Abstract Fermionic Gaussian states (FGSs) are eigenstates of quadratic Hamiltonians and widely used in quantum many-body problems. We propose a highly efficient algorithm that converts FGSs to matrix product states (MPSs). It can be formulated for finite-size systems without translation invariance, but becomes particularly appealing when applied to infinite systems with translation invariance. If the ground states of a topologically ordered system on infinite cylinders are expressed as MPSs, then the fixed points of the transfer matrix can be harnessed to filter out the anyon eigenbasis, also known as minimally entangled states. This allows for efficient computation of universal properties such as entanglement spectrum and modular matrices. The potential of our method is demonstrated by numerical calculations in two chiral spin liquids that have the same topological orders as the bosonic Laughlin and Moore–Read states, respectively. The anyon eigenbasis for the first one has been worked out before and serves as a useful benchmark. The anyon eigenbasis of the second one is, however, not transparent and its successful construction provides a nontrivial corroboration of our method.
Abstract Fermionic Gaussian states (FGSs) are eigenstates of quadratic Hamiltonians and widely used in quantum many-body problems. We propose a highly efficient algorithm that converts FGSs to matrix product states (MPSs). It can be formulated for finite-size systems without translation invariance, but becomes particularly appealing when applied to infinite systems with translation invariance. If the ground states of a topologically ordered system on infinite cylinders are expressed as MPSs, then the fixed points of the transfer matrix can be harnessed to filter out the anyon eigenbasis, also known as minimally entangled states. This allows for efficient computation of universal properties such as entanglement spectrum and modular matrices. The potential of our method is demonstrated by numerical calculations in two chiral spin liquids that have the same topological orders as the bosonic Laughlin and Moore–Read states, respectively. The anyon eigenbasis for the first one has been worked out before and serves as a useful benchmark. The anyon eigenbasis of the second one is, however, not transparent and its successful construction provides a nontrivial corroboration of our method.
Bosonic Gaussian states are ubiquitous in quantum optics and condensed matter physics. While they are efficiently handled within the Gaussian formalism, sampling requires calculating amplitudes in the boson occupation basis. This step, however, is hindered by a significant bottleneck due to the hafnian. We present an efficient algorithm that converts pure bosonic Gaussian states into matrix product states (MPSs), thereby establishing a versatile tool for probing bosonic Gaussian systems in settings where direct Gaussian-formalism-based calculations become inefficient. Our method combines a Gaussian singular value decomposition with a projected-creation-operator mapping that constructs local MPS tensors without computing hafnians. Benchmarking on covariance matrices from the Jiuzhang 2.0 and Jiuzhang 4.0 Gaussian boson sampling experiments demonstrates substantial speedups over previous tensor-network approaches in the low-entanglement regime relevant to lossy devices. The method provides a scalable classical simulation framework for bosonic Gaussian states with limited entanglement. In this regime, a target accuracy can be achieved with a bond dimension that remains computationally tractable, thereby extending the applicability of MPS-based methods to a broad range of bosonic systems.
Here we present an efficient and numerically stable procedure for compressing a correlation matrix into a set of local unitary single-particle gates, which leads to a very efficient way of forming the matrix product state (MPS) approximation of a pure fermionic Gaussian state, such as the ground state of a quadratic Hamiltonian. The procedure involves successively diagonalizing subblocks of the correlation matrix to isolate local states which are purely occupied or unoccupied. A small number of nearest-neighbor unitary gates isolate each local state. The MPS of this state is formed by applying the many-body version of these gates to a product state. We treat the simple case of compressing the correlation matrix of spinless free fermions with definite particle number in detail, although the procedure is easily extended to fermions with spin and more general BCS states (utilizing the formalism of Majorana modes). We also present a density matrix renormalization group--like algorithm to obtain the compressed correlation matrix directly from a hopping Hamiltonian. In addition, we discuss a slight variation of the procedure which leads to a simple construction of the multiscale entanglement renormalization ansatz of a fermionic Gaussian state, and present a simple picture of orthogonal wavelet transforms in terms of the gate structure we present in this paper. As a simple demonstration, we analyze the Su-Schrieffer-Heeger model (free fermions on a one-dimensional lattice with staggered hopping amplitudes).
Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry nonlocal degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of the anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite one dimensional critical system of (a) Ising anyons and (b) Fibonacci anyons, both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.
Fermionic non-Gaussianity is a key resource for universal quantum computation, and its behavior in quantum many-body systems has attracted growing interest. Recently, a convolution-based measure, which we call the magic Rényi entropy (MRE), has been proposed as a resource measure of non-Gaussianity, but its evaluation is limited to small systems due to the computational cost growing exponentially with system size. In this work, we develop a perfect-sampling method to calculate the second-order MRE from matrix product states (MPSs). Our method uses a bounded estimator to control sampling fluctuations and a recursive sweep to draw the samples directly from the input MPS, which allows us to generate each sample with O(nD3) time, where n and D denote the number of modes and MPS bond dimension, respectively. We benchmark our method on the XXZ chain with up to 128 sites, demonstrating that the MRE captures intricate higher-order correlations at large scales, which are inaccessible to covariance-based measures. These results provide a computational tool that quantitatively evaluates many-body fermionic non-Gaussianity by accounting for higher-order correlations.
We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga, Phys. Rev. B 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological F symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the F symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of Z3, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S3 and Rep(S3) string-net models.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.