Measurement-induced phase transitions in d 1-dimensional stabilizer circuits Piotr Sierant1Marco Schir o2Maciej Lewenstein1 3and Xhek Turkeshi2 1ICFO-Institut de Ci encies Fotoniques The Barcelona Institute of Science and Technology

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Measurement-induced phase transitions in (d+ 1)-dimensional stabilizer circuits

Piotr Sierant,1Marco Schir`o,2Maciej Lewenstein,1, 3 and Xhek Turkeshi2

1ICFO-Institut de Ci`encies Fot‘oniques, The Barcelona Institute of Science and Technology,

Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain

2JEIP, USR 3573 CNRS, Coll`ege de France, PSL Research University,

11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France

3ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain

(Dated: October 24, 2022)

The interplay between unitary dynamics and local quantum measurements results in unconven-

tional non-unitary dynamical phases and transitions. In this paper we investigate the dynamics

of (d+ 1)-dimensional hybrid stabilizer circuits, for d= 1,2,3. We characterize the measurement-

induced phases and their transitions using large-scale numerical simulations focusing on entangle-

ment measures, puriﬁcation dynamics, and wave-function structure. Our ﬁndings demonstrate the

measurement-induced transition in (d+ 1) spatiotemporal dimensions is conformal and close to the

percolation transition in (d+ 1) spatial dimensions.

I. INTRODUCTION

Understanding the propagation of quantum informa-

tion in a many-body quantum system coupled to an en-

vironment is a central problem at the interface between

the quantum foundation, statistical mechanics, high en-

ergy physics, and condensed matter theory [1,2]. In

the most straightforward formulation, the environment

is a measurement apparatus that continuously probes

the system’s local degrees of freedom – a setup that

plays a pivotal role in comprehending noisy intermediate-

scale quantum devices [3] and, in general, for quantum

simulation [4]. The dynamics combine quantum uni-

tary evolution and measurements in such settings, re-

sulting in stochastic quantum trajectories [5–15]. The

unitary evolution introduces coherences and scrambles

quantum information throughout the system while local

quantum measurements disentangle and localize the de-

grees of freedom according to the Born-Von Neumann

postulate. For a many-body system, this competition re-

sults in measurement-induced phase transitions (MIPTs)

for the states along quantum trajectories, visible in non-

linear functionals of the state [16–50].

Random unitary circuits are minimal models for in-

vestigating the dynamical phases of monitored quantum

systems [51–53]. In such circuits, the unitary part of

the dynamics is built out of random unitary gates drawn

from a speciﬁc distribution — hence lacking the struc-

tural constraints present in realistic Hamiltonians [54–

58]. The non-unitary dynamics stem from local pro-

jective (or generalized) measurements that act stochas-

tically at every space-time point with a speciﬁed rate.

In (1+1)-dimensional systems, intensive analytical and

numerical investigations for Haar and stabilizer random

circuits lead to a well-understood phase diagram [59–

77]. At a low-measurement rate, quantum correlations

are resilient to the disturbing action of local measure-

ments resulting in a quantum error-correcting (QEC)

phase [78,79]. In contrast, frequent measurements dete-

riorate the quantum correlations and freeze the dynam-

ics onto a reduced manifold, yielding a quantum Zeno

(QZ) phase [17,80]. A measurement-induced critical

point separates these dynamical phases, exhibiting a rich

non-unitary conformal ﬁeld theory (CFT) with geometric

features [16,81–83].

Diﬀerent but related aspects of these dynamical phases

have been identiﬁed depending on the initial condition

and the probes used to investigate the transition. Entan-

glement measures [16–18] show that the entanglement en-

tropy of a pure state in the QEC phase follows a volume

law akin to ergodic systems [84]. In contrast, in the QZ

phase, the entanglement entropy obeys an area law simi-

lar to ground states [85] or eigenstates in many-body lo-

calized systems [86]. In contrast, when the system starts

evolving from a mixed state, the environment monitoring

reduces the entropy of the system density matrix. This

allows identifying the QEC phase with a mixed phase

with exponentially divergent puriﬁcation time, distinct

from a pure phase with polynomial puriﬁcation time, cor-

responding to the QZ phase. From that point of view, the

measurement-induced phase transition becomes a puriﬁ-

cation transition [87]. Lastly, Ref. [88] has characterized

the measurement-induced criticality by directly inspect-

ing the structure of wave functions on individual quan-

tum trajectories. In the QEC phase, the participation

entropy develops a non-zero sub-leading term determined

by details of the unitary part of dynamics, which, in con-

trast, vanishes in the QZ phase.

Beyond one spatial dimension, quantum correlations

and information propagation are less comprehended. For

random Haar circuits that contain generic local unitary

gates, numerical methods are prohibitive due to the ex-

ponential (classical) resources needed for their simula-

tion. Ref. [16] proposed a phenomenological geometric

minimal-cut picture for entanglement spreading in this

generic case in the presence of random measurements

(see also Ref. [54]). Conversely, the unitary gates in sta-

bilizer circuits are chosen from the Cliﬀord group which

is a discrete sub-group of the full unitary group, allow-

ing for simulation of such circuits with classical polyno-

mial resources even in the presence of projective mea-

arXiv:2210.11957v1 [cond-mat.stat-mech] 21 Oct 2022

surements [89,90]. This enables numerical investigation

of the transition between QEC and QZ phases in higher

dimensions. In particular, Ref. [91–93] studied the (2+1)-

dimensional systems but found conﬂicting results, likely

due to the limited system sizes in their computations or

the choice of observable used to locate the transition.

Therefore, it is desirable to employ large-scale numeri-

cal simulations and clarify the phase diagram in higher

dimensions for stabilizer circuits.

This paper revisits the MIPT in (d+ 1)-dimensional

stabilizer circuits. We utilize large-scale numerics up to

N= 32768 qubits to unveil a precise characterization of

the critical exponents of the MIPT for d= 1,2, as well

as consider the previously unexplored case d= 3. We

provide a coherent picture of measurement-induced crit-

icality by probing the three aspects of dynamics of the

circuits that change between QEC and QZ phases: the

entanglement of the pure state, the puriﬁcation of ini-

tially maximally mixed state as well as the structure of

the wave function along quantum trajectories. We test

the robustness of our results by considering diﬀerent ob-

servables and various spatiotemporal architectures of the

circuits. Furthermore, we investigate the bulk-boundary

critical properties by considering ancilla qubits coupled

to our circuits. In the spirit of Ref. [94], we suggest that

the resulting statistical ﬁeld theory at the transition be-

tween QEC and QZ phases is that of a perturbed perco-

lation model, and hence the critical point is conformal.

This ansatz is compatible with our numerical ﬁndings.

We show that the critical exponents for (1+1)D circuits

are close but distinct from the percolation transition in

2dimensions, whereas the results for (d+ 1)-dimensional

circuits for d= 2,3are compatible with the percolation

transition in (d+ 1) spatial dimensions, see Table. Ifor

the summary of the results.

The rest of this work is structured as follows. First,

in Sec. II we give a brief overview of our approach and

summarize the obtained critical exponents for MIPT in

(d+ 1)-dimensional circuits. Sec. III describes the ar-

chitectures of stabilizer circuits that are employed in our

investigations of MIPTs. In Sec. IV we detail the entan-

glement entropy measures and extract the universal prop-

erties of the MIPT through ﬁnite-size scaling. Sec. Vcon-

tains results about the puriﬁcation aspect of MIPT while

Sec. VI describes the wave-function structure change at

the transition between QEC and QZ phases. Sec. VII

is devoted to the investigation of the bulk-boundary

features of the MIPT through coupling with ancillary

qubits. Finally, Sec. IV discusses our results and illus-

trates the conformal character of the critical point, while

Sec. IX concludes the manuscript with some outlooks. In

the Appendices, we detail the most technical aspects of

our work.

II. OVERVIEW OF THE RESULTS

Let us ﬁrst present the rationale of our analysis

and summarize our main ﬁndings. After motivating

the choice of observables for the quantum trajectories,

we brieﬂy discuss how they describe the measurement-

induced phases and transitions, highlighting the key re-

sults. In summary, we establish that (d+ 1) spacetime

dimensional stabilizer circuits have a MIPT with univer-

sal behavior close to a percolation ﬁeld theory in (d+ 1)

spatial dimensions. In particular, for d= 1 the stabilizer

circuit critical exponents are distinct and within ﬁve er-

ror bars from the 2D percolation exact results, while for

d= 2 (d= 3)all our estimates are compatible within one

error bar with those of 3D (4D) percolation I. In addition,

we report evidence suggesting an emergence of a confor-

mal ﬁeld theory right at the MIPT in (d+1)-dimensional

stabilizer circuits for d= 2 and d= 3.

a. Observables of interest and quantum trajectories

This manuscript considers quantum circuits build from:

(i) random unitary gates (drawn uniformly from the Clif-

ford group), (ii) projective measurements onto the local

qubit basis. In particular, a circuit realization Km(with

implicit depth t, also referred to as time) stems from

the choice of Cliﬀord gates, the spacetime location of

measurements, and the measurement outcomes. Start-

ing from the initial state ρ0, the ﬁnal state is given by

ρm=Kmρ0K†

tr(K†

mKmρ0).(1)

We denote the collective average by Em. The trajectory

nature of Eq. (1) translates to statistical properties of an

observable Ξon the state ρm. If Ξis linear in the density

matrix, it is easy to see that

Em[Ξ[ρm]] = Ξ[Em(ρm)].(2)

Notably, the above equation does not hold when Ξis a

non-linear function of the density matrix. Consider for

instance the purity Ξ[ρ] = tr(ρ2), and a pure state quan-

tum trajectory. Since the average state ρ=Em(ρm)is

mixed, we have 1 = Em[Ξ[ρm]] 6= Ξ[ρ]≤1. In particular,

the average state evolves through a quantum channel and

is insensible of the trajectory registry m. In other words,

the statistical structure of quantum trajectories contains

more information (including measurement-induced criti-

cality) than the average state. We consider entropic ob-

servables: being non-linear in the density matrix, they

allow us to investigate the various aspects of the MIPT.

Crucially, as discussed in the remaining, they are ef-

ﬁciently computable for stabilizer states, allowing ex-

tensive numerical characterization of MIPTs in (d+ 1)-

dimensional stabilizer circuits. In Sec. III we detail the

properties of stabilizer circuits and specify the circuital

architectures treated in this manuscript.

b. Entanglement transition. For a given density ma-

trix ρ, the von Neumann entropy is deﬁned as

S(ρ)≡ −tr(ρlog2ρ)(3)

Stabilizer circuits Classical percolation

Exponent 1+1D 2+1D 3+1D 2D 3D 4D

ν1.265(15) 0.87(2) 0.68(2) 1.333 0.8774(13) 0.686(2)

η0.212(4) -0.02(3) -0.04(5) 0.208 −0.03(1) -0.084(4)

ηk0.70(2) 0.99(4) 1.5(2) 0.667 1.08(10) 1.37(13)

η⊥0.461(8) 0.43(5) 0.42(10) 0.438 0.5(1) 0.65(7)

β0.129(8) 0.44(1) 0.60(3) 0.139 0.429(4) 0.658(1)

βs0.46(2) 0.86(2) 1.14(9) 0.444 0.854(2) 1.09(8)

z1.00(1) 1.01(2) 1.02(4) 1 1 1

Table I. Summary of the critical exponents for the stabilizer hybrid quantum circuits and comparison with classical percolation

theory. Systems in (d+ 1) space-time dimensions should be compared with (d+ 1) spatial dimensions classical percolation

theory. The critical exponents of 2D percolation are exact and follow by conformal ﬁeld theory [95], and here we report the

truncation of their rational values to facilitate the comparison. The value z= 1 arises from the conformal invariance of the

percolation ﬁeld theory. For the 3D and 4D percolation critical exponents, we refer to Ref. [96–98] for the bulk properties, and

for Ref. [99–102] for the surface properties.

Figure 1. Cartoon of the partitions considered for the en-

tanglement entropy and for the tripartite quantum mutual

information (TQMI) for (a) (1+1)-dimensional, (b) (2+1)-

dimensional, and (c) (3+1)-dimensional systems (we illustrate

only the dspatial dimensions of the circuits). If not speciﬁed,

we infer periodic boundary conditions in every direction. For

the analysis in Sec. IV we consider LA=LB=LC=LD=

Lx/4, and Lx=Ly=Lz. Instead, in Sec. VIII we consider

Ly=Lzwhile varying Lxand LA.

where the trace is over the available degrees of freedom.

The entanglement entropy is deﬁned for a pure state ρ=

|ΨihΨ|and a bipartition A∪Acof the system as

SA≡S(ρA), ρA= trAcρ, (4)

where trAcis the trace over degrees of freedom of subsys-

tem Ac. Within these conditions, the entanglement en-

tropy is a measure of distillable quantum information [2].

We consider a qubit system on a d-dimensional hyper-

parallelepiped Λand a subsystem Ais extensive (|A| ∼

|Λ|, where |X|is the total number of sites in X. See

Fig. 1, where Ac=B∪C∪D). Due to the trajectory na-

ture of the system, in the following, we refer to the entan-

glement entropy as its average value SA=Em[S(ρm,A)],

with ρm,X = trXc(ρm).

Then, the MIPT manifests itself by an abrupt change

in the scaling of the entanglement entropy as the mea-

surement rate pis tuned across the critical measurement

rate pc[16–18,91–93]. For p < pc, the system is in the

QEC phase, and the quantum information encoded in the

system is resilient to the local measurements, yielding a

volume-law stationary entanglement entropy SA∝ |A|.

Conversely, in the QZ phase (p>pc), the measurements

deteriorate quantum correlations restricting the creation

of quantum entanglement. As a result, the entanglement

entropy follows an area-law [85]SA∝ |∂A|, with ∂X be-

ing the boundary of X. In Sec. IV, we indeed observe

numerically the volume-law and area-law scaling regimes

for the entanglement entropy SAin stabilizer circuits in

(d+ 1)-dimensions for d= 1,2,3.

The entanglement entropy at the transition p=pctyp-

ically presents non-trivial system size dependence (for

instance, a logarithmic correction for d= 1). A mea-

sure better suited for the analysis of the entanglement

transition is the tripartite quantum mutual information

(TQMI) [81] deﬁned as

I3=SA+SB+SC−SA∪B−SA∪C−SB∪C+SA∪B∪C.(5)

Here {A, B, C, D}is the quadripartition of |Λ|in Fig. 1,

where A, B, C, D are neighboring regions of equal size.

Then, the scaling of the entanglement entropy SAin QEC

and QZ phases implies the following scaling for the TQMI

I3(p, L) = 









O(Ld), p < pc,

O(1), p =pc,

∼e−γ(p)L, p > pc,

(6)

where γ(p)>0is a pdependent constant. In particular,

in the scaling limit, the curves I3(p)for diﬀerent system

sizes Lcross at criticality, allowing us to pinpoint the

transition point. To analyze the properties of the transi-

tion, we assume an emergence of a power-law divergent

length scale at the transition

ξ∼1

|p−pc|ν,(7)

where pcis the critical measurement rate, and νis a

critical exponent. This leads us to the scaling ansatz

around the transition

I3(p, L)'F[(p−pc)L1/ν ],(8)

where F(x)is a system size independent function.

As we detail in Sec. IV, Eq. (8) allows us to obtain

excellent data collapses for (1+1)D, (2+1)D and (3+1)D

stabilizer circuits, with the critical exponents νsumma-

rized in Table I[103]. While we postpone a more detailed

discussion of those results to Sec. VIII, we note that the

critical exponent νfor (1+1)D is close but diﬀerent from

2D classical percolation, whereas the exponents νde-

scribing entanglement transition in (2+1)D and (3+1)D

circuits are compatible with the exponents of 3D and 4D

classical percolation.

c. Puriﬁcation transition. Another facet of

measurement-induced criticality reveals itself when

one initializes the system in a mixed state ρ[87]. In

such a case, discussed in Sec. V, the measurements

will increasingly resolve the state and lead to complete

puriﬁcation [74]. However, the timescale at which the

puriﬁcation happens varies considerably between the

QEC and the QZ phases. In the former, the coherences

obstruct the environment from the complete resolution

of the information content, and the state puriﬁes at

times of the order the Hilbert space dimension (meaning

that the environment has to probe virtually all available

degrees of freedom). In the latter case, the measure-

ments resolve information in a polynomial timescale in

system size in the QZ phase [104].

Starting from a fully mixed state ρ0∝11, we calculate

the von Neumann entropy S(ρm)of the time-evolved den-

sity matrix (cf. Eq. (3)), and average the results obtain-

ing the average entropy Spur =Em[S(ρm)]. In Sec. V,

we observe that at the MIPT, p=pc, the entropy Spur

becomes a universal function of t/Lzwhere tis the cir-

cuit depth and zis a dynamical critical exponent and

ﬁnd that, within estimated error bars, z= 1 for (1+1)D,

(2+1)D and (3+1)D stabilizer circuits, as shown in Ta-

ble I. This value of the dynamical critical exponent stems

from scaling invariance, hence it is compatible with a

conformal invariance at the critical point. Moreover, we

demonstrate that data for the average entropy Spur at

circuit depth τ=αL (with αis a constant) can be col-

lapsed using the ansatz

Spur =G((p−pc)L1/ν , t/L),(9)

with the same values of the critical exponents pc,νas

the ones obtained in the analysis of the tripartite mutual

information (cf. Eq. (8)), and with the dynamical criti-

cal exponent assumed to be z= 1, see Sec. V. The above

conclusions, and the coincidence of puriﬁcation and en-

tanglement transition, hold for all the spacetime dimen-

sions considered. This phenomenon is non-trivial as a

class of circuits exhibits qualitatively diﬀerent behavior

between the mixed puriﬁcation dynamics and the pure

entanglement dynamics [105–108].

d. Wave-function structure across MIPT. We study

the structural properties of stabilizer circuits employ-

ing participation entropy. This quantity is a measure

of localization of the system wave function in a given

basis of many-body Hilbert space and captures changes

in wave-function structure across quantum phase tran-

sitions [109–114], as well as in non-equilibrium settings

such as many-body localization [115–118]. The partic-

ipation entropy was also recently shown to successfully

capture the universal behavior of MIPTs in (1+1)D hy-

brid quantum circuits [88].

Given a pure state ρ=|ΨihΨ|the participation en-

tropy is deﬁned as

Spart(ρ) = −X

hσ|ρ|σilog2hσ|ρ|σi,(10)

where the sum extends over the full basis |σiof 2|Λ|-

dimensional Hilbert space. We consider the wave func-

tion at individual quantum trajectories and consider the

average value Spart =Em[Spart(ρm)]. It exhibits the scal-

ing

Spart =D|Λ|+c, (11)

where Dand care the fractal dimension and the frac-

tal sub-leading term, respectively. The coeﬃcients D

and ccan be used to analyze further the transition be-

tween QEC and QZ phases. In particular, for (1+1)D

systems, the fractal subleading term cencodes the uni-

versal content of the measurement-induced phase transi-

tion [88,119], while Dis non-universal and phase depen-

dent. In Sec. VI we show that the same conclusion holds

for (2+1)D and (3+1)D systems and perform a ﬁnite size

scaling analysis of the fractal sub-leading term with the

ansatz

c=Y((p−pc)L1/ν ),(12)

to extract the critical measurement strength pcand the

correlation length exponent ν. Our estimates of the expo-

nent νand the value of pcare consistent with the results

obtained for the entanglement and puriﬁcation transi-

tions (cf. Table I).

e. Bulk-boundary properties at MIPT. To further

characterize measurement-induced criticality in (d+ 1)-

dimensional stabilizer circuits we obtain, in Sec. VII, the

bulk and boundary critical exponents by investigating the

puriﬁcation of ancilla qubits entangled with the system

at a certain moment of time. Notably, these local order

parameters are experimentally accessible through a few

site full tomography [65,66].

We consider the combined framework |Φi≡|Ψi ⊗ |ai,

where |Ψiis the system pure state and |aiis a refer-

ence state of an ancilla qubit. The system is entangled

with the ancilla at time t0through a certain unitary op-

eration and the combined setup evolves under the quan-

tum circuit which acts non-trivially only on the system

qubits. To characterize the entanglement between the

ancilla qubit and the system, we compute the average

entanglement entropy of the ancilla qubit Sanc. The an-

cilla qubit gets quickly disentangled with the system in

the QZ phase, yielding a vanishing value for Sanc at suf-

ﬁciently large circuit depths. In contrast, in QEC phase,

for p < pc, we ﬁnd that Sanc ∝ |pc−p|˜

β. Depending on

the choice of the initial state of the system, the behavior

of Sanc determines the exponents ˜

β≡βs, i.e. the bound-

ary order parameter, and ˜

β≡β, i.e. the bulk order

parameter (as we discuss in details in Sec. VII). Table I

presents our results for the bulk and boundary order pa-

rameters βand βsfor (1+1)D, (2+1)D and (3+1)D sta-

bilizer circuits. Besides the small deviations for (1+1)D

case, the exponents β, βsfor the (d+ 1)-dimensional sta-

bilizer circuits are compatible with percolation theory in

d+ 1 spatial dimensions.

Furthermore, we set the measurement rate to be equal

to its critical value, p=pc, and use z= 1 estimated from

the entanglement and puriﬁcation observables to charac-

terize the bulk-bulk (η), bulk-boundary (η⊥), boundary-

boundary (ηk) critical exponents [81,92]. In this case,

we consider the combined framework |Ψi⊗|a, biwhere

|Ψiis the system pure state and |a, biare two ancilla

qubits. We entangle the latter at time t0with two sys-

tem qubits distant r, and, for t>t0, we study the behav-

ior of the average bipartite quantum mutual information

I2=Em[I2(ρm,a,b)] where

I2(ρm,ab)≡S(ρm,a) + S(ρm,b)−S(ρm,ab).(13)

Using I2, we access all the anomalous critical exponents

˜η={η, η⊥, ηk}by considering circuits with diﬀerent

boundary conditions and diﬀerent starting times t0. The

exponents ˜ηare determined by the behavior of the bipar-

tite quantum mutual information

I2(t, r) = 1

rd−1+˜ηG((t−t0)/r)(14)

where ˜η∈ {η, η⊥, ηk}depends on the setup, as speciﬁed

in Sec. VII. Table Ipresents our results for the various ˜η

in (1+1)D, (2+1)D and (3+1)D stabilizer circuits. Be-

sides the small deviations for (1+1)D case, for the (d+1)-

dimensional stabilizer circuits, these exponents are com-

patible with percolation theory in d+ 1 spatial dimen-

sions.

f. Entanglement entropy behavior at the transition.

Lastly, we study the entanglement entropy at the transi-

tion and report evidence of a conformal critical behavior.

Speciﬁcally, we consider (2+1)D and (3+1)D circuits and

calculate the torus entanglement entropy introduced in

Ref. [120] for (2+1)D and (3+1)D systems with confor-

mal invariance.

Assuming that a conformal ﬁeld theory describes the

critical point, the scaling of the entanglement entropy at

p=pcfor region Ain d > 1dimensions is given by

S(A) = B|∂A| − χ+O(1/|∂A|),(15)

where Bis a non-universal coeﬃcient, and χis a universal

constant. The latter can be analytically computed from

an exemplary model with conformal invariance at d > 1,

i.e. the Extensive Mutual Information model [121,122].

Following the guideline of Ref. [120], in Sec. VIII, we nu-

merically compute the torus entanglement entropy ﬁxing

the aspect ratio b=Lx/Ly(Ly=Lz) (see Fig. 1) for

the (2+1)D and (3+1)D circuits, and comparing it with

the analytical predictions within the CFT. We ﬁnd that

the analytical predictions of CFT match the numerical

data, and observe diminishing discrepancies with increas-

ing system size.

III. STABILIZER CIRCUITS

Stabilizer states and circuits are pivotal in quantum

error correction [2,89] and in the study of MIPT [17,

94,123]. This is due to a combination of their classical

simulability via the Gottesman-Knill theorem [90], abil-

ity to host an extensive entanglement [124,125] as well

as the fact that they form a unitary-2 design [126] (see

also [127,128]). This section reviews stabilizer circuits

and deﬁnes the circuital architectures considered in the

manuscript. The system is deﬁned on the d-dimensional

hyper-parallelepiped lattice Λ, with i∈Λits sites. We

shall consider both periodic and open boundary condi-

tions, depending on the observables of interest.

In our numerical calculations, we employ the state-

of-the-art package Stim [129] which relies on the ideas

of [89,90,130]. The package Stim allows for an eﬃ-

cient numerical simulation of stabilizer circuits with time

and memory complexity scaling polynomially in system

volume |Λ|. Below, we outline the main ideas underly-

ing simulation of stabilizer circuits and provide details of

spatiotemporal architectures of circuits employed in this

work.

A. Round-up on stabilizer circuits

Here, for self-consistency and completeness, we brieﬂy

review the crucial properties of stabilizer circuits and how

they are eﬃciently simulatable via the Gottesman-Knill

theorem [90].

Apure stabilizer state |Ψion N=|Λ|qubits is char-

acterized by a group of Pauli strings GΨ={g}such that

g|Ψi=|Ψiand |GΨ|= 2N. In this paper, we shall con-

sider GΨto be a subgroup of all the Pauli strings acting

on the lattice Λ, whose elements are in the form

g=eiπφ Y

i∈Λ

(XniZmi).(16)

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Measurement-inducedphasetransitionsin(d+1)-dimensionalstabilizercircuitsPiotrSierant,1MarcoSchiro,2MaciejLewenstein,1,3andXhekTurkeshi21ICFO-InstitutdeCienciesFot`oniques,TheBarcelonaInstituteofScienceandTechnology,Av.CarlFriedrichGauss3,08860Castelldefels(Barcelona),Spain2JEIP,USR3573CNRS,Collegede...

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Measurement-induced phase transitions in d 1-dimensional stabilizer circuits Piotr Sierant1Marco Schir o2Maciej Lewenstein1 3and Xhek Turkeshi2 1ICFO-Institut de Ci encies Fotoniques The Barcelona Institute of Science and Technology.pdf

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Measurement-induced phase transitions in d 1-dimensional stabilizer circuits Piotr Sierant1Marco Schir o2Maciej Lewenstein1 3and Xhek Turkeshi2 1ICFO-Institut de Ci encies Fotoniques The Barcelona Institute of Science and Technology

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