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:2406.18458 · Physical Review A

Scalable tomography of many-body quantum environments with low temporal entanglement

I. A. Luchnikov, Michael Sonner, Dmitry A. Abanin

Published Mar 13, 2026• 0 citations
arXivHTMLPDF

Abstract

Describing dynamics of a quantum system coupled to a complex many-body environment is a ubiquitous problem in quantum science. General non-Markovian environments are characterized by their influence matrix~(IM) -- a multi-time tensor arising from repeated interactions between the system and environment. While complexity of the most generic IM grows exponentially with the evolution time, recent works argued that for many instances of physical many-body environments, the IM is significantly less complex. This is thanks to area-law scaling of temporal entanglement, which quantifies the correlations between the past and the future states of the system. However, efficient classical algorithms for computing IM are only available for non-interacting environments or certain interacting 1D environments. Here, we study a learning algorithm for reconstructing IMs of large many-body environments simulated on a quantum processor. This hybrid algorithm involves experimentally collecting quantum measurement results of auxiliary qubits which are repeatedly coupled to the many-body environment, followed by a classical machine-learning construction of a matrix-product (MPS) representation of the IM. Using the example of 1D spin-chain environments, with a classically generated training dataset, we demonstrate that the algorithm allows scalable reconstruction of IMs for long evolution times. The reconstructed IM can be used to efficiently model quantum transport through an impurity, including cases with multiple leads and time-dependent controls. These results indicate the feasibility of characterizing long-time dynamics of complex environments using a limited number of measurements, under the assumption of a moderate temporal entanglement.

DOI
10.1103/78jc-wcqb

Related papers

Temporal entanglement transition in chaotic quantum many-body dynamics

Ilya Vilkoviskiy, Michael Sonner, Qi Camm Huang +3 more

Temporal entanglement (TE) of an influence matrix (IM) has been proposed as a measure of complexity of simulating dynamics of local observables in a many-body system. Foligno et al. [Phys. Rev. X 13, 041008 (2023)] recently argued that the TE in chaotic 1d quantum circuits obeys linear (volume-law) scaling with evolution time. To reconcile this apparent high complexity of IM with the rapid thermalization of local observables, here we study the relation between TE, non-Markovianity, and local temporal correlations for chaotic quantum baths. By exactly solving a random-unitary bath model, and bounding distillable entanglement between future and past degrees of freedom, we identify a regime where TE is extensive and reflects genuine non-Markovianity. This memory, however, is entirely contained in highly complex temporal correlations, and its effect on few-point temporal correlators is negligible. An IM coarse-graining procedure, reducing the allowed frequency of measurements of the probe system, results in a transition from volume- to {subvolume- (area or logarithmic)} law TE scaling. We demonstrate the generality of this TE transition in 1d circuits by analyzing the kicked Ising model analytically at dual-unitary points, as well as numerically away from them. This finding indicates that dynamics of local observables are fully captured by a subvolume-law IM. We provide evidence that the compact IM MPS obtained via standard compression algorithms accurately describes local~evolution.

quant-pharXiv:2511.03846•Nov 7, 2025•0 citations

Efficient construction of time-invariant process tensors for simulating high-dimensional non-Markovian open quantum systems

Émile Cochin, Jonathan Keeling, Brendon W. Lovett +1 more

Numerical methods for obtaining exact dynamics of non-Markovian open quantum systems are mostly limited to either small systems or to short-time evolution only. Here, we propose a new algorithm for computing process tensors--matrix product operator (MPO) representations that capture the environment influence--which achieves greatly enhanced computational scalings with system size, while maintaining linear scaling with simulation length. We build on recent developments in the field which allow for long-time evolutions through process tensors which have a time-translational invariance. These can be built for general Gaussian environments and generic coupling operators with the system using infinite time-evolving block decimation (iTEBD). We introduce a modified iTEBD algorithm using intermediate compression steps which bring down the computation time scaling with system size $d$ from $\mathcal{O}(d^8)$ to $\mathcal{O}(d^4)$, as well as significantly lowering the required memory. To illustrate the power of this method, we apply it to the problem of dispersive qubit readout in circuit QED, which was previously out-of-reach numerically. The full treatment of the measurement resonator, which requires a large system space, combined with the long simulation times precipitated by the separation of timescales between the measurement drive and the environment dissipation, is now possible. The algorithm we introduce not only allows for capturing non-Markovian dynamics in large open quantum systems, but also further extends all the existing capabilities of process tensors, for example in quantum optimal control, or in computation of multi-time correlations or of steady states, to more complex systems with tens of levels.

quant-pharXiv:2603.06840•Mar 10, 2026•0 citations

Overcoming the entanglement barrier in quantum many-body dynamics via space-time duality

Alessio Lerose, Michael Sonner, Dmitry A. Abanin

Describing nonequilibrium properties of quantum many-body systems is challenging due to high entanglement in the wave function. We describe the evolution of local observables via the influence matrix (IM), which encodes the effects of a many-body system as an environment for local subsystems. Recent works found that in many dynamical regimes the IM of an infinite system has low temporal entanglement and can be efficiently represented as a matrix-product state (MPS). Yet, direct iterative constructions of the IM encounter highly entangled intermediate states---a temporal entanglement barrier (TEB). We argue that TEB is ubiquitous, and elucidate its physical origin via a semiclassical quasiparticle picture that exactly captures the behavior of integrable spin chains. Further, we show that a TEB also arises in chaotic spin chains, which lack well-defined quasiparticles. Based on these insights, we formulate an alternative light-cone growth algorithm, which provably avoids TEB, thus providing an efficient construction of the thermodynamic-limit IM as a MPS. This work uncovers the origin of the efficiency of the IM approach for studying thermalization and transport.

arXiv:2201.04150•Physical review. B./Physical review. B•Feb 15, 2023•33 citations

Efficient time-evolution of matrix product states using average Hamiltonians

Belal Abouraya, Jirawat Saiphet, Fedor Jelezko +1 more

Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.

quant-pharXiv:2602.04955•Feb 6, 2026•0 citations

Practical learning of multi-time statistics in open quantum systems

Gregory A. L. White, Lloyd C. L. Hollenberg, Charles D. Hill +1 more

Randomised measurements can efficiently characterise many-body quantum states by learning the expectation values of observables with low Pauli weights. In this paper, we generalise the theoretical tools of classical shadow tomography to the temporal domain to explore multi-time phenomena. This enables us to efficiently learn the features of multi-time processes such as correlated error rates, multi-time non-Markovianity, and temporal entanglement. We test the efficacy of these tools on a noisy quantum processor to characterise its noise features. Implementing these tools requires mid-circuit instruments, typically slow or unavailable in current quantum hardware. We devise a protocol to achieve fast and reliable instruments such that these multi-time distributions can be learned to a high accuracy. This enables a compact matrix product operator representation of large processes allowing us to showcase a reconstructed 20-step process (whose naive dimensionality is that of a 42-qubit state). Our techniques are pertinent to generic quantum stochastic dynamical processes, with a scope ranging across condensed matter physics, quantum biology, and in-depth diagnostics of noisy intermediate-scale quantum devices.

arXiv:2412.17862•Dec 20, 2024•0 citations

Reconstructing non-Markovian open quantum evolution from multiple time measurements

Chu Guo

For a quantum system undergoing non-Markovian open quantum dynamics, we demonstrate a tomography algorithm based on multitime measurements of the system, which reconstructs a minimal environment coupled to the system, such that the system plus environment undergoes unitary evolution and that the reduced dynamics of the system is identical to the observed dynamics of it. The reconstructed open quantum evolution model can be used to predict any future dynamics of the system when it is further assumed to be time-independent. We define the memory size and memory complexity for the non-Markovian open quantum dynamics which characterize the complexity of the reconstruction algorithm.

arXiv:2205.06521•Physical Review A•Aug 15, 2022•7 citations
Describing dynamics of a quantum system coupled to a complex many-body environment is a ubiquitous problem in quantum science. General non-Markovian environments are characterized by their influence matrix~(IM) -- a multi-time tensor arising from repeated interactions between the system and environment. While complexity of the most generic IM grows exponentially with the evolution time, recent works argued that for many instances of physical many-body environments, the IM is significantly less complex. This is thanks to area-law scaling of temporal entanglement, which quantifies the correlations between the past and the future states of the system. However, efficient classical algorithms for computing IM are only available for non-interacting environments or certain interacting 1D environments. Here, we study a learning algorithm for reconstructing IMs of large many-body environments simulated on a quantum processor. This hybrid algorithm involves experimentally collecting quantum measurement results of auxiliary qubits which are repeatedly coupled to the many-body environment, followed by a classical machine-learning construction of a matrix-product (MPS) representation of the IM. Using the example of 1D spin-chain environments, with a classically generated training dataset, we demonstrate that the algorithm allows scalable reconstruction of IMs for long evolution times. The reconstructed IM can be used to efficiently model quantum transport through an impurity, including cases with multiple leads and time-dependent controls. These results indicate the feasibility of characterizing long-time dynamics of complex environments using a limited number of measurements, under the assumption of a moderate temporal entanglement.
Temporal entanglement (TE) of an influence matrix (IM) has been proposed as a measure of complexity of simulating dynamics of local observables in a many-body system. Foligno et al. [Phys. Rev. X 13, 041008 (2023)] recently argued that the TE in chaotic 1d quantum circuits obeys linear (volume-law) scaling with evolution time. To reconcile this apparent high complexity of IM with the rapid thermalization of local observables, here we study the relation between TE, non-Markovianity, and local temporal correlations for chaotic quantum baths. By exactly solving a random-unitary bath model, and bounding distillable entanglement between future and past degrees of freedom, we identify a regime where TE is extensive and reflects genuine non-Markovianity. This memory, however, is entirely contained in highly complex temporal correlations, and its effect on few-point temporal correlators is negligible. An IM coarse-graining procedure, reducing the allowed frequency of measurements of the probe system, results in a transition from volume- to {subvolume- (area or logarithmic)} law TE scaling. We demonstrate the generality of this TE transition in 1d circuits by analyzing the kicked Ising model analytically at dual-unitary points, as well as numerically away from them. This finding indicates that dynamics of local observables are fully captured by a subvolume-law IM. We provide evidence that the compact IM MPS obtained via standard compression algorithms accurately describes local~evolution.
Numerical methods for obtaining exact dynamics of non-Markovian open quantum systems are mostly limited to either small systems or to short-time evolution only. Here, we propose a new algorithm for computing process tensors--matrix product operator (MPO) representations that capture the environment influence--which achieves greatly enhanced computational scalings with system size, while maintaining linear scaling with simulation length. We build on recent developments in the field which allow for long-time evolutions through process tensors which have a time-translational invariance. These can be built for general Gaussian environments and generic coupling operators with the system using infinite time-evolving block decimation (iTEBD). We introduce a modified iTEBD algorithm using intermediate compression steps which bring down the computation time scaling with system size ddd from O(d8)\mathcal{O}(d^8)O(d8) to O(d4)\mathcal{O}(d^4)O(d4), as well as significantly lowering the required memory. To illustrate the power of this method, we apply it to the problem of dispersive qubit readout in circuit QED, which was previously out-of-reach numerically. The full treatment of the measurement resonator, which requires a large system space, combined with the long simulation times precipitated by the separation of timescales between the measurement drive and the environment dissipation, is now possible. The algorithm we introduce not only allows for capturing non-Markovian dynamics in large open quantum systems, but also further extends all the existing capabilities of process tensors, for example in quantum optimal control, or in computation of multi-time correlations or of steady states, to more complex systems with tens of levels.
Describing nonequilibrium properties of quantum many-body systems is challenging due to high entanglement in the wave function. We describe the evolution of local observables via the influence matrix (IM), which encodes the effects of a many-body system as an environment for local subsystems. Recent works found that in many dynamical regimes the IM of an infinite system has low temporal entanglement and can be efficiently represented as a matrix-product state (MPS). Yet, direct iterative constructions of the IM encounter highly entangled intermediate states---a temporal entanglement barrier (TEB). We argue that TEB is ubiquitous, and elucidate its physical origin via a semiclassical quasiparticle picture that exactly captures the behavior of integrable spin chains. Further, we show that a TEB also arises in chaotic spin chains, which lack well-defined quasiparticles. Based on these insights, we formulate an alternative light-cone growth algorithm, which provably avoids TEB, thus providing an efficient construction of the thermodynamic-limit IM as a MPS. This work uncovers the origin of the efficiency of the IM approach for studying thermalization and transport.
Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.
Randomised measurements can efficiently characterise many-body quantum states by learning the expectation values of observables with low Pauli weights. In this paper, we generalise the theoretical tools of classical shadow tomography to the temporal domain to explore multi-time phenomena. This enables us to efficiently learn the features of multi-time processes such as correlated error rates, multi-time non-Markovianity, and temporal entanglement. We test the efficacy of these tools on a noisy quantum processor to characterise its noise features. Implementing these tools requires mid-circuit instruments, typically slow or unavailable in current quantum hardware. We devise a protocol to achieve fast and reliable instruments such that these multi-time distributions can be learned to a high accuracy. This enables a compact matrix product operator representation of large processes allowing us to showcase a reconstructed 20-step process (whose naive dimensionality is that of a 42-qubit state). Our techniques are pertinent to generic quantum stochastic dynamical processes, with a scope ranging across condensed matter physics, quantum biology, and in-depth diagnostics of noisy intermediate-scale quantum devices.
For a quantum system undergoing non-Markovian open quantum dynamics, we demonstrate a tomography algorithm based on multitime measurements of the system, which reconstructs a minimal environment coupled to the system, such that the system plus environment undergoes unitary evolution and that the reduced dynamics of the system is identical to the observed dynamics of it. The reconstructed open quantum evolution model can be used to predict any future dynamics of the system when it is further assumed to be time-independent. We define the memory size and memory complexity for the non-Markovian open quantum dynamics which characterize the complexity of the reconstruction algorithm.