# Sequential Quantum Circuit

Entanglement in many-body quantum systems is notoriously hard to characterize due to the exponentially many parameters involved to describe the state. On the other hand, we are usually not interested in all the microscopic details of the entanglement pattern but only some of its global features. It turns out, quantum circuits of different levels of complexity provide a useful way to establish a hierarchy among many-body entanglement structures. A circuit of a finite depth generates only short range entanglement which is in the same gapped phase as an unentangled product state. A linear depth circuit on the other hand can lead to chaos beyond thermal equilibrium. We find that, to reach the interesting regime in between that contains nontrivial gapped orders, we need the Sequential Quantum Circuit — a circuit of linear depth but with each layer acting only on one subregion in the system. We showed how the Sequential Quantum Circuit can be used to generate nontrivial gapped states with long range correlation or long range entanglement, perform renormalization group transformation in foliated fracton order, and create defect excitations inside the bulk of a higher dimensional topological state. It provides a lattice interpretation of notions like gauging, duality, and topological defect (and its related higher category theory).

### Related Publications

- "Local Unitary Transformation, Long-Range Quantum Entanglement, Wave Function Renormalization, and Topological Order", Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen,
*Phys. Rev. B*82, 155138 (2010).

(This is an old work. It is the prequel of the Sequential Circuit project. It defines an equivalence relation between gapped ground states using finite depth quantum circuits.) - "Renormalization of Ising cage-net model and generalized foliation", Zongyuan Wang, Xiuqi Ma, David T. Stephen, Michael Hermele, Xie Chen,
*Phys. Rev. B*108, 035148 (2023).

(The renormalization group transformation of the Ising cage-net model uses in-plane sequential quantum circuit for each step of transformation.) - "Sequential Quantum Circuits as Maps between Gapped Phases", Xie Chen, Arpit Dua, Michael Hermele, David T. Stephen, Nathanan Tantivasadakarn, Robijn Vanhove, Jing-Yu Zhao,
*Phys. Rev. B***109**, 075116 (2024).

(Explicit construction of SQC to map between gapped (nonchiral) phases.) - "String operators for Cheshire strings in topological phases", Nathanan Tantivasadakarn, Xie Chen, arXiv:2307.03180.

(Shows that SQC can generate topological defects in toric code while the fusion of the defects is realized with finite depth circuit.) - "Sequential Adiabatic Generation of Chiral Topological States", Xie Chen, Michael Hermele, David T. Stephen, arXiv:2402.03433.

(Constructed a sequential adiabatic evolution process for generating chiral models like the integer quantum Hall and p+ip superconductor. Also demonstrated that coupling to discrete gauge field can be realized with a sequential circuit.) - "Fusion of one-dimensional gapped phases and their domain walls", David T. Stephen, Xie Chen, arXiv:2403.19068.

(Studied how 1D gapped phases fuse using explicit lattice models and quantum circuits, especially how fusion works in the presence of domain walls.)