Measurement-Induced Complexity Transition

• Measurement-induced complexity transition is characterized by a sudden change in classical simulation and state complexity as measurement parameters or circuit architecture are varied.
• Key methodologies include varying graph regularity, measurement rate, and circuit depth to delineate regimes from classically tractable to #P-hard simulation and decode hidden quantum information.
• Practical insights extend to learnability, entanglement width analysis, and the design of efficient quantum decoding strategies for diverse experimental and theoretical frameworks.

Searching arXiv for recent and foundational papers on measurement-induced complexity transitions and closely related formulations. Measurement-induced complexity transition denotes a sharp change in the classical or operational complexity associated with monitored quantum systems as a measurement control parameter is varied. Across the literature, the term does not refer to a single invariant definition; rather, it encompasses several rigorously formulated transitions in which measurements reorganize simulability, learnability, observability, or state complexity. In one line of work, the transition is a sharp change in the classical simulation complexity of single-qubit measurements on $k$-regular graph states, with easy regimes at $k\le 2$ and $k\ge n-3$ and a hard regime for $3\le k\le n-4$ (Ghosh et al., 2022). In monitored random circuits, related transitions arise when changing measurement rate or strength, producing entangling versus disentangling phases and corresponding changes in classical decoding, learnability, or exact state complexity (Suzuki et al., 2023, Agrawal et al., 2023, Feng et al., 3 Feb 2025). Other formulations treat measurement complexity itself—the circuit depth of a readout—as the control parameter governing a hidden-to-visible transition in accessible Fisher information (Du et al., 8 Jun 2026). A unifying theme is that measurements do not merely reduce entanglement; depending on architecture, measurement type, and complexity notion, they can induce sharp thresholds in computational hardness, inferential power, and effective classical description.

1. Graph-state formulation and sharp regularity thresholds

A precise and fully characterized instance appears in the study of locally rotated $k$-regular graph states, where the task is to simulate arbitrary single-qubit measurements on graph states associated to $k$-regular graphs on $n$ qubits (Ghosh et al., 2022). An $n$-qubit graph state $\lvert G\rangle$ is defined by

$$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$

with stabilizer generators

$k\le 2$0

The simulation problem considers a final layer of arbitrary single-qubit rotations $k\le 2$1 followed by computational-basis measurements, with output probabilities

$k\le 2$2

Within this formulation, the paper proves a sharp complexity transition as the regularity parameter $k\le 2$3 is varied from $k\le 2$4 to $k\le 2$5 (Ghosh et al., 2022). The easy regime theorem states that for $k\le 2$6 and $k\le 2$7, locally rotated $k\le 2$8-regular graph states have constant entanglement width and admit polynomial-time classical simulation, including exact probability computation up to multiplicative error and sampling. The hard regime theorem states that for every $k\le 2$9, there exist locally rotated $k\ge n-3$0-regular graph states such that approximating output probabilities $k\ge n-3$1 up to constant multiplicative error is #P-hard, and the entanglement width satisfies $k\ge n-3$2 (Ghosh et al., 2022).

This yields a sharply delineated phase structure: easy at $k\ge n-3$3, hard for $k\ge n-3$4, and easy again at $k\ge n-3$5 (Ghosh et al., 2022). The critical points are identified as $k\ge n-3$6, which marks the onset of MBQC universality and intractability, and $k\ge n-3$7, which marks the return to bounded-width tractability. A central structural ingredient is a duality between low and high regularity: taking graph complements maps $k\ge n-3$8 to $k\ge n-3$9, and bounded rank width is preserved under complement, explaining why complements of $3\le k\le n-4$0- and $3\le k\le n-4$1-regular graphs are again classically easy (Ghosh et al., 2022).

In this setting, “measurement-induced” refers to the computational task induced by arbitrary single-qubit product measurements on a static entangled resource state. The transition is not driven by measurement rate, but by the entanglement structure that measurements probe and exploit. The paper explicitly contrasts this with monitored random-circuit transitions: here one fixes a family of graph states and varies regularity, while the measurements remain arbitrary single-qubit product measurements (Ghosh et al., 2022).

2. Entanglement width, tensor-network cost, and MBQC hardness

The graph-state formulation admits an exact entanglement-theoretic characterization. For a bipartition $3\le k\le n-4$2 of the vertex set, with $3\le k\le n-4$3 the cross-adjacency submatrix over $3\le k\le n-4$4, the Schmidt rank and von Neumann entropy are

$3\le k\le n-4$5

Thus, bipartite entanglement in graph states is determined exactly by the $3\le k\le n-4$6-rank of the cross-adjacency (Ghosh et al., 2022).

The complexity-relevant quantity is entanglement width,

$3\le k\le n-4$7

defined via binary tree decompositions. For graph states, $3\le k\le n-4$8, the graph rank width, and also equals the Schmidt-rank width (Ghosh et al., 2022). Tree-tensor-network contraction then computes probabilities and samples in time $3\le k\le n-4$9, so bounded entanglement width implies polynomial-time classical simulation (Ghosh et al., 2022).

This directly explains the easy regimes. For $k$0, the graph is a disjoint union of edges; for $k$1, it is a disjoint union of cycles. These have bounded tree width, hence bounded rank width and $k$2 (Ghosh et al., 2022). By complement duality, the same bounded-width conclusion extends to $k$3 (Ghosh et al., 2022). For the complete graph $k$4, the paper also gives a specialized polynomial-time algorithm based on Hamming-weight symmetry, reducing amplitude evaluation to a recursion with $k$5 paths (Ghosh et al., 2022).

The hard regime is established through MBQC resource-state constructions and postselection-based hardness reductions. For $k$6 and $k$7, toric hexagonal and toric square lattices furnish universal MBQC resource states. For $k$8, the paper constructs explicit $k$9-regular parent graphs by connecting two toric square lattices with a bipartite gadget whose existence is guaranteed by the Gale–Ryser theorem; deleting one torus reduces to a toric square lattice, preserving #P-hardness of multiplicative probability approximation (Ghosh et al., 2022). For $k$0, complement duality combined with local complementation and vertex deletion extends hardness to high regularity (Ghosh et al., 2022).

The complexity-theoretic consequence is explicit: approximating certain output probabilities is #P-hard, and by Stockmeyer’s theorem an efficient classical sampler would imply a collapse of the polynomial hierarchy (Ghosh et al., 2022). The constructions are worst-case over graphs, while average-case hardness over local rotations follows through worst-to-average-case reductions cited there (Ghosh et al., 2022). Under additional assumptions, entanglement-width lower bounds strengthen from $k$1 to $k$2 assuming $k$3, and further to $k$4 or $k$5 under ETH or SETH, respectively (Ghosh et al., 2022).

3. Monitored random circuits: entanglement phases and exact state complexity

A distinct formulation studies monitored random circuits on a one-dimensional chain with nearest-neighbor brickwork Haar-random two-qubit gates and probabilistic $k$6-basis measurements at rate $k$7 (Suzuki et al., 2023). The output is the normalized conditional state of the monitored circuit, conditioned on the full measurement record. Two exact state-complexity notions are defined: exact $k$8-complexity, the minimal number of two-qubit gates required to prepare a state from $k$9, and exact $n$0-complexity, which also allows single-qubit $n$1-basis measurements with post-selection while counting only two-qubit gates (Suzuki et al., 2023).

In this setting, the main theorem establishes a sharp complexity phase transition at

$n$2

For $n$3, with probability $n$4, $n$5 grows linearly in time and $n$6 grows at least linearly in time, up to exponential times $n$7; beyond that, both saturate at $n$8 (Suzuki et al., 2023). For $n$9, with probability at least $n$0,

$n$1

and saturation to $n$2 occurs after time at most $n$3 (Suzuki et al., 2023).

The proof maps the monitored circuit to bond percolation on a tilted square lattice, with open edges corresponding to unmeasured bonds and open-edge probability $n$4 (Suzuki et al., 2023). Since the square-lattice bond-percolation threshold is $n$5, one obtains $n$6 (Suzuki et al., 2023). Below threshold, there are $n$7 disjoint open paths traversing the circuit for times up to $n$8, allowing the embedding of exponentially long unitary computations using teleportation through non-causal turns and vertical bridges between paths (Suzuki et al., 2023). Above threshold, dual-lattice cuts sever long-range connectivity, resetting history and confining dynamics to logarithmically sized open clusters (Suzuki et al., 2023).

The algebraic-geometric core of the lower bound is the accessible dimension $n$9 of the monitored circuit image. The paper proves the bridge

$\lvert G\rangle$0

so linear growth of accessible dimension implies linear growth of exact state complexity (Suzuki et al., 2023). This transition is explicitly distinguished from the usual entanglement transition: the critical point matches the Rényi-$\lvert G\rangle$1 percolation threshold, whereas for Rényi-$\lvert G\rangle$2 entropies with $\lvert G\rangle$3 the critical point is typically smaller (Suzuki et al., 2023).

A closely related but entanglement-centric monitored-circuit literature identifies a measurement-induced entanglement transition in $\lvert G\rangle$4 dimensions. For Haar-random circuits, analysis based on tripartite mutual information gives a critical measurement rate

$\lvert G\rangle$5

with bulk critical exponents $\lvert G\rangle$6 and $\lvert G\rangle$7, and dynamic exponent $\lvert G\rangle$8 (Zabalo et al., 2019). In that formulation, the volume-law phase is associated with hard-to-simulate states, while the area-law phase is associated with efficient classical representations such as finite-bond-dimension matrix-product states (Zabalo et al., 2019). This provides a broader entanglement-based backdrop for later learnability and decoding formulations.

4. Learnability, decoding, and postselection-free experimental formulations

Another major usage of the term concerns learnability rather than direct state simulation. In trapped-ion experiments on the Quantinuum H1-1 processor, monitored charge-conserving circuits exhibit a learnability phase transition in which an observer attempts to infer a conserved total charge $\lvert G\rangle$9 from the measurement record alone (Agrawal et al., 2023). The setting uses chains of length $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$0, with circuit depth $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$1, random charge-conserving two-qubit gates, and tunable-strength weak measurements implemented through Kraus operators

$$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$2

The control parameter is the measurement strength $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$3 (Agrawal et al., 2023).

Three decoders are compared: an optimal PostBQP decoder, a statistical-mechanics decoder based on a noisy symmetric exclusion process, and an LSTM-based neural-network decoder (Agrawal et al., 2023). The trajectory-level metrics are decoder accuracy and credence, with criticality diagnosed using $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$4 and an ancilla-based sharpening entropy $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$5 together with a Binder cumulant (Agrawal et al., 2023). Across hardware and simulation, the learnability transition occurs near

$$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$6

with $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$7 and $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$8 (Agrawal et al., 2023).

Below $$\lvert G\rangle \;=\; \prod_{(i,j)\in E} \mathrm{CZ}_{ij} \; \lvert +\rangle^{\otimes n},$$9, average credence remains significantly below $k\le 2$00 and only weakly depends on system size; above $k\le 2$01, average credence increases with $k\le 2$02 toward $k\le 2$03, consistent with asymptotically perfect learnability (Agrawal et al., 2023). The same transition coincides with the collapse of the per-trajectory uncertainty $k\le 2$04 to zero in the QND setting (Agrawal et al., 2023). The statistical-mechanics decoder is efficient and shares the same universality class as the optimal decoder, described as a modified Kosterlitz–Thouless transition, but with a higher critical point (Agrawal et al., 2023).

The paper explicitly separates learnability complexity from entanglement-based classical simulability. In the realized qubit-only model, the learnability transition occurs in an area-law entangled regime where matrix-product simulations remain feasible; in related variants, learnability transitions can occur in volume-law regimes believed to be classically non-simulable, while the stat-mech decoder remains efficient (Agrawal et al., 2023). This suggests that inferability of conserved observables and amplitude-level state simulation define distinct complexity notions.

A related postselection-free experimental framework uses monitored tree circuits with universal gates (Feng et al., 3 Feb 2025). There, the MIPT is diagnosed via purification of a probe qubit initially entangled with the root qubit, and the same entangling-to-disentangling transition becomes a sharp change in the predictability of the root bit from mid-circuit outcomes (Feng et al., 3 Feb 2025). The single-qubit weak-measurement Kraus operators are

$k\le 2$05

with $k\le 2$06 (Feng et al., 3 Feb 2025). Because the tree is acyclic, exact bottom-up decoding is linear in the number of qubits $k\le 2$07 (Feng et al., 3 Feb 2025).

In that model, the order parameter is the smallest eigenvalue $k\le 2$08 of the probe reduced state, and the exact critical measurement strength is

$k\le 2$09

Below criticality, the typical order parameter obeys

$k\le 2$10

while at criticality

$k\le 2$11

(Feng et al., 3 Feb 2025). Here, the computational cost remains $k\le 2$12 throughout because of the tree architecture; what changes sharply is the information complexity of inference rather than asymptotic decoder runtime (Feng et al., 3 Feb 2025).

5. Measurement-only scrambling, emergent symmetries, and analytic thresholds

A broader analytic framework for random circuits maps purity dynamics to an effective spin problem and studies both hybrid unitary-plus-measurement circuits and measurement-only models (Tang et al., 22 Jun 2025). The degrees of freedom are $k\le 2$13 qudits of local dimension $k\le 2$14, evolving under Brownian $k\le 2$15-body Hamiltonians and/or projective Haar-random rank-1 measurements on $k\le 2$16-qudit subsets (Tang et al., 22 Jun 2025). The averaged doubled-state dynamics closes on subsystem purities and can be Hermitized in a permutation-symmetric sector, reducing the problem to an $k\le 2$17-dimensional effective description (Tang et al., 22 Jun 2025).

The main observable is the second Rényi entropy proxy

$k\le 2$18

with purity dynamics

$k\le 2$19

(Tang et al., 22 Jun 2025). In the large-$k\le 2$20, large-$k\le 2$21 limit, the Hermitian generator becomes a classical rotor whose ground states encode steady-state entanglement structure (Tang et al., 22 Jun 2025).

For hybrid circuits with $k\le 2$22-body unitaries and single-body measurements, the analytic threshold is

$k\le 2$23

Below $k\le 2$24, the steady state is globally scrambled, with spontaneous $k\le 2$25 symmetry breaking and a Page curve with a cusp at half system size; above $k\le 2$26, the phase is purified, with a unique symmetric minimum and a smooth entropy profile (Tang et al., 22 Jun 2025).

The same framework shows that measurements alone can generate a globally scrambled phase provided the measurement range exceeds a threshold. For all-to-all measurement-only models,

$k\le 2$27

Thus the minimal measurement range required for a global volume-law phase is $k\le 2$28 (Tang et al., 22 Jun 2025). For qubits, $k\le 2$29, so one needs at least five-body projective measurements to enter the globally scrambled phase; for large $k\le 2$30, $k\le 2$31, so three-body measurements suffice (Tang et al., 22 Jun 2025).

Near this threshold, the Liouvillian gap obeys

$k\le 2$32

yielding “critical purification” with purification time diverging linearly in $k\le 2$33 (Tang et al., 22 Jun 2025). The paper also identifies emergent $k\le 2$34 symmetry for two-body measurements in the large-$k\le 2$35 limit and emergent $k\le 2$36 symmetry for certain one-dimensional spatially local measurement-only circuits, with CFT-like or Haldane-like consequences depending on the effective spin parity (Tang et al., 22 Jun 2025).

This formulation broadens the meaning of measurement-induced complexity transition. The complexity growth is governed by the competition between unitary spreading, local purification, and frustration-induced re-entangling due to non-commuting many-body projectors (Tang et al., 22 Jun 2025). The control parameters are measurement rate, measurement range, qudit dimension, and geometry, rather than regularity of a static resource state or decoder class (Tang et al., 22 Jun 2025).

6. Measurement complexity as a resource and other extensions

A further generalization shifts the control parameter from measurement rate or basis to measurement complexity itself. In “hidden-to-visible” transitions in quantum observation, the observable of interest is the readout capability

$k\le 2$37

or, in worst-case form for architecture $k\le 2$38 and depth $k\le 2$39,

$k\le 2$40

(Du et al., 8 Jun 2026). The measurement model is a depth-$k\le 2$41 unitary pre-processing circuit followed by computational-basis measurement of all qubits (Du et al., 8 Jun 2026).

The main theorems establish architecture-dependent critical depths. For a $k\le 2$42-dimensional local architecture,

$k\le 2$43

while for all-to-all connectivity,

$k\le 2$44

Below these thresholds, there exist pure-state encodings with unit QFI such that every allowed readout has exponentially small observability,

$k\le 2$45

for some architecture-dependent $k\le 2$46 (Du et al., 8 Jun 2026). Immediately above threshold, randomized measurements forming approximate unitary $k\le 2$47-designs recover a constant fraction of the QFI: $k\le 2$48 with

$k\le 2$49

For fixed $k\le 2$50, this yields an $k\le 2$51 readout fraction independent of system size (Du et al., 8 Jun 2026).

The lower bound mechanism is locality and bounded Heisenberg light cones: shallow circuits cannot convert globally encoded phase information into locally accessible measurement sensitivity (Du et al., 8 Jun 2026). The visible regime is enabled by approximate unitary $k\le 2$52-designs, with explicit constructions in depth $k\le 2$53 in one dimension, $k\le 2$54 for all-to-all connectivity, and $k\le 2$55 in higher-dimensional local architectures with clean ancillas (Du et al., 8 Jun 2026). This formulation makes “measurement-induced complexity transition” refer literally to the complexity of the measurement circuit.

Several additional extensions use still different operational meanings. In random Clifford circuits doped with one-shot single-qubit non-Clifford measurements, injecting $k\le 2$56 such measurements is both necessary and sufficient to drive the variance of subsystem purity from Clifford scaling $k\le 2$57 to universal Haar-like scaling $k\le 2$58 (Oliviero et al., 2021). The key formula is

$k\le 2$59

so $k\le 2$60 one-shot non-Clifford measurements induce a transition from $k\le 2$61-design to $k\le 2$62-design entanglement-complexity behavior (Oliviero et al., 2021).

In data-driven settings, one can define a learnability transition for measurement-induced entanglement between distant target qubits $k\le 2$63 and $k\le 2$64 by learning two-sided bounds from measurement records alone. The uncertainty metric is

$k\le 2$65

and a finite-depth threshold separates a regime where $k\le 2$66 decreases with polynomial resources from one where it saturates near $k\le 2$67 despite increasing shots and model parameters (Qian et al., 1 Dec 2025). The same paper reports this transition for one-dimensional all-to-all and two-dimensional nearest-neighbor random circuits, and verifies robustness on IBM devices (Qian et al., 1 Dec 2025).

Other works connect measurement-induced entanglement transitions to sampling complexity of Ising partition functions encoded in two-dimensional cluster states (Liu et al., 2023), organize hybrid-circuit phase boundaries by gate entangling power and gate typicality (Manna et al., 2024), or identify measurement-induced complexity jumps in subregion holographic complexity through RT-surface and entanglement-wedge transitions in AdS/BCFT constructions (Jian et al., 2023).

7. Conceptual scope, distinctions, and recurring mechanisms

The literature uses the phrase “measurement-induced complexity transition” for several non-equivalent but structurally related phenomena. One recurring misconception is that all such transitions are simply entanglement transitions. This is not supported by the cited works. In the graph-state formulation, the transition is between classically easy and hard simulation of product measurements on fixed resource states, quantified through entanglement width and MBQC universality (Ghosh et al., 2022). In monitored random circuits, the complexity notion can be exact state complexity (Suzuki et al., 2023), entanglement-based classical simulability (Zabalo et al., 2019), or decoding and learnability of observables from measurement records (Agrawal et al., 2023, Feng et al., 3 Feb 2025). In observation-driven formulations, the transition concerns the fraction of QFI recovered by shallow versus sufficiently complex readouts (Du et al., 8 Jun 2026).

A second misconception is that measurements are always disentangling and therefore always simplify dynamics. Several papers explicitly show otherwise. Measurement-only circuits can enter globally scrambled phases when the joint measurement range exceeds $k\le 2$68 (Tang et al., 22 Jun 2025). Projection measurements on one subsystem can amplify the holographic subregion complexity of another by reconnecting its entanglement wedge to a measurement-induced brane (Jian et al., 2023). Bulk measurements on two-dimensional cluster states can induce long-range entanglement in an effective boundary state, causing a transition from area-law to volume-law boundary entanglement and a corresponding change in tensor-network contraction cost (Liu et al., 2023).

Despite these differences, several mechanisms recur. One is a competition between local purification and nonlocal resource generation: measurements can erase coherence locally while either revealing or generating nonlocal structure through frustration, postselection, graph connectivity, or transfer-matrix effects (Tang et al., 22 Jun 2025, Liu et al., 2023). Another is a threshold in the geometry of information propagation: bounded light cones below a readout-depth threshold hide globally encoded information (Du et al., 8 Jun 2026), whereas percolating unmeasured paths below a measurement-rate threshold enable exponentially long computations (Suzuki et al., 2023). A third is that efficient classical descriptions often survive only while an entanglement-width, bond-dimension, or open-cluster parameter remains bounded (Ghosh et al., 2022, Suzuki et al., 2023, Zabalo et al., 2019).

Taken together, these works indicate that “measurement-induced complexity transition” is best understood as a family of sharp thresholds in the operational consequences of quantum measurement. Depending on the model, the order parameter may be entanglement width, probe mixedness, decoder credence, Liouvillian gap, purity fluctuations, or readout capability. The control parameter may be graph regularity, measurement rate, measurement strength, measurement range, circuit depth, or the number of non-Clifford measurements. What unifies the subject is the appearance of sharply separated regimes—easy versus hard simulation, hidden versus visible information, purified versus scrambled steady states, or low versus exponential exact complexity—created or exposed by the structure of quantum measurement itself (Ghosh et al., 2022, Suzuki et al., 2023, Tang et al., 22 Jun 2025, Du et al., 8 Jun 2026).