Disordered purification phase transition in hybrid random circuits

Noise is inevitable in realistic quantum circuits. It arises randomly in space. Inspired by the spatial nonuniformity of the noise, we investigate the effects of spatial modulation on quantum phase transitions in a hybrid random Clifford circuit with a mixed initial state. As an efficient observable for extracting quantum entanglement in mixed states, we employ many-body negativity. The behavior of the many-body negativity well characterizes the presence of the phase transitions and its criticality. We find the effect of spatial nonuniformity in measurement probability on the phase transition. The criticality of the phase transition changes from that of uniform probability, which is elucidated by the argument of the Harris criterion. The critical correlation length exponent <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>ν</a:mi> </a:math> changes from <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>ν</b:mi> <b:mo>&lt;</b:mo> <b:mn>2</b:mn> </b:mrow> </b:math> for uniform probability to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>ν</c:mi> <c:mo>&gt;</c:mo> <c:mn>2</c:mn> </c:mrow> </c:math> for spatially modulated probability. We further investigate a setting where a two-site random Clifford gate becomes spatially (quasi)modulated. We find that the modulation induces a phase transition, leading to a different pure phase where a short-range quantum entanglement remains.