for ground state energy computations; b) show how to decrease the complexity and boost the speed of convergence of the semidefinite programming hierarchies described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of finite-dimensional quantum correlations.
In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.
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