Noisy Monitored Quantum Circuits

• Noisy monitored quantum circuits are systems that combine random unitary gates, projective measurements, and local noise to study entanglement structure and information flow.
• They exhibit noise-induced transitions from volume-law to area-law entanglement, featuring universal scaling behaviors such as a KPZ-type q⁻¹ᐟ³ decay in mutual information.
• Their mapping to classical spin models enables efficient simulation, hardware diagnostics, and development of error mitigation strategies for NISQ devices.

Noisy monitored quantum circuits are quantum circuits that explicitly model both intrinsic quantum noise and active projective measurements interleaved with unitary gates. This framework unifies the study of quantum entanglement, quantum information flow, error correction, and practical computation in decohering many-body systems. Their dynamics are governed by a delicate interplay between information-scrambling random quantum gates, decohering quantum noise channels, and measurement-induced state collapse, producing a rich phenomenology that includes noise-induced phase transitions, novel mixed-state phases, information protection thresholds, and universal scaling behaviors. Noisy monitored circuits have become pivotal both as experimental testbeds for noisy intermediate-scale quantum (NISQ) devices and as theoretical models for understanding the statistical mechanics of open quantum systems (Liu et al., 21 Dec 2025).

1. Circuit Architecture and Noise Modeling

A standard noisy monitored quantum circuit comprises three components acting sequentially on a chain of $L$ qudits (or qubits):

• Random two-qudit unitaries (often Haar random or from the Clifford group) arranged in a brick-wall pattern.
• Projective (e.g., $Z$-basis) measurements applied to each qudit with independent probability $p_{m}$ per time step.
• Local noise channels (such as depolarizing, dephasing, or reset) applied independently with probability $q$ per qudit per layer.

A single layer has the structure: $$\cdots \xrightarrow{U_{t,1},U_{t,2},\ldots} \xrightarrow{\text{projective measurements (}p_{m})} \xrightarrow{\text{noise (}q\text{)}}$$ Each operation is formalized as a completely positive trace-preserving (CPTP) map. The explicit implementation of noise channels via Kraus operators and their decomposition into coherent and incoherent parts is crucial for capturing the effect of error types and symmetry breaking in real hardware (Ivaki et al., 18 Oct 2024).

2. Entanglement Scaling and Phase Transitions

The interplay of scrambling, monitoring, and decohering yields a sharp transition in entanglement structure—most prominently, the measurement-induced phase transition (MIPT):

• At low measurement rate $p_{m}$, random unitary gates dominate, stabilizing a volume-law entangled phase: $S(L)\sim s_\infty L$.
• Above a critical $p_{m}^c$, measurements dominate, yielding an area-law phase $S(L)\sim O(1)$ (Liu et al., 21 Dec 2025, Liu et al., 30 Jan 2024).
• In the presence of noise, the sharp MIPT is generically replaced by a crossover: bulk noise acts as a permutation symmetry-breaking field in the effective classical model, suppressing criticality and rounding the transition (Dias et al., 2022).

A universal noise-induced entanglement scaling emerges in the crossover regime: $I_{A:B}(q)\propto q^{-1/3}$ where $I_{A:B}$ is the bipartite mutual information and $q$ is the noise probability. This scaling arises from Kardar–Parisi–Zhang-type (KPZ) fluctuations in the domain wall structure of the statistical mechanics mapping (Liu et al., 21 Dec 2025).

Introducing size-dependent noise rates $q=p/L^{\alpha}$, with $\alpha=1$, restores true first-order phase transitions from volume- to area- (or subvolume-) law entanglement, accompanied by a discontinuous jump in order parameters such as mutual information and coding capacity. This phenomenon distinguishes bulk from boundary noise and reveals a profound connection to coding transitions and information-protection thresholds (Liu et al., 30 Jan 2024).

3. Classical Statistical Mechanics Mappings

Central to the theoretical understanding of noisy monitored circuits is their mapping to classical statistical mechanics problems:

• The average Rényi entropy or mutual information is recast as the free energy of classical spin models with domain walls (permutation spin models or Potts models).
• Quantum noise is mapped to an explicit "magnetic field" term that favors the identity permutation, breaking the spontaneous symmetry responsible for entangled phases (Dias et al., 2022).
• Measurements act as random pinning potentials for domain walls.
• This statistical mapping predicts universal scaling forms, finite-size crossovers, and the critical exponents of the transition.

The effective Hamiltonian includes contributions from domain wall tension, noise-induced symmetry-breaking fields, and KPZ interface fluctuation terms. The structure and scaling of these terms determine the observed entanglement and information phases (Liu et al., 21 Dec 2025).

4. Multipartite Entanglement and Mixed-State Phases

Noisy monitored circuits can host phases characterized by multipartite entanglement and mixed-state order:

• In generic (unstructured) monitored random circuits, even at criticality, the quantum Fisher information (QFI) density remains bounded, and no genuine macroscopic multipartite entanglement is observed (Lira-Solanilla et al., 20 Dec 2024).
• Special constructions that incorporate symmetry protection (e.g., global $\mathbb{Z}_2$ parity) or protected two-site measurements (such as $\sigma^x_i \sigma^x_{i+1}$) can realize GHZ-like phases with extensive QFI and critical scaling $f_Q\sim N^{1/3}$, provided the protection mechanism remains unbroken by the noise.
• The mixed-state analog of symmetry breaking, such as spontaneous strong-to-weak symmetry breaking (SW-SSB), characterizes fuzzy vs. sharp information-theoretic phases in monitored noisy circuits with strong global symmetries (Singh et al., 13 Mar 2025).
• Logarithmic negativity serves as an efficient entanglement monotone for quantifying genuine mixed-state entanglement in circuits subject to both monitoring and decoherence (Anzai et al., 17 Jul 2025).

Spatial disorder in measurement or gate error rates alters the universality class of purification transitions, consistent with the Harris criterion, and gives rise to new forms of short-range entangled, "pure-like" phases in the presence of quasi-periodic or random spatial modulation (Anzai et al., 17 Jul 2025).

5. Information Protection, Coherent Information, and Noise-Induced Transitions

Noisy monitored circuits provide a natural setting for exploring quantum channel capacity, information protection, and error correction:

• The dynamical transition between recoverable (positive coherent information) and irrecoverable (negative coherent information) phases is controlled by the balance between noise and quantum-enhanced operations. This transition is first-order and features nontrivial finite-size scaling consistent with a biased 2D Ising model (Qian et al., 29 Aug 2024).
• Coding transitions, characterized by a sudden loss of mutual information between a region and a reference, coincide with sharp entanglement transitions at specific system-size-dependent noise rates (Liu et al., 30 Jan 2024).
• Protocols for characterizing quantum capacity transitions exploit cross-entropy-like proxies and avoid post-selection, enabling benchmarking of information retention in realistic NISQ hardware (Qian et al., 29 Aug 2024).
• Timescales for information protection depend on the temporal and spatial structure of noise; for uncorrelated noise, the protection time obeys $t_{\text{protect}}\sim q^{-1/2}$, while for temporally correlated noise, $t_{\text{protect}}\sim q^{-2/3}$ (Liu et al., 21 Dec 2025).

6. Classical Simulation, Complexity, and Quantum Advantage

The area-law regime induced by monitoring and decoherence greatly enhances the classical simulability of noisy quantum circuits:

• Trajectory sampling techniques (unravelings) and tensor network (TEBD, MPO, MPS) approaches efficiently capture the dynamics up to critical noise strengths and circuit depths (Cheng et al., 2023).
• Efficient sampling of noisy shallow circuits is possible up to critical depth $T_c(\varepsilon)$ determined by the noise rate and system geometry, with practical thresholds matching current device parameters ($T_c \simeq 5$ iSWAP layers for $\varepsilon\approx0.02$) (Cheng et al., 2023).
• Below the entanglement/complexity transition, the output distribution is efficiently accessible and the circuit is classically tractable; above it, classical sampling costs become exponential in system size and depth.

These findings quantify the requirements for demonstrating genuine quantum advantage in the presence of noise and inform circuit design for near-term experiments.

7. Noise Diagnosis, Mitigation, and Practical Calibration

Noisy monitored circuits also enable hardware diagnostics and noise mitigation strategies:

• Continuous monitoring protocols infer gate and measurement noise parameters from arbitrary circuit workloads without requiring dedicated calibration runs (Zolotarev et al., 2022).
• Tools such as 2MC-OBPPP provide polynomial-time, hardware-agnostic pre-execution diagnostics for parameterized circuits, measuring noise robustness, trainability (barren plateaus), and expressibility. The method locates spatiotemporal noise hotspots in large-scale circuits, enabling targeted error suppression that is exponentially more efficient than global error-mitigation strategies (Shao et al., 14 Sep 2025).
• Adaptive protocols and postselection can stabilize symmetry-protected phases and absorbent-state transitions against symmetry-breaking noise, permitting robust order parameters and direct fidelity benchmarking even in the presence of strong decoherence (Ivaki et al., 18 Oct 2024).

Key Topic	Primary Reference(s)	Highlighted Contributions
Classical-statistics mapping	(Liu et al., 21 Dec 2025, Dias et al., 2022)	Mapping to spin models, symmetry breaking by noise
$q^{-1/3}$ scaling, KPZ	(Liu et al., 21 Dec 2025)	Universal scaling of mutual information with noise
Noise-induced MIPT	(Liu et al., 30 Jan 2024, Dias et al., 2022)	Volume/area-law transition, coding transition
Mixed-state entanglement	(Lira-Solanilla et al., 20 Dec 2024, Singh et al., 13 Mar 2025)	GHZ protection, multipartite QFI, SW-SSB
Coherent information PT	(Qian et al., 29 Aug 2024)	Channel capacity transitions, post-selection-free probe
Efficient simulation	(Cheng et al., 2023)	Noisy-SEBD, MPS/MPO sampling, noisy complexity phase
Hardware noise monitoring	(Zolotarev et al., 2022, Shao et al., 14 Sep 2025)	Continuous calibration, noise-robustness metrics
Symmetry-protected order	(Ivaki et al., 18 Oct 2024)	Restoration via adaptive feedback or postselection
Spatial disorder	(Anzai et al., 17 Jul 2025)	Harris criterion, universality class shift

Noisy monitored quantum circuits thus serve as a central platform for probing the structure of quantum many-body dynamics in realistic settings, guiding both theoretical developments and experimental strategies for control, diagnostics, and information protection in open quantum systems.