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.
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.
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.
Nonstabilizerness, or “magic,” is a critical quantum resource that, together with entanglement, characterizes the nonclassical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random matrix product states (RMPSs). RMPSs represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension χ . We demonstrate that the stabilizer Rényi entropies converge to that of Haar-random states as N / χ α , where N is the system size and the α are integer exponents. This indicates that MPSs with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford-enhanced matrix product states ( C MP Ss ), built by the action of Clifford unitaries on RMPSs. Leveraging our previous result, we show that C MP Ss can approximate quantum state 4-designs with arbitrary accuracy. Specifically, for a constant N , C MP Ss become close to 4-designs, with a scaling as χ − 2 . Our findings indicate that combining Clifford unitaries with polynomially complex tensor-network states can generate highly nontrivial quantum states.
Anticoncentration describes how an ensemble of quantum states spreads over the allowed Hilbert space, leading to statistically uniform output probability distributions. In this work, we investigate the anticoncentration of random Clifford circuits toward the overlap distribution of random stabilizer states. Using exact analytical techniques and extensive numerical simulations based on Clifford replica tensor networks, we demonstrate that random Clifford circuits fully anticoncentrate in logarithmic circuit depth; namely, higher-order moments of the overlap distribution converge to those of random stabilizer states. Moreover, we investigate the effect of introducing a controlled number of non-Clifford (magic) resources into Clifford circuits. We show that inserting a polylogarithmic in qudit number of T states is sufficient to drive the overlap distribution toward the Porter-Thomas statistics, effectively recovering full quantum randomness. In short, this fact presents doped tensor networks and shallow Clifford circuits as pseudomagic quantum states. Our results clarify the interplay between Clifford dynamics, magic-state injection, and quantum complexity, with implications for quantum circuit sampling, many-body quantum physics, and the benchmarking of quantum computational advantage.
Unitary randomness underpins both fundamental tasks in quantum information and the modern theory of quantum chaos. On one side, a central concept is that of approximate unitary designs: circuits that look random according to small moments and for forward-in-time protocols. In a distinct setting, out-of-time-ordered correlators (OTOCs), intensely studied as a measure of information scrambling, have recently been shown to probe freeness between Heisenberg operators, the noncommutative generalization of statistical independence. Bridging these two concepts, we study the emergence of freeness in a random matrix product unitary ensemble. We prove that, with only polynomial bond dimension, these unitaries reproduce Haar values of higher-order OTOCs for local, finite-trace observables, while traceless observables instead require exponential resources. Indeed, local observables are precisely those predicted to thermalize in chaotic many-body systems according to the eigenstate thermalization hypothesis. Moreover, adding to previous literature, we show how random matrix product unitaries constitute approximate designs: we exactly compute the frame potential of the ensemble, showing convergence to the Haar value with polynomial deviations and so indicating that global observables are freely independent on-average. Our results highlight the need to refine previous notions of unitary design in the context of operator dynamics, guiding us towards protocols for quantum advantage and shedding light on the emergent complexity of chaotic many-body systems.
We establish universal behavior in the anticoncentration properties of random quantum circuits, demonstrating its broad independence from the circuit architecture. Specifically, universality emerges in a certain scaling limit and extends beyond the leading order, incorporating subleading corrections arising from the finite system size N. We compute these corrections through exact calculations on ensembles of random tensor network states and corroborate the results with analytical findings in the random phase model. We then identify a heuristic framework for generic brickwork circuits, conjecturing the universality of these corrections. We further support our claim of anticoncentration universality through extensive numerical simulations, capturing the distribution of overlaps for systems up to N=64 qudits and computing collision probabilities for systems up to N≤1024. Collectively, our results highlight the critical role of finite-size corrections and lead to a thorough understanding of the core phenomenology governing anticoncentration in quantum circuits.
The chaotic dynamics of quantum many-body systems are expected to quickly randomize any structured initial state, delocalizing it in the Fock space. In this Letter, we study the spreading of an initial product state in Hilbert space under dual-unitary dynamics, captured by the inverse participation ratios and the distribution of overlaps (bit-string probabilities). We consider the self-dual kicked Ising model, a minimal model of many-body quantum chaos that can be seen as either a periodically driven Floquet model or a dual-unitary quantum circuit. Both analytically and numerically, we show that the inverse participation ratios rapidly approach their ergodic values, corresponding to those of Haar random states, and establish the emergence of the Porter-Thomas distribution for the overlap distribution. Importantly, this convergence happens exponentially fast in time, with a timescale that is independent of system size. We inspect the effect of local perturbations that break dual unitarity and show a slowdown of the spreading in Fock space, indicating that dual-unitary circuits are maximally efficient at preparing random states. Our Letter establishes bridges between the dynamics of many-body systems and random matrix theory through the time evolution of structured initial states and finds natural applications in demonstrating a quantum advantage in random sampling and in benchmarking quantum devices.