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arXiv:2212.03901 · Physical review. B./Physical review. B

Universal Kardar-Parisi-Zhang scaling in noisy hybrid quantum circuits

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

Published May 22, 2023• 36 citations
arXivHTMLPDF

Abstract

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 $q$ 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 $p<{p}_{c}$. 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.

DOI
10.1103/physrevb.107.l201113

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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{q}^{\ensuremath{-}1/3}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{q}^{\ensuremath{-}1/3}q\ensuremath−1/3 scaling can be understood as the result of Kardar-Parisi-Zhang fluctuations of directed polymer with an effective length scale Leff\ensuremath∼q\ensuremath−1{L}_{\mathrm{eff}}\ensuremath{\sim}{q}^{\ensuremath{-}1}Leff​\ensuremath∼q\ensuremath−1 in a random environment.
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.
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.
Scrambling dynamics induced by random unitary gates can protect information from low-rate measurements, which underpins the phenomenon known as the measurement-induced phase transition (MIPT). However, typical decoherence noise disrupts the volume-law phase, complicating the observation of the MIPT on noisy intermediate-scale quantum devices. Here, we demonstrate that incorporating quantum-enhanced operations can effectively protect the MIPT from environmental noise, thereby enabling its detection in experiment. The transition is characterized by conditional entanglement entropy, which is associated with a statistical mechanics model wherein noise and quantum-enhanced operations act as competing external random fields. When the net external field is zero, a ferromagnetic-paramagnetic phase transition is expected, resulting in the MIPT. This zero-field condition also ensures an average apparatus-environment symmetry, making conditional entanglement entropy a valid probe of entanglement and establishing the transition as a genuine entanglement phase transition. Additionally, we provide numerical results that demonstrate the MIPT in a (2+1)-dimensional quantum circuit under dephasing noise. We also propose a method to estimate the noise rate, enabling the zero-field condition to be achieved experimentally and ensuring the feasibility of our protocol. Our result serves as a concrete example of the power of quantum enhancement in combating noise.
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.
We study entanglement fluctuations and quantum error correction in the weakly monitored volume-law phase of quantum automaton circuits subject to repeated local measurements. We numerically observe that the entanglement entropy exhibits strong fluctuation with the exponent close to the ``growth exponent'' of the Kardar-Parisi-Zhang (KPZ) universality class, the same as other local random circuits studied previously. We also investigate the dynamically generated quantum error correction code in the purification process and show that this model has different contiguous code distances for two types of errors that exhibit similar sublinear power-law scaling. We give an interpretation of these results by mapping them to various quantities in a classical particle model. We demonstrate that the subleading correction term of the entanglement entropy and the sublinear power-law scaling of the contiguous code distance in the volume-law phase are both the emergent phenomena of the hybrid random dynamics. Finally, we show that this classical particle dynamics itself has a type of error correction ability and can dynamically generate a classical linear code.
We investigate the distribution of entanglement entropy in hybrid quantum circuits consisting of random unitary gates and local measurements applied at a finite rate. We demonstrate that higher moments of the entanglement entropy distribution, such as the ratio between the variance and the mean and the skewness, capture nontrivial features of the measurement-induced dynamics that are invisible to the mean entropy alone. We demonstrate that these quantities exhibit distinct and robust behaviors across the volume-law and area-law phases, and can serve as effective diagnostics of measurement-induced entanglement transitions. We propose a phenomenological model describing the effect of measurements in the area-law regime, which, when combined with the directed polymer in a random environment description of the volume-law phase, well matches numerical simulations across the entire phase diagram.