Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals

Jørgensen, Mathias R.; Pollock, Felix A.

doi:10.1103/physrevlett.123.240602

Cited by 131 publications

(120 citation statements)

References 54 publications

(81 reference statements)

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“…Cf. the recent works [38,28]. Given the usages of CPTP maps and temporally extended causal structure in this paper, it is interesting to find possible relations between different formalisms.…”

Section: Discussionmentioning

confidence: 99%

“…Secondly, the choice of the entangled pairs is not unique, and it is event possible to consider only quantum coherences between two histories. The choice of maximally entangled pairs in (28) is again motivated by the results in loop quantum gravity. The kinematical Hilbert space of loop quantum gravity is H kin = e v H v,e , where v and e are vertices and edges of a spin network graph.…”

Section: Entangled Quantum Causal Historiesmentioning

confidence: 99%

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Tensor networks for quantum causal histories

Guo¹

2020

J. Phys. A: Math. Theor.

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In this paper, we construct a tensor network representation of quantum causal histories, as a step towards directly representing states in quantum gravity via bulk tensor networks. Quantum causal histories are quantum extensions of causal sets in the sense that on each event in a causal set is assigned a Hilbert space of quantum states, and the local causal evolutions between events are modelled by competely positive and trace-preserving maps. Here we untilize the channel-state duality of competely positive and trace-preserving maps to transform the causal evolutions to bipartite entangled states. We construct the matrix product state for a single quantum causal history by projecting the obtained bipartite states onto the physical states on the events. We also construct the two dimensional tensor network states for entangled quantum causal histories in a restricted case with compatible causal orders. The possible holographic tensor networks are explored by mapping the quantum causal histories in a way analogous to the exact holographic mapping. The constructed tensor networks for quantum causal histories are exemplified by the non-unitary local time evolution moves in a quantum system on temporally varying discretizations. Finally, we comment on the limitations of the constructed tensor networks, and discuss some directions for further studies aiming at applications in quantum gravity. *

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Section: Discussionmentioning

confidence: 99%

Section: Entangled Quantum Causal Historiesmentioning

confidence: 99%

Tensor networks for quantum causal histories

Guo¹

2020

J. Phys. A: Math. Theor.

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“…The TRN is closely related to the recent reformulations of the Feynman-Vernon path integrals [63,[70][71][72][73] and the process tensor [68,[74][75][76]…”

Section: The Total Hamiltonian Readsmentioning

confidence: 99%

“…[70][71][72] the contraction calculation is approximated by fixing a finite memory depth K = T /τ . References [63,73] further use MP approximation of the influence functional (with rank λ max and accuracy λ c ), which allows to deal with longer memory depths. Since λ max is d 2 ER in our model, we actually estimate the complexity (λ max ) of the algorithm in Ref.…”

Section: The Total Hamiltonian Readsmentioning

confidence: 99%

Simulation Complexity of Open Quantum Dynamics: Connection with Tensor Networks

et al. 2019

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The difficulty to simulate the dynamics of open quantum systems resides in their coupling to many-body reservoirs with exponentially large Hilbert space. Applying a tensor network approach in the time domain, we demonstrate that effective small reservoirs can be defined and used for modeling open quantum dynamics. The key element of our technique is the timeline reservoir network (TRN), which contains all the information on the reservoir's characteristics, in particular, the memory effects timescale. The TRN has a one-dimensional tensor network structure, which can be effectively approximated in full analogy with the matrix product approximation of spinchain states. We derive the sufficient bond dimension in the approximated TRN with a reduced set of physical parameters: coupling strength, reservoir correlation time, minimal timescale, and the system's number of degrees of freedom interacting with the environment. The bond dimension can be viewed as a measure of the open dynamics complexity. Simulation is based on the semigroup dynamics of the system and effective reservoir of finite dimension. We provide an illustrative example showing scope for new numerical and machine learning-based methods for open quantum systems.

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“…However, the environments of many physical systems are rather complex and structured [7][8][9][10][11][12][13][14][15][16][17][18]. A model of the systemenvironment interaction is often heuristic and oversimplified (e.g., a harmonic environment), but even in this case the analysis is rater complicated and requires some elaborated analytical and numerical methods [19][20][21]. A theoretical model may also neglect some additional sources of decoherence and relaxation.…”

mentioning

confidence: 99%

Machine Learning Non-Markovian Quantum Dynamics

et al. 2020

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Machine learning methods have proved to be useful for the recognition of patterns in statistical data. The measurement outcomes are intrinsically random in quantum physics, however they do have a pattern when the measurements are performed successively on an open quantum system. This pattern is due to the system-environment interaction and contains information about the relaxation rates as well as non-Markovian memory effects. Here we develop a method to extract the information about the unknown environment from a series of single-shot measurements on the system (without resorting to the process tomography). The method is based on embedding the non-Markovian system dynamics into a Markovian dynamics of the system and the effective reservoir of finite dimension. The generator of Markovian embedding is learned by the maximum likelihood estimation. We verify the method by comparing its prediction with an exactly solvable non-Markovian dynamics. The developed algorithm to learn unknown quantum environments enables one to efficiently control and manipulate quantum systems. Introduction. Quantum systems are never perfectly isolated which makes the study of open quantum dynamics important for various disciplines including solid state physics [1], quantum chemistry [2], quantum sensing [3], quantum information transmission [4], and quantum computing [5]. Open quantum dynamics is a result of interaction between the system of interest and its environment. It is usually assumed that the environment is an infinitely large reservoir in statistical equilibrium, which has a well-defined interaction with the system [6]. However, the environments of many physical systems are rather complex and structured [7][8][9][10][11][12][13][14][15][16][17][18]. A model of the systemenvironment interaction is often heuristic and oversimplified (e.g., a harmonic environment), but even in this case the analysis is rater complicated and requires some elaborated analytical and numerical methods [19][20][21]. A theoretical model may also neglect some additional sources of decoherence and relaxation. The experimental analysis of the environmental degrees of freedom is difficult because of their inaccessibility in practice. In fact, one can only get some information about the actual environment by probing the system [22]. Therefore, one faces an important problem to learn the unknown environment and its interaction with the quantum system by probing and affecting the system only.

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Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals

Cited by 131 publications

References 54 publications

Tensor networks for quantum causal histories

Tensor networks for quantum causal histories

Simulation Complexity of Open Quantum Dynamics: Connection with Tensor Networks

Machine Learning Non-Markovian Quantum Dynamics

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