Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBond dimension witnesses and the structure of homogeneous matrix product states
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
Suppose we would like to approximate all local properties of a quantum many-body state to accuracy $delta$. In one dimension, we prove that an area law for the Renyi entanglement entropy $R_alpha$ with index $alpha<1$ implies a matrix product state representation with bond dimension $mathrm{poly}(1/delta)$. For (at most constant-fold degenerate) ground states of one-dimensional gapped Hamiltonians, it suffices that the bond dimension is almost linear in $1/delta$. In two dimensions, an area law for $R_alpha(alpha<1)$ implies a projected entangled pair state representation with bond dimension $e^{O(1/delta)}$. In the presence of logarithmic corrections to the area law, similar results are obtained in both one and two dimensions.
We numerically study the single-flavor Schwinger model with a topological $theta$-term, which is practically inaccessible by standard lattice Monte Carlo simulations due to the sign problem. By using numerical methods based on tensor networks, especially the one-dimensional matrix product states, we explore the non-trivial $theta$-dependence of several lattice and continuum quantities in the Hamiltonian formulation. In particular, we compute the ground-state energy, the electric field, the chiral fermion condensate, and the topological vacuum susceptibility for positive, zero, and even negative fermion mass. In the chiral limit, we demonstrate that the continuum model becomes independent of the vacuum angle $theta$, thus respecting CP invariance, while lattice artifacts still depend on $theta$. We also confirm that negative masses can be mapped to positive masses by shifting $thetarightarrow theta +pi$ due to the axial anomaly in the continuum, while lattice artifacts non-trivially distort this mapping. This mass regime is particularly interesting for the (3+1)-dimensional QCD analog of the Schwinger model, the sign problem of which requires the development and testing of new numerical techniques beyond the conventional Monte Carlo approach.
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].
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
