Strong-to-weak symmetry breaking states in stochastic dephasing stabilizer circuits

TL;DR

This paper investigates strong-to-weak symmetry breaking in mixed quantum states under decoherence, using stabilizer formalism and numerical methods to analyze phase transitions and critical phenomena in quantum circuits.

Contribution

It introduces a scheme to describe SSSB phenomena in mixed states and demonstrates its application through numerical studies of 2D circuits and SPT order under decoherence.

Findings

Clear observation of SSSB phase transition in 2D stochastic Ising decoherence

Calculation of Rényi-2 correlations and critical exponents for SSSB transition

Insight into subgroup SSSB and percolation viewpoint for understanding decoherence effects

Abstract

Discovering mixed state quantum orders is an on-going issue. Recently, it has been recognized that there are (at least) two kinds of symmetries in the mixed state; strong and weak symmetries. Under symmetry-respective decoherence, spontaneous strong-to-weak symmetry breaking (SSSB) can occur. This work provides a scheme to describe SSSB and other decoherence phenomena in the mixed state by employing the stabilizer formalism and the efficient numerical algorithm of Clifford circuits. We present two systematic numerical studies.In a two-dimensional (2D) circuit with a stochastic Ising type decoherence, an SSSB phase transition is clearly observed and its criticality is elucidated by the numerical methods. In particular, we calculate R\'{e}nyi-2 correlations and estimate critical exponents of the SSSB transition. For the second system, we introduce an idea of subgroup SSSB. As an example, we study a system with symmetry-protected-topological (SPT) order provided by both one-form and zero-form symmetries, and observe how the system evolves under decoherence. After displaying numerical results, we show that viewpoint of percolation is quite useful to understand the SSSB transition, which is applicable for a wide range of decohered states. Finally, we comment on SSSB of one-form-symmetry exemplifying toric code.