Demons on a Budget: Adaptive Measurement Placement at the Entanglement Phase Transition

Rohan Pandey

quant-phAug 16, 2026

Demons on a Budget: Adaptive Measurement Placement at the Entanglement Phase Transition

Authors: Rohan Pandey

Organizations: Independent researcher

Abstract

Monitored quantum circuits exhibit a measurement-induced phase transition between volume-law and area-law entanglement as a function of the measurement rate $p$ . Prior work places measurements at random locations and treats the rate as the control parameter. We instead fix the measurement budget and vary the placement process, comparing random placement against hand-designed and learned policies in brickwork random Clifford circuits at matched budget. First, placement geometry matters more than placement information. A deterministic contiguous sweep cuts the half-cut entropy by a factor of 3.4 relative to random placement, while equal-coverage unstructured placement and a greedy policy with full state access do far worse. The effect is carried by spatial order alone: measuring the $k$ least recently measured sites gives $4.14 \pm 0.06$ bits with random tie-breaking and $1.29 \pm 0.04$ bits with position-ordered tie-breaking. Second, the sweep eliminates the transition rather than shifting it. Tripartite mutual information crossings recede as $p^* \propto 1/L$ , the steady-state entropy saturates at an $L$ -independent ceiling near $0.46/p$ , and data for $64 \le L \le 512$ collapse onto the form $S = p^{-1} f(pL)$ predicted by a ballistic regrowth argument. Third, in stabilizer dynamics every outcome is deterministic or a fair coin flip, so the record's Shannon entropy is exactly countable; the sweep dominates the entropy-versus-record-cost frontier while paying the same roughly one bit per measurement as random placement. Policies trained by cross-entropy and proximal policy optimization do not find the sweep: score-based policies parameterize which sites to measure, not the order in which degenerate scores are resolved, and the effect lives in that order. The phase diagram of monitored dynamics is a property of the placement process, not only of the measurement rate.

Figures & tables

Figure 1: Random placement baseline. Tripartite mutual information versus measurement rate for $L=64$ – $512$ at depth $T=4L$ . Inset: the crossing region. The dashed line and shaded band give the mean and spread of the four locally significant pairwise crossings, $p_{c}=0.158\pm 0.003$ , consistent with the established value $p_{c}\approx 0.16$ for this model [ 21 ] .

Figure 2: Identical budget, different geometry. (a) Sweep placement (dashed, orange) drives $\mathcal{I}_{3}$ to zero one to two decades in $p$ below the random-placement transition (solid, blue); shade encodes $L$ . Under random placement the curves steepen with $L$ and cross at a fixed rate; under the sweep they separate and recede. (b) At fixed $p=0.1<p_{c}$ the random-placement entropy grows linearly in $L$ while the sweep entropy is flat.

Figure 3: Ballistic scaling test for sweep placement, $L=64$ – $512$ . (a) Raw half-cut entropy versus system size. Curves peel away from the volume law $s_{v}L/2$ (dotted) and flatten. (b) The same data rescaled: $p\,S$ against $pL$ collapses onto a single curve for $p\leq 0.035$ (colored), with small-argument slope $s_{v}/2$ and plateau $f_{\infty}\simeq 0.46$ (dotted and dashed guides). Rates above $0.035$ (grey) fall below the plateau, as expected from lattice-scale corrections once $1/p$ approaches the gate spacing.

Figure 4: Entanglement profile

S([0,x))

as a function of cut position

x

and layer

t

under five placement policies at matched budget (

L=24

k=3

per layer, one trajectory each, shared color scale). Blue dots mark measurement events; titles give the ensemble mean over

400

episodes. Panels (d) and (e) apply the same rule, “measure the

k

least recently measured sites,” and differ only in how ties are broken: at random in (d), in position order in (e). Random tie-breaking freezes a scattered permutation and performs like random placement; ordered tie-breaking produces the advancing front of the sweep and the lowest entropy of any policy studied. The learned linear policy (c) pins measurements at two fixed sites and fences entanglement into the unmeasured region.

Policy	Information	$\langle S(L/2)\rangle$ [bits]	$H_{\rm rec}$
Random placement	none	$4.36\pm 0.06$	$0.95$
Shuffled coverage	none	$4.08\pm 0.06$	$0.96$
Oracle greedy	quantum	$3.56\pm 0.06$	$0.99$
Learned linear (CEM)	classical	$3.26\pm 0.09$	$0.88$
Stalest, random tie-break	classical	$4.14\pm 0.06$	$0.97$
Stalest, ordered tie-break

Table 1: Final half-cut entropy at matched budget (

L=24

k=3

per layer,

T=96

; mean

\pm

s.e.m. over

400

episodes).

H_{\rm rec}

is the exact record entropy per measurement, from the lemma of Sec. III . The last two rows apply the same staleness rule and differ only in how ties are broken.

Figure 5: The information ledger at $L=64$ . (a) The sweep dominates the entropy-versus-record-cost frontier: at every budget of recorded bits it reaches a lower entanglement entropy than random placement. Labels give the corresponding measurement rate. (b) Both placements pay between $0.85$ and $0.99$ bits of record entropy per measurement over the whole range of rates, so the advantage in (a) is not cheaper information.

Policy	Information	$\langle S(L/2)\rangle$ [bits]	$H_{\rm rec}$
Random placement	none	$5.27\pm 0.08$	$0.96$
Shuffled coverage	none	$4.94\pm 0.08$	$0.97$
Stalest, random tie-break	classical	$4.87\pm 0.08$	$0.97$
PPO (stochastic)	classical	$5.38\pm 0.08$	$0.96$
PPO (greedy decoding)	classical	$5.87\pm 0.10$	$0.87$
Oracle greedy	quantum	$4.24\pm 0.08$	$0.99$

Table 2: Final half-cut entropy at matched budget (

L=32

k=4

per layer,

T=128

; mean

\pm

s.e.m. over

300

episodes).

H_{\rm rec}

is the exact record entropy per measurement.

Phase Transitions

Phase transitions are abrupt changes in collective behavior as temperature, density, noise, or other control parameters vary.

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Quantum Machine Learning

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Topic pending

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Abstract

Phase Transitions

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