Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2505.09714

Overcoming the entanglement barrier with sampled tensor networks

Stefano Carignano, Guglielmo Lami, Jacopo De Nardis, Luca Tagliacozzo

Published May 14, 2025• 2 citations
arXivHTMLPDF

Abstract

The rapid growth of entanglement under unitary time evolution is the primary bottleneck for modern tensor-network techniques--such as Matrix Product States (MPS)--when computing time-dependent expectation values. This {entanglement barrier} restricts classical simulations and, conversely, underpins the quantum advantage anticipated from future devices. Here we show that, for one-dimensional Hamiltonian dynamics, the spatio-temporal tensor network encoding the evolved wave function amplitudes can be contracted efficiently along the left-right (spatial) direction. Exploiting this structure, we develop a hybrid Tensor-Network/Monte-Carlo (TN-MC) algorithm that samples the wave function and evaluates expectation values of generic local operators with computational cost that scales only polynomially in time. The accurate contraction of the wave function amplitudes is a consequence of the favorable scaling with time of the generalised temporal entropies. We find that their real part either saturates or, at most, grows logarithmically with time, revealing new instances of continuous dynamical quantum phase transitions (DQPTs) which we characterize. Our results therefore show that, when computing expectation values of local operators, the entanglement barrier in one-dimensional Hamiltonian evolution can be bypassed with a TN-MC blend.

DOI
10.48550/arxiv.2505.09714

Related papers

On the out-of-equilibrium dynamics with tensor networks

Ramos Marimón, Carlos

[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.

•Sep 10, 2025•0 citations

Time Evolution on Hybrid Tensor Networks -- A Novel and Parallelizable Algorithm

Tobias V. Bauer, Richard M. Milbradt, Christian B. Mendl

We develop a novel time-evolution algorithm for matrix product states based on the recently introduced hybrid tensor network (hTN) framework. We retain the tensors close to the boundary on the classical computer and offload the highly entangled inner ones to the quantum computer. In our variant, we employ the Basis Update and Galerkin (BUG) integrator to time-evolve the classical tensors, and we develop a coupling scheme between the classical and quantum parts. Our framework admits modular combination with any quantum time-evolution method, such as (classically pre-optimized) Trotterization. The ratio of classical and quantum tensor degrees of freedom can be dynamically adjusted during the time evolution, which can be advantageous when the classical memory requirements become prohibitive. The quantum and classical components can run in parallel during a single time step and are not constrained by synchronization barriers or mid-circuit measurements. We describe the detailed steps and pseudocode for our algorithm specialized for tensor networks originating from the matrix product state Ansatz.

quant-pharXiv:2606.28169•Jun 29, 2026•0 citations

Reweighted time-evolving block decimation for improved quantum dynamics simulations

Sayak Guha Roy, Kevin Slagle

We introduce a simple yet significant improvement to the time-evolving block decimation (TEBD) tensor network algorithm for simulating the time dynamics of strongly correlated one-dimensional (1D) mixed quantum states. The efficiency of 1D tensor network methods stems from using a product of matrices to express either: the coefficients of a wavefunction, yielding a matrix product state (MPS); or the expectation values of a density matrix, yielding a matrix product density operator (MPDO). To avoid exponential computational costs, TEBD truncates the matrix dimension while simulating the time evolution. However, when truncating an MPDO, TEBD does not favor the likely more important low-weight expectation values, such as \langle c_i^\dagger c_j \rangle ⟨ c i † c j ⟩ , over the exponentially many high-weight expectation values, such as \langle c_{i_1}^\dagger c^\dagger_{i_2} \cdots c_{i_n} \rangle ⟨ c i 1 † c i 2 † ⋯ c i n ⟩ of weight n n , despite the critical importance of the low-weight expectation values. Motivated by this shortcoming, we propose a reweighted TEBD (rTEBD) algorithm that deprioritizes high-weight expectation values by a factor of \gamma^{-n} γ − n during the truncation. This modification makes rTEBD significantly more accurate than the TEBD time-dependent simulation of an MPDO, and competitive with and sometimes better than TEBD using MPS. Furthermore, by prioritizing low-weight expectation values, rTEBD preserves conserved quantities to high precision.

arXiv:2412.08730•SciPost Physics•Jul 21, 2026•0 citations

Tensor-network Monte Carlo approach based on time-evolving block decimation

Shimpei Goto, Hidemaro Suwa, Synge Todo

We propose a tensor-network Monte Carlo (TNMC) approach for unitary evolution following the compression sequence of the time-evolving block decimation (TEBD) algorithm. In the TNMC approach, the obtained results contain evaluable statistical errors rather than truncation errors, unlike ordinary singular-value-decomposition-based methods such as the TEBD algorithm. Consequently, one can estimate unbiased expectation values within statistical errors even with a finite bond dimension. Since the sampling scheme is introduced in the simulations of unitary evolution, the proposed Monte Carlo scheme may suffer from a sign problem. We observe that the sign problem can be mitigated by increasing the bond dimension. We apply the proposed TNMC approach to the Hamiltonian and the Floquet dynamics. Numerical experiments show that the TNMC approach can estimate accurate expectation values of observables even when the TEBD method with the same bond dimension cannot. The proposed approach can be a new direction for improving the classical simulatability of unitary evolution.

cond-mat.stat-mechCross-listedarXiv:2609.12466•Sep 14, 2026•0 citations

Temporal entropy and the complexity of computing the expectation value of local operators after a quench

Stefano Carignano, Carlos Ramos Marimón, Luca Tagliacozzo

We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states(MPSs) and argue that it is intimately related to that of encoding temporal transition matrices and their partial traces. We show that we can upper-bound the rank of these reduced transition matrices by one of the Heisenberg evolution of local operators, thus making aconnection between two apparently different quantities, the temporal entanglement and the local operator entanglement(OE). As a result, whenever the local OE grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal MPSs is likely advantageous with respect to computing the same quantities using standard MPS techniques. Published by the American Physical Society 2024

arXiv:2307.11649•Physical Review Research•Jul 3, 2024•18 citations

Quantum-inspired classical simulation through randomized time evolution

Fredrik Hasselgren, Bálint Koczor

Tensor-network simulations of quantum many-body dynamics are fundamentally limited by entanglement build-up, which leads to exponentially growing computational costs. Furthermore, these classical simulation algorithms are inherently sequential as typically a tensor network representation of the quantum state is updated incrementally at each time step. We build on recently introduced randomized quantum algorithms for time evolution (TE-PAI), and adapt them to the classical simulation context with the purpose of enabling massive parallelisation. Our MPS TE-PAI approach achieves exact time evolution on average (unbiased estimator) and proceeds by representing an ensemble of randomized shallow Trotter-variant circuits as tensor networks. As each circuit instance yields a deterministic quantum state (or observable expecation value), the only source of randomness is the sampling of circuit variants; the absence of shot noise therefore yields a reduced estimator variance relative to quantum hardware implementations of TE-PAI. We simulate representative disordered one-dimensional spin-ring Hamiltonians, and numerically observe reductions in the per-sample gate-count by a factor of up to $10^3$ relative to Trotterized MPS evolution, yielding orders of magnitude reduction in the time-to-solution under realistic levels of parallelisation. Finally, we numerically observe that MPS TE-PAI is substantially more robust against severe bond-dimension truncation than product formulas, potentially making it useful for the simulation of strongly correlated systems where truncation is necessary in practice. We also demonstrate that the approach can be used naturally in combination with existing time evolution algorithms, effectively extending their time depth via parallelisation.

quant-pharXiv:2604.13144•Apr 15, 2026•0 citations
The rapid growth of entanglement under unitary time evolution is the primary bottleneck for modern tensor-network techniques--such as Matrix Product States (MPS)--when computing time-dependent expectation values. This {entanglement barrier} restricts classical simulations and, conversely, underpins the quantum advantage anticipated from future devices. Here we show that, for one-dimensional Hamiltonian dynamics, the spatio-temporal tensor network encoding the evolved wave function amplitudes can be contracted efficiently along the left-right (spatial) direction. Exploiting this structure, we develop a hybrid Tensor-Network/Monte-Carlo (TN-MC) algorithm that samples the wave function and evaluates expectation values of generic local operators with computational cost that scales only polynomially in time. The accurate contraction of the wave function amplitudes is a consequence of the favorable scaling with time of the generalised temporal entropies. We find that their real part either saturates or, at most, grows logarithmically with time, revealing new instances of continuous dynamical quantum phase transitions (DQPTs) which we characterize. Our results therefore show that, when computing expectation values of local operators, the entanglement barrier in one-dimensional Hamiltonian evolution can be bypassed with a TN-MC blend.
[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.
We develop a novel time-evolution algorithm for matrix product states based on the recently introduced hybrid tensor network (hTN) framework. We retain the tensors close to the boundary on the classical computer and offload the highly entangled inner ones to the quantum computer. In our variant, we employ the Basis Update and Galerkin (BUG) integrator to time-evolve the classical tensors, and we develop a coupling scheme between the classical and quantum parts. Our framework admits modular combination with any quantum time-evolution method, such as (classically pre-optimized) Trotterization. The ratio of classical and quantum tensor degrees of freedom can be dynamically adjusted during the time evolution, which can be advantageous when the classical memory requirements become prohibitive. The quantum and classical components can run in parallel during a single time step and are not constrained by synchronization barriers or mid-circuit measurements. We describe the detailed steps and pseudocode for our algorithm specialized for tensor networks originating from the matrix product state Ansatz.
We introduce a simple yet significant improvement to the time-evolving block decimation (TEBD) tensor network algorithm for simulating the time dynamics of strongly correlated one-dimensional (1D) mixed quantum states. The efficiency of 1D tensor network methods stems from using a product of matrices to express either: the coefficients of a wavefunction, yielding a matrix product state (MPS); or the expectation values of a density matrix, yielding a matrix product density operator (MPDO). To avoid exponential computational costs, TEBD truncates the matrix dimension while simulating the time evolution. However, when truncating an MPDO, TEBD does not favor the likely more important low-weight expectation values, such as \langle c_i^\dagger c_j \rangle ⟨ c i † c j ⟩ , over the exponentially many high-weight expectation values, such as \langle c_{i_1}^\dagger c^\dagger_{i_2} \cdots c_{i_n} \rangle ⟨ c i 1 † c i 2 † ⋯ c i n ⟩ of weight n n , despite the critical importance of the low-weight expectation values. Motivated by this shortcoming, we propose a reweighted TEBD (rTEBD) algorithm that deprioritizes high-weight expectation values by a factor of \gamma^{-n} γ − n during the truncation. This modification makes rTEBD significantly more accurate than the TEBD time-dependent simulation of an MPDO, and competitive with and sometimes better than TEBD using MPS. Furthermore, by prioritizing low-weight expectation values, rTEBD preserves conserved quantities to high precision.
We propose a tensor-network Monte Carlo (TNMC) approach for unitary evolution following the compression sequence of the time-evolving block decimation (TEBD) algorithm. In the TNMC approach, the obtained results contain evaluable statistical errors rather than truncation errors, unlike ordinary singular-value-decomposition-based methods such as the TEBD algorithm. Consequently, one can estimate unbiased expectation values within statistical errors even with a finite bond dimension. Since the sampling scheme is introduced in the simulations of unitary evolution, the proposed Monte Carlo scheme may suffer from a sign problem. We observe that the sign problem can be mitigated by increasing the bond dimension. We apply the proposed TNMC approach to the Hamiltonian and the Floquet dynamics. Numerical experiments show that the TNMC approach can estimate accurate expectation values of observables even when the TEBD method with the same bond dimension cannot. The proposed approach can be a new direction for improving the classical simulatability of unitary evolution.
We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states(MPSs) and argue that it is intimately related to that of encoding temporal transition matrices and their partial traces. We show that we can upper-bound the rank of these reduced transition matrices by one of the Heisenberg evolution of local operators, thus making aconnection between two apparently different quantities, the temporal entanglement and the local operator entanglement(OE). As a result, whenever the local OE grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal MPSs is likely advantageous with respect to computing the same quantities using standard MPS techniques. Published by the American Physical Society 2024
Tensor-network simulations of quantum many-body dynamics are fundamentally limited by entanglement build-up, which leads to exponentially growing computational costs. Furthermore, these classical simulation algorithms are inherently sequential as typically a tensor network representation of the quantum state is updated incrementally at each time step. We build on recently introduced randomized quantum algorithms for time evolution (TE-PAI), and adapt them to the classical simulation context with the purpose of enabling massive parallelisation. Our MPS TE-PAI approach achieves exact time evolution on average (unbiased estimator) and proceeds by representing an ensemble of randomized shallow Trotter-variant circuits as tensor networks. As each circuit instance yields a deterministic quantum state (or observable expecation value), the only source of randomness is the sampling of circuit variants; the absence of shot noise therefore yields a reduced estimator variance relative to quantum hardware implementations of TE-PAI. We simulate representative disordered one-dimensional spin-ring Hamiltonians, and numerically observe reductions in the per-sample gate-count by a factor of up to 10310^3103 relative to Trotterized MPS evolution, yielding orders of magnitude reduction in the time-to-solution under realistic levels of parallelisation. Finally, we numerically observe that MPS TE-PAI is substantially more robust against severe bond-dimension truncation than product formulas, potentially making it useful for the simulation of strongly correlated systems where truncation is necessary in practice. We also demonstrate that the approach can be used naturally in combination with existing time evolution algorithms, effectively extending their time depth via parallelisation.