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.
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.
We analyze the entanglement structure of states generated by random constant-depth two-dimensional quantum circuits, followed by projective measurements of a subset of sites. By deriving a rigorous lower bound on the average entanglement entropy of such postmeasurement states, we prove that macroscopic long-ranged entanglement is generated above some constant critical depth in several natural classes of circuit architectures, which include brickwork circuits and random holographic tensor networks. This behavior had been conjectured based on previous works, which utilize nonrigorous methods such as replica theory calculations, or work in regimes where the local Hilbert space dimension grows with system size. To establish our lower bound, we develop new replica-free theoretical techniques that leverage tools from multiuser quantum information theory, which are of independent interest, allowing us to map the problem onto a statistical mechanics model of self-avoiding walks without requiring large local Hilbert space dimension. Our findings have consequences for the complexity of classically simulating sampling from random shallow circuits and of contracting tensor networks. First, we show that standard algorithms based on matrix product states which are used for both these tasks will fail above some constant depth and bond dimension, respectively. In addition, we also prove that these random constant-depth quantum circuits cannot be simulated by any classical circuit of sublogarithmic depth.
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.
Recent studies of "deep thermalization" have revealed universal physics in quantum many-body dynamics beyond equilibration towards Gibbs states: maximally random quantum state ensembles can emerge on local subsystems, generated by measurements on their complement. In this work, we further identify a new form of universality exhibited in the "finer fingerprints" of quantum dynamics for local subsystems, induced by global unitary time-evolution. Specifically, we consider the projected Kraus ensemble, an ensemble of Kraus operators obtained by unraveling the quantum channel on a small subsystem with respect to knowledge of the classical configurations of its complement. Our central result is a one-parameter random matrix Ansatz that captures the ensemble's emergent statistical behavior along particular spacetime scalings, valid for generic 1D circuit dynamics without conservation laws: the ensemble is described by the product of a complex Ginibre random matrix and an independent log-normal random real scalar. The former encodes scrambling within the subsystem, captured by rotational invariance of the Ginibre measure, whereas the latter encodes fluctuations in the Born probabilities, arising from locality of the underlying dynamics. Our Ansatz can be established in the special cases of dynamics generated by global Haar random unitaries and dual-unitary circuits, while we motivate it for generic circuits using arguments of spacetime duality and the multiplicative ergodic theorem on long products of spatial transfer matrices. Extensive numerical simulations for random and Floquet circuit models verify these predictions. This universality in Kraus operators also provides a microscopic mechanism for deep thermalization in generic 1D quantum circuit dynamics, and has implications for local quantum information recoverability involving classical side information tied to knowledge of the bath.
Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary k-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on n qudits form approximate 2-designs in O(n) depth, as is known. Furthermore, we argue that random circuits form approximate unitary k-designs in O(nk) depth and are thus essentially optimal in both n and k. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.
Classical simulations of quantum circuits play a vital role in the development of quantum computers and in taking the temperature of the field. Here, we classically simulate various physically-motivated circuits using 2D tensor network ansätze for the many-body wavefunction, which match the geometry of the underlying quantum processor. We employ a generalized boundary Matrix Product State contraction algorithm to controllably generate samples from the resultant tensor network states. Our approach systematically converges both the quality of the final state and the samples drawn from it to the true distribution defined by the circuit, with GPU hardware providing significant speedups over CPU hardware. Using these methods, we simulate the largest local unitary Jastrow ansatz circuit from recent IBM experiments to numerical precision. We also study a domain-wall quench in a 2D discrete-time Heisenberg model on large heavy-hex and rotated square lattices, reflecting IBM’s and Google’s latest processors, respectively. We observe a rapid buildup of complex loop correlations on the Google Willow geometry, which significantly impacts local properties of the system. Meanwhile, loop correlations build up extremely slowly on heavy-hex processors, with almost negligible impact even at large circuit depths. Our results underscore the role processor geometry plays in classical simulability.
We investigate quantum random tensor network states in which the dimension of the bond scales polynomially with the size of the system N. Specifically, we examine the delocalization properties of random matrix product states (RMPS) in the computational basis by deriving an exact analytical expression for the inverse participation ratio (IPR) of any degree, applicable to both open and closed boundary conditions. For bond dimensions χ∼γN, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution, characteristic of Haar-random states, as γ increases. Additionally, we provide numerical evidence for the frame potential, measuring the 2-distance from the Haar ensemble, which confirms the convergence of random MPS to Haar-like behavior for χ≫sqrt[N]. We extend this analysis to two-dimensional systems using random projected entangled pair states, where we similarly observe the convergence of IPRs to their Haar values for χ≫sqrt[N]. These findings demonstrate that random tensor networks with bond dimensions scaling polynomially in the system size are fully Haar anticoncentrated and approximate unitary designs, regardless of the spatial dimension.