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 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.
In this work, we develop a stochastic matrix product state (stoMPS) approach that combines the MPS technique and Monte Carlo samplings and can be applied to simulate quantum lattice models down to low temperature. In particular, we exploit a procedure to unbiasedly sample the local tensors in the matrix product states, which has one physical index of dimension d and two geometric indices of dimension D, and find the results can be continuously improved by enlarging D. We benchmark the methods on small system sizes and then compare the results to those obtained with minimally entangled typical thermal states, finding that stoMPS has overall better performance with finite D. We further exploit the MPS sampling to simulate long spin chains, as well as the triangular and square lattices with cylinder circumference W up to 4. Our results showcase the accuracy and effectiveness of stochastic tensor networks in finite-temperature simulations.
Abstract We present several features of a thermal pure quantum matrix product state (TPQ-MPS). TPQ-MPS can simulate finite temperature quantum many-body systems with a numerical cost comparable to the matrix product algorithm for the ground state in one-dimensional systems. Starting from a random matrix product state with auxiliary sites attached to the edges of the system, we successively operate the Hamiltonian with a shift-constant and obtain a series of TPQ-MPS that represents the microcanonical pure state at the highest temperature to the one at the lowest available temperature. How the parameters of the TPQ-MPS protocol influence the effective bond dimension χ of the matrix and the temperature distribution of these microcanonical states is examined.
Despite early developments on the foundations of quantum mechanics concern the wave function, quantum statistics has been developed with the density matrix formalism, leading to very important results in explaining molecular observations. Only recently, several authors argued a new interpretation by focusing on the wave function representing the quantum state of an isolated system, showing how a single wave function can exhibit statistical properties and generate the same results expected in the standard quantum statistical framework. Starting from these results, investigation on the foundations of quantum statistical mechanics has gained recently a renewed interest. As a matter of fact, the possibility of studying single molecule properties as well as the need of a better understanding of the effect of quantum dynamics, in order to develop new nanoscaled materials suitable to quantum computing tasks, have opened new intriguing questions leading to quantum statistical approaches far from being well understood and accepted. In this framework the behaviour of a single realization of quantum systems has gained a central role in the description of molecular systems. Furthermore, in recent years, an increasing number of studies has been presented on quantum dynamics through the numerical solution of the Schrödinger equation for systems of interacting components. These studies demonstrates that Quantum Dynamics Simulations could be a practicable route. In order to study phenomena such as dissipation, relaxation and thermalization, the focus has to be moved from isolated molecules to modular systems made of mutual interacting components, with model Hamiltonians possessing a sufficiently low dimensional representation. An important issue concerns the rules to be employed for the choice of the initial quantum state for the simulation of isolated systems. As long as one considers molecular degrees of freedom interacting with a (model) environment, there are no reasons to select a particular quantum state for the overall system and, therefore, a random choice has to be performed amongst a well defined statistical ensemble of pure states. Furthermore, one would like to operate a choice assuring that the simulation of the system is in a well defined thermal state with given temperature. This necessarily calls for a statistical description like for classical systems. A useful parametrization of the wave function will be presented in order to highlight the most appropriate variable for the statistical analysis. In particular some of them, called phases, retain all the dynamical information whereas the others, called populations, are the constants of motion. The latter, in particular are very important because they describe the equilibrium properties strictly related to the thermodynamic description. Since the dynamics of wave function does not supply any information about populations, the definition of a probability distribution on these variables is required. Different probability distributions on populations have been proposed, only on the bases of reasonable assumptions and they validation has been performed only with a posteriori considerations. In particular Fresch and Moro have demonstrated how the agreement with thermodynamics can be employed to discriminate different probability distributions on pure states. This had led the uniform ensembles to be, up to now, the most self-consistent models for quantum pure states. However, in this thesis I will highlight a drawback of the uniform distribution ensemble that can be described as follow: if we bring into contact two systems, even through a perturbative interaction, we are not able to describe the equilibrium properties after the interaction within the uniform distribution statistics, since the uniform character is lost. It represents a severe shortcoming of the statistical ensemble from a methodological point of view, since closed systems can be always considered as the result of interaction among previously isolated systems. On the other hand this drawback introduces a further requirement of a different nature that can be used for the definition of a new statistical ensemble. In this work I intend to find and characterize a statistical ensemble for populations that overcomes the drawbacks of the uniform distribution of pure states. The invariance of the thermal state in the coupling of identical systems will be used as a guideline in the definition af a new probability distribution on populations. Such an ensemble for pure states, called Thermalization Resilient Ensemble, provides a convenient framework for treating the interactions between quantum systems, as long as the structure of the statistical distribution is preserved and the identification of thermodynamic properties is assured. In perspective it should be the privileged statistical ensemble to implement Quantum Dynamics Simulations. Once the average properties of the Thermalization Resilient Ensemble have been introduced, I will obtain a probability distribution on pure states with the use of a geometrical analysis on the Hilbert space. The surface elements of an ellipsoidal manifold will be related to the probability density on populations. As a matter of fact the explicit form of the probability distribution is a prerequisite in order to perform Quantum Dynamical Simulations. However the results obtained through the geometrical analysis cannot be easily extended to systems with unbounded energy spectrum and an alternative strategy has been developed. A scaling algorithm on the basis of the uniform statistical ensemble will be described and this allows a well defined sampling of a probability distribution with desired averages. In this framework I demonstrate the emergence of thermodynamic behavior in the limit of macroscopic systems. In the last part of the thesis I consider the dynamical features of the thermalization experiment. Two identical systems, initially at different temperature, will be brought in interaction and the analysis of the final equilibrium state will be performed for two different generic forms of the interaction Hamiltonian, highlighting how the statistical approach can be very useful in the definition of the equilibrium in complex quantum systems.
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
For quantum many-body systems in one dimension, computational complexity theory reveals that the evaluation of ground-state energy remains elusive on quantum computers, contrasting the existence of a classical algorithm for temperatures higher than the inverse logarithm of the system size. This highlights a qualitative difference between low- and high-temperature states in terms of computational complexity. Here, we describe finite-temperature states using the matrix product state formalism. Within the framework of random samplings, we derive an analytical formula for the required number of samples which provides both quantitative and qualitative measures of computational complexity. At high and low temperatures, its scaling behavior with system size is linear and quadratic, respectively, demonstrating a distinct crossover between these numerically difficult regimes of quantitative difference.
Thermal pure state algorithms, which employ pure quantum states representing thermal equilibrium states instead of statistical ensembles, are useful both for numerical simulations and for theoretical analysis of thermal states. However, their inherently large entanglement makes it difficult to represent efficiently and limits their use in analyzing large systems. Here we propose a tensor network algorithm for constructing thermal pure states for systems with certain antiunitary symmetries, such as time-reversal or complex-conjugate symmetry. Our method utilizes thermal pure states that, while exhibiting volume-law entanglement, can be mapped to tensor network states through simple transformations. Furthermore, our approach does not rely on random sampling and thus avoids statistical uncertainty. Moreover, we can compute not only thermal expectation values of local observables but also thermodynamic quantities. We demonstrate the validity and utility of our method by applying it to the one-dimensional X Y model and the two-dimensional Ising model on a triangular lattice. Our results suggest a class of variational wave functions for volume-law states that are not limited to thermal equilibrium states. Published by the American Physical Society 2024