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Register a new userMatrix product state algorithms for Gaussian fermionic states
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While general quantum many-body systems require exponential resources to be simulated on a classical computer, systems of non-interacting fermions can be simulated exactly using polynomially scaling resources. Such systems may be of interest in their own right, but also occur as effective models in numerical methods for interacting systems, such as Hartree-Fock, density functional theory, and many others. Often it is desirable to solve systems of many thousand constituent particles, rendering these simulations computationally costly despite their polynomial scaling. We demonstrate how this scaling can be improved by adapting methods based on matrix product states, which have been enormously successful for low-dimensional interacting quantum systems, to the case of free fermions. Compared to the case of interacting systems, our methods achieve an exponential speedup in the entanglement entropy of the state. We demonstrate their use to solve systems of up to one million sites with an effective MPS bond dimension of 10^15.
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We prove that ground states of gapped local Hamiltonians on an infinite spin chain can be efficiently approximated by matrix product states with a bond dimension which scales as D~(L-1)/epsilon, where any local quantity on L consecutive spins is approximated to accuracy epsilon.
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 isolates 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, though the procedure is easily extended to fermions with spin and more general BCS states (utilizing the formalism of Majorana modes). We also present a DMRG-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 (MERA) 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 1D lattice with staggered hopping amplitudes).
We quantify how well matrix product states approximate exact ground states of 1-D quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms, and justifies their use even in the case of critical systems.
A generic method to investigate many-body continuous-variable systems is pedagogically presented. It is based on the notion of matrix product states (so-called MPS) and the algorithms thereof. The method is quite versatile and can be applied to a wide variety of situations. As a first test, we show how it provides reliable results in the computation of fundamental properties of a chain of quantum harmonic oscillators achieving off-critical and critical relative errors of the order of 10^(-8) and 10^(-4) respectively. Next, we use it to study the ground state properties of the quantum rotor model in one spatial dimension, a model that can be mapped to the Mott insulator limit of the 1-dimensional Bose-Hubbard model. At the quantum critical point, the central charge associated to the underlying conformal field theory can be computed with good accuracy by measuring the finite-size corrections of the ground state energy. Examples of MPS-computations both in the finite-size regime and in the thermodynamic limit are given. The precision of our results are found to be comparable to those previously encountered in the MPS studies of, for instance, quantum spin chains. Finally, we present a spin-off application: an iterative technique to efficiently get numerical solutions of partial differential equations of many variables. We illustrate this technique by solving Poisson-like equations with precisions of the order of 10^(-7).
We characterize the conditions under which a translationally invariant matrix product state (MPS) is invariant under local transformations. This allows us to relate the symmetry group of a given state to the symmetry group of a simple tensor. We exploit this result in order to prove and extend a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the context of MPS. We illustrate the results with an exhaustive search of SU(2)--invariant two-body Hamiltonians which have such MPS as exact ground states or excitations.
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