Entanglement Transitions in Random Circuits

Characterizing entanglement phase transitions in qudit circuits.

Duration : April 2024 - April 2025
Supervisor : Prof. Dr. Frank Pollmann
Institute : Technical University of Munich

Here you can find the complete thesis published on the TUM library database, as well as a poster summarizing it.

Abstract

Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution [1]. Bipartite entanglement measures, such as Rényi entropies and mutual information, can be used to characterize the dynamical phases that emerge in these systems [2].

Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions [3], with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators.

In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism, and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions [4, 5]. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.

References:

1. M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annual Review of Condensed Matter Physics 14, 335 (2023).
2. Y. Li, X. Chen, and M. P. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Physical Review B 100, 134306 (2019).
3. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Physical Review X 7, 031016 (2017).
4. D. Gottesman, Fault-tolerant quantum computation with higher-dimensional systems, in NASA International Conference on Quantum Computing and Quantum Communications (Springer, 1998) pp. 302–313.
5. M. Heinrich, On stabiliser techniques and their application to simulation and certification of quantum devices, Ph.D. thesis, Universit¨at zu K¨oln (2021).
6. A. Lavasani, Y. Alavirad, and M. Barkeshli, Topological order and criticality in (2+ 1) d monitored random quantum circuits, Physical review letters 127, 235701 (2021).

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