Monitored Quantum Circuits

• Monitored quantum circuits are a non-equilibrium framework combining discrete unitary operations with probabilistic measurements to induce entanglement phase transitions.
• They exhibit a transition from volume-law to area-law entanglement, characterized by critical exponents (e.g., ν ≈ 4/3 in 1D) and classical statistical mappings.
• These circuits serve as dynamic quantum error-correcting codes, enabling measurement-induced teleportation and advanced noise diagnostics in synthetic quantum matter.

Monitored quantum circuits are a non-equilibrium framework in which unitary quantum dynamics is interleaved with in-circuit projective measurements or weak measurements. This hybrid dynamics gives rise to rich information-theoretic phenomena, including entanglement phase transitions, emergent error-correcting properties, unconventional scaling laws, and symmetry-protected phases that have no closed-system analog. Their study has become central to the theory of synthetic quantum matter, quantum information processing, and the characterization of noise and decoherence in realistic quantum devices.

1. Foundations and Universal Models

Monitored quantum circuits (MQCs) consist of qubits (or qudits) arranged on a lattice, with time evolution in discrete steps. Each time step typically comprises a layer of local unitary gates, often sampled from Haar-random, Clifford, or circular ensembles (CUE/COE/CSE), followed by a layer in which each qubit (or small subset) is measured projectively with probability $p$ or undergoes weak measurement with continuous strength (Li et al., 2021, Sang et al., 2022). The outcome is a quantum trajectory, and ensemble characteristics emerge from averaging over both the unitary randomness and measurement outcomes.

The measurement rate $p$ is a key control parameter, tuning the system from predominantly quantum-coherent, high-entanglement dynamics (volume-law entangled phase) to measurement-dominated, locally-decohered dynamics (area-law or trivial phase). The precise scheme—single-qubit versus multi-qubit measurements, measurement bases, adaptive feedback—strongly influences qualitative behavior (Lira-Solanilla et al., 20 Dec 2024).

2. Entanglement and Measurement-Induced Phase Transitions

One of the hallmark phenomena in MQCs is the measurement-induced entanglement phase transition (MIPT). As $p$ increases, entanglement scaling in the steady state changes from volume-law ($S_A \propto |A|$) to area-law ($S_A=O(1)$), with critical scaling at a threshold $p_c$ (Li et al., 2021, Kalsi et al., 2022). The transition is characterized by critical exponents—correlation length $\xi \sim |p-p_c|^{-\nu}$ with $\nu \approx 4/3$ (1D), and dynamic scaling that can mimic conformal invariance at criticality.

In many cases, the entanglement structure and subleading corrections are mapped to classical statistical mechanics problems: specifically, the dynamics of entanglement domain walls, whose statistics correspond to directed polymers in a random environment (KPZ/DPRE universality) (Li et al., 2021). For example, the roughness exponent $\zeta=2/3$ and free-energy fluctuation exponent $\chi=1/3$ control subleading entropy growth and error-correction properties (Li et al., 2021, Sang et al., 2022).

In monitored circuits with additional symmetries—such as $U(1)$ charge conservation or dipole conservation—not only does the MIPT persist, but new sharpness or sharpening transitions emerge, separating volume-law phases whose dynamics differ in how quickly global conserved quantities can be inferred from measurement outcomes (e.g., charge-sharp/fuzzy, dipole-sharp/fuzzy) (Agrawal et al., 2021, Zerba et al., 16 Dec 2025, Barratt et al., 2021). Dynamically, these can exhibit emergent Lorentz invariance ($z=1$) or quantum Lifshitz scaling ($z=2$), with subleading entanglement corrections proportional to $\log |A|$ or to system size (Agrawal et al., 2021, Klocke et al., 3 Sep 2024).

3. Operator Dynamics and Information Spreading

MQCs exhibit information dynamics that transcend conventional Lieb-Robinson bounds. In the weakly monitored regime ($p < p_c$), entanglement and information can propagate faster than ballistically. Specifically, the stabilizer length distribution for Clifford circuits grows as $h(\ell, t) \sim \ell^{-5/3} \exp[-\ell/t^{3/2}]$, and the front of information spreading (as measured by mutual information or operator support) advances as $t^{3/2}$, rather than linearly. Two distant regions can be entangled in time $t \sim \ell^{2/3}$—sublinear in separation (Sang et al., 2022).

This ultrafast propagation is due to measurement-induced teleportation: local measurements collapse and reestablish correlations nonlocally, enabling super-ballistic growth of code distance and operator support (Sang et al., 2022, Shkolnik et al., 5 Nov 2024). At disorder-dominated critical points (e.g., temporally fluctuating measurement rates), one can observe “infinitely fast” (activated) scaling—$\log x \sim t^{\psi_\tau}$—at criticality, and temporal Griffiths phases (Shkolnik et al., 5 Nov 2024).

4. Error Correction and Coding Theory

The volume-law phase in MQCs is more than just delocalized—it acts as a dynamical quantum error-correcting code (QECC). The code distance—the minimal region size required to destroy logical information—scales sublinearly with system size, typically as $d_{\mathrm{code}} \sim L^{1/3}$ (in 1D) (Li et al., 2021, Yoshida, 2021). The structure of such a code can be mapped to a classical linear code associated with the commutation relations of local operators and prior measurement history. Recoverability (hence entanglement) is linked to successful decoding in this dual code (Yoshida, 2021).

In “randomly monitored quantum codes,” rigorous necessary and sufficient conditions for information destruction under random Pauli measurements are established: logical information survives unless a nontrivial logical is measured. In topological codes (toric/color), measurement thresholds can reach $p_m^{\mathrm{th}}=1$ under random single-qubit measurements, exceeding the erasure threshold ($p_e^{\mathrm{th}}\leq 1/2$) (Lee et al., 31 Jan 2024). This robustness allows for measurement-induced teleportation of logical qubits and connects to holographic properties in AdS/CFT (Lee et al., 31 Jan 2024).

5. Symmetries, Topology, and Protected Phases

Symmetry plays a crucial role in shaping monitored circuit phases. Local conservation laws (Abelian/non-Abelian) fragment circuit dynamics, leading to distinct scaling of entanglement and purification (e.g., the emergence of sharp vs. fuzzy charge/dipole phases, Kosterlitz-Thouless–type criticality) (Agrawal et al., 2021, Barratt et al., 2021, Zerba et al., 16 Dec 2025). In monitored Kitaev and Yao–Kivelson circuits, measurement-only limits admit Majorana loop model mappings, with clear area-law, $L\ln L$, and critical phases classified by symmetry class (BDI/D). These admit efficient numerical simulation on unprecedented system scales and are described by nonlinear sigma models with quantum Lifshitz scaling (Klocke et al., 3 Sep 2024, Vijayvargia et al., 20 Sep 2025).

Non-Hermitian topology is accessible in MQCs by emulating, e.g., the non-Hermitian Su–Schrieffer–Heeger chain via measurement-induced anti-Hermitian terms; monitored dynamics provide direct access to biorthogonal topological invariants and physical realization of non-Hermitian phases without engineered external baths (Fleckenstein et al., 2022).

Multipartite entanglement structure, quantified by quantum Fisher information (QFI), reveals that—without symmetry or protection—even at the bipartite critical point no extensive multipartite entanglement is present (QFI density remains bounded) (Lira-Solanilla et al., 20 Dec 2024). Introducing two-qubit measurement schemes with global $Z_2$ symmetry allows stabilization of genuine macroscopic cate states (GHZ), with QFI scaling as $L$ and critical exponents set by percolation universality (Lira-Solanilla et al., 20 Dec 2024).

6. Monitoring, Noise, and Pragmatic Aspects

Monitored circuit architectures have inspired new methodologies for error diagnostics and device characterization. Continuous monitoring frameworks for NISQ devices extract Markovian noise channel estimates from arbitrary user circuit outputs using tensor network simulation and Riemannian optimization, outperforming standard calibration and benchmarking in resource usage and predictive accuracy (Zolotarev et al., 2022).

Advanced debugging protocols—such as QMON—insert mid-circuit measurement and reset operations while reconstructing logical pathways to guarantee output-state preservation (fidelity-1), enabling effective bug localization and runtime monitoring with minimal disturbance (Ma et al., 15 Dec 2025). Fidelity and coverage remain high unless circuit-wide entanglement prohibits in-circuit monitoring.

Symmetry-protected phases in the presence of noise can be resilient: feedback and postselection can restore absorbing/charge-sharp phases, and symmetry-based benchmarking provides a scalable, macroscopic alternative to randomized benchmarking (Ivaki et al., 18 Oct 2024). Analytical noise models show coherent symmetry-breaking turns sharp transitions into crossovers, but essential features persist far from phase boundaries (Ivaki et al., 18 Oct 2024).

7. Postselection, Rare Events, and Efficient Protocols

Postselection—conditioning on rare patterns of measurement outcomes—is vital for probing high-order correlations and rare-event statistics in MQCs. For light-cone dynamical correlators, there exist efficient “rare-to-typical” mappings: by tilting the measurement-weighted channel with counting fields and applying a generalized Doob transform, one obtains an auxiliary unitary circuit whose typical correlations reproduce the postselected correlations of the original monitored circuit, dramatically reducing computational cost (Li et al., 23 Aug 2024). This rare–typical correspondence relies on symmetry and explicit channel structure and generalizes to higher-order and more general dynamical observables.

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References:

• (Li et al., 2021) Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment
• (Agrawal et al., 2021) Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits
• (Barratt et al., 2021) Field theory of charge sharpening in symmetric monitored quantum circuits
• (Yoshida, 2021) Decoding the Entanglement Structure of Monitored Quantum Circuits
• (Kalsi et al., 2022) Three-fold way of entanglement dynamics in monitored quantum circuits
• (Sang et al., 2022) Ultrafast Entanglement Dynamics in Monitored Quantum Circuits
• (Lee et al., 31 Jan 2024) Randomly Monitored Quantum Codes
• (Zolotarev et al., 2022) Continuous monitoring for noisy intermediate-scale quantum processors
• (Lira-Solanilla et al., 20 Dec 2024) Multipartite entanglement structure of monitored quantum circuits
• (Shkolnik et al., 5 Nov 2024) Infinitely fast critical dynamics: Teleportation through temporal rare regions in monitored quantum circuits
• (Ma et al., 15 Dec 2025) QMon: Monitoring the Execution of Quantum Circuits with Mid-Circuit Measurement and Reset
• (Ivaki et al., 18 Oct 2024) Noise resilience in adaptive and symmetric monitored quantum circuits
• (Li et al., 23 Aug 2024) Efficient post-selection in light-cone correlations of monitored quantum circuits
• (Vijayvargia et al., 20 Sep 2025) Error stabilized logical qubits in qudit generalizations of the monitored Kitaev model
• (Klocke et al., 3 Sep 2024) Entanglement dynamics in monitored Kitaev circuits: loop models, symmetry classification, and quantum Lifshitz scaling
• (Fleckenstein et al., 2022) Non-Hermitian topology in monitored quantum circuits
• (Zerba et al., 16 Dec 2025) Strong-to-weak symmetry breaking in monitored dipole conserving quantum circuits
• (Bulchandani et al., 2023) Random-matrix models of monitored quantum circuits