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Accessing scrambling using matrix product operators


Scrambling, a process in which quantum information spreads over a complex quantum system, becoming inaccessible to simple probes, occurs in generic chaotic quantum many-body systems, ranging from spin chains to metals and even to black holes. Scrambling can be measured using out-of-time-ordered correlators (OTOCs), which are closely tied to the growth of Heisenberg operators. We present a general method to calculate OTOCs of local operators in one-dimensional systems based on approximating Heisenberg operators as matrix product operators (MPOs). Contrary to the common belief that such tensor network methods work only at early times, we show that the entire early growth region of the OTOC can be captured using an MPO approximation with modest bond dimension. We analytically establish the goodness of the approximation by showing that, if an appropriate OTOC is close to its initial value, then the associated Heisenberg operator has low entanglement across a given cut. We use the method to study scrambling in a chaotic spin chain with \(201\) sites. On the basis of these data and previous results, we conjecture a universal form for the dynamics of the OTOC near the wavefront. We show that this form collapses the chaotic spin chain data over more than 15 orders of magnitude.

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Fig. 1: Sketch of the emergent light cone and entanglement structure of local Heisenberg operators in tensor network forms.
Fig. 2: The operator entanglement entropy for the mixed-field Ising model.
Fig. 3: The early growth regime of the squared commutator for the mixed-field Ising model from MPO dynamics.
Fig. 4: The late-time regime of the squared commutator from the MPO dynamics with time splitting.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

Code availability

The code used to perform the numerical simulation within this paper is available from the corresponding author on reasonable request.


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We thank J. Garrison, D. Roberts, A. Kitaev and D. Stanford for interesting discussions. This material is based on work supported by the Simons Foundation via the It from Qubit Collaboration, by the Air Force Office of Scientific Research under award number FA9550-17-1-0180 and by the NSF Physics Frontier Center at the Joint Quantum Institute (PHY-1430094).

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S.X. and B.S. initiated the study. S.X. developed the computational tool and performed the numerical simulations. S.X. and B.S. analysed the results and wrote the manuscript.

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Correspondence to Brian Swingle.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Beni Yoshida and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Xu, S., Swingle, B. Accessing scrambling using matrix product operators. Nat. Phys. 16, 199–204 (2020).

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