Entanglement dynamics in monitored Kitaev circuits: loop models, symmetry classification, and quantum Lifshitz scaling

TL;DR

This paper develops a framework to understand entanglement dynamics in monitored Kitaev circuits, revealing distinct phases and phase transitions characterized by quantum Lifshitz scaling and symmetry classification.

Contribution

It introduces Majorana loop models for symmetry classes BDI and D, providing analytical and numerical tools to study entanglement phases and criticality in monitored quantum circuits.

Findings

Identified localized and delocalized Majorana loop phases.

Discovered continuous phase transitions with quantum Lifshitz scaling.

Developed efficient simulation techniques for large circuits with up to 10^8 qubits.

Abstract

Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of non-equilibrium quantum matter. One principal example are measurement-induced phase transitions arising from non-unitary dynamics in monitored circuits, which employ mid-circuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of dynamics in generic circuits is still evolving, we contend that monitored quantum circuits yield robust phases of dynamic matter, which -- akin to Hamiltonian ground state phases -- can be categorized based on symmetries and spatial dimensionality. To illustrate this concept, we focus on quantum circuits within symmetry classes BDI and D, which are measurement-only adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time-reversal. We establish a general framework -- Majorana loop models -- for both symmetry classes to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of non-linear sigma models, but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to $10^8$ qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.