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arXiv:1504.07701 · Physical Review B

Compression of correlation matrices and an efficient method for forming matrix product states of fermionic Gaussian states

Matthew Fishman, Steven R. White

Published Aug 21, 2015• 78 citations
arXivHTMLPDF

Abstract

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).

DOI
10.1103/physrevb.92.075132

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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).
Tensor network quantum states are powerful tools for strongly correlated systems, tailored to capture local correlations such as in ground states with entanglement area laws. When applying tensor network states to interacting fermionic systems, a proper choice of the basis or orbitals can reduce the bond dimension of tensors and provide physically relevant orbitals. We introduce such a change of basis with unitary gates obtained from compressing fermionic Gaussian states into quantum circuits corresponding to various tensor networks. These circuits can reduce the ground-state entanglement entropy and improve the performance of algorithms such as the density matrix renormalization group. We study the Anderson impurity model with one and two impurities to show the potential of the method for improving computational efficiency and interpreting impurity physics. Furthermore, fermionic Gaussian circuits can also suppress entanglement during the time evolution out of low-energy state. Last, we consider Gaussian multiscale entanglement renormalization ansatz (GMERA) circuits which compress fermionic Gaussian states hierarchically. The emergent coarse-grained physical models from these GMERA circuits are studied in terms of their entanglement properties and suitability for performing time evolution.
The truncation or compression of the spectrum of Schmidt values is inherent to the matrix product state (MPS) approximation of one-dimensional quantum ground states. We provide a renormalization group picture by interpreting this compression as an application of Wilson's numerical renormalization group along the imaginary time direction appearing in the path integral representation of the state. The location of the physical index is considered as an impurity in the transfer matrix and static MPS correlation functions are reinterpreted as dynamical impurity correlations. Coarse-graining the transfer matrix is performed using a hybrid variational ansatz based on matrix product operators, combining ideas of MPS and the multiscale entanglement renormalization ansatz. Through numerical comparison with conventional MPS algorithms, we explicitly verify the impurity interpretation of MPS compression, as put forward by V. Zauner et al. [New J. Phys. 17, 053002 (2015)] for the transverse-field Ising model. Additionally, we motivate the conceptual usefulness of endowing MPS with an internal layered structure by studying restricted variational subspaces to describe elementary excitations on top of the ground state, which serves to elucidate a transparent renormalization group structure ingrained in MPS descriptions of ground states.
Gaussian fermionic matrix product states (GfMPS) form a class of ansatz quantum states for 1d systems of noninteracting fermions. We show, for a simple critical model of free hopping fermions, that (i) any GfMPS approximation to its ground state must have bond dimension scaling superpolynomially with the system size, whereas (ii) there exists a non-Gaussian fermionic MPS approximation to this state with polynomial bond dimension. This proves that, in general, imposing Gaussianity at the level of the tensor network may significantly alter its capability to efficiently approximate critical Gaussian states. We also provide numerical evidence that the required bond dimension is subexponential, and thus can still be simulated with moderate resources.
The fertile new field of quantum information theory is inspiring new ways to study correlated quantum systems by providing fresh insights into the structure of their Hilbert spaces. One of the latest developments in this direction was the extension of the ubiquitous matrix-product-state constructions, epitomized by the density-matrix renormalization-group algorithm, to continuous space-time; so as to be able to describe low-dimensional field theories within a variational approach. Following the earlier success achieved for bosonic theories, we present the first implementation of a continuous matrix product state (cMPS) for spinfull non-relativistic fermions in 1D. We propose a construction of variational matrices with an efficient parametrization that respects the translational symmetry of the problem (without being overly constraining) and readily meets the regularity conditions that arise from removing the ultraviolet divergences in the kinetic energy. We tested the validity of our approach on an interacting spin-1/2 system with spin imbalance. We observe that the ansatz correctly predicts the ground-state magnetic properties for the attractive spin-1/2 Fermi gas, including a phase-oscillating pair correlation function in the partially polarized regime (the 1D correlate of the FFLO state). We shall also discuss how to generalize the cMPS ansatz to other situations.
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.
The most efficient known quantum circuits for preparing unitary coupled cluster states and applying Trotter steps of the arbitrary basis electronic structure Hamiltonian involve interleaved sequences of Fermionic Gaussian circuits and Ising interaction-type circuits. These circuits arise from factorizing the two-body operators generating those unitaries as a sum of squared one-body operators that are simulated using product formulas. We introduce a numerical algorithm for performing this factorization that has an iteration complexity no worse than single particle basis transformations of the two-body operators and often results in many times fewer squared one-body operators in the sum of squares, compared to the analytical decompositions. As an application of this numerical procedure, we demonstrate that our protocol can be used to approximate generic unitary coupled cluster operators and prepare the necessary high-quality initial states for techniques (like ADAPT-VQE) that iteratively construct approximations to the ground state.