Hilbert Space Delocalization under Random Unitary Circuits

The unitary dynamics of a quantum system initialized in a selected basis state yield, generically, a state that is a superposition of all the basis states. This process, associated with the quantum information scrambling and intimately tied to the resource theory of coherence, may be viewed as a gradual delocalization of the system’s state in the Hilbert space. This work analyzes the Hilbert space delocalization under the dynamics of random quantum circuits, which serve as a minimal model of the chaotic dynamics of quantum many-body systems. We employ analytical methods based on the replica trick and Weingarten calculus to investigate the time evolution of the participation entropies which quantify the Hilbert space delocalization. We demonstrate that the participation entropies approach, up to a fixed accuracy, their long-time saturation value in times that scale logarithmically with the system size. Exact numerical simulations and tensor network techniques corroborate our findings.

Natural quantifiers for localization and delocalization properties of a state ρ = |Ψ⟩⟨Ψ| over the many-body Hilbert space are the inverse participation ratios (IPR) and the participation entropies, that measure the spreading of the state distribution p n ≡ ⟨n|ρ|n⟩ over the basis B = {|n⟩}.The IPR and participation entropy are given respectively, by1 Despite the explicit basis dependence, extensive studies demonstrated that the IPR captures structural properties of quantum matter, including universal behavior at phase transitions [26,45].However, the investigations of IPR and participation entropies have focused so far on the equilibrium and stationary state features, leaving the question of the time evolution of quantum coherence and, hence, of Hilbert space delocalization unresolved.In this work, we investigate Hilbert space delocalization under the dynamics of (1 + 1)D quantum circuits comprising 2-qudit Haar random gates arranged in a brick-wall pattern [23].The locality and unitarity of this setup constitute minimal requirements for chaotic evolution in many-body systems , hence allow us for a phenomenological understanding of the Hilbert space delocalization under generic quantum dynamics.Our analysis combines rigorous analytical methods, based on the mapping between average of IPR over the random circuits and a statistical mechanics model, together with exact numerical simulations, including tensor network techniques [68][69][70][71] 2 .We find that the saturation of participation entropy S q to the long-time stationary value, equal to the random Haar state average SH q , occurs exponentially quickly in time (i.e. in circuit depth) where K d > 0 is a constant characterizing the properties of 2-qudit gates and α q > 0 is a constant.We find an exact expression for the constant K d for Haar random 2-qudit gates, and provide analytic arguments for the value of α q in the limit of large on-site Hilbert space dimension.In other words, Eq. ( 2) implies that the participation entropy S q approaches its stationary value, up to a fixed accuracy, at time scaling logarithmically with the system size, τ HSD ∼ ln(N).The paper is structured as follows.In Sec. 2, we outline the toolbox employed in the rest of the paper, while in Sec. 3, we discuss the stationary value of participation entropy that the deep quantum circuit reaches.Sec. 4 presents the core of our analytical approach based on the mapping of IPR calculation to a statistical mechanics problem.We resolve the Rényi-2 participation entropy evolution in Sec. 5, where we present the analytic prediction.These findings are further corroborated by the numerical analysis in Sec.6 for generic Rényi index q.We present an outlook of our work and a discussion of its further implications in Sec. 7. The more technical parts of the paper are detailed in the Appendix.In App.A we present a self-contained discussion about Haar averages, in App.B we review the interface problem related to the entanglement propagation, and in App.C briefly discuss the numerical implementations.

Methods
We begin presenting the analytical toolbox we will employ in the rest of the paper.The latter is based on the replica trick, the superoperator formalism [74], and the Weingarten calculus [75][76][77][78], see also Ref. [51,58].Since ⟨n| • |m⟩ = tr(|m⟩⟨n|(•)) and tr(AB) q = tr(A ⊗q B ⊗q ), the IPR definition (1) implies that I q = tr(Λ B ρ ⊗q ), where Λ B = ∑ n∈B |n⟩⟨n| ⊗q is a replica operator acting on H ⊗q N .Throughout this manuscript we will study qudits with local Hilbert space dimension d, hence . For concreteness, we will fix B to be the computational basis, where |n⟩ = |b 1 , . . ., b N ⟩ with b j = 0, . . ., d − 1 for each j.Nevertheless, due to the invariance of the considered circuits under local basis rotations, our results remain valid for any basis of H N obtained by a unitary transformation U = U 1 ⊗ ⋯ ⊗ U N of the computational basis, where U k belongs to the unitary group U(d) for a single qudit.
For the computational basis, q which defines the "book" boundary condition for each qudit [79,80] Λ where the superscript in Λ (j) q bookkeeps the number of the affected qudit.For later convenience, we recast the problem in the superoperator formalism.For any operator U and A n,m |n, m⟫, and U AU † ↦ (U ⊗ U * )|A⟫ 3 .In this representation ⟪B|A⟫ = tr(B † A), so the IPR can be written as For convenience, we introduce a graphical notation.We implicitly define it for the replica boundary Λ q and the replica state ρ ⊗q via the IPR q , . . ., Λ (N) Lastly, since we will consider the unitary evolution |ρ⟫ = (U ⊗ U * )|ρ 0 ⟫ for some |ρ 0 ⟫ selected as the initial state and unitary operation U acting on the Hilbert space H N , we introduce the graphical notation for U and U * , respectively, [U] We will omit the legs indices, which can be inferred from the context.Moreover, we will typically reserve thick (thin) lines for multi-(single)-replica objects.

Delocalization properties of random Haar states
We begin our discussion reviewing the Hilbert space delocalization of random states [21,79] which correspond to the stationary ensemble of states obtained under the action of sufficiently deep random circuits.Uniformly distributed random states in the Hilbert space |ρ Haar ⟫ = (U ⊗ U * )|ρ 0 ⟫ are obtained from a reference state |ρ 0 ⟫ = |Ψ 0 , Ψ 0 ⟩ acting with a Haar distributed unitary U ∈ U(d N ), where d is the qudit local Hilbert space dimension and N is the total number of qudits.Let us compute the average IPR over the Haar ensemble, which reads, using the graphical notation introduced in Sec. 2, where E Haar denotes the expectation value over the unitary group U(d N ) taken with the Haar measure.Here, we have defined, via the linearity of the average and of the expectation value, the q-replica transfer matrix on N qudits The explicit formula for W N can be obtained by a direct evaluation of the Haar average, as detailed in App. A. The final result is expressed in terms of the permutation operators |σ⟫ j acting on each qudit j, with the matrix elements in the superoperator computational basis given as fixed by the permutation σ ∈ S q .Employing the Weingarten function Wg(D; σ) 4 allows us to define the dual states | σ⟫ 1,2,...,N = ∑ τ∈S q Wg(d N ; στ −1 )|τ⟫ 1 ⊗ ⋯ ⊗ |τ⟫ N acting on the whole system of N qudits.With these states, the transfer matrix reads We are now in position to compute the IPR.Since the initial state ρ 0 is pure, ⟪σ, . . ., σ|ρ [77] we have where we used |S q | = q! and ⟪Λ q |τ⟫ = d for all τ ∈ S q .From this calculation, it follows that the annealed averaged Rényi entropy for the Haar states is given by As expected and already discussed in [21], the Haar random states are (almost) fully delocalized over the many-body basis B. Indeed, (13) shows that SH q differs only by a sub-leading constant term (q − 1) −1 ln[q!] from the maximal value of participation entropy N ln(d) achieved for a unifromly distributed state with ⟨n|ρ|n⟩ = d −N .Since the logarithm is non-linear, in principle one should expect corrections when considering the quenched average of the participation entropy S H q ≡ E Haar [S q ].We briefly estimate these fluctuations via the variance 5 4 Denoting the projectors of the irreducible representations of the permutation group S k onto the D dimensional Hilbert space by Π λ , their dimensions by d λ and their character functions by χ λ , the Weingarten function can be written as sums over the integer partitions of k see Ref. [75,78] fur further details. 5 For the explicit expression for I r q , see Ref. [79].
For this expression, we need to compute I 2 q , which requires the use of 2q replicas of H N , and the boundary conditions ⟪Λ q ⊗ Λ q | acting on each site.Algebraic manipulations analogous to the ones described above give It follows that std(I q ) ∼ O(d (1−2q)N/2 ), and in the thermodynamic limit, these fluctuations are irrelevant.In particular, in the scaling limit N → ∞, the annealed and the quenched averages coincide, SH q − S H q N→∞ → 0, and with probability equal to unity, approximate the value of participation entropy calculated for a single Haar random state with an arbitrarily small fixed accuracy.

Hilbert space delocalization in brick-wall quantum circuits
After this preliminary discussion, relevant to the deep circuit limit, we now discuss the Hilbert space delocalization under random quantum circuits.As the initial state we fix |Ψ 0 ⟩ = |0⟩ ⊗N , and study the approach of the annealed and quench average participation entropy to the asymptotic value SH q = (1 − q) −1 ln[I H q ], with I H q ≡ q!(d N )!/(d N + q − 1)!, cf.Eqs. ( 12) and ( 13).The evolution operator corresponding to the brick-wall quantum circuit of depth t (referred to also as time) is given by where ⌊•⌋ is the integer part of •.Here, each U i,j is an independent identically distributed (i.i.d.) Haar random unitary from U(d 2 ) acting on the i-th and j-th qudit.Our goal is to compute the where the N copies of |ψ 0 ⟫ = |0⟩ ⊗2q implement the initial condition |Ψ 0 ⟩ on the replica space.We note that the E Haar denotes here the average over the realizations of the two-body gates U i,j which comprise the considered brick-wall circuit.We adapt the graphical notation of Sec. 2 to the two-body gates and recall that the two-body gates U i,j are i.i.d.variables.It follows that their averages factorize and the transfer matrix corresponding to the circuit of depth t reads Contracting with the replica boundaries, and recalling the decomposition for W (q) 2 in Eq. ( 18), we get the final expression (20) where in the last step we used that ⟪σ|ψ 0 ⟫ = 1 for each site, and we defined |+⟫ i,i+1 = ∑ σ∈S q |σ⟫ i ⊗ |σ⟫ i+1 and | +⟫ i,i+1 = ∑ σ∈S q | σ⟫ i,i+1 .Expression Eq. ( 20) is one of the main results of this work, as it already allows for efficient tensor network implementations [72].At the same time, Eq. ( 20) can be viewed as the partition function of a statistical mechanics model once the W (q) 2 are replaced with their decomposition Eq. ( 18).To highlight this interpretation, we can unravel the upper boundary condition as It follows that W t is a sum of backward paths from each initial state |Q in ({σ j })⟫ to each We conclude this section highlighting the similarities and the differences of the IRP calculation with the computations for propagation of entanglement entropy [51,52], defined as S ent q = − ln[tr(ρ q A )], where ρ A = tr A c (ρ) is the reduced density matrix of the pure state ρ = |Ψ⟩⟨Ψ| and A ∪ A c a bipartition of the system.As for the participation entropy, we focus on the annealed averages, determined by the purities, i.e., the reduced density matrix moments P q ≡ E Haar [tr(ρ q A )]. Denoting as I ∈ S q the identity permutation () and S ∈ S q the cyclic permutation (12 . . .q), in the superoperator formalism we have where ⟫ is a product of identity (cyclic) operators on the qudit j, for j ∈ A c (j ∈ A).The statistical mechanics model corresponding to the calculation of P q (t; A) is exactly the same of the Eq. ( 20), but with different boundary conditions: for the IPR we have free boundaries, cf.Eq. ( 21), while for the P q (t; A) the boundary conditions are of the domain wall type.This fact leads to substantial differences between the Hilbert space delocalization and the entanglement propagation in the considered random circuits, which we will discuss in the next section focusing on the two-replica limit.We will further corroborate these findings with a numerical analysis for various replica number (Rényi index) q.

Two-replica computations
In the two-replica limit, the parallelism between the inverse participation ratios and the reduced density matrix moments is more direct.Since S 2 = {I, S}, Eq. ( 21) translates to the sum over all domain wall boundary conditions (22).Therefore, the IPR, up to the overall constant, is given by the sum of P 2 over all possible choices of subsets A of contiguous pairs of sites Q ≡ {{1, 2}, {3, 4}, . . ., {N − 1, N}}, cf.Eqs.(20) and Eq. ( 21), resulting in where the shift to time t + 1 comes from the simplification of one layer in Eq. ( 20).The time-evolution of the purities [51,52] can be split into two terms where the first time describes the long-time saturation value of the purity, the second term parametrizes the approach to the saturation value, and the factors 2 two in the exponents arise because the set Q contains N/2 pairs of neighboring lattice sites.Combining the above Eqs.( 23) and ( 24) leads to where we used that the first term in Eq. ( 24) is independent of the partition 6 .Converting to participation entropies, we have We are interested in identifying the leading terms governing the decay of SH 2 − S2 (t) at long times.Fixing the system size N and considering the limit of d ≫ N, the factor in Eq. ( 26) simplifies to unity.We focus first on the single domain wall configurations, corresponding to bipartitions of the system into A = {1, 2, . . ., 2N A } and A c = {2N A + 1, . . ., N}.The exact solution for the single domain wall case, detailed in App.B, shows that f A (t) ∝ (2K d ) t for domains localized in the bulk of the system, where The time evolution of the purities P 2 (t; A) where A corresponds to a configuration with n d domains is suppressed by the factor (2K d ) n d t , which vanishes at any t > 0 in the large d limit.Therefore, the leading contributions to the SH 2 − S2 (t) arise due to the single domain configurations.The number of the relevant single domain wall configurations scales proportionally to the system size N, translating Eq. ( 26) to where the coefficient α d d≫1 → 1/2 at sufficiently large system size N.While we have used the large on-site Hilbert space dimension limit to derive Eq. ( 28), we anticipate that this equation applies to any d ≥ 2 with a properly chosen factor α d .
We are unable to analytically demonstrate the validity of Eq. ( 28) at fixed finite value of d and for N ≫ 1.Nevertheless, we outline the underlying calculations which allow us to pin-point the leading factors in the time dependence of SH 2 − S2 (t).First, we observe that at 6 This fact implies that ].Performing tensor network contractions, we find for a generic initial configuration that f A (t) = a A (2K d ) t in the long time limit.Interpreting the tensor network contraction as the wandering domain walls problem (cf.App.B), the factor (2K d ) t corresponds to the single domain wall configuration which dominates f A (t) at long times, while the factor a A depends on the processes required to reach the single domain wall configuration from the initial condition.The summation over all initial conditions yields an overall factor (2K d ) t which describes the behavior of SH 2 − S2 (t) at any d ≥ 2. We conjecture that the competition of the renormalizing factors a A with the exponential terms appearing in the summation over A ⊂ Q yields a factor proportional to the system size N at any d ≥ 2. This conjecture is equivalent to the validity of Eq. ( 28) at any d ≥ 2.
To corroborate our analytical considerations, we compare our predictions with the numerical simulations of the brick-wall random circuits varying the on-site Hilbert space dimension d.A brief summary of our numerical implementations is detailed in App. C. First, we numerically calculate the exact time evolution of the system's state |Ψ⟩ up to the system size N = 24.Our findings are reported in Fig. 1(Left).We observe that already at the time scale O(1) the difference between the annealed average S 2 and the quenched average S2 of the participation entropy is negligible.Thus, the self-averaging properties of the IPR and participation entropies derived in the long-time limit in Sec. 3 apply also for the relatively shallow circuits allowing us to use the S2 (t) = − ln[I 2 (t)] as an accurate proxy for the circuit averaged participation entropy S 2 (t).
To test our analytical prediction (28), we focus on the difference between the stationary Haar value SH 2 and the annealed average S2 (t).We implement Eq. ( 20) as a tensor network contraction, which allows us to reach system sizes N ≤ 1024 for any on-site Hilbert space dimension d.In Fig. 1(Right) we observe that the difference SH 2 − S2 (t) decreases exponentially in time (circuit dept) t proportionally to (2K d ) t−1 or at slightly larger pace at the smallest considered system sizes N = 8, 16.The prefactor of the exponential decay increases monotonically with system size N.The growth of the prefactor becomes clearly linear beyond N = 64.At N > 64 the behavior of SH 2 − S2 (t) at longer times is captured with a good accuracy by Eq. 28 with α 2 = 0.291 (5).
Performing tensor network calculations for the on-site Hilbert space dimension d = 3, . . ., 13 and for system sizes up to N = 256, we verify that SH 2 − S2 (t) decreases exponentially in time matching the prediction in Eq. ( 28), and reproducing the analytically found value of the constant K d .We also find that the prefactor α d increases with d, as shown in the inset in Fig. 1(Right).In particular, the numerical results indicate that α d d→∞ → 1/2, in accordance with the analytical prediction for the large on-site Hilbert space dimension limit.
The results discussed in this section demonstrate that the Hilbert space delocalization under the brick-wall random quantum circuits is an abrupt process.Indeed, for a given tolerance error ε ≪ 1, the participation entropy reaches S2 (t) = SH 2 − at time t HSD ∼ ln(N) which scales logarithmically with the system size N.

Numerical results for any replica
For larger values of the Rényi index q ≥ 3, the analysis presented above cannot be trivially extended.First, the calculation of I q requires all domain wall configurations |Q 0 ({σ j })⟫ with σ j ∈ S q .Thus, the calculation of IPR involves a sum over many more initial configurations than just the configurations appearing in the computation of the purities P q which include only the identity permutation I and the cyclic permutation S.Moreover, W 2 involves contributions with negative signs and additional weights which affect the value of K d , resulting in more complex analytical analysis.
Nonetheless, the main features of participation entropy growth under brick-wall quantum circuits discussed in Sec. 5 may be expected to hold for a generic replica number q.This leads us to the following conjecture SH q − Sq (t) ≃ α d,q Nβ t d,q SH q − Sq (t) = ε at t HSD ∼ ln(N) , (29) where α d,q is a coefficient such that α d,2 = α d in Eq. ( 28), and β d,q < 1 is a constant with β d,2 = 2K d and ε ≪ 1 is a fixed tolerance.The heuristic idea behind the above conjecture is that the limit d ≫ 1 leads to positive weights in W (q) 2 , and up to corrections in O(1/d), the main argument leading to Eq. ( 28) should apply for Eq.(29).We corroborate this hypothesis analyzing the dynamics for various values of q.The results are summarized in Fig. 2.
For qubits (d = 2), we first benchmark the self-averaging properties of the circuit, as shown in Fig. 2(a) for q = 3.The self-averaging holds similarly as for q = 2, i.e. the annealed S3 and the quenched average S 3 of participation entropy rapidly approach each with the increase of the circuit's depth.We find analogous results for q = 1, 4 (data not shown), and expect that a similar phenomenology will arise for any Rényi index q.We therefore limit our discussion to the difference between the saturation value and the annealed average of the participation entropy, i.e., SH q − Sq (t).In panels (b,c,d) of Fig. 2 we demonstrate that the exponential decay of SH q − Sq (t) with the circuit depth t is observed for all considered values of the Rényi index q.For d = 2 and q = 3, see Fig. 2(b), our tensor network implementation allows us to reach N ≤ 512.Surprisingly, we find that the exponential decay of SH q − Sq (t) is very accurately fitted with α 2,3 N(2K 2 ) t , despite we could have expected a decay β t 2,3 with a different coefficient β 2,3 fixed by the weights in | σ⟫ i,i+1 , cf.Eq. ( 29) and Sec. 4.
For larger values of q ≥ 4, as well as for the Von Neumann entropy limit q = 1, we are limited to the relatively small system sizes N ≤ 24 accessible with the exact numerical simulation of the brick-wall random circuits.Time evolution of SH q − Sq (t) for q = 1, 4 approaches, with increasing system size N, the exponential decay (2K 2 ) t .The separation of SH q − Sq (t) at a fixed time for different system sizes N suggests that the prefactor in front of presents the approach to SH 3 of the annealed average S3 (blue symbols) calculated with the exact numerical simulation compared with the tensor network contraction results (purple lines), and the fitted formula SH 3 − S3 (t) ≈ 0.76N(2K d ) t−1 (red dashed line) consistent with Eq. (29).Panels (c) and (d) show the approach of the circuit averaged participation entropy Sq to the random Haar state value SH q respectively for q = 1 and q = 4.
the exponential decay (2K 2 ) t may be scaling proportionally to the system size N.However, the range of available system sizes does not allow us to fully confirm this observation.Finally, performing exact numerical calculations for d = 3 (up to N = 14) and d = 4 (up to N = 10) we verify that the decay of SH q − Sq (t) is accurately fitted with (2K d ) t for q = 1 and q = 4 (data not shown).This finding is analogous to the results for qubits (d = 2) and corroborates further the observation that the base 2K d characterizing the exponential decay of SH q − Sq (t) is independent of the Rényi index q.
Overall, the numerical results for the Rényi index q ≠ 2 presented in this section support our analytical analysis confirming that the approach of the participation entropy Sq (t) to its long-time saturation value SH q occurs exponentially in time and with the prefactor that scales extensively with the number of qubits.Hence, also for q ≠ 2 the approach, up to a fixed tolerance, of the participation entropy to its saturation value ε occurs at time t HSD ∼ ln(N) which scales logarithmically with the system size N.

Discussion and conclusion
We have investigated how the dynamics of random unitary circuits delocalize an initially localized state over a basis of the many-body Hilbert space.We combined analytical and numerical methods to calculate the time evolution of the participation entropies (1) that characterize the spread of state ρ = |Ψ⟩⟨Ψ| over a basis B of many qudit systems under dynamics of brick-wall random circuits.Our main finding is that the process of Hilbert space delocalization occurs abruptly so that the long-time saturation value of participation entropies is approached up to a fixed tolerance at times t HSD ∼ ln(N) scaling logarithmically with the system size N.These results may appear surprising from the point of view of the close relations between the participation entropy and the entanglement entropy [26,36,43,81].Indeed, entanglement entropy under local circuits grows linearly in time [46], saturating at times scaling proportionally to the system size N.While our analytical considerations show close parallels between entanglement entropy and participation entropy calculation, the boundary conditions of the relevant statistical mechanics model are different.This difference amounts to the uncovered contrasts between the time evolution of participation entropy and entanglement entropy.We focused on times short compared to the system size.A complementary path, considering random circuits in the limit of t → ∞ and N → ∞ and demonstrating the onset of universal features in distributions of the overlaps between the output states generalizing the Porter-Thomas distribution was considered in Ref. [82].
Random quantum circuits may be perceived as minimal models of local unitary dynamics.Hence, similarly to the entanglement entropy case [83], we expect that our results about Hilbert space delocalization extend to generic chaotic non-integrable many-body systems.Moreover, our results may provide a relevant reference point for understanding the Hilbert space delocalization in non-ergodic many-body systems, cf.[84][85][86].The statistical invariance of the considered circuits under a product of arbitrary on-site unitary transformations shows that our results about the growth of participation entropies hold for any basis B U obtained from the computational basis B by a product of independent on-site rotations.
The participation entropies considered here are not only directly available in numerical simulations of many-body systems but are also of experimental relevance.Indeed, recent progress in stochastic sampling of many-body wave functions in ultracold atomic and solid state experiments [87,88] allows, in principle, for their direct experimental evaluation.Similarly, the quantum processors realizing quantum circuits enable high frequency sampling of the output state [89,90], which is a basis of the cross-entropy benchmarking [91], and could enable a direct measuring of the participation entropies.Finally, the process of estimation of IPR, which, in general requires resources scaling exponentially with the system size N, may be simplified by use of appropriate quantum algorithms [92].
This work has identified the leading terms relevant to the participation of entropy growth in random unitary dynamics.There are several interesting directions for further research.Analogously to the case of entanglement entropies [46,63], one may ask a question about fluctuations of the participation entropies around the identified mean values.Properties of the dynamics of participation entropies under higher dimensional circuits are another unresolved problem.Moreover, stabilizer Rényi entropies [93][94][95][96][97], which determine the amount of beyond-classical (non-Clifford) operations needed to perform a quantum task are related to participation entropies [79].The long long time limit of Stabilizer Rényi entropies under local unitary dynamics has been recently understood in [98].However, understanding the properties of growth of Stabilizer Rényi entropies is an exciting direction for further research, facilitated by the results of this work.

Figure 1 .
Figure 1.Participation entropy for the Rényi index q = 2.The left panel shows the quenched S 2 (in red) and the annealed S2 (in blue) average of the participation entropy rescaled by the system size N and plotted as a function of circuit depth t.The inset shows the exponential in time decay of the difference between the average participation entropy SH 2 for random Haar states and the averages S 2 , S2 as a function of time.The right panel shows approach to SH 2 of the annealed average S2 (blue symbols), the exact solution obtained with the tensor network approach (solid purple lines), compared with the asymptotic formula SH 2 − S2 (t) = α d N(2K d ) t−1 (red dashed line), where, for qubits, i.e. d = 2, α 2 ≈ 0.291(5) and K 2 = 2 5 .The inset shows the prefactor α d as function of the on-site Hilbert space dimension d the α d extrapolated from tensor network simulations, while α d→∞ → 1 2 is shown by the black dashed line.

Figure 2 .
Figure 2. Participation entropy growth for various Rényi indices q.Panel (a) shows the approach of the quenched S 3 (in red) and annealed S3 (in blue) average of the participation entropy to the random Haar state value SH 3 as a function of the circuit depth t.Panel (b)presents the approach to SH 3 of the annealed average S3 (blue symbols) calculated with the exact numerical simulation compared with the tensor network contraction results (purple lines), and the fitted formula SH 3 − S3 (t) ≈ 0.76N(2K d ) t−1 (red dashed line) consistent with Eq.(29).Panels (c) and (d) show the approach of the circuit averaged participation entropy Sq to the random Haar state value SH q respectively for q = 1 and q = 4.