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arXiv:2401.01593 · Physical Review Letters

Entanglement Structure and Information Protection in Noisy Hybrid Quantum Circuits

Shuo Liu, Ming-Rui Li, Shi‐Xin Zhang, Shao-Kai Jian

Published Jun 13, 2024• 32 citations
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Abstract

In the context of measurement-induced entanglement phase transitions, the influence of quantum noises, which are inherent in real physical systems, is of great importance and experimental relevance. In this Letter, we present a comprehensive theoretical analysis of the effects of both temporally uncorrelated and correlated quantum noises on entanglement generation and information protection. This investigation reveals that entanglement within the system follows q^{-1/3} scaling for both types of quantum noises, where q represents the noise probability. The scaling arises from the Kardar-Parisi-Zhang fluctuation with effective length scale L_{eff}∼q^{-1}. More importantly, the information protection timescales of the steady states are explored and shown to follow q^{-1/2} and q^{-2/3} scaling for temporally uncorrelated and correlated noises, respectively. The former scaling can be interpreted as a Hayden-Preskill protocol, while the latter is a direct consequence of Kardar-Parisi-Zhang fluctuations. We conduct extensive numerical simulations using stabilizer formalism to support the theoretical understanding. This Letter not only contributes to a deeper understanding of the interplay between quantum noises and measurement-induced phase transition but also provides a new perspective to understand the effects of Markovian and non-Markovian noises on quantum computation.

DOI
10.1103/physrevlett.132.240402

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In the context of measurement-induced entanglement phase transitions, the influence of quantum noises, which are inherent in real physical systems, is of great importance and experimental relevance. In this Letter, we present a comprehensive theoretical analysis of the effects of both temporally uncorrelated and correlated quantum noises on entanglement generation and information protection. This investigation reveals that entanglement within the system follows q^{-1/3} scaling for both types of quantum noises, where q represents the noise probability. The scaling arises from the Kardar-Parisi-Zhang fluctuation with effective length scale L_{eff}∼q^{-1}. More importantly, the information protection timescales of the steady states are explored and shown to follow q^{-1/2} and q^{-2/3} scaling for temporally uncorrelated and correlated noises, respectively. The former scaling can be interpreted as a Hayden-Preskill protocol, while the latter is a direct consequence of Kardar-Parisi-Zhang fluctuations. We conduct extensive numerical simulations using stabilizer formalism to support the theoretical understanding. This Letter not only contributes to a deeper understanding of the interplay between quantum noises and measurement-induced phase transition but also provides a new perspective to understand the effects of Markovian and non-Markovian noises on quantum computation.
Measurement-induced phase transitions (MIPTs) have attracted increasing attention due to the rich phenomenology of entanglement structures and their relation with quantum information processing. Since physical systems are unavoidably coupled to environment, quantum noise, which can qualitatively modify or even destroy certain entanglement structure, needs to be considered in analyzing a system with MIPT. In this Letter, we investigate the effect of quantum noise modeled by a reset quantum channel acting on each site with a probability qqq on MIPT. Based on the numerical results from Clifford circuits, we show that the quantum noise can qualitatively change the entanglement properties---the entanglement obeys ``area law'' instead of ``volume law'' with a measurement rate p<pcp<{p}_{c}p<pc​. In the noise-induced area law phase, the entanglement exhibits a novel ${q}^{\ensuremath{-}1/3}$ power-law scaling. Using an analytic mapping from the quantum model to a classical statistical model, we further show that the area law entanglement is the consequence of noise-driven symmetry-breaking field, and the ${q}^{\ensuremath{-}1/3}$ scaling can be understood as the result of Kardar-Parisi-Zhang fluctuations of directed polymer with an effective length scale ${L}_{\mathrm{eff}}\ensuremath{\sim}{q}^{\ensuremath{-}1}$ in a random environment.
Noisy monitored quantum circuits have emerged as a versatile and unifying framework connecting quantum many-body physics, quantum information, and quantum computation. In this review, we provide a comprehensive overview of recent advances in understanding the dynamics of such circuits, with an emphasis on their entanglement structure, information-protection capabilities, and noise-induced phase transitions. A central theme is the mapping to classical statistical models, which reveals how quantum noise reshapes dominant spin configurations. This framework elucidates universal scaling behaviors, including the characteristic q−1/3q^{-1/3}q−1/3 entanglement scaling with noise probability qqq and distinct timescales for information protection. We further highlight a broad range of constructions and applications inspired by noisy monitored circuits, spanning variational quantum algorithms, classical simulation methods, mixed-state phases of matter, and emerging approaches to quantum error mitigation and quantum error correction. These developments collectively establish noisy monitored circuits as a powerful platform for probing and controlling quantum dynamics in realistic, decohering environments.
The presence of quantum noises inherent to real physical systems can strongly impact the physics in hybrid quantum circuits with local random unitaries and midcircuit measurements. The quantum noises with a size-independent occurring probability can lead to the disappearance of a measurement-induced entanglement phase transition and the emergence of a single area-law phase. In this work, we investigate the effects of quantum noises with size-dependent probabilities $q=p/{L}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ represents the scaling exponent. We have identified a noise-induced entanglement phase transition from a volume law to a power (area) law in the presence (absence) of measurements as ppp increases when $\ensuremath{\alpha}=1$. With the help of an effective statistical model, we reveal that the phase transition is of first order arising from the competition between two types of spin configurations and shares the same analytical understanding as the noise-induced coding transition. This unified picture further deepens the understanding of the connection between entanglement behavior and the capacity of information protection. When $\ensuremath{\alpha}\ensuremath{\ne}1$, one spin configuration always dominates regardless of ppp and thus the phase transition disappears. Moreover, we highlight the difference between the effects of size-dependent bulk noise and boundary noises. We validate our analytical predictions with extensive numerical results from stabilizer circuit simulations.
Generic many-body systems coupled to an environment lose their quantum entanglement due to decoherence and evolve to a mixed state with only classical correlations. Here, we show that measurements can stabilize quantum entanglement within open quantum systems. Specifically, in random unitary circuits with dephasing at the boundary, we find both numerically and analytically that projective measurements performed at a small nonvanishing rate result in a steady state with an L^{1/3} power-law scaling entanglement negativity within the system. Using an analytical mapping to a statistical mechanics model of directed polymers in a random environment, we show that the power-law negativity scaling can be understood as Kardar-Parisi-Zhang fluctuations due to the random measurement locations. Further increasing the measurement rate leads to a phase transition into an area-law negativity phase, which is of the same universality as the entanglement transition in monitored random circuits without decoherence.
We study theoretically entanglement and operator growth in a spin system coupled to an environment, which is modeled with classical dephasing noise. Using exact numerical simulations we show that the entanglement growth and its fluctuations are described by the Kardar-Parisi-Zhang equation. Moreover, we find that the wavefront in the out-of-time ordered correlator (OTOC), which is a measure for the operator growth, propagates linearly with the butterfly velocity, and broadens diffusively with a diffusion constant that is larger than the one of spin transport. The obtained entanglement velocity is smaller than the butterfly velocity for finite noise strength, yet both of them are strongly suppressed by the noise. We calculate perturbatively how the effective timescales depend on the noise strength, both for uncorrelated Markovian and for correlated non-Markovian noise.
The interplay of entanglement, measurement, and noise in near-term quantum devices may lead to novel emergent phenomena. This work presents a theoretical framework to understand collective effects in the dynamics of quantum entanglement and information, using the tools of statistical mechanics. The new effective description of generic quantum circuits lends insight into a measurement-induced phase transition in the information content of the system and points toward novel schemes to identify this transition in experiments.