Noisy Monitored Quantum Circuits

TL;DR

This work reviews noisy monitored quantum circuits as a unifying framework linking quantum dynamics, information protection, and computation through a mapping to higher-dimensional classical statistical models. It highlights the universal $q^{-1/3}$ entanglement scaling, KPZ-type domain-wall fluctuations, and distinct information-protection timescales under temporally correlated vs uncorrelated noise, all within a cohesive statistical-model picture. The review also details noise-induced phase transitions (entanglement, coding, and complexity) and explores broad applications in variational algorithms, error mitigation, and mixed-state physics, underscoring practical pathways for robust quantum technologies. Together, these results position noisy monitored circuits as a versatile platform for understanding and harnessing decoherence in complex quantum systems and devices.

Abstract

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 $q^{-1/3}$ entanglement scaling with noise probability $q$ 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.

Figures (14)

• Figure 1: Setup of noisy monitored quantum circuits. (a) Temporally uncorrelated and (b) correlated quantum noises (red circles). Projective measurements are indicated by green circles. In the steady state encoding scheme, after the system reaches its steady state, a reference qudit (blue circle) is introduced and maximally entangled with the central qudit to form a Bell pair, thereby encoding one-qudit quantum information into the circuit. Reprinted with permission from Ref. PhysRevLett.132.240402, Copyright (2024) by the American Physical Society.
• Figure 2: Outline of the mapping to the effective statistical model. Starting from the bulk circuit (left), averaging over the unitary gates in Eq. \ref{['eq:Haaraverage']} introduces two permutation spins, $\sigma$ and $\tau$, for each two-qubit gate (center). Vertical bonds are weighted by the Weingarten function in Eq. \ref{['eq:Wg']}, while diagonal bonds are weighted according to Eq. \ref{['eq:diagonalbond']} in the absence of measurements. When a measurement occurs, the corresponding diagonal bonds are removed. After integrating out the $\tau$ spins, we obtain a model with three-body weights on downward-facing triangles (right) in the absence of measurements; these three-body weights reduce to two-body weights when one of the bonds is measured. Reprinted with permission from Ref. PhysRevLett.129.080501, Copyright (2022) by the American Physical Society.
• Figure 3: $q^{-1/3}$ entanglement scaling in noisy monitored quantum circuits. (a) $I_{A:B}(q)$ and (b) $E_{N}(q)$ of the steady state of noisy monitored quantum circuits. The measurement probability is $p_{m} = 0.1<p_{m}^{c}$. A characteristic $q^{-1/3}$ scaling emerges for the entanglement in the presence of noise. The insets show the dependence of $I_{A:B}$ and $E_{N}$ on the system size $L$ for various noise rates. Without noise, the entanglement exhibits a volume-law scaling since $p_{m} < p_{m}^{c}$, whereas any finite noise probability drives the system into an area-law regime. Reprinted with permission from Ref. PhysRevB.107.L201113, Copyright (2023) by the American Physical Society.
• Figure 4: Statistical model understanding of entanglement structure and information protection timescale. The schematic dominant spin configurations are illustrated here. The upper panel corresponds to the entanglement-generation setup, while the lower panel depicts the information-protection setup, where an additional Bell pair is introduced at $(x_{0}, t_{0})$. The horizontal ($x$) and vertical ($y$) axes represent spatial and temporal directions, respectively. Different colors denote distinct permutation-spin configurations in the effective classical model. The symbols $N$ and $R$ mark the locations of quantum-noise events and the inserted Bell pair, respectively; for clarity, additional noise events inside the $\mathbb{I}$ domain are omitted. Panel (a) shows the dominant configuration contributing to $F^{(n,k)}_{A}$. The presence of a midpoint defect on the top boundary modifies the local length scale of the adjacent domain-wall segment. Panel (b) depicts the dominant configuration relevant for $F^{(n,k)}_{AB}$. Panels (c) and (d) show the dominant configurations for $F^{(n,k)}_{AB}$ and $F^{(n,k)}_{AB \cup R}$, respectively, where the permutation spin at $(x_{0}, t_{0})$ is fixed to $\mathbb{I}$ or $\mathbb{C}$ depending on the boundary condition imposed by the Bell-pair encoding. In the presence of projective measurements, the domain wall exhibits fluctuations around its noise-determined trajectory. Reprinted with permission from Ref. PhysRevLett.132.240402, Copyright (2024) by the American Physical Society.
• Figure 5: Measurement-induced KPZ entanglement scaling. (a) Noisy monitored quantum circuits with quantum noise located at the boundary. (b) The entanglement for the steady state scales as $L^{1/3}$. Reprinted with permission from Ref. PhysRevLett.129.080501, Copyright (2022) by the American Physical Society.