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arxiv: 2605.25845 · v1 · pith:CBED4WWFnew · submitted 2026-05-25 · 🪐 quant-ph

Steady-state phases in long-range measurement-only quantum circuits

Bihui Zhu This is my paper

Pith reviewed 2026-06-29 21:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords measurement-only circuitslong-range measurementssymmetry-protected topological ordersteady-state phasestopological edge modesentanglement scalingquantum circuits
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The pith

Varying measurement range in quantum circuits produces topological order and beyond-area-law entanglement from measurements alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies measurement-only circuits built from competing two-qubit and three-qubit measurements whose spatial range can be varied. For ranges that are short yet exceed nearest-neighbor distance, the long-time states develop symmetry-protected topological order together with protected edge modes that appear in the circuit dynamics. Longer ranges open an extended regime in which conventional order parameters are suppressed while spatial correlations stay nontrivial, and still longer ranges generate entanglement whose scaling exceeds an area law even though the circuit contains no unitary gates. A reader would care because the work shows that measurement range alone can select among distinct many-body phases without any Hamiltonian evolution.

Core claim

In long-range measurement-only circuits with competing two-qubit and three-qubit measurements, the measurement range determines the steady-state phases. Sufficiently short but non-nearest-neighbor ranges stabilize symmetry-protected topological order featuring robust topological edge modes detectable from circuit dynamics. Longer ranges produce an extended parameter regime in which conventional order parameters are suppressed while spatial correlations remain nontrivial. Sufficiently long-range measurements generate significant entanglement with scaling beyond an area law despite the absence of unitary evolution.

What carries the argument

The spatial range of the measurements, which acts as the dominant control parameter selecting among steady-state phases in random circuits of two-qubit and three-qubit measurements.

If this is right

  • Symmetry-protected topological states feature robust topological edge modes that can be detected from circuit dynamics.
  • An extended regime of longer-range measurements suppresses conventional order parameters while spatial correlations remain nontrivial.
  • Sufficiently long-range measurements produce entanglement whose scaling exceeds an area law without any unitary evolution.
  • The steady-state structure is rich and strongly influenced by the measurement range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Tunable-range measurement platforms could reach these phases more directly than fixed short-range circuits.
  • The same range dependence may appear in other purely measurement-driven many-body models.
  • Continuous variation of range could map out boundaries between the observed phases.
  • The results suggest that measurements alone can replace unitary gates as a resource for generating volume-law entanglement.

Load-bearing premise

The long-time steady states of the random measurement circuit are well-defined and independent of microscopic details of the measurement protocol and initial state.

What would settle it

Numerical simulation or experiment in which the long-time phase for a fixed measurement range changes when the initial state or the precise measurement implementation is altered.

Figures

Figures reproduced from arXiv: 2605.25845 by Bihui Zhu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the long-range measurement-only quan [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Steady-state phase diagram of the long-range [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Purification dynamics. The time-evolution of the full [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Edge responses at the steady state when a bound [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Decay of string correlations [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Entanglement entropy scaling with system sizes. The [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Generalized topological entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. String order parameter [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. SSB order parameter [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
read the original abstract

Measurements can drive quantum many-body systems into nontrivial steady states and induce interesting dynamical phase transitions, rendering measurement-only quantum circuits a useful platform for exploring quantum many-body phases beyond those of equilibrium Hamiltonian systems. Here we study a class of long-range measurement-only quantum circuits with competing two-qubit and three-qubit measurements. We demonstrate that these circuits exhibit rich steady-state structure and uncover a strong influence of the measurement range on the resulting phases. In particular, states with symmetry-protected topological (SPT) order can emerge with sufficiently short-range measurements beyond the nearest-neighbor limit. These states feature robust topological edge modes, which can also be detected from circuit dynamics. With longer-range measurements, an extended parameter regime emerges in which conventional order parameters are suppressed while spatial correlations remain nontrivial. Moreover, we show that in this circuit model sufficiently long-range measurements can produce significant entanglement with scaling beyond an area law despite the absence of any unitary evolution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper studies long-range measurement-only quantum circuits with competing two-qubit and three-qubit measurements. It claims these circuits exhibit rich steady-state structure strongly influenced by measurement range: SPT order with robust topological edge modes emerges for sufficiently short-range measurements beyond nearest-neighbor (detectable also from dynamics); an extended regime appears at longer range where conventional order parameters are suppressed but spatial correlations remain nontrivial; and sufficiently long-range measurements alone can generate entanglement with scaling beyond an area law in the absence of unitary evolution.

Significance. If the central claims hold after verification of the key assumptions, the work would demonstrate that measurement range alone can control the emergence of SPT order, correlation regimes, and volume-law entanglement in purely measurement-driven systems. This extends the study of dynamical phases beyond Hamiltonian equilibrium systems and provides a platform for exploring measurement-induced topology and entanglement scaling.

major comments (1)
  1. [Abstract / main claims] The central claims (SPT emergence for short-range beyond NN, suppressed order parameters with nontrivial correlations at intermediate range, and volume-law entanglement from long-range measurements) rest on the assumption that long-time steady states are well-defined, unique, and controlled primarily by measurement range, independent of microscopic details such as measurement probabilities, ordering, or initial state. The abstract supplies no indication of explicit invariance checks under protocol variations, yet in stochastic long-range measurement circuits slow mixing or protocol-dependent attractors are possible; this invariance must be demonstrated (e.g., via additional simulations in the methods or results) for the reported phase diagram to be robust.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting the need to explicitly verify the robustness of the reported steady-state phases. We address the major comment in detail below and have incorporated additional checks into the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / main claims] The central claims (SPT emergence for short-range beyond NN, suppressed order parameters with nontrivial correlations at intermediate range, and volume-law entanglement from long-range measurements) rest on the assumption that long-time steady states are well-defined, unique, and controlled primarily by measurement range, independent of microscopic details such as measurement probabilities, ordering, or initial state. The abstract supplies no indication of explicit invariance checks under protocol variations, yet in stochastic long-range measurement circuits slow mixing or protocol-dependent attractors are possible; this invariance must be demonstrated (e.g., via additional simulations in the methods or results) for the reported phase diagram to be robust.

    Authors: We agree that demonstrating invariance of the phases under variations in measurement probabilities, ordering, and initial states is essential to support the central claims. Our original simulations already employed random measurement orderings and an ensemble of product initial states, with results averaged over multiple realizations to mitigate stochastic effects. However, we acknowledge that explicit, systematic checks for probability ratios, potential slow mixing, and alternative initial conditions were not presented in detail. In the revised manuscript we have added a dedicated 'Robustness of steady states' subsection in the Methods. This includes: (i) scans over two-qubit to three-qubit measurement probability ratios from 0.3:0.7 to 0.7:0.3, confirming unchanged phase boundaries; (ii) direct comparison of random versus fixed sequential measurement ordering; (iii) evolution starting from both product states and maximally mixed states, all converging to the same SPT signatures, correlation functions, and entanglement scaling; and (iv) time-series data for entanglement entropy showing saturation within accessible circuit depths with no indication of slow mixing or protocol-dependent attractors for the system sizes studied. These results confirm that the phases are controlled primarily by measurement range. We have also added a brief statement on this robustness to the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from circuit definition

full rationale

The paper defines a concrete model of long-range measurement-only circuits (competing two- and three-qubit measurements with tunable range) and extracts steady-state phases, SPT edge modes, correlation behavior, and entanglement scaling directly from the stochastic projector dynamics. No equation or claim reduces by construction to a fitted parameter, self-citation, or renamed input; the measurement-range parameter is an explicit control variable, not derived from the target observables. The independence assumption on long-time attractors is an explicit modeling choice rather than a hidden definitional loop. The analysis therefore remains non-circular and externally falsifiable against the stated circuit rules.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identifiable. The work rests on standard assumptions of quantum circuit theory such as projective measurements and random application of gates.

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