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arXiv:1003.5253 · Physical Review A

Statistical properties of random matrix product states

Silvano Garnerone, Thiago Rodrigues de Oliveira, Stephan Haas, Paolo Zanardi A

Published Nov 15, 2010• 43 citations
arXivHTMLPDF

Abstract

We study the set of random matrix product states (RMPS) introduced by Garnerone, de Oliveira, and Zanardi [S. Garnerone, T. R. de Oliveira, and P. Zanardi, Phys. Rev. A 81, 032336 (2010)] as a tool to explore foundational aspects of quantum statistical mechanics. In the present work, we provide an accurate numerical and analytical investigation of the properties of RMPS. We calculate the average state of the ensemble in the nonhomogeneous case, and numerically check the validity of this result. We also suggest using RMPS as a tool to approximate properties of general quantum random states. The numerical simulations presented here support the accuracy and efficiency of this approximation. These results suggest that any generalized canonical state can be approximated with high probability by the reduced density matrix of a RMPS, if the average matrix product states coincide with the associated microcanonical ensemble.

DOI
10.1103/physreva.82.052312

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We study the set of random matrix product states (RMPS) introduced by Garnerone, de Oliveira, and Zanardi [S. Garnerone, T. R. de Oliveira, and P. Zanardi, Phys. Rev. A 81, 032336 (2010)] as a tool to explore foundational aspects of quantum statistical mechanics. In the present work, we provide an accurate numerical and analytical investigation of the properties of RMPS. We calculate the average state of the ensemble in the nonhomogeneous case, and numerically check the validity of this result. We also suggest using RMPS as a tool to approximate properties of general quantum random states. The numerical simulations presented here support the accuracy and efficiency of this approximation. These results suggest that any generalized canonical state can be approximated with high probability by the reduced density matrix of a RMPS, if the average matrix product states coincide with the associated microcanonical ensemble.
Recent results suggest that the use of ensembles in statistical mechanics may not be necessary for isolated systems, since typically the states of the Hilbert space would have properties similar to those of the ensemble. Nevertheless, it is often argued that most of the states of the Hilbert space are nonphysical and not good descriptions of realistic systems. Therefore, to better understand the actual power of typicality it is important to ask if it is also a property of a set of physically relevant states. Here we address this issue, studying if and how typicality emerges in the set of matrix product states. We show analytically that typicality occurs for the expectation value of subsystems' observables when the rank of the matrix product state scales polynomially with the size of the system with a power greater than 2. We illustrate this result numerically and present some indications that typicality may appear already for a linear scaling of the rank of the matrix product state.
Method of random phase product state (RPPS) is proposed to calculate canonical ensemble average of quantum systems described with matrix product states and also with tensor network states in general. The RPPS method is an extension of the method of random phase state for full Hilbert space representation. The validity of the method is confirmed by comparing the average energy of N-site antiferromagnetic spin-1/2 Heisenberg chain model with open boundary conditions with the result of direct method (for up to N=14) and minimally entangled typical thermal state (METTS) method (for N=100). Numerical advantages of the RPPS such as parallelization, combined calculation of thermal averages at different temperatures, parameters for controlling error are discussed. View point of self-averaging for the super-convergence of random state method is emphasized in addition to that of typicality.
Magic, or nonstabilizerness, characterizes how far away a state is from the stabilizer states, making it an important resource in quantum computing, under the formalism of the Gotteman-Knill theorem. In this paper, we study the magic of the one-dimensional (1D) random matrix product states (RMPSs) using the L1{L}_{1}L1​-norm measure. We first relate the L1{L}_{1}L1​ norm to the L4{L}_{4}L4​ norm. We then employ a unitary four-design to map the L4{L}_{4}L4​ norm to a 24-component statistical physics model. By evaluating partition functions of the model, we obtain a lower bound on the expectation values of the L1{L}_{1}L1​ norm. This bound grows exponentially with respect to the qudit number nnn, indicating that the 1D1\mathrm{D}1D RMPS is highly magical. Our numerical results confirm that the magic grows exponentially in the qubit case.
We extend the formalism of pure state thermodynamics to matrix product states. In pure state thermodynamics finite-temperature properties of quantum systems are derived without the need of statistical mechanic ensembles, but instead using typical properties of random pure states. We show that this formalism can be useful from a computational point of view when combined with tensor network algorithms. In particular, a recently introduced Monte Carlo algorithm is considered which samples matrix product states at random for the estimation of finite-temperature observables. Here we analytically justify the effectiveness of this algorithm, and we prove that sampling one single state is in principle sufficient to obtain a very good estimation of finite-temperature expectation values. These results provide a substantial computational improvement with respect to similar algorithms for one-dimensional quantum systems based on uniformly distributed pure states. The analytical calculations are numerically supported, simulating finite-temperature interacting spin systems of size up to 100 qubits.
We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.
The study of generic properties of quantum states has led to an abundance of insightful results.A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by matrix product states.In this work, we introduce a picture of generic states within the trivial phase of matter with respect to their nonequilibrium and entropic properties.We do so by rigorously exploring nontranslation-invariant matrix product states drawn from a local independent and identically distributed Haar measure.We arrive at these results by exploiting techniques for computing moments of random unitary matrices and by exploiting a mapping to partition functions of classical statistical models, a method that has lead to valuable insights on local random quantum circuits.Specifically, we prove that such disordered random matrix product states equilibrate exponentially well with overwhelming probability under the time evolution of Hamiltonians featuring a nondegenerate spectrum.Moreover, we prove two results about the entanglement Rényi entropy: the entropy with respect to sufficiently disconnected subsystems is generically extensive in the system size, and for small connected systems, the entropy is almost maximal for sufficiently large bond dimensions.