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arXiv:1302.3980 · Physical Review B

Generalized quantum microcanonical ensemble from random matrix product states

Silvano Garnerone, Thiago Rodrigues de Oliveira

Published Jun 25, 2013• 13 citations
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Abstract

We propose a tensor network algorithm for the efficient sampling of quantum pure states belonging to a generalized microcanonical ensemble. The algorithm consists in an adaptation of the power method to a recently introduced ensemble of random matrix product states. The microcanonical ensemble that we consider is characterized by the fact that the participating energy eigenstates are not required to have identical statistical weight. To test the method we apply it to the Heisenberg model with an external magnetic field, and we find that the magnetization curves, due to the microcanonical constraint, are qualitatively different from those obtained in the canonical ensemble. Possible future applications include the study of isolated quantum systems evolving after a quantum quench.

DOI
10.1103/physrevb.87.214426

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We propose a tensor network algorithm for the efficient sampling of quantum pure states belonging to a generalized microcanonical ensemble. The algorithm consists in an adaptation of the power method to a recently introduced ensemble of random matrix product states. The microcanonical ensemble that we consider is characterized by the fact that the participating energy eigenstates are not required to have identical statistical weight. To test the method we apply it to the Heisenberg model with an external magnetic field, and we find that the magnetization curves, due to the microcanonical constraint, are qualitatively different from those obtained in the canonical ensemble. Possible future applications include the study of isolated quantum systems evolving after a quantum quench.
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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 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.
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.
Strong entanglement limits the reach of tensor-network descriptions of quantum many-body systems. We study Gaussian-augmented matrix product states (GAMPS), formed by applying a free-fermionic (Gaussian) unitary to a matrix product state. To explore the capabilities of this family, we consider random GAMPS}, a minimally structured ensemble obtained by independently sampling random matrix product states and Gaussian unitaries. Exploiting the algebraic structure of Gaussian unitaries, we develop a replica tensor-network method that computes averages over the random GAMPS ensemble for hundreds of qubits. Random GAMPS exhibit nearly maximal volume-law entanglement already at modest bond dimension, while other quantum resource quantifiers rapidly approach Haar-random state values as the bond dimension increases. The entanglement scaling suggests an exponential reduction in the bond dimension needed to approach the volume-law entropy compared with unaugmented matrix product states. We find a state two-design behavior of random GAMPS at fixed accuracy with a bond dimension independent of system size. Our results reveal how Gaussian augmentation expands the expressive power of tensor networks.
We propose a variational quantum algorithm for estimating microcanonical expectation values in models obeying the eigenstate thermalization hypothesis. Using a relaxed criterion for convergence of the variational optimization loop, the algorithm generates weakly entangled superpositions of eigenstates at a given target energy density. An ensemble of these variational states is then used to estimate microcanonical averages of local operators, with an error whose dominant contribution decreases initially as a power law in the size of the ensemble and is ultimately limited by a small bias. We apply the algorithm to the one-dimensional mixed-field Ising model, where it converges for ansatz circuits of depth roughly linear in system size. The most accurate thermal estimates are produced for intermediate energy densities. In our error analysis, we find connections with recent works investigating the underpinnings of the eigenstate thermalization hypothesis. In particular, the failure of energy-basis matrix elements of local operators to behave as independent random variables is a potential source of error that the algorithm can overcome by averaging over an ensemble of variational states.