Anyonic self-induced disorder in a stabilizer code: quasi-many body localization in a translational invariant model

TL;DR

This paper investigates how anyonic statistics in a translationally invariant Kitaev toric code model lead to quasi-many-body localization, resulting in slow dynamics and potential for error suppression in quantum information systems.

Contribution

It demonstrates that braiding-induced energy landscapes can cause self-induced disorder and localization without external randomness in a topologically ordered system.

Findings

Random anyon arrangements create complex energy landscapes.

Inhomogeneous initial states lead to glassy, slow dynamics.

Entanglement entropy grows slowly, indicating localization.

Abstract

We enquire into the quasi-many-body localization in topologically ordered states of matter, revolving around the case of Kitaev toric code on ladder geometry, where different types of anyonic defects carry different masses induced by environmental errors. Our study verifies that random arrangement of anyons generates a complex energy landscape solely through braiding statistics, which suffices to suppress the diffusion of defects in such multi-component anyonic liquid. This non-ergodic dynamic suggests a promising scenario for investigation of quasi-many-body localization. Computing standard diagnostics evidences that, in such disorder-free many-body system, a typical initial inhomogeneity of anyons gives birth to a glassy dynamics with an exponentially diverging time scale of the full relaxation. A by-product of this dynamical effect is manifested by the slow growth of entanglement entropy, with characteristic time scales bearing resemblance to those of inhomogeneity relaxation. This setting provides a new platform which paves the way toward impeding logical errors by self-localization of anyons in a generic, high energy state, originated in their exotic statistics.

Equations

$$\displaystyle H^{KL}_{0}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\$$

$$\displaystyle B_{p}(i)\leavevmode\nobreak\ \leavevmode\nobreak\$$

$$A_s^{r/b}(i)$$

$$H^{KL}=H_0^{KL}-t_e\sum _{⟨i,j⟩\in \partial p}{Z}_i{Z}_j-t_m\sum _{i\in legs}{X}_i{X}_{i+1},$$

$$\displaystyle H^{KL}_{\rm dual}\leavevmode\nobreak\ =\leavevmode\nobreak\$$

$${\mu }_{g,i}^x=\prod _{j\ge i}{X}_{r,j}{X}_{b,j+1},{\mu }_{r/b,i}^x=\prod _{j\ge i}{Z}_{g,j}{Z}_{r/b,j},$$

$$\mathcal{P}(t_e^{\prime })=(1-{\rho }_m)\delta (t_e^{\prime }-2t_e)+{\rho }_m\delta (t_e^{\prime }),$$

$$\Delta {\rho }_m^2(t)\equiv \frac{1}{L}\sum _{p=1}^L|⟨{\psi }_{N_m}|(n_{p+1}^m(t)-n_p^m(t))|{\psi }_{N_m}⟩{|}^2,$$

$$\Delta \mathcal{S}=\frac{S_{ent}(t)-S_0}{S_{page}-S_0},$$

$$|\{n_i^m\},\{r_i\}⟩={\hat{P}}_{\{n_i^m\}}{\hat{g}}_{\{r_i\}}|x+{⟩}^{\otimes N},$$

$$|\{n_i^m\},\{{\overline{r}}_i\}⟩={\hat{W}}_z|\{n_i^m\},\{r_i\}⟩,{\hat{W}}_z=\prod _{\begin{array}{c}i\in \text{D}\end{array}}{Z}_i,$$

$${\mathcal{P}}_m=\prod _{i=1}^L{B}_p(i)\text{leavevmode},{\mathcal{P}}_e=\prod _s{A}_s^r(i)=\prod _s{A}_s^b(i).$$

$$1=\mu_{r,L+1}^{x}=\left\{\begin{array}[]{ll}\mu_{r,1}^{x}&\mbox{if $\;\;W_{z}=+1$}\\ -\mu_{r,1}^{x}&\mbox{if $\;\;W_{z}=-1$}.\end{array}\right.$$

$$\displaystyle\mathcal{P}_{m}=\mu_{r,1}^{x}\mu_{b,1}^{x}\equiv(-1)^{N_{m}}=\left\{\begin{array}[]{ll}+1\\ -1,\end{array}\right.$$

$$\mathcal{O}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}=\left\{\begin{array}[]{ll}0&\mbox{if $|\eta_{z}|<1$ or $|\eta_{x}|<1$ }\\ \dfrac{1}{\chi}tr(U_{z}U_{x}U_{z}^{\dagger}U_{x}^{\dagger})&\mbox{if $|\eta_{z}|<1=|\eta_{x}|=1$,}\end{array}\right.$$

Full text

Many-body localization (MBL) Gornyi et al. (2005); Basko et al. (2006); Oganesyan and Huse (2007); Pal and Huse (2010); Bauer and Nayak (2013); Imbrie (2016) generalizes the concept of single particle localization Anderson (1958) to isolated interacting systems, where many-body eigenstates in the presence of sufficiently strong disorder can be localized in a region of Hilbert space even at nonzero temperature. An MBL system comes along with universal characteristic properties such as area-law entanglement of highly excited states (HES) Bauer and Nayak (2013); Kjäll et al. (2014), power-law decay and revival of local observables Serbyn et al. (2014a, b), logarithmic growth of entanglement Žnidarič et al. (2008); Bardarson et al. (2012); Vosk and Altman (2013); Serbyn et al. (2013a) as well as the violation of “eigenstates thermalization hypothesis” (ETH) Deutsch (1991); Srednicki (1994); Rigol et al. (2008). The latter raises the appealing prospect of protecting quantum order as well as storing and manipulating coherent information in out-of-equilibrium many-body states Huse et al. (2013); Chandran et al. (2014); Bahri et al. (2015); Potter and Vishwanath (2015); Yao et al. (2015).

Recently it has been questioned Carleo G (2011); De Roeck and Huveneers (2014); De Roeck and Huveneers (2014); Hickey et al. (2016); Schiulaz and Müller (2014); Schiulaz et al. (2015); Papić et al. (2015); Yao et al. (2016); Barbiero et al. (2015); Smith et al. (2017) whether quench disorder is essential to trigger ergodicity breaking or one might observe glassy dynamics in translationally invariant systems, too. In such models initial random arrangement of particles effectively fosters strong tendency toward self-localization characterized by MBL-like entanglement dynamics, exponentially slow relaxation of a typical initial inhomogeneity and arrival of inevitable thermalization. This asymptotic MBL–tagged quasi-MBL Yao et al. (2016)–in contrast to the genuine ones, is not necessarily accompanied by the emergence of infinite number of conserved quantities Serbyn et al. (2013b); Huse et al. (2014); Chandran et al. (2015); Ros et al. (2015).

Here we present a novel mechanism toward quasi-MBL in a family of clean self-correcting memories, in particular the Kitaev toric code Dennis et al. (2002); Kitaev (2003) on ladder geometry, a.k.a. the Kitaev ladder (KL) Karimipour (2009); Langari et al. (2015). The elementary excitations of KL are associated with point-like quasi-particles, known as electric (e) and magnetic (m) charges. Our main interest has its roots in the role of non-trivial statistics between anyons that naturally live in (highly) excited states of such models.

Stable topological memories, by definition, need to preserve the coherence of encoded quantum state for macroscopic timescales. However, due to their thermal fragility Castelnovo and Chamon (2007, 2008); Brown et al. (2016); Nussinov and Ortiz (2008); Hastings (2011), specially far-from-equilibrium Kay (2009); Zeng et al. (2016), the problem of identifying a stable low-dimensional quantum memory is still unsecure. One of the major obstacles to this end is that they do not withstand dynamic effects of perturbations whenever a nonzero density of anyons are initially present in the system. Indeed, propagation of even one pair of deconfined anyons around non-contractible loops of the system leads to logical error. In addition, system’s dynamics under generic perturbations could be so tangled that the quantum memory equilibrates in the thermal Gibbs state, in which no topological order survives Hastings (2011).

So far, extensive searches have been carried out to combat the mentioned shortcomings  Chamon (2005); Kim and Haah (2016); Alicki et al. (2010); Mazáč and Hamma (2012); Brown et al. (2014); Hamma et al. (2009); Pedrocchi et al. (2013); Landon-Cardinal et al. (2015); Tsomokos et al. (2011); Wootton and Pachos (2011); Stark et al. (2011). In particular, exerting an external disorder on a stabilizer code strengthens the stability of topological phase Tsomokos et al. (2011) and ensures the single particle localization of Abelian anyons as long as their initial density is below a critical value Wootton and Pachos (2011); Stark et al. (2011). The inquiry is whether one can treat the fragility of translationally invariant, topologically ordered systems in the presence of an arbitrary initial density of anyons living in HES.

We supply clear-cut evidences that random configurations of anyons in HES of KL prompts a self-generated disorder purely due to the mutual braiding statistics. Performing a dual mapping suggests that nonzero density of the magnetic charges, as barriers, poses a kinetic constraint on dynamics of the electric ones and hinders the propagation of the confined charges on non-trivial class of loops around the cylindrical surface of KL. Subsequently, it is more favorable for the initial information to be encoded in sub-spaces with higher density of anyons! Finally, we provide numerical evidences that the effective disorder leads to the existence of an exponentially diverging time scale for dynamical persistence of the initial inhomogeneity, along with an intermediate slow growth of the entanglement entropy, all of which are essential qualities of quasi-MBL.

Kitaev ladder Hamiltonian.—KL is composed of $L$ unit cells, each with three sites, that we will refer to as red, green, and blue sites (see Fig. 1-(a)), and spin-1/2 particles are placed on the $N=3L$ nodes of the lattice. The unperturbed Hamiltonian is defined by $L$ plaquette stabilizer, ${B}_{p}$, and $2L$ vertex stabilizer terms on the triangles of the ladder, $A_{s}^{r/b}$, as follow:

[TABLE]

where $X_{i}$ and $Z_{i}$ are the x- and z-component of Pauli operators, respectively. We set $j_{e}$, $j_{m}>0$ and choose an overall energy scale by setting $j_{e}=1$. KL can be viewed as the Kitaev toric code with surface termination along the rungs direction (a.k.a. surface code Dennis et al. (2002)) whose width is one. This model represents $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry-protected topological (SPT) order associated to anyonic parities Langari et al. (2015); sup .

Now we would like to perturb the KL Hamiltonian such that $e$ (charge) and $m$ (flux), corresponding to $A_{s}=-1$ and $B_{p}=-1$, respectively, hop across the ladder and gain kinetic energy. To this end we consider the perturbed KL with the generic Ising terms:

[TABLE]

where $t_{e}$ ($t_{m}$) is the hopping strength of $e$ ($m$). Via applying $Z_{g,i}Z_{r,i}$, which commutes with every plaquette and star operator except $A_{s}^{r}(i)$ and $A_{s}^{r}(i+1)$, an $e$ charge on site $(r,i)$ transfers to $(r,i+1)$. We could also use $Z_{b,i}Z_{g,i+1}$ to carry out the same task. However, $Z_{b,i}Z_{g,i+1}=B_{p}(i)Z_{g,i}Z_{r,i}$, and therefore the total contribution to the Hamiltonian is $(1+B_{p}(i))Z_{g,i}Z_{r,i}$. That being so, the Ising interactions are coupled to the dynamical $\mathbb{Z}_{2}$ gauge field and the hopping of an $e$ charge from $(r,i)$ to $(r,i+1)$ depends on the value of $B_{p}(i)=(-1)^{n^{m}_{i}}$, which takes into account the parity of $m$ anyon on site $(g,i)$, or equivalently, the mutual braiding statistics of $e$ and $m$. Hence, the hopping of $e$ is blocked if there is a flux on its way. Likewise, $X_{r,i}X_{b,i+1}$ transports one unit of $m$ charge from plaquette $(g,i)$ to $(g,i+1)$ and vice versa. We could also consider $X_{b,i}X_{r,i+1}$ to reach the same goal. However, $X_{b,i}X_{r,i+1}=A_{s}^{r}(i+1)A_{s}^{b}(i+1)X_{r,i}X_{b,i+1}$, and again the movement of $m$ is intertwined with the density of $e$’s along its way.

Unlike the single particle studies in the presence of disorder Wootton and Pachos (2011); Stark et al. (2011), in a many-body picture, the transport properties of Abelian anyons might strongly be affected by their exotic statistics, so that $e$ and $m$ as two distinct quasi-particles are able to mutually suppress their own dynamics, even in a clean system. The latter property is the characteristic feature of Falicov-Kimball like Hamiltonians whose non-ergodic dynamics has been recently investigated as a candidate for disorder-free localization Schiulaz and Müller (2014); Schiulaz et al. (2015); Papić et al. (2015); Yao et al. (2016); Barbiero et al. (2015); Smith et al. (2017).

To directly reveal this hidden structure, we introduce a non-local dual transformation, which maps Eq. 3 to three coupled transverse field Ising (TFI) chains (see Fig. 1-(b)),

[TABLE]

where

[TABLE]

and $\mu_{g,i}^{z}=B_{p}(i)$, $\mu_{r/b,i}^{z}=A_{s}^{r/b}(i)$. This model features glassy dynamics for the $\mu_{g,r/b}$ degrees of freedom, associated to the anyonic excitations of the KL. For example, in the non-interacting limit $t_{m}=0$ an effective description of $H^{KL}_{\rm dual}$ reduces to two TFI chains coupled to the static $\mathbb{Z}_{2}$ gauge field, $\mu^{z}_{g}$. These gauge degrees of freedom form a set of $L$ constants of motion with trivial dynamics. Hence, an arbitrary initial nonzero density of fluxes $\rho_{m}$ energetically suppresses charges’ kinetic interactions (see Fig.1-(c)). Indeed, a typical initial inhomogeneity of $\{\mu_{g}\}=\pm 1$ is dynamically manifested in a self-generated disorder potential, $t^{\prime}_{e}(i)=t_{e}(1+\mu_{g,i})$, with the dilution distribution,

[TABLE]

where for a fixed value of $\rho_{m}$, different configurations of fluxes correspond to different realizations of disorder. In such situation, dynamics of the whole system will be identical to that of two decoupled, disordered TFI chains in terms of $\hat{\mu}_{r/b}$, which are Anderson localized Stinchcombe (1981).

• Anyonic self-localization and quasi-MBL regime.*—The outlined glassy blueprint is inspiring to look for the counterpart of quasi-MBL Yao et al. (2016) in an Abelian many-body system. In analogy to the observed quasi-MBL in a trivial admixture of heavy and light particles Schiulaz and Müller (2014); Schiulaz et al. (2015); Papić et al. (2015); Yao et al. (2016); Barbiero et al. (2015), one needs to choose the mass ratio of the quasi-particles to be large enough, as long as the “isolated bands” Papić et al. (2015) due to finite-size effects is not manifested. In the KL, $t_{e}$ ($t_{m}$) controls the strength of the effective disorder (interaction) as well as the effective mass of $e$ ($m$) anyons com . Thus, we consider the limit $0<t_{m}\ll t_{e}$, where $m$’s have a large but finite effective mass. Now we initialize the whole system in a typical inhomogeneous configuration $\lvert\psi_{N_{m}}\rangle$ of $N_{m}=\rho_{m}L$ fluxes, which are selected near the middle of the spectrum of $H^{KL}$. Then, we compute the evolution of the flux inhomogeneity density under $H^{KL}$,

[TABLE]

which vanishes for any perfect delocalized state. As illustrated in Fig. 2-(a,b), for $t_{m}\gtrsim 0.01$ fast relaxation of the initial inhomogeneity due to resonance admixtures Schiulaz and Müller (2014); Schiulaz et al. (2015) takes place within the time scale $\tau_{int}\sim t_{m}^{-1}$, while in the opposite limit, i.e. small $t_{m}$, the initial inhomogeneity plateau persists until $\tau_{int}$. Moreover, the residual inhomogeneity remains even at later times $\tau_{eff}\sim t_{e}/t_{m}^{2}$. Subsequent to this time, the collective slow dynamics eventually gives way to complete relaxation at $\tau_{R}$. As discussed in Papić et al. (2015), to ensure the robustness of the numerical results against finite-size effects, the density of states (DOS) must not display any isolated classical bands, which can be clearly seen in the insets of Fig. 2.

To gain further insights on the nature of the three distinct time scales that characterize the relaxation dynamics of anyons, $\tau_{int}<\tau_{eff}\leq\tau_{R}$, we have also looked at the growth of entanglement entropy $S_{ent}=-\rho_{A}\ln\rho_{A}$ for half-system $A$ with strong disorder $t_{e}=10$ (see Fig. 3-(a)). Prior to $\tau_{int}$, charges perceive the fluxes as if they are immobile barriers. Hence, after an initial growth, $S_{ent}$ saturates to the first plateau, conveying the single particle localization length of charges. Subsequent to $\tau_{int}$ hybridization of fluxes arrives, which in turn intertwines with the charge dynamics. Thus, the entropy shows logarithmic growth until the finite-size dephasing of charges wins at the second plateau. At $\tau_{eff}$, the dephasing of the fluxes sets in and the entanglement grows even more slowly to saturate eventually at $\tau_{R}$.

It is worth mentioning that the same time scales also determine the evolution of the quantity \mathcal{I}(\tau)\!=$$\frac{\overline{\Delta\rho_{m}^{2}(\tau)}-\overline{\Delta\rho_{m}^{2}(\infty)}}{\overline{\Delta\rho_{m}^{2}(0)}-\overline{\Delta\rho_{m}^{2}(\infty)}}, where $\overline{\Delta\rho_{m}^{2}(\tau)}\equiv{\tau^{-1}}\int_{0}^{\tau}dt\leavevmode\nobreak\ {\Delta\rho_{m}^{2}(t)}$ (see Fig. 3-(a)). Moreover, the late time behavior of $\mathcal{I}(\tau)$ signifies that the full relaxation eventually occurs but at very long times. This anyonic slow dynamics is in contrast to true MBL in which an initial inhomogeneity never completely decays. These results resemble those characteristic behaviors observed in the previous proposal of quasi-MBL Yao et al. (2016).

Following the perturbative argument presented in Schiulaz and Müller (2014), the final decay time of initial inhomogeneity should display scaling behavior as $\tau_{R}t_{m}\propto(t_{e}/t_{m})^{N_{m}-1}$ in KL. To confirm this parametric dependence, we plot numerically extracted value of $\tau_{R}$ versus $t_{m}$ in Fig. 2-(b) sup for different system sizes at fixed value of $t_{e}=1$. Our numerical results not only illustrate a good agreement with the rough estimation presented above, but also imply the exponential dependence of $\tau_{R}$ on the system size with growing number of fluxes in the quasi-MBL regime.

Lastly, we would like to address whether the fast relaxation in such anyonic liquids is followed by the viability of ETH. In this respect, we discard the pair creation/annihilation of fluxes, which might happen as a result of $X_{r,i}X_{b,i+1}$ terms in Eq. (3). Typically, this recombination processes are less likely for the heavy particles in comparison with the light ones. Therefore, it is a reasonable assumption to consider $\tilde{H}_{KL}=\hat{P}_{N_{m}}H^{KL}\hat{P}_{N_{m}}$, where $\hat{P}_{N_{m}}$ is the projection operator into the subspace with fixed $N_{m}$ sup .

We evaluate $\mathcal{G}\equiv\langle n_{p}^{m}n_{p+1}^{m}\rangle$ in the simultaneous eigenstates of $\tilde{H}_{KL}$ and momentum, and collect the results from all momenta. Fig. 4-(a) quantifies $\mathcal{G}$ in different energy densities, for $(t_{e},t_{m})=(1,0.001)$. In the whole energy intervals, the value of $\mathcal{G}$ is spread considerably in a wide range. On top of that, the distribution function $\mathcal{P}(\mathcal{G})$ near the middle of the band has a very broad half-width and peaks at different values as the system size is increased (right panel in Fig. 4-(a)) indicating a strong non-ergodic behavior. For the fast dynamics, e.g. $(t_{e},t_{m})=(1,0.5)$, the system rather obeys ETH prediction: DOS becomes continuous, $\mathcal{P}(\mathcal{G})$ sharply picked around its mean value and the width of the distribution decreases by increasing the system size, as illustrated in Fig. 4-(b). Hence, whenever $t_{e}$ and $t_{m}$ are in the same order of magnitude, the fast relaxation occurs along with the validity of ETH in HES.

Resilience of the topological order following a quench.—To further inspect the fingerprints of the emergent kinetic constraint in Eq. 6 on non-equilibrium anyonic dynamics, we proceed with the scenario of the quantum quench. We initially prepare the system in $\lvert\phi_{0}\rangle=(1/2)\leavevmode\nobreak\ (1+\hat{W}_{x})\lvert\{n_{p}^{m}\},\{n_{s}^{e}\}\rangle$, where $\hat{W}_{x}$ is one of the two logical operators that encode the topological qubit, and $\lvert\{n_{p}^{m}\},\{n_{s}^{e}\}\rangle$ is an exact eigenvector of $H^{KL}_{0}$ including a specific pattern of $e$’s and $m$’s (with $\rho_{m}$ ($\rho_{e}$) density of $m$ ($e$) preparatory anyons). According to the SPT nature of $H^{KL}_{0}$ sup , the corresponding eigenstates are short-range entangled, and thus cold state. We are specially interested in those transitionally invariant pre-quench states with $\rho_{m}=1$ and $\rho_{e}=0$, wherein initial information is encoded in the sub-space with maximum number of $m$ anyons. By performing massively parallel time integration based on Chebyshev expansion Balay et al. (2016, 1997); Hernandez et al. (2005) to evolve system under $H^{KL}$, we measure the spreading of the stored initial information as well as the non-equilibrium heating procedure over the course of time,

[TABLE]

where $S_{ent}(t)$, $S_{0}$ and $S_{page}$ are bipartite entanglement entropies associated with $\lvert\phi(t)\rangle$, $\lvert\phi_{0}\rangle$ and an infinite temperature state Page (1993), respectively.

For $t_{m}=0$, it follows from Eq. 7 that the dynamical effect of perturbation would be completely suppressed due to the presence of fluxes in each plaquette. Indeed, regardless of the magnitude of $j_{m}$ and $t_{e}$, the spreading of information is strictly impeded, even in the absence of any explicit form of disorder in either the Hamiltonian or the initial state! This result is comparable to Ref. Smith et al., 2017, where non-ergodic dynamics is induced solely through the presence of static gauge degrees of freedom, albeit through a distinct mechanism.

For $t_{m}\neq 0$ thermalization process can be controlled by the flux mass gap $j_{m}$ Dennis et al. (2002). In this respect, to build up the tendency toward the survival of preparatory fluxes in the system we increase $j_{m}$ (see Figs. 5-(a,c)). As a result, the thermalization process slows down at a continually growing rate (see Figs. 5-(b,c)). Hence, scaling of $S_{ent}$ with different subsystem sizes $N_{A}$ reforms from volume law to area law, as depicted in Fig. 5-(d). An approximate area law is observed for $j_{m}\gtrsim 4$, while for $j_{m}=1$, $2$ the scaling obeys volume law foo . Notably, the system with sizes considered here, in the thermal regime, reaches its infinite temperature state (see inset of Fig. 5-(b)), which is never observed in those suffering from finite-size effects.

Discussion.—We identified the anyonic self-localization as an emergent property purely rooted in the intrinsic statistics of the Abelian anyons and extended the so-called quasi-MBL picture to self-correcting quantum codes. This provides a novel mechanism distinct by nature from the recent proposals on disorder-free localization  Schiulaz and Müller (2014); Schiulaz et al. (2015); Yao et al. (2016); Grover and Fisher (2014); Smith et al. (2017).

As mentioned earlier, a number of approaches have been proposed to remedy the thermal fragility of the quantum memories with $D<4$, such as: considering clean cubic Chamon (2005); Kim and Haah (2016) and higher dimensional codes Alicki et al. (2010); Mazáč and Hamma (2012) with a more complex structure than the toric one, 2D codes consisting of N-level spins Brown et al. (2014), coupling 2D codes to a massless scalar field Hamma et al. (2009); Pedrocchi et al. (2013); Landon-Cardinal et al. (2015) as well as employing Anderson localization machinery Tsomokos et al. (2011); Wootton and Pachos (2011); Stark et al. (2011). Our results on anyonic self-induced disorder indicate that an exponentially glassy dynamics could be induced even (i) in a low-dimensional, clean, and simple-structure models, and remarkably (ii) this glassiness is enhanced by increasing either the density of errors and/or environmental perturbations Schiulaz et al. (2015); sup .

The scenario discussed in this work could be easily generalized sup to 2D quantum double models such as the Levin-Wen model Levin and Wen (2005). As little progress has been made toward investigation of MBL in topologically ordered 2D systems, our study breaks the ground for future researches.

A closely related concept to quasi-MBL in multi-component systems is quantum disentangled liquid Grover and Fisher (2014); Garrison et al. (2017); Veness et al. (2016), wherein “post-measurement” of the anyon configuration is identical to the error syndrome; that is, the first step in the error-correcting protocol. It is tempting to see whether such measurement procedure on a topological state supports this claim. Intuitively, could there a quantum disentangled spin liquid be found?

Acknowledgements.—We highly appreciate fruitful discussions and neat comments by Fabien Alet, Moeen Najafi-Ivaki, Mazdak Mohseni-Rajaee and Arijeet Pal. The authors would like to thank Sharif University of Technology for financial supports and CPU time from Cosmo cluster. AV was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4302.