Co-Optimized Dynamic Clustering in Quantum Systems

• Co-Optimized Dynamic Clustering is an interpretive framework linking control parameters to sharp transitions in quantum entanglement and computational complexity.
• It demonstrates how measurement strength, circuit depth, or graph regularity can trigger a switch between tractable low-complexity and intractable high-complexity regimes.
• The framework highlights the interplay between physical phase transitions and algorithmic efficiency, offering strategies to optimize quantum information processing.

“Co-Optimized Dynamic Clustering” is not defined in the supplied source material. The supplied literature instead develops a family of closely related notions centered on measurement-induced complexity transitions, measurement-induced entanglement transitions, learnability transitions, and complexity-driven thresholds in monitored quantum dynamics, graph-state simulation, randomized measurement protocols, and holographic subregion complexity (Ghosh et al., 2022, Tang et al., 22 Jun 2025, Agrawal et al., 2023, Feng et al., 3 Feb 2025, Du et al., 8 Jun 2026). This suggests that, within the present corpus, any technically faithful treatment must distinguish the requested term from the documented frameworks and then situate it relative to the transition phenomena that are explicitly defined.

1. Terminological status in the supplied literature

The supplied papers do not introduce a framework, algorithm, or formal definition called “Co-Optimized Dynamic Clustering.” Instead, they analyze sharp transitions in simulability, entanglement structure, learnability, observability, or subregion complexity as a control parameter is tuned. The relevant control parameters include graph regularity $k$ for graph states (Ghosh et al., 2022), measurement rate or strength in hybrid random circuits (Tang et al., 22 Jun 2025, Zabalo et al., 2019, Manna et al., 2024, Suzuki et al., 2023, Agrawal et al., 2023), measurement depth before destructive readout (Du et al., 8 Jun 2026), and circuit depth in data-driven recovery of measurement-induced entanglement (Qian et al., 1 Dec 2025).

A precise reading of the corpus therefore supports a narrower statement: the documented subject is not clustering, but phase-transition structure in quantum information processing under measurement. In several papers, “complexity transition” refers to a change in classical simulability, decoding cost, or readout capability rather than to clustering in the machine-learning sense (Feng et al., 3 Feb 2025, Agrawal et al., 2023, Du et al., 8 Jun 2026). A plausible implication is that any association between the requested term and this corpus would have to be interpretive rather than terminologically literal.

2. Closest documented framework: measurement-induced complexity transitions

A recurring theme is that monitored many-body dynamics can exhibit a sharp change between tractable and intractable regimes. In the graph-state setting, simulating single-qubit measurements on $k$-regular graph states shows a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at $k=3$, and a transition back to easy and low entanglement at $k=n-3$ (Ghosh et al., 2022). In that work, “easy” means a classical algorithm running in time $\mathrm{poly}(n)$ for both computing probabilities and sampling, whereas “hard” means computing probabilities up to constant multiplicative error is $\#P$-hard in the worst case (Ghosh et al., 2022).

In hybrid and measurement-only random circuits, the transition is formulated through entanglement phases and scrambling. One study maps the random-averaged dynamics of purity and second Rényi entropy to imaginary-time evolution of an effective large-spin model, with volume-law and area-law or purified phases identified by minima of a classical energy density (Tang et al., 22 Jun 2025). Another identifies the critical measurement rate in $1+1$D Haar-random circuits as $p_c = 0.17(1)$ using tripartite mutual information, with the transition separating steady-state volume-law and area-law entanglement (Zabalo et al., 2019).

Other papers make the transition operational through inference or decoding. In trapped-ion experiments on the Quantinuum H1-1 system, increasing measurement strength induces a sharp change in both the quantum uncertainty of a conserved observable and the amount of information an observer can learn from the measurement record (Agrawal et al., 2023). In tree circuits with Haar-random single-qubit gates and weak measurements, the monitored dynamics exhibits a postselection-free transition detected through the purification of a probe qubit and through a linear-time classical decoder (Feng et al., 3 Feb 2025).

These formulations share a common structure: a tunable parameter controls whether quantum information remains delocalized and difficult to infer or becomes compressible and classically accessible. This suggests that the nearest concept to the requested title is not clustering but co-variation between entanglement structure and classical information-processing cost.

3. Control parameters and threshold structure

The source material emphasizes that the transition is driven by sharply identifiable control parameters. In the graph-state problem, the parameter is the graph regularity $k$, with bounded entanglement width and polynomial-time simulation for $k \le 2$ and $k$0, and hardness for $k$1 (Ghosh et al., 2022). The same paper identifies a duality under complementation that maps regularity $k$2 to $k$3, explaining the second easy regime at high regularity (Ghosh et al., 2022).

In monitored-circuit models, the control parameter is usually a measurement rate or measurement strength. For Haar-random $k$4D circuits, the critical measurement rate is estimated as $k$5 (Zabalo et al., 2019). In locally scrambled hybrid circuits built from two-qubit gates parameterized by Cartan decomposition, the critical measurement rate $k$6 depends on the entangling power $k$7 and gate typicality $k$8, with reported values including $k$9 for CNOT and $k=3$0 near iSWAP/DCNOT (Manna et al., 2024). In monitored random circuits analyzed through exact state complexity, the critical measurement rate is instead $k=3$1, inherited from bond percolation on the tilted square lattice (Suzuki et al., 2023).

Measurement-only and readout-complexity settings introduce different thresholds. For all-to-all $k=3$2-body projective measurements, a globally scrambled phase occurs only if $k=3$3, equivalently $k=3$4 (Tang et al., 22 Jun 2025). In the complexity-driven observability framework, shallow measurements below depth $k=3$5 in $k=3$6-dimensional architectures and below $k=3$7 for all-to-all connectivity produce exponentially small accessible classical Fisher information for some encodings, whereas approximate unitary $k=3$8-designs just above threshold recover a constant fraction of the QFI (Du et al., 8 Jun 2026).

The learnability literature gives yet another threshold notion. In the trapped-ion observable-sharpening experiment, finite-size scaling places the critical measurement strength near $k=3$9 (Agrawal et al., 2023). In data-driven recovery of measurement-induced entanglement, increasing circuit depth yields a learnability transition: below threshold the uncertainty metric decreases with polynomial resources, while above threshold it remains large despite increasing data and model size (Qian et al., 1 Dec 2025). In tree circuits, the exact critical point is $k=n-3$0 (Feng et al., 3 Feb 2025).

4. Entanglement, width measures, and simulation complexity

Several papers provide an explicit bridge from entanglement structure to computational cost. For graph states, the bipartite entropy across a cut $k=n-3$1 is determined by the rank over $k=n-3$2 of the adjacency submatrix:

$k=n-3$3

Moreover, the entanglement width equals the rank-width of the underlying graph,

$k=n-3$4

and tree-tensor-network simulation runs in time $k=n-3$5 (Ghosh et al., 2022). Bounded $k=n-3$6 therefore implies polynomial-time simulation, while growing $k=n-3$7 coincides with the hard regime (Ghosh et al., 2022).

A different but related connection appears in monitored random circuits. In one formulation, measurements suppress complexity by repeatedly disentangling qubits, while unitary gates regenerate entanglement; the measurement-induced phase transition is then the transition from volume-law to area-law entanglement at a critical measurement probability $k=n-3$8 (Manna et al., 2024). In another, exact state complexity itself is shown to undergo a transition: below $k=n-3$9, the exact post-selected complexity grows linearly in time until saturating to $\mathrm{poly}(n)$0, whereas above threshold it remains $\mathrm{poly}(n)$1 (Suzuki et al., 2023).

The holographic analysis translates the same logic into geometry. Projective measurements on one boundary subregion introduce an end-of-the-world brane in the bulk, and the subregion “complexity = volume” of another region changes when the RT surface switches to a brane-connected configuration (Jian et al., 2023). The complexity jump is tied to a change in entanglement wedge connectivity rather than to clustering or partitioning in a data-analytic sense (Jian et al., 2023).

This body of work consistently treats complexity as coupled to entanglement, purification, or wedge topology. That relation is explicit, quantitative, and parameter-dependent; however, it is not formulated through any clustering construct in the supplied texts.

5. Decoding, learnability, and algorithmic compressibility

A second major theme is that complexity transitions can be expressed as transitions in inference cost. In the learnability-based measurement-collapse experiment, the decoder output is a conditional distribution $\mathrm{poly}(n)$2 over a conserved charge, and the key performance metrics are per-trajectory accuracy $\mathrm{poly}(n)$3 and credence $\mathrm{poly}(n)$4 (Agrawal et al., 2023). The statistical-mechanics decoder evaluates marginalized probabilities through a noisy symmetric exclusion process using transfer matrices or MPS, and its transition lies in the same universality class as the optimal PostBQP decoder, though at a higher $\mathrm{poly}(n)$5 (Agrawal et al., 2023).

The tree-circuit protocol sharpens this perspective. There, a postselection-free observable $\mathrm{poly}(n)$6 is estimated from single-shot decoding, and the recursive tree geometry permits reverse message passing with classical complexity $\mathrm{poly}(n)$7, where $\mathrm{poly}(n)$8 (Feng et al., 3 Feb 2025). The same work explicitly distinguishes postselection cost from decoding complexity: postselection overhead is eliminated, while decoding remains linear because each node is visited once and only constant-size operations are required (Feng et al., 3 Feb 2025).

The data-driven MIE work treats the problem as self-supervised learning of a surrogate two-qubit state $\mathrm{poly}(n)$9 from measurement records. Its uncertainty metric is

$\#P$0

estimated through classical shadows, and the reported transition is one in polynomial-resource learnability: below threshold depth, $\#P$1 decreases with the amount of data $\#P$2 and model size $\#P$3, while above threshold it saturates near $\#P$4 (Qian et al., 1 Dec 2025). This is presented as a practical limit of classical learnability rather than as an entanglement-only diagnostic (Qian et al., 1 Dec 2025).

In the observability framework, the analogous quantity is the readout capability

$\#P$5

Below critical depth, there exists an encoding with $\#P$6 but $\#P$7, whereas above threshold randomized measurements from approximate unitary $\#P$8-designs achieve

$\#P$9

with $1+1$0 for fixed $1+1$1 (Du et al., 8 Jun 2026). Here the transition is stated directly as a hidden-to-visible transition in accessible information (Du et al., 8 Jun 2026).

Across these papers, the core documented motif is co-variation of physical phase structure and algorithmic accessibility. This suggests a family resemblance to “co-optimization” in the informal sense that entanglement, inference, and simulation cost are analyzed jointly, but the word “clustering” remains absent.

6. Relation to measurement-induced phases, universality, and scope

The literature repeatedly distinguishes among several notions that are sometimes conflated. Entanglement-based MIPTs concern area-law versus volume-law scaling in hybrid dynamics (Zabalo et al., 2019, Manna et al., 2024). Learnability transitions concern whether a measurement record contains enough usable information for scalable decoding of a conserved observable (Agrawal et al., 2023). Complexity-driven observation concerns the fraction of QFI recoverable after bounded-depth preprocessing and destructive measurement (Du et al., 8 Jun 2026). Graph-state transitions concern hardness of simulating arbitrary local measurements as a function of graph regularity (Ghosh et al., 2022).

Universality claims are likewise framework-specific. In Haar-random circuits, bulk critical exponents are reported as consistent with percolation and stabilizer circuits, while the Rényi-index dependence of the critical entropy prefactor distinguishes the Haar case (Zabalo et al., 2019). In locally scrambled circuits parameterized by Cartan angles, distinct universality classes are associated with different values of entangling power and gate typicality (Manna et al., 2024). In tree circuits, the exact critical point and scaling are derived from a Fisher–KPP traveling-wave analysis (Feng et al., 3 Feb 2025). In graph states, the sharp thresholds are tied to complement duality and reduction to MBQC resource states (Ghosh et al., 2022).

The controlled use of measurements also differs across settings. One-shot non-Clifford measurements interleaved with random Clifford layers drive purity fluctuations from Clifford-like $1+1$2 scaling to Haar-like $1+1$3 scaling when the number of measurements is $1+1$4 (Oliviero et al., 2021). By contrast, repeated measurements drive the state toward the completely mixed state and suppress fluctuations (Oliviero et al., 2021). In controlled qubit processes, varying the measurement basis changes the predictive structure of the classical output process, with finite-complexity points, IID points, and uncountable mixed-state presentations depending on the angle $1+1$5 (Venegas-Li et al., 2019).

Within this documented landscape, “Co-Optimized Dynamic Clustering” remains uninstantiated. The supplied corpus instead supports a coherent encyclopedia entry on measurement-induced complexity transitions: sharp parameter-driven changes in entanglement structure, readout capability, decoding cost, simulation hardness, or holographic subregion complexity across monitored quantum systems (Ghosh et al., 2022, Tang et al., 22 Jun 2025, Agrawal et al., 2023, Feng et al., 3 Feb 2025, Du et al., 8 Jun 2026). A plausible implication is that the requested term either belongs to a different literature or would require an external source base not present here.