Measurement-Induced Phase Transitions in Sparse Nonlocal Scramblers

Tomohiro Hashizume,1 Gregory Bentsen,2 and Andrew J. Daley1

1Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom2Martin A. Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02465, USA

(Dated: September 24, 2021)

Measurement-induced phase transitions arise due to a competition between the scrambling ofquantum information in a many-body system and local measurements. In this work we investigatethese transitions in different classes of fast scramblers, systems that scramble quantum information asquickly as is conjectured to be possible – on a timescale proportional to the logarithm of the systemsize. In particular, we consider sets of deterministic sparse couplings that naturally interpolatebetween local circuits that slowly scramble information and highly nonlocal circuits that achievethe fast-scrambling limit. We find that circuits featuring sparse nonlocal interactions are able towithstand substantially higher rates of local measurement than circuits with only local interactions,even at comparable gate depths. We also study the quantum error-correcting codes that support thevolume-law entangled phase and find that our maximally nonlocal circuits yield codes with nearlyextensive contiguous code distance. Use of these sparse, deterministic circuits opens pathwaystowards the design of noise-resilient quantum circuits and error correcting codes in current andfuture quantum devices with minimum gate numbers.

I. INTRODUCTION

Measurement-induced phase transitions (MIPT) [1–4]are transitions driven by a competition between scram-bling dynamics, which tends to generate many-body en-tanglement across all degrees of freedom of a quan-tum many-body system, and local measurements, whichtend to destroy this entanglement. Measurement-driventransitions of this type have garnered an extraordinaryamount of theoretical interest recently [5–20], and havebeen studied in a wide range of physical systems, includ-ing in random quantum circuits, free fermions [21–23],and non-Hermitian Brownian models [24]. They havealso been extended to systems with long-range interac-tions [25–28], including models with all-to-all coupling[29, 30], and non-Hermitian Sachdev-Ye-Kitaev (SYK)models [31–33]. MIPTs have also been related to purifi-cation transitions [34], where entanglement of the systemundergoing MIPT with an ancilla can be used to detectthe transition [35], as recently demonstrated in experi-ments with trapped ions [36]. Importantly, these tran-sition have also been related to the properties of quan-tum error-correcting codes (QECC) [11, 34]. These codesare dynamically generated during the evolution and areresponsible for protecting the extensive many-body en-tanglement present in the system despite repeated localmeasurements.

The durability of this many-body entangled phase, de-spite the presence of repeated projective measurements,is particularly compelling from the perspective of exper-iments with modern quantum technologies. In this con-text, the goal is often to generate interesting or usefulentangled quantum states, while always contending withnoise and dissipation which inevitably degrade this en-tanglement [4]. In this context, it is therefore of con-siderable value to identify mechanisms for building uprobust patterns of entanglement as quickly and as effi-ciently as possible before noise has a chance to degrade

the computation or simulation being carried out. Par-ticularly efficient quantum systems that build up many-body entanglement as rapidly as possible are known asfast scramblers [37, 38]. Fast scrambling was first dis-cussed in the context of information spreading in blackholes, and refers to systems that can scramble quantuminformation on timescales t∗ ∼ logN that grow only log-arithmically with the system size N [37–41]. Random all-to-all circuits [42], and other disordered all-to-all coupledmodels such as the SYK model [43, 44] are good examplesof fast scramblers, but recently, a number of determin-istic and experimentally feasible systems exhibiting fastscrambling have also been explored [45–48]. These showsimilar dynamics, but often on sparse nonlocal couplinggraphs. In this context, we are led to the natural ques-tions: how do these deterministic fast scrambling circuitsperform when subjected to random measurements? Doesfast scrambling dynamics uniformly enhance the stabilityof the mixed phase as a general principle, and can thislead to improved quantum error-correcting code proper-ties?

In this work we explore these connections by studyinghow the strength of scrambling affects the properties ofthe MIPT mixed phase by considering sparsely-coupledquantum circuits with tunable nonlocal interactions. Byadjusting the range of interactions in these sparse cir-cuits, we can tune from local spreading of information togenuine fast scrambling. Our analysis demonstrates thatsparse nonlocal interactions can significantly improve aquantum system’s robustness to local measurements. Weshow that only a few additional layers of nonlocal gatessubstantially increase the critical measurement rate pc.We further demonstrate that highly nonlocal interactionsimprove the properties of the quantum error-correctingcodes that stabilize the mixed phase.

As specific examples, we choose sparse nonlocal cir-cuits that can be realised in the laboratory by combin-ing nearest-neighbor Rydberg interactions with nonlocal

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FIG. 1. Sparse fast scrambling PWR2k circuits with tunablenonlocality k subjected to random projective measurements.(a,b) Two-qubit entangling gates Qij (green, turquoise, blue,purple) are applied between qubits i, j if and only if theyare separated by an integer power of 2, |i− j| = 2m−1 form = 1, . . . , k. These gates are applied in consecutive interac-tion layers at times t = 1, . . . , T in a nonlocal bricklayer pat-tern arranged into alternating even and odd blocks. Projec-tive single-qubit measurements (red crosses) occur with prob-ability p between each interaction layer. The degree k of theresulting interaction graph (b) controls the nonlocality of in-teractions where the longest-range interactions occur betweenqubits initially separated by dmax = |i− j| = 2k−1. (c) Thecritical measurement rates for the purification transition pcp(red) and entanglement transition pce (blue) increase signifi-cantly with k, interpolating between the critical points foundfor random nearest neighbor (NN) models for k = 1 and ran-dom all-to-all (AA) models for larger k. Haar-random circuitswith identical coupling patterns show a similar improvementin robustness to measurement as k increases (green). (d) Inthe mixed phase for N = 256 qubits, the maximum code dis-tance dcode improves with k at fixed code rate rcode. Errorbars are shown or are smaller than datapoints; lines are guidesto the eye.

shuffling operations [48]. The strength of scrambling inthese circuits is governed by a nonlocality parameter k.In experimental implementations with Rydberg atomsthis parameter is controlled by the number of shuffle op-erations applied, thereby providing a natural tunable pa-rameter controlling the strength of scrambling in thesecircuits. In particular, the case k = 1 corresponds toa nearest neighbour (NN) circuit with slow scrambling,while k ∼ log2N yields highly nonlocal circuits whichrapidly scramble quantum information. In the follow-

ing we study how these circuits perform when subjectedto random projective measurements with a view towardunderstanding how nonlocal interactions can improve thestability of the MIPT mixed phase.

The rest of this paper is organised as follows: InSection II we introduce the deterministic PWR2k cir-cuit models that we study in this work. In Section IIIwe study measurement-induced transitions in PWR2kcircuits featuring Haar-random gates, which allows theproblem to be mapped to a classical percolation prob-lem. In Section IV we study the entanglement transitionnumerically in PWR2k circuits featuring Clifford gatesand characterize the transition in terms of the tripartitemutual information I(A : B : C) between three consecu-tive regions of the output qubits. In Section V we studythe purification transition numerically for the same Clif-ford circuits as characterized by the purification time τof a single qubit. In Section VI we numerically study theproperties of the dynamically-generated error-correctingcode in the mixed phase, and demonstrate that nonlo-cal interactions allow for improved code distance at fixedcode rate. In Section VII we consider the ‘complete’PWR2k circuit with k = log2N and compare its behaviorto the random all-to-all model. Finally, we discuss out-standing issues and prospects for future work in SectionVIII.

II. MODELS

The models we study here feature sparse nonlocal in-teractions in which spins residing at sites i, j = 1, . . . , Nof a 1d lattice are coupled if and only if the distancebetween them is an integer power of 2: |i− j| = 2m−1

for m = 1, . . . , k as shown in Fig. 1a. Sparse nonlo-cal interactions of this type have been shown to gener-ate fast scrambling dynamics [45], and are inspired bythe nonlocal coupling patterns that can be engineered insingle-mode cavities [49] or in Rydberg arrays with theaid of tweezer-assisted shuffling operations [48]. We ex-pect the fast scrambling dynamics in these nonlocal cir-cuits to rapidly generate many-body entanglement thatis increasingly robust to the destructive effects of localmeasurements relative to circuits featuring only local in-teractions.

In the following we demonstrate these expectationsexplicitly by studying sparse nonlocal Floquet circuitsconsisting of consecutive layers of pairwise nonlocal two-qubit gates Qij = Qji. The gates are arranged in a non-local bricklayer pattern, as illustrated in Fig. 1b, whereeach timestep t = 1, . . . , T represents a single interac-tion layer consisting of exactly N/2 gates. These inter-action layers are stacked into an alternating sequence ofeven and odd interaction blocks. During the even blockt = m = 1, . . . , k we place gates Qij between qubits i < jif and only if |i− j| = 2m−1 and mod(bi/2m−1c, 2) = 0.During the subsequent odd block t = k+m = k+1, . . . , 2kwe place gates according to the same rules but with the

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odd-bricklayer condition mod(bi/2m−1c, 2) = 1. Finally,we apply a layer of single-qubit phase gates Pi to allqubits and repeat the entire sequence until a final timet = T . This alternating sequence of even and odd nonlo-cal interaction blocks ensures that quantum informationrapidly spreads through the system via nonlocal inter-actions as rapidly as possible within a single block, butis not siloed into one of the hierarchical branches of thenonlocal circuit that occurs if, for example, the circuitconsists only of even blocks. Unless otherwise stated, themodels which appear in this paper always assume peri-odic boundary conditions.

Following each interaction layer t we randomly ap-ply projective measurements in the Pauli-z basis to in-dividual qubits with probability p as illustrated by thered crosses in Fig. 1b. These projective measurementstend to destroy the long-range entanglement built up bythe previous interaction layers of the PWR2k circuit.The competition between these two effects leads to ameasurement-induced phase transition in the entangle-ment structure of the output state [7, 29, 34]. In thispaper, we are particularly interested in how this transi-tion changes as we add more nonlocal interactions.

In the PWR2k family, this degree of nonlocality is con-trolled by the parameter k ≤ log2N , which determinesthe distance of the longest-range interactions dmax =|i− j|max = 2k−1. Thus, PWR2k circuits with k ∼ O(1)consist only of short-range couplings which restrict thespread of quantum information [50, 51] and thereforegenerate only slow information scrambling. By contrast,circuits with k ∼ log2N consist of highly nonlocal in-teractions that span the entire system and are known torapidly scramble quantum information [45, 48, 52]. Thenonlocality parameter k therefore provides a convenientmeans by which to interpolate between the local and non-local limits of the PWR2k family.

These two extreme cases can be compared totwo prototypical scrambling circuits known to exhibitmeasurement-induced transitions and that have beenstudied extensively in existing literature. For local in-teractions k ∼ O(1) we compare to a random 1+1Dnearest-neighbor (NN) circuit [34] in which the scram-bling of quantum information is highly restricted by localLieb-Robinson bounds [50–52]. For highly nonlocal in-teractions k ∼ log2N we compare to a random all-to-all(AA) circuit in which qubits interact pairwise at randomwithout regard to their spatial location [29, 34]. Thesemaximally nonlocal AA models were first proposed bySekino and Susskind [37] as prototypical examples of fastscramblers, systems which scramble quantum informa-tion at the fastest rate allowed by quantum mechanics.

In the remainder of this paper, we study the entangle-ment properties of these monitored PWR2k circuits as afunction of measurement rate p and nonlocality k usinga combination of analytical and numerical methods. Weconsider two types of gates Qij in this paper. In Sec.III we take Qij to be Haar-random unitary gates actingbetween pairs of qudits with Hilbert space dimension q,

which maps to a classical bond percolation problem ona degree-4 network in the limit q → ∞. In Secs. IV- VI we take Qij = CZijHiHj , where CZij = CZji isthe controlled-Z gate on qubits i, j and Hi, Pi are theHadamard and Phase gates on qubit i, respectively. Inthis case the entire circuit is composed of Clifford gatesand can therefore be efficiently simulated on a classicalcomputer [53, 54]. In each of these cases we find that thecritical measurement rate pc significantly improves withk, indicating that nonlocal interactions substantially im-prove a deterministic quantum circuit’s ability to with-stand local measurements by leveraging nonlocal scram-bling interactions.

III. PERCOLATION TRANSITION INHAAR-RANDOM PWR2k CIRCUITS

To demonstrate that nonlocal sparse interactions canimprove a system’s robustness to local measurements, wefirst study PWR2k circuits consisting of Haar-randomnonlocal gates Qij acting between pairs of qudits withlocal Hilbert space dimension q. In the limit q → ∞,the circuit and the randomly applied projective mea-surements can be mapped to a bond percolation problem[2, 29, 34]. We illustrate this mapping in Fig. 2a, whereeach two-qubit gate Qij (green) corresponds to a ver-tex in the percolation network while the qubit worldlines(black) correspond to four bonds connected to this ver-tex. Projective measurements (red) correspond to cut-ting these bonds. We discuss this mapping more explic-itly in Appendix A [55]. Applying this mapping to theentire Floquet circuit illustrated in Fig. 1b yields a non-local bond percolation network whose bonds are cut withprobability p.

For sufficiently small p this nonlocal network perco-lates, in the sense that the network contains at least oneconnected component that extends from the input qubitbonds at t = 0 to the output bonds at t = T . In thethermodynamic limit N → ∞ and for extensive timeT = O(N), this corresponds to a network with at leastone connected component of infinite extent. The perco-lation critical point, pc,Haar, is defined as the maximummeasurement rate p for which the lattice percolates inthe thermodynamic limit. For p < pc,Haar, successfulpercolation implies that the output qubits at t = T re-tain partial information about the input state at t = 0.By contrast, for p > pc,Haar, failure to percolate indi-cates that any information embedded in the initial stateat t = 0 is lost due to the proliferation of projective mea-surements in the circuit.

Classical bond-percolation transitions of this type arewell-studied in the literature [56], including on a widevariety of local and nonlocal networks. In the extremelocal case of nearest-neighbor interactions (i.e. k = 1in the PWR2k family), the percolation mapping yields abond percolation problem on a two dimensional squarelattice, whose critical properties are known analytically

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[56, 57]. In this limit, one can naturally define a corre-lation length ξ which captures the typical size of con-nected components in the network. Near the criticalpoint pNN

c,Haar = 1/2 the correlation length diverges al-

gebraically like ξ ∼ |p− pc,Haar|−ν , where ν = 4/3 is thecritical exponent [2, 56, 57].

The classical bond-percolation transition has also beenstudied in the opposite nonlocal limit where interactionsbecome arbitrarily long-range. The percolation criticalpoint of the random AA model was studied early on byGullans and Huse [34] and more extensively by Nahum etal. [29]. The critical point of the AA network is knownto be at pAA

c,Haar = 2/3. Such a high critical point comesfrom the system’s locally tree-like structure in the limitof large system size, where the probability of the networkhaving a local loop vanishes as 1/N [29].

Here we interpolate between these two extreme limitsusing the nonlocality parameter k in the Haar-randomPWR2k circuit family, where k = 1 corresponds to theNN case and k ∼ log2N approaches the AA case. Inorder to find the critical point pc,Haar for each value of k,we numerically simulate bond percolation on the corre-sponding network using the Newman-Ziff Algorithm [58].The critical point pc,Haar and the critical exponent ν ofthe correlation length can be identified by calculating thebinder cumulant [59, 60],

b(p) =1

2

(3−

⟨C4

max(p)⟩

〈C2max(p)〉2

)(1)

where Cmax is the maximum cluster size in the nonlocalpercolation network and 〈. . . 〉 denotes the averaging overdifferent configurations of randomly-cut bonds. Near thecritical point, we expect the Binder cumulant to obey thescaling law [59–61]

b(p) = f(

(p− pc,Haar)N1/ν), (2)

which is governed by the same critical exponent ν thatcontrols the divergence of the correlation length ξ nearthe critical point.

We plot the results of these numerical simulationsin Fig. 2b for k = 1, 3, 5 across system sizes N =64, . . . , 1024 (light to dark blue). To extract the criticalpoint pc,Haar we plot the Binder cumulant b(p) (insets ofFig. 2b) and locate the crossing point as a function ofsystem size N . At each fixed k in this analysis, we onlyconsider the crossing point for sufficiently large systemsizes N > 2k, such that the longest-range interactionsdmax = 2k−1 are never extensive. We then fit the scalingform (2) to the data near the critical point and use thisto extract an estimate of the critical exponent ν for eachvalue of k. Our resulting estimates for the critical pointand critical exponent can be used to collapse the Bindercumulant b(p) near the critical point to a single universalcurve (2) as shown in the main panels of Fig. 2b.

We plot the resulting critical points pc,Haar and criticalexponents ν as a function of k in Fig. 2c. The critical

FIG. 2. Measurement-induced transitions in the PWR2k cir-cuit family with Haar-random gates. (a) Nonlocal circuits fea-turing Haar-random gates can be mapped to a classical per-colation network, where gates in the original circuit (green)correspond to sites in the percolation network, and projec-tive measurements in the Haar-random circuit (red crosses)correspond to cut bonds in the percolation network. (b) TheBinder cumulant b(p) of the maximum cluster size in the clas-sical percolation network for k = 1, 3, 5 (top to bottom) andsystem sizes N = 256, . . . , 2048 (light to dark blue). The crit-ical point pc,Haar is extracted from the crossing point of b(p)across finite-size systems (insets). We observe good scalingcollapse with critical exponent ν = 4/3 expected for a 1+1DNN circuit even for large nonlocality k. (c) The percolationcritical point pc,Haar (green) increases with k; the critical ex-ponent ν (black) is nearly constant as k varies, indicatingthat these nonlocal circuits near criticality likely belong tothe same universality class for all k < log2N . Error bars areshown or are smaller than markers; lines are guides to theeye.

points clearly increase with nonlocality k; for k ≥ 5, thecritical point is closer to the AA limit than the NN limit.The critical exponent ν, on the other hand, appears tobe largely independent of k and is consistent with thecritical exponent ν = 4/3 expected for a 1+1 NN model(Fig. 2c, dotted black line). These results suggest thatall Haar-random PWR2k models with k < log2N fallinto the same universality class as the local 1+1D modelin the thermodynamic limit. Nevertheless, these resultsalso demonstrate that the critical point pc,Haar, a non-universal parameter, significantly increase with nonlocal-ity k.

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FIG. 3. Measurement-induced entanglement transition inthe PWR2k circuit family with Clifford gates. (a) The circuitis initialized with a separable pure input state, while the out-put is divided into four equal regions A,B,C,D. (b) Tripar-tite mutual information I(A : B : C) for consecutive regionsA,B,C in the PWR2k circuit for k = 1, 3, 5 (top to bot-tom) and system sizes N = 64, . . . , 2048 (light to dark blue).The critical point pce is extracted from finite-size scaling forN = 64, . . . , 2048 (insets); only sufficiently large system sizesN > 2k are used to estimate the critical point. Near pce weobserve a scaling collapse with the critical exponent ν ≈ 1.28.(c) The critical measurement rate pce (blue) increases signif-icantly with k while the critical exponent ν (black) is consis-tent with the 1+1D critical exponent (dotted black) to withinstatistical fluctuations for all values of k. Error bars are shownor are smaller than datapoints; lines are guides to the eye.

IV. ENTANGLEMENT TRANSITION INCLIFFORD PWR2k CIRCUITS

The nonlocal classical percolation network we stud-ied in the previous section demonstrated that just afew additional layers of nonlocal interactions can sig-nificantly increase the critical point of the Haar-randomPWR2k circuit. In the following sections we turn our at-tention to deterministic nonlocal circuits composed en-tirely of Clifford gates. We choose interaction gatesQij = Qij = CZijHiHj and show that the circuit’s abil-ity to withstand the destructive effects of local measure-ment is improved as a function of the nonlocality param-eter k.

We begin our study of these nonlocal Clifford circuitsby characterizing the entanglement at the output of the

circuit as a function of the measurement rate p. We ini-tialize the monitored PWR2k circuit with a pure, sepa-rable z-polarized state, and find a phase transition in thetripartite mutual information I(A : B : C) between threeequal-size consecutive regions A,B,C of the output stateas shown in Fig 3a. The tripartite mutual information

I(A : B : C) = I(A,B) + I(A,C)− I(A,BC) (3)

is defined in terms of the mutual information I(A,B) =

S(2)A +S

(2)B −S

(2)AB , where S

(2)A = − ln Tr

[ρ2A

]is the Renyi

entropy of the subsystem A. Because the circuit con-

sists entirely of Clifford gates, the Renyi entropy S(2)A is

always equal to the conventional von Neumann entropySA = −Tr [ρA ln ρA] and we may therefore characterizethe entanglement of the system entirely in terms of the

Renyi entropies S(2)A .

To extract the entanglement critical point pce, we per-form a finite-size scaling analysis similar to the previoussection. In Fig. 3b, we plot the tripartite mutual infor-mation as a function of measurement rate p for varioussystem sizes N = 64, . . . , 1024 (light to dark blue). Thecritical point pce is determined by the crossing point ofthe tripartite mutual information across system sizes asshown in the insets of Fig. 3b. Only sufficiently large sys-tem sizes N > 2k are used to extract the critical point.Near the critical point pce we expect the tripartite mutualinformation to obey the universal scaling law

I(A : B : C) = f(

(p− pce)N1/ν)

(4)

where f is a universal function and ν is the critical expo-nent of the correlation length ξ. After determining thecritical point pce we fit this scaling form to each curve inFig. 3b and use this to extract an estimate for the criticalexponent ν. The resulting estimates for pce, ν allow usto collapse the original data down to a universal curve asshown in the main panels of Fig. 3b [34].

We plot the resulting estimates for the critical pointpce and critical exponent ν in Fig. 3c for k = 1, . . . , 6.Similar to our findings in the previous section, the non-universal critical point pce increases significantly with k.The critical exponent, however, is nearly constant acrossk, and is consistent with the critical exponent ν ≈ 1.28(2)found numerically for 1+1D NN models [34] to withinstatistical fluctuations (Fig. 3c). This indicates that themeasurement-induced transitions in these models likelyfall into the same universality class as the completelylocal NN models. Nevertheless, the fact that the criti-cal point pce increases with the nonlocality parameter kdemonstrates that just a few additional layers of nonlocalinteractions can substantially improve a circuit’s abilityto retain complex many-body entanglement even in thepresence of local measurements, similar to our observa-tions in previous sections.

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V. SINGLE-QUBIT PURIFICATION INCLIFFORD PWR2k CIRCUITS

FIG. 4. Single-qubit purification transition in the PWR2kcircuit family with Clifford gates. (a) Schematic diagram forsingle-qubit purification. (b) Finite-size scaling for single-qubit purification time τ in the PWR2k circuit for k = 1, 3, 5(top to bottom) and system sizes N = 64, . . . , 2048 (light todark blue). Main figures show scaling collapse with criticalexponent ν ∼ 1.30 and dynamical critical exponent z = 1 fork = 1, 3, 5. The critical point pcp is determined by the cross-ing point of τ/Lz=1 across system sizes N > 2k (insets). (c)The purification critical point pcp increases as a function ofk, closely mirroring the increase in the entanglement criticalpoint pce found in the previous section. The critical expo-nent, on the other hand, agrees with the ν for k = 1 found inthe entanglement criticality in the previous section within theerror (green solid line, with 1-sigma fluctuations indicated bythe green dashed line). Error bars are shown or are smallerthan the data points; lines are guides to the eye.

Another measure of a circuit’s robustness to measure-ments is the timescale τ required to purify a single qubitthat has been entangled with the system [26, 35]. Absentany measurements such a qubit will remain entangledwith the system forever, but carefully-placed projectivemeasurements can destroy the entanglement between thesystem and qubit, causing the qubit to collapse into apure state. The typical timescale τ required for this pu-rification process to occur can be used to characterize ourcircuit’s ability to retain quantum information encodedin the initial state when subjected to noise. In particu-lar, below a critical measurement rate p < pcp we expectthe purification time to become extensive τ ∼ Nz, indi-

cating that quantum information can be robustly storedin the circuit despite the presence of repeated projectivemeasurements at a rate p.

Although for nearest-neighbor circuits the purificationcritical point pcp coincides with the entanglement criticalpoint pce, we emphasize that these transitions can gener-ically be different, as has been pointed out in the liter-ature [34]. We are especially interested whether highlynonlocal circuits support an intermediate phase in be-tween the two critical points pcp, pce where the referencequbit has been purified, but where there is neverthelessstill volume-law entanglement in the final state.

Here we study the possibility of such an intermediatephase in our nonlocal PWR2k circuits by determiningthe single-qubit purification time τ . To determine thepurification critical point pcp, we maximally entangle asingle reference qubit R with the system Q and apply t =4N layers of a unitary NN circuit to the system such thatthe qubit of information is scrambled deeply within thesystem [26] (Fig. 4a, Uthermalizer). Then the monitoredPWR2k circuit is applied as illustrated in Fig. 4a. Atthe end of each timestep t we compute the Renyi entropy

S(2)Q of the system. The single qubit purification time τ is

the number of timesteps required for the Renyi entropy

to vanish from its initial value of S(2)Q (0) = ln 2.

To extract the critical point pcp for each k < log2N weperform a finite-size scaling analysis as shown in Fig. 4bfor k = 1, 3, 5. Similar to the analysis of previous sec-tions, the critical point is determined by the location ofthe crossing point of τ(p) as the system size N is var-ied, as shown in the insets of Fig. 4b. The critical pointspcp obtained from this analysis are plotted in Fig. 4c andgrow significantly with k = 1, . . . , 6 in agreement withearlier analysis. In each of these cases the critical pointpcp of the purification transition agrees with the entan-glement critical point pce within error bars. This stronglysuggests that for k < log2N the critical points pcp, pceare in fact identical and that there is no intermediatephase between the purification and entanglement phases[34].

Near the critical point, we expect the purification timeto obey the universal scaling law

τ(p) = Nzf(

(p− pcp)N1/ν)

(5)

where z is the dynamical exponent and ν is the critical ex-ponent of the correlation length ξ. Based on our findingsin previous sections, we expect the purification transitionstudied here for k < log2N to be in the same universalityclass as the purification transition in 1+1D NN models,which are believed to be governed by a conformal fieldtheory with dynamical exponent z = 1 [2, 26, 34]. Wetherefore assume z = 1 and use the scaling form (5) tofit the data plotted in Fig. 4b. These fits yield estimatesfor the critical exponent ν, which we plot in Fig. 4c. Theresulting critical exponents ν = 1.30 ± 0.02 are largelyindependent of k and agree with the critical exponentof the entanglement transition of the 1+1D NN model.

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This consistent with previous results showing that thecritical behaviour of the PWR2k with fixed k < log2N isin the same universality class as the conventional 1+1DMIPT transition.

VI. QUANTUM ERROR-CORRECTING CODEPROPERTIES

Below the critical measurement rate p < pc, the mixedphase is underpinned by a dynamically-generated quan-tum error-correcting code [8, 9, 11, 17]. The improve-ment in the critical measurement rates pc,Haar, pcp, pce asa function of k observed in the previous sections suggeststhat circuits with highly nonlocal interactions generateimproved quantum error-correcting codes in the mixedphase. An important characteristic of any QECC is itscode distance, which is the smallest number of single-qubit errors required to transform any code state intoany other – equivalently, the code distance is the minimalweight of all nontrivial logical operators. Here we esti-mate the contiguous code distance dcode for our nonlocalClifford circuits, and show that the improved robustnessto local measurements observed in the previous sectionsalso generates QECCs with improved code distance.

To characterize the QECC in the mixed phase, wemaximally entangle the system Q with a reference sys-

tem R and study the Renyi entropy S(2)R of the reference

as well as the mutual information I(A,R) between thereference and a subset A ⊂ Q of the output qubits asshown in Fig. 5a. In the language of quantum error-correction, the Renyi entropy determines the code rate

rcode = S(2)R /N ln 2 of the PWR2k Clifford circuit, or the

number of logical qubits that are encoded within the N -qubit systemQ. We plot rcode as a function of p in Fig. 5bfor N = 256 and find that the code rate substantiallyimproves with nonlocality k = 1, . . . , 8 at any fixed mea-surement rate p, consistent with previous results. FromFig. 5b it is clear that for any fixed k, our choice of mea-surement rate p < pc below the critical point determinesthe code rate rcode of the underlying QECC.

We can also study the contiguous code distance dcode

of the QECC in the mixed phase by analyzing the mu-tual information I(A,R) between the reference R andsubregions A ⊂ Q of the system as shown in Fig. 5a [17].For sufficiently small subregions A in the mixed phasep < pc, one generically finds vanishing mutual informa-tion I(A,R) = 0, indicating that the region A containsno information about the reference R. In this circum-stance we may safely discard any of the qubits in A andstill reliably recover the quantum information shared be-tween the reference and the system. In the language oferror correction, we can view the subregion A as a setof qubits that has possibly been corrupted by errors. Solong as I(A,R) = 0 we may simply discard these cor-rupted qubits and still recover the information containedin R. On the other hand, sufficiently large subregionsA will ultimately yield I(A,R) > 0, implying that suffi-

FIG. 5. Code rate and code distance for the PWR2k Cliffordcircuit family at time t = 8N . (a) To determine rcode, dcode,

we examine the entropy S(2)R of a maximally-entangled refer-

ence R and the mutual information I(A,R) between the ref-erence and a subregion A ⊂ Q of the output qubits. (b) Thecode rate in the PWR2k circuit for N = 256 as a functionof measurement rate p improves significantly with k (greenthrough red). (c) Normalized mutual information as a func-tion of subregion size for p = 0.12, deep in the mixed phase.The effective contiguous code distance dcode is determined byfinding the minimum size of the linear bipartition |A| to havethe mutual information I(A,Ref.) of ln 2. (d) Code distanceversus k at fixed code rates rcode = 0.1, 0.2, 0.3 (black, red,orange) for the system size N = 256. Error bars are shownor are smaller than data points; lines are guides to the eye.

ciently large errors can degrade the correlations betweenthe system and reference. The crossover point |A|∗ atwhich the mutual information becomes nonzero providesan estimate of the system’s code distance dcode [17]. Wereview these arguments in more technical detail in Ap-pendix E [55].

To extract an estimate of the contiguous code dis-tance dcode, we plot the mutual information I(A,R) asa function of the subregion size |A| /N and look for thecrossover point |A|∗ where the mutual information in-crease by one bit ∆I(A∗, R) = ln 2 (Fig. 5c). The ef-fective code distance is given by dcode = 〈|A|∗〉 where〈. . . 〉 is an average over different realizations of projec-tive measurements in the circuit. We plot the resultingeffective contiguous code distance as a function of k inFig. 5d, which shows a striking improvement of dcode

with increasingly nonlocal interactions k. Moreover, bytuning the measurement rate p < pc and the nonlocal-

8

ity parameter k we can obtain quantum error-correctingcodes with a variety of code rates and code distances.For fixed k we find the expected tradeoff between coderate and code distance that is typical in quantum error-correcting codes. By increasing the nonlocality parame-ter k we generate codes with significantly improved codedistance dcode for any fixed code rate rcode. In this sense,the nonlocal interactions for k ∼ log2N substantiallyimprove the quantum error-correcting code properties inthe mixed phase.

VII. FULLY NONLOCAL AND ALL-TO-ALLMODELS

So far, we have considered PWR2k circuits at fixed k <log2N where the longest-range interactions dmax = 2k−1

have been strictly smaller than the system size N . Inthese cases we found that nonlocal interactions could sub-stantially improve both the critical measurement rate pcand the code distance dcode in the mixed phase. We nowconsider what happens in the ‘complete’ PWR2k circuitwith k = log2N , where the longest-range interactionsdmax = N/2 are extensive, and show that the behaviorradically changes relative to the cases previously stud-ied. In particular, the complete PWR2k circuit withoutmeasurements is known to be a fast scrambler capableof generating system-wide volume-law entanglement af-ter only t∗ ∝ logN interaction layers [48]. In this sectionwe demonstrate that the fast scrambling dynamics in thiscircuit leads to markedly improved code properties, in-cluding a nearly extensive contiguous code distance dcode.We also observe many similarities in this section betweenthe complete PWR2k circuit and a maximally nonlocalrandom AA model, another known fast scrambler whichhas no spatial geometry whatsoever. These similaritieshighlight the central role played by fast scrambling in de-termining the physics of the mixed phase in these circuits,and suggest that fast scrambling circuits may be governedby the same universal physics near the measurement-induced critical point.

We first study the complete PWR2k circuit with Haar-random gates, and numerically simulate the correspond-ing classical percolation network similar to section III.In that prior analysis we computed the Binder cumulantb(p) of the network and estimated the critical point byfixing the parameter k and performing a finite-size scalinganalysis in the system size N . In the present case, finite-size scaling is difficult to define consistently because theconnectivity k = log2N of the graph is coupled to thesystem sizeN . Instead, we estimate a finite-size crossoverpoint by computing the susceptibility

χ(p) = 〈C2max(p)〉 − 〈Cmax(p)〉2 (6)

of the resulting classical percolation network. In the ther-modynamic limit N → ∞ the susceptibility diverges atthe critical point; here we estimate the finite-size transi-

FIG. 6. Maximally nonlocal ‘complete’ PWR2k circuits withk = log2N and random AA circuits. (a) Crossover pointsppeak,Haar as a function of system size N for the completeHaar-random PWR2k circuit (purple) and for the randomAA model (light blue). (b) The maximum cluster size Cmax

at the peak ppeak,Haar grows as a power law Nµ with systemsize for the complete PWR2k circuit (purple) and the AAcircuit (light blue), with exponents df = 1.809 ± 0.005 anddf = 1.786± 0.005 respectively. Black dashed line shows thefractal dimension, df = 91/48, of the cluster at the criticalpoint of two-dimensional system. (c) The code distance forthe complete PWR2k Clifford circuit (top) and the randomAA Clifford circuit (bottom) for code rates rcode = 0.1, 0.2, 0.3(black, red, orange). In both cases, the code distance isnearly extensive, scaling like dcode ∝ Nβ with β = 0.96±0.04(PWR2) and β = 0.97±0.02 (AA). (d) Scaling collapse of thenormalized single-qubit purification time τ(p) in the completePWR2k Clifford circuit (top) and the random AA Clifford cir-cuit (bottom). In both cases the data exhibits strong collapseaccording to the nonstandard scaling form in Eq. (7).

tion point by locating the peak ppeak,Haar of the suscep-tibility χ(p) for each N .

The resulting finite-size crossover points are plotted inFig. 6a, where we compare to the crossover points fora random AA model analyzed using the same methods.The crossover points noticeably increase with N in bothcases, and should therefore be considered as finite-sizecrossover points and not bona fide critical points. We alsoplot the value of the maximum cluster size at the peakCmax(ppeak,Haar) in Fig. 6b, and find that it increases as apower law with system size Cmax(ppeak,Haar) ∝ Nµ withfractal dimension df = 1.809 ± 0.005 for the completePWR2k circuit and 1.786±0.005 for the random AA cir-cuit. The similarity in the fractal dimension df betweenthese two circuits suggests that they may be governed by

9

the same universal physics near the critical point. More-over, the fractal dimension in both cases differs from theanalytical prediction df = 91/48 ≈ 1.896 obtained fromthe two-dimensional percolation universality class. Thissuggests that there is an abrupt change in the universal-ity class from the k < log2N circuits to the completePWR2k circuit.

Next, we turn our attention to circuits composed ofClifford circuits with two-qubit gates Qij = CZijHiHj

and study the quantum error-correcting codes that sup-port the mixed phase. Again, we find striking similaritieswith the random AA circuit, suggesting that these mod-els may be governed by the same universal physics. Us-ing the methods discussed in Sec. VI, we extract thecode distance dcode of the QECC in the mixed phaseat fixed code rate rcode. We plot the results in Fig. 6cfor code rates rcode = 0.1, 0.2, 0.3. Linear fits indicate anearly extensive code distance, where dcode ∝ Nβ withβ = 0.98 ± 0.05 for the PWR2 circuit and 0.96 ± 0.01for the random AA circuit. The fast scrambling dynam-ics common to both of these models apparently gener-ates similar quantum error-correcting codes with excel-lent properties deep in the mixed phase.

Finally, we study the single-qubit purification time τin the complete PWR2k circuit with k = log2N . Insteadof the conventional scaling law (5) near the critical point,here we empirically find strong scaling collapse of τ onlyafter normalizing by the number of interaction layers k =log2N within each even (or odd) block. This leads to anonstandard scaling law

τ/ log2N = τ(p) = N z f(

(p− pc)N1/ν), (7)

which yields strong data collapse for both the completePWR2 model and the random AA model as shown inFig. 6d. This suggests that the critical point in the fast-scrambling limit may be governed by a logarithmic scal-ing law (7) as opposed to the conventional scaling (5)that governs the phase transition in general short- andlong-range models [26, 62]. The origin of this scaling be-haviour is an interesting topic for further investigation.

VIII. DISCUSSION AND OUTLOOK

In this paper we studied the role of sparse nonlocalinteractions in protecting quantum information againstthe destructive effects of local measurements. We char-acterized the protection afforded by these nonlocal in-teractions by numerically studying measurement-inducedphase transitions in a class of sparse, nonlocal PWR2kcircuits with tunable nonlocality k. These studies demon-strated that non-universal properties of the transition,such as the critical points pc,Haar, pce, pcp, can be signif-icantly improved by the nonlocality of the interactiongraph. At the same time, we found that the universalproperties of the transition such as the critical exponentν are largely unaffected by the presence of these nonlocal

interactions so long as k < log2N . In the language ofthe renormalization group, it appears that sparse non-local interactions are an irrelevant perturbation to the1+1D fixed point. It would be interesting to make thisnotion of irrelevancy more precise within a renormaliza-tion group framework.

While the phase transition for fixed k < log2N ap-pears to be governed by the same universality class as the1+1D model, we find that the properties of the systemdeep in the mixed phase can change considerably in thepresence of nonlocal interactions. In particular, we havedemonstrated that the code distance dcode (at fixed coderate rcode) can be significantly improved by the presenceof sparse, nonlocal interactions. By varying the param-eter k, we can thus tune between the code properties ofa NN circuit and the improved code properties of an AAcircuit.

We also studied ‘complete’ PWR2k circuits with k =log2N and observed that these circuits are closely com-parable to known fast scramblers such as the random AAcircuit. Although a proper finite-size scaling analysis isnot available for these circuits, we numerically studiedfinite-size crossover points and showed that the criticalproperties of the complete PWR2k circuit are markedlysimilar to those of a random AA circuit. We also stud-ied the code properties of the complete PWR2k and AAClifford circuits, and found codes with nearly extensivecode distance dcode ∝ Nβ with β ≈ 1 in both cases. Fi-nally, we observed a non-standard scaling collapse in thesingle-qubit purification time τ for both models, anothersignal that the critical properties of these models may bedescribed by the same universality class. In future workwe hope to make these connections more explicit.

One of the most exciting prospects for engineering fastscrambling circuits in the lab is the ability to generategood quantum error-correcting codes using the fewestgates possible. Here we have demonstrated that de-terministic fast scrambling circuits with local projectivemeasurements are capable of generating good quantumerror-correcting codes, and that the properties of thesecodes can be easily tuned. It is natural to ask whetherone could reasonably use these monitored circuits togenerate useful codes in near-term experiments. Oneoutstanding problem from an experimental perspective,however, is the post-selection problem. Because quantumprojective measurements are inherently probabilistic, onecan only force a particular outcome by repeating the pro-tocol ad infinitum until the correct measurement record isobtained. This necessarily requires the experiment typi-cally be repeated an exponentially large number of times,which is impractical. In the case of Clifford circuits itis known in principle how to apply feedback to effec-tively correct for undesired measurement outcomes [36].It would be interesting to apply these ideas to our fastscrambling Clifford circuits to understand whether single-shot experiments with suitable feedback can be used togenerated useful quantum error-correcting codes in near-term experiments with cold neutral atoms or trapped

10

ions. The data for this manuscript is available in openaccess at [63].

ACKNOWLEDGMENTS

We thank Hans Peter Buchler and Nicolai Lang forhelpful discussions. GSB is supported by the DOE Ge-

oFlow program (DE-SC0019380). Work at the Universityof Strathclyde was supported by the EPSRC ProgrammeGrant DesOEQ (EP/P009565/1), the EPSRC QuantumTechnologies Hub for Quantum Computing and simu-lation (EP/T001062/1), the European Union’s Horizon2020 research and innovation program under grant agree-ment No. 817482 PASQuanS, and AFOSR grant numberFA9550-18-1-0064.

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FIG. 7. (a) Haar-random two-site gates typically generate nearly maximal entanglement. In this case the zeroth Renyi entropyS0 of site i = 0 is proportional to the minimum number of cuts to required to isolate it, S0/ ln 2 = 1. (b) Haar-random gatesapplied between sites i, j = 1, 2 and sites i, j = 0, 3. In this case S0 takes a value of S0/ ln 2 = 2 for the subsystem consistingof sites i = 0, 1.

Appendix A: Haar-Random Circuits and Bond Percolation

1. Percolation Mapping

The zeroth order Renyi entropy is S0 = logM , where M is the rank of the reduced density matrix ρA of thesubsystem A. When an independent qubit is entangled with A, the rank of ρA increases by a factor of 2. A projectivemeasurement, on the other hand, isolates the measured qubit from the rest of the system, typically undoing the effectof the entangling gates applied previously to the measurement. In the tensor network picture, the calculation ofS0/ ln 2 maps to a classical minimal cut problem. In this picture the zeroth Renyi entropy is given by the minimumnumber of legs one must cut in order to isolate the qubits in A.

By considering the two extreme limits of the probability of the projective measurement p, one finds that in thelimit of p = 0 the initial state and final state is guaranteed to be connected, and in the limit of p = 1, there is noconnection between them. Between these two limits there must be a critical probability p = pc,Haar which separatesthe two phases. For p < pc,Haar, there always exist a path between qubits in the initial state and the final state, andfor pc,Haar < p, they are independent.

If two qubit gates are taken from the Haar-random distribution and applied to a pair of independent qubits, thentwo qubit gates are almost always guaranteed to be entangled. In this case, the circuit can be drawn as a tensornetwork diagram consists of entangling tensors (Haar-random gates) which act on the pairs of qubits accordingly tothe interaction graph, with the legs being cut with probably p due to the projective measurements. This is equivalentto the bond-percolation with the bond occupation probability q = 1− p. The critical point, pc,Haar, therefore, can bedetermined by solving the classical bond percolation problem of the mapped network.

2. Numerical Method for Simulating Bond Percolation

The Newman-Ziff algorithm [58] is used for simulating the critical behaviour of bond percolation. This algorithmcalculates the various observables at different values of the bond occupation probability q = 1− p in a computationaltime which scales only linearly with the number of bonds M in the network.

The algorithm proceeds by generating random configurations of a given network with m = 1, 2, . . . ,M bonds beingoccupied; where this is done by adding one random bond at each step starting from an empty network. At eachstep of this process, one can keep track of the sizes of the connected components (clusters) in the network with theUnion-Find algorithm [64, 65]. Observables such as cluster sizes and their moments can then be stored at the endof each step of the algorithm. The observable Q as a function of the bond occupation probability q can then becalculated by taking the microcanonical ensemble of the different configurations:

Q(p) =∑m

B(m,M, q)Qm =∑m

(Mm

)qmqM−mQm (A1)

where Qm is the value of observable Q with m occupied bonds and

(Mm

)= M !

m!(M−m)! is the usual binomial coefficient.

The calculation of the binomial distribution B(m,N, p) is numerically unstable because one requires evaluations ofthe factorials of large numbers and addition of numbers which differ largely in the order of the magnitudes. Here weadopt a more stable method for evaluating the microcanonical ensemble introduced by Newman and Ziff [58]. If we

normalize the binomial distribution by its peak value mmax = pM , this normalized binomial distribution B(m,M, p)

13

FIG. 8. Block-decimation procedure for the k = 2 PWR2k circuit. The entire structure on the left is reduced down to a singleribbon-like structure on the right via the renormalization transformation.

is defined recursively as

B(m,M, q) =

{B(m− 1,M, q)M−m+1

mq

1−q (m > mmax)

B(m+ 1,M, q) m+1M−m

1−qq (m < mmax)

(A2)

Therefore the observable of interest can be calculating the following

Q(q) =

∑m′ B (m′,M, q)Qm∑m′ B (m′,M, q)

. (A3)

3. Renormalization Group Solution for full PWR2

The PWR2 circuit with size N0 can be constructed from two sub-systems of size N0/2. This can be done by inter-leaving the degrees of freedom of the two sub-systems and coupling them together with nearest neighbour interactions.

This self-similar structure allows us to define a renormalization transformation. This transformation consists of twosteps: First, a large block as depicted in Fig. 8 (left), is renormalized to a ribbon-like structure in Fig. 8 (left). Herethe renormalized is done by collapsing the bonds with different colors in Fig. 8 (left) to the bonds of the correspondingcolors in Fig. 8 (right). This step reduces the size of the vertical dimension by half. In order to reduce the size of thehorizontal direction, two of the ribbon-like structure is merged into one. After these steps, the size of the network ofN/2 by T is reduced to N/4 by T/2, where N is the number of qubits and T is the number of layers in the originalcircuit.

Let q = 1 − p be the probability of a bond being present in the network. The renormalized probability R(q) interms of the probability which a ribbon structure (Fig. 8, right) is spanned (Qribbon) is

R(q) = Q2ribbon + 2Qribbon(1−Qribbon). (A4)

Now we define the condition of the ribbon to be spanned when there exists at least one bond coming out of each thenodes. In terms of the probability of at least a bond being spanned qbond, Qribbon is

Qribbon = q4bond + 4q3

bond(1− qbond) + 2q2bond(1− qbond)2. (A5)

Finally, the probability of the bond being spanned, qbond, as a function of q is

q′ = q4 + 4q3(1− q) + 4q2(1− q)2. (A6)

The equation R(q∗) = q∗ has a non-trivial fixed point solution in the real domain of 0 < q∗ < 1, which is q∗ ≈ 0.335 . . .or equivalently p∗ ≈ 0.665 . . . .

14

Appendix B: Clifford Circuits

1. Stabilizer States

The stabilizer formalism is a powerful tool for understanding a special class of many-body quantum states andquantum error-correcting codes whose dynamics can be efficiently simulated on a classical computer [53, 54]. Considerthe Pauli group P(Q) of all Pauli strings acting on a set of qubits Q. We define an Abelian stabilizer subgroupS ≤ P(Q) generated by a set G = {g1, g2, . . . , gm} of linearly-independent, mutually-commuting Pauli strings [g`, g`′ ] =0 ∀`, `′. A stabilizer group with m ≤ N = |Q| independent generators has order |S| = 2m. Given a stabilizer groupS we define a stabilizer state (also called a code state)

ρQ(S) = 2−|Q|∑g∈S

g (B1)

which is the unique density matrix stabilized by all elements g ∈ S. If we specify a ‘complete’ set of m = N stabilizers,then (B1) is a rank-1 projector onto the unique pure state |Σ〉 stabilized by the group S (i.e. g |Σ〉 = + |Σ〉 for allg ∈ S).

Quantum error correction is particularly easy to understand in the stabilizer formalism. Suppose we prepare a codestate (B1) defined by a stabilizer group S, and consider disturbing it with an error operator e ∈ P(Q). There arethree possibilities: first, if e ∈ S then the error is trivial because it leaves the state (B1) unchanged. Second, if e /∈ Sfails to commute with one or more stabilizers g then this is a detectable error because we can detect (and subsequentlycorrect) the error by measuring the stabilizer g. The final possibility is if e /∈ S but e ∈ C(S) is in the centralizer ofS. These errors are undetectable because they modify the quantum state but cannot be detected by measuring anyof the stabilizers g. The collection of all undetectable errors gives the logical operator group e ∈ L ≡ C(S)/S.

2. Classical Simulation of Clifford Circuits

Clifford group on N qubits, CN , is a group formed by unitary operators which transforms the elements of N qubitPauli group to other elements in the same group. The action of those unitaries on a stabilizer of O|Σ〉 thereforetransforms the stabilizer to other stabilizer O|Σ′〉. This transformation is equivalent to the transformation of state|Σ〉 to |Σ′〉. Therefore, the evolution of a stabilizer state under the unitaries from the Clifford group can be trackedby keeping track of how the initial stabilizer transforms.

The evolution with unitaries from the Clifford group is proven to be able to be computed classically in polynomialtime by Gottesman and Knill [53, 54]. This is done with a binary matrix M of size N by 2N . In this representation,Pauli strings in the stabilizer are mapped to each rows of M and the actions of Clifford gates are mapped as series oflogical operations[54].

In this simulation a Pauli strings are mapped to a row in the following fashion. A Pauli operators which correspondsto the lth qubit is encoded by mapping the number of Pauli-X to lth column and the number of Pauli-Z to N + lth

column. If lth operator is Pauli Y , then because of the identity, Y = iXZ, the values of the lth and N + lth columnsis registered as 1, 1. If lth operator is an identity, both of the values are registered as 0.

To keep track of the evolution by the unitaries in the Clifford group, it is useful to identify the transformation ruleof the binary matrix for the generators of the Clifford group. The generator of N -qubit Clifford group are Hadamard(H), Phase (P ) and Controlled-NOT (C-NOT) gates [53, 54, 66]. Hadamard gate acting on a site, l, maps operatorsZl to Xl and Xl to Zl while keeping Yl to be Yl. This is equivalent to swapping the columns l and N + l. Phase gateacting on a site, l, on the other hand, maps Xl to Yl and Yl to Xl while keeping Zl unchanged. This is equivalent tosetting the column N + l as the modulo 2 sum of columns l and N + l. Let qubit l be a control and qubit m be thetarget, then the Controlled-NOT gate acting on those qubits flips the target basis whenever the state of the l is |1〉.By transforming all the elements in the two-qubit Pauli group, one finds that there are only four non-trivial rules,which are : XlIm toXlXm, IlXm to IlXm, ZlIm to ZlIm, and IlZm to ZlZm. Since H, P , and C-NOT gates are thegenerators of the Clifford group, one may systematically generate all the operators in n-qubit group by following thealgorithms such as the one proposed by Selinger [66].

3. Entanglement Entropy

The stabilizer is equivalently the density matrix of the stabilizer state because the stabilizer is a projector whichprojects the state onto the stabilizer state. With the density matrix ρQ(S), entanglement entropy of a subregion Acan be calculated [34, 67].

15

FIG. 9. (a) The binder cumulant b(p) of the percolation network of PWR2k for k = 1, 2, t.s6 for the system sizes N =28, 29, 210, 211 (light to dark blue). The solid vertical lines are the estimated values of the pc,Haar and the shaded regions are thecorresponding 1-sigma errors. (b) Collapsed Binder cumulant for k = 1, 2, . . . 6 for the system sizes N = 28, 29, 210, 211 (lightto dark blue).

The reduced density matrix on the subregion A, ρA (A ∈ Q) is obtained by tracing out A. This is equivalent totracing out the Pauli operators which belong to A. However, Pauli operators are traceless, therefore

ρA = TrA{ρQ} =2|A|

2Q

∑gA∈GA

gA =1

2|A|

∑gA∈GA

gA, (B2)

where GA ∈ G is a set of all g where trace over A is non-zero. [67]. Let NA be the number of linearly independentPauli strings that generate gA, then

∑gA∈GA gA is also a projector of rank 2|A|−NA because this projects out the −1

eigenstates of its generators. The von Neumann or Renyi entropy SA is

SA = (|A| −NA) ln 2 (B3)

From the identities 2NA+NA = 2N and NA = rankbinaryMA, where M is the binary matrix of the stabilizer state,rankbinary is the binary rank, and MA is the matrix which its columns corresponds to the Pauli operators of region Ain M ,

SA =(rankbinary(MA)− |A|

)ln 2. (B4)

Using SA = SA, we obtain

SA = (rankbinary(MA)− |A|) ln 2. (B5)

Appendix C: Finite-Size Scaling

1. Finite Size Scaling of the Percolation Critical Point for k = 1, . . . , 6

Finite-size scaling analysis for the percolation transition of the PWR2k circuit for k = 1, 3, 5 is presented in Fig. 2 b.Here we show the finite-size scaling and the crossings of the binder cumulants for all values of k = 1, 2, . . . , 6 for thesystem sizes N = 28, 29, 210, 211.

The error bars are estimated by taking the standard error of the 4000 realizations of the Newman-Ziff algorithm(Appendix A). The critical points and their errors are estimated by computing the average intersection points of thecurves with 5000 different realizations of the fluctuations added to each of the points. Fluctuations are drawn fromthe appropriate distribution with the standard error as the standard deviation at each point (Fig. 9 a). The criticalexponents and their errors are also estimated by collapsing the 5000 different realizations of the fluctuations addedfrom the same distribution (Fig. 9 b).

16

FIG. 10. (a) The tripartite mutual information I(A;B;C) of initial z-polarized state evolved under a deterministic CliffordPWR2k circuit until t = 8N for k = 1, 2, . . . 6 for the system sizes N = 26, . . . , 211 (light to dark blue). The solid vertical linesare the estimated value of the pce and the shaded region is the corresponding 1-sigma errors. (b) Collapsed tripartite mutualinformation for k = 1, 2, . . . 6 for the system sizes N = 26, . . . , 211 (light to dark blue).

2. Finite Size Scaling of the Entanglement Critical Point for k = 1, . . . , 6

The finite-size scaling analysis of the entanglement criticality of PWR2k circuit for k = 1, 3, 5 are presented inFig. 3 b. Here we show the finite-size scaling and the crossings of the tripartite mutual information I(A;B;C) for allthe values of k = 1, 2, . . . , 6 for system sizes N = 26, . . . , 211 (the system sizes that are smaller than 2k+2 are not usedin the analysis).

The error bars are estimated by taking the standard error of up to 1000 realizations of the random projectivemeasurement of the evolution to t = 8N . The critical points and their errors are calculated by computing the averageintersection points of the curves with 5000 different realizations of the fluctuations added to each of the points that aredrawn from the appropriate distribution with the standard error as the standard deviation at the point (Fig. 10 a). Thecritical exponents and their errors are also estimated by collapsing the 5000 different realizations of the fluctuationsadded to the data points from the same distribution (Fig. 10 b).

3. Finite Size Scaling of the Purification Critical Point for k = 1, . . . , 6

The finite-size scaling analysis of the purification criticality of PWR2k circuit for k = 1, 3, 5 are presented inFig. 4 b. Here we show the finite-size scaling and the crossings of the single-qubit purification time for all the valuesof k = 1, 2, . . . , 6 for system sizes N = 26, . . . , 211 (the system sizes that are smaller than 2k+2 are not used in theanalysis).

The error bars are estimated by taking the standard error of up to 1000 realizations of the random projectivemeasurement of the evolution until the state is purified. The critical points and their errors are estimated by computingthe average intersection points of the curves with 5000 different realization of the fluctuations added to each of thepoints that are drawn from the appropriate distribution with standard error as its standard deviation at each points(Fig. 11 a). The critical exponents and their errors are also estimated by collapsing the 5000 different realizations ofthe fluctuations added to the data points from the same distribution (Fig. 11 b).

17

FIG. 11. (a) The single-qubit purification time for k = 1, 2, . . . 6 for the system sizes N = 26, . . . , 211 (light to dark blue).The initial stat is prepared such that there is a qubit entangled to one of the N qubits (system), then the system qubits arescrambled with nearest neighbour random Clifford circuit up to t = 4N . The solid vertical lines are the estimated value of thepcp and the shaded region is the corresponding 1-sigma errors. (b) Collapsed tripartite mutual information for k = 1, 2, . . . 6for the system sizes N = 26, 211 (light to dark blue).

FIG. 12. (a) Susceptibility, χ(p), of the complete PWR2k (left) and AA (right) percolation networks for the system sizesN = 26, 27, . . . , 210. Although there exists sharp peaks, the positions of the peaks depends significantly on the system size.The size of the circuits in this calculations is taken to be N × T . The observables of the bond percolation of the networks arecalculated with Newman-Ziff algorithm by taking the microcanonical ensemble of over 4000 trajectories. (b) τ /N z as a functionof p without the scaling collapse for complete PWR2k (left) and AA (right) circuits for the system sizes N = 26, 27, . . . , 210

with z = 0.21 ± 0.01 (PWR2) and z = 0.170 ± 0.006 (AA). The critical point determined from the scaling collapse is markedby the black line and the 1-sigma error is marked by the grey shade. The critical points and their error are pc = 0.267± 0.001and pc = 0.312± 0.001 respectively.

4. Finite Size Scaling of the Full PWR2 and AA circuits

The finite-size scaling of the MIPTs of the complete PWR2k circuit is difficult due to the long-range interactionswhich give rise to strong boundary effects and the loss of locality. In the percolation transition, this appears as thestrong system size dependence of the ppeak,Haar as shown in the figures Fig. 6 a. The positions of the peaks aredetermined from the χ(p) plotted in Fig. 12 a.

The loss of locality makes the determination of the entanglement transition impossible. Due to the coupling ofdistance N/2, even in the regime near p = 1, the entanglement entropy of the region of size A < N/2 is almostguaranteed to be ∼ A(1 − p) at the end of the application of 2 log2N − 1 layers of the gates. Therefore we do notargue on the existence of the entanglement transition in the complete PWR2k and AA circuits.

The percolation transition suffers from the similar boundary problem. However, the transition does not ask thegeometry of the circuit as it only asks the entropy of the reference qubits that are entangled to the system at t = 0.The problem in the determination of the critical properties of the purification transition is the number of layers of theeven (or odd) blocks which depends on the system size. In this paper, we proposed the potential workaround, which

18

is normalizing the time t by the number of layers in the even (or odd) blocks by defining t = t/ log2N . This lead tothe empirical scaling law of the form in Eq. (7).

As shown in Fig. 6 d, the single-qubit purification time τ(p) collapsed surprisingly well with Eq. (7) with thepc = 0.27 ± 0.01,νc = 2.08 ± 0.08, and zc = 0.21 ± 0.01 (PWR2) and pc = 0.31 ± 0.01,νc = 2.29 ± 0.05, andzc = 0.170 ± 0.006 (AA). For this simulation, τ(p) is estimated by taking the average over up to 500 realizationsof the random projective measurements. The errors in the critical exponents and critical points are estimated byperforming the scaling collapse on the 5000 different realizations of the data set with random fluctuations. Therandom fluctuations are taken from the appropriate distribution with standard deviation of the standard error at eachpoint.

Appendix D: The Gap Between Entanglement (pce) and Purification (pcp) Critical Points of PWR2k

FIG. 13. The difference between entanglement and purification critical points of PWR2k circuit are plotted (red) for k =1, . . . , 6. The blue line shows the error of entanglement critical points centered at the expected value. There is no statisticallysignificant deviation observed between the entanglement and purification critical points.

Whether there exists a gap between the entanglement and purification critical points is a topic of an ongoing debate.Gullans and Huse[34] show that in 1+1D, they occur at the same point in general. Here we show that such a gap isnot observed in the PWR2k circuits for the values of k = 1, . . . , 6. No statistically significant deviations between theentanglement and the purification critical points are observed (Fig. 13).

Appendix E: Quantum Error-Correcting Code Properties

1. Code Distance for Stabilizer Circuits

Here we review an argument of Li and Fisher that connects the code distance dcode of a stabilizer circuit to themutual information I(A : R) between a subregion A ⊂ Q of the output qubits and a maximally-entangled referenceR [17]. In this work we are primarily interested in contiguous regions A, but the arguments in this Appendix holdequally well for arbitrary, possibly disconnected regions. For any bipartition A ∪ A = Q of the system we define thequotient group

LA ≡{g ∈ C(S)|projA(g) ∈ projA(S)}

S. (E1)

Here projA is the group homomorphism from P(Q) to P(A) defined by

projA : gA ⊗ gA → gA (E2)

which projects any Pauli string g = gA ⊗ gA down to its support on the region A. The group LA is the group ofundetectable errors that are localizable on A. In other words, any g in this group can be localized to A simply bymultiplying by a suitable choice of g′ ∈ S. The order |LA| = 2`A of this group gives the number of independent

19

undetectable errors that can be localized to the region A. We must have `A = 0 in order for a stabilizer code tosuccessfully protect quantum information from errors acting on A.

In the following we compute the order |LA| of the group LA and show that `A ≡ log2 |LA| is proportional to themutual information I(A : R). Starting from the definition (E1), we find that the order of the group can be written

|LA| =|C(S)| · |projA(S)||S| · |projA(C(S))|

(E3)

(see Appendix A of [17] for a rigorous group-theoretic derivation). Our goal is to convert the quantities on the RHSinto Renyi entropies, allowing us to relate `A to I(A : R).

As a preliminary step, we first show how to compute the Renyi entropy S(2)(ρA(S)) of a reduced density matrixρA(S) for a subregion A ⊂ Q of the full system. The Renyi entropy of such a region A is given by:

ρA(S) = TrA [ρQ(S)]

= 2−|Q|∑g∈S

TrA [g]

= 2−|A|∑

g∈S∩Ker projA

g (E4)

where in the third line we used the fact that TrA [g] = 0 except when projA(g) = IA (i.e. g ∈ Ker projA). Introducingthe group SA ≡ S ∩ Ker projA

∼= S/projA(S) consisting of all stabilizer operators that act trivially outside thesubregion A, we can simply write the reduced density matrix as

ρA(S) = 2−|A|∑g∈SA

g (E5)

which is a natural generalization of (B1). Finally, it is easy to show that the Renyi entropy of the reduced state ρA(S)is given by

(ln 2)−1S(2)(ρA(S)) = |A| − log2 |SA|= |A| − log2 |S|+ log2 |projA(S)| (E6)

where we have used |SA| = |S| / |projA(S)|.Finally, combining (E3) and (E6) we find:

(ln 2)−1I(A : R) = (ln 2)−1 (SR + SA − SAR)

= |R|+ (ln 2)−1SA − (ln 2)−1SAR + (log2 |C(S)| − log2 |C(S)|) + (log2 |S| − log2 |S|)= log2 |C(S)| − log2 |S|+

((ln 2)−1SA − |A|+ log2 |S|

)−((ln 2)−1SAR − |AR|+ log2 |C(S)|

)= log2 |C(S)| − log2 |S|+ log2 |projA(S)| − log2 |projA(C(S))|= log2 |LA| = `A

where we have used the shorthand SA ≡ S(2)(ρA(S)) to shorten notation, and in going from the third to fourth line wehave used the identity (E6) and its corresponding generalization for the Renyi entropy SAR of the combined regionsA,R [17].

2. Quantum Hamming Bound

Here we review the quantum Hamming bound, which places a fundamental bound on achievable code rates andcode distances for correctable nondegenerate QECCs. Suppose we have a QECC with rate rcode = k/N that can

correct all errors of weight less than or equal to w. There are a total of∑wi=0 3i

(Ni

)such errors (Pauli strings). For

nondegenerate codes, we must have 〈ψ|EE′ |ψ′〉 = 0 for all errors E,E′ and all code states |ψ〉 , |ψ′〉. That is, everycode state |ψ〉 and all of its erroneous versions must be orthogonal to every other code state and all of its erroneousversions [68]. The only way all of these orthogonal states fit into the same 2N -dimensional Hilbert space is when

2kw∑i=0

3i(N

i

)≤ 2N (E7)

20

For large N and fixed k/N,w/N this is equivalent to

k

N≤ 1− w

Nlog2 3−H(w/N) (E8)

where H(x) = −x log x− (1− x) log(1− x) is the classical Shannon entropy.One can derive a similar bound for the case where errors only occur contiguously. In this case, there are a total of∑wi=0 3i(N + 1− i) contiguous errors of weight less than or equal to w. For large N and fixed k/N,w/N this yields a

bound

k

N≤ 1− w

Nlog2 3− 1

Nlog2(N − w) (E9)

where the final term vanishes in the strict limit N →∞, but is still relevant for large but finite size systems.