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Theory of the phase transition in random unitary circuits with measurements

Yimu Bao, Soonwon Choi, Ehud Altman

arXiv:1908.04305v3cond-mat.stat-mechcond-mat.dis-nncond-mat.str-elhep-thquant-ph

TL;DR

The paper asks how measurement affects entanglement and information retention in random unitary circuits. It uses replica quantities mapped to classical spin models to analyze entanglement and Fisher information, finding a large-q transition at pc = 1/2 associated with bond percolation and no transition for arbitrary nonlocal measurement bases.

Abstract

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We present a theory of the entanglement transition tuned by measurement strength in qudit chains evolved by random unitary circuits and subject to either weak or random projective measurements. The transition can be understood as a nonanalytic change in the amount of information extracted by the measurements about the initial state of the system, quantified by the Fisher information. To compute the von~Neumann entanglement entropy $S$ and the Fisher information $\mathcal{F}$, we apply a replica method based on a sequence of quantities $\tilde{S}^{(n)}$ and $\mathcal{F}^{(n)}$ that depend on the $n$-th moments of density matrices and reduce to $S$ and $\mathcal{F}$ in the limit $n\to 1$. These quantities with $n\ge 2$ are mapped to free energies of a classical spin model with $n!$ internal states in two dimensions with specific boundary conditions. In particular, $\tilde{S}^{(n)}$ is the excess free energy of a domain wall terminating on the top boundary, and $\mathcal{F}^{(n)}$ is related to the magnetization on the bottom boundary. Phase transitions occur as the spin models undergo ordering transitions in the bulk. Taking the limit of large local Hilbert space dimension $q$ followed by the replica limit $n\to 1$, we obtain the critical measurement probability $p_c=1/2$ and identify the transition as a bond percolation in the 2D square lattice in this limit. Finally, we show there is no phase transition if the measurements are allowed in an arbitrary nonlocal basis, thereby highlighting the relation between the phase transition and information scrambling. We establish an explicit connection between the entanglement phase transition and the purification dynamics of a mixed state evolution and discuss implications of our results to experimental observations of the transition and simulability of quantum dynamics.

I. INTRODUCTION

The paper develops a replica-based statistical-mechanics framework for measurement-induced entanglement transitions and relates the transition to information extracted about initial states. It analyzes weak and projective measurements, identifies critical behavior, and discusses experimental accessibility and measurement-basis dependence.

I. INTRODUCTION

  • Local measurements compete with unitary entanglement generation, producing a transition between volume-law and area-law steady-state entanglement as measurement frequency increases.The transition addresses how robust entanglement generation is against measurement-induced nonunitary evolution.Source: §I · p. 1

A. Replica method: conditional entropy

  • The replica method maps conditional entanglement quantities to two-dimensional classical spin models with n! permutation states and measurement-strength-dependent ordering.The generalized entropy is represented by domain-wall free energy, with ferromagnetic and paramagnetic phases corresponding to volume-law and area-law entanglement.Source: §I · p. 1, A. Replica method: conditional entropy · p. 3
  • In the large-q limit, the critical measurement probability is pc = 1/2; for qubits, simulations give pc = 0.26 ± 0.02.The large-q spin model becomes an n!-state Potts model, whose replica-limit transition is associated with bond percolation.Source: A. Replica method: conditional entropy · p. 3

B. Alternative signature of the transition

  • Fisher information provides an experimentally relevant signature because it measures how much outcome distributions reveal about the initial state.Unlike conditional entropy, outcome-based information avoids exponentially unlikely postselection and additional entropy-estimation repetitions.Source: §II · p. 2, B. Alternative signature of the… · p. 3
  • The Fisher information undergoes the same transition as entanglement: it reaches its maximal value above the critical measurement rate but falls below maximum below pc.This behavior reflects complete versus incomplete information extraction about the initial state in the long-time limit.Source: B. Alternative signature of the… · p. 4
  • Information scrambling acts as natural quantum error correction: sparse local measurements cannot reveal nonlocal correlations, whereas higher rates or nonlocal measurements can defeat this protection.The framework therefore connects the transition to both simulability questions and the information accessible to experiments.Source: §I · p. 2, B. Alternative signature of the… · p. 4, §III · p. 5

III. RANDOM UNITARY CIRCUITS WITH WEAK MEASUREMENTS

  • Weak measurements are implemented through ancillas and reproduce probabilistic projective measurements, with p = sin2 α for the chosen coupling.Ancilla outcomes other than |0⟩m reveal a system state, while |0⟩m corresponds to no projective measurement.Source: §III · p. 4

A. Entanglement entropy within the system

The paper connects averaged entanglement and measurement-outcome information to replica quantities whose n→1 limits recover von Neumann entropy and Fisher information, revealing shared phase-transition signatures.

A. Entanglement entropy within the system

  • The averaged subsystem entropy is exactly a conditional entropy of the system on measurement ancillas for the weak-measurement evolution.Source: A. Entanglement entropy within the… · p. 5
  • The replica quantities ˜S^(n) are analytically tractable n-th-moment objects that recover the averaged conditional entropy as n→1.They are not themselves Rényi conditional entropies, because averaging over unitaries occurs inside the logarithm.Source: A. Entanglement entropy within the… · pp. 5–6
  • For n≥2, these generalized entropies map to two-dimensional classical spin models with n! internal states and distinct boundary conditions.In the large-q limit, the models reduce to standard n!-state Potts models.Source: A. Entanglement entropy within the… · p. 6

B. Kullback-Leibler divergence and Fisher information from measurement outcomes

  • The Fisher information measures how much measurement outcomes reveal about a small perturbation of the initial state, using the ancilla outcome distribution.The perturbation is implemented locally as δU(θ)=e^−i X̂₁θ with θ≪1.Source: B. Kullback-Leibler divergence and… · p. 6
  • The KL divergence is nondecreasing in time but saturates finitely, and a nonanalytic change in its long-time saturation signals the phase transition.Fisher information is obtained from the second derivative with respect to perturbation strength.Source: B. Kullback-Leibler divergence and… · p. 6
  • The generalized divergence maps to free-energy differences with different bottom boundaries, while generalized Fisher information is proportional to bottom-boundary magnetization.Thus entropy and information extraction detect the same transition through boundary conditions of one bulk spin model.Source: B. Kullback-Leibler divergence and… · p. 7

IV. MAPPING TO SPIN MODELS

This section maps generalized entropies and divergences in weakly measured random unitary circuits to two-dimensional classical spin models with measurement-dependent boundary conditions. It then uses the Ising-model representation to relate entropy and Fisher-information quantities to boundary domain-wall and magnetization costs across the ferromagnetic and paramagnetic phases.

IV. MAPPING TO SPIN MODELS

  • Generalized entropies and divergences map to the same bulk classical spin model, differing only through top and bottom boundary conditions.The spin model has n! internal states; identical initial states give an open bottom boundary, while distinct states induce a bottom magnetic field.Source: §IV · pp. 7–8
  • For n ≥ 3 and sufficiently large q, the mapped model has n! states on a triangular lattice and becomes the standard n!-state Potts model on a square lattice as q →∞.For n = 2, integrating out auxiliary variables instead yields a two-dimensional triangular-lattice Ising model.Source: §IV · p. 8

A. Mapping between purity and partition function of classical Ising model

  • For n = 2, measurement strength tunes an anisotropic triangular-lattice Ising model between ferromagnetic and paramagnetic phases.Large q and weak measurement correspond to the ferromagnetic low-temperature regime, whereas α → π/2 gives the paramagnetic high-temperature regime.Source: A. Mapping between purity and… · p. 10
  • The generalized entropy is the excess free energy of an Ising domain wall terminating on the top boundary, whose pinned spins encode the subsystem.The top boundary fixes σ = −1 in subsystem A and σ = +1 in subsystem B.Source: A. Mapping between purity and… · p. 11
  • The second KL divergence is the free-energy cost of applying a boundary magnetic field at the perturbed bottom site, and its derivative yields the second Fisher information.For the perturbed product state, the field is h = −log cos θ, with cos θ = |⟨ψ0|δU(θ)|ψ0⟩|.Source: A. Mapping between purity and… · p. 11
  • In the ferromagnetic phase associated with weak or sparse measurements, the top boundary induces nonzero bottom-boundary magnetization.This phase has ⟨σx0,1⟩ > 0.Source: A. Mapping between purity and… · p. 11
  • F^(2) remains below 1 in the volume-law phase, increases continuously with measurement strength, and saturates nonanalytically at F^(2) = 1 at and beyond the critical point.The ferromagnetic regime has F^(2) < 1, while the paramagnetic regime has F^(2) = 1.Source: A. Mapping between purity and… · p. 11
  • In the paramagnetic phase associated with stronger or more frequent measurements, bottom-boundary magnetization vanishes as T →∞.The vanishing occurs in the thermodynamic limit.Source: A. Mapping between purity and… · p. 11

B. Generalization to the n-th moment

The n-th-moment construction maps random-unitary-circuit quantities to permutation-valued classical spin models, whose positivity and ordering behavior determine the analytic framework for the transition.

B. Generalization to the n-th moment

  • Haar averaging introduces emergent spin variables labeled by permutations in P_n, giving n! possible internal states for each replicated degree of freedom.The associated tensors encode alternative contractions of replicated indices.Source: B. Generalization to the n-th moment · pp. 11–12
  • Applying the averaged tensor decomposition across the circuit produces an intermediate spin model on a honeycomb lattice with local diagrammatic weights.The weights depend on permutation relations and closed-loop contributions.Source: B. Generalization to the n-th moment · p. 12
  • Integrating out auxiliary τ variables yields a triangular-lattice spin model, while top and bottom boundary contractions encode the replicated observables and initial-state overlaps.Cyclic and identity contractions distinguish subsystem boundaries in the tensor network.Source: B. Generalization to the n-th moment · pp. 11–12
  • For fixed n, all weights are nonnegative in a finite parameter regime; in particular, positivity is ensured when q²>n and the stated inequality holds.At fixed κ and large q, the inequality is generally satisfied because its right-hand side can become arbitrarily small.Source: B. Generalization to the n-th moment · p. 12
  • The model has P_n×P_n symmetry and changes from a high-temperature paramagnet to a lower-temperature ferromagnet as q and measurement parameters vary.As q→∞, the symmetry enlarges to Potts symmetry, allowing quantitative analysis of the ordering transition.Source: B. Generalization to the n-th moment · p. 12

1. The n-th generalized conditional entropy

The generalized conditional entropy is represented as a domain-wall free-energy excess: ordered and disordered spin phases respectively produce volume-law and area-law entanglement scaling.

1. The n-th generalized conditional entropy

  • ˜S^(n) is the excess free energy of a domain wall terminating at the top-boundary interface between subsystems A and B.The construction compares free energies under cyclic-permutation and identity boundary conditions.Source: §1 · p. 13
  • In the ferromagnetic phase, pinned inhomogeneous boundaries create an interface whose length scales with subsystem size, producing volume-law entanglement.The same domain-wall interpretation connects this transition to random-tensor-network holographic entanglement entropy [16].Source: §1 · p. 13
  • In the paramagnetic phase, domain walls are condensed and cost at most order-one free energy, yielding area-law scaling of ˜S^(n).Source: §1 · p. 13

2. The n-th generalized KL divergence and Fisher information

The generalized Fisher information is represented by boundary magnetization and changes nonanalytically across the bulk spin-model transition. In the long-time limit, measurements fully extract locally accessible information in the paramagnetic phase, while scrambling preserves some information in the volume-law phase.

Boundary interpretation

  • The n-th generalized KL divergence maps to a free-energy difference between spin models with identical top boundaries and distinct bottom boundary conditions.The bottom distinction arises from initializing one replicated state differently and acts as a boundary magnetic field distinguishing spin types.Source: §2 · p. 13
  • The Fisher information is proportional to the density of down-type boundary spins, which is independent of position after averaging over the circuit ensemble.The down-type fraction is defined relative to permutations that do not preserve the first copy.Source: §2 · p. 13

Long-time behavior

  • In the ferromagnetic phase, top-boundary order keeps the bottom boundary correlated with the initial condition, whereas the paramagnetic phase reaches the statistical down-spin average.This distinction produces a nonanalytic change in the long-time Fisher information at the bulk critical point.Source: §2 · p. 13
  • The Fisher information reaches F^(n) = 2/n only in the paramagnetic phase and becomes F = 2 in the replica limit n →1.The volume-law phase instead saturates below the maximum because some information remains inaccessible after infinitely many measurements.Source: §2 · pp. 13–14

Information interpretation

  • Saturation at the maximum means local measurement devices obtain all information accessible from local measurements, while scrambling acts as an effective quantum error-correcting mechanism.The latter prevents complete extraction in the volume-law phase.Source: §2 · pp. 13–14

V. PHASE TRANSITIONS

The phase-transition analysis is exactly solvable for n = 2 at arbitrary q through an emergent triangular-lattice Ising model. Its transition has Ising conformal-field-theory universality.

V. PHASE TRANSITIONS

  • The phase-transition program treats n = 2 at arbitrary q, large-q Potts models, analytic continuation to n = 1, and finite-q numerical simulations.For n = 2, the emergent triangular-lattice Ising model is exactly solvable; large q yields standard Potts models, while n = 1 leads to bond percolation.Source: §V · p. 14

A. Classical Ising model (n = 2 with an arbitrary q)

  • For n = 2, the replicated purity maps to a two-dimensional classical Ising model on a triangular lattice with couplings determined by q and measurement strength.The diagonal interaction is ferromagnetic, the horizontal interaction antiferromagnetic, and their sum satisfies Jd + Jh ≤ 0.Source: A. Classical Ising model (n = 2 with… · p. 14
  • The Ising model undergoes a paramagnetic-to-ferromagnetic transition at the exactly solvable coupling condition.The critical point can therefore be computed for arbitrary q.Source: A. Classical Ising model (n = 2 with… · p. 14
  • The phase transition is described by the Ising conformal-field-theory universality class and has critical exponents ν = 1 and β = 1/8.Source: A. Classical Ising model (n = 2 with… · p. 14

B. Standard Potts model (n ≥2 with q →∞)

In the large-q limit, the replicated spin models reduce to standard Potts models, and analytic continuation to n →1 yields a square-lattice bond-percolation transition. The percolation result is exact only at q →∞, while qubit numerics give a distinct finite-q estimate.

B. Standard Potts model (n ≥2 with q →∞)

  • At q →∞, the spin model acquires P_n! symmetry and reduces to the standard n!-state Potts model on a square lattice.The reduction follows by taking α →π/2 while holding κ = q^(n−1) cot^(2n)(α) fixed; horizontal couplings vanish and diagonal couplings are ferromagnetic.Source: B. Standard Potts model (n ≥2 with q →∞) · pp. 14–15, Figure 6 · p. 15
  • For integer n ≥3, the resulting n!-state Potts model has a first-order phase transition, while its critical coupling is obtained exactly through Kramers-Wannier duality.The n = 2 critical point agrees with the arbitrary-q solution in the large-q limit.Source: B. Standard Potts model (n ≥2 with q →∞) · p. 15

C. Bond Percolation (n →1 and q →∞)

  • For q = 2, the von Neumann entanglement entropy has a numerically extracted critical point pc = 0.26 ± 0.02, while the approximation may not be valid as q decreases.Source: C. Bond Percolation (n →1 and q →∞) · p. 15
  • In the replica limit n →1, the Potts model becomes bond percolation on the two-dimensional square lattice, with activated bonds percolating at f_c = 1/2.This continuation gives the critical measurement probability p_c = 1/2 in the large-q limit.Source: C. Bond Percolation (n →1 and q →∞) · p. 15
  • The large-q entanglement transition has bond-percolation universality, with critical exponents ν = 4/3 and β = 5/36.The percolation description applies to the von Neumann transition in the q →∞ limit.Source: C. Bond Percolation (n →1 and q →∞) · p. 15
  • The analytic continuation and percolation prediction are strictly controlled only at q →∞, where 1/q corrections can reduce the Potts symmetry and perturb the percolation fixed point.The finite-q critical behavior, especially for qubits, therefore remains an open quantitative issue.Source: C. Bond Percolation (n →1 and q →∞) · p. 16

D. Exact Numerical Simulations

Exact simulations find a qubit critical probability near 0.26, while unrestricted nonlocal measurements eliminate the transition by transferring all initial-state information to ancillas.

D. Exact Numerical Simulations

  • 0.26±0.02 is the extracted critical measurement probability for von Neumann entropy at q=2, consistent with prior numerical results but below the large-q prediction.The estimate uses finite-size scaling for 10≤N≤30 and indicates significant 1/q corrections.Source: D. Exact Numerical Simulations · p. 16
  • Critical probabilities show little dependence on Rényi order k from 0.2 to 5, suggesting simultaneous transitions and classical simulability in the von Neumann area-law phase.The results are summarized in Fig. 9, whose large-q reference is pc=1/2.Source: D. Exact Numerical Simulations · p. 16, Figure 9 · p. 17

VI. ABSENCE OF PHASE TRANSITION FOR UNRESTRICTED MEASUREMENTS

  • Unrestricted measurements can access nonlocal correlations, and the resulting dynamics has no phase transition because information scrambling no longer protects the volume-law phase.Removing ancilla dephasing breaks permutation and Potts symmetries, preventing an ordering transition.Source: §VI · pp. 16–17

A. Dynamics of quantum relative entropy of ancilla qudits

  • The unrestricted-measurement spin models lose permutation symmetry, so their bulk interactions cannot support the ferromagnetic-to-paramagnetic transition.For n=2, the modified interaction explicitly breaks the relevant symmetry and prevents ordering.Source: A. Dynamics of quantum relative… · p. 17
  • 0.02–0.16 measurement strengths yield a down-spin density that rapidly approaches unity after a long transient, with curves collapsing under the rescaled time T sin^2 α.The saturation occurs on the timescale T* sin^2 α=O(1), when each system qudit has been measured once on average.Source: A. Dynamics of quantum relative… · p. 18

B. Dynamics of the system qudits

  • The system density matrix loses all dependence on its initial state, while the ancillas recover the full information on the same timescale.The system approaches a maximally mixed steady state, and the corresponding relative entropy decays to zero as T→∞.Source: B. Dynamics of the system qudits · pp. 18–19

VII. DISCUSSION AND OUTLOOK

The paper identifies purification dynamics as identical to the entanglement transition and discusses experimental observables, simulability, imperfections, and links to quantum chaos.

A. Relation to purification phase transition

  • The purification transition matches the entanglement transition: area-law phases purify, whereas volume-law phases retain entropy proportional to system size.The spin-model description represents purification through a domain wall traversing the system.Source: A. Relation to purification phase… · p. 19

B. Experimental considerations

  • Experimental detection is difficult because direct entropy estimation requires exponentially many repetitions and, for projective dynamics, trajectory postselection adds further overhead.The KL divergence and empirical distribution tests can reduce these costs, although faithful detection by local observables remains open.Source: B. Experimental considerations · p. 20
  • Imperfections explicitly break permutation symmetry and become significant when ηNT ≳ O(N), potentially hindering a sharp transition.Susceptibility to these perturbations is enhanced near the critical point, making the transition itself a possible probe of experimental imperfections.Source: B. Experimental considerations · p. 20

C. Implications to the simulability of open system dynamics

  • Area-law entanglement often supports classical simulation, but it is not sufficient in every case because small Schmidt coefficients can strongly affect observable estimates.The paper notes that area-law Rényi entropies with k > 1 can provide pragmatic simulability criteria for many physical states.Source: C. Implications to the simulability of… · pp. 20–21

D. Outlook

  • The paper proposes KL divergence, local observables, interferometric methods, and complexity bounds as directions for detecting the transition and connecting it to quantum chaos.It also relates the stability of the volume-law phase to information scrambling and effective quantum error correction.Source: D. Outlook · p. 21
  • The mapping may help assess experimental imperfections in quantum-supremacy tests by evaluating discrepancies between ideal and realized sampling distributions.This could estimate tolerable imperfections or characterize near-term quantum devices.Source: D. Outlook · p. 21

Appendix A: Details of the mapping to classical spin models

The appendix derives measurement-dependent tensor-network weights and the n = 2 three-body Ising weight used in the classical spin-model mapping.

1. Derivation of w(n)

  • Dephasing removes off-diagonal ancilla density-matrix elements, producing index constraints that simplify the replicated tensor contraction.The contraction yields loop factors q^#cycle(στ^-1), recovering the no-measurement result at α = 0.Source: §1 · p. 23
  • The weak-measurement tensor network couples each qudit to an ancilla, applies dephasing, and contracts ancilla degrees of freedom across replicas.The resulting diagonal weight w^(n)_d depends on the replicated system tensors and ancilla contraction.Source: Figure 12 · p. 23, §1 · p. 23

2. Derivation of ¯w(2)

  • For n = 2, the three-body weight is symmetric under exchange of its first two spins and obeys Ising symmetry, so three configurations determine its values.The complementary spin is defined by ¯σ = −σ.Source: §2 · p. 24
  • Unlike the unmeasured case, the configuration ¯w^(2)(σ,σ,¯σ) is nonzero for α > 0, reflecting the effect of measurement coupling.The appendix attributes this difference to the role of unitarity in the measured model.Source: §2 · p. 24

3. Derivation of Ising couplings Jh and Jd

The appendix derives the effective Ising couplings and establishes a finite large-q regime where replicated weights are nonnegative for n ≥ 3.

3. Derivation of Ising couplings Jh and Jd

  • The three-body weight factorizes into pairwise Ising couplings, with J_d ≤ 0 ferromagnetic, J_h ≥ 0 antiferromagnetic, and J_d + J_h ≤ 0.These signs follow from the ordering u_2 ≥ u_1.Source: §3 · p. 24
  • At large q, |J_d| dominates |J_h| and the transition occurs when J_d is O(1), placing α_c close to π/2.The relevant regime is q ≫ 1 with π/2 − α ≪ 1.Source: §3 · p. 24

4. A sufficient condition for nonnegative weights for n ≥3

  • Negative replicated weights can arise for general q and α, so the classical-spin interpretation requires a sufficient nonnegativity condition.The appendix derives this condition after integrating out τ variables.Source: §4 · pp. 24–25
  • For fixed κ = q^(n−1) cot^(2n) α and sufficiently large finite q, the derived inequality can be satisfied, proving a finite parameter region with nonnegative weights.The leading correction on the right-hand side scales as 2n^2/q^2.Source: §4 · p. 25

Appendix B: Spin model for quantum relative entropy

The appendix constructs the spin-model weight for quantum relative entropy and shows that removing ancilla dephasing breaks replica permutation symmetry.

Appendix B: Spin model for quantum relative entropy

  • The relative-entropy mapping modifies the two-body weight by omitting the ancilla dephasing channel.The resulting tensor contraction uses ancilla density matrices without dephasing.Source: Appendix B · p. 25, §1 · p. 25

1. Derivation of two-body weight

  • Without dephasing, v^(n)_d lacks permutation symmetry for every integer n ≥ 2 because the cyclic permutation contracts the replicated ancillas at the top.The transformation σ, τ → ξ_1 ◦ σ ◦ ξ_2, ξ_1 ◦ τ ◦ ξ_2 is therefore not invariant.Source: §1 · p. 25
  • When either spin is the cyclic permutation C^(n), the weight becomes independent of α and reduces to q^#cycle(στ^-1), matching the unmeasured random-unitary result.The maximum value q^n occurs when both spins equal C^(n).Source: §1 · p. 26

2. Derivation of three-body weight

The second-moment mapping produces a nonnegative three-body Ising model, while unitarity imposes a special weight identity that measurements generally break. The associated Potts-model duality determines the exact transition condition.

2. Derivation of three-body weight

  • The second-moment weights form a nonnegative three-body Ising model on a triangular lattice, enabling a classical partition-function interpretation.The model favors σ = −1 spins, whose density increases toward the bottom boundary and polarizes there as T →∞.Source: §2 · p. 26

Appendix C: Implication of unitarity in classical spin models

  • Without measurements, unitarity gives ¯w(n)(σ, σ, σ′) = δσσ′ when q^2 ≥ n; weak measurements generally violate this identity.The identity is explicitly associated with unitarity and does not hold once weak measurements break unitarity.Source: Appendix C · p. 26

Appendix D: Duality in standard Potts models

  • The mapped Potts model has Q = n! local states, and its high- and low-temperature expansions are related by a duality.The self-dual point is where the free energy becomes singular, providing the transition criterion.Source: Appendix D · pp. 26–27
  • The critical coupling is obtained from the self-dual relation ˜κ = κ, reducing at Q = 2 to Kramers-Wannier duality for the square-lattice Ising model.The Q = 2 case connects the Potts-model construction to the standard 2D Ising transition.Source: Appendix D · p. 27

Appendix E: Monte-Carlo simulation of the transition in F (2)

Monte Carlo simulations represent the second Fisher information through bottom-boundary spin density and identify paramagnetic and ferromagnetic regimes separated by the classical Ising critical line. The appendix also describes the numerical algorithm, finite-size scaling, and the Fisher-information bound for optimized measurements.

Appendix E: Monte-Carlo simulation of the transition in F (2)

  • F^(2) is proportional to the bottom-layer density of σ = −1 spins, F^(2) = 2⟨m−1⟩, which grows initially before saturating according to α and q.Above the critical measurement strength, the density saturates at 1/2; below it, it remains below 1/2 and signals long-range boundary correlations.Source: Appendix E · p. 27
  • The phase boundary of F^(2) matches the exact classical Ising critical point, with large α and small q giving a paramagnet and small α and large q a ferromagnet.Figure 14 maps the transition as a function of measurement strength α and local dimension q.Source: Appendix E · p. 27, Figure 14 · p. 27

1. Algorithm

  • The simulation algorithm stores only entangled qubits while tracking disentangled ones, applying disjoint two-qubit gates and projective measurements sequentially.On average, pN qubits are projected and remain disentangled at each time step, allowing simulations up to N = 30 qubits.Source: §1 · p. 28

2. Finite-size scaling

  • Finite-size scaling extracts p_c and ν by optimizing a data-collapse quality function over measurement probabilities and system sizes.Uncertainties are estimated by repeating the analysis on randomly selected subsets of measurement probabilities and using the resulting standard deviations.Source: §2 · p. 28

Appendix G: The Fisher information for nonlocal measurements

  • The KMB Fisher information provides an attainable upper bound on Fisher information optimized over all POVMs, although the optimal POVM may depend on each unitary circuit.The paper computes the KMB quantity after averaging over unitary gates, while the optimizing measurement need not be common across gates.Source: Appendix G · pp. 28–29
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