$\displaystyle\mathcal{U}_{b,j}=\exp\left[-\frac{J_{b}\delta
t}{2}\bigg{(}\sum_{\alpha,\sigma}\sigma\,\mathrm{i}\gamma_{2j,\alpha\sigma}\gamma_{2j+1,\alpha\sigma}\bigg{)}^{2}\right],$
(54)

where we have denoted the variance $\overline{\theta_{b,j}^{2}}\equiv
J_{b}\delta t$. The averaged measurement in the duplicated Hilbert space takes
the form

$\displaystyle\mathcal{M}_{a,j}=(1-\Gamma_{a}\delta t)+\Gamma_{a}\delta
t\sum_{m=\pm}P_{a,j,m}^{\otimes 4},$ (55)

where $a\in\\{s,b\\}$ indicates the site and bond measurements implemented by
the projections $P_{s,j,\pm}=(1\pm\hat{\pi}_{s,j})/2$ and
$P_{b,j,\pm}=(1\pm\hat{\pi}_{b,j})/2$ on measurement outcomes $m=\pm$.
$\Gamma_{a}$ is the measurement rate so that the measurement probability in a
time step is $p_{a}=\Gamma_{a}\delta t$. Crucially, the evolution operators
$\mathcal{U}_{b,j}$ and $\mathcal{M}_{a,j}$ are both Hermitian. Thus in the
limit $\delta t\to 0$ they describe imaginary time evolution over an
infinitesimal time step, generated by an effective quantum Hamiltonian
$H_{\textrm{eff}}=H_{\mathcal{U}}+H_{\mathcal{M}}$.

We could have also included site unitary gates

$\displaystyle\mathcal{U}_{s,j}=\overline{\exp\left[\sum_{\alpha,\sigma}\theta_{s,j}\sigma\mathrm{i}\gamma_{2j-1,\alpha\sigma}\gamma_{2j,\alpha\sigma}\right]}$
(56)

on the same footing in this program, averaging over $\theta_{s,j}$ drawn from
a Gaussian distribution of coupling constants. This would result in a complete
effective Hamiltonian written in terms of the Majorana operators on the four
chains. However, to obtain a simpler effective model without changing the
essential structure and symmetries of the problem, we choose the site coupling
constants $\theta_{s,j}$ to be uniformly distributed on $[0,2\pi)$. In this
case, exactly as in Section II, averaging over the site unitaries implements a
projection on a six-dimensional local Hilbert space on physical sites (i.e.
Majorana sites $2j-1,2j$). However, in the fermionic system, it is
advantageous to use a different basis of the reduced Hilbert space, which
makes use of the fermionic symmetries.

We recall the above observation that unitary gates, including the single-site
gates, commute with generators of the $SU(2)$ spin ($\Sigma$) and $SU(2)$
charge ($\eta$) symmetry. Therefore, the basis of the projected Hilbert space
can be organized into multiplets of the $\Sigma$ and $\eta$ spins on each
physical site $j$. Three of the basis states
$\mbox{$|m\rangle\rangle$}_{\eta}$ with $m=\pm 1,0$ form the $\eta$ triplet
which transform as the $\Sigma$ singlet. The other three states
$\mbox{$|m\rangle\rangle$}_{\Sigma}$ with $m=\pm 1,0$ form the $\Sigma$
triplet and transform as the $\eta$ singlet. These basis states can be written
explicitly in terms of the second quantized conserved fermion operators
defined in Eq. (52):

$\displaystyle\mbox{$|-\rangle\rangle$}_{\eta}=\mbox{$|\text{vac}\rangle\rangle$},$
(57a)
$\displaystyle\mbox{$|\,0\,\rangle\rangle$}_{\eta}=\frac{1}{\sqrt{2}}\left(c_{2j-1,\uparrow}^{\dagger}c_{2j-1,\downarrow}^{\dagger}-c_{2j,\uparrow}^{\dagger}c_{2j,\downarrow}^{\dagger}\right)\mbox{$|\text{vac}\rangle\rangle$},$
(57b)
$\displaystyle\mbox{$|+\rangle\rangle$}_{\eta}=c_{2j-1,\uparrow}^{\dagger}c_{2j,\uparrow}^{\dagger}c_{2j-1,\downarrow}^{\dagger}c_{2j,\downarrow}^{\dagger}\mbox{$|\text{vac}\rangle\rangle$},$
(57c)
$\displaystyle\mbox{$|-\rangle\rangle$}_{\Sigma}=c_{2j-1,\downarrow}^{\dagger}c_{2j,\downarrow}^{\dagger}\mbox{$|\text{vac}\rangle\rangle$},$
(57d)
$\displaystyle\mbox{$|\,0\,\rangle\rangle$}_{\Sigma}=\frac{1}{\sqrt{2}}\left(c_{2j-1,\downarrow}^{\dagger}c_{2j,\uparrow}^{\dagger}+c_{2j-1,\uparrow}^{\dagger}c_{2j,\downarrow}^{\dagger}\right)\mbox{$|\text{vac}\rangle\rangle$},$
(57e)
$\displaystyle\mbox{$|+\rangle\rangle$}_{\Sigma}=c_{2j-1,\uparrow}^{\dagger}c_{2j,\uparrow}^{\dagger}\mbox{$|\text{vac}\rangle\rangle$},$
(57f)

where $|\text{vac}\rangle\rangle$ is defined by the annihilation
$c_{j,\sigma}\mbox{$|\text{vac}\rangle\rangle$}=0$. Intuitively we can view
the system as hosting on each physical site a single spin-1 particle, which
can be either an $\eta$ or a $\Sigma$ spin. The occupation of $\eta$ or
$\Sigma$ triplet at site $j$ corresponds to distinct eigenvalues of the local
conserved quantities: $\mathcal{R}_{2j-1}=\mathcal{R}_{2j}=\pm 1$ for $\eta$
and $\Sigma$, respectively.

The effective Hamiltonian that generates the imaginary time evolution directly
follows from Eqs. (54) and (55) projected on the reduced six-dimensional local
Hilbert space. After straightforward algebra, we find that $H_{\text{eff}}$
simplifies to

$\displaystyle H_{\text{eff}}=H_{\text{eff},\eta}+H_{\text{eff},\Sigma},$ (58)

where each term in $H_{\text{eff},\eta}$ ($H_{\text{eff},\Sigma}$) acts
nontrivially only for $\eta$-spins ($\Sigma$-spins) and otherwise annihilates
the wave function. We can write this explicitly as

$\displaystyle H_{\text{eff},\kappa}$
$\displaystyle=\sum_{j}\Delta\left(S_{\kappa,j}^{z}\right)^{2}$ (59)
$\displaystyle-\sum_{j}J_{\perp}\left(S_{\kappa,j}^{x}S_{\kappa,j+1}^{x}+S_{\kappa,j}^{y}S_{\kappa,j+1}^{y}\right)+J_{z}S_{\kappa,j}^{z}S_{\kappa,j+1}^{z},$

where the spin-1 operators
$\vec{S}_{\kappa,j}=(S_{\kappa,j}^{x},S_{\kappa,j}^{y},S_{\kappa,j}^{z})$
annihilate the singlet state of the species $\kappa=\eta,\Sigma$. The coupling
constants in this Hamiltonian are determined from the parameters of the
unitary gates and the measurements as follows: $\Delta=\Gamma_{s}$,
$J_{\perp}=2J_{b}+\Gamma_{b}/4$, and $J_{z}=2J_{b}-\Gamma_{b}/4$.

Let us make a few remarks on the effective Hamiltonian. First,
$H_{\text{eff}}$ conserves the total $z$ components as well as the parities
associated with $S_{\kappa,j}^{z}\mapsto-S_{\kappa,j}^{z}$ transformation of
$\eta$ and $\Sigma$ spins, separately. Besides, $H_{\text{eff}}$ is the same
in $\eta$ and $\Sigma$ sector and therefore invariant under the exchange of
$\eta$ and $\Sigma$ spins. Thus, $H_{\text{eff}}$ respects the symmetry
$[(U(1)\rtimes\mathbb{Z}_{2})\times(U(1)\rtimes\mathbb{Z}_{2})]\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$
as expected. Second, the occupation of $\eta$ and $\Sigma$ triplet states is
locally conserved at every site. Note that the number of local integrals of
motion is reduced from $2L$ to $L$ upon averaging over the unitary gates and
projection unto the six-dimensional local Hilbert space
$\mathcal{P}\mathcal{R}_{2j-1}\mathcal{P}=\mathcal{P}\mathcal{R}_{2j}\mathcal{P}$.
Accordingly, the dynamics is partitioned into $2^{L}$ sectors corresponding to
configurations of the $\eta$ and $\Sigma$ occupations.

In our case, the physical circuit is initialized in the vacuum state of the
fermions $f_{j}$. When extended to the duplicated Hilbert space, this state
can be written using the conserved fermions as

$\displaystyle\mbox{$|\rho_{0}\rangle\rangle$}_{j}$
$\displaystyle=\frac{1}{2}\left(c_{2j-1,\uparrow}^{\dagger}+c_{2j,\uparrow}^{\dagger}\right)\left(c_{2j-1,\downarrow}^{\dagger}-c_{2j,\downarrow}^{\dagger}\right)\mbox{$|\text{vac}\rangle\rangle$}_{j}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mbox{$|0\rangle\rangle$}_{\eta}-\mbox{$|0\rangle\rangle$}_{\Sigma}\right).$
(60)

We see that the initial state is already in the reduced Hilbert space and
equally populates both $\eta$ and $\Sigma$ triplet states at every site.

In the long imaginary time evolution, the wave function is dominated by the
global ground states of the Hamiltonian (58) among all $2^{L}$ sectors. The
ground states are found in the two uniform sectors, i.e. the $\eta$ sector
with all sites occupied by $\eta$ spins ($\mathcal{R}_{\ell}=1$) and the
$\Sigma$ sector with all sites occupied by $\Sigma$ spins
($\mathcal{R}_{\ell}=-1$). Thus, the
$\lim_{t\rightarrow\infty}\mbox{$|\rho{(t)}\rangle\rangle$}\approx\mbox{$|\psi_{gs}\rangle\rangle$}_{\eta}+\mbox{$|\psi_{gs}\rangle\rangle$}_{\Sigma}$

The steady state phases of the circuit can be classified based on the reduced
effective symmetry within either one of the two relevant sectors. The symmetry
is reduced because in each sector the two $U(1)$ charges are not independent.
In the $\eta$ sector every site is a sigma singlet with
$N_{\uparrow,j}=N_{\downarrow,j}$, so only $N_{\uparrow}+N_{\downarrow}$ is a
non trivial charge. In the $\Sigma$ sector two fermions occupy each site, so
$N_{\uparrow,j}+N_{\downarrow,j}=2$, so only $N_{\uparrow}-N_{\downarrow}$ is
a non trivial charge. Hence, either sector only forms a faithful
representation of
$U(1)\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\mathbb{T}})$, which is a
subgroup of $\mathcal{G}_{f,\text{eff}}^{(2)}$. One can verify
$U(1)\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\mathbb{T}})$ is exactly the
symmetry of $H_{\text{eff}}$ in either sector. For vanishing measurement
rates, owing to local conserved quantities, each sector only forms a faithful
representation of either $SU(2)$ spin or charge symmetry, reducing the
effective symmetry down to
$O(3)=SO(3)\rtimes\mathbb{Z}_{2}^{\mathbb{T}}$.666The symmetry is $SO(3)$
rather than $SU(2)$ because the ground state occupies an integer
representation. Indeed in this limit the effective Hamiltonian in each of the
two sectors is the Heisenberg ferromagnet, which manifests the $O(3)$
symmetry.

Finally, we establish the correspondence between the ground state properties
of $H_{\textrm{eff}}$ and the steady states of quantum circuits by mapping
various physical observables measured on the circuit to the matrix elements of
different boundary operators in the effective spin model. As explained in
Section II, properties related to the second moments of the density matrix
(conditional on the measurement device) generally take the form

$O^{(2)}\leftrightarrow{\mbox{$\langle\langle\mathcal{I}|$}\mathcal{O}\mbox{$|\rho^{(2)}\rangle\rangle$}\over\mbox{$\langle\langle\mathcal{I}|\rho^{(2)}\rangle\rangle$}},$
(61)

where $\langle\langle\mathcal{I}|$ is the reference state that implements the
(doubled) trace operation in the duplicated Hilbert space, i.e.
$\mbox{$\langle\langle\mathcal{I}|\rho^{(2)}\rangle\rangle$}=\mathrm{tr}\rho^{(2)}$.

Within the effective description, the state
$\mbox{$\langle\langle\mathcal{I}|$}\mathcal{O}$ can be viewed as the boundary
condition to the imaginary time evolution at the latest time. The action of
$\mathcal{O}$ to the left transforms the boundary conditions set by the
reference state $\langle\langle\mathcal{I}|$ in the region on which
$\mathcal{O}$ is supported. The kind of boundary condition imposed is dictated
by the symmetries of $\langle\langle\mathcal{I}|$. We show in appendix F that
the reference state breaks the $U(1)$ symmetry in both the $\eta$ and $\Sigma$
sector, exhibiting the long range order
$\mbox{$\langle\langle\mathcal{I}|$}\eta^{y}_{i}\eta^{y}_{j}\mbox{$|\mathcal{I}\rangle\rangle$}\to
c$ and
$\mbox{$\langle\langle\mathcal{I}|$}\Sigma^{y}_{i}\Sigma^{y}_{j}\mbox{$|\mathcal{I}\rangle\rangle$}\to
c$ as $|i-j|\to\infty$. Thus, $\langle\langle\mathcal{I}|$ effectively imposes
a symmetry breaking boundary condition on the imaginary time evolution by
$H_{\text{eff}}$ [see Fig. 5(a)].

Figure 5: Boundary conditions imposed on the $U(1)$ phase by the boundary
operators corresponding to different probes. (a) The reference state
$|\mathcal{I}\rangle\rangle$ imposes a symmetry breaking state with $U(1)$
phase locked to zero (blue arrows). (b) The site parity variance
$\hat{\Pi}_{s,A}$ rotates the phase by $\pi$-phase in region $A$, which
creates a vortex anti-vortex pair at the edges of the region. (c) The bond
parity variance $\hat{\Pi}_{b,A}$ similarly affects a $\pi$ phase rotation,
creating a vortex anti-vortex pair at the edges of $A$. However, in this case,
the vortex creation operators are bound to a $\mathbb{Z}_{2}$ charge (of the
symmetry $\phi\to-\phi$). (d) The swap operator $\mathcal{C}_{\ell,A}$ rotates
the phase in $A$ by $\pi/2$ (orange arrows), creating a pair of half-vortices
at edges of the region.

The various operators we use to probe the circuit transform the broken
symmetry boundary conditions in different ways. Consider first the site parity
variance $\Pi^{(2)}_{s,A}$ on subsystem $A$, which is mapped to the boundary
matrix element (61) of the operator

$\displaystyle\hat{\Pi}_{s,A}$ $\displaystyle=\prod_{j\in
A}(-\mathrm{i}\gamma_{2j-1,1,\uparrow}\gamma_{2j,1,\uparrow})(-\mathrm{i}\gamma_{2j-1,2,\uparrow}\gamma_{2j,2,\uparrow})$
$\displaystyle=\prod_{j\in A}e^{-\mathrm{i}\pi(\eta^{z}_{j}+\Sigma^{z}_{j})}.$
(62)

This operator rotates the $U(1)$ order parameters for both $\eta$ and $\Sigma$
spins by $\pi$ everywhere in subsystem $A$. The ensuing boundary condition is
illustrated schematically in Fig. 5(b). We’ll see that this is equivalent to
inserting a pair of topological defects (vortices) at the edges of the region
$A$.

The bond parity variance operator $\hat{\Pi}_{b,A}$ can be obtained by simply
translating $\hat{\Pi}_{s,A}$ by half of a physical site (one Majorana site).
Thus, the operator leads to the same $\pi$ rotation of the $U(1)$ orders of
the boundary state within region $A$, but it is bound to a $\mathbb{Z}_{2}$
charge (of the symmetry $\phi\to-\phi$) on each edge of the region [see Fig.
5(c)]. As we will see in the next section, this difference has nontrivial
implications on the behavior of the bond parity variance in the different
phases.

The entanglement entropy of a subsystem is associated with the matrix element
of a swap operator $\mathcal{C}_{\ell,A}$ as discussed in Section II. Because
$\mathcal{C}_{\ell,A}$ is nonlocal when written in terms of fermionic
operators, we replace it with a local operator
$\mathcal{\widetilde{C}}_{\ell,A}$ which has exactly the same action as
$\mathcal{C}_{\ell,A}$ on the reference state, namely
$\mathcal{\widetilde{C}}_{\ell,A}\mbox{$|\mathcal{I}\rangle\rangle$}=\mathcal{C}_{\ell,A}\mbox{$|\mathcal{I}\rangle\rangle$}$
(see Appendix G). The operator $\mathcal{\widetilde{C}}_{\ell,A}$ can be
simply expressed in terms of the $f$-fermions defined on the forward branches
and in terms of the $\eta$ and $\Sigma$ spins,

$\displaystyle\widetilde{\mathcal{C}}_{\ell,A}=$ $\displaystyle\prod_{j\in
A}f^{\dagger}_{j,1,\uparrow}f_{j,2,\uparrow}+f_{j,1,\uparrow}f_{j,2,\uparrow}^{\dagger}+\frac{1}{2}\left(1+\hat{\Pi}_{j,1,\uparrow}\hat{\Pi}_{j,2,\uparrow}\right)$
$\displaystyle=$ $\displaystyle\prod_{j\in
A}e^{-\mathrm{i}\frac{\pi}{2}(\Sigma_{j}^{z}+\eta_{j}^{z})},$ (63)

where
$\hat{\Pi}_{j,\alpha,\uparrow}=1-2f_{j,\alpha,\uparrow}^{\dagger}f_{j,\alpha,\uparrow}$
is the fermion parity in the forward branch of copy $\alpha$ at site $j$.
Since $\mathcal{\widetilde{C}}_{\ell,A}$ only operates on the forward
branches, it naturally breaks the time-reversal symmetry of spin-$1/2$
fermions. Specifically, the unitary operator
$\widetilde{\mathcal{C}}_{\ell,A}$ rotates both $U(1)$ order parameters in the
region $A$ by an angle $\pi/2$. We’ll see below that this is equivalent to
inserting a pair of half vortices at the edges of region $A$ leading to the
boundary state illustrated in Fig. 5(d).

###  IV.4 Phases of the effective model

The effective Hamiltonian $H_{\text{eff}}$ consists of two identical and
decoupled spin-1 Hamiltonians $H_{\text{eff},\kappa}$, each acts on the
$\kappa=\eta,\Sigma$ sectors. The ground states and phase transitions of
$H_{\text{eff},\kappa}$ have been extensively investigated numerically Chen
_et al._ (2003) and admit a simple long-wavelength description Schulz (1986).

In the physically relevant parameter regimes, $\Gamma_{b},\Gamma_{s},J_{b}\geq
0$, the ground state phase diagram of $H_{\text{eff},\kappa}$ contains three
phases, as illustrated schematically in Fig. 6(a). For vanishing measurement
rates, $H_{\text{eff},\kappa}$ is a $O(3)$ ferromagnet, which admits a
fluctuation-free broken symmetry state. This is a singular point in the phase
diagram corresponding to the volume-law state established for purely unitary
dynamics of free fermions. For any non-zero measurement rate, the $O(3)$
symmetry is broken down to
$U(1)\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\mathbb{T}})$ leading to a
critical phase with algebraic long-range order. Increasing the measurement
rates $\Gamma_{s}$ and $\Gamma_{b}$ beyond a critical threshold leads to two
possible gapped phases, depending on the ratio between the two measurement
processes. The site measurements $\Gamma_{s}$ contribute to the $\Delta$-term
(also known as single-ion anisotropy), leading to a trivial gapped phase when
they are dominant. The bond measurements $\Gamma_{b}$, on the other hand,
leads to the Haldane gapped phase, which is an SPT phase of the spin-1 model.

The above phases and phase transitions can be captured within a long
wavelength theory. First, note that the imaginary time evolution with the
effective Hamiltonian has a coarse-grained description in terms of the 2d XY
model $S_{XY}={K\over 2}\int dxd\tau(\nabla\theta)^{2}$ with
$\theta=\theta+2\pi$. We can gain even more insight, however, by framing the
long-wavelength theory of the XY model in the form of the one-dimensional
sine-Gordon Hamiltonian

$H={1\over 2}\int dx\left[K(\nabla{\hat{\theta}})^{2}+{1\over
K}(\nabla\hat{\phi})^{2}\right]-g\int dx\cos(2\hat{\phi}).$ (64)

Here, $\hat{\theta}$ is related to the $U(1)$ phase in the $XY$ model (though
it is not compact). $\nabla\hat{\phi}/\pi$ is the long wavelength fluctuation
of the conserved charge conjugate to $\hat{\theta}$, so that
$[\hat{\theta}(x),\hat{\phi}(x^{\prime})]=\mathrm{i}\pi\Theta(x-x^{\prime})$.
The coupling $g$ is related to the vortex fugacity implicit in the XY model.
This model is invariant to a shift of $\hat{\phi}\to\hat{\phi}+\pi$, which
corresponds to a translation by a lattice constant, and to
$\hat{\phi}\to-\hat{\phi}$ related to the $\mathbb{Z}_{2}$ “particle-hole”
symmetry.

In the critical phase of the spin-1 Hamiltonian, the renormalized Luttinger
parameter $K>2$ and the coupling $g$ is irrelevant in the long wavelength
limit. As the microscopic parameters are varied, the system eventually
undergoes a Kosterlitz-Thouless transition into a gapped phase at a critical
universal value $K_{c}=2$. The long wavelength theory distinguishes the two
gapped phases through the sign of the coupling $g$. If $g>0$, the dual field
$\hat{\phi}$ is locked to $\phi=0$ giving rise to a trivial gapped phase. If
$g<0$, then $\hat{\phi}$ is locked to $\phi=\pi/2$, which corresponds to the
SPT (Haldane) phase. Note that the two phases are related through a shift by
half a lattice constant (equivalently, one Majorana site).

We now turn to discuss how the different phases of the effective model are
probed by the boundary operators. As noted in the previous section, the action
of these operators on the boundary state imposes the boundary conditions
depicted in Fig. 5. Hence, they correspond to boundary condition changing
operators in the long wavelength theory. The logarithm of the boundary overlap
(61) is directly related to the excess free energy of the effective 2d XY
model with the modified boundary conditions relative to the free energy with
the uniform boundary conditions set by the reference state.

Consider first the parity string operators $\hat{\Pi}_{s,A}$ and
$\hat{\Pi}_{b,A}$, which enter the calculation of the parity variances. These
operators affect a $\pi$-phase flip on region $A$, which creates a pair of
vortices on the boundary of the 2d XY model. In the long wave length limit,
the vortex insertion is affected by the operators
$e^{\pm\mathrm{i}\hat{\phi}}$. To compute the site parity string
$\hat{\Pi}_{s,A}$, we must choose the symmetric combination
$\cos\hat{\phi}(x)$ at each end because the site parity string preserves the
particle-hole symmetry on every site. Thus, in the long wavelength limit we
can write $\hat{\Pi}_{s,A}\sim\cos\phi(x_{\ell})\cos\phi(x_{r})$, where
$x_{\ell},x_{r}$ are the two ends of the region $A$. The bond parity string is
obtained from the site parity string through translation by half a physical
site (one Majorana site), implemented by a shift
$\hat{\phi}\to\hat{\phi}+\pi/2$ in the long wavelength theory. Thus,
$\hat{\Pi}_{b,A}\sim\sin\phi(x_{\ell})\sin\phi(x_{r})$. This is consistent
with the fact that, on each end of the region $A$, the bond parity string
carries a residual parity operator of half a physical site, which is anti-
symmetric with respect to the particle-hole symmetry. The swap string, which
enters the calculation of the entanglement entropy, affects a $\pi/2$ rotation
of the $U(1)$ phase, which creates a pair of half-vortices on the boundary of
the 2d XY model. In the long wavelength theory, this is achieved by the
operators $\mathcal{\widetilde{C}}_{\ell,A}\sim
e^{-\mathrm{i}\phi(x_{\ell})/2}e^{\mathrm{i}\phi(x_{r})/2}$.

Having established the long wavelength form of the operators associated with
the different boundary observables, we can determine the behavior of the
boundary matrix element in the different phases. In the critical phase, the
distinction between the site and bond parity strings is not important. Both
decay as a power law, dictated by the scaling dimensions of boundary vortex
insertions $\Pi_{a,A}^{(2)}\sim|x_{\ell}-x_{r}|^{-K/4}$ Giamarchi (2003). The
behavior of the entanglement entropy on the other hand is dictated by a pair
of half-vortex insertions at the boundary, each having half the scaling
dimension of a vortex insertion. Thus, we expect $e^{-S_{A}^{(2)}}\sim\lvert
x_{\ell}-x_{r}\rvert^{-K/16}$. So, while the decay exponents vary continuously
in the the long wavelength theory, the theory predicts a universal ratio of
$4$ between the decay exponent of the parity variance and that of
$\exp(-S^{(2)}_{A})$ in the critical phase. Furthermore, the KT transition is
expected to occur at a universal value of the stiffness $K_{c}=2$, which
implies a critical exponent $\alpha^{c}_{\Pi}=1/2$ for the decay of the parity
variance and $\alpha^{c}_{S}=1/8$ for $\exp(-S_{A}^{(2)})$.

In the low energy fixed points corresponding to gapped ground states,
$\hat{\phi}(x)$ is nonfluctuating; it is locked to $\phi=0$ in the trivial
phase and to $\phi=\pi/2$ in the SPT phase. Hence, we can immediately deduce
the action of parity and swap strings on the ground states. The swap does not
vanish in either phase, thus giving $\exp(-S^{(2)}_{A})\to\text{const}$ at
long distances in both the trivial and SPT state. The two phases are
distinguished by the action of the parity strings. The site parity string,
which involves $\cos\hat{\phi}$, acts as a constant on the trivial ground
state and vanishes on the SPT ground state at the fixed point. Conversely, the
bond parity string acts as a constant on the SPT ground state and vanishes on
the trivial ground state. Therefore, in the trivial phase, we expect
$\Pi_{s,A}^{(2)}\to\text{const}$ and
$\Pi_{b,A}^{(2)}\to\exp(-|x_{\ell}-x_{r}|/\xi)$. In the SPT phase, this is
inverted, $\Pi_{b,A}^{(2)}\to\text{const}$, whereas
$\Pi_{s,A}^{(2)}\to\exp(-|x_{\ell}-x_{r}|/\xi)$. In the following section, we
examine the predictions of the long wave-length theory for the various phases
by numerically simulating the quantum circuit.

###  IV.5 Numerical simulation in Gaussian fermionic circuits

Figure 6: Entanglement entropy and subsystem parity variances in three phases
of Gaussian fermionic circuits. (a) Schematic phase diagram in the space of
bond measurement versus site measurement probabilities ($p_{b}$ v.s. $p_{s}$).
Yellow markers represent numerically extracted critical points along marked
cuts. (b-d) Results for representative points of the three phases
$R_{1}:(0.2,0.2)$, $R_{2}:(0.6,0.8)$, $R_{3}:(0.8,0.4)$ shown on the phase
diagram. (b) Entanglement entropy $e^{-S_{A}}$ shows power-law decay
(logarithmic entanglement scaling) with subsystem size $L_{A}$ in the critical
phase ($R_{1}$, red diamonds) and a constant (area law) in the trivial and SPT
phases. (c) Subsystem site parity variance $\Pi_{s,A}$ shows a power-law decay
with $L_{A}$ in the critical phase (red diamonds), faster than power-law in
the SPT phase (green squares) and is nondecaying in the trivial phase (blue
triangles). (d) Subsystem bond-parity variance $\Pi_{b,A}$ shows a power-law
decay in the critical phase (red diamonds), exponential decay in the trivial
phase and is nondecaying in the SPT phase. The numerical results are obtained
with system size $L=160$ and averaged over $400$ random circuit realizations
and measurement outcomes.

The fermions circuit defined in Section IV.1 can be simulated efficiently
because it preserves the Gaussianity of the wave function in each quantum
trajectory Terhal and DiVincenzo (2002). This allows us to test the
predictions of the effective model by direct numerical calculation of the
circuit dynamics. Technically, it is enough to propagate the $O(N^{2})$ two
point functions
$G_{ij}\equiv\langle\Psi|\mathrm{i}\gamma_{i}\gamma_{j}|\Psi\rangle-\mathrm{i}\delta_{ij}$,
which fully determine the Gaussian wave function. Indeed, all quantities of
interest, including the subsystem entanglement entropy $S_{A}$ and parity
variances $\Pi_{s,A}$ and $\Pi_{b,A}$, can be directly read out from $G_{ij}$.
We perform the calculation with system sizes of up to $L=160$ sites using
periodic boundary conditions to facilitate better finite size scaling. We
operate the circuit to depth $3L$ to achieve a steady state.

We explore a two-dimensional phase space defined by the site and bond
measurement probabilities as illustrated in Fig. 6(a). The behavior of the
three observables we extract indicates the establishment of three phases.
Panels (b-d) of the figure demonstrate this for three representative points in
the phase diagram. At the point $(p_{b},p_{s})=(0.2,0.2)$ (red diamond), we
observe the expected behavior of the critical phase with entanglement entropy
scaling as $\log L_{A}$ and power-law decay of the parity variances. The point
$(p_{b},p_{s})=(0.6,0.8)$ (blue triangle) exhibits the behavior we expect in
the trivial phase with entanglement entropy and the site parity variance
saturating to a constant, while the bond parity variance decaying
exponentially. Finally, the point $(p_{b},p_{s})=(0.8,0.4)$ (green square)
shows a constant entanglement entropy and bond parity variance as well as an
exponential decaying site parity variance, as expected in the SPT phase.

Figure 7: Tuning across the critical points on the three cuts shown in Fig.
6(a): cut 1 ($p_{b}=0$), 2 ($p_{s}=0.02$), 3 ($p_{b}+p_{s}=1.35$). (a, b)
Fitted power-law decay exponent of $e^{-S_{A}}$ ($\alpha_{S}$, blue diamonds),
site parity variance ($\alpha_{\Pi,s}$, red triangles), and bond parity
variance ($\alpha_{\Pi,b}$, yellow squares) along cut 1 [panel (a)] and cut 2
[panel (b)]. The horizontal axis represents the tuning parameter along each
cut. The exponents are extracted from simulations with the system size $L=160$
averaged over $400$ random circuit realizations and measurement outcomes. (c)
Ratio between exponents $\alpha_{\Pi,s}/\alpha_{S}$ and
$\alpha_{\Pi,b}/\alpha_{S}$ along cut 1. Results for
$\alpha_{\Pi,s}/\alpha_{S}$ (purple crosses) and $\alpha_{\Pi,b}/\alpha_{S}$
(green “+”). The red dashed line represents the prediction of the two-replica
model for the critical phase: $\alpha_{\Pi}/\alpha_{S}=4$. (d) Parity
variances (blue, left vertical axis) and normalized entanglement entropy (red,
right vertical axis) along cut 3 from $(p_{b},p_{s})=(0.4,0.95)$ to
$(0.95,0.4)$. $\Pi_{s,A}$ (blue dashed line) and $\Pi_{b,A}$ (blue dotted
line) are for $L_{A}=L/2$. The plotted entanglement entropy is taken for
subsystem size $L_{A}=L/2$ and normalized by $\log L_{A}$. The three red lines
correspond to system sizes $L=40,80,160$; the peak gets sharper with
increasing system size and indicates logarithmic entanglement scaling only at
the critical point. The numerical results are averaged over $400$ random
circuit realizations and measurement results.

Having confirmed these basic behaviors, we turn to investigate the critical
phase and the transitions from this phase into the two gapped states. To this
end, we vary the measurement probabilities $p_{s}$ and $p_{b}$ along the two
cuts depicted as cut 1 and cut 2 in Fig. 6(a). Along these cuts, we aim to fit
the subsystem size dependence of $\Pi_{s,A}$, $\Pi_{b,A}$ and $\exp(-S_{A})$
to a power law of $L_{A}$. More precisely, since we anticipate that the
critical phase has a conformal field theory description, we can account for
the finite system size and periodic boundary conditions by fitting the decay
of these objects to a power law of the conformal coordinate $z_{A}=L\sin(\pi
L_{A}/L)$. Figure 7(b,c) shows the variation of the fitted exponents
$\alpha_{\Pi,s}$, $\alpha_{\Pi,b}$, and $\alpha_{S}$ along the cuts. We see
that the exponents $\alpha_{\Pi,s}$ and $\alpha_{\Pi,b}$ coincide and decrease
together on increasing the measurement rate toward the transition, as expected
in the critical phase. The exponents then start to deviate, presumably near
the critical point, in order to match the opposite behavior of the two
parities in the gapped phases. For example, in the trivial phase, we expect
$\alpha_{\Pi,s}\to 0$, which encodes $\Pi_{s,A}\to\text{const}$, and
$\alpha_{\Pi,b}\to\infty$ in order to mimic $\Pi_{b,A}\sim\exp(-L_{A}/\xi)$.

In the range of tuning parameter, where the two parity exponents coincide, we
find that the ratio between these exponents and the one controlling the decay
of $\exp(-S_{A})$ is pinned to a constant value. The ratio, as seen in Fig.
7(c), is pinned to $\alpha_{\Pi}/\alpha_{S}\approx 3$, which deviates from the
ratio of $4$ predicted by the low energy effective theory. This discrepancy
could be related to the difference between the von Neumann entropy computed
here and the second conditional Rényi entropy calculated in the effective
model.

We now focus on the phase transition from the critical phase to the trivial
phase along cut 1 in Fig. 6(a). To establish the universality class, we
analyze the behavior of the correlation length and compare it to the hallmark
exponential divergence associated with a KT transition.

We extract the correlation length $\xi$ from the behavior of the entanglement
entropy in the area-law phase near the transition since it gives much cleaner
data than the parity variances. On distances small compared to the correlation
length, i.e. $L_{A}\ll\xi$, we expect the critical behavior

$\displaystyle S(z_{A},p_{s},L)=b(p_{s})+\alpha_{S}(p_{s})\log z_{A},$ (65)

while for $L_{A}\gg\xi$ we should see a saturation to
$b(p_{s})+\alpha_{S}(p_{s})\log(z_{\xi})$. Here, $z_{\xi}=L\sin(\pi\xi/L)$ is
the conformal coordinate for the correlation length $\xi$.

To accurately obtain $\xi$, we first fit $b(p_{s})$ and $\alpha_{S}(p_{s})$
from the short distance behavior of $S(z_{A},p_{s},L)$. We then use the
interpolating function

$\displaystyle
S(z_{A},p_{s},L)=\frac{\alpha_{S}(p_{s})}{\beta}\log\left(z_{\xi}^{\beta}\tanh\frac{z_{A}^{\beta}}{z_{\xi}^{\beta}}\right)+b(p_{s})$
(66)

to obtain $\xi(p_{s},L)$ for different values of $p_{s}$ and system sizes
$L=40,80,160$. The parameter $\beta$ is added to improve the fit and details
of the procedure are given in Appendix H.2.

The fitted values of the correlation length are presented in Fig. 8 at all
points for which $\xi\leq L/2$. We find that $\xi(p_{s},L)$ fall close to the
same curve for the different values of $L$. The figure shows a fit of the data
for the largest length $L=160$ to the KT form of the correlation length
$\xi(p_{s})=\exp(A/\sqrt{p_{s}-p_{s,c}}+B)$.

The results are consistent with a KT transition within the range of system
sizes reached and also provide an estimate of the critical point
$p_{s,c}=0.17$. We note that the finite size corrections to the critical
stiffness $K_{c}$ in the KT transition exhibit a slow logarithmic decay with
system size. Therefore, it is not surprising that the critical exponents
governing the decay of the various boundary correlations are larger than the
expected thermodynamic value.

Having analyzed the critical phase and the phase transition into the area-law
phases, we turn to the phase transition between the two area-law states along
cut 3 in Fig. 6(a). The variation of the half system entanglement entropy
normalized by $\log(L/2)$ is shown by the red curves in Fig. 6(d) for the
three system sizes $L=40,80,160$. The normalized value is seen to be size
independent at a single point and the curve appears to sharpen with increasing
system size. This gives strong evidence for single critical point along the
cut as expected from the $O(2)$ long-wavelength description (64). Thus we do
not find an indication of a generic critical phase intervening between the two
area-law phases as suggested based on a relation to the completely packed loop
model Sang  _et al._ (2021).

The details of the transition, however, also do not precisely match the
predictions of the effective theory (64). On the critical line separating the
two area-law phases, the effective theory predicts a stiffness constant $K<2$,
which is continuously decreasing with increasing measurement rate along this
line. Accordingly the coefficient of the logarithmic half-system entropy is
expected to be $\alpha_{S}<1/8$, which is violated by the observed value of
$\alpha_{S}\approx 0.43$.

The variation of the half-system site and bond parity variances across the
transition behave broadly as expected, but again show a mismatch with the
effective theory upon more detailed comparison. These variations are shown by
the blue curves in the figure for a single system size of $L=160$. The two
string “order parameters” are seen to switch roles between the two phases, as
expected. However, the sharp suppression of both order parameters at the
critical point is unexpected. The effective model implies a slow decay of the
parity variance with an exponent $\alpha_{\Pi}<1/2$. This should produce a
rather broad finite size coexistence region of the two “order parameters” Berg
_et al._ (2008), which is not observed in the numerical results.

###  IV.6 Beyond the two-replica theory

In our analysis of the Gaussian fermion circuits we focused on the two-replica
theory without taking the replica limit $n\to 1$. Nonetheless, several key
predictions of this model are supported by the numerical results obtained from
direct simulation of fermion circuits. These includes the establishment of a
critical phase characterized by logarithmic entanglement entropy with variable
coefficient $\alpha_{S}$, the connection between $\alpha_{S}$ and the power-
law decays of the site and bond parity variances, and the KT transition
separating the critical phase from the area-law states. At the same time, the
numerical results suggest that the phase transition between the two area-law
states is not captured correctly within the two-replica model.

To gain insights into the success and the limitations of this model, we
consider an approximate scheme to evaluate $O_{2}$ in Eq. (13) in terms of a
statistical mechanics model with quenched disorder. The second moment $O_{2}$
is again given by

$O_{2}=\overline{\sum_{m}p_{m}(U)\left(\frac{\text{tr}\left({\hat{O}\,\tilde{\rho}}_{m}(U)\right)}{\text{tr}\left(\tilde{\rho}_{m}(U)\right)}\right)^{2}}.$
(67)

Here $U$ runs over the different realizations of unitary gates, and we
explicitly note the dependence of the trajectory state $\tilde{\rho}_{m}(U)$
on $U$.

Now, if $p_{m}(U)\equiv\text{tr}\left(\tilde{\rho}_{m}(U)\right)$ has a narrow
distribution over $U$, we can approximate it by its average
$p_{m}=\overline{p_{m}(U)}$ and perform the average over unitaries in Eq. (67)
before averaging over measurement outcomes to obtain

$O_{2}\approx\sum_{m}p_{m}\frac{\mbox{$\langle\langle{\mathcal{I}}|$}\mathcal{O}^{(2)}\mbox{$|\rho^{(2)}_{m}\rangle\rangle$}}{\langle\langle{\mathcal{I}}\mbox{$|\rho^{(2)}_{m}\rangle\rangle$}}.$
(68)

In this expression the measurements enter as quenched disorder. Typical
trajectories $m$ consist of imaginary time evolution with an effective
Hamiltonian generated by the averaged unitary dynamics, interrupted by
isolated measurements located in random space-time positions. The isolated
measurements reduce the $O(3)$ symmetry of the Heisenberg ferromagnet down to
$O(2)$. We note that a similar scheme, which avoids the replica limit at the
expense of introducing disorder, was considered in Ref. Zhou and Nahum (2019)
for purely unitary circuits.

Such a model with quenched disorder has a true symmetry $O(2)$. In the case of
low measurement rates, the stiffness is large enough such that the only
relevant effect of the disorder is the reduction of the symmetry from $O(3)$
to $O(2)$; the space-time randomness is expected to be irrelevant in the
ensuing $O(2)$ critical phase. In contrast the disorder does become relevant
beyond the KT transition. Thus, taking a proper replica limit may be important
for describing the transition between the two area-law states, which explains
the failure of the two-replica theory to describe this critical behavior
correctly.

It remains an interesting challenge to evaluate the second moment $O_{2}$ in
the proper replica limit $n\to 1$. One perturbative approach to the problem
suggested in Ref. Fu and Kane (2012) in the context of disordered fermion
models, is through an expansion of the $O(n)$ nonlinear sigma model in
$\epsilon=2-n$. Alternatively, Sang et al. Sang  _et al._ (2021) proposed to
utilize a mapping of certain Majorana fermion circuits to the completely
packed loop model with crossings Nahum  _et al._ (2013). Although the more
generic model we considered does not map exactly to the loop model, it is
possible that the two models are in the same universality class. The numerical
results at the available system sizes do not appear consistent with this
possibility. These results indicate a direct transition between the two area-
law states, while in the loop model the two area-law states are generically
separated by a finite region of the critical phase.

Figure 8: Correlation length $\xi$ in the trivial area-law phase along cut 1
indicated on Fig. 6(a). The green line represents the fitted correlation
length according to the scaling $\log\xi(p_{s})=A/\sqrt{p_{s}-p_{s,c}}+B$ in
the KT transition with $A=0.58$, $p_{s,c}=0.17$, and $B=0.34$. The results for
$L=40,80,160$ are denoted by blue, red, and yellow diamonds, respectively. The
data used to extract $\xi(p_{s})$ is presented in Appendix H.2.

##  V Discussion

The quantum circuits we have considered in this paper generate the time
evolution of many-body wave-functions or density matrices through a series of
unitary gates and local measurements. Such a time evolution gives rise to a
special kind of ensemble, whose members are the quantum states corresponding
to all possible histories of measurement outcomes. Our goal has been to
characterize the phases that can emerge as steady states in this new kind of
ensemble. A key observation of this work is that, unlike in thermal
equilibrium or in quantum ground states, the symmetry that governs the
classification of phases in quantum circuits is not simply the symmetry of the
physical interactions imposed by the circuit elements. Rather, many
statistical properties of the ensemble in steady states are dictated by an
enlarged symmetry group, which combines the physical circuit symmetry with
intrinsic dynamical symmetries of the problem.

The enlarged symmetry emerges because distinctive properties of the circuit
steady state can be seen only in the fluctuations of physical observables over
the different measurement histories, while a simple average over all possible
measurement outcomes looks trivial. In particular, the variance of physical
observables, which is the minimal nontrivial fluctuation, is encoded in the
time evolution of two identical copies of the density matrix. A symmetry to
permute quantum states between the two copies naturally arises and combines
with the physical circuit symmetries.

This symmetry enlargement has an illuminating analogy in Anderson’s pioneering
work on spin glasses (see also Ref. Nahum  _et al._ (2021)). Edwards and
Anderson famously pointed out that spin glass order is characterized by the
random frozen magnetic moment, which can be diagnosed by the variance of the
spin moment over the ensemble of random samples $q_{EA}=\overline{\langle
s_{i}\rangle^{2}}$ Edwards and Anderson (1975). In close analogy with the
diagnostics, we considered for the quantum circuits, the EA order parameter
can be expressed as a standard average (correlation function) in an effective
probability distribution constructed from two identical copies of the system.
This doubling naturally gives rise to a permutation symmetry between replicas,
which is reminiscent of the permutation symmetries in our system.

The analogy with the spin glass also extends to information theoretic aspect
of the problem. We have shown that broken symmetry phases of the circuit are
states that encode and protect information on symmetry breaking in the initial
state. Similarly, the onset of an EA order parameter in a spin glass signals
the emergence of an error correcting code, allowing to encode information in
random broken symmetry spin configurations Sourlas (1989); Nishimori (2001).

There is however a fundamental difference between the quantum measurement
ensemble and the spin glass problem. Due to the quantum nature of the circuit,
fluctuations in the measurement ensemble are obtained from the time evolution
of doubled density operators rather than from (static) probability
distributions. This leads to a richer symmetry, since the forward (ket) and
backward (bra) evolution of the density operators are associated with
independent permutation symmetries. Spontaneous breaking of these inherently
quantum symmetries signals a capacity to encode quantum information Choi  _et
al._ (2019); Gullans and Huse (2019); Bao  _et al._ (2020); Fan  _et al._
(2020); Ippoliti  _et al._ (2020).

To facilitate the classification of phases, we formulated a mapping between
the steady states of the circuit ensemble and the ground states of an
effective Hamiltonian, which inherits the enlarged dynamical symmetry. This
leads to establishment of phases that would not have been possible in presence
of the physical circuit symmetry alone. Here is a summary of the most striking
consequences. (i) A topological area-law phase can form in a one-dimensional
model with only $\mathbb{Z}_{2}$ physical circuit symmetry, owing to
protection from an enlarged symmetry $\mathbb{Z}_{2}\times\mathcal{S}_{2}$
that combines the circuit symmetry and the dynamical permutation symmetry
between trajectories. We argued that this state admits an efficient
experimental detection scheme using the Fisher information associated with
perturbations applied at the edge. Demonstrating the existence of this state
in a numerical simulation of a concrete model, or alternatively showing a
fundamental obstruction to realize it, remains a challenge for future work.
(ii) Several distinct volume-law phases, characterized by different broken
symmetry patterns, arise as ground states of a circuit with $\mathbb{Z}_{2}$
symmetry. We demonstrate the existence of three of these volume-law entangled
states in numerical simulations of a one-dimensional model. An interesting
conclusion from this result is that the quantum measurements allow to
stabilize orders, which could not have existed in a finite entropy equilibrium
state. We argue, as a corollary, that measurements can even facilitate the
establishment of topological order and protected edge modes in a state with
volume-law entanglement. (iii) A Gaussian fermionic circuit with only
$\mathbb{Z}_{2}$ fermion parity conservation exhibits a critical phase
facilitated by a $U(1)$ dynamical symmetry of two copies of the density
matrix. We predict a Kosterlitz-Thouless transition to area-law states above a
critical rate of measurement. We show numerical evidence supporting this
transition.

It is worth noting that the predictions of the effective model rely on
analysis of quantum dynamics with two replicas without taking the formal
replica limit $n\to 1$. The rather detailed agreement with exact numerical
results may suggest that two replicas play a special role in this problem
because the minimal diagnostics of the ensemble are fundamentally second
moments. It is worth exploring alternative schemes to average over the
ensemble, which avoid introducing auxiliary replicas, possibly at the expense
of retaining space-time disorder Zhou and Nahum (2019). We also remark that
our classification of phases in this paper was based on symmetry alone. It
would be important to understand if additional constraints on the dynamics,
such as complete positivity of quantum channels, could inhibit the
establishment of certain states.

Arguably the most pressing challenge for the theory right now is to connect
with the rapidly advancing experimental work on quantum circuits. An important
first step would be to extend the theoretical framework to more realistic
models, which include, at a minimum, decoherence and possibly other
dissipative processes in addition to unitary gates and measurements. Such an
extension would also be interesting from the theoretical perspective because
decoherence changes the symmetry of the problem. Any weak coupling to the
environment explicitly breaks the independent permutation symmetries among
replicated quantum states propagating forward or backward, and it only
respects simultaneous permutations of forward and backward states. We have
noted that the separation of the two permutation symmetries is a signature of
quantumness that is absent in classical settings, such as the replica theory
of a spin glass. Nonetheless, the universal behavior and encoding power may
not immediately collapse to that of a classical circuit. There is another
quantum symmetry associated with the exchange between forward and backward
propagating states, namely the hermiticity of the density matrix, which is
neither removed nor trivialized immediately by the addition of decoherence. A
nontrivial hermiticity symmetry, in principle distinguishes the quantum
dynamics of density matrices from classical probability distributions even in
the presence of decoherence. It would be interesting to explore under what
conditions the hermiticity symmetry can be spontaneously broken and to
understand the implications of such order on the capacity for encoding quantum
information.

We close by recalling a dialogue that had not occurred in Paris in the 1920s.
Hopefully it sums up our message clearly:

FITZGERALD: Quantum circuits are different from spin glasses.

HEMINGWAY: Yes, they have more symmetry.

###### Acknowledgements.

We thank David Huse, Michael Gullans and Zack Weinstein for useful
discussions. We acknowledge support from the NSF QLCI program through grant
number OMA-2016245. SC acknowledges support from the Miller Institute for
Basic Research in Science. EA is supported in part by the Mike Gyorgy Chair in
Physics at UC Berkeley.

## References

  * Landsman  _et al._ (2019) K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Nature 567, 61 (2019).
  * Arute  _et al._ (2019) F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, _et al._ , Nature 574, 505 (2019).
  * Zhong  _et al._ (2020) H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, _et al._ , Science 370, 1460 (2020).
  * Skinner  _et al._ (2018) B. Skinner, J. Ruhman, and A. Nahum, arXiv preprint arXiv:1808.05953 (2018).
  * Li  _et al._ (2018) Y. Li, X. Chen, and M. P. Fisher, Physical Review B 98, 205136 (2018).
  * Chan  _et al._ (2019) A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Phys. Rev. B 99, 224307 (2019).
  * Choi  _et al._ (2019) S. Choi, Y. Bao, X.-L. Qi, and E. Altman, arXiv preprint arXiv:1903.05124 (2019).
  * Gullans and Huse (2019) M. J. Gullans and D. A. Huse, arXiv preprint arXiv:1905.05195 (2019).
  * Fan  _et al._ (2020) R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, arXiv preprint arXiv:2002.12385 (2020).
  * Ippoliti  _et al._ (2020) M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, arXiv preprint arXiv:2004.09560 (2020).
  * Anderson (1972) P. W. Anderson, Science 177, 393 (1972).
  * Bao  _et al._ (2020) Y. Bao, S. Choi, and E. Altman, Physical Review B 101, 104301 (2020).
  * Jian  _et al._ (2020a) C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. Ludwig, Physical Review B 101, 104302 (2020a).
  * Nahum  _et al._ (2021) A. Nahum, S. Roy, B. Skinner, and J. Ruhman, PRX Quantum 2, 010352 (2021).
  * Hayden  _et al._ (2016) P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, Journal of High Energy Physics 2016, 9 (2016).
  * Nahum  _et al._ (2018) A. Nahum, S. Vijay, and J. Haah, Physical Review X 8, 021014 (2018).
  * Vasseur  _et al._ (2018) R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. Ludwig, arXiv preprint arXiv:1807.07082 (2018).
  * Zhou and Nahum (2019) T. Zhou and A. Nahum, Physical Review B 99, 174205 (2019).
  * Sang and Hsieh (2020) S. Sang and T. H. Hsieh, arXiv preprint arXiv:2004.09509 (2020).
  * Pollmann  _et al._ (2010) F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Physical review b 81, 064439 (2010).
  * Chen  _et al._ (2011) X. Chen, Z.-C. Gu, and X.-G. Wen, Physical review b 83, 035107 (2011).
  * Edwards and Anderson (1975) S. F. Edwards and P. W. Anderson, Journal of Physics F: Metal Physics 5, 965 (1975).
  * Terhal and DiVincenzo (2002) B. M. Terhal and D. P. DiVincenzo, Physical Review A 65, 032325 (2002).
  * Cao  _et al._ (2018) X. Cao, A. Tilloy, and A. De Luca, arXiv preprint arXiv:1804.04638 (2018).
  * Fidkowski  _et al._ (2020) L. Fidkowski, J. Haah, and M. B. Hastings, arXiv preprint arXiv:2008.10611 (2020).
  * Alberton  _et al._ (2020) O. Alberton, M. Buchhold, and S. Diehl, arXiv preprint arXiv:2005.09722 (2020).
  * Sang  _et al._ (2021) S. Sang, Y. Li, T. Zhou, X. Chen, T. H. Hsieh, and M. P. Fisher, PRX Quantum 2, 030313 (2021).
  * Chen  _et al._ (2020) X. Chen, Y. Li, M. Fisher, and A. Lucas, arXiv preprint arXiv:2004.09577 (2020).
  * Liu  _et al._ (2020) C. Liu, P. Zhang, and X. Chen, measurement (2020).
  * Jian  _et al._ (2020b) C.-M. Jian, B. Bauer, A. Keselman, and A. W. Ludwig, arXiv preprint arXiv:2012.04666 (2020b).
  * Nahum and Skinner (2020) A. Nahum and B. Skinner, Physical Review Research 2, 023288 (2020).
  * Ryu  _et al._ (2007) S. Ryu, C. Mudry, H. Obuse, and A. Furusaki, Phys. Rev. Lett. 99, 116601 (2007).
  * Fu and Kane (2012) L. Fu and C. L. Kane, Phys. Rev. Lett. 109, 246605 (2012).
  * Lang and Büchler (2020) N. Lang and H. P. Büchler, Physical Review B 102, 094204 (2020).
  * Li  _et al._ (2019) Y. Li, X. Chen, and M. Fisher, arXiv preprint arXiv:1901.08092 (2019).
  * Chen  _et al._ (2014) X. Chen, Y.-M. Lu, and A. Vishwanath, Nature communications 5, 1 (2014).
  * Lavasani  _et al._ (2021) A. Lavasani, Y. Alavirad, and M. Barkeshli, Nature Physics , 1 (2021).
  * Aaronson and Gottesman (2004) S. Aaronson and D. Gottesman, Physical Review A 70, 052328 (2004).
  * Chandran and Laumann (2015) A. Chandran and C. Laumann, Physical Review B 92, 024301 (2015).
  * Webb (2015) Z. Webb, arXiv preprint arXiv:1510.02769 (2015).
  * Nahum  _et al._ (2017) A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Physical Review X 7, 031016 (2017).
  * von Keyserlingk  _et al._ (2018) C. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. Sondhi, Physical Review X 8, 021013 (2018).
  * Nahum  _et al._ (2013) A. Nahum, P. Serna, A. Somoza, and M. Ortuno, Physical Review B 87, 184204 (2013).
  * Yang and Zhang (1990) C. N. Yang and S. Zhang, Modern Physics Letters B 4, 759 (1990).
  * Chen  _et al._ (2003) W. Chen, K. Hida, and B. Sanctuary, Physical Review B 67, 104401 (2003).
  * Schulz (1986) H. Schulz, Physical Review B 34, 6372 (1986).
  * Giamarchi (2003) T. Giamarchi,  _Quantum physics in one dimension_ , Vol. 121 (Clarendon press, 2003).
  * Berg  _et al._ (2008) E. Berg, E. G. Dalla Torre, T. Giamarchi, and E. Altman, Physical Review B 77, 245119 (2008).
  * Sourlas (1989) N. Sourlas, Nature 339, 693 (1989).
  * Nishimori (2001) H. Nishimori,  _Statistical physics of spin glasses and information processing: an introduction_ , 111 (Clarendon Press, 2001).

##  Appendix A Local symmetries

###  A.1 Qubit circuit

Here we show that, for the circuits considered in Section II and III, the
average hybrid dynamics in $\mathcal{H}^{(n)}$ exhibits local symmetries
generated by $\mathcal{X}_{j}=\prod_{a=1}^{n}(X_{a}X_{\bar{a}})_{j}$,
$\mathcal{Y}_{j}=\prod_{a=1}^{n}(Y_{a}Y_{\bar{a}})_{j}$, and
$\mathcal{Z}_{j}=\prod_{a=1}^{n}(Z_{a}Z_{\bar{a}})_{j}$. Our proof relies on
the facts: (i) unitary gates $U=e^{-\mathrm{i}\theta O_{P}}$ are generated by
Pauli string operators $O_{P}$, e.g.
$O_{P}=\tau_{i}^{\alpha}\tau_{j}^{\beta}$; (ii) the random couplings $\theta$
are drawn from a symmetric distribution to zero; (iii) measurement operators
are Pauli strings. We note that, as a special case discussed in Section II,
the average hybrid dynamics in $\mathcal{H}^{(2)}$ generated by uniformly
random single-qubit phase rotations $e^{-\mathrm{i}\theta_{i}Z_{i}}$ exhibits
these local symmetries. In the following, we demonstrate the symmetries for
the average dynamics in $\mathcal{H}^{(n)}$ generated by unitary gates and
projective measurements separately.

Without lose of generality, we consider a unitary gate
$U_{i}=e^{-\mathrm{i}\theta^{\alpha\beta}_{i}\tau^{\alpha}_{i}\tau^{\beta}_{i+1}}$.
$U_{i}$ acts on $\mathcal{H}^{(n)}$ after averaging over the random coupling
as

$\displaystyle\mathcal{U}_{i}^{(n)}=\overline{e^{-\mathrm{i}\theta_{i}^{\alpha\beta}H_{i}^{(n)}}}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}\overline{\left(\theta_{i}^{\alpha\beta}\right)^{2k}}\left(H_{i}^{(n)}\right)^{2k},$
(69)

where
$H_{i}^{(n)}=\sum_{a=1}^{n}\tau_{i,a}^{\alpha}\tau_{i+1,a}^{\beta}-\tau_{i,\bar{a}}^{\alpha}\tau_{i+1,\bar{a}}^{\beta}$.
Here, we use the fact that $\theta_{i}^{\alpha\beta}$ is drawn from a
symmetric distribution to zero, and odd powers of $H_{i}^{(n)}$ in the
expansion of $\mathcal{U}_{i}^{(n)}$ vanish. A special case of
$\mathcal{U}_{i}^{(n)}$ with $n=2$ and Gaussian random
$\theta_{i}^{\alpha\beta}$ is given in Eq. (33). To demonstrate the local
symmetries in the averaged dynamics $\mathcal{U}_{i}^{(n)}$, it suffices to
show

$\displaystyle\left[\left(H_{i}^{(n)}\right)^{2},\mathcal{R}_{j}\right]=0$
(70)

for any $i,j$ and
$\mathcal{R}_{j}\in\\{\mathcal{X}_{j},\mathcal{Y}_{j},\mathcal{Z}_{j}\\}$.
Each nontrivial term in $(H_{i}^{(n)})^{2}$ contains a product of Pauli
matrices $\tau^{\alpha}_{i,a}\tau^{\alpha}_{i,a^{\prime}}$ on site $i$ and
$\tau^{\beta}_{i+1,a}\tau^{\beta}_{i+1,a^{\prime}}$ on site $i+1$, which
commutes with $\mathcal{R}_{j}$. Hence, $(H_{i}^{(n)})^{2}$ commutes with
$\mathcal{R}_{j}$, and $\mathcal{R}_{j}$ are local symmetries of
$\mathcal{U}_{i}^{(n)}$. The generalization to other unitary gates generated
by Pauli string operators is straightforward.

The probabilistic application of a projective measurement averaged over
measurement results acts on $\mathcal{H}^{(n)}$ as

$\displaystyle\mathcal{M}_{\nu}^{(n)}=(1-\Gamma_{\nu}\delta
t)\mathds{1}^{\otimes 2n}+\Gamma_{\nu}\delta
t\sum_{m_{\nu}=\pm}P_{m_{\nu}}^{\otimes 2n}.$ (71)

A special case of $\mathcal{M}_{\nu}^{(2)}$ is given in Eq. (32). Here,
$P_{m_{\nu}}=(1\pm M_{\nu})/2$ are the projection operators associated with
the measurements on the Pauli string $M_{\nu}$. We note that measurement
results for any Pauli string can only be $m_{\nu}=\pm 1$. To demonstrate the
local symmetries of $\mathcal{M}_{\nu}^{(n)}$, it suffices to show

$\displaystyle[P_{+}^{\otimes 2n}+P_{-}^{\otimes 2n},\mathcal{R}_{j}]=0$ (72)

for any $j$ and
$\mathcal{R}_{j}\in\\{\mathcal{X}_{j},\mathcal{Y}_{j},\mathcal{Z}_{j}\\}$. To
verify that Eq. (72) is satisfied, we expand $P_{+}^{\otimes
2n}+P_{-}^{\otimes 2n}$ according to the power of operator $M_{\nu}$. Each
nonvanishing term contains a product of even number of operator $M_{\nu}$
(acting on different branches) and therefore commute with
$\mathcal{X}_{j},\mathcal{Y}_{j}$, and $\mathcal{Z}_{j}$.

###  A.2 Fermionic circuit

For the Gaussian fermionic circuits considered in Section IV, the averaged
dynamics preserves local symmetries generated by
$\mathcal{R}_{\ell}=\prod_{\alpha=1}^{n}\mathrm{i}\gamma_{\ell,\alpha\uparrow}\gamma_{\ell,\alpha\downarrow}$
at every Majorana site $\ell$. The local symmetry $\mathcal{R}_{\ell}$
commutes with any Majorana bilinear defined on the same site, i.e.
$[\mathcal{R}_{\ell},\gamma_{\ell^{\prime},\alpha\sigma}\gamma_{\ell^{\prime},\beta\sigma^{\prime}}]=0$.
Using this property, we can verify, for any replica index $n$, both the
averaged unitary gates $\mathcal{U}_{s/b,j}^{(n)}$ and the averaged
measurements $\mathcal{M}_{s/b,j}^{(n)}$ commute with $\mathcal{R}_{\ell}$.

##  Appendix B Effective quantum Hamiltonian for $\mathbb{Z}_{2}$ symmetric
quantum circuits

To demonstrate our method, we present an explicit formula of $H_{\text{eff}}$
for a simple paradigmatic $1$D circuit model with $\mathbb{Z}_{2}$ symmetry
that conserves the total parity of Pauli-Z operator. Our model follows the
structure depicted in Fig. 1. The inter-layer hybrid dynamics involves the
following unitary gate and measurements: (i) random
$\theta_{ij}^{xx}X_{i}X_{j}$ interactions with Gaussian random
$\theta_{ij}^{xx}$ of zero mean, (ii) random $\theta_{ij}^{zz}Z_{i}Z_{j}$
interactions with Gaussian random $\theta_{ij}^{xx}$ of zero mean, (iii)
measurements of $Z_{j}$ with probability $\Gamma^{z}_{j}\delta t$, and (iv)
measurements of $X_{i}X_{j}$ with probability $\Gamma^{xx}_{ij}\delta t$. All
unitary couplings and measurement projections in this model respect the
$\mathbb{Z}_{2}$-parity symmetry generated by $\hat{\pi}=\prod_{j}Z_{j}$. As
explained in Appendix A, $\mathcal{X}_{j}$ are local conserved quantities, and
$s_{j}$ variables denote local integrals of motion.

Using the framework developed in Section II, we can show the effective
Hamiltonian $H_{\text{eff},+}$ for the dynamics of two copies of density
matrices in the even sector of local parities is of the form

$\displaystyle H_{\text{eff},+}=H_{JXX}+H_{JZZ}+H_{PZ}+H_{PXX},$ (73)

where $H_{JXX}$ and $H_{JZZ}$ arise from unitary gates with random couplings
$\theta^{xx}_{ij}$ and $\theta^{zz}_{ij}$, and $H_{PZ}$ and $H_{PXX}$ arise
from $Z_{j}$ and $X_{i}X_{j}$ measurements. The Hamiltonian terms can be
written in terms of spin-1 operators

$\displaystyle H_{JZZ}=\sum_{i<j}$ $\displaystyle
J^{zz}_{ij}\left(M_{i,+}M_{j,-}+M_{i,-}M_{j,+}\right),$ (74) $\displaystyle
H_{JXX}=\sum_{i<j}$
$\displaystyle-J^{xx}_{ij}\Big{(}S_{i}^{x}S_{j}^{x}+S_{i}^{y}e^{\mathrm{i}\pi
S_{i}^{x}}e^{\mathrm{i}\pi S_{j}^{x}}S_{j}^{y}$ $\displaystyle\quad\quad\quad-
L_{i}L_{j}\Big{)},$ (75) $\displaystyle H_{PXX}=\sum_{i<j}$
$\displaystyle-\frac{\Gamma^{xx}_{ij}}{4}\Big{(}S_{i}^{x}S_{j}^{x}+S_{i}^{y}e^{\mathrm{i}\pi
S_{i}^{x}}e^{\mathrm{i}\pi S_{j}^{x}}S_{j}^{y}$
$\displaystyle\quad\quad\quad+L_{i}L_{j}\Big{)},$ (76) $\displaystyle
H_{PZ}=\sum_{j}$ $\displaystyle\Gamma_{j}^{z}\left(S_{j}^{z}\right)^{2},$ (77)

where the coupling $J^{zz}_{ij}\delta t=8\overline{(\theta^{zz}_{ij})^{2}}$,
$J^{xx}_{ij}\delta t=2\overline{(\theta^{xx}_{ij})^{2}}$, the operator
$L_{i}=[(S_{i}^{+})^{2}+(S_{i}^{-})^{2}]/2$ with $S_{i}^{\pm}$ being the spin
ladder operators, and $M_{j,\pm}=\mbox{$|m=\pm
1\rangle\rangle$}\mbox{$\langle\langle m=\pm 1|$}_{j}$ are projectors to the
state $|m=\pm 1\rangle\rangle$ at site $j$.

The effective Hamiltonian exhibits a $D_{4}\times\mathbb{Z}_{2}^{\mathbb{H}}$
symmetry, where
$D_{4}=(\mathbb{Z}_{2}^{\Pi_{L}}\times\mathbb{Z}_{2}^{\Pi_{1}})\rtimes\mathcal{S}_{2}$.
In the spin-1 Hilbert space, the symmetry generators are given by
$\hat{\Pi}_{L}=\prod_{j}\mathrm{diag}(-1,1,-1)_{j}$,
$\hat{\Pi}_{1}=\prod_{j}\mathrm{diag}(1,1,-1)_{j}$, and
$\mathcal{C}_{\ell}=\prod_{j}-\exp(\mathrm{i}\pi\sum_{j}S_{j}^{x})$. The
Hermitian conjugate $\mathbb{H}$ manifests as the complex conjugation
$\mathcal{K}$. This $D_{4}\times\mathbb{Z}_{2}^{\mathbb{H}}$ symmetry agrees
with the analysis in Section III.1.

##  Appendix C Exchange symmetry and hermiticity symmetry in the $n=2$ qubit
model

In this section, we derive the exchange operation
$\mathcal{C}_{\ell}\mathcal{C}_{r}$ and Hermitian conjugation $\mathbb{H}$ in
the representation of $\mathcal{G}^{(2)}_{\text{eff}}$ on the even parity
sector of the $n=2$ qubit model. In particular, we show the exchange operation
acts trivially as an identity matrix in the even parity sector and therefore
cannot be broken. This proof only relies on the presence of local conserved
quantities $\mathcal{X}_{j}$, $\mathcal{Y}_{j}$, and $\mathcal{Z}_{j}=1$ in
the qubit model and does not require the physical circuit symmetry.

To start with, the exchange operation $\mathcal{C}_{\ell}\mathcal{C}_{r}$ is
given by a product of single-site exchange operations
$\mathcal{C}_{\ell,i}\mathcal{C}_{r,i}$. The single-site left and right
permutation can be written in terms of Pauli operators as

$\displaystyle\mathcal{C}_{\ell,i}$
$\displaystyle=(IIII+XIXI+YIYI+ZIZI)_{i}/2,$ (78)
$\displaystyle\mathcal{C}_{r,i}$ $\displaystyle=(IIII+IXIX+IYIY+IZIZ)_{i}/2.$
(79)

In the even parity sector, one can show the product
$\mathcal{C}_{\ell,i}\mathcal{C}_{r,i}$ acts as an identity matrix regardless
of the physical symmetry, and so is $\mathcal{C}_{\ell}\mathcal{C}_{r}$.
Hence, $\mathcal{S}_{2}^{X}$ is in the kernel of the representation and cannot
be broken.

The hermiticity symmetry can be written as a combination of transpose
$\mathcal{T}$ and complex conjugation $\mathcal{K}$. The transpose operation
$\mathcal{T}=\mathcal{T}_{1}\mathcal{T}_{2}$ is a product of transpose for the
first and second copy. $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ can be further
written respectively as a product of single-site transpose operations
$\mathcal{T}_{1,i}$ and $\mathcal{T}_{2,i}$ which take the form

$\displaystyle\mathcal{T}_{1,i}$ $\displaystyle=(IIII+XXII+YYII+ZZII)_{i}/2,$
(80) $\displaystyle\mathcal{T}_{2,i}$
$\displaystyle=(IIII+IIXX+IIYY+IIZZ)_{i}/2.$ (81)

We can verify the product $\mathcal{T}_{1,i}\mathcal{T}_{2,i}$ is identity in
the even parity sector, and so is
$\mathcal{T}=\prod_{i}\mathcal{T}_{1,i}\mathcal{T}_{2,i}$. Hence, the
Hermitian conjugate $\mathbb{H}=\mathcal{K}$ in the even parity sector.

##  Appendix D Higher replicas ($n\geq 3$)

Having shown the existence of a broad array of symmetry enriched phases in the
case of two replica copies $n=2$, we here discuss the correspondence of each
phase in the higher replica models ($n\geq 3$). We verify that two area-law
and three volume-law phases found in the two-chain model in Section III.3 have
natural generalizations in the higher replica models with identical physical
signatures. This provides evidence that these phases can be extrapolated to
the replica limit $n\to 1$, predicting distinct phases of quantum circuits.

Among the rest five (two area-law and three volume-law) phases which haven’t
been realized in a concrete model, we show that only the composite volume-law
phase has a higher replica generalization. The coexistence of EA order and
parity variance is generically not allowed in the higher replica models; four
coexistence phases (two area-law and two volume-law phases) do not have
natural higher replica generalizations. However, we note this does not exclude
the possibility of the realizing these phases in the replica limit. Whether
there is fundamental physical obstruction to realizing these phases in a
concrete quantum circuit remains an open question.

According to the analysis in Section II.2, the dynamics of $n$ replicas
exhibits an enlarged symmetry $\mathcal{G}^{(n)}=(B_{n}\times
B_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$, where the hyperoctahedral group
$B_{n}=\mathbb{Z}_{2}\wr\mathcal{S}_{n}=\mathbb{Z}_{2}^{\otimes
n}\rtimes\mathcal{S}_{n}$ is the symmetry group of an $n$-dimensional
hypercube. The dynamics also conserves local parities $\mathcal{X}_{j}$,
$\mathcal{Y}_{j}$, and $\mathcal{Z}_{j}$ (see Appendix A).

Similar to the case of two replicas $n=2$, the effective symmetry that
determines the ground state phases is reduced due to the conservation of local
parities. We are interested in the ground state in the same sector of local
parities as the reference state, i.e. $\mathcal{X}_{j}=\mathcal{Z}_{j}=+1$ and
$\mathcal{Y}_{j}=(-1)^{n}$. In this sector, the nonvanishing expectation
values of local parities necessitate the breaking of single-branch symmetries.
Hence, we are left with an effective global symmetry
$\mathcal{G}^{(n)}_{\text{eff}}$ generated by the elements of the left and
right permutation in $\mathcal{S}_{n}$, $\mathbb{H}$ together with
$\hat{\Pi}_{1}\equiv\prod_{j}(Z_{1}Z_{\bar{1}})_{j}$ and
$\hat{\Pi}_{12}\equiv\prod_{j}(Z_{1}Z_{2})_{j}$. Distinct phases of quantum
circuits are characterized by the residual symmetry which is a subgroup of
$\mathcal{G}^{(n)}_{\text{eff}}$.

We note that, in contrast to the case of two replicas ($n=2$), the generators
of the $\mathcal{S}_{n}^{X}$ exchange symmetry, i.e.
$\mathcal{C}_{\ell,\xi}\mathcal{C}_{r,\xi}$ for $\xi\in\mathcal{S}_{n}$, act
nontrivially in the even parity sector; $\mathcal{S}_{n}^{X}$ can in principle
be broken for $n\geq 3$.

_Area-law phases_.— The area-law phase is again characterized by the unbroken
$\mathcal{S}_{n}$ symmetry or the composite symmetry of the cyclic permutation
$\mathcal{C}_{\ell}$ and $g_{\mathcal{I}}$ present in the reference state
$|\mathcal{I}\rangle\rangle$. The presence of the hermiticity symmetry
necessitates the dynamical symmetry
$(\mathcal{S}_{n}\times\mathcal{S}_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$.

Starting from the fully symmetric area-law phase with residual symmetry
$\mathcal{G}^{(n)}_{\text{eff}}$, we can condense the charge
$\mathcal{Q}_{1}\equiv X_{1}X_{\bar{1}}$ together with all operators related
to it by the
$(\mathcal{S}_{n}\times\mathcal{S}_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$
symmetry. This leads to a broken symmetry area-law phase with the residual
symmetry
$(\mathcal{S}_{n}\times\mathcal{S}_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$. It
is easily verified that both the symmetric and the broken symmetry state have
exactly the same physical signatures as their corresponding state in the $n=2$
model.

The coexistence of EA order and parity variance is generically not allowed in
the model of higher replicas. To obtain a nonvanishing EA order, one needs to
condense $\mathcal{Q}_{ab}\equiv X_{a}X_{b}$ for $a,b=1,2,\cdots,n$. Such a
condensate breaks not only
$\hat{\Pi}_{a}\equiv\prod_{j}(Z_{a}Z_{\bar{a}})_{j}$ but also
$\hat{\Pi}_{ab}\equiv\prod_{j}(Z_{a}Z_{b})_{j}$, indicating an exponentially
decaying parity variance. One can also verify that condensing
$X_{a}Z_{b}Y_{c}$ and its related charges by
$(\mathcal{S}_{n}\times\mathcal{S}_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$ in
the composite area-law phase breaks both $\hat{\Pi}_{a}$ and $\hat{\Pi}_{ab}$,
giving a vanishing parity variance. Hence, two area-law phases with coexisting
orders do not have a higher replica generlation.

The symmetric SPT phase, generalized to $n\geq 3$, is found starting from the
broken symmetry area-law phase. We can restore the full symmetry by condensing
domain walls of $\hat{\Pi}_{12}$ and $\hat{\Pi}_{1}$ as well as all domain
walls related to them by the symmetry
$(\mathcal{S}_{n}\times\mathcal{S}_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$.
The symmetric SPT phase is obtained by condensing $\hat{\Pi}_{ab}$ and
$\hat{\Pi}_{a}$ domain walls bound to charges of the $\mathcal{S}_{n}$
symmetries. The dual picture of the same phase is a condensate of the
$\mathcal{S}_{n}$ domain walls bound to the charge $\mathcal{Q}_{1}$ and
symmetry related operators.

_Volume-law phases_.— We now turn to the volume-law regime. One correspondence
of the symmetric volume-law phase in the higher replicas is the phase with
fully broken left and right $\mathcal{S}_{n}$, while retaining the symmetry
$\hat{\Pi}_{a}\equiv\prod_{j}(Z_{a}Z_{\bar{a}})_{j}$,
$\hat{\Pi}_{ab}\equiv\prod_{j}(Z_{a}Z_{b})_{j}$, and
$\mathbb{Z}_{2}^{\mathbb{H}}$.777We note that the volume-law phase in the
replica limit is believed to be a condensate of elementary domain walls Jian
_et al._ (2020a), which breaks the full $\mathcal{S}_{n}$ symmetry. Hence, in
higher replica models, we only consider the volume-law phase with fully broken
$\mathcal{S}_{n}$. One can check that the physical quantities in this phase
behave the same as in its $n=2$ correspondence.

Starting from the symmetric volume-law phase, we can condense
$\mathcal{Q}_{a}=X_{a}X_{\bar{a}}$, leading to a featureless phase preserving
symmetry $\hat{\Pi}_{a}$, while breaking $\hat{\Pi}_{ab}=Z_{a}Z_{b}$. We can
further condense $\mathcal{Q}_{ab}$ in the featureless phase to break the
symmetry $\hat{\Pi}_{a}$, leading to the broken symmetry volume-law phase with
$\mathbb{Z}_{2}^{\mathbb{H}}$ symmetry. Two coexistence volume-law phases do
not have natural generalizations in the higher replicas as condensing either
$\mathcal{Q}_{ab}$ or $\mathcal{Q}_{a\bar{b}}\equiv X_{a}X_{\bar{b}}$ for all
$a,b=1,2,\ldots,n$ breaks $\hat{\Pi}_{a^{\prime}b^{\prime}}$ leading to
vanishing parity variance.

A correspondence of the composite volume-law phase can be shown to exist in
the model of even number of replicas ($n$ being even). This phase features a
residual symmetry group
$(\mathbb{Z}_{n}\times\mathbb{Z}_{n})\rtimes\mathbb{Z}_{2}^{\mathbb{H}}$. Two
$\mathbb{Z}_{n}$ subgroups are generated respectively by the composite
symmetry $\mathcal{C}_{\ell}\hat{\Pi}_{L}$ and
$\mathcal{C}_{\ell}\mathcal{C}_{r}$, where
$\hat{\Pi}_{L}\equiv\prod_{j}(\prod_{a=1}^{n}Z_{a})_{j}$. Starting from the
symmetric area-law phase, we can obtain this phase by condensing
$(X_{1}Y_{\bar{1}}Z_{2})_{j}$ symmetrized under the residual symmetry. We note
that the symmetrized operator is nonvanishing for the even replicas.

##  Appendix E Fermionic states and operators in the duplicated Hilbert space

Here we detail the correspondence between second quantized fermionic operators
$f_{j}$, acting in the Hilbert space of a single copy of quantum circuit, and
the $f_{j,\alpha,\sigma}$ acting in the duplicated Hilbert space of the
effective Hamiltonian.

We start from the Fock states
$\mbox{$|\vec{n}\rangle$}=\prod_{j}(f^{\dagger}_{j})^{n_{j}}\mbox{$|0\rangle$}$
in the Hilbert space of quantum circuit. The space of double density matrices
is accordingly spanned by the states:
$\mbox{$|\vec{n}_{1}\rangle$}\mbox{$\langle\vec{m}_{1}|$}\otimes\mbox{$|\vec{n}_{2}\rangle$}\mbox{$\langle\vec{m}_{2}|$}$.
To define a proper fermionic Fock space for the duplicated system, we make the
correspondence:

$\displaystyle\mbox{$|\vec{n}_{1}\rangle$}\mbox{$\langle\vec{m}_{1}|$}\otimes\mbox{$|\vec{n}_{2}\rangle$}\mbox{$\langle\vec{m}_{2}|$}\leftrightarrow$
(82)
$\displaystyle\prod_{i}(f^{\dagger}_{i,1,\uparrow})^{n_{1i}}\prod_{j}(f^{\dagger}_{j,1,\downarrow})^{m_{1j}}\prod_{k}(f^{\dagger}_{k,2,\uparrow})^{n_{2k}}\prod_{l}(f^{\dagger}_{l,2,\downarrow})^{m_{2l}}\mbox{$|\text{vac}\rangle\rangle$}_{f}$
$\displaystyle\equiv\mbox{$|\vec{n}_{1},\vec{m}_{1},\vec{n}_{2},\vec{m}_{2}\rangle\rangle$}.$

Here, the second quantized operators $f^{\dagger}_{i,\alpha,\sigma}$ create a
fermion in the $\sigma$ branch (ket or bra) of copy $\alpha$.

The above correspondence between states gives a simple representation of the
boundary state $\langle\langle\mathcal{I}|$, which implements the doubled
trace operation:

$\mbox{$\langle\langle\mathcal{I}|$}=\sum_{\vec{n}_{1},\vec{n}_{2}}\mbox{$\langle\langle\vec{n}_{1},\vec{n}_{1},\vec{n}_{2},\vec{n}_{2}|$}.$
(83)

Indeed we can check that
$\mbox{$\langle\langle\mathcal{I}|\rho^{(2)}\rangle\rangle$}=\mathrm{tr}\rho^{(2)}$.

The above correspondence between states implies a correspondence between
operators

$\displaystyle
f^{\dagger}_{j}\otimes\mathds{1}\otimes\mathds{1}\otimes\mathds{1}\leftrightarrow
f^{\dagger}_{j,1,\uparrow},$ (84a) $\displaystyle\mathds{1}\otimes
f_{j}\otimes\mathds{1}\otimes\mathds{1}\leftrightarrow
f^{\dagger}_{j,1,\downarrow}(-1)^{N_{1\uparrow}},$ (84b)
$\displaystyle\mathds{1}\otimes\mathds{1}\otimes
f^{\dagger}_{j}\otimes\mathds{1}\leftrightarrow
f^{\dagger}_{j,2,\uparrow}(-1)^{N_{1\uparrow}+N_{1\downarrow}},$ (84c)
$\displaystyle\mathds{1}\otimes\mathds{1}\otimes\mathds{1}\otimes
f_{j}\leftrightarrow
f^{\dagger}_{j,2,\downarrow}(-1)^{N_{1\uparrow}+N_{1\downarrow}+N_{2\uparrow}}.$
(84d)

The sign factors on the right are needed to ensure that operators on the two
sides of the correspondence have identical action on the respective states.
$N_{\alpha\sigma}$ denote the total number in copy $\alpha$ and “branch”
$\sigma$ (ket $\uparrow$ or bra $\downarrow$). We make the following remarks.
First, with this correspondence, all the fermion parity conserving local
operators, acting on a single circuit state (in any copy or branch) remain
local and the sign factors cancel out of them. The extension to any number of
copies is obvious. Second, due to the additional sign factors, the hermiticity
symmetry transformation appears to be nonlocal in the duplicated Hilbert
space, namely
$\mathbb{H}f^{\dagger}_{j,\alpha,\uparrow}\mathbb{H}^{-1}=(-1)^{N_{\alpha,\uparrow}}f^{\dagger}_{j,\alpha,\downarrow}$.
However, $\mathbb{H}$ is local for any bosonic operator made of fermionic
bilinears and therefore becomes a local symmetry in the effective Hilbert
space.

Having defined the operators in the duplicated Hilbert space
$\mathcal{H}^{(2)}$, we can write the effective Hamiltonian in a Fock basis
using any ordering convention of the fermions. In particular, the form of the
effective spin-1 Hamiltonian in Section IV.3 implicitly assumes that it
operates between Fock states with a site-local ordering convention:

$\displaystyle\bigotimes_{j}\mbox{$|n_{j,1,\uparrow},n_{j,1,\downarrow},n_{j,2,\uparrow},n_{j,2,\downarrow}\rangle\rangle$}\equiv$
(85)
$\displaystyle\prod_{j}\Big{[}(f^{\dagger}_{j,1,\uparrow})^{n_{j,1,\uparrow}}(f^{\dagger}_{j,1,\downarrow})^{n_{j,1,\downarrow}}(f^{\dagger}_{j,2,\uparrow})^{n_{j,2,\uparrow}}(f^{\dagger}_{j,2,\downarrow})^{n_{j,2,\downarrow}}\Big{]}\mbox{$|\text{vac}\rangle\rangle$}_{f}.$

The same Hamiltonian will take a different form if written in terms of Fock
states with copy-local ordering
$|\vec{n}_{1\uparrow},\vec{n}_{1\downarrow},\vec{n}_{2\uparrow},\vec{n}_{2\downarrow}\rangle\rangle$.
In particular, the $U(1)$ symmetries will become nonlocal and not be
explicitly apparent because they are generated by conservation of the
$c^{\dagger}_{\ell,\sigma}$, which are not local to one copy.

The price we pay for working with the site-local Fock states
$|{\bf{n}}\rangle\rangle$ in which the effective spin Hamiltonian is simple
and the $U(1)$ symmetry apparent is that we also need to rewrite the boundary
state $\langle\langle\mathcal{I}|$ with the same Fock states. This in order to
compute its overlap with the state generated by the effective imaginary time
evolution (together with any boundary operators we want to measure). We have
seen that $\langle\langle\mathcal{I}|$ has a simple representation in terms of
the copy-local Fock states in Eq. (83). The Fock states in the two ordering
conventions are related to each other by a sign factor which is a function of
the occupation numbers in the state. These factors take a simple form for the
basis states included in the boundary state, giving:

$\displaystyle\mbox{$\langle\langle\mathcal{I}|$}=\sum_{\vec{n}_{1},\vec{n}_{2}}g(N_{1},N_{2})\bigotimes_{j}\mbox{$\langle\langle
n_{j,1},n_{j,1},n_{j,2},n_{j,2}|$},$ (86)

where the factor $g(N_{1},N_{2})=(-1)^{{1\over 2}N_{1}(N_{1}-1)+{1\over
2}N_{2}(N_{2}-1)}$.

##  Appendix F $U(1)$ symmetry breaking in $|\mathcal{I}\rangle\rangle$

In this section, we show the boundary state $|\mathcal{I}\rangle\rangle$
breaks both $U(1)$ symmetries corresponding to the conservation of
$\Sigma^{z}$ and $\eta^{z}$. We demonstrate the long-range order in
$|\mathcal{I}\rangle\rangle$ manifested by nondecaying correlation functions
of order parameters
$\mbox{$\langle\langle\mathcal{I}|$}\eta_{i}^{y}\eta_{j}^{y}\mbox{$|\mathcal{I}\rangle\rangle$}/4^{L}=\mbox{$\langle\langle\mathcal{I}|$}\Sigma_{i}^{y}\Sigma_{j}^{y}\mbox{$|\mathcal{I}\rangle\rangle$}/4^{L}=1/2$.

To evaluate the correlation of order parameters in
$|\mathcal{I}\rangle\rangle$, we write the boundary state
$|\mathcal{I}\rangle\rangle$ [Eq. (83)] in terms of second quantized fermionic
operators $f_{j,\alpha,\sigma}$ acting on the Hilbert space of branch $\sigma$
of copy $\alpha$

$\displaystyle\mbox{$|\mathcal{I}\rangle\rangle$}=$
$\displaystyle\prod_{i=1}^{N}\left[1+f_{i,1,\uparrow}^{\dagger}f_{i,1,\downarrow}^{\dagger}e^{\mathrm{i}\pi\sum_{i^{\prime}=i+1}^{N}n_{1i^{\prime}}}\right]$
$\displaystyle\left[1+f_{i,2,\uparrow}^{\dagger}f_{i,2,\downarrow}^{\dagger}e^{\mathrm{i}\pi\sum_{i^{\prime}=i+1}^{N}n_{2i^{\prime}}}\right]\mbox{$|\text{vac}\rangle\rangle$}_{f}.$
(87)

Here, the state $|\mathcal{I}\rangle\rangle$ has a definite $f$-fermion parity
on every site in the Hilbert space of forward and backward branch of each
copy, i.e.
$\mbox{$\langle\langle\mathcal{I}|$}\hat{\Pi}_{j,\alpha,\uparrow}\hat{\Pi}_{j,\alpha,\downarrow}\mbox{$|\mathcal{I}\rangle\rangle$}=1$,
where
$\hat{\Pi}_{j,\alpha,\sigma}=1-2f_{j,\alpha,\sigma}^{\dagger}f_{j,\alpha,\sigma}$
is the $f$-fermion parity.

In terms of $f$-fermion operators, the order parameters $\Sigma^{\pm}_{j}$ and
$\eta^{\pm}_{j}$ take the form

$\displaystyle\Sigma_{j}^{+}=$
$\displaystyle-\mathrm{i}\mathcal{O}_{j,1\uparrow,1\downarrow}-\mathcal{O}_{j,1\uparrow,2\downarrow}-\mathcal{O}_{j,2\uparrow,1\downarrow}+\mathrm{i}\mathcal{O}_{j,2\uparrow,2\downarrow},$
(88) $\displaystyle\Sigma_{j}^{-}=$
$\displaystyle\mathrm{i}\mathcal{O}_{j,1\uparrow,1\downarrow}-\mathcal{O}_{j,1\uparrow,2\downarrow}-\mathcal{O}_{j,2\uparrow,1\downarrow}-\mathrm{i}\mathcal{O}_{j,2\uparrow,2\downarrow},$
(89) $\displaystyle\eta_{j}^{+}=$
$\displaystyle\mathrm{i}\mathcal{O}_{j,1\uparrow,1\downarrow}-\mathcal{O}_{j,1\uparrow,2\downarrow}+\mathcal{O}_{j,2\uparrow,1\downarrow}+\mathrm{i}\mathcal{O}_{j,2\uparrow,2\downarrow},$
(90) $\displaystyle\eta_{j}^{-}=$
$\displaystyle-\mathrm{i}\mathcal{O}_{j,1\uparrow,1\downarrow}-\mathcal{O}_{j,1\uparrow,2\downarrow}+\mathcal{O}_{j,2\uparrow,1\downarrow}-\mathrm{i}\mathcal{O}_{j,2\uparrow,2\downarrow},$
(91)

where $\mathcal{O}_{j,\alpha\sigma,\beta\sigma^{\prime}}$ is the fermion
bilinear defined as

$\displaystyle\mathcal{O}_{j,\alpha\sigma,\beta\sigma^{\prime}}=\frac{\mathrm{i}}{2}\left(f_{j,\alpha,\sigma}^{\dagger}f_{j,\beta,\sigma^{\prime}}^{\dagger}-f_{j,\beta,\sigma^{\prime}}f_{j,\alpha,\sigma}\right).$
(92)

Operator $\mathcal{O}_{j,\alpha\sigma,\beta\sigma^{\prime}}$ flips the local
$f$-fermion parity in Hilbert space of copy $\alpha$ and $\beta$ on site $j$.
Therefore, the correlation function of
$\mathcal{O}_{j,\alpha\sigma,\beta\sigma^{\prime}}$ in the boundary state
$|\mathcal{I}\rangle\rangle$ is non-vanishing only when $\alpha=\beta$. We can
further show the only nonvanishing two-point correlations are given by

$\displaystyle\frac{1}{4^{L}}\mbox{$\langle\langle\mathcal{I}|$}\mathcal{O}_{i,\alpha\uparrow,\alpha\downarrow}\mathcal{O}_{j,\alpha\uparrow,\alpha\downarrow}\mbox{$|\mathcal{I}\rangle\rangle$}=\frac{1}{4},$
(93)

where $1/4^{L}$ is the normalization of $|\mathcal{I}\rangle\rangle$, i.e.
$\mbox{$\langle\langle\mathcal{I}|\mathcal{I}\rangle\rangle$}=4^{L}$. Using
Eq. (93), we can show the order parameter correlation functions in the
boundary state $|\mathcal{I}\rangle\rangle$ is nondecaying

$\displaystyle\frac{1}{4^{L}}\mbox{$\langle\langle\mathcal{I}|$}\Sigma_{i}^{y}\Sigma_{j}^{y}\mbox{$|\mathcal{I}\rangle\rangle$}=\frac{1}{4^{L}}\mbox{$\langle\langle\mathcal{I}|$}\eta_{i}^{y}\eta_{j}^{y}\mbox{$|\mathcal{I}\rangle\rangle$}=\frac{1}{2}.$
(94)

Hence, the boundary state exhibits long-range orders and breaks both the
$U(1)$ symmetries generated by $\eta^{z}$ and $\Sigma^{z}$. Furthermore, we
note that the expectation value of order parameters in
$|\mathcal{I}\rangle\rangle$ is vanishing, i.e.
$\mbox{$\langle\langle\mathcal{I}|$}\Sigma^{\pm}_{j}\mbox{$|\mathcal{I}\rangle\rangle$}=\mbox{$\langle\langle\mathcal{I}|$}\eta^{\pm}_{j}\mbox{$|\mathcal{I}\rangle\rangle$}=0$.
For this reason, the boundary state $|\mathcal{I}\rangle\rangle$ can be
understood as a superposition of $U(1)$ symmetry breaking states with the
ordering of $U(1)$ phase in different directions.

##  Appendix G Swap operator in fermionic duplicated Hilbert space

The swap operator $\mathcal{C}_{\ell,A}$ is in general nonlocal in the
fermionic duplicated Hilbert space $\mathcal{H}^{(2)}$. In this section, we
show that $\mathcal{C}_{\ell,A}$ bears a simple local description when acting
on the boundary state $|\mathcal{I}\rangle\rangle$. This allows a long
wavelength description for the swap boundary state
$\mbox{$|\mathcal{C}_{\ell,A}\rangle\rangle$}\equiv\mathcal{C}_{\ell,A}\mbox{$|\mathcal{I}\rangle\rangle$}$
in Section IV.4.

To start with, we consider the swap operator in the Hilbert space of two
copies of quantum circuits

$\displaystyle\mathcal{C}_{A}=\sum_{\vec{n}_{1},\vec{n}_{2}}\mbox{$|\vec{n}_{2A}\vec{n}_{1B}\rangle$}\mbox{$\langle\vec{n}_{1A}\vec{n}_{1B}|$}\otimes\mbox{$|\vec{n}_{1A}\vec{n}_{2B}\rangle$}\mbox{$\langle\vec{n}_{2A}\vec{n}_{2B}|$},$
(95)

where $\vec{n}_{\alpha}=(\vec{n}_{\alpha,A},\vec{n}_{\alpha,B})$ is the
occupation of fermionic modes in the subsystem $A$ and its complement $B$. The
purity of subsystem can be obtained by applying the permutation operator to
the double density matrix either from left and then taking the trace, i.e.
$\mathrm{tr}\rho^{2}_{A}=\mathrm{tr}(\mathcal{C}_{A}\,\rho\otimes\rho)$.

Using the correspondence established in Appendix E, we can formulate the
purity as an overlap between the swap boundary state
$\langle\langle\mathcal{C}_{\ell,A}|$ and $|\rho\rangle\rangle$, where
$|\mathcal{C}_{\ell,A}\rangle\rangle$ takes the form

$\displaystyle\mbox{$|\mathcal{C}_{\ell,A}\rangle\rangle$}=\sum_{\vec{n}_{1},\vec{n}_{2}}\mbox{$|\vec{n}_{2A}\vec{n}_{1B},\vec{n}_{1A}\vec{n}_{1B},\vec{n}_{1A}\vec{n}_{2B},\vec{n}_{2A}\vec{n}_{2B}\rangle\rangle$}.$
(96)

Alternatively, we can write $|\mathcal{C}_{\ell,A}\rangle\rangle$ in the site-
local ordering convention as

$|\mathcal{C}_{\ell,A}\rangle\rangle$
$\displaystyle=\sum_{\vec{n}_{1},\vec{n}_{2}}g(N_{1},N_{2})$ (97)
$\displaystyle\bigotimes_{j\in
A}\mbox{$|n_{j,2},n_{j,1},n_{j,1},n_{j,2}\rangle\rangle$}\bigotimes_{j\in
B}\mbox{$|n_{j,1},n_{j,1},n_{j,2},n_{j,2}\rangle\rangle$}.$

We note that the boundary state $|\mathcal{C}_{\ell,A}\rangle\rangle$ [Eq.
(97)] and $|\mathcal{I}\rangle\rangle$ [Eq. (86)] in the site-local convention
only differ by a local exchange of the occupation in the forward branches of
copy 1 and 2 in subsystem $A$. This allows one to define an effective swap
operator $\mathcal{\widetilde{C}}_{\ell,A}=\prod_{j\in
A}\mathcal{\widetilde{C}}_{\ell,j}$ such that
$\mathcal{\widetilde{C}}_{\ell,A}\mbox{$|\mathcal{I}\rangle\rangle$}=\mbox{$|\mathcal{C}_{\ell,A}\rangle\rangle$}$,
where $\mathcal{\widetilde{C}}_{\ell,j}$ is the single-site swap operator
satisfying

$\displaystyle\mathcal{\widetilde{C}}_{\ell,j}\mbox{$|n_{j,1},n_{j,1},n_{j,2},n_{j,2}\rangle\rangle$}=\mbox{$|n_{j,2},n_{j,1},n_{j,1},n_{j,2}\rangle\rangle$}.$
(98)

The single-site swap operator $\mathcal{\widetilde{C}}_{\ell,j}$ can be
written in terms of $f$-fermionic operators and further in terms of $U(1)$
symmetry generators

$\displaystyle\widetilde{\mathcal{C}}_{\ell,j}=$ $\displaystyle
f^{\dagger}_{j,1,\uparrow}f_{j,2,\uparrow}+f_{j,1,\uparrow}f_{j,2,\uparrow}^{\dagger}+\frac{1}{2}\left(1+\hat{\Pi}_{j,1,\uparrow}\hat{\Pi}_{j,2,\uparrow}\right)$
$\displaystyle=$ $\displaystyle
e^{-\mathrm{i}\frac{\pi}{2}\left(\Sigma_{j}^{z}+\eta^{z}_{j}\right)},$ (99)

where
$\hat{\Pi}_{j,\alpha,\uparrow}=1-2f_{j,\alpha,\uparrow}^{\dagger}f_{j,\alpha,\uparrow}$
is the fermion parity in the forward branch of copy $\alpha$ at site $j$.
Therefore, the swap operator $\mathcal{C}_{\ell,A}$, when acting on the
boundary state $|\mathcal{I}\rangle\rangle$, is equivalent to
$\mathcal{\widetilde{C}}_{\ell,A}$, which rotates both $U(1)$ order parameters
($\eta^{+}_{j}$ and $\Sigma^{+}_{j}$) counterclockwise by an angle $\pi/2$ in
region $A$. In the long-wavelength description developed in Section IV.4,
$\mathcal{\widetilde{C}}_{\ell,A}$ inserts a pair of half vortices at the
edges of region $A$ on the boundary of (1+1)d XY model. Here, we note again
that $\mathcal{\widetilde{C}}_{\ell,A}$ is not the swap operator when acting
on a general state and $\mathcal{\widetilde{C}}_{\ell,A}^{2}\neq\mathds{1}$.
We also note that the swap operator breaks the time-reversal symmetry for the
spin-$1/2$ fermions as it only operates on the forward branches
($\sigma=\uparrow$).

##  Appendix H Details of the numerical simulation in Gaussian fermionic
circuits

###  H.1 Power-law decay exponents of correlations in the critical phase

Figure 9: Phase transition in Gaussian fermionic circuits along Cut 1 in Fig.
6. Entanglement entropy $\exp(-S_{A})$ [panel (a)], parity variances
$\Pi_{s,A}$ [panel (b)] and $\Pi_{b,A}$ [panel (c)] as function of the
conformal coordinate $z_{A}$ for various measurement probabilities $p_{s}$.
Data points for the same $p_{s}$ are marked in the same color in different
panels. $p_{s}$ ranges from $0.02$ to $0.25$ and increases from the bottom to
the top in panel (a). The numerical results are obtained with system size
$L=160$, averaged over $400$ realizations, and plotted in a log-log scale.

In this section, we present the numerical results of $e^{-S_{A}}$, $\Pi_{s,A}$
and $\Pi_{b,A}$ in Gaussian fermionic circuits along cut 1. We use the
numerical results to extract power-law exponents presented in Fig. 7(a-c).

Figure 9 presents $e^{-S_{A}}$ [panel(a)], $\Pi_{s,A}$ [panel(b)] and
$\Pi_{b,A}$ [panel(c)] as a function of the conformal coordinate $z_{A}$ for
various measurement probabilities $p_{s}$. We show that these quantities
exhibit power-law decay in the critical phase (when $p_{s}<p_{s,c}$ on cut 1).
The scaling deviates from power laws when increasing the measurement
probability $p_{s}$. Numerical results show an area-law $S_{A}$, a constant
$\Pi_{s,A}$, and an exponentially decaying $\Pi_{b,A}$ in the trivial area-law
phase (when $p_{s}>p_{s,c}$ on cut 1), which verifies our theoretical
predictions. Our theoretical understanding also explains the non-monotonic
behavior in the bond parity variance $\Pi_{b,A}$ as a function of $p_{s}$. In
the critical phase, $\Pi_{b,A}$ increases with $p_{s}$ owing to the decreasing
Luttinger parameter $K$ towards the critical point. On the other hand, in the
trivial area-law phase, $\Pi_{b,A}$ decreases with $p_{s}$ due to the
decreasing correlation length. We note that the values of $\Pi_{b,A}$ are
small along cut 1 owing to the vanishing bond measurement probability, i.e.
$p_{b}=0$.

###  H.2 Correlation length in the area-law phase of Gaussian fermionic
circuit

Figure 10: Subsystem entanglement entropy as a function of conformal
coordinate $z_{A}=L\cos(\pi L_{A}/L)$ for various different measurement
probabilities $p_{s}$ along cut 1. $p_{s}$ ranges from $0.05$ to $0.35$ and
increases in the data points from the top to the bottom. (Inset) Logarithmic
coefficient $\alpha(p_{s})$ of the entanglement entropy as a function of
$p_{s}$.

We present the details on numerically extracting correlation length in the
area-law phase of the Gaussian fermionic circuit. Specifically, we show the
entanglement entropy data used to extract the correlation length and provide
the procedure to determine the logarithmic coefficient $\alpha_{S}(p_{s})$ and
constant $b(p_{s})$ in Eq. (65).

To start with, we show in Fig. 10 the entanglement entropy along cut 1
exhibits distinct behaviors across the phase transition. When measurement rate
$p_{s}$ is small, the entanglement exhibits a critical scaling as a function
of the subsystem size $L_{A}$ (linear in $\log z_{A}$). Increasing $p_{s}$
along cut 1, the circuit enters an area-law phase, which is characterized by
the saturation of entanglement entropy as a function of $L_{A}$.

Here, we focus on the area-law phase and use the entanglement data to extract
the finite correlation length $\xi$. The entanglement exhibits distinct
scaling for different subsystem sizes $L_{A}$. When $L_{A}\ll\xi$, the
entanglement exhibits critical scaling as Eq. (65). Increasing the subsystem
size $L_{A}$, the entanglement scaling changes to Eq. (66). We extract $\xi$
in two steps: (1) extract $\alpha(p_{s})$ and $b(p_{s})$ using the
entanglement data for small subsystem sizes, i.e. $L_{A}\ll\xi$; (2) extract
$\xi$ using the entanglement data for larger subsystem sizes by fitting to Eq.
(66).

To extract $\alpha(p_{s})$ and $b(p_{s})$, we use the data for various $L_{A}$
and $L$ in the following way: (1) For a given $p_{s}$, we plot
$S(z_{A},p_{s},L)$ as a function of $\log z_{A}$ for various $L$; (2) We
choose at least seven consecutive data points of $z_{A}$ and perform the
linear regression according to Eq. (65). We optimize the $r^{2}$ of the linear
regression over all possible choices to obtain the best $\alpha(p_{s})$ and
$b(p_{s})$. The optimized $\alpha(p_{s})$ is shown in the inset of Fig. 10.

Using the optimized $\alpha(p_{s})$ and $b(p_{s})$, we perform the least
square fitting for $S(z_{A},p_{s},L)$ as a function of $z_{A}$ according to
Eq. (66) for every $p_{s}$ and $L$ to extract $\xi(p_{s},L)$. The result is
presented in Fig. 8.