Emergent statistical mechanics of entanglement in random unitary circuits

By Tianci Zhou, A. Nahum. Physical review B, 2018

251 citations recorded by the source corpus.

We map the dynamics of entanglement in random unitary circuits, with finite on-site Hilbert space dimension $q$, to an effective classical statistical mechanics. We demonstrate explicitly the emergence of a `minimal membrane' governing entanglement growth, which in 1+1D is a directed random walk in spacetime (or a variant thereof). Using the replica trick to handle the logarithm in the definition of the $n$th R\'enyi entropy $S_n$, we map the calculation of the entanglement after a quench to a problem of interacting random walks. For the second R\'enyi entropy, $S_2$, we are able to take the replica limit explicitly. This gives a mapping between entanglement growth and a directed polymer in a random medium at finite temperature (confirming KPZ scaling for entanglement growth in generic noisy systems). We find that the entanglement growth rate (`speed') $v_n$ depends on the R\'enyi index $n$, and we calculate $v_2$ and $v_3$ in an expansion in the inverse local Hilbert space dimension, $1/q$. These rates are determined by the free energy of a random walk, and of a bound state of two random walks, respectively, and include contributions of `energetic' and `entropic' origin. We give a combinatorial interpretation of the Page-like subleading corrections to the entanglement at late times and discuss the dynamics of the entanglement close to and after saturation. We briefly discuss the application of these insights to time-independent Hamiltonian dynamics.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/42249b2ea700fa766aa681a1f59c89bf2ad77d64

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Measurement-Induced Phase Transitions in the Dynamics of Entanglement (2018, methodology): “If the minimal-cut formula holds exactly in the limit q → ∞ for random unitary dynamics plus measurements, as it does for random unitary dynamics at infinite q [9,18] and for tensor networks with infinite bond dimension [28], then taking q → ∞ is an alternative way to obtain the effective classical ”
Unitary designs from statistical mechanics in random quantum circuits. (2019, methodology): “These techniques for random circuits were studied in [12,16], and we make extensive use of the ideas developed there.”
Unitary circuits of finite depth and infinite width from quantum channels (2019, background): “This observation is implicit in the fact that different Sn have different growth rates [18].”
Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions (2021, background): “[13] Tianci Zhou and Adam Nahum, “Emergent statistical mechanics of entanglement in random unitary circuits,” Phys.”
Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, methodology): “[35], for Haar circuits in [33], and for circuits with measurement in Refs.”
Measurement-driven entanglement transition in hybrid quantum circuits (2019, background): “cal model? A simplification occurs in the random Haar circuit with large onsite Hilbert space dimension q →∞, which can be mapped to an effective spin model [59], again described in terms of bond percolation.”
Statistical mechanics of quantum error correcting codes (2020, background): “We note that similar scaling behaviors have been found in random unitary circuits without measurements [26– 28, 31].”
Entanglement Membrane in Chaotic Many-Body Systems (2019, background): “The additional structure we examine is instead associated with pairings between spacetime histories in the ‘worlds’ associated with different layers [14].”
Random Quantum Circuits (2022, methodology): “Domain walls have an interesting combinatorial structure for N > 2: each domain wall is labelled by a permutation group element, and domain walls can split and recombine in accordance with the rules for composing permutations (113, 134, 95, 81).”
Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits (2021, background): “[68] DWs are restricted by unitarity to only make certain “turns” (See [10] for details).”