We report a way of wave function estimation for the density matrix renormalization group (DMRG) method applied to quantum systems, which has 2-site modulation, when the system size extension is necessary in both the finite and the infinite algorithms. The estimation is performed by renormalization group (RG) transformation applied to the ground state wave function, which is represented as the matrix product. This RG scheme is known as the product wave function renormalization group (PWFRG) method. In order to treat the 2-site modulation, the operation of the RG transformation is shifted by amount of 2 lattice sites. It turns out that this 2-site shift algorithm provides better wave function estimation in the thermodynamic limit, compared with the previously known PWFRG algorithm.
We show how to extract the scaling behavior of quantum walks using the renormalization group (RG). We introduce the method by efficiently reproducing well-known results on the one-dimensional lattice. As a nontrivial model, we apply this method to the dual Sierpinski gasket and obtain its exact, closed system of RG-recursions. Numerical iteration suggests that under rescaling the system length, $L^{\prime}=2L$, characteristic times rescale as $t^{\prime}=2^{d_{w}}t$ with the exact walk exponent $d_{w}=\log_{2}\sqrt{5}=1.1609\ldots$. Despite the lack of translational invariance, this is very cl…
The density matrix renormalization group (DMRG) of White 1992 remains to this day an integral component of many state-of-the-art methods for efficiently simulating strongly correlated quantum systems. In quantum chemistry, QC-DMRG became a powerful tool for ab initio calculations with the non-relativistic Schrödinger equation. An important issue in QC-DMRG is the so-called ordering problem -- the optimal ordering of DMRG sites corresponding to electronic orbitals that produces the most accurate results. To this end, a commonly used heuristic is the grouping of strongly correlated orbitals as m…
This work is concerned with phrasing the concepts of fault-tolerant quantum computation within the framework of disordered systems, Bernoulli site percolation in particular. We show how the so-called "threshold theorems" on the possibility of fault-tolerant quantum computation with constant error rate can be cast as a renormalization (coarse-graining) of the site percolation process describing the occurrence of errors during computation. We also use percolation techniques to derive a trade-off between the complexity overhead of the fault-tolerant circuit and the threshold error rate.
Recent experiments with cold atoms on the impurity motion in one-dimensional liquids of interacting bosons have revealed an interesting interplay between the polaronic impurity mass shift and the renormalization of the optical potential. We show that the influence of the external trap on the Bose gas leads to a steeper effective potential for the impurity. We propose a framework in which this potential renormalization and the mass shift can be quantitatively understood by combining a semi-classical theory of density wave excitations in the Luttinger liquid with the non equilibrium formalism of…
It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of …
In the context of real-space renormalization group methods, we propose a novel scheme for quantum systems defined on a D-dimensional lattice. It is based on a coarse-graining transformation that attempts to reduce the amount of entanglement of a block of lattice sites before truncating its Hilbert space. Numerical simulations involving the ground state of a 1D system at criticality show that the resulting coarse-grained site requires a Hilbert space dimension that does not grow with successive rescaling transformations. As a result we can address, in a quasi-exact way, tens of thousands of qua…
Two-dimensional electron gas coupled to adjacent impurity sites in high-frequency out-of-plane ac control electric field is investigated. Modification of tunneling rates as a function of the field amplitude is calculated. Nonlinear dependence on the ac field strength is reported for the conductivity of two-dimensional electron gas. It develops a periodic peak structure.
We investigate the topic of renormalization in the theory of weakly interacting open quantum systems. Our starting point is an open quantum system interacting with a single heat bath. For a given setup, we discuss that the stationary state of the Davies-GKSL equation is thermodynamically inconsistent with the presence of the Lamb-Stark shift term. For this reason, we postulate the self-consistency condition for the dynamical equations. The condition fixes the renormalization procedure and recovers the thermodynamical consistency. In this way, we rederive the cumulant equation to illustrate how…
Entanglement renormalization group flow of the Haah cubic code produces another fracton model with 4 qubits per lattice site, dubbed as the Haah B-code. We provide a schema that generalizes both models to stabilizer codes on any finite group with 2q qubits per site and labeled by multi-subsets of the finite group and a symmetric binary matrix.
As quantum technologies develop, we acquire control of an ever-growing number of quantum systems. Unfortunately, current tools to detect relevant quantum properties of quantum states, such as entanglement and Bell nonlocality, suffer from severe scalability issues and can only be computed for systems of a very modest size, of around $6$ sites. In order to address large many-body systems, we propose a renormalisation-type approach based on a class of local linear transformations, called connectors, which can be used to coarse-grain the system in a way that preserves the property under investiga…
We demonstrate the coherent transport of two-dimensional (2D) arrays of small ensembles of neutral atoms in a shift register architecture based on 2D arrays of microlenses. We show the scalability of the transport process by presenting the repeated hand-over of atoms from site to site. We prove the conservation of coherence during transport, reloading, and a full shift register cycle. This shows that the fundamental shift sequence can be cascaded and thus scaled to complex and versatile 2D architectures for atom-based quantum information processing, quantum simulation, and the investigation of…
We study the time development of correlation functions at both zero and finite temperature with Shibata-Hashitsume's projection operator method and carry out the renormalization of ultraviolet divergence that appears in a time-dependent frequency shift, using a mass counter term. A harmless divergence of log t-type remains at an initial time t=0 after the lowest order renormalization.
The Aharonov-Bohm elastic scattering with incident particles described by plane waves is revisited by using the phase-shifts method. The formal equivalence between the cylindrical Schrödinger equation and the one-dimensional Calogero problem allows us to show that up to two scattering phase-shifts modes in the cylindrical waves expansion must be renormalized. The renormalization procedure introduces new length scales giving rise to spontaneous breaking of the conformal symmetry. The new renormalized cross-section has an amazing property of being non-vanishing even for a quantized magnetic flux…
The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of one-dimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the su…
