Absorbing state transitions with long-range annihilation
Nicholas O'Dea Sayak Bhattacharjee Sarang Gopalakrishnan Vedika Khemani






Department of Physics, Stanford University, Stanford, CA 94305, USA

Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA

Department of Physics, Stanford University, Stanford, CA 94305, USA






§ ABSTRACT 
We introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint. The probability for a pair-annihilation event decays as a power-law in the distance between particles, with a tunable exponent. Such long-range processes arise naturally in various classical settings, such as chemical reactions involving reagents with long-range electromagnetic interactions. They also increasingly play a role in the study of quantum dynamics, in which certain quantum protocols can be mapped to classical stochastic processes with long-range interactions: for example,  state preparation or error correction processes aim to prepare ordered ground states, which requires removing point-like excitations in pairs via non-local feedback operations conditioned on a global set of measurement outcomes. We analytically and numerically describe features of absorbing phases and phase transitions in this family of classical models as pairwise annihilation is performed at larger and larger distances. Notably, we find that the two canonical absorbing-state universality classes – directed-percolation and parity-conserving – are  endpoints of a line of universality classes with continuously interpolating critical exponents. 



Absorbing state transitions with long-range annihilation
    Vedika Khemani
    Received 16 July 2024; accepted 04 September 2024
========================================================



Introduction–
Absorbing state transitions from a fluctuating active phase to an inactive absorbing state are widespread in non-equilibrium classical settings, from chemical reactions to epidemic spreading <cit.>. These transitions also play a growing role in quantum interactive dynamics in which a classical observer monitors a quantum system and performs feedback conditioned on measurement outcomes <cit.>. 
A canonical task in quantum information theory is to prepare a desired many-body entangled state, such as a topologically ordered ground state of the 2D toric code <cit.> or a Schrodinger-cat ground state of the 1D Ising model. A typical “low-entropy" configuration around the desired state often looks like a dilute gas of point-like excitations  (such as anyons in the 2D toric code or domain-walls in a 1D Ising model). In the simplest cases, these excitations may only by annihilated or created in pairs, and they may be separated freely without energy penalty.   Canonical measurement-based state-preparation protocols make measurements everywhere in the system to deduce the locations of excitations, and apply non-local “string-like" recovery operations to remove excitations in pairs <cit.>. This process can be mapped to a classical stochastic process describing the dynamics of excitations <cit.>. However, crucially,  the long-range nature of feedback means that pairs of particles can effectively instantaneously annihilate over long distances. 

Long-range interactions are also natural in classical models showing absorbing state transitions, in settings ranging from chemical reactions involving charged particles, to predator-prey models incorporating movement patterns of different species (see Ref. <cit.> for a review discussing long-range hopping and branching, and Refs. <cit.> for recent work on biased hopping). However, models with long-range annihilation have largely been neglected in the classical absorbing state literature. An exception is Ref. <cit.> by Park and Deem, which, however, does not consider branching processes and does not study the transition out of the absorbing phase.





To address this hole in the literature, we introduce a family of stochastic reaction-diffusion processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint which enforces that particles are created and annihilated in pairs (Fig. <ref>(a)-(c)). By tuning a long-range exponent κ, we analytically and numerically probe how the features of the absorbing phase and the transition change as pairwise annihilation is successfully performed at larger and larger distances[We note that a key distinction between these absorbing state models and conventional descriptions of quantum error correction is that a noisy quantum system allows defects to be created in pairs, so that the defect-free (empty) state is not an absorbing state. This is discussed further in <cit.>, where we also discuss various other classes of quantum models that do display the same universality classes as the classical models we study.].

Our model uncovers a continuum of unusual universality classes. Two paradigmatic universality classes of absorbing state transitions are directed percolation (DP) and parity-conserving (PC), respectively corresponding to local models without and with a parity constraint. Having a parity constraint or not naively seems like a binary question, making DP and PC discrete and ostensibly unconnected universality classes. However, our model uncovers a line of universality classes with exponents that continuously interpolate between DP and PC despite a global parity constraint (Fig. 1d)), greatly enriching the set of known critical phenomena in non-equilibrium phase transitions. 



In long-range equilibrium and non-equilibrium systems, continuously varying exponents are usually associated with an approach to mean-field physics under sufficiently-long-range interactions (e.g. section 3 of Ref. <cit.>). However, we find in numerics on a chain that the transition in the long-range limit of our model corresponds to (1+1)d DP, not mean-field DP. Below, we do not interpret the long-range interactions as changing the dimensionality of the system but rather as weakening the parity constraint to no longer have local consequences.

Our numerics are for one dimension, but we discuss scaling arguments that should be valid in the absorbing phase in all dimensions. We probe the transition numerically, and we outline a field theory program to understand its properties. We derive the field theory describing our model in the Supplementary Material <cit.> and discuss some of its interesting features in the main text. Though field theories usually have actions that are polynomial in the fields, we have a nontrivial exponential term in our action. We show that the resulting mean-field theory in the absorbing phase crucially relies on this exponential term and matches the predictions of our simple scaling arguments in d ≥ 2. 




Model and phase diagram– 
We study a classical stochastic process describing a system of particles undergoing diffusion, branching and annihilation subject to a parity constraint which conserves the number of particles modulo two. Our model consists of a mixture of deterministic and stochastic steps, illustrated in Fig. <ref>(a)-(c) for a system in one spatial dimension. Each time step begins by selecting a particle at random, illustrated by the green outline. With probability , we continue to the annihilation stage in (a), where the particle's distance r to the nearest excitation is calculated and with probability 1/r^ the two particles are annihilated. Thus,  sets the rate at which pairing is attempted, while  controls the nonlocality of the annihilation. Because we only attempt to annihilate a particle with the nearest particle rather than attempting pairwise annihilation with all other particles, the thermodynamic limit is already well-defined without any system-size dependent rescaling of the attempted annihilation rate (e.g. as in Ref. <cit.>), even for the extreme long-range case of κ=0.

If, with probability 1-, annihilation is not selected, then (b) with probability 
the particle branches (produces two offspring on two different adjacent sites) or (c) with probability 1-, it attempts to hop to a neighboring site. A hard-core constraint is enforced by aborting the attempt to branch or hop if more than one particle would occupy the same site. At the end of each time step, in a system with N particles, the time increments by 1/N to model these processes occurring in parallel; the more particles, the more of these processes happen during a given time interval. Note that a system with no particles is an absorbing state without dynamics; this state can be reached but never left.  In our numerics, we work in one spatial dimension and fix = .2, while we explore the phase diagram shown in Fig. <ref>d) with parameters  and . Throughout, we consider initial states with an even number of particles so that in principle the absorbing state can be reached; in particular, we choose even-length chains and initialize in the fully occupied state. 

When the annihilation rate  is low, the density of particles n(t) is asymptotically constant up to a time growing exponentially in system size. This “active" phase is denoted by “I" in Fig. <ref>d). When  is sufficiently large, the density of particles decays to zero in a time at most polynomial in system size. The functional form of the decay rate n(t) ∼ t^- in these “absorbing" phases is set by  (see phases II, III, IV in Fig. <ref>d)). Between the active and absorbing phases is a critical line of absorbing state transitions where the density decays as n(t) ∼t^-; we denote the critical decay exponent  to differentiate it from the absorbing phase decay exponent of . We summarize our results for the phase diagram in Fig. <ref>d), and provide a detailed exposition below.

Absorbing phase analytics and numerics–
The properties of the absorbing phases are controlled by the pure annihilation fixed point and hence can be understood through scaling arguments in the limit of no branching. 
Though our main focus in this work is the case of d=1, we also consider the characteristics of the absorbing phase in higher dimensions. 

We note that section 3.1 of Ref. <cit.> discusses a related model with both long-range annihilation and diffusion, and in particular notes an analogous phase to our PC absorbing phase II in Fig. <ref>. However, that section restricts to κ > min(2,d) (and the paper restricts to κ > d more generally), and so does not obtain our regimes III and IV. 
Furthermore, the existence and properties of the absorbing state transition (our next section) require branching and are not considered in Ref. <cit.>.
Our model also has an emergent cutoff on the range of the interaction. By only pairing a particle to the particle closest to it, we avoid diverging annihilation rates in the thermodynamic limit even when the nonlocality of the annihilation is maximal. Together, branching and the natural cutoff give us access to a novel regime of the phase diagram; this otherwise hidden part of the phase diagram contains the novel phase transitions that interpolate all the way between the PC and DP universality classes. 


First, consider the short-range limit of = ∞ where particles must be neighboring in order for annihilation to proceed. Annihilation is then diffusion-limited, as particles must effectively collide in order to annihilate. In this limit, there is a standard set of arguments for the decay of n(t) in terms of the number of unique sites visited by a random walker in a time t. The number of unique sites visited controls the number of particles a given particle can encounter as it diffuses and hence the number of annihilation events (see chapter 13 of Ref. <cit.> for more details, and Ref. <cit.> for alternative arguments using field theory).
In particular, in d dimensions

    n(t) ∼t^-d/2    d < 2 
        
     t^-1log(t)    d = 2
        
    t^-1    d > 2


On the other hand, for < ∞, there will be a steady background decay of particles from the long-range annihilation. Suppose  is small so that the particle annihilation rate is dominated by events where the particles are not necessarily close. In d dimensions, the typical interparticle distance will scale on the order of r∼n(t)^-1/d and hence the typical probability for a pairing to result in annihilation will scale as n(t)^/d. The number of annihilation events per unit time will scale as the number of particles times this probability, giving an effective rate equation ṅ(t) ∼ n(t)^1+/d. Solving this equation gives

    n(t) ∼t^-d/    >0
        
     e^-c t    = 0

With diffusion turned off, this result (for κ>0) has been rigorously proven in Ref. <cit.>, and further discussed in a renormalization group context in Ref. <cit.> (for κ > d).

We now make the simple assumption that the process (short-range annihilation induced by diffusion vs. long-range annihilation between distant particles) which leads to a faster decay is the one that (asymptotically) dominates the dynamics. Comparing Eqs. <ref> and  <ref>, we get in d=1:

    n(t)  ∼t^-1/2   ≥ 2                (II)
    t^-1/    0 < ≤ 2     (III)
     e^-c t    = 0                (IV)

The parenthetical labels (II/III/IV) are shown in the corresponding regions of the phase diagram in Fig. <ref>d). 

We numerically confirm this picture in Fig. <ref> via linear fits of log(n(t)) vs log(t). There is good agreement except for a small, systematic discrepancy around ≈ 2.6. We believe this discrepancy is caused by asymptotically sub-leading terms from the background long-range decay from  in this otherwise diffusion-limited regime, see <cit.>. For example, =3 in the inset shows an ultimately-transient slower decay at early times. 


Similarly, for d ≥ 2,

    n(t) ∼t^-1   ≥ d
        
     t^-d/    0 < ≤ d
        
     e^-c t    = 0

We save the numerical verification of this prediction in d ≥ 2 for future work. To summarize, for > max(2,d), the model is effectively short-ranged and the absorbing phase is diffusion-limited, while for < max(2,d), absorption occurs  faster and is dominated by long-range annihilation. In particular, in d=1, the decay in the absorbing phases at ≥ 2 and =0 are consistent with the PC and DP absorbing phases of n(t) ∼ t^-1/2 and n(t) ∼ e^-ct respectively.

We can interpret these results in terms of the parity constraint. Conventionally, 
the DP/ PC universality are classes distinguished by the absence/ presence of a discrete parity symmetry.  In our model, the global parity constraint exactly holds, but it no longer has consequences locally when κ=0. In particular, looking at just a local subsystem, long-range annihilation will violate the parity of the number of particles in the subsystem even as it preserves the global parity. This allows a model with a global parity constraint to be described by symmetry-free DP rather than PC physics. Thus we expect that the κ = 0 model is described by DP, including at the phase transition. We confirm this numerically in the next section.

Transition numerics–
In Fig. <ref>, we find that the decay exponent  is consistent with DP's ≈ .159 at =0 and PC's ≈ .285 at ≥ 2, and the exponents for 0 <  < 2 appear to monotonically interpolate between the two, indicating a line of critical fixed points with continually varying exponents.  From our numerics, we cannot rule out that the critical exponents at the transition begin to change from  at a _c slightly different from 2. However, our numerics are consistent with the idea that the  at which the phase behavior changes (= 2 from the previous section) is the same  at which the critical behavior changes. 

In the inset of Fig. <ref> we show an example of our method of extracting  and  from the dynamics. At large system sizes and for  close to , the dynamics will appear to follow the algebraic decay of the critical model. However, when  is detuned to be above or below the transition, the dynamics at late times will respectively drift to the different asymptotic behaviors of the absorbing and active phases. By calculating a “running" (t) ≡log_10[n_d(t) / n_d(10t)], we see a horizontal separatrix between the absorbing and active phases; the  and  (after finite-time transients) of the separatrix give our estimates for  and . 
See Ref. <cit.> for a succinct discussion of this and other numerical probes of absorbing state critical behavior and chapter 4 of Ref. <cit.> for a general discussion. Note that both DP and PC have a large z critical exponent (and so should the interpolating universality classes), meaning that systematic late-time finite-size effects controlled by t/L^z should only appear at much later times than we probe.

In the following sections, we briefly outline a field theory program to analytically estimate the critical exponents and critical dimensions of these newly uncovered universality classes. 

Field theory action–
Cardy and Tauber gave a systematic field theory treatment of critical phenomena and phase behavior in models with local, pairwise annihilation processes <cit.> (see also the review <cit.>). The starting point is the derivation of the field theory directly from the stochastic process <cit.> by treating the real-time stochastic master equation as an imaginary-time Schrodinger equation. The new degrees of freedom are complex fields ϕ(x,t) and (x,t); statistical expectation values of n(x,t) can be computed via expectation values of ϕ(x,t) in the field theory (e.g. 𝔼[n(x,t)] = ⟨ϕ(x,t)⟩, 𝔼[n^2(x,t)] = ⟨ϕ^2(x,t) + ϕ(x,t)⟩). 

Approximations like derivative expansions and the continuum limit for field theories near criticality simplify the field theory while maintaining the universality of the phase transition and phases, allowing for renormalization group calculations of critical exponents. However, the field theory keeps a few peculiar signatures of its origins as a classical stochastic process: conservation of probability ultimately forces the action to vanish whenever = 1, whatever the value of ϕ; processes changing the number of particles make the theory explicitly non-Hermitian, which manifests as terms in the action carrying an unbalanced number of ϕ and . It is useful to identify ϕ(x,t) with n(x,t) at the mean-field level and to build intuition for ϕ(x,t) from n(x,t), but this identification cannot be made rigorously beyond the mean-field level <cit.>. 


Using a method pioneered in <cit.>, we extend their results to derive a field theory for our nonlocal annihilation process in <cit.>. Our action density ℒ includes standard diffusion (∂_t-D ∇^2)ϕ and branching B(1-^2)ϕ terms, but it also contains an annihilation term that is exponentially suppressed in particle density:

    ℒ(𝐱,t)    ∋  A ∫d^d𝐲/|𝐱 - 𝐲|^[  (1-(𝐱,t)(𝐲,t))ϕ(𝐱,t)ϕ(𝐲,t) 
       exp(-∫_r=0^r=|𝐱-𝐲| d^d 𝐫 (𝐱+𝐫,t)ϕ(𝐱+𝐫,t) ) ]

The exponential suppression comes directly from the constraint that a particle can only be paired with the closest particle <cit.>; heuristically, the exponential term is accounting for the low likelihood that two far away particles have no particles in between them.

We believe this exponential factor plays a fundamental role in both the critical and absorbing phase behavior at small . This is straightforward to see at the level of the saddle-point approximation or Euler-Lagrange equations, which we will refer to as mean-field theory.

Mean-field theory
The Euler-Lagrange equations of this action, under assumptions and identifications noted in <cit.>, reduce to 

    ṅ(𝐱,t) = D ∇^2 n(𝐱,t) + 2B n(𝐱,t) - A n(𝐱,t)  [ 
       ∫ d^d y n(𝐲,t)/|𝐱-𝐲|^( e^-∫_r=0^|𝐱-𝐲| d^dr n(𝐱+𝐫, t) + 𝐱↔𝐲) ]

We will further assume a uniform initial condition; this equation is translationally invariant, and so n(x,t) will then be independent of x at all times. Introducing 𝐳 = 𝐱-𝐲 and the volume of a d-dimensional unit ball V_d, the dynamics further simplify to

    ṅ(t) = 2B n(t) - 2 A n(t)^2 ∫ d^d z 1/z^ e^-V_d z^d n(t)

For ≤ d, the integral needs the infrared-regulator provided by the decaying exponential, which forces the resulting integral to depend strongly on n. When > d, there is no longer an infrared divergence, but there is an ultraviolet divergence; this small-z divergence in the integral is regulated by the lattice spacing, and so ∫ d^d z 1/z^ e^-V_d z^d ρ∼const + O(n(t)). Together, these give (with non-universal constant c depending on d,, a)

    ṅ(t) ∼ 2B n(t) - cA  n(t)^1+/d   ≤ d 
        
     n(t)^2 log(1/n(t))     = d
        
      n(t)^2     > d

To reiterate, at the level of mean-field theory, the exponential in the annihilation term prevents the infrared divergence of the long-range annihilation double-integral at small  by introducing a density-dependent cutoff length.

Note that mean field theory only holds in sufficiently high dimensions, if at all. Indeed, our mean field theory does not predict an absorbing phase for any > 0 at any positive branching B; a positive coefficient on the dominant term n(t) forces a non-vanishing steady state n(t) ∼const. This was argued (in the case of κ = ∞) to be an artifact of the infinite-dimensional nature of mean field theory <cit.>; a special upper critical dimension d_c was identified below which the absorbing phase is indeed stable. For =∞, d_c was found to be 4/3 at one-loop order <cit.>; on the other hand, our mean-field equations at =0 show a transition at nonzero branching and are hence consistent with d_c = ∞. We thus anticipate that d_c will be a function of κ. 

Despite the omission of the absorbing phase at nonzero branching for κ>0, the mean-field predictions for the absorbing phase – i.e. solving Eq. <ref> when B=0 – match that of our scaling arguments when d ≥ 2 in Eq. <ref> (up to logarithmic corrections when = d).

Discussion–
Our work mapped out the phase diagram of diffusive particles undergoing branching and long-range annihilation, and we uncovered a new line of universality classes connecting the naively unrelated DP and PC classes. 
We characterized the phases and phase transition by how the density of the particles decays with time starting from a completely occupied state.  Other probes of critical exponents are associated with survival time and growth of small seeds of particles in otherwise empty systems; we leave these complementary probes to future work.
    
We also derived a field theory describing the unusual transitions and absorbing phases and showed that the resulting mean field theory reproduces our simple scaling arguments for absorbing phase behavior in d ≥ 2. We view this agreement in high dimensions as signaling that field theory is an appropriate tool for understanding properties of the absorbing phase and the transition. Further studies of this unusual field theory is interesting in its own regard. However, to study low dimensions and go beyond mean-field theory, it is necessary to consider renormalization. 
We note that both perturbative RG (once appropriate counterterms are identified for the exponential factor) in the style of Cardy and Tauber <cit.>, and non-perturbative RG in the style of section 3.4.2 of the review <cit.> appear promising.  

Finally, we note that our study was inspired by considering quantum error correction processes which often require completely non-local feedback operations (κ =0) in various canonical cases. Despite this broad similarity, there are also crucial differences between descriptions of error correction and absorbing state transitions <cit.>, and it would be interesting to explore how the different universality classes of absorbing state transitions as a function of κ may inform error correction protocols with varying degrees of non-locality in feedback.  

Acknowledgements– We are grateful to Su-Chan Park for useful discussions about scaling arguments and mean field theory,  Alan Morningstar for prior collaboration on related work, and Pavel Nosov and Samuel Alipour-fard for discussions about field theory.
This work was supported by the Office of Naval Research Young Investigator Program (ONR YIP) under Award Number N00014-24-1-2098 (S.B and V.K.). 
V.K. also acknowledges support from the Alfred P. Sloan Foundation through a Sloan Research Fellowship and the Packard Foundation through a Packard Fellowship in Science and Engineering. N.O.D.  acknowledges support
from the ARCS Foundation for ARCS Scholar funding.
Numerical simulations were performed on Stanford Research Computing Center's Sherlock cluster. Formulating the models was supported in part by the US Department of Energy Co-design Center for Quantum Advantage (C2QA)
under contract number DE-SC0012704 (S.G.).














§ FIELD THEORETIC ANALYSIS




 §.§ Derivation of the field theory

In this appendix, we provide explicit steps for the derivation of the field theory described in the main text. We follow the treatment in Refs. <cit.> and <cit.>; much of this derivation is standard, but we find it worthwhile to spell out relevant details. The main new contribution in this appendix is our treatment of the “only annihilate with the nearest particle" constraint on long-range annihilation, for which we adapt results of Ref. <cit.>. The results of  Ref. <cit.> were originally intended to realize hard-core constraints, and we use them both for their original purpose and in realizing our annihilation constraint.


The starting point is a master equation that describes the reaction-diffusion system under consideration; in particular, we consider diffusion, branching and long-range annihilation processes with rates denoted by D, B and A respectively. We will define the master equation in d spatial dimensions on a hypercubic lattice for concreteness. For the stochastic process in the main text, D_0=1/2d(1-)(1-), B_0=(1-) and A_0=.

We define an occupation vector 𝐧 where n_i is the number of particles at site i. We also define unit vectors 𝐞_i for ease in describing changes in 𝐧. As an example, if the number of particles on site i increases by 1, then 𝐧→𝐧 + 𝐞_i. Sometimes it is useful to directly refer to the position vector of site i, which we will denote as 𝐫_i, such as in the power-law decay of the annihilation rate 1/|𝐫_i - 𝐫_j|^κ.

When indexing sums, we will use the notation ∑_j ∼ i to mean the sum over the set of j such that j is a nearest neighbor to i. ∑_j,k ∼ i will be the sum over the set of j and k such that j ≠ k and j and k are both nearest neighbors of i. ∑_j ≠ i will denote the sum over the set of j with j ≠ i.


The function P(𝐦;t) will denote the probability that the state of the system is represented by the occupation vector 𝐦 at time t. The master equation for the reaction-diffusion process can be decomposed into parts representing diffusion, branching, and annihilation

    ∂_t P(𝐧;t)=P_diff+P_branch+P_ann

with

    P_diff:=   D_0[∑_i ∑_j ∼ i[(n_i+1)P(𝐧 +𝐞_i - 𝐞_j;t )δ_n_j,1-n_iP(𝐧;t)δ_n_j,0]]
     P_branch :=   (B_0/ )[∑_i∑_j,k ∼ i n_i[P(𝐧 - 𝐞_j - 𝐞_k; t)δ_n_j,1δ_n_k,1- P(𝐧;t)δ_n_j,0δ_n_k,0]]
    
        P_ann:=    A_0 ∑_i∑_j ≠ i1/|𝐫_i-𝐫_j|^κ[(n_i+1)(n_j+1)P(𝐧+𝐞_i + 𝐞_j; t)-n_i n_jP(𝐧;t)]{∏_k: |𝐫_k - 𝐫_i|<|𝐫_j - 𝐫_i|δ_n_k,0}

We briefly explain the above construction below. Each of these expressions P_diff,  P_branch and P_ann is the difference of two terms corresponding to the rate that the configuration 𝐧 is entered minus the rate that the configuration 𝐧 is left. The coefficients like (n_i+1) or n_i and the like are combinatorial factors that account for choosing arbitrarily any of the particles at that site to diffuse, branch or annihilate.  The diffusion and branching rates are normalized by 2d and  to account for the fact that diffusion can cause hopping of the particle i to any of its 2d neighbors and branching can birth particles on any of the  pairs of sites neighboring i.

The Kronecker deltas multiplying the rates in P_diff and P_branch implement a hard-core constraint for the particles, restricting to at most one particle at a site. In particular, an initial configuration that respects this constraint will continue to respect this constraint under the dynamics. We also note that we do not expect changes in universality from loosening the hard-core constraint. (More precisely, we don't expect changes as long as the on-site annihilation rate is <∞. See Ref. <cit.> for a brief discussion of certain processes without hard-core constraints always being absorbing when the onsite annihilation is infinite.)

The product of Kronecker deltas in the curly brackets in P_ann implements the nearest-particle constraint, ensuring that there are no particles closer to i than j when annihilating the pair; this is the factor that gives rise to the exponential term in the field theory. As noted in the main text, we do expect this term to affect the physics, particularly at small κ, since it is responsible for regulating the long-range annihilation.

The next step in deriving the field theory in the Doi-Peliti formalism <cit.> is to write an equivalent bosonic Hamiltonian whose imaginary-time quantum dynamics will exactly correspond to the real time dynamics described by the stochastic master equation. Occupation numbers n_i can be interpreted as n_i many bosonic particles at site i, and the corresponding state vector 𝐧 can be represented as an appropriate Fock state. We introduce bosonic operators _̱i, _̱i^† obeying the usual algebra [_̱i,_̱j]=[_̱i^†, _̱j^†]=0 and [_̱i, _̱j^†]=δ_ij. We introduce the states |𝐧⟩:= ⊗_i (^̱† _i)^n_i|0⟩ (this choice of normalization is standard and aids in the calculations) with the action |̱%̱s̱⟩̱n=n|n-1⟩ and ^̱†|n⟩=|n+1⟩. The occupation vector 𝐧 can then be written as a state vector |𝐧⟩:= ⊗_i (^̱† _i)^n_i|0⟩. 

We can then define a wavefunction corresponding to the vector of probabilities 

    |Ψ(t)⟩:= ∑_𝐧 P(𝐧;t)|𝐧⟩

Note that this state is not normalized according to ⟨Ψ(t) | Ψ(t)⟩ = 1; it is instead normalized according to the rule that the probabilities sum to one, which corresponds to the normalization

    ⟨ 0 | e^∑_i b̂_i |Ψ(t) ⟩ = 1

Expectation values in the stochastic process then become matrix elements of operators in the Hilbert space

    𝔼[f(n(t))] =  ⟨ 0 | e^∑_i b̂_i f(n̂) |Ψ(t) ⟩

The time evolution of |Ψ(t) ⟩ takes the form of Schrodinger equation in imaginary time t of the form

    -  ∂/∂ t|Ψ(t)⟩= |Ψ(t)⟩

for bosonic  computed below. We emphasize again that the imaginary time dynamics with a quantum Hamiltonian actually corresponds to the real time dynamics of the classical stochastic process. 

For the master equation in Eqs. <ref> and <ref>, the corresponding quantum Hamiltonian is 

    = _diff+_branch+_ann

with

    _diff :=    -D_0∑_i∑_j ∼ i(^̱†_j_̱i-_̱i^†_̱i)δ_n̂_j,0
    _branch:=   -(B_0/)∑_i∑_j,k∼ i_̱i^†_̱i(_̱j^†_̱k^†-1)δ_n̂_j,0δ_n̂_k,0
    _ann:=   -A_0∑_i∑_j ≠ i1/|𝐫_i-𝐫_j|^κ[_̱i_̱j-_̱i^†_̱i_̱j^†_̱j]{_k: |𝐫_k - 𝐫_i|<|𝐫_j - 𝐫_i|δ_n̂_k,0}.

    
The bosonic Hamiltonians are intuitive—diffusion manifests as hopping from site i to j, branching manifests by creation operators given a particle is present at site i, and annihilation manifests as annihilation operators at site i and j. The branching and annihilating operators are manifestly non-Hermitian, implying the irreversibility of the processes. The second term in each Hamiltonian (i.e. the bosonic density operators acting as onsite energies or density-density interactions) follow from the rates of leaving configurations in the master equation; these are effectively enforcing conservation of probability. The exclusion constraints can be explicitly represented in terms of the operator n̂ by δ_n̂_i,m=∫_-π^π (du/2π) e^i u (n̂_i-m) <cit.>.

Note that the equality of statistical expectation values with matrix elements given in Eq. <ref> can be re-written as 

    𝔼[f(n(t))] =  ⟨ 0 | e^∑_i b̂_i f(n̂) e^-Ĥt|Ψ(0) ⟩

By inserting resolutions of the identity made of coherent states, we can transform this expression into a coherent state path integral. There are subtleties, especially with time ordering and boundary conditions, and we direct the interested reader to section 3.3 of Ref. <cit.> and references therein for a detailed discussion of the coherent state path integral in this context.


We consider coherent states labeled by a vector of complex numbers ϕ and denoted by |ϕ⟩. These states are defined as eigenstates of the annihilation operator, b̂_i |ϕ⟩ = ϕ_i |ϕ⟩, and constructed as |ϕ⟩=e^∑_iϕ_i_̱i^†|0⟩. For this choice of normalization, the resolution of the identity is 

    =∫(∏_j(dϕ_jd_j/2π i))e^-∑_jϕ_j_j|ϕ⟩⟨ϕ|.

The integral is to be interpreted as follows. We can decompose ϕ into its real and imaginary parts ϕ= R + i I. The integral in Eq. <ref> becomes 

    =∫(∏_j(d R_jdI_j/π))e^-∑_j(R_j^2 + I_j^2)|R + i I⟩⟨R + i I|.

where all integrals run from -∞ to ∞.

As noted, we can compute a path integral by inserting resolutions of the identity into time-slices of Eq. <ref>. e^-Ĥ t will become a weighting e^-S with S a functional of ϕ(t) and e^-S understood perturbatively <cit.>. It is nice to write H in a normal-ordered form, as this allows the b's of the Hamiltonian to become the ϕ's of one time slice and the b^†'s to be the ϕ^*'s of the next time slice. Besides a time derivative term and boundary terms, S can then be written down by computing ⟨ϕ|H| ϕ⟩/⟨ϕ|ϕ⟩:

    S[ϕ] = boundary terms + ∫_0^t_f dt [ ⟨ϕ|H| ϕ⟩/⟨ϕ|ϕ⟩ + ∑_i _i ∂_t ϕ_i ]

where ϕ is a function of the integration variable t. It is nice to use t as the dummy integration variable; when calculating observables at time t, t_f will then be t in a slight abuse of notation. As noted, we direct the interested reader to section 3.3 of Ref. <cit.> and references therein for more details.

The only non-trivial normal-ordering is of the exclusion constraint operators δ_n̂_i,0 and δ_n̂_i,1. The normal ordering can be computed <cit.> using the integral representation mentioned before. For our purposes, we only need the following identity, which holds for integers m≥ 0, 

    ϕδ_n̂,mϕ/⟨ϕ||ϕ⟩   =1/m!(ϕ)^me^-ϕ

from which related identities can be derived straightforwardly, including ϕb^†δ_n̂,mϕ/⟨ϕ||ϕ⟩ = ϕ^*  ϕδ_n̂,mϕ/⟨ϕ||ϕ⟩ or ϕb δ_n̂,mϕ/⟨ϕ||ϕ⟩ = ϕδ_n̂,m-1 bϕ/⟨ϕ||ϕ⟩ = ϕϕδ_n̂,m-1ϕ/⟨ϕ||ϕ⟩. Note that ϕδ_n̂,mϕ/⟨ϕ||ϕ⟩=0 for m<0. 

For diffusion:

    ⟨ϕ|H_ diff| ϕ⟩/⟨ϕ|ϕ⟩   = -D_0∑_i ∑_j ∼ i (_j ϕ_i - _i ϕ_i)e^-_j ϕ_j
       = -D_0∑_i ∑_j ∼ i (_j ϕ_i - _i ϕ_i) +  subleading

Near the critical point and in the absorbing phase, we can Taylor-expand the exponential. The contributions labeled subleading in Eq. <ref> will ultimately be subleading in the RG sense after taking the continuum limit, as they will contain too many fields and derivatives. 
Similarly, for annihilation:

    ⟨ϕ|H_ ann| ϕ⟩/⟨ϕ|ϕ⟩=   -A_0∑_i∑_j ≠ i1/|𝐫_i-𝐫_j|^κ[ϕ_iϕ_j-_i ϕ_i_jϕ_j] e^-∑_k: |𝐫_k - 𝐫_i|<|𝐫_j - 𝐫_i|_kϕ_k

Note that we do not attempt to Taylor expand the argument of the exponential. The reason is that although each individual _kϕ_k may be small close to the transition and in the absorbing phase, their sum might be considerable if sites i and j are far apart. We will indeed find that this is the case when we consider the saddlepoint equations.
For branching:

    ⟨ϕ|H_ branch| ϕ⟩/⟨ϕ|ϕ⟩   = -(B_0/)∑_i∑_j,k∼ i_i ϕ_i (_j _k-1) e^- _j ϕ_j - _j ϕ_k
       = -(B_0/)∑_i∑_j,k∼ i_i ϕ_i ( (_i + (_j - _i)) (_i + (_k - _i))-1) e^ - _j ϕ_j - _j ϕ_k
       = -(B_0/)∑_i∑_j,k∼ i_i ϕ_i ( _i^2-1)  +  subleading
       = -B_0 ∑_i _i ϕ_i ( _i^2-1) +  subleading

For branching, almost all of the terms labeled subleading are indeed subleading in the RG sense, as they contain products of many fields and/or derivatives. However, one of the terms coming from the linear order of the Taylor expansion of the exponential is _i ϕ_i_jϕ_j with j and i nearest neighbors, which is comparable to the analogous term in the annihilation process. We neglect this contribution for simplicity, as we view it as effectively already contained in the annihilation process. Additionally, when considering the saddlepoint equations, such a contribution will enter at O(n^2).

As noted above, these expectation values will ultimately yield the action S of the field theory, up to a time derivative term and boundary terms. f(n̂) will give rise to fields multiplying e^-S in the path integral. Writing f(n̂) explicitly in terms of b̂ and b̂^† in a normal-ordered form, the b̂^† acting to the left on ⟨ 0 | e^∑_i b̂_i will be just 1, leaving only products of b̂_i terms. After inserting resolutions of the identity to construct the path integral, b̂_i →ϕ_i(t). |Φ(0)⟩ and ⟨ 0 | e^∑_i b̂_i will give boundary conditions of the path integral; the ⟨ 0 | e^∑_i b̂_i boundary condition is especially straightforward, as this is in fact the coherent state ⟨1|.

For example, following this procedure, 𝔼[n_i(t)] = ⟨ϕ_i(t) ⟩ and 𝔼[(n_i(t))^2] = ⟨ (ϕ_i(t))^2 + ϕ_i(t) ⟩, where ⟨⟩ means expectation value with respect to the above-mentioned action and boundary conditions. If we refrain from Taylor expanding the exponentials implementing the hard-core constraint and then dropping subleading terms, the coherent state path integral at this stage will still reproduce the statistical expectation values. However, we indeed drop terms and will subsequently take a continuous space limit. These approximations will maintain universality but will no longer allow for exactly reproducing statistical expectation values.

We will focus on the action in the following, and we will not further discuss boundary conditions. Taking the continuous space limit of Eq. <ref> using Eqs. <ref>, <ref>, and <ref> (with the choice ϕ_i(t) → a^d ϕ(𝐱,t) and _i(t) →(𝐱,t) for lattice spacing a), we obtain the action

    S:= ∫_t_1^t_2d t [ ∫d^d𝐱[∂_tϕ+ℋ_local(;ϕ)]+∫d^d 𝐱d^d𝐲 ℋ_non-local(;ϕ)]
 



    ℋ_local   :=-D ∇^2ϕ +B(1-^2)ϕ
    ℋ_non-local   :=- A_κ(|𝐱-𝐲|)(1-(𝐱,t)(𝐲,t))ϕ(𝐱,t)ϕ(𝐲,t)  exp(-∫_r=0^r=|𝐱-𝐲|d^d 𝐫 (𝐱+𝐫,t)ϕ(𝐱+𝐫,t) )
 
where A_κ(|𝐱-𝐲|):=A/|𝐱-𝐲|^κ  Eqs. <ref> and <ref> describe the field theory referenced in the main text. D,B,A are equal to D_0, B_0, A_0 up to factors of the lattice spacing. We will generically take t_1 = 0 and t_2 = t.

Before concluding this subsection, we highlight certain salient features of this field theory. (1) We have obtained a non-relativistic field theory (since the kinetic term is given by (∂_t-D∇^2)ϕ) for a complex scalar field ϕ. (2) The branching term contributes to a mass B for the fields and can be appended to the Gaussian part of the action. The branching term also contributes an interaction term of the form ^3ϕ; the annihilation term completely contributes to interaction terms, of which key to the annihilation is ϕ(𝐱,t)ϕ(𝐲,t) while the other term is a non-local density-density interaction, and both of these terms are weighted by the nearest-particle exponential factor. We also note that the field theory is non-local only in space, and remains local in time. (3) The lack of particle number conservation manifests in the non-Hermitian nature of the field theory and lack of U(1) symmetry. The explicit parity conservation of the dynamics corresponds to ℤ_2 symmetry of ϕ↦ -ϕ and ↦ -. (4) The scaling dimensions of the coupling constants can be straightforwardly computed using [length] =-1 and [time]=-2 so that [momentum]=1; under renormalization, there will be anomalous dimensions ascribed to fluctuation effects over mean-field level.



 §.§ Euler-Lagrange minimization

In this subsection, we provide additional details about the mean-field analysis described in the main text. The mean-field rate equation is obtained by Euler-Lagrange minimization of the action derived in the previous subsection for the fields ϕ and . The Euler-Lagrange equations we obtain are as follows,

    
    [∂_t-D ∇^2+B-3B^2(𝐱,t)+∫d^d𝐳[A_κ(|𝐱-𝐳|)(𝐳,t)ϕ(𝐳,t)(( 𝐳, 𝐱;t )+( 𝐱, 𝐳;t ))]]ϕ(𝐱,t)-
       [∫d^d𝐳_1d^d𝐳_2  A_κ(|𝐳_1-𝐳_2|)(1-(𝐳_1,t)(𝐳_2,t))ϕ(𝐳_1,t)ϕ(𝐳_2,t)( 𝐳_1, 𝐳_2;t )1_0≤ |𝐳_1-𝐱|≤ |𝐳_1-𝐳_2|]ϕ(𝐱,t)=0
       [-(∂_t +D∇^2)+B-B^2(𝐱,t)-∫d^d𝐳  A_κ(|𝐱-𝐳|)(1-(𝐱,t)(𝐳,t))ϕ(𝐳,t)(( 𝐳, 𝐱;t )+( 𝐱, 𝐳;t ))](𝐱,t)+
        [∫d^d𝐳_1d^d𝐳_2  A_κ(|𝐳_1-𝐳_2|)(1-(𝐳_1,t)(𝐳_2,t))ϕ(𝐳_1,t)ϕ(𝐳_2,t)( 𝐳_1, 𝐳_2;t )1_0≤ |𝐳_1-𝐱| ≤ |𝐳_1-𝐳_2|](𝐱,t)=0


where (𝐱,𝐲;t):=exp(-∫_r=0^r=|𝐱-𝐲|d^d 𝐫 (𝐱+𝐫,t)ϕ(𝐱+𝐫,t) ) is a shorthand notation for the exponential factor implementing the nearest-particle constraint and 1_x ∈support is the indicator function on the real line. While the Euler-Lagrange equations look complicated, a remarkable simplification is obtained by noting that =1 is a solution of Eq. <ref> (this occurs generically <cit.>). Using this solution in Eq. <ref>, and identifying the mean-field ϕ as the local density n, we obtain the rate equation,

    ṅ(𝐱,t) = D ∇^2 n(𝐱,t) + 2B n(𝐱,t) - A n(𝐱,t)   ∫ d^d 𝐲n(𝐲,t)/|𝐱-𝐲|^( e^-∫_r=0^|𝐱-𝐲| d^dr n(𝐱+𝐫, t) + e^-∫_r=0^|𝐱-𝐲| d^dr n(𝐲+𝐫, t))

This is the rate equation discussed in the main text. We made several approximations in the field theory; explicitly including the exponentials from the hard-core constraints (particularly on the branching term, see discussion below Eq. <ref>) will give corrections of size at most O(n^2).



§ SUBLEADING CORRECTIONS IN THE DIFFUSION-DOMINATED ABSORBING PHASE


In the main text, we discuss the behavior of absorbing phase decay exponent α (n(t) ∼ t^-α) as a function of . One of our predictions in d=1 is that for every ≥ 2, = 1/2. This is borne out in our numerics except for a small discrepancy that is most pronounced when ≈ 2.6. In this brief appendix, we discuss this discrepancy in more detail, and why we believe it  comes from finite-time transients in the decay.

There are several ways to describe the density decay n(t) ∼ t^-1/2 of a diffusion-limited pure annihilation process in d=1. As noted in chapter 13 of Ref. <cit.>, one method is to use an effective rate equation of the form ṅ(t) ∼ -n(t)^3 which, despite being deterministic and having no diffusion term, yields the correct exponent of the decay. We will combine this effective rate equation with the contribution n(t)^κ+1 to the decay from the long-range annihilation, giving 

    ṅ(t) ∼ -c_d n(t)^3 - c_l n(t)^κ + 1

where c_d and c_l are κ-dependent constants multiplying the terms corresponding to diffusion and long-range annihilation.
We emphasize that this effective rate equation is heuristic and likely will not capture the quantitative forms of the subleading asymptotics, but we expect this equation will successfully describe their qualitative features.

For κ≥ 2, the long-range term n(t)^κ + 1 is subleading relative to the leading diffusive term n(t)^3. It is useful to consider different regimes of κ≥ 2.

When κ is sufficiently large, the long-range term rapidly vanishes relative to the diffusive term and will not have an effect on the dynamics outside of a short-lived transient. This would explain why κ≥ 3 does not show a noticeable discrepancy besides early times (the κ=3 data are shown in the inset of Fig. <ref>). 

When κ is close to 2, κ = 2 + ϵ with ϵ very small, the long-range term is comparable in magnitude to the diffusive term for a time that diverges as ϵ→ 0. However, despite being long-lived, the subleading term is so similar to n(t)^3 for small ϵ that it then cannot noticeably change the functional form of the decay. This would explain why we do not see a discrepancy for κ=2.

To summarize, it is only for κ close but not too close to 2 that the subleading term will lead to a long-lived transient that is also sufficiently different from the asymptotic decay to be noticeable. This explains why the discrepancy goes to zero at κ=2 and κ≥ 3; the subleading term needs to strike the right balance. 

Eq. <ref> would also predict subleading transients for κ < 2. However, it is likely that the correct effective rate equation will have a rather suppressed diffusive term n(t)^3. The reason is that in the long-range-dominated regime of κ < 2, long-range annihilation will with very high probability destroy far-away pairs of particles before they can diffuse near each other, removing much of the diffusive contribution to annihilation. This should be contrasted to what happens in the diffusion-limited regime. Although particles coming close to one another from diffusion is what sets the asymptotic decay rate, long-range annihilation events will still occur. They will be subleading, rather than both subleading and suppressed.



§ RELATED MODELS FROM RANDOM CIRCUITS WITH FEEDBACK

We discuss a few ways for random quantum circuits to realize models in the same universality class as the one studied in the main text. Mapping quantum circuits to classical stochastic processes is a standard technique; for a review, see <cit.> and references therein. Appendix I of Ref. <cit.> is also useful for a review of mappings in cases involving measurement and feedback, and we summarize the main idea here. Averaging over block-Haar gates gives rise to a stochastic process described by block-diagonal stochastic matrices satisfying detailed balance, while feedback based on measurement outcomes can violate detailed balance and induce interesting non-equilibrium behavior like absorbing state transitions. 

The average over random circuit realizations can be realized as a channel acting on the averaged density matrix of the system. As noted in Ref. <cit.>, for appropriate choices of random gates and feedback, the diagonal and off-diagonal of the density matrix decouple. The diagonal of the density matrix can be viewed as a 2^L vector of probabilities that is time-evolved by a stochastic matrix. This time-evolution is equivalent to the stochastic evolution of a particle system on L sites, allowing the dynamics of certain observables to be described by a simple classical Markov process. In the case of a direct realization of the stochastic process on the sites, the time evolution of all diagonal observables are encoded in the classical process.

We can consider cases where the particles of the stochastic process don't live directly on the sites; for example, they might correspond to domain walls living halfway between sites in a 1d system, or anyons living on the centers of plaquettes in a 2d system. The domain walls would then be probed through σ^z_i σ^z_i+1 measurements and the anyons through plaquette measurements. However, for simplicity, we will walk through the simpler special case where the particles of the stochastic process live directly on sites. The main difference in the examples with domain walls and anyons is a redundancy of the classical description, where a given description in terms of classical particles may correspond to a handful of different quantum density matrices. Despite this redundancy, appropriate local observables can still be extracted from the classical stochastic process, such as ℤ_2-symmetric diagonal operators in the case of a 1d system with domain walls.

We will consider a realization of our 1d process directly on a lattice of qubits. All operations on the quantum system will be in the computational basis generated by |↑⟩ and |↓⟩, while the corresponding operations on the related classical process will be described in terms of the action on bitstrings generated by 0 and 1 corresponding respectively to |↑⟩⟨↑| and |↓⟩⟨↓|. That is, spin-down on a single site will ultimately correspond to a particle in the classical stochastic process occupying that site, while spin-up corresponds to an empty site.

A two-site particle number conserving block-diagonal Haar matrix U_h gives rise to a hopping term in the classical process. In particular, averaging U_h ⊗ U^*_h gives rise to a matrix that is block-diagonal between the diagonals and off-diagonals of the density matrix. In particular, there is a block corresponding to |↑↑⟩⟨↑↑|,|↑↓⟩⟨↑↓|,|↓↑⟩⟨↓↑|,|↓↓⟩⟨↓↓|, which we will identify with the bitstring basis {00, 01, 10, 11}. In this basis, the block takes the form 

    [   1   0   0   0;   0 1/2 1/2   0;   0 1/2 1/2   0;   0   0   0   1 ]

corresponding to unbiased hopping between 01 and 10. 

Similarly, a two by two block in a three-site Haar matrix between |↑↓↑⟩ and |↓↓↓⟩ gives rise to processes taking 010 ↔ 111 and back corresponding to branching and local annihilation. Alternatively, if there are relevant non-unitary processes, one could have the processes 010 ↔ 111 occur with unequal probabilities. In our model, we assume this to be the case, taking all of our annihilation to come from feedback conditioned on measurements rather than additionally from local unitary processes. However, we do not find significant differences between models where 010 ↔ 111 occurs locally with equal probabilities and models where only 010 → 111 occurs (modulo the feedback steps in these models). 

Feedback allows for nonlocal annihilation processes. In our model, measurements are assumed to be taken everywhere so that all defect positions are known. A particle is selected at random for its dynamics, including whether or not to attempt to pair it with the closest particle to it and attempt annihilation. Alternatively, using the ingredients described above, we could have constructed models with parallel update rules, involving parallelized feedback and brickwork circuits of unitary gates to induce hopping and branching. 

Our serial model has some differences relative to the parallel models; the parallel models may be more natural, given that defects are paired all at once in minimum-weight perfect matching decoders. We chose our serial model for our numerics because we found that it has better finite-size and -time effects, making it easier to illustrate our results.

We also found that some choices of parallel feedback could induce long-range attraction between the defects, even in the ostensibly short-range limit of = ∞. The idea is that if defects are always paired to minimize the sum total of the weights between pairs, there are certain processes occurring at nonzero probability that effectively induce biased random walks. For example, consider the following hypothetical process where two far-away particles both branch and then undergo the annihilation feedback step:

00000100000100000

00001110001110000

00000010001000000

Because the feedback minimizes the sum of the distances, the two particles at the edge of the cluster of 1s closest to the opposite cluster always pair. Though this pair won't annihilate, this pairing forces the pairing and annihilation of the remaining two pairs of adjacent particles in the clusters, leaving just those “inner" two 1's at the final time step. This is effectively inducing a biased motion between the 1s closer to each other; while there are other possible processes or orderings of processes that could occur, there is a net bias even when taking those into account. Biased walks that favor attraction between particles can be relevant for the universality class of the transition <cit.>. Given our interest in interpolating between a long-range directed percolation and a short-range parity conserving limit, we also chose our serial process to avoid these biased walks. However, we believe it would be interesting in future work to compare the bias induced from the above parallel process to bias induced in other ways in the literature.

In this appendix, we have outlined several ingredients in quantum circuits with measurements and feedback that give rise to models similar to the one numerically probed in the main text. While many of these models fall in the same universality class as the model in the main text, there are some that may fall outside, particularly those with biased random walks attracting particles towards one another. We note that there are further variations on measurement and feedback, including using “flags" both for measurement outcomes and for error locations; these introduce further classical degrees of freedom which may themselves undergo absorbing state transitions.



 §.§ Differences from quantum error correction

While our “attempt annihilation only with the closest particle" rule was inspired by quantum error correction, we note some key differences between our absorbing state transition and thresholds in the literature. 

In particular, our absorbing state dynamics do not keep track of whether a “logical error" occurred (the primary object of interest in quantum error correction). Furthermore, in quantum error correction, errors may occur everywhere, but our dynamics instead only allow the spawning of new defects via local particle branching. This restriction allows for the existence of an absorbing state. 

There are similarities, in that the proliferation of defects is what can induce a logical error when attempting to decode in quantum error correction, but the two problems are ultimately concerned with different quantities. 

We do note that the two can be intimately tied together, particularly when classical flags are used. In Ref. <cit.>, the classical flags can reach an absorbing state. Decoding operations are conditional on the classical flags, and the absorbing state for the classical flags halts all these decoding operations. This effectively forces the logical failure rate to be O(1) in the classical flag absorbing phase, tying the logical information and absorbing state transitions together.


