# Fate of Algebraic Many-Body Localization under driving

Heiko Burau  Max Planck Institute for the Physics of Complex Systems, 01187
Dresden, Germany  Markus Heyl  Max Planck Institute for the Physics of Complex
Systems, 01187 Dresden, Germany  Giuseppe De Tomasi  T.C.M. Group, Cavendish
Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

In this work we investigate the stability of an algebraically localized phase
subject to periodic driving. First, we focus on a non-interacting model
exhibiting algebraically localized single-particle modes. For this model we
find numerically that the algebraically localized phase is stable under
driving, meaning that the system remains localized at arbitrary frequencies.
We support this result with analytical considerations using simple
renormalization group arguments. Second, we inspect the case in which short-
range interactions are added. By studying both, the eigenstates properties of
the Floquet Hamiltonian and the out-of-equilibrium dynamics in the interacting
model, we provide evidence that ergodicity is restored at any driving
frequencies. In particular, we observe that for the accessible system sizes
localization sets in at driving frequency that are comparable with the many-
body bandwidth and thus it might be only transient, suggesting that the system
might thermalize in the thermodynamic limit.

##  I Introduction

Understanding the breakdown of ergodicity in the quantum realm is an active
and fast growing front of research, motivated by recent developments of
controllable quantum simulations which allows one to access out-of-equilibrium
dynamics [1, 2]. In particular, many-body localization (MBL) describes the
paradigm of ergodicity breaking in quantum phases of matter, generalizing the
concept of Anderson localization to the interacting case [3, 4, 5, 6].

An MBL phase is described by an emergent form of integrability, i.e., the
existence of a complete set of quasi-local integrals of motion (LIOMs) [7, 8,
9, 10]. These LIOMs are adiabatically connected to the integrals of motion of
the non-interacting model and thus are typically exponentially localized. As a
consequence, transport is absent, some memory of the local structure of its
initial state is retained during the quantum evolution and the interactions
between the LIOMs allow a slow information propagation through the system [11,
12, 13]. The MBL phase should be distinguished from an ergodic/thermal one, in
which eigenstates are chaotic and are expected to obey the eigenstate
thermalization hypothesis (ETH) [14, 15, 16, 17].

Several works have studied the stability of the MBL phase in the case of
interacting one-dimensional short-range systems subject to strong disorder
[12, 18, 19, 20, 21, 22]. Recently, the possible existence of MBL has been
shown in systems in which its single-particles are power-law localized, dubbed
algebraic MBL [23, 24, 23]. Algebraic MBL generalizes the paradigm of power-
law localization to the interacting case and therefore its LIOMs develop
algebraically decaying tails. An immediate consequence of having power-law
localized LIOMs is that the bipartite entanglement entropy after a quantum
quench has an unbounded algebraic growth in time [23, 25, 26], unlike the case
with exponentially localized single-particles for which the entanglement
propagates logarithmically slowly [11, 13].

Figure 1: Schematic phase diagram of an algebraically localized phase under
driving. $\alpha$ is the rate of decay of the hopping amplitudes and $\omega$
is the frequency of the driving. The non-interacting localized phase is robust
under driving (left). Instead ergodicity is restored at any driving frequency
for the interacting case (right). In the limiting case
$\alpha\rightarrow\infty$ (short-range), the system undergoes an MBL
transition at a finite driving frequency.

Furthermore, an increasing amount of efforts has been devoted to study the
fate of MBL under periodic driving [27, 28, 29, 30, 31, 32, 33, 34, 35]. In
Refs. 27, 28 it was shown that at high frequency an MBL phase evades the fate
of thermalization remaining localized, while at sufficiently low frequency the
system is delocalized. As a consequence of the robustness of an MBL phase
under driving, it has been possible to explore novel non-equilibrium quantum
phases of matter, i.e., discrete time-crystals which are characterized by a
spontaneous breaking of discrete time translation symmetry [29, 30, 31, 32,
33].

The aim of this work is to investigate the stability of an algebraic MBL under
driving. In particular, we periodically drive (Floquet) a system of spinless
fermions with long-range hopping known to have algebraically localized single-
particle wavefunctions ($|\psi(x)|\sim 1/|x|^{\alpha}$) [36, 37, 38, 39, 40,
41].

First, we study the stability of a non-interacting algebraically localized
phase by numerically inspecting several probes to distinguish a localized from
an extended phase. We conclude that for the non-interacting case the localized
phase does not break down under periodically driving. We support this claim by
analytical considerations based on strong-disorder renormalization group
techniques. Next, we focus on the interacting case. By studying both the
eigenstates properties of the Floquet Hamiltonian and the out-of-equilibrium
dynamics, we provide numerical evidence that the algebraic MBL phase breaks
down under driving. We surrogate this conjecture with simple analytical
considerations. Thus, ergodicity is restored at any finite drive’s frequency
in the thermodynamic limit. This result might imply the non-existence of time-
crystals with algebraically localized LIOMs. Figure 1 summarizes schematically
the phase diagram of an (non-)interacting algebraically localized phase under
driving.

The rest of the work is organized as follows. In Sec. II we introduce both,
the non-interacting and the interacting model and we discuss the different
probes that have been used to detect localization/delocalization. Section
III.1 is devoted to probe the stability of a non-interacting case. We support
these findings using a renormalization group approach, which is explained in
Sec. III.2. Finally, in Sec. III.3 we study numerically the stability of an
algebraic MBL phase subject to driving.

##  II Model $\&$ Methods

We consider a periodically driven system of spinless fermions on an one-
dimensional lattice with long-range hopping

$\displaystyle H(t)=\sum_{i\neq j}\quantity(J_{ij}c_{i}^{\dagger}c_{j}+h.c.)$
$\displaystyle+\quantity(1+\delta(t))\sum_{i}h_{i}n_{i}$
$\displaystyle+V\sum_{i}n_{i}n_{i+1},$ (1)

where $J_{ij}=J_{ji}=\mu_{ij}/\sqrt{1+|i-j|^{2\alpha}}$ describes a disordered
long-range hopping involving all pairs of sites, with an algebraically
decaying amplitude w.r.t. their distance according to an exponent $\alpha>0$
[42, 43, 36, 37, 38, 39, 40, 41]. Quenched hopping disorder is given by
$\\{\mu_{ij}\\}$, which is uniformly distributed between $[-1,1]$, and on-site
disorder is added by $\\{h_{i}\\}$ which are random fields uniformly
distributed between $[-W,W]$. $V$ is the interaction strength and $L$ is the
number of sites and we indicate with $N=L/2$ (half-filling) the number of
particles.

We choose a binary driving protocol of period $T$, i.e., $\delta(t)=-\delta$
for $0<t<T/2$ and $\delta(t)=+\delta$ for $T/2<t<T$ and
$\delta(t+nT)=\delta(t),n\in\mathbb{Z}$. As an energy reference we define the
hopping unit $J=\sqrt{\frac{1}{2L}\sum_{i\neq j}\langle{J_{ij}^{2}\rangle}}$,
where $\langle\cdot\rangle$ indicates the average over disorder and by
$\omega_{D}=2\pi/T$ we denote the driving frequency in the following.

We now briefly summarize the dynamical phase diagram of the non-driven
Hamiltonian $H(\delta=0)$ in Eq. (II). The non-interacting case without
driving ($V,\delta=0$), referred as power-law banded random matrix (PLBRM), is
known to have a metal-insulator transition at $\alpha=1$ from an extended
($\alpha<1$) to an algebraically localized phase ($\alpha>1$) [44, 45, 46].
Thus, for $\alpha>1$ the single-particle orbitals are algebraically localized
meaning that they possess power-law decaying tails around their centers of
localization $i_{c}$, $\psi(i)\sim|i-i_{c}|^{-\alpha}$. In this work, we are
interested in the robustness of this localized phase under driving, therefore
we mostly consider $\alpha>1$.

As interactions are turned on ($V>0$), the non-driven system goes through an
MBL transition as a function of the disorder strength $W$ [23, 47]. For $V=1$
and $\alpha=3$, the critical point has been estimated to be $W_{c}\approx 3$,
ergodic for $W<W_{c}$ and localized for $W>W_{c}$. The MBL phase of $H$ in Eq.
(II) is described by a complete set of LIOMs $\\{\tau_{i}\\}_{i}^{L}$
111$[H,\tau_{i}]=0$ $\forall i$. which have algebraically decaying tails, i.e,
$\text{Tr}[\tau_{i}A_{j}]\sim 1/|i-j|^{\alpha_{1}}$, where $A_{j}$ is a
generic local observable with support at site $j$ [23].

It is important to point out, that the question whether MBL is stable in
higher dimensions or for long-range Hamiltonians, is still under debate [49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. Recent experiments in fairly
large system sizes and for relevant time-scales, have shown evidence of an MBL
phase for both, two-dimensional systems and long-range Hamiltonians [62, 63,
64, 65]. However, in quantum-avalanche theory the proposed mechanism for
delocalization invokes the existence of rare entropic thermal bubbles, which
will have effect at very long time scales and large system-sizes [49, 50, 51,
66, 67].

We will numerically show that the algebraic MBL phase is vastly unstable
against external driving. This fragility appears to be independent of the
driving frequency, although partially accompanied with finite-size effects
depending on the domain of frequency. As a result, we observe a qualitative
difference with respect to the behavior from short-ranged, or a ”conventional”
MBL phase under driving, in which the LIOMs are exponentially localized. For
the latter, it is believed that a system-size independent critical driving
frequency $\omega_{c}$ exists, above which the system remains localized [27,
28, 34].

Throughout this manuscript, numerical results were obtained using free-
fermions techniques for the non-interacting case and exact diagonalization for
the interacting one. In order to establish the existence of a localized phase,
we use a variety of probes, each giving insights to the dynamical phase
structure from different facets. In particular, for each quantity we average
over at least $10^{2}$ random configurations and over energy spectrum.

In the remainder of this section we describe all utilized probes in detail.

###  II.1 Level spacing statistics

The distribution of level spacing has been found to be an useful probe to
distinguish an ergodic phase from an MBL one [17, 68, 69]. In an ergodic phase
at infinite temperature, the level spacing is expected to be distributed as
the one of a random matrix belonging to the same universality class of the
Hamiltonian, namely the Gaussian Orthogonal Ensemble (GOE). However, in an MBL
phase, due to the existence of LIOMs, energy levels are weakly correlated and
the level spacings are Poisson distributed.

Specifically, the degree of proximity to a random matrix ensemble can be
extracted from the distribution of level spacing through the so called
$r$-value, or gap-ratio parameter, which is defined as

$r=\langle\min(\delta_{n}/\delta_{n+1},\delta_{n+1}/\delta_{n})\rangle,$ (2)

where $\delta_{n}$ are level spacing of two consecutive level energies and
$\langle\cdot\rangle$ refers to the disorder and spectrum average [68, 69].
The $r$-value in the ergodic phase is $r_{\text{GOE}}\approx 0.531$, while it
is $r_{\text{Poisson}}=2\log{2}-1$ in a localized one.

For the periodically driven system in Eq. (II), we consider the quasi-energies
of the Floquet Hamiltonian $H_{\text{F}}$ defined by

$e^{-iH_{\text{F}}T}=U(T)=\mathcal{T}e^{-i\int_{0}^{T}\differential{t^{\prime}}H(t^{\prime})},$
(3)

which serves as generator for the discrete translation in time of period $T$
222$\mathcal{T}$ denotes the time-ordered product..

###  II.2 Energy propagation

In driven systems the energy is not conserved during time evolution and at any
discrete time step an extensive amount of energy is pumped up in the system.
Thus, in an ergodic phase the long-time asymptotic steady state is expected to
be a structureless infinite temperature state. However, in an MBL phase the
expected heating is effectively prevented at high driving frequencies. This
can be understood from the emergence of an extensive number of locally
conserved quantities, which effectively hinder the system to exchange energy
beyond a characteristic localization length [34, 27, 28].

In order to study the energy absorbed through the quantum time evolution, we
define the dimensionless parameter

$\epsilon(\tau)=\frac{\expectationvalue{H(\tau=0)}{\psi(\tau)}-E_{\rm
min}}{E_{\rm max}-E_{\rm min}},$ (4)

where $\tau=t/T$ and $\ket{\psi(\tau)}$ is the time-evolved state, starting
from the ground-state of $H(\tau=0)$. Here, $E_{\rm min}$ and $E_{\rm max}$
refer to the lowest and highest eigenvalues of $H(\tau=0)$, respectively. In
an ergodic phase in the thermodynamic limit the asymptotic long-time value of
$\epsilon(\tau\to\infty)=1/2$ (middle of the spectrum), while in an MBL phase
$\epsilon(\tau\to\infty)<1/2$.

###  II.3 Entanglement entropy

The MBL transition can be understood in terms of a rearrangement of the local
structure of entanglement in the system [18, 21, 19]. In particular, the half-
chain bipartite entanglement entropy (EE) measures the amount of entanglement
through the cut and can be used, both to detect the MBL transition and to
characterize the two phases. The EE ($S$) is defined through the half-
partition reduced density matrix $\rho_{L/2}$, where the sites belonging to
one half of the system have been traced out

$S=-\text{Tr}(\rho_{L/2}\ln\rho_{L/2}).$ (5)

In an ergodic phase at infinite temperature, the eigenstates behave like
random vectors and therefore the EE reaches the so called Page value $S_{\rm
Page}=L/2\log{2}-1/2$ [71], 333It is important to point out that however also
fully ergodic states can reach the Page value [86].

For the out-of-equilibrium dynamics, we compute the EE following a quantum
quench starting from the Néel state
$|\psi(0)\rangle=\prod_{i}^{N}c_{2i}^{\dagger}|0\rangle$. In an ergodic system
driven in time, $S(\tau)$ increases up to the value of an infinite temperature
thermal state, $S_{\rm Page}=L/2\log{2}-1/2$. In an MBL phase, on the other
hand, $S(\tau)$ is known to saturate to an extensive value which is not
thermal [11, 13], thus we expect $\lim_{t\to\infty}S(\tau)/S_{\rm Page}<1$.

Furthermore, we compute the EE for the eigenstates of the effective
Hamiltonian $H_{\text{F}}$, which will have a volume-law $S\sim O(L)$ in the
ergodic phase and a sub-volume scaling in the localized one
$\lim_{L\rightarrow\infty}S/L\rightarrow 0$ [73].

###  II.4 Inverse partition ratio

Finally, for the non-interacting case ($V=0$), we compute the inverse
partition ratio ($IPR$) [74] for the single-particle eigenfunctions of the
Floquet Hamiltonian $H_{\text{F}}$, which is defined as

$IPR(\epsilon_{\nu})=\sum_{i}|\phi_{\nu}(i)|^{4},$ (6)

where $i$ indicates the lattice site and $\phi_{\nu}(i)$ refers to the single-
particle wavefunction with single-particle energy $\epsilon_{\nu}$. The $IPR$
separates a delocalized phase from a localized one from its scaling with $L$,
$IPR\sim O(L^{-1})$ in a delocalized phase and $IPR\sim O(L^{0})$ in a
localized one.

We emphasize that in driven systems the density of states (DOS) is constant,
up to finite-size fluctuations, due to the lack of energy ordering, therefore
we can safely average $IPR(\epsilon_{\nu})$ over all energies.

##  III Results

###  III.1 Non-interacting case

In this section, we inspect the eigenstate properties of the Floquet
Hamiltonian $H_{\text{F}}$ for the non-interacting case. In this case,
$H_{\text{F}}$ is still a quadratic Hamiltonian
$H_{\text{F}}=\sum_{i,j}h^{\text{F}}_{ij}c_{i}^{\dagger}c_{j}$. Thus, we can
simply focus on the single-particle energies $\\{\epsilon_{\nu}\\}$ and
eigenstates $\\{\phi_{\nu}\\}$ of $H_{\text{F}}$.

Figure 2: The $r$-statistics parameter for the quasi-eigenenergies of the non-
interacting model with for $\alpha=1.25$ (a), (c) and $\alpha=2$ (b), (d) for
several system sizes $L$. The panels (c), (d) show the collapse of $r$ with
$L$, $r=r(\omega/J\cdot L^{2(\alpha-1)},\delta,W)$. In all panels $W=6$ and
$\delta=0.5$. The horizontal dashed lines indicate the GOE value $\approx
0.531$ and the Poissonian one $\approx 0.386$.

We start our analysis by inspecting the level statistics of the single-
particle Floquet Hamiltonian $h^{\text{F}}$. Fig. 2 shows the level spacings
parameter $r$ computed with $\\{\epsilon_{\nu}\\}$ as a function of the
driving frequency $\omega_{D}=2\pi/T$. As expected, at low frequencies
($\omega_{D}\ll J$) the system is more delocalized than higher ones
($\omega_{D}\gg J$). Indeed, at low frequencies and relatively small system
sizes $L$, the $r$-value is close to the one of a random matrix
($r_{GOE}\approx 0.52$), see Fig. 2 (a). However, an overall scaling towards
localization (Poissonian statistics) with increasing $L$ is visible in Fig. 2
(a),(b). As shown in Fig. 2 (c), (d), we found the scaling law which describes
the crossover from delocalization to localization with increasing $L$

$r=r(\omega/J\cdot L^{2(\alpha-1)},\delta,W).$ (7)

Thus, only in the $L$-dependent frequency regime $\omega_{D}\ll
JL^{-2(\alpha-1)}$ the single-particle wavefunction are delocalized.
Importantly, the above scaling law breaks down at the metal-insulator
transition ($\alpha=1$) of the PLBRM. Indeed, for $\alpha<1$ we expect level
energies to show repulsion, as visible in Fig. 3 (a), where the $r$-value is
shown as a function of $\alpha$ at fixed frequency for several system sizes
$L$. Finite-size scaling analyse shows that the phase diagram of the PLBRM,
delocalized for $\alpha<1$ and localized for $\alpha>1$, remains unchanged
under driving, see inset in Fig. 3 (a).

Thus, we conclude that the localized phase is robust under driving, meaning
that for $\alpha>1$ and at any finite frequency $\omega_{D}$ the spectrum of
the single-particle quasi-energy remains Poissonian in the thermodynamic limit
($L\rightarrow\infty$). For the sake of completeness, we further confirm this
result by inspecting the $IPR$ in Eq. (6) for the single-particle
wavefunctions $\\{\phi_{\nu}\\}$ of the Floquet Hamiltonian, which quantifies
their degree of localisation in real space. In Fig. 4 we present the averaged
$IPR$ for two values of $\alpha$ and fixed $\omega_{D}/J=0.1$. In agreement
with the analysis of the $r$ level statistics, the $IPR$ converges to a finite
value with $L$ ($\lim_{L\rightarrow\infty}IPR(L)>0$) and therefore the system
is localized (see inset in Fig. 4 (a) for $\alpha=2.5$).

Figure 3: The $r$-statistics of quasi-eigenenergies (a) and long-time
saturation value of the absorbed energy density $\epsilon(\tau)$ (b) as
function of $\alpha$ for the non-interacting case. The system is driven with a
frequency $\omega_{D}=J(\alpha)$ in (a) and $\omega_{D}=J(\alpha)/10$ in (b),
both with $W=6$ and $\delta=0.5$. The inset in panel (a) shows the finite-size
scaling proving evidence of the transition at $\alpha=1$. In panel (a) the
horizontal dashed lines indicate the GOE value $\approx 0.531$ and the
Poissonian one $\approx 0.386$. In both panels, the vertical dashed line at
$\alpha=1$ markers the delocalization-localization transition for the PLBRM

We complete our analysis for the non-interacting case by investigating the
behavior of two dynamical indicators, the energy transport and the information
propagation quantified by $\epsilon(\tau)$ in Eq. (4) and $S(\tau)$ in Eq.
(5), respectively. As explained in Sec. II, the energy density
$\epsilon(\tau)$ is computed starting from the ground-state of $H(\tau=0)$ and
for the EE from the Néel state $\prod_{i}^{N}c_{2i}^{\dagger}|0\rangle$.

As expected, in the localized phase the system does not heat up to infinite
temperature, meaning that $\epsilon(\tau)<1/2$ and the saturation value of EE
is sub-volume ($\lim_{L\rightarrow\infty}S(\infty)/S_{\rm Page}\rightarrow 0$)
[73], as one can see in Fig. 5. In particular, we study the long-time
saturation value of $\epsilon(\tau)$ as function of the parameter $\alpha$. As
one can see in Fig. 3 (b), $\epsilon(\infty)\approx 1/2$ for $\alpha\leq 1$,
meaning that in the delocalized phase energy diffuses to its maximum value.

To summarize, by inspecting several indicators, i.e., level spacing, $IPR$,
entanglement and transport properties, we have provided numerical evidence
that the non-interacting localized phase remains localized once the system is
driven in time.

Figure 4: The inverse partition ratio ($IPR$) as function of $L$ for the
single-particle wavefunctions for $\alpha=1.25$, $\alpha=2$ in panels (a) and
(b), respectively. The inset in panel (a) shows a finite-size scaling of $IPR$
with $L$, indicating that $\lim_{L\rightarrow\infty}IPR(L)\sim O(L^{0})$.
Figure 5: Absorbed energy-density $\epsilon(\tau)$ in Eq. (4) in (a),(b) and
the half-chain bipartite entanglement entropy in (c),(d) as function of the
period driving period $\tau$. The inset in (a) shows the finite-size scaling
of the saturation value at asymptotic large times of $\epsilon(\tau)$.
$S(\tau)$ is normalized with the value of the entanglement entropy of a
random-state $S_{\rm Page}=L/2\log{2}-1/2$.

###  III.2 Driven flow equations

In the previous section, we have shown numerically that the non-interacting
localized phase is robust under driving. We now present an analytical argument
supporting our results.

For this purpose, we closely follow a recent Renormalization Group (RG)
approach [45, 75] which uses flow-equations [76] to diagonalize a static
Hamiltonian. After briefly reviewing this RG approach, we extend it to the
periodic driving case, $\delta(\tau)\neq 0$, and show that the driving becomes
irrelevant in the RG sense. Thus, we will conclude that external driving does
not destroy the phase structure of the non-interacting case, in agreement with
our numerical results.

In Ref. 45 Quito et al. have shown that the diagonalization of Eq. (II), with
$V,\delta(\tau)=0$, can essentially be decomposed into a sequence of two-body
problems, i.e., a successive decimation of bonds, whereby the level-spacing
statistics of the entire Hamiltonian is maintained. This strictly holds as
long as $\alpha>1/2$ and becomes asymptotically exact in the limit
$\alpha\to\infty$ [45]. Let us consider a single RG-step, or in other words,
the decimation of a bond between two lattice sites $i,j$. Here it is
convenient to define new variables, $\mathcal{J}=J_{ij}$ and $x=h_{i}-h_{j}$.
Now, the Wegner-Wilson flow [77, 78, 79, 80] defines a continuous scale
$\Gamma$ together with unitary transformations
$U(\Gamma)=\mathcal{T}_{\Gamma}\exp(\int^{\Gamma}d\Gamma^{\prime}S(\Gamma^{\prime}))$
which diagonalizes the problem in the limit $\Gamma\rightarrow\infty$.
$S(\Gamma)$ is the instantaneous generator $S(\Gamma)$ and satisfies the
condition $[H_{0}(\Gamma),S(\Gamma)]=H_{1}(\Gamma)$, where
$H_{0}=\frac{x}{2}(n_{i}-n_{j})$ refers to the diagonal and
$H_{1}=\mathcal{J}c_{i}^{\dagger}c_{j}+h.c.$ to the off-diagonal part of the
two-body problem (for details see Ref. 45). Here, the explicit
$\Gamma$-dependency is dropped for simplicity. Applying $U(\Gamma)$ to the
Hamiltonian, or equivalently, solving the Heisenberg equation
$\partial_{\Gamma}H(\Gamma)=[S(\Gamma),H(\Gamma)]$, with
$H(\Gamma)=H_{0}(\Gamma)+H_{1}(\Gamma)$, leads to a continuous decay of the
off-diagonal matrix elements $H_{1}(\Gamma)$ such that $H(\Gamma)$ becomes
diagonal for $\Gamma\to\infty$. With the explicit form of
$S=\mathcal{J}xc_{i}^{\dagger}c_{j}-h.c.$ the above Heisenberg equation can be
straightforwardly expressed by the following coupled differential
(flow-)equations,

$\begin{cases}\frac{\partial}{\partial\Gamma}\mathcal{J}&=-x^{2}\mathcal{J},\\\
\frac{\partial}{\partial\Gamma}x&=4x\mathcal{J}^{2},\end{cases}$ (8)

where $\mathcal{J}\to 0$ for $\Gamma\to\infty$ with arbitrary initial
conditions. A characteristic RG time scale on which $\mathcal{J}$ decays, is
given by $\tau_{\Gamma}=1/r^{2}$ with $r^{2}=4\mathcal{J}^{2}+x^{2}$ being an
integral of motion w.r.t. $\Gamma$ in Eq. (8). The time scale $\tau_{\Gamma}$
sets an order of the RG-steps beginning with the largest $r_{max}^{2}$. The
fact that $r^{2}$ is conserved gives rise to an intuitive picture of the
renormalization process: each RG-step moves one point (bond) within the
$x-\mathcal{J}$ plane towards the $x$-axis on a circle of radius $r$. Thereby,
$x^{2}_{\Gamma\approx\tau_{\Gamma}}\to
x^{2}_{\Gamma=0}+4\mathcal{J}^{2}_{\Gamma=0}$ gets renormalized. As a side
note it should be mentioned that a renormalization of adjacent bonds affected
by the flow can be neglected here as the distribution $P(\mathcal{J})$ behaves
invariant under the full RG-flow [45]. The above picture of renormalization
therefore connects the initial distribution $P_{\Gamma=0}(\mathcal{J},x)$ to
the final one ($P_{\Gamma\to\infty}(x)$). The latter is equivalent to the
level-spacing distribution $P(s)$ in the limit $x\to 0$.

With this picture in mind, of the RG accumulating decimations on a circular
shell, two phases can be immediately identified: (i) For an initially uniform
distribution $P_{\Gamma=0}(\mathcal{J},x)\sim{\rm const}$, one obtains an
extended phase, exhibiting level-repulsion, i.e., $P_{\Gamma\to\infty}(x)\sim
x$. This is because the number of accumulated decimations $N_{\rm dec}$ within
a shell $[r-dr,r]$ is $N_{\rm dec}\propto 2\pi rdr$ in this case [45]. (ii)
For any kind of short-ranged distribution, $P_{\Gamma=0}(\mathcal{J},x)\sim
e^{-\mathcal{J}/\xi}$, where $\xi\ll r_{max}$ is a characteristic length
scale, the RG leads to a localized phase, since
$P_{\Gamma\to\infty}(x)\sim\rm{const}$ (no level-repulsion).

The initial marginal distribution $P(\mathcal{J})$ in our system consists of
two regimes separated by a critical $\mathcal{J}_{c}\sim\frac{1}{L^{\alpha}}$
[45], set by the typical hopping amplitude at maximum distance $L$,

$P_{\Gamma=0}(\mathcal{J})\sim\begin{cases}1/(J^{1+1/\alpha})&,\mathcal{J}>\mathcal{J}_{c}\\\
L^{\alpha}&,\mathcal{J}<\mathcal{J}_{c}.\end{cases}$ (9)

As described above by means of the RG, these two regimes map to two different
phases: For $\mathcal{J}<\mathcal{J}_{c}$, the distribution in Eq. (9) is
constant, leading to level-repulsion, while for $\mathcal{J}>\mathcal{J}_{c}$
it is concentrated near $\mathcal{J}=0$ and thus avoids level-repulsion. At
the crucial scale of the mean level spacing $\overline{\delta}\sim\frac{1}{L}$
(bandwidth can roughly be considered size-independent), the normalized
critical $j_{c}$ defined by
$j_{c}=\mathcal{J}_{c}/\overline{\delta}\sim\frac{1}{L^{\alpha-1}}$ then gives
a critical point at $\alpha=1$.

####  III.2.1 Dynamical extension

In this section we derive a dynamical extension of Eqs. (8) that accounts for
a periodic driving term $\delta(t)\neq 0$ in the Hamiltonian. As a
simplification we start with a harmonic driving ansatz,

$H_{0}(t)=\frac{x}{2}\quantity(1+\delta e^{i\Omega t}+h.c.)(n_{i}-n_{j}),$
(10)

where $\Omega$ denotes a driving frequency and $\delta$ a dimensionless
driving strength. A time-dependent change of basis is generally given by
$H^{\prime}(t)=e^{-S(\tau)}H(t)e^{S(\tau)}-i\partial_{t}S(\tau)$, with the
Schrieffer-Wolff (SW) generator $S(\tau)=-S^{\dagger}(t)$. In order to
continuously dampen the off-diagonal hopping terms of $H_{1}$, we use the
following generator

$S(\tau)=\mathcal{J}x\quantity(c_{i}^{\dagger}c_{j}+\quantity(\frac{\delta}{1-\omega}c_{j}^{\dagger}c_{i}-\frac{\delta}{1+\omega}c_{i}^{\dagger}c_{j})e^{i\Omega
t})-h.c.,$ (11)

where $\omega=\Omega/x$ represents a normalized driving frequency. For
simplicity, this generator is chosen such that it does not create a time-
dependence for the off-diagonal hopping terms during the flow. It can be seen
that the continuous change of basis with $S(\tau)$ can be expressed in closed
form by the following, ”driven” flow equations, where we omit the explicit
$\Gamma$-dependencies

$\begin{cases}\frac{\partial}{\partial\Gamma}\mathcal{J}&=-x^{2}\mathcal{J}\quantity(1+2\delta^{2}\frac{\omega}{1-\omega^{2}}),\\\
\frac{\partial}{\partial\Gamma}x&=4x\mathcal{J}^{2},\\\
\frac{\partial}{\partial\Gamma}(x\delta)&=-4h\delta\mathcal{J}^{2}\frac{1}{1-\omega^{2}}.\end{cases}$
(12)

A few considerations are in order. (i) We neglected higher-harmonic terms
$\propto\delta^{4}$, which naturally arise from a time-dependent $S(\tau)$.
(ii) The driving strength $\delta$ enters quadratically and therefore explains
the need for a relatively large value of $\delta=1/2$ to see significant
effects from driving in our numerics. (iii) In the infinite frequency limit
$\omega\to\infty$ the flow equations just reduce to the static ones,
consistent with the universal scaling law of Eq. (7).

For extraction of the phase structure we are only interested in the behavior
at large RG times $\tau_{\Gamma}$, since they correspond to small values of
$x$ of the level-spacing. This implies that large RG times also correspond to
high values of $\omega=\Omega/x$. The above set of equations thus can be
further simplified by considering only terms up to leading order in
$1/\omega$,

$\begin{cases}\frac{\partial}{\partial\Gamma}\mathcal{J}&=-x^{2}\mathcal{J}\quantity(1-\frac{2\delta^{2}}{\omega}),\\\
\frac{\partial}{\partial\Gamma}x&=4x\mathcal{J}^{2},\\\
\frac{\partial}{\partial\Gamma}(x\delta)&=0.\end{cases}$ (13)

Within this approximation we find a first-order correction of the integral of
motion $r^{2}$ to $\tilde{r}^{2}=4J^{2}(1+\frac{2\delta^{2}}{\omega})+x^{2}$.
Hence, a typical level spacing from the static case gets scaled by
$\frac{2\delta^{2}}{\omega}$. By identifying $\tau_{\Gamma}\approx 1/x^{2}$,
the driving correction becomes manifestly irrelevant in the course of the RG,
$\frac{2\delta^{2}}{\omega}\approx\frac{1}{\sqrt{\tau_{\Gamma}}}\frac{2\delta^{2}}{\Omega}$.
As a result, the non-interacting algebraically localized phase is robust
against driving.

###  III.3 Algebraic MBL

Having established that the non-interacting algebraic phase is stable under
periodic driving, we now turn to the interacting case by setting $V=1$ in Eq.
(II). With the aim to understand the robustness of an algebraic MBL phase, we
fix the parameters $\alpha$ and $W$ of $H$ in Eq. (II) such that the non-
driven system is localized [81, 47, 73]. The numerical results have been
computed using exact diagonalization techniques, thus we will be limited by
relatively small systems $L\in[6,16]$. However, by studying several probes to
distinguish an ergodic phase from an MBL one, even if affected by finite-size
effects, we provide numerical evidence that at any finite frequency,
ergodicity is expected to be restored in the thermodynamic limit.

Figure 6: $r$-statistics for the interacting case ($V=1$) as a function of the
driving frequency $\omega_{D}$, for two values of the hopping decaying-rate
$\alpha=2.5,3$ in panel (a) and (b), respectively. The dashed horizontal lines
indicate the GOE $r$-value $\approx 0.531$ and the Poissonian one $\approx
0.386$, whereas the vertical lines indicate the spectral width
$\Lambda(L=16)$. In panels (c),(d), we show the $r$-value as function of
$\omega_{D}/\Lambda(L)$. In all panels $W=6$ and $\delta=0.5$. Figure 7: The
averaged entanglement entropy $S$ of the Floquet Hamiltonian for $V=1$ and the
variance of the entanglement entropy $\delta S^{2}$ in panels (a)-(b) and
(c)-(d), respectively. The inset in (a) shows $S$ at smaller frequencies for
several $L$ with respective Page values. $\delta S^{2}$ at smaller frequencies
for $\alpha=3$ is shown in the inset of panel (d). The dashed vertical lines
indicate the bandwidth $\Lambda(L)$. Figure 8: In panels (a)-(b) are shown the
energy absorption rate $\epsilon(\tau)$ in Eq. (4) starting from the ground-
state of the interacting model for two values of $\alpha$ for the interacting
case. The inset in panel (b) shows the finite-size scaling for the long-time
saturation value of $\epsilon(\tau)$. In panels (c)-(d) the long-time
saturation point for the bipartite entanglement entropy with the initial state
given by the Néel state ($\prod_{i}^{N}c^{\dagger}_{2i}|0\rangle$). The dashed
vertical lines indicate the bandwidth $\Lambda(L)$.

As well as for the non-interacting case, we start our analysis by inspecting
the spectrum of the effective Hamiltonian $H_{\text{F}}$ in Eq. 3. Fig. 6
shows the $r$-value as function of the driving frequency $\omega_{D}$, for
several $L$. A clear trend towards delocalization ($r\approx 0.531$) with
increasing $L$ is visible in Fig. 6 (a)-(b). The $r$-value change their
behavior at high-frequency $\omega_{D}\gg J$, where a frequency-independent
plateau is formed close to the Poissonian value $r\approx 0.386$. This plateau
belongs to the finite size perturbative regime, in which the system is close
to the non-driven case $U(T)\approx\mathbb{I}-\frac{iT}{2}(H(T/2)+H(T))$,
which disappears in the thermodynamic limit. Generically, a driving localized
phase remains localized at enough high frequencies. In particular, this will
happen if the driving frequency $\omega_{D}$ exceed the many-body bandwidth
$\Lambda(L)$ of the energy spectrum of the non-driven Hamiltonian [27]. We
compute the averaged density of states of the non-driven Hamiltonian and using
a Gaussian fit we extrapolate its typical bandwidth $\Lambda(L)$ (vertical
dashed line in Fig. 6 for the largest $L$).

Figure 9: $r$-statistics as a function of the driving frequency $\omega_{D}$,
for the hopping decaying rate $\alpha=2.5$ in panel (a) and
$\alpha\rightarrow\infty$ in (b). The dashed horizontal lines indicate the GOE
$r$-value $\approx 0.531$ and the Poissonian one $\approx 0.386$, whereas the
vertical lines indicate the spectral width ($\Lambda(L=16)$). In the inset of
panel (a) is shown the $r$-value as a function of $\omega_{D}/\Lambda(L)$ for
$\alpha=2.5$. In both panels $W=6$ and $\delta=0.5$.

Thus, for $\omega_{D}\geq\Lambda$ our numerical result cannot be used to
understand the dynamical phase diagram of $H(t)$. By safely ignoring the high-
frequency regime ($\omega_{D}\geq\Lambda$), the $r$-value of $H_{\text{F}}$
shows a clear tendency towards delocalization (rigid spectrum), allowing us to
claim that ergodicity might be restored in the thermodynamic limit. This is
visible in Fig. 6 (c),(d), where $r$-value is showed as function of
$\omega_{D}/\Lambda(L)$, providing evidence that the critical frequency
$\omega_{c}$ for which localization sets on, scales as
$\omega_{c}\sim\Lambda(L)\sim\sqrt{L}$.

With the aim to further support the claim of delocalization of an algebraic
MBL phase under driving, we study the entanglement properties of the
eigenstates of the Floquet Hamiltonian ($H_{\text{F}}$). Fig. 7 shows the
bipartite EE averaged over disorder and eigenstates and its fluctuations
$\delta S^{2}$ as function of the driving frequency [19, 21]. The EE has a
similar behavior as the $r$ level statistics parameter, meaning that low
frequency EE reaches the ergodic values (see inset in Fig. 7 (a)) and then the
curves indicate a shift to delocalization (volume law scaling) before
approaching the natural cut-off $\Lambda$. Furthermore, we study fluctuations
around the mean value $S$, which are quantified by the variance ($\delta
S^{2}$) over random samples and eigenstates [19, 21]. At low frequencies
$\delta S^{2}$ decays with $L$ in agreement with the ETH (see inset in Fig. 7
(d)). Meanwhile at moderate frequencies $\delta S^{2}$ develops a growing peak
with $L$, a feature which has been used as a fingerprint for the MBL
transition [19, 21]. However, as one can observe, the peak shifts inexorably
with system size, getting closer to $\Lambda$.

Finally, we inspect some dynamical properties of the interacting driven
system, i.e., energy and entanglement propagation.

As for the non-interacting case, we start with the propagation of energy. Fig.
8 (a)-(b) shows the energy density $\epsilon(\tau)$ at moderate frequency
$\omega_{D}\approx 4J$. For both values, $\alpha=2.5,3$ and for our available
system sizes, the infinite temperature value ($\varepsilon=1/2$) is not
reached yet. However, in long-time limit, $\epsilon(\tau)$ tends to
$\varepsilon\approx 1/2$, which it also supported by the finite-size scaling
(see inset in Fig. 8). The remaining two panels, Fig. 8 (c), (d), are
dedicated to the asymptotic long-time saturation value of $S(\tau)$. As a
function of the driving frequency, we compute $S(\tau\rightarrow\infty)/S_{\rm
Page}$, where $S(\tau\rightarrow\infty)$ is determined by averaging over a
long-time window in which $S(\tau)$ reaches its saturation value. The
numerical analysis of $S(\tau\rightarrow\infty)/S_{\rm Page}$ in Fig. 8 (c),
(d), in which is visible the tendency $S(\tau\rightarrow\infty)/S_{\rm
Page}\rightarrow 1$ as $L$ increases. Thus, in agreements with the results of
the spectrum and eigenstates of $H_{\text{F}}$, the finite size analysis of
$S(\tau\rightarrow\infty)$ further supports the conjecture that ergodicity is
restored in the thermodynamic limit.

Unlike the case in which the LIOMs have exponentially decaying tails
($\alpha\rightarrow\infty$), the algebraic MBL phase shows a clear tendency
towards delocalization at any finite-driving frequencies. To better pin-down
this fundamental difference, we study a Floquet minimal MBL model [23, 81,
82], which can be considered as the strong disorder limit of $H$ in Eq. (II)

$U(T)=e^{-iH^{+}T/2}e^{-iH^{-}T/2},$ (14)

with

$H^{\pm}=\sum_{l}\epsilon_{l}^{\pm}I_{l}^{\pm}+V\sum_{p,q}B_{pq}^{\pm}I_{p}^{\pm}I_{q}^{\pm},$
(15)

where $\\{\epsilon_{l}^{\pm}\\}$ and $\\{I_{l}^{\pm}\\}$ are the single-
particle energies and LIOMs of the non-interacting model ($V=0$) in Eq. (II)
with $\pm\delta$, respectively [23, 81]. The couplings
$B_{pq}=\sum_{i}\\{|\phi_{p}^{\pm}(i)|^{2}|\phi_{q}^{\pm}(i+1)|^{2}-\phi_{p}^{\pm}(i)\phi_{p}^{\pm}(i+1)\phi_{q}^{\pm}(i)\phi_{q}^{\pm}(i+1)\\}$
are obtained using first order perturbation theory in $H$ in the limit $V/W\ll
1$ [23, 81]. By construction, $H^{\pm}$ is localized meaning that it has
Poissonian spectrum and integrals of motion are algebraically localized for
$\alpha<\infty$ and exponential as $\alpha\rightarrow\infty$. As a further
consequence, for $\alpha<\infty$ the couplings $B_{pq}$ decay algebraically
w.r.t. the distance between the centers of localisation of $\phi_{p}$ and
$\phi_{q}$, while for $\alpha=\infty$ the decay is exponentially fast. The
spatial structure of the decay of $B_{pq}$ determinants the growth in time of
the EE after a global quantum quench with $H^{\pm}$. Thus, for $\alpha<\infty$
the EE has an unbounded algebraic growth, while for $\alpha=\infty$ the growth
is logarithmic slow.

Returning back to the driven case, in Fig. 9 (a)-(b) is shown the $r$-value as
function of the driving frequency $\omega_{D}$ computed with the eigenvalues
of the Floquet Hamiltonian of the minimal model in Eq. (15) for $\alpha=2.5$
(algebraically LIOMs) and $\alpha\rightarrow\infty$ (exponentially LIOMs). The
crossing between curves for different system sizes, which may demarcate the
existence of a putative MBL transition as function of $\omega_{D}$, is located
in proximity of the many-body width cut-off for $\alpha=2.5$ and at smaller
frequencies for $\alpha\rightarrow\infty$. Considering only values
$\omega_{D}\leq\Lambda$ (see Fig. 9 (a) and its inset), and the tendency
towards delocalization for the algebraic case, we may conclude that the
localization will be completely destroyed in the thermodynamic limit. Indeed,
the crossing between curves for different $L$ happens for frequencies
$\omega_{D}\geq\Lambda$. However, in the limiting case of exponential LIOMs
($\alpha\rightarrow\infty$), we observe a much more robust and defined cross-
over between the two regimes which occurs before the natural bound
$\omega_{D}\sim\Lambda(L)$, and therefore it may culminate to an MBL
transition at a finite frequency strength [34, 27, 28].

Now, we provide an analytical argument based on Ref. 34 to support the
delocalization of an algebraic MBL phase under periodic drive. For the sake of
simplicity, consider the case where the driving terms contain only of one
harmonic and the time-dependent Hamiltonian can be separated in two parts
$H(t)=H_{\ell-\text{bit}}(W)+VH_{1}(t)$. The Hamiltonian
$H_{\ell-\text{bit}}(W)$ is algebraic MBL and $H_{1}(t)\sim\cos(2\pi t/T)$ is
the time-depended part which allows jumps between the localized eigenstates of
$H_{\ell-\text{bit}}(W)$. $W$ and $V$ are the disorder and interaction
strength, respectively. We assume that the disorder is strong enough to
localize the instantaneous eigenstates. If $T$ is large enough, we can
estimate the probability of mixing the instantaneous eigenstates
$|E_{\alpha}(t)\rangle$, $|E_{\beta}(t)\rangle$ using the Landau-Zener formula
[83, 84, 85]

$P_{\alpha\rightarrow\alpha}=e^{-C_{\alpha,\beta}},\quad
C_{\alpha,\beta}=\pi\frac{|M_{\alpha,\beta}|^{2}}{v_{\alpha,\beta}}$ (16)

where $M_{\alpha,\beta}=\langle E_{\alpha}|H_{1}(t_{c})|E_{\beta}\rangle$ is
the matrix element coupling the eigenstates, the Landau–Zener velocity
$v_{\alpha,\beta}=T\partial_{t}(E_{\alpha}-E_{\beta})|_{t=t_{c}}$ and $t_{c}$
the time at which the crossing takes place. In the adiabatic
($C_{\alpha,\beta}\approx 0$) and diabatic ($C_{\alpha,\beta}\gg 1$) limits
the system remains localized after the crossing. In the intermediate situation
eigenstates are characterized by different values of LIOMs mixing together and
starting to thermalize. As argued in Ref. 34, there exist two length scales
$x_{1}$ and $x_{2}$, which determine the behavior of
$P_{\alpha\rightarrow\alpha}$. Consider two eigenstates
$|E_{\alpha}(t)\rangle$, $|E_{\beta}(t)\rangle$ which differs from the values
of LIOMs only in a region of size $x$. Thus, these eigenstates are the same
outside a region of size $x$. There are $\sim 2^{x}$ of these eigenstates
which typically belong to an energy shell with bandwidth $\sim W\sqrt{x}$ and
the mean level spacing

$\Delta(x)=W\sqrt{x}/2^{x}.$ (17)

On the other hand, the typical gap at the crossing is controlled by
$VH_{1}(t_{c})$

$\delta E_{\alpha,\beta}(x)\sim V(\langle
E_{\alpha}|H_{1}|E_{\alpha}\rangle-\langle
E_{\beta}|H_{1}|E_{\beta}\rangle)\sim V\sqrt{x}.$ (18)

In the limit $\delta E_{\alpha,\beta}\gg\Delta$ multiple crossing between
eigenstates which differ only in the region of size $x$ happen. This sets the
length scale $x_{1}$ by comparing $\delta
E_{\alpha,\beta}(x_{1})\sim\Delta(x_{1})$

$x_{1}=\log_{2}\frac{W}{V}.$ (19)

The length $x_{2}$ determinants the character of the transition, which is
directly connected to
$C_{\alpha,\beta}\propto|M_{\alpha,\beta}|^{2}/v_{\alpha,\beta}$. The matrix
element between two algebraically localized states can be estimated by

$M_{\alpha,\beta}\sim V/x^{\gamma},$ (20)

which decays as a power-law in distance $x$ since the LIOMs are algebraically
localized. Instead, the Landau–Zener velocity is

$v_{\alpha,\beta}\sim\Delta E_{\alpha,\beta}/T\sim V\sqrt{x}/T.$ (21)

Thus, the length $x_{2}$ is defined by $C_{\alpha,\beta}(x_{2})\sim 1$ which
separate the adiabatic limit from the diabatic one,

$x_{2}=(VT)^{\frac{1}{2\gamma+1/2}}.$ (22)

In the regime $x_{1}/x_{2}\ll 1$, the system undergoes multiples crossing in
an intermediate regime $C_{\alpha,\beta}\sim 1$ which implies the
thermalization of the region of size $x\ll x_{1}\ll x_{2}$. The relation
$x_{1}/x_{2}\ll 1$ implies that

$\frac{W}{V}\ll 2^{(VT)^{\frac{1}{2\gamma+1/2}}}.$ (23)

Although this simple argument (Eq. (23)) does not strictly prove the
delocalization of the system at any frequency $\omega_{D}=\frac{2\pi}{T}$, it
shows that the delocalization is parametrically favorable, meaning that it is
exponentially sensitive to $\omega_{D}\sim 1/T$.

##  IV Conclusion

In this work we have investigated the stability of an algebraically localized
phase subject to periodically global driving for both, the non-interacting and
the interacting case. For sufficiently strong disorder, the non-interacting
algebraically localized phase is robust once short-range interactions are
turned on, generalizing the paradigm of algebraic localization to the
interacting case (algebraic MBL). As a consequence, in an algebraic MBL phase,
ergodicity is broken and the phase is fully characterized by an extensive
number of integrals of motion with algebraic decaying tails, which are
believed to be adiabatically connected to the algebraic decaying integrals of
motion of the non-interacting case.

In particular, we studied a periodically driven (Floquet) spinless fermionic
chain with long-range random hopping and density-density short-range
interactions. The non-driven non-interacting limit is known as the power-law
random banded matrix model and hosts an algebraically localized phase, meaning
that its single-particle eigenstates are localized with power-law tails.

First, we considered the non-interacting case. With a combination of
analytical arguments and exact numerical simulations, we provided evidence
that the algebraically localized phase is stable under driving. The spectrum
of the signle-particle Floquet Hamiltonian shows Poissonian statistics and its
eigenstates are localized. We supported this claim by using analytical
arguments based on strong disorder renormalization group techniques.

Second, we have tackled the interacting case by turning on short-range
density-density type interactions. At strong disorder and for sufficiently
large decay rate of the hopping the non-driven case is believed to be in an
algebraic MBL phase. Several MBL markers have been used to numerically inspect
the stability of the algebraic MBL phase under driving. After a careful
analysis of the finite-size effects, we argue that ergodicity might be
restored in the thermodynamic limit for any driving frequencies. For finite-
size systems we showed that localization sets in for driving frequency that
are comparable with the many-body bandwidth. This indicates that, the finite-
size localized phase might disperse in the thermodynamic limit. Analytical
arguments were made to support our numerical results by showing that
delocalization appears parametrically favorable.

Thus, unlike the non-interacting case, we showed that the algebraic MBL phase
is unstable once the system is periodically driven. This result should be
compared to the case in which the LIOMs are exponentially localized. For
conventional MBL phases, it is believed that a finite, critical frequency
exists, which separates an ergodic phase from an MBL one. To further point out
this difference, in the final part of the work we inspected a toy Floquet
model, in which we can tune the type of localization, from algebraic to
exponential. We confirmed the existence of a critical driving frequency in a
conventional MBL phase and showed the breakdown of an algebraic MBL phase
under driving.

Our work imposes strong constraints on the possible existence of a genuine MBL
phase for long-range models under driving. This result could be relevant for
trapped-ions experiments, where long-range interactions are present. It is
important to point out that even though we do not expect the existence of an
MBL phase, interesting non-ergodic behavior could persist for fairly long time
scales and large systems. Thus, understanding the time scales for
thermalization is an important question which is left for future research.

###### Acknowledgements.

We would like to thank S. Bera and I. Khaymovich for helpful discussions. We
also express our gratitude to S. Woo Kim for a critical reading of the
manuscript. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No. 853443), and M.H. further acknowledges support by
the Deutsche Forschungs- gemeinschaft (DFG) via the Gottfried Wilhelm Leibniz
Prize program.

## References

  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * Gross and Bloch [2017] C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  * Abanin  _et al._ [2019] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  * Nandkishore and Huse [2015] R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annual Review of Condensed Matter Physics 6, 15 (2015).
  * Alet and Laflorencie [2018] F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, Comptes Rendus Physique 19, 498 (2018), quantum simulation / Simulation quantique.
  * Basko  _et al._ [2006] D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Physics 321, 1126 (2006).
  * Serbyn  _et al._ [2013a] M. Serbyn, Z. Papić, and D. A. Abanin, Local conservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013a).
  * Huse  _et al._ [2014] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phenomenology of fully many-body-localized systems, Phys. Rev. B 90, 174202 (2014).
  * Imbrie [2016] J. Z. Imbrie, On many-body localization for quantum spin chains, Journal of Statistical Physics 163, 998 (2016).
  * Ros  _et al._ [2015] V. Ros, M. Müller, and A. Scardicchio, Integrals of motion in the many-body localized phase, Nuclear Physics B 891, 420 (2015).
  * Bardarson  _et al._ [2012] J. H. Bardarson, F. Pollmann, and J. E. Moore, Unbounded growth of entanglement in models of many-body localization, Phys. Rev. Lett. 109, 017202 (2012).
  * Žnidarič  _et al._ [2008] M. Žnidarič, T. c. v. Prosen, and P. Prelovšek, Many-body localization in the heisenberg $xxz$ magnet in a random field, Phys. Rev. B 77, 064426 (2008).
  * Serbyn  _et al._ [2013b] M. Serbyn, Z. Papić, and D. A. Abanin, Universal slow growth of entanglement in interacting strongly disordered systems, Phys. Rev. Lett. 110, 260601 (2013b).
  * Deutsch [1991] J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  * Srednicki [1994] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  * Rigol  _et al._ [2008] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008).
  * D’Alessio  _et al._ [2016] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics 65, 239 (2016), https://doi.org/10.1080/00018732.2016.1198134 .
  * Pal and Huse [2010] A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  * Kjäll  _et al._ [2014] J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Many-body localization in a disordered quantum ising chain, Phys. Rev. Lett. 113, 107204 (2014).
  * Bera  _et al._ [2015] S. Bera, H. Schomerus, F. Heidrich-Meisner, and j. H. Bardarson, Many-body localization characterized from a one-particle perspective, Phys. Rev. Lett. 115, 046603 (2015).
  * Luitz  _et al._ [2015] D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field heisenberg chain, Phys. Rev. B 91, 081103 (2015).
  * De Tomasi  _et al._ [2017] G. De Tomasi, S. Bera, J. H. Bardarson, and F. Pollmann, Quantum mutual information as a probe for many-body localization, Phys. Rev. Lett. 118, 016804 (2017).
  * De Tomasi [2019] G. De Tomasi, Algebraic many-body localization and its implications on information propagation, Physical Review B 99, 054204 (2019).
  * Botzung  _et al._ [2019] T. Botzung, D. Vodola, P. Naldesi, M. Müller, E. Ercolessi, and G. Pupillo, Algebraic localization from power-law couplings in disordered quantum wires, Phys. Rev. B 100, 155136 (2019).
  * Deng  _et al._ [2020] X. Deng, G. Masella, G. Pupillo, and L. Santos, Universal algebraic growth of entanglement entropy in many-body localized systems with power-law interactions, Phys. Rev. Lett. 125, 010401 (2020).
  * Safavi-Naini  _et al._ [2019] A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey, and R. M. Nandkishore, Quantum dynamics of disordered spin chains with power-law interactions, Phys. Rev. A 99, 033610 (2019).
  * Lazarides  _et al._ [2015] A. Lazarides, A. Das, and R. Moessner, Fate of many-body localization under periodic driving, Physical review letters 115, 030402 (2015).
  * Ponte  _et al._ [2015] P. Ponte, Z. Papić, F. Huveneers, and D. A. Abanin, Many-body localization in periodically driven systems, Physical review letters 114, 140401 (2015).
  * Wilczek [2012] F. Wilczek, Quantum time crystals, Phys. Rev. Lett. 109, 160401 (2012).
  * Sacha [2015] K. Sacha, Modeling spontaneous breaking of time-translation symmetry, Phys. Rev. A 91, 033617 (2015).
  * Else  _et al._ [2016] D. V. Else, B. Bauer, and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117, 090402 (2016).
  * Yao  _et al._ [2017] N. Y. Yao, A. C. Potter, I.-D. Potirniche, and A. Vishwanath, Discrete time crystals: Rigidity, criticality, and realizations, Phys. Rev. Lett. 118, 030401 (2017).
  * Khemani  _et al._ [2016] V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, Phase structure of driven quantum systems, Phys. Rev. Lett. 116, 250401 (2016).
  * Abanin  _et al._ [2016] D. A. Abanin, W. De Roeck, and F. Huveneers, Theory of many-body localization in periodically driven systems, Annals of Physics 372, 1–11 (2016).
  * Bordia  _et al._ [2017] P. Bordia, H. Lüschen, U. Schneider, M. Knap, and I. Bloch, Periodically driving a many-body localized quantum system, Nature Physics 13, 460 (2017).
  * Deng  _et al._ [2018] X. Deng, V. E. Kravtsov, G. V. Shlyapnikov, and L. Santos, Duality in power-law localization in disordered one-dimensional systems, Phys. Rev. Lett. 120, 110602 (2018).
  * Nosov  _et al._ [2019] P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, Correlation-induced localization, Phys. Rev. B 99, 104203 (2019).
  * Levitov [1989] L. S. Levitov, Absence of localization of vibrational modes due to dipole-dipole interaction, Europhysics Letters (EPL) 9, 83 (1989).
  * Mirlin and Fyodorov [1993] A. D. Mirlin and Y. V. Fyodorov, The statistics of eigenvector components of random band matrices: analytical results, Journal of Physics A: Mathematical and General 26, L551 (1993).
  * Levitov [1990] L. S. Levitov, Delocalization of vibrational modes caused by electric dipole interaction, Phys. Rev. Lett. 64, 547 (1990).
  * Mirlin  _et al._ [1996a] A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices, Phys. Rev. E 54, 3221 (1996a).
  * Tran  _et al._ [2019] M. C. Tran, A. Ehrenberg, A. Y. Guo, P. Titum, D. A. Abanin, and A. V. Gorshkov, Locality and heating in periodically driven, power-law-interacting systems, Physical Review A 100, 052103 (2019).
  * Kravtsov  _et al._ [2015] V. Kravtsov, I. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New Journal of Physics 17, 122002 (2015).
  * Mirlin  _et al._ [1996b] A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices, Physical Review E 54, 3221 (1996b).
  * Quito  _et al._ [2016] V. L. Quito, P. Titum, D. Pekker, and G. Refael, Localization transition in one dimension using wegner flow equations, Physical Review B 94, 104202 (2016).
  * Khatami  _et al._ [2012a] E. Khatami, M. Rigol, A. Relano, and A. M. García-García, Quantum quenches in disordered systems: Approach to thermal equilibrium without a typical relaxation time, Physical Review E 85, 050102 (2012a).
  * Khatami  _et al._ [2012b] E. Khatami, M. Rigol, A. Relaño, and A. M. García-García, Quantum quenches in disordered systems: Approach to thermal equilibrium without a typical relaxation time, Phys. Rev. E 85, 050102 (2012b).
  * Note [1] $[H,\tau_{i}]=0$ $\forall i$.
  * Thiery  _et al._ [2018] T. Thiery, F. m. c. Huveneers, M. Müller, and W. De Roeck, Many-body delocalization as a quantum avalanche, Phys. Rev. Lett. 121, 140601 (2018).
  * Gopalakrishnan and Huse [2019] S. Gopalakrishnan and D. A. Huse, Instability of many-body localized systems as a phase transition in a nonstandard thermodynamic limit, Phys. Rev. B 99, 134305 (2019).
  * De Roeck and Huveneers [2017] W. De Roeck and F. m. c. Huveneers, Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B 95, 155129 (2017).
  * Wahl  _et al._ [2019] T. B. Wahl, A. Pal, and S. H. Simon, Signatures of the many-body localized regime in two dimensions, Nature Physics 15, 164 (2019).
  * De Roeck and Imbrie [2017] W. De Roeck and J. Z. Imbrie, Many-body localization: stability and instability, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160422 (2017).
  * Kennes [2018] D. M. Kennes, Many-body localization in two dimensions from projected entangled-pair states (2018), arXiv:1811.04126 [cond-mat.dis-nn] .
  * Théveniaut  _et al._ [2020] H. Théveniaut, Z. Lan, G. Meyer, and F. Alet, Transition to a many-body localized regime in a two-dimensional disordered quantum dimer model, Physical Review Research 2, 10.1103/physrevresearch.2.033154 (2020).
  * Potirniche  _et al._ [2019] I.-D. Potirniche, S. Banerjee, and E. Altman, Exploration of the stability of many-body localization in $d>1$, Phys. Rev. B 99, 205149 (2019).
  * Chertkov  _et al._ [2021] E. Chertkov, B. Villalonga, and B. K. Clark, Numerical evidence for many-body localization in two and three dimensions, Physical Review Letters 126, 180602 (2021).
  * Pietracaprina and Alet [2021] F. Pietracaprina and F. Alet, Probing many-body localization in a disordered quantum dimer model on the honeycomb lattice, SciPost Phys. 10, 44 (2021).
  * Thomson and Schiró [2018] S. J. Thomson and M. Schiró, Time evolution of many-body localized systems with the flow equation approach, Phys. Rev. B 97, 060201 (2018).
  * Tang  _et al._ [2021] H.-K. Tang, N. Swain, D. C. W. Foo, B. J. J. Khor, F. F. Assaad, S. Adam, and P. Sengupta, Evidence of many-body localization in 2D from quantum Monte Carlo simulation, arXiv e-prints , arXiv:2106.08587 (2021), arXiv:2106.08587 [cond-mat.str-el] .
  * Decker  _et al._ [2021] K. S. C. Decker, D. M. Kennes, and C. Karrasch, Many-body localization and the area law in two dimensions, arXiv e-prints , arXiv:2106.12861 (2021), arXiv:2106.12861 [cond-mat.dis-nn] .
  * Choi  _et al._ [2016] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
  * Smith  _et al._ [2016] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Many-body localization in a quantum simulator with programmable random disorder, Nature Physics 12, 907 (2016).
  * Xu  _et al._ [2018] K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song, W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, K. Huang, H. Wang, X. Zhu, D. Zheng, and H. Fan, Emulating many-body localization with a superconducting quantum processor, Phys. Rev. Lett. 120, 050507 (2018).
  * Guo  _et al._ [2020] Q. Guo, C. Cheng, H. Li, S. Xu, P. Zhang, Z. Wang, C. Song, W. Liu, W. Ren, H. Dong, R. Mondaini, and H. Wang, Stark many-body localization on a superconducting quantum processor, arXiv e-prints , arXiv:2011.13895 (2020), arXiv:2011.13895 [quant-ph] .
  * De Tomasi  _et al._ [2020] G. De Tomasi, I. M. Khaymovich, F. Pollmann, and S. Warzel, Rare thermal bubbles at the many-body localization transition from the Fock space point of view, arXiv e-prints , arXiv:2011.03048 (2020), arXiv:2011.03048 [cond-mat.dis-nn] .
  * Dumitrescu  _et al._ [2019] P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran, M. Serbyn, and R. Vasseur, Kosterlitz-thouless scaling at many-body localization phase transitions, Phys. Rev. B 99, 094205 (2019).
  * Oganesyan and Huse [2007] V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  * Atas  _et al._ [2013] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).
  * Note [2] $\mathcal{T}$ denotes the time-ordered product.
  * Page [1993] D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71, 1291 (1993).
  * Note [3] It is important to point out that however also fully ergodic states can reach the Page value [86].
  * Singh  _et al._ [2017] R. Singh, R. Moessner, and D. Roy, Effect of long-range hopping and interactions on entanglement dynamics and many-body localization, Phys. Rev. B 95, 094205 (2017).
  * Evers and Mirlin [2008] F. Evers and A. D. Mirlin, Anderson transitions, Reviews of Modern Physics 80, 1355–1417 (2008).
  * Mard  _et al._ [2014] H. J. Mard, J. A. Hoyos, E. Miranda, and V. Dobrosavljević, Strong-disorder renormalization-group study of the one-dimensional tight-binding model, Physical Review B 90, 125141 (2014).
  * Kehrein [2007] S. Kehrein,  _The flow equation approach to many-particle systems_ , Vol. 217 (Springer, 2007).
  * Wegner [1994] F. Wegner, Flow-equations for hamiltonians, Annalen der physik 506, 77 (1994).
  * Głazek and Wilson [1993] S. D. Głazek and K. G. Wilson, Renormalization of hamiltonians, Physical Review D 48, 5863 (1993).
  * Glazek and Wilson [1994] S. D. Glazek and K. G. Wilson, Perturbative renormalization group for hamiltonians, Physical Review D 49, 4214 (1994).
  * Pekker  _et al._ [2017] D. Pekker, B. K. Clark, V. Oganesyan, and G. Refael, Fixed points of wegner-wilson flows and many-body localization, Physical review letters 119, 075701 (2017).
  * De Tomasi  _et al._ [2019] G. De Tomasi, F. Pollmann, and M. Heyl, Efficiently solving the dynamics of many-body localized systems at strong disorder, Phys. Rev. B 99, 241114 (2019).
  * Wu  _et al._ [2019] L.-N. Wu, A. Schnell, G. D. Tomasi, M. Heyl, and A. Eckardt, Describing many-body localized systems in thermal environments, New Journal of Physics 21, 063026 (2019).
  * Zener [1932] C. Zener, Nonadiabatic crossing of energy levels, Proc. Roy. Soc. Lond. A 137, 696 (1932).
  * Lacki  _et al._ [2009] J. Lacki, H. Ruegg, and G. Wanders, eds., Theorie der unelastischen stösse zwischen atomen, in  _E.C.G. Stueckelberg, An Unconventional Figure of Twentieth Century Physics: Selected Scientific Papers with Commentaries_ (Birkhäuser Basel, Basel, 2009) pp. 117–171.
  * Majorana [1932] E. Majorana, Atomi orientati in campo magnetico variabile, Il Nuovo Cimento (1924-1942) 9, 43 (1932).
  * De Tomasi and Khaymovich [2020] G. De Tomasi and I. M. Khaymovich, Multifractality meets entanglement: Relation for nonergodic extended states, Phys. Rev. Lett. 124, 200602 (2020).


#  An Evolutionary Pathway for the Quantum Internet Relying on Secure
Classical Repeaters

Gui-Lu Long, Dong Pan, Yu-Bo Sheng, Qikun Xue, Jianhua Lu, and Lajos Hanzo
Gui-Lu Long is with the State Key Laboratory of Low-Dimensional Quantum
Physics and Department of Physics, Tsinghua University, and the Frontier
Science Center for Quantum Information, Beijng 100084, China, and the Beijing
Academy of Quantum Information Sciences, Beijing 100193, China.Dong Pan is
with the Beijing Academy of Quantum Information Sciences, Beijing 100193,
China and the State Key Laboratory of Low-Dimensional Quantum Physics and
Department of Physics, Tsinghua University, Beijing 100084, China.Yu-Bo Sheng
is with the College of Electronic and Optical Engineering, Nanjing University
of Posts and Telecommunications, Nanjing, 210003, China. Qikun Xue is with the
State Key Laboratory of Low-Dimensional Quantum Physics and Department of
Physics, Tsinghua University, the Frontier Science Center for Quantum
Information, Beijng 100084, China, the Beijing Academy of Quantum Information
Sciences, Beijing 100193, China, and the Southern University of Science and
Technology, Shenzhen 518055, China.Jianhua Lu is with the School of
Information Science and Technology, Tsinghua University, the Beijing National
Research Center for Information Science and Technology, and the Frontier
Science Center for Quantum Information, Beijing 100084, China.Lajos Hanzo is
is with the School of Electronics and Computer Science, University of
Southampton, Southampton SO17 1BJ, United Kingdom.

###### Abstract

Until quantum repeaters become mature, quantum networks remain restricted
either to limited areas of directly connected nodes or to nodes connected to a
common node. We circumvent this limitation by conceiving quantum networks
using secure classical repeaters combined with the quantum secure direct
communication (QSDC) principle, which is a compelling form of quantum
communication that directly transmits information over quantum channel. The
final component of this promising solution is our classical quantum-resistant
algorithm. Explicitly, in these networks, the ciphertext gleaned from a
quantum-resistant algorithm is transmitted using QSDC along the nodes, where
it is read out and then transmitted to the next node. At the repeaters, the
information is protected by our quantum-resistant algorithm, which is secure
even in the face of a quantum computer. Hence, our solution offers secure end-
to-end communication across the entire network, since it is capable of both
eavesdropping detection and prevention in the emerging quantum internet. It is
compatible with operational networks, and will enjoy the compelling services
of the popular Internet, including authentication. Hence, it smoothens the
transition from the classical Internet to the Quantum Internet (Qinternet) by
following a gradual evolutionary upgrade. It will act as an alternative
network in quantum computing networks in the future. We have presented the
first experimental demonstration of a secure classical repeater based hybrid
quantum network constructed by a serial concatenation of an optical fiber and
free-space communication link. In conclusion, secure repeater networks may
indeed be constructed using existing technology and continue to support a
seamless evolutionary pathway to the future Qinternet of quantum computers.

##  I Introduction

The Quantum internet (Qinternet) links nodes such as quantum computers,
quantum sensors as well as other quantum devices, and it will support
fascinating functions and tasks that are impossible in the classical Internet
[1]. A quantum network includes nodes, routers, repeaters, and quantum
channels [2]. A quantum link includes fiber or free-space optical channels,
and the specific type of communications protocol adopted, such as quantum
secure direct communication (QSDC) [3], quantum key distribution (QKD) [4], or
quantum teleportation [5].

The roadmap to build the fully-fledged Qinternet has been envisioned in [2].
According to its functionality, six stages lie ahead. At the time of writing,
the first three stages are in the experimental phase, and the trusted-repeater
networks concept has the potential of extending the network to large areas. It
forms an important stepping stone towards the Qinternet [2], although it does
not provide end-to-end security. Prepare-and-measure networks provide links
for directly-linked nodes, while entanglement-distribution networks link up
nodes connected to a common intermediate node, where the qubits received from
the other two nodes are measured using the measurement-device-independent
technique of [6]. Thus, the scale of quantum networks classified as prepare-
and-measure networks and entanglement-distribution networks remain limited.
Satellite-based repeaters are capable of increasing the distance between the
directly connected nodes, but they remain limited to 3 nodes, so they are also
at the entanglement-distribution network stage.

Once having practical quantum memories becomes a reality, quantum repeaters
will be used in quantum networks for extending their scale. However, it is
still a tremendous open challenge to fabricate practical quantum memory.
Furthermore, the incompatibility of quantum communication with the existing
classical Internet presents another challenge. Therefore, the Qinternet
remains unable to exploit the vast resources and powerful facilities of the
conventional Internet.

Against this background, we propose the secure-repeater networks (SRNs)
concept by intrinsically combining QSDC [3, 7, 8] and classical cryptographic
algorithms, in particular, the family of quantum-resistant algorithms [9]. In
these networks, the ciphertext gleaned from a quantum-resistant algorithm is
transmitted using QSDC step-by-step along the nodes, where it is read out and
retransmitted to the next node.

In contrast to relying solely on the trust in the nodes of trusted-repeater
networks, computational security 111A cryptosystem can be said to be
computationally secure when the computational requirements become so excessive
that it becomes practically infeasible to break the cryptosystem in
’reasonable time’. is provided at the classical repeater nodes. Thus it
provides secure end-to-end communication for the entire network. Communication
satisfying information-theoretic security 222A cryptosystem can be said to be
information-theoretically secure when its security can be purely proven by
information theory, and the cryptosystem cannot be broken even if the
adversary has unlimited computing power.  is provided using QSDC for directly
connected nodes, or when using measurement-device-independent QSDC [6] for the
nodes connected to a common node, respectively. Because the nodes of SRNs are
linked by QSDC, both eavesdropping detection and prevention capabilities are
inherent in SRNs.

In contrast to the conventional Internet, SRN replaces classical communication
by quantum communication. Since SRNs are fully compatible with the existing
Internet, they circumvent the incompatibility challenge in the development of
the Qinternet. As a benefit, SRNs can take full advantage of the compelling
facilities and services provided by the operational Internet, above all, its
salient authentication function. The compatibility of SRNs creates an
evolutionary path for the Qinternet. However, it is worth noting that SRNs are
not merely a provisional measure conceived for circumventing the lack of
quantum memory. Quite the opposite, the services of SRNs are still required
even at further evolved stages of the Qinternet, making it a potent
alternative network for the future Qinternet.

This paper is organized as follows. In Sec. II, we introduce the basic concept
of QSDC, which is an essential ingredient of SRNs. In Sec. III, we describe
the SRN stage, while in Sec. IV, we present a roadmap of constructing the
Qinternet in 7 evolutionary stages. The corresponding proof of concept
demonstration is provided in Sec. V. Finally, a summary is given in Sec. VI.

##  II Quantum secure direct communication

QSDC was originally proposed in 2000 [3], which transmits information directly
through a quantum channel, while maintaining information-theoretic security
without a prior secret key distribution stage. QSDC can be realized using
entanglement [3, 7] or using single-photons [8]. Here we briefly highlight the
first QSDC protocol [3], where information is mapped to the quantum states of
the Einstein-Podolsky-Rosen (EPR) pairs. The illustration of the protocol is
shown in Fig. 1, which contains the following steps:

(1) Alice prepares an ordered sequence of EPR-pairs containing both the
message bits and check bits. Explicitly, first she prepares a sequence of EPR-
pairs according to the message. Then she also chooses some random numbers
mapped to the checking bits and prepares the corresponding EPR-pairs, which
are inserted into the message EPR-pair sequence at random positions. As shown
in Fig. 1, the pair of spheres labeled by red, green, black and purple
represent the entangled state that carries information or checking bits 00,
01, 10 and 11, respectively. Then Alice chooses from each EPR-pair one qubit
to form the ordered sequence (b), and the remaining qubits form another
ordered single qubit sequence (a).

(2) Alice sends the (b)-sequence to Bob.

(3) Alice measures some of the checking qubits using either the Z or the X
bases randomly. She then informs Bob of the positions, of the basis and the
results of these measurements through a classical channel. Bob then measures
the corresponding particles in the (b)-sequence at his site. He then continues
by comparing his results with Alice’s results and estimates the error rate. If
it is below a threshold, then they proceed to the next step. Otherwise the
process is abandoned and restarted. The pair of dotted circles in Fig. 1
represents the checking qubits.

(4) In case of successful confirming the security of the transmission of the
(b)-sequence, Alice sends the (a)-sequence to Bob. He applies Bell-basis
measurements to the received EPR-pairs and reads out both the message bits and
the remaining checking bits. Alice proceeds by announcing the states of the
remaining checking bits, followed by Bob comparing them to his results. If the
error rate is below the tolerable threshold, they conclude that the
transmission is successful.

This scheme is deemed to be an efficient-QSDC protocol, since it harnesses all
EPR pairs for transmitting information - except for those selected for
checking the presence of eavesdroppers [3].

Figure 1: Procedure of the efficient-QSDC protocol [3].

In the protocol of Fig. 1, information is mapped to the quantum states of the
EPR-pairs, while in [7], information is mapped to the dense coding operations
applied to the EPR-pairs. A single-photon QSDC protocol has also been proposed
in [8]. However, the original QSDC protocols still rely on quantum memory to
wait for the results of eavesdropping checking. As a remedy, recently,
quantum-memory-free QSDC protocols were proposed, where the information is
protected by classical coding, which serendipitously plays the role of quantum
memory [10]. The security analysis of QSDC has been completed using Wyner’s
wiretap channel theory [11, 12]. As an evolution from classical
communications, where infinitesimally low error probability is guaranteed by
Shannon’s theory in noisy channels, QSDC [3, 7, 8] simultaneously guarantees
both reliable and secure communication in the presence of both noise and
eavesdropping. Recently, a practical QSDC prototype has been constructed [12],
and its upgraded version is capable of communicating at 4 kbps over a 10 km
fiber link, which is eminently suitable for texts, images, and telephone
calls.

Table I provides a comparison of QSDC and QKD from the perspectives of their
history, functionality, security, communication pattern, and their best
performance at the time of writing. QKD was proposed earlier than QSDC and it
has been fully developed. It can be seen that the practical performance of
QSDC is below that of QKD at the time of writing. Hence, the engineering of
QSDC must be accelerated. The SRN network concept proposed here can be relied
upon for mitigating the security risks imposed by trusted repeater, since it
is capable of supporting end-to-end secure communication at computational
security relying on quantum-resistant algorithms.

TABLE I: Comparison between QKD and QSDC. Features Schemes | QKD | QSDC
---|---|---
History |  The first QKD protocol was proposed by C. H. Bennett and G. Brassard in 1984. |  The first QSDC protocol was proposed by G. L. Long and X. S. Liu in 2000.
Functionality |  Negotiating secret key between a pair of legitimate users. $\bullet$ Pros: Easy to implement with privacy amplification for generation of the final secret key. $\bullet$ Cons: Limited application; Need key management. |  Realizing reliable and secure communication directly between a pair of legitimate users. $\bullet$ Pros: Wide application, secure communication of information and key without explicit encryption. $\bullet$ Cons: More complicated to implement compared to QKD.
Security |  Security is assured by detecting eavesdropping and discarding transmitted data are leaked. $\bullet$ Pros: Information-theoretically secure; Supporting measurement-device-independent protocol and device-independent protocol; Eavesdropping detection. $\bullet$ Cons: Lack of industrial standards for engineering; Security risks in trusted repeater. |  Security is assured by detecting and preventing eavesdropping. $\bullet$ Pros: Information-theoretically secure; Supporting measurement-device-independent protocol and device-independent protocol; Eavesdropping detection; Eavesdropping prevention; Secure repeater offers secure end-to-end communication. $\bullet$ Cons: Lack of industrial standards for engineering.
Pattern |  QKD negotiates secure key first, information is transmitted through another classical communication. |  QSDC completes reliable and secure communication in a single transmission.
Best performance |  $\bullet$ Secret key rate: $>$2 Mbps @10 km fiber link. |  $\bullet$ Information rate: 4.85 kbps @ 12.04 km fiber link. $\bullet$Theoretically, it has the same distance and rate as QKD using current technology.

##  III secure repeater aided quantum networks

The SRN is a quantum network relying on QSDC transmitting ciphertext generated
by a classical cryptographic algorithm, such as a quantum-resistant algorithm
[9]. At a node of the SRN, the ciphertext is read out and then routed to the
next node using QSDC again. The information is protected by classical
cryptography benefiting from computational security both at the node as well
as for end-to-end communication across the whole network. For directly-linked
nodes, communication capable of maintaining information-theoretic security is
provided using QSDC. For a pair of nodes connected to a common node,
measurement-device-independent QSDC [6] can be used, which also provides
information-theoretic security. Naturally, in the case of two nodes connected
to a common node, a secure classical repeater can also be used for maintaining
computational security, which is easier to implement than measurement-device-
independent QSDC.

The security principles of quantum and classical cryptography are distinctly
different. Instead of viewing them as competing technologies, they may be
viewed as complementary to each other in SRNs. QSDC provides security by
denying the eavesdropper’s chance to intercept information, either plain text
or ciphertext. Hence an eavesdropper fails to decrypt any useful information,
even if he/she has abundant computing power. By contrast, classical
cryptography relies on solving excessively complex mathematical problems -
such as the factorization of large numbers - for ensuring that Eve cannot
solve the problem in a reasonably short time. The combination of these two
technologies in SRNs is capable of further increasing the security level.
Explicitly, this limits the potential acquisition of ciphertext from the
entire Internet to some repeater nodes. Furthermore, the eavesdropping
detection capability of legitimate users will spot any illegal eavesdropping
activities, and will enhance protection in those areas, where eavesdropping is
frequently detected.

Figure 2: The model of (a) trusted repeater network and (b) secure repeater
network.

SRN provides secure end-to-end communication, in contrast to trusted-repeater
network, where the latter provides no end-to-end security. Although both
trusted-repeater network and SRN use classical repeaters, there is an
essential difference between them. As shown in Fig. 2 (a), in trusted-repeater
network, the key ’key1’ is negotiated between Alice and node 1, key ’key2’
between node 1 and node 2, and so on until the destination is reached. Key
’key1’ is used between the transmitter and the receiver. In trusted-repeater
network, ’key1’ will be relayed from node 1 encrypted using a classical
cryptographic algorithm, say the one-time-pad using key ’key2’, to node 2,
then to node 3 encrypted with ’key3’ until the receiver of Bob is reached.
Thus, ’key1’ is known to all the intermediate repeater nodes, and they must be
trusted for ensuring that the key ’key1’ will not leak. Compared to existing
classical networks, where the information is protected by classical
cryptography in both the communication channels and repeater nodes, the
security of transmission in trusted-repeater network is enhanced. However, the
security level at the intermediate nodes is reduced to relying solely on the
trust of these nodes. In contrast to trusted-repeater networks, in SRNs, the
information is encrypted using a classical cryptographic algorithm, and it is
protected by relying on computational security at the repeater nodes, as shown
in Fig. 2 (b). It should be pointed out that the same strategy cannot be used
in trusted-repeater networks, because the random nature of the negotiated key
prohibits the pre-encryption of the key.

The substitution of classical communication across the classical Internet by
employing QSDC in SRNs has the benefit of both eavesdropping detection and
prevention capabilities, which are unique to QSDC. In SRNs, at each channel
transmission of QSDC between a pair of nodes, we may incorporate a random bit
sequence as the label of each transmitted packet, and publicize them after the
packet reaches the end user. By comparing the positions of bit errors, the
legitimate users can readily spot the specific channel, where eavesdropping
takes place. By exploiting these capabilities, both the users and service
providers can deploy enforced protection in geographic areas, where
eavesdropping is frequently detected. This beneficial capability will increase
the security level of SRNs. This is particularly important, because the
unconditional security of classical cryptographic algorithms, including
quantum-resistant algorithms, has not been unequivocally proven. By contrast,
the unconditional security of quantum communication has been proven in
principle. When quantum computing networks have finally been realized,
information-theoretic security will be realized across the entire Qinternet
and SRNs will pave the way for this process. Both trusted-repeater network and
SRN require an authenticated classical channel for exchanging data concerning
eavesdropping detection in QKD and QSDC, respectively, which were omitted in
Fig. 2. The classical channel required for quantum-resistant algorithm also
has been omitted in Fig. 2 (b), because it constitutes a pre-processing stage
for SRN-based communication.

##  IV Seven-stage roadmap of the Qinternet evolution

Let us now evolve the Qinternet roadmap outlined by S. Wehner et al. [2] a
little further by incorporating the SRN stage, as seen in Fig. 3. The
definitions and the specific types of tasks in each of the other six stages
were detailed in [2].

Figure 3: The seven-stage Qinternet evolved from the six-stage scheme of S.
Wehner et al. [2] by adding secure repeater networks.

Each stage has its own application examples, which are listed below.

  * Stage 1 - trusted repeater networks: QKD with no end-to-end security can be constructed relying on the trusted-repeater networks.

  * Stage 2 - Prepare-and-measure networks: The prepare-and-measure networks support QKD, QSDC and secure identification.

  * Stage 3 - entanglement-distribution networks: The entanglement-distribution networks constitute the foundations of device independent protocols.

  * Stage 4 - secure repeater networks: The SRNs are compatible with the operational Internet and provide secure end-to-end communication relying on computational security. However, compared to the Internet purely relying on quantum-resistant algorithms, an SRN adds both eavesdropping detection as well as prevention for the evolved Internet. Furthermore, it can serve as an auxiliary network of the ultimate fully-fledged Qinternet.

  * Stage 5 - quantum memory networks: Blinding quantum computing becomes available by relying on quantum memory networks.

  * Stage 6 - fault-tolerant few qubit networks: Both clock synchronization and distributed quantum computation are the products of the fault-tolerant few qubit networks.

  * Stage 7 - quantum computing networks: The quantum computing networks afford leader election and prompt Byzantine agreement.

Long-distance end-to-end communication could also be achieved using hop-by-hop
QSDC in prepare-and-measure networks relying on trusted intermediate nodes in
the same way as in trusted repeater networks. This evolutionary step has been
facilitated by the proposed SRN concept for the first time. SRNs fill the gap
between entanglement-distribution networks, and quantum memory networks, which
might need decades to develop, hence SRNs accelerate the evolution of quantum
networks. In the context of the seven-stage roadmap, SRN has a pair of
important roles.

To elaborate, SRN only requires replacing classical communication by quantum
communication. The immediate advantage for SRN users is that they can enjoy
all the services of the operational Internet, such as authentication and
packet switching based transmission. Asymmetric cryptography such as quantum-
resistant algorithm provides efficient authentication in the operational
Internet at a complexity which increases linearly with the number of users. By
contrast, in quantum cryptography, any pair of users has to pre-share a common
secret in order to perform authentication. Hence the associated complexity
becomes intractable, because it increases quadratically with the number of
users.

Given the huge benefits of interconnecting quantum computers only capable of
processing a modest number of qubits for constructing more powerful quantum
computers, the Qinternet must be rolled out in the near future. However, the
time-scale of rolling out the Qinternet is uncertain. Because SRNs are
compatible with the classical Internet, we can start from SRN at the earliest
and then may flexibly update parts of the SRN upon replacing the secure
classical repeaters by quantum repeaters. While updating the system, the SRN
and the accompanying Qinternet operating either in fault-tolerant few qubit
networks or in quantum computing networks will continue to operate
simultaneously, making a gradual and smooth transition to the full Qinternet.
It is expected that this kind of hybrid quantum networks will exist for
several decades.

The SRN will continue to exist as an alternative network even at the quantum
computing network stage, because it is capable of supporting the
functionalities of the classical Internet, such as authentication,
certification and so on. The DL04 QSDC-based SRN is also capable of exchanging
an arbitrary known single qubit. Both quantum repeaters and classical secure
repeaters co-exist at the nodes. SRNs are also capable of finding the route
from the source to the destination, while relying on QSDC.

##  V Proof of concept demonstration of SRN

As shown in Fig. 4, a three-node SRN is constructed, where nodes Alice and R
are connected through a 10 km fiber link, while nodes R and Bob are connected
by a 2-meter free-space link [13]. Node R plays the role of a secure-classical
repeater. We have adopted a lattice based cryptosystem, LAC, a quantum
resistant algorithm designed by X. H. Lu et al., a candidate quantum-resistant
algorithm in the 2nd round post quantum cryptography competition of NIST [9].
Lattice-based cryptographic enjoys very strong security based on worst-case
hardness. In LAC, a 32-byte information string is encrypted into 1024 bytes of
ciphertext. In this demonstration an image file is transformed into an
800-kbyte encrypted file and transmitted from Alice to the Relay R and then to
Bob. The DL04 QSDC [8] is adopted for both hops transmissions. Explicitly, the
single photons transmitted from Alice are detected at the receiver module of R
for recovering the classical bits of the ciphertext, which are then mapped
again to new single photons sent to Bob by using the QSDC transmitter module.
Upon reception it is then decrypted to recover the original image. At the
repeater node, the encrypted image is undecipherable by the repeaters, as it
transpires from Fig. 4. These experiments demonstrate the reliable operation
of the SRN.

Figure 4: The SRN demonstration system. LAC is the abbreviation of a LAttice
based Cryptosystem.

The pair of QSDC links are fully functional, relying on low density parity
check coding and secure transmission. Due to concatenating a pair of QSDC
links, it takes slightly longer to transmit the encrypted file from R to Bob,
than from Alice to R. The channel’s quantum bit error rates are also different
and both of them are below the error rate threshold 5.7%, as shown in Fig. 5.
The higher quantum bit error rate of fiber channel than that of the free space
result arises from its higher length-related attenuation. There are a whole
host of interesting networking issues set aside for further study, such as the
associated throughputs and delays, just to name a few.

Figure 5: The channel quantum bit error rates during image file transmission.

##  VI Conclusions and outlooks

In a nutshell, we have proposed an evolutionary pathway for the Qinternet
relying on secure classical repeaters, the SRN. It relies on QSDC that
transmits ciphertext encrypted by classical quantum resistant algorithms. In
contrast to purely relying on the trust of the repeater nodes in trusted-
repeater networks, it guarantees having computational security. It provides
secure end-to-end communication for the entire quantum network right away at
the current state-of-the-art. It has eavesdropping detection and prevention
capabilities for the Internet, which mitigates the risk of future retroactive
decryption currently faced by classical cryptographic algorithms, when higher
computing power becomes available. It is compatible with the existing Internet
and hence it can exploit all the functions provided by the Internet, which
paves the way for a smooth and gradual transition to the Qinternet. It will
serve as an ancillary network for the ultimate Qinternet, providing
authentication, routing and other Internet services. In conclusion, we have
constructed a three-node SRN testbed and demonstrated the secure transmission
of an image encrypted by the quantum-resistant algorithm over our concatenated
fiber and free-space quantum channel.

Finally, it is worth mentioning that QSDC - which is the central core of SRN -
is gradually developing into a perfectly secure practical application. The SRN
will secure next-generation communications and will support the creation of
long-haul quantum networks, once QSDC becomes fully developed [14]. A working
QSDC prototype that transmits 4 kbps of secure information through a 10 km
fiber link was reported in [12]. The distance and transmission rate can be
further increased by using the technology of increasing capacity using masking
[15], and it is estimated that a 300 kbps information rate can be achieved
over a 50 km fiber link in the near future, and the technology is also
suitable for satellite applications, making SRN concept an implementation-
technology at the time of writing.

## Acknowledgement

The authors would like to thank Dr. Zengrong Zhou as well as Dr. Shijie Wei
for their help in the quantum-resistant algorithm LAC and acknowledge the
helpful discussions with Prof. Liuguo Yin. This work was supported in part by
the National Natural Science Foundation of China (Grant No. 11974205 and No.
11974189), in part by the National Key Research and Development Program of
China under Grant No. 2017YFA0303700 and in part by the Key R&D Program of
Guangdong Province (Grant No. 2018B030325002). L. Hanzo would like to
acknowledge the financial support of the Engineering and Physical Sciences
Research Council projects EP/P034284/1 and EP/P003990/1 (COALESCE) as well as
of the European Research Council’s Advanced Fellow Grant QuantCom (Grant No.
789028).

## References

  * [1] S. Lloyd, J. H. Shapiro, F. N. Wong, P. Kumar, S. M. Shahriar, and H. P. Yuen, “Infrastructure for the quantum internet,” _ACM SIGCOMM Computer Communication Review_ , vol. 34, no. 5, pp. 9–20, 2004.
  * [2] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” _Science_ , vol. 362, no. 6412, 2018.
  * [3] G.-L. Long and X.-S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” _Physical Review A_ , vol. 65, no. 3, p. 032302, 2002, arXiv preprint quant-ph/0012056, 2000.
  * [4] C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in _Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India_ , IEEE, 1984, pp. 175–179.
  * [5] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” _Physical Review Letters_ , vol. 70, no. 13, p. 1895, 1993.
  * [6] Z.-R. Zhou, Y.-B. Sheng, P.-H. Niu, L.-G. Yin, G.-L. Long, and L. Hanzo, “Measurement-device-independent quantum secure direct communication,” _Science China Physics, Mechanics & Astronomy_, vol. 63, no. 3, p. 230362, 2020.
  * [7] F.-G. Deng, G. L. Long, and X.-S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” _Physical Review A_ , vol. 68, no. 4, p. 042317, 2003.
  * [8] F.-G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” _Physical Review A_ , vol. 69, no. 5, p. 052319, 2004.
  * [9] G. Alagic, J. Alperin-Sheriff, D. Apon, D. Cooper, Q. Dang, J. Kelsey, Y.-K. Liu, C. Miller, D. Moody, R. Peralta _et al._ , “Status report on the second round of the post-quantum cryptography standardization process,” _US Department of Commerce, National Institute of Standards and Technology_ , 2020.
  * [10] Z. Sun, L. Song, Q. Huang, L. Yin, G. Long, J. Lu, and L. Hanzo, “Toward practical quantum secure direct communication: A quantum-memory-free protocol and code design,” _IEEE Transactions on Communications_ , vol. 68, no. 9, pp. 5778–5792, 2020.
  * [11] A. D. Wyner, “The wire-tap channel,” _Bell System Technical Journal_ , vol. 54, no. 8, pp. 1355–1387, 1975.
  * [12] R. Qi, Z. Sun, Z. Lin, P. Niu, W. Hao, L. Song, Q. Huang, J. Gao, L. Yin, and G.-L. Long, “Implementation and security analysis of practical quantum secure direct communication,” _Light: Science & Applications_, vol. 8, no. 1, p. e22, 2019.
  * [13] D. Pan, Z. Lin, J. Wu, H. Zhang, Z. Sun, D. Ruan, L. Yin, and G. Long, “Experimental free-space quantum secure direct communication and its security analysis,” _Photonics Research_ , vol. 8, no. 9, pp. 1522–1531, 2020.
  * [14] X. You, C.-X. Wang, J. Huang, X. Gao, Z. Zhang, M. Wang, Y. Huang, C. Zhang, Y. Jiang, J. Wang _et al._ , “Towards 6G wireless communication networks: Vision, enabling technologies, and new paradigm shifts,” _Science China Information Sciences_ , vol. 64, no. 1, pp. 1–74, 2021.
  * [15] G.-L. Long and H. Zhang, “Drastic increase of channel capacity in quantum secure direct communication using masking,” _Science Bulletin_ , vol. 66, no. 13, pp. 1267–1269, 2021.

Gui-Lu Long [M’16] (gllong@mail.tsinghua.edu.cn), FAPS, FIOP, received his
B.S. degree in 1982 and his Ph.D. degree in 1987. He is a professor at
Tsinghua University. He served as President of AAPPS (2017-2019), and Vice-
chair of C13 of IUPAP (2015-2017). He published 300+ refereed papers, and has
21000+ citations.
---
Dong Pan<EMAIL_ADDRESS>received his B.S. degree from the Northwest
University in 2016 and his Ph.D. degree from the Tsinghua University in 2021.
From 2018 to 2019, he was a Visiting Student with the University of
Southampton. He is an assistant research scientist at Beijing Academy of
Quantum Information Sciences.
---
Yu-bo Sheng<EMAIL_ADDRESS>received his Ph.D. degree from Beijing
Normal University in 2009 and engaged in postdoctoral research in Tsinghua
University from 2009 to 2011. He is a professor at Nanjing University of Posts
and Telecommunications. His research interests are quantum communication,
quantum repeater, and quantum computation.
---
Qikun Xue<EMAIL_ADDRESS>FAPS, received his B.S. degree in 1984 and
his Ph.D. degree in 1994. He is a member of Chinese Academy of Sciences. He
won the Fritz London Memorial Prize for low-temperature physics in 2020. He
has authored/coauthored ~370 papers.
---
Jianhua Lu [F’15] (lhh-dee@mail.tsinghua.edu.cn), received his B.S. and M.S.
degrees from Tsinghua University in 1986 and 1989, respectively, and his Ph.D.
degree from the Hong Kong University of Science & Technology in 1998. He is
now a vice president of NSFC, a member of Chinese Academy of Sciences.
---
Lajos Hanzo [F’04]<EMAIL_ADDRESS>(http://www-mobile.ecs.soton.ac.uk,
https://en.wikipedia.org/wiki/Lajos_Hanzo), received Honorary Doctorates from
the Technical University of Budapest and Edinburgh University. He is a Foreign
Member of the Hungarian Science-Academy, Fellow of the Royal Academy of
Engineering (FREng), of the IET, of EURASIP and holds the Eric Sumner Field
Award.
---