Phase transitions in ( 2 + 1 ) D subsystem-symmetric monitored quantum circuits

Abstract

The interplay of unitary evolution and projective measurements is a modern interest in the study of many-body entanglement. On the one hand, the competition between these two processes leads to the recently discovered measurement-induced phase transition (MIPT). On the other hand, measurement-based quantum computation (MBQC) is a well-known computational paradigm where measurements simulate unitary evolution by utilizing the entanglement of special resources such as the two-dimensional (2D) cluster state. The entanglement properties enabling MBQC may be attributed to symmetry-protected topological (SPT) orders, particularly subsystem-symmetry-protected topological (SSPT) orders. It was recently found that the one-dimensional cluster state may be associated with an SPT phase in random circuits respecting a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, and furthermore that all phase transitions in this scenario belong to the same universality class. As resources with greater computational power feature greater symmetry, it is fruitful to investigate further any relationship between levels of symmetry in MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels of symmetry-respecting unitary evolution interspersed by measurements. Although we find two area-law phases and one volume-law phase with distinct entanglement structures for each ensemble, the phase transition from the volume-law phase to the area-law phase associated with the 2D SSPT cluster state has variable correlation length exponent $\ensuremath{\nu}$. Whereas $\ensuremath{\nu}\ensuremath{\approx}0.90$ for unconstrained Clifford unitaries and $\ensuremath{\nu}\ensuremath{\approx}0.83$ for globally symmetric Cliffords, subsystem-symmetric Cliffords feature a much smaller value $\ensuremath{\nu}\ensuremath{\approx}0.38$. We discuss how these distinct $\ensuremath{\nu}$'s quantify spacetime response scales where quantum information is manipulated by single-qubit measurements as in MBQC.