We introduce Gaussian Matrix Product States (GMPS), a generalization of Matrix Product States (MPS) to lattices of harmonic oscillators. Our definition resembles the interpretation of MPS in terms of projected maximally entangled pairs, starting from which we derive several properties of GMPS, often in close analogy to the finite dimensional case: We show how to approximate arbitrary Gaussian states by MPS, we discuss how the entanglement in the bonds can be bounded, we demonstrate how the correlation functions can be computed from the GMPS representation, and that they decay exponentially in one dimension, and finally relate GMPS and ground states of local Hamiltonians.
Continuous Matrix Product States (cMPS) are powerful variational ansatz states for ground states of continuous quantum field theories in (1+1) dimension. In this paper we introduce a novel parametrization of the cMPS wave function based on basis-spline functions, which we coin spline-based MPS (spMPS), and develop novel regauging techniques for inhomogeneous cMPS. We extend a recently developed ground-state optimization algorithm for translational invariant cMPS [M. Ganahl, J. Rincón, G.Vidal. Phys.Rev.Lett. 118,220402 (2017)] to the case of inhomogeneous cMPS and, as proof-of-principle, use it to obtain the ground-state of a gas of Lieb-Liniger bosons in a periodic potential. The proposed method provides a first working implementation of a cMPS optimization for non-translational invariant continuous Hamiltonians.
Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors
{Xj[k]}j=1D which are homogeneously distributed at every site
k in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.
Abstract. In this work, we investigate the universal representational capacity of matrix product states (MPSs) in the context of Boolean and continuous functions. We show that MPSs can precisely realize arbitrary Boolean functions by presenting a constructive method for designing MPS representations of Boolean gates. Furthermore, we prove that the function space of MPSs, equipped with scale-invariant sigmoidal activation functions, is dense in the space of continuous functions on compact subsets of [Formula: see text]. By examining the connection between MPSs and neural networks (NNs), we establish that an MPS with scale-invariant sigmoidal activation is equivalent to a one-hidden-layer NN equipped with kernel functions. For specific MPS models, we construct equivalent NNs and show that nonlinear kernels, such as polynomial kernels coupling input components, arise naturally in these representations. Finally, we analyze the realization of Gaussian processes via infinitely wide MPSs, leveraging their equivalence to NNs. Our analysis of MPSs highlights the critical role of the kernel function in MPSs, as it encodes the correlations between different components of the input data. Its careful design directly influences the model’s representational capacity.