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  • Open Access

Flow to Nishimori Universality in Weakly Monitored Quantum Circuits with Qubit Loss

Malte Pütz1, Romain Vasseur2, Andreas W.W. Ludwig3, Simon Trebst1, and Guo-Yi Zhu4,*

  • *Contact author: guoyizhu@hkust-gz.edu.cn
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PRX Quantum 6, 040372 – Published 26 December, 2025

DOI: https://doi.org/10.1103/ygfz-crvp

Abstract

In circuit-based quantum state preparation, qubit loss and coherent errors are circuit imperfections that imperil the formation of long-range entanglement beyond a certain threshold. The critical theory at the threshold is a continuous entanglement transition known to be described by a (2+0)-dimensional nonunitary conformal field theory which, for the two types of imperfections of certain circuits, is described by either percolation or Nishimori criticality, respectively. Here we study the threshold behavior when the two types of errors simultaneously occur and show that, when moving away from the Clifford regime of projective stabilizer measurements, the percolation critical point becomes unstable and the critical theory flows to Nishimori universality. We track this critical renormalization group (RG) crossover flow by mapping out the entanglement phase diagrams, parametrized by the probability and strength of random weak measurements, of two dual protocols preparing surface code or GHZ-class cat states from a parent cluster state via constant-depth circuits. Extensive numerical simulations, using hybrid Gaussian fermion and tensor network/Monte Carlo sampling techniques on systems with more than a million qubits, demonstrate that an infinitesimal deviation from the Clifford regime leads to a sudden, strongly nonmonotonic entanglement growth at the incipient nonunitary RG flow. We argue that spectra of scaling dimensions of both the percolation and Nishimori fixed points exhibit multifractality. For percolation, we provide the exact (nonquadratic) multifractal spectrum of exponents, while for the Nishimori fixed point we show high-precision numerical results for five leading exponents characterizing multifractality.

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In the quest to build robust quantum computers, it is essential that macroscopically entangled quantum states withstand common errors like qubit loss and imperfect measurements. When both error types strike together, multiple thresholds and associated phase transitions compete to give rise to universal critical phenomena.

In this paper, the authors show that even a tiny deviation due to imperfect measurement from the ideal “Clifford” regime—a special, simpler class of quantum operations—triggers a dramatic shift in the system’s behavior. The critical point, originally described by a geometric percolation model, becomes unstable and immediately flows to a different universal class of criticality known as “Nishimori criticality,” which is argued to be an intrinsic fixed point for monitored quantum critical points. Using extensive numerical simulations on systems with over a million qubits, the researchers tracked this renormalization group flow and revealed a surprising, nonmonotonic surge in entanglement along the critical line. They further showed that both critical theories exhibit a rich “multifractal” structure, implying that an infinite spectrum of scaling dimensions emerges from the inherent randomness of quantum measurements.

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PRX Quantum 6, 040372– Published 26 December, 2025
  • Received 25 July 2025
  • Revised 8 October 2025
  • Accepted 19 November 2025
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