The complexity of quantum many-body systems is manifested in the vast diversity of their correlations, making it challenging to distinguish the generic from the atypical features.This can be addressed by analyzing correlations through ensembles of random states, chosen to faithfully embody the relevant physical properties.Here, we focus on spins with local interactions, whose correlations are extremely well captured by tensor network states.Adopting an operational perspective, we define ensembles of random tensor network states in one and two spatial dimensions that admit a sequential generation.As such, they directly correspond to outputs of quantum circuits with a sequential architecture and random gates.In one spatial dimension, the ensemble explores the entire family of matrix product states, while in two spatial dimensions, it corresponds to random isometric tensor network states.We extract the scaling behavior of the average correlations between two subsystems as a function of their distance.Using elementary concentration results, we then deduce the typical case for measures of correlation such as the von Neumann mutual information and a measure arising from the Hilbert-Schmidt norm.We find for all considered cases that the typical behavior is an exponential decay (for both one and two spatial dimensions).We observe the consistent emergence of a correlation length that depends only on the underlying spatial dimension and not the considered measure.Remarkably, increasing the bond dimension leads to a higher correlation length in one spatial dimension but has the opposite effect in two spatial dimensions.
The complexity of quantum many-body systems is manifested in the vast diversity of their correlations, making it challenging to distinguish the generic from the atypical features.This can be addressed by analyzing correlations through ensembles of random states, chosen to faithfully embody the relevant physical properties.Here, we focus on spins with local interactions, whose correlations are extremely well captured by tensor network states.Adopting an operational perspective, we define ensembles of random tensor network states in one and two spatial dimensions that admit a sequential generation.As such, they directly correspond to outputs of quantum circuits with a sequential architecture and random gates.In one spatial dimension, the ensemble explores the entire family of matrix product states, while in two spatial dimensions, it corresponds to random isometric tensor network states.We extract the scaling behavior of the average correlations between two subsystems as a function of their distance.Using elementary concentration results, we then deduce the typical case for measures of correlation such as the von Neumann mutual information and a measure arising from the Hilbert-Schmidt norm.We find for all considered cases that the typical behavior is an exponential decay (for both one and two spatial dimensions).We observe the consistent emergence of a correlation length that depends only on the underlying spatial dimension and not the considered measure.Remarkably, increasing the bond dimension leads to a higher correlation length in one spatial dimension but has the opposite effect in two spatial dimensions.
We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural conditions on the associated transfer operators, we prove the existence of thermodynamic limits of expectations of local observables and establish almost-sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites, with a deterministic rate. In the i.i.d. case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance. For strictly stationary ensembles with decaying spatial dependence, the correlation decay quantitatively reflects the mixing profile: (ρ)-mixing yields polynomial bounds with high probability, while stretched-exponential (resp. exponential) decay in (ρ) (resp. (β)) yields stretched-exponential (resp. exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic and Gaussian translation-invariant ensembles, providing a transfer-operator route to typical correlation decay in random MPS.
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.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Tensor networks have emerged as a powerful theoretical and computational framework for understanding and simulating quantum many-body systems. Their efficiency in representing quantum states, particularly those satisfying area laws of entanglement, is intimately linked to the structure of their constituent tensors. A fundamental aspect of these networks is their spectral properties, which manifest in various forms, such as the singular values of bond tensors or the eigenvalues of associated transfer matrices. Concurrently, entanglement measures provide quantitative insights into the quantum correlations present within these systems. This paper explores the intricate relationship between the spectra arising from tensor network representations and the different entanglement measures used to characterize quantum states. We delve into how the decay rates of singular values directly impact entanglement entropy, how entanglement spectra derived from Schmidt decompositions can reveal topological order, and how the spectral gaps of transfer matrices are connected to correlation lengths. By analyzing various tensor network architectures, including Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), we elucidate the mechanisms through which their spectral features encode profound information about quantum entanglement. The study highlights how a deeper understanding of these spectral properties can not only guide the construction of more efficient tensor network algorithms but also provide fundamental insights into the nature of quantum phases and phase transitions.
Solvable matrix product and projected entangled pair states evolved by dual and ternary-unitary quantum circuits have analytically accessible correlation functions. Here, we investigate the influence of disorder. Specifically, we compute the average behavior of a physically motivated two-point equal-time correlation function with respect to random disordered solvable tensor network states arising from the Haar measure on the unitary group. By employing the Weingarten calculus, we provide an exact analytical expression for the average of the kth moment of the correlation function. The complexity of the expression scales with k! and is independent of the complexity of the underlying tensor network state. Our result implies that the correlation function vanishes on average, while its covariance is nonzero.
Randomness is a fundamental aspect of quantum mechanics, arising from the measurement process that collapses superpositions into definite outcomes according to Born's rule. Generating large-scale random quantum states is crucial for quantum computing and many-body physics, yet it remains a key challenge. We present a practical method based on local measurements of random Tensor Networks, focusing on random Matrix Product States (MPS) generated by two distinct quantum circuit architectures, both feasible on near-term devices. We certify the emergent quantum randomness using the frame potential and establish a mapping between its behavior and the statistical mechanics of a domain wall particle model. In both architectures, the effect of quantum measurements induces a nontrivial confinement mechanism, where domain walls are either trapped by an external potential or bound in pairs to form meson-like excitations. Our results, supported by both exact analytical calculations and numerical simulations, suggest that confinement is a general mechanism underlying random state generation in broader settings with local measurements, including quantum circuits and chaotic dynamics.