# General terms of all almost balancing numbers of first and second type

Ahmet Tekcan and Alper Erdem  Ahmet Tekcan and Alper Erdem– Bursa Uludag
University, Faculty of Science, Department of Mathematics, Bursa, Turkiye
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we determined the general terms of all almost balancing numbers
of first and second type in terms of balancing numbers and conversely we
determined the general terms of all balancing numbers in terms of all almost
balancing numbers of first and second type. We also set a correspondence
between all almost balancing numbers of first and second type and Pell
numbers.

###### keywords:

Balancing number, almost balancing number, Pell number.

11B37, 11B39, 11D25. 31 2023 1 https://doi.org/10.46298/cm.10318

##  1  Introduction

Behera and Panda ([B-Pa]) defined that a positive integer $n$ is called a
balancing number if the Diophantine equation

$1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r)$ (1)

holds for some positive integer $r$ which is called balancer corresponding to
$n$. If $n$ is a balancing number with balancer $r$, then from (1) they get

$r=\frac{-2n-1+\sqrt{8n^{2}+1}}{2}.$ (2)

So from (2), they noted that $n$ is a balancing number if and only if
$8n^{2}+1$ is a perfect square. Though the definition of balancing numbers
suggests that no balancing number should be less than $2$. But from (2), they
noted that $8(0)^{2}+1=1$ and $8(1)^{2}+1=3^{2}$ are perfect squares. So they
accepted that $0$ and $1$ to be balancing numbers. Let $B_{n}$ denote the
$n^{\text{th}}$ balancing number. Then $B_{0}=0,B_{1}=1,B_{2}=6$ and
$B_{n+1}=6B_{n}-B_{n-1}$ for $n\geq 2$. Later Panda and Ray ([pa-ray]) defined
that a positive integer $n$ is called a cobalancing number if the Diophantine
equation

$1+2+\cdots+n=(n+1)+(n+2)+\cdots+(n+r)$ (3)

holds for some positive integer $r$ which is called cobalancer corresponding
to $n$. If $n$ is a cobalancing number with cobalancer $r$, then from (3) they
get

$r=\frac{-2n-1+\sqrt{8n^{2}+8n+1}}{2}.$ (4)

So from (4), they noted that $n$ is a cobalancing number if and only if
$8n^{2}+8n+1$ is a perfect square. Since $8(0)^{2}+8(0)+1=1$ is a perfect
square, they accepted $0$ to be a cobalancing number just like Behera and
Panda accepted $0$ and $1$ to be balancing numbers. Let $b_{n}$ denote the
$n^{\text{th}}$ cobalancing number. Then $b_{0}=b_{1}=0,b_{2}=2$ and
$b_{n+1}=6b_{n}-b_{n-1}+2$ for $n\geq 2$. It is clear from (1) and (3) that
every balancing number is a cobalancer and every cobalancing number is a
balancer, that is, $B_{n}=r_{n+1}$ and $R_{n}=b_{n}$ for $n\geq 1$, where
$R_{n}$ is the $n^{\text{th}}$ the balancer and $r_{n}$ is the $n^{\text{th}}$
cobalancer. Since $R_{n}=b_{n}$, we get from (1) that

$b_{n}=\frac{-2B_{n}-1+\sqrt{8B_{n}^{2}+1}}{2}\text{ \ and\ \
}B_{n}=\frac{b_{n}+1+\sqrt{8b_{n}^{2}+8b_{n}+1}}{2}.$ (5)

Thus from (5), $B_{n}$ is a balancing number if and only if $8B_{n}^{2}+1$ is
a perfect square and $b_{n}$ is a cobalancing number if and only if
$8b_{n}^{2}+8b_{n}+1$ is a perfect square. So

$C_{n}=\sqrt{8B_{n}^{2}+1}\ \ \text{and}\ \ c_{n}=\sqrt{8b_{n}^{2}+8b_{n}+1}$
(6)

are integers which are called the $n^{\text{th}}$ Lucas-balancing number and
$n^{\text{th}}$ Lucas-cobalancing number, respectively. Let
$\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$ be the roots of the characteristic
equation for Pell numbers $P_{n}$. Then Binet formulas for balancing numbers,
cobalancing numbers, Lucas-balancing numbers and Lucas-cobalancing numbers are

$B_{n}=\frac{\alpha^{2n}-\beta^{2n}}{4\sqrt{2}},b_{n}=\frac{\alpha^{2n-1}-\beta^{2n-1}}{4\sqrt{2}}-\frac{1}{2},C_{n}=\frac{\alpha^{2n}+\beta^{2n}}{2}\
\ \text{and}\ \ c_{n}=\frac{\alpha^{2n-1}+\beta^{2n-1}}{2}$

for $n\geq 1$, respectively (see also [tkcn1], [olajas1], [pa-ray12],
[raytez], [tkcn2]). Balancing numbers and their generalizations have been
investigated by several authors from many aspects. In [lip], Liptai proved
that there is no Fibonacci balancing number except $1$ and in [lip1] he proved
that there is no Lucas balancing number. In [szalay], Szalay considered the
same problem and obtained some nice results by a different method. In [tunde],
Kovács, Liptai and Olajos extended the concept of balancing numbers to the
$(a,b)$-balancing numbers defined as follows: Let $a>0$ and $b\geq 0$ be
coprime integers. If

$(a+b)+\cdots+(a(n-1)+b)=(a(n+1)+b)+\cdots+(a(n+r)+b)$

for some positive integers $n$ and $r$, then $an+b$ is an $(a,b)$-balancing
number. The sequence of $(a,b)$-balancing numbers is denoted by
$B_{m}^{(a,b)}$ for $m\geq 1$. In [akos], Liptai, Luca, Pintér and Szalay
generalized the notion of balancing numbers to numbers defined as follows: Let
$y,k,l\in\mathbb{Z}^{+}$ such that $y\geq 4$. Then a positive integer $x$ with
$x\leq y-2$ is called a $(k,l)$-power numerical center for $y$ if

$1^{k}+\cdots+(x-1)^{k}=(x+1)^{l}+\cdots+(y-1)^{l}.$

They studied the number of solutions of the equation above and proved several
effective and ineffective finiteness results for $(k,l)$-power numerical
centers. For positive integers $k,x$, let

$\Pi_{k}(x)=x(x+1)\dots(x+k-1).$

Then it was proved in [tunde] that the equation $B_{m}=\Pi_{k}(x)$ for fixed
integer $k\geq 2$ has only infinitely many solutions and for $k\in\\{2,3,4\\}$
all solutions were determined. In [tengely] Tengely considered the case

$B_{m}=x(x+1)(x+2)(x+3)(x+4)$

for $k=5$ and proved that this Diophantine equation has no solution for $m\geq
0$ and $x\in\mathbb{Z}$. In [komats], Panda, Komatsu and Davala considered the
reciprocal sums of sequences involving balancing and Lucas-balancing numbers.
In [patel1], Patel, Irmak and Ray considered incomplete balancing and Lucas-
balancing numbers and in [raysums], Ray considered the sums of balancing and
Lucas-balancing numbers by matrix methods. In [erdem], Tekcan and Erdem
considered the $t$-cobalancing numbers and $t$-cobalancers, in [ayd], Tekcan
and Aydın considered the $t$-balancers, $t$-balancing numbers and Lucas
$t$-balancing numbers and in [meryem], Tekcan and Yıldız considered the
balcobalancing numbers and balcobalancers.

##  2 Results

In this section we determine the general terms of almost balancing numbers,
almost cobalancing numbers, almost Lucas-balancing numbers and almost Lucas-
cobalancing numbers of first and second type. Almost balancing numbers first
defined by Panda and Panda in [almost]. A positive integer $n$ is called an
almost balancing number if the Diophantine equation

$\left|[(n+1)+(n+2)+\cdots+(n+r)]-[1+2+\cdots+(n-1)]\right|=1$ (7)

holds for some positive integer $r$ which is called the almost balancer. From
(7), they have two cases: If
$[(n+1)+(n+2)+\cdots+(n+r)]-[1+2+\cdots+(n-1)]=1$, then $n$ is called an
almost balancing number of first type and $r$ is called an almost balancer of
first type and in this case

$r=\frac{-2n-1+\sqrt{8n^{2}+9}}{2}.$ (8)

If $[(n+1)+(n+2)+\cdots+(n+r)]-[1+2+\cdots+(n-1)]=-1$, then $n$ is called an
almost balancing number of second type and $r$ is called an almost balancer of
second type and in this case

$r=\frac{-2n-1+\sqrt{8n^{2}-7}}{2}.$ (9)

Let $B_{n}^{\ast}$ and $B_{n}^{\ast\ast}$ denote the $n^{\text{th}}$ almost
balancing number of first type and of second type, respectively. Then from
(8), $B_{n}^{\ast}$ is an almost balancing number of first type if and only if
$8(B_{n}^{\ast})^{2}+9$ is a perfect square and from (9), $B_{n}^{\ast\ast}$
is an almost balancing number of second type if and only if
$8(B_{n}^{\ast\ast})^{2}-7$ is a perfect square. Thus

$C_{n}^{\ast}=\sqrt{8(B_{n}^{\ast})^{2}+9}\text{ \ and\ \
}C_{n}^{\ast\ast}=\sqrt{8(B_{n}^{\ast\ast})^{2}-7}$ (10)

are integers which are called the $n^{\text{th}}$ almost Lucas-balancing
number of first type and of second type, respectively. Later in [almosttez],
Panda defined that a positive integer $n$ is called an almost cobalancing
number if the Diophantine equation

$\left|[(n+1)+(n+2)+\cdots+(n+r)]-(1+2+\cdots+n)\right|=1$ (11)

holds for some positive integer $r$ which is called an almost cobalancer. From
(11), they have two cases: If $[(n+1)+(n+2)+\cdots+(n+r)]-(1+2+\cdots+n)=1$,
then $n$ is called an almost cobalancing number of first type and $r$ is
called an almost cobalancer of first type and in this case

$r=\frac{-2n-1+\sqrt{8n^{2}+8n+9}}{2}.$ (12)

If $[(n+1)+(n+2)+\cdots+(n+r)]-(1+2+\cdots+n)=-1$, then $n$ is called an
almost cobalancing number of second type and $r$ is called an almost
cobalancer of second type and in this case

$r=\frac{-2n-1+\sqrt{8n^{2}+8n-7}}{2}.$ (13)

Let $b_{n}^{\ast}$ and $b_{n}^{\ast\ast}$ denote the $n^{\text{th}}$ almost
cobalancing number of first type and of second type, respectively. Then from
(12), $b_{n}^{\ast}$ is an almost cobalancing number of first type if and only
if $8(b_{n}^{\ast})^{2}+8b_{n}^{\ast}+9$ is a perfect square and from (13),
$b_{n}^{\ast\ast}$ is an almost cobalancing number of second type if and only
if $8(b_{n}^{\ast\ast})^{2}+8b_{n}^{\ast\ast}-7$ is a perfect square. Thus

$c_{n}^{\ast}=\sqrt{8(b_{n}^{\ast})^{2}+8b_{n}^{\ast}+9}\text{ \ and\ \
}c_{n}^{\ast\ast}=\sqrt{8(b_{n}^{\ast\ast})^{2}+8b_{n}^{\ast\ast}-7}$ (14)

are integers which are called the $n^{\text{th}}$ almost Lucas-cobalancing
number of first type and of second type, respectively. Like in balancing
numbers, we notice that every almost balancing number is an almost cobalancer
and every almost cobalancing number is an almost balancer, that is,
$B_{n}^{\ast}=r_{n+1}^{\ast\ast},$
$B_{n}^{\ast\ast}=r_{n}^{\ast},b_{n}^{\ast}=R_{n+2}^{\ast}$ and
$b_{n}^{\ast\ast}=R_{n}^{\ast}$ for $n\geq 1$, where $R_{n}^{\ast}$ is the
$n^{\text{th}}$ almost balancer of first type, $R_{n}^{\ast\ast}$ is the
$n^{\text{th}}$ almost balancer of second type, $r_{n}^{\ast}$ is the
$n^{\text{th}}$ almost cobalancer of first type and $r_{n}^{\ast\ast}$ is the
$n^{\text{th}}$ almost cobalancer of second type.

###  2.1 Almost Balancing and Almost Lucas-Balancing Numbers of First and
Second Type.

We see in (8) that $x$ is an almost balancing number of first type if and only
if $8x^{2}+9$ is a perfect square and in (9), $x$ is an almost balancing
number of second type if and only if $8x^{2}-7$ is a perfect square. Let
$8x^{2}+9=y^{2}$ and let $8x^{2}-7=w^{2}$ for some positive integers $y$ and
$w.$ Then we get the Pell equations ([barb], [Jacob])

$8x^{2}-y^{2}=-9\text{ \ and \ }8x^{2}-w^{2}=7.$ (15)

For the set of all (positive) integer solutions of (15), we need some
notations: Let $\Delta$ be a non-square discriminant. Then the $\Delta$-order
$O_{\Delta}$ is defined to be the ring

$O_{\Delta}=\\{x+y\rho_{\Delta}:x,y\in\mathbb{Z\\}},$

where $\rho_{\Delta}=\sqrt{\frac{\Delta}{4}}$ if $\Delta\equiv 0(\text{mod}\
4)$ or $\frac{1+\sqrt{\Delta}}{2}$ if $\Delta\equiv 1(\text{mod}\ 4)$. So
$O_{\Delta}$ is a subring of
$\mathbb{Q(}\sqrt{\Delta})\mathbb{=}\\{x+y\sqrt{\Delta}:x,y\in\mathbb{Q}\\}$.
The unit group $O_{\Delta}^{u}$ is defined to be the group of units of the
ring $O_{\Delta}$. We can rewrite an integral indefinite quadratic form
([flath]) $F(x,y)=ax^{2}+bxy+cy^{2}$ of discriminant $\Delta$ to be

$F(x,y)=\frac{(xa+y\frac{b+\sqrt{\Delta}}{2})(xa+y\frac{b-\sqrt{\Delta}}{2})}{a}.$

So the module $M_{F}$ of $F$ is
$M_{F}=\\{xa+y\frac{b+\sqrt{\Delta}}{2}:x,y\in\mathbb{Z}\\}\subset\mathbb{Q}(\sqrt{\Delta}).$
Therefore we get
$(u+v\rho_{\Delta})(xa+y\frac{b+\sqrt{\Delta}}{2})=x^{\prime}a+y^{\prime}\frac{b+\sqrt{\Delta}}{2}$,
where

$[x^{\prime}\ \ y^{\prime}]=\left\\{\begin{array}[]{cc}[x\ \
y]\left[\begin{array}[]{cc}u-\frac{b}{2}v&av\\\
-cv&u+\frac{b}{2}v\end{array}\right]\text{ \ \ }&\text{if }\Delta\equiv
0(\text{mod}\ 4)\\\ [x\ \ y]\left[\begin{array}[]{cc}u+\frac{1-b}{2}v&av\\\
-cv&u+\frac{1+b}{2}v\end{array}\right]&\text{if }\Delta\equiv 1(\text{mod}\
4).\end{array}\right.$ (16)

Let $m$ be any integer and let $\Omega$ denote the set of all integer
solutions of $F(x,y)=m$, that is, $\Omega=\\{(x,y):F(x,y)=m\\}$. Then there is
a bijection $\Psi:\Omega\rightarrow\\{\gamma\in M_{F}:N(\gamma)=am\\}.$ The
action of $O_{\Delta,1}^{u}=\\{\alpha\in O_{\Delta}^{u}:N(\alpha)=1\\}$ on the
set $\Omega$ is most interesting when $\Delta$ is a positive non-square since
$O_{\Delta,1}^{u}$ is infinite. Therefore the orbit of each solution will be
infinite and so the set $\Omega$ is either empty or infinite. Since
$O_{\Delta,1}^{u}$ can be explicitly determined, the set $\Omega$ is
satisfactorily described by the representation of such a list, called a set of
representatives of the orbits. Let $\varepsilon_{\Delta}$ be the smallest unit
of $O_{\Delta}$ that is grater than $1$ and let
$\tau_{\Delta}=\varepsilon_{\Delta}$ if $N(\varepsilon_{\Delta})=1$ or
$\varepsilon_{\Delta}^{2}$ if $N(\varepsilon_{\Delta})=-1$. Then every
$O_{\Delta,1}^{u}$ orbit of integral solutions of $F(x,y)=m$ contains a
solution $(x,y)\in\mathbb{Z}\times\mathbb{Z}$ such that $0\leq y\leq U$, where
$U=\left|\frac{am\tau_{\Delta}}{\Delta}\right|^{\frac{1}{2}}(1-\frac{1}{\tau_{\Delta}})$
if $am>0$ or
$U=\left|\frac{am\tau_{\Delta}}{\Delta}\right|^{\frac{1}{2}}(1+\frac{1}{\tau_{\Delta}})$
if $am<0$. So for finding a set of representatives of the $O_{\Delta,1}^{u}$
orbits of integral solutions of $F(x,y)=m$, we must find for each integer
$y_{0}$ in the range $0\leq y_{0}\leq U$, whether $\Delta y_{0}^{2}+4am$ is a
perfect square or not since

$ax_{0}^{2}+bx_{0}y_{0}+cy_{0}^{2}=m\Leftrightarrow\Delta
y_{0}^{2}+4am=(2ax_{0}+by_{0})^{2}.$

If $\Delta y_{0}^{2}+4am$ is a perfect square, then
$x_{0}=\frac{-by_{0}\pm\sqrt{\Delta y_{0}^{2}+4am}}{2a}.$ So there is a set of
representatives $\text{Rep}=\\{[x_{0}\ \ y_{0}]\\}$. Thus for the matrix $M$
defined in (16), the set of all integer solutions of $F(x,y)=m$ is
$\Omega=\\{\pm(x,y):[x\ \ y]=[x_{0}\ \ y_{0}]M^{n},n\in\mathbb{Z}\\}$. For the
set of all integer solutions of (15), we can can give the following theorem.

######  Theorem 2.1.

The set of all integer solutions of $8x^{2}-y^{2}=-9$ is
$\Omega=\\{(3B_{n},3C_{n}):n\geq 1\\},$ and the set of all integer solutions
of $8x^{2}-w^{2}=7$ is

$\Omega=\\{(B_{n-1}+C_{n-1},8B_{n-1}+C_{n-1}):n\geq
1\\}\cup\\{(-B_{n}+C_{n},8B_{n}-C_{n}):n\geq 1\\}.$

######  Proof 2.2.

For the Pell equation $8x^{2}-y^{2}=-9$, we have $F(x,y)=8x^{2}-y^{2}$ of
discriminant $\Delta=32$. So we get $\tau_{32}=3+\sqrt{8}$. Thus the set of
representatives is $\text{Rep}=\\{[0\ \ 3]\\}$ and
$M=\left[\begin{array}[]{cc}3&8\\\ 1&3\end{array}\right]$ by (16). Here we
notice that $[0\ \ 3]M^{n}$ generates all integer solutions $(x_{n},y_{n})$ of
$8x^{2}-y^{2}=-9$ for $n\geq 1$. It can be easily seen that the
$n^{\text{th}}$ power of $M$ is

$M^{n}=\left[\begin{array}[]{cc}C_{n}&8B_{n}\\\ B_{n}&C_{n}\end{array}\right]$

for $n\geq 1$. Thus the set of all integer solutions is
$\Omega=\\{(3B_{n},3C_{n}):n\geq 1\\}$. For the second Pell equation
$8x^{2}-w^{2}=7$, we get $\tau_{32}=3+\sqrt{8}$. So the set of representatives
is $\text{Rep}=\\{[\pm 1\ \ 1]\\}$ and in this case $[1\ \ \ 1]M^{n-1}$
generates all integer solutions $(x_{2n-1},w_{2n-1})$ and $[1\ \ -1]M^{n}$
generates all integer solutions $(x_{2n},w_{2n})$ for $n\geq 1$. Thus the
result is obvious.

From Theorem 2.1, we can give the following theorem.

######  Theorem 2.3.

The general terms of almost balancing and almost Lucas-balancing numbers of
first type are

$B_{n}^{\ast}=3B_{n},\ C_{n}^{\ast}=3C_{n}$

for $n\geq 1,$ and the general terms of almost balancing and almost Lucas-
balancing numbers of second type are

$\displaystyle B_{2n-1}^{\ast\ast}=B_{n-1}+C_{n-1},\
B_{2n}^{\ast\ast}=-B_{n}+C_{n}$ $\displaystyle
C_{2n-1}^{\ast\ast}=8B_{n-1}+C_{n-1},\ C_{2n}^{\ast\ast}=8B_{n}-C_{n}$

for $n\geq 1.$

######  Proof 2.4.

We proved in Theorem 2.1 that the set of all integer solutions of
$8x^{2}-y^{2}=-9$ is $\Omega=\\{(3B_{n},3C_{n}):n\geq 1\\}$. Since
$x=B_{n}^{\ast}$, we get $B_{n}^{\ast}=3B_{n}$. So from (10), we deduce that

$C_{n}^{\ast}=\sqrt{8(B_{n}^{\ast})^{2}+9}=\sqrt{8(3B_{n})^{2}+9}=3\sqrt{8B_{n}^{2}+1}=3C_{n}.$

Similarly since the set of all integer solutions of $8x^{2}-w^{2}=7$ is

$\Omega=\\{(B_{n-1}+C_{n-1},8B_{n-1}+C_{n-1}):n\geq
1\\}\cup\\{(-B_{n}+C_{n},8B_{n}-C_{n}):n\geq 1\\},$

we get $B_{2n-1}^{\ast\ast}=B_{n-1}+C_{n-1},$
$C_{2n-1}^{\ast\ast}=8B_{n-1}+C_{n-1},B_{2n}^{\ast\ast}=-B_{n}+C_{n}$ and
$C_{2n}^{\ast\ast}=8B_{n}-C_{n}$ for $n\geq 1$.

Here we note that
$B_{0}^{\ast}=0,C_{0}^{\ast}=3,B_{0}^{\ast\ast}=1,C_{0}^{\ast\ast}=-1$. Also
since $8(1)^{2}-7=1$ and $8(2)^{2}-7=5^{2}$ are perfect squares by (9), we
accept $1$ and $2$ be almost balancing numbers of second type.

###  2.2 Almost Cobalancing and Almost Lucas-Cobalancing Numbers of First and
Second Type.

In this subsection, we will determine the general terms of almost cobalancing
and almost Lucas-cobalancing numbers of first and second type. Since $n$ is an
almost cobalancing number of first type if and only if $8n^{2}+8n+9$ is a
perfect square by (12) and $n$ is an almost cobalancing number of second type
if and only if $8n^{2}+8n-7$ is a perfect square by (13), we set
$8n^{2}+8n+9=y^{2}$ and $8n^{2}+8n-7=w^{2}$ for some positive integers $y$ and
$w$. Then we get the equations $2(2n+1)^{2}-y^{2}=-7$ and
$2(2n+1)^{2}-w^{2}=9$. Taking $2n+1=x$, we get the Pell equations

$2x^{2}-y^{2}=-7\text{ \ and \ }2x^{2}-w^{2}=9.$ (17)

For the set of all integer solutions of (17), we can can give the following
theorem.

######  Theorem 2.5.

The set of all integer solutions of $2x^{2}-y^{2}=-7$ is

$\Omega=\\{(6B_{n-1}+C_{n-1},4B_{n-1}+3C_{n-1}):n\geq
1\\}\cup\\{(6B_{n}-C_{n},-4B_{n}+3C_{n}):n\geq 1\\},$

and the set of all integer solutions of $2x^{2}-w^{2}=9$ is

$\Omega=\\{(6B_{n-1}+3C_{n-1},12B_{n-1}+3C_{n-1}):n\geq 1\\}.$

######  Proof 2.6.

For the Pell equation $2x^{2}-y^{2}=-7$, we get $F(x,y)=2x^{2}-y^{2}$ of
discriminant $\Delta=8$. So $\tau_{8}=3+2\sqrt{2}$ and hence the set of
representatives is $\text{Rep}=\\{[\pm 1\ \ 3]\\}$ and
$M=\left[\begin{array}[]{cc}3&4\\\ 2&3\end{array}\right]$. Here $[1\ \
3]M^{n-1}$ generates all integer solutions $(x_{2n-1},y_{2n-1})$ and $[-1\ \
3]M^{n}$ generates all integer solutions $(x_{2n},y_{2n})$ for $n\geq 1$.
Since the $n^{\text{th}}$ power of $M$ is

$M^{n}=\left[\begin{array}[]{cc}C_{n}&4B_{n}\\\
2B_{n}&C_{n}\end{array}\right]$

for $n\geq 1$, we deduce that the set of all integer solutions is

$\Omega=\\{(6B_{n-1}+C_{n-1},4B_{n-1}+3C_{n-1}):n\geq
1\\}\cup\\{(6B_{n}-C_{n},-4B_{n}+3C_{n}):n\geq 1\\}.$

For the second Pell equation $2x^{2}-w^{2}=9$, we get $\tau_{8}=3+2\sqrt{2}$
and the set of representatives is $\text{Rep}=\\{[\pm 3\ \ 3]\\}$. In this
case $[3\ \ 3]M^{n-1}$ generates all integer solutions $(x_{n},w_{n})$ for
$n\geq 1.$ Thus the result is obvious.

From Theorem 2.5, we can give the following theorem.

######  Theorem 2.7.

The general terms of almost cobalancing and almost Lucas-cobalancing numbers
of first type are

$\displaystyle b_{2n}^{\ast}=2b_{n+1}-b_{n},\
b_{2n-1}^{\ast}=4b_{n}-b_{n-1}+1,$ $\displaystyle
c_{2n}^{\ast}=c_{n+2}-4c_{n+1},\ c_{2n-1}^{\ast}=c_{n+1}-2c_{n}$

for $n\geq 1,$ and the general terms of almost cobalancing and almost Lucas-
cobalancing numbers of second type are

$b_{n}^{\ast\ast}=3b_{n}+1,\ c_{n}^{\ast\ast}=3c_{n}$

for $n\geq 1.$

######  Proof 2.8.

We proved in Theorem 2.5 that the set of all integer solutions of
$2x^{2}-y^{2}=-7$ is $\Omega=\\{(6B_{n-1}+C_{n-1},4B_{n-1}+3C_{n-1}):n\geq
1\\}\cup\\{(6B_{n}-C_{n},-4B_{n}+3C_{n}):n\geq 1\\}$. Since $x=2n+1$, we get

$\displaystyle b_{2n}^{\ast}$ $\displaystyle=\frac{6B_{n}+C_{n}-1}{2}$
$\displaystyle=\frac{6(\frac{\alpha^{2n}-\beta^{2n}}{4\sqrt{2}})+\frac{\alpha^{2n}+\beta^{2n}}{2}-1}{2}$
$\displaystyle=\frac{\alpha^{2n}(2\alpha-\alpha^{-1})+\beta^{2n}(-2\beta+\beta^{-1})}{4\sqrt{2}}-\frac{1}{2}$
$\displaystyle=2(\frac{\alpha^{2n+1}-\beta^{2n+1}}{4\sqrt{2}}-\frac{1}{2})-(\frac{\alpha^{2n-1}-\beta^{2n-1}}{4\sqrt{2}}-\frac{1}{2})$
$\displaystyle=2b_{n+1}-b_{n}.$

Thus from (14), we get

$\displaystyle c_{2n}^{\ast}$
$\displaystyle=\sqrt{8(b_{2n}^{\ast})^{2}+8b_{2n}^{\ast}+9}$
$\displaystyle=\sqrt{8(2b_{n+1}-b_{n})^{2}+8(2b_{n+1}-b_{n})+9}$
$\displaystyle=\sqrt{\alpha^{4n}(\frac{11+6\sqrt{2}}{4})+\beta^{4n}(\frac{11-6\sqrt{2}}{4})+\frac{7}{2}}$
$\displaystyle=\sqrt{\begin{array}[]{c}(\frac{\alpha^{2n+3}+\beta^{2n+3}}{2})^{2}-4(\frac{\alpha^{2n+3}+\beta^{2n+3}}{2})(\alpha^{2n+1}+\beta^{2n+1})+4(\alpha^{2n+1}+\beta^{2n+1})^{2}\end{array}}$
$\displaystyle=\sqrt{[(\frac{\alpha^{2n+3}+\beta^{2n+3}}{2}-4(\frac{\alpha^{2n+1}+\beta^{2n+1}}{2})]^{2}}$
$\displaystyle=\frac{\alpha^{2n+3}+\beta^{2n+3}}{2}-4(\frac{\alpha^{2n+1}+\beta^{2n+1}}{2})$
$\displaystyle=c_{n+2}-4c_{n+1}.$

The others can be proved similarly.

Here we note that $b_{0}^{\ast}=0,c_{0}^{\ast}=3,$ $b_{0}^{\ast\ast}=1$ and
$c_{0}^{\ast\ast}=3$. Also since $8(1)^{2}+8(1)$ $-7=3^{2}$ is a perfect
square by (13), we accept $1$ to be an almost cobalancing number of second
type. In Theorems 2.3 and 2.7, we deduce the general terms of all almost
balancing numbers of first and second type in terms of balancing numbers.
Conversely, we can deduce the general terms of all balancing numbers in terms
of all almost balancing numbers of first and second type as follows:

######  Theorem 2.9.

The general terms of all balancing numbers are

$B_{n}=\frac{B_{n}^{\ast}}{3},\
b_{n}=\frac{b_{2n-1}^{\ast}-b_{2n-2}^{\ast}-1}{2},\
C_{n}=\frac{C_{n}^{\ast}}{3},\
c_{n}=\frac{c_{2n-1}^{\ast}-c_{2n-2}^{\ast}}{2}$

for $n\geq 1,$ or

$B_{n}=\frac{B_{2n+1}^{\ast\ast}-B_{2n}^{\ast\ast}}{2},\
b_{n}=\frac{b_{n}^{\ast\ast}-1}{3},\
C_{n}=\frac{C_{2n+1}^{\ast\ast}-C_{2n}^{\ast\ast}}{2},\
c_{n}=\frac{c_{n}^{\ast\ast}}{3}$

for $n\geq 1.$

######  Proof 2.10.

The result is obvious from Theorems 2.3 and 2.7.

Thus we construct a one-to-one correspondence between all balancing numbers
and all almost balancing numbers of first and second type. Moreover, the
general terms of all almost balancing numbers of first type can be given in
terms of all almost balancing numbers of second type and conversely the
general terms of all almost balancing numbers of second type can be given in
terms of all almost balancing numbers of first type as follows.

######  Theorem 2.11.

The general terms of all almost balancing numbers of first type are

$\displaystyle
B_{n}^{\ast}=\frac{3B_{2n+1}^{\ast\ast}-3B_{2n}^{\ast\ast}}{2},\
C_{n}^{\ast}=\frac{3C_{2n+1}^{\ast\ast}-3C_{2n}^{\ast\ast}}{2},$
$\displaystyle
b_{2n-1}^{\ast}=\frac{4b_{n}^{\ast\ast}-b_{n-1}^{\ast\ast}}{3},\
b_{2n}^{\ast}=\frac{2b_{n+1}^{\ast\ast}-b_{n}^{\ast\ast}-1}{3},$
$\displaystyle
c_{2n-1}^{\ast}=\frac{c_{n+1}^{\ast\ast}-2c_{n}^{\ast\ast}}{3},\
c_{2n}^{\ast}=\frac{c_{n+2}^{\ast\ast}-4c_{n+1}^{\ast\ast}}{3}$

for $n\geq 1,$ and the general terms of all almost balancing numbers of second
type are

$\displaystyle B_{2n-1}^{\ast\ast}=\frac{B_{n-1}^{\ast}+C_{n-1}^{\ast}}{3},\
B_{2n}^{\ast\ast}=\frac{-B_{n}^{\ast}+C_{n}^{\ast}}{3},$ $\displaystyle
b_{n}^{\ast\ast}=\frac{3b_{2n-1}^{\ast}-3b_{2n-2}^{\ast}-1}{2},\
c_{n}^{\ast\ast}=\frac{3c_{2n-1}^{\ast}-3c_{2n-2}^{\ast}}{2},$ $\displaystyle
C_{2n-1}^{\ast\ast}=\frac{8B_{n-1}^{\ast}+C_{n-1}^{\ast}}{3},\
C_{2n}^{\ast\ast}=\frac{8B_{n}^{\ast}-C_{n}^{\ast}}{3}$

for $n\geq 1$.

######  Proof 2.12.

Since $B_{n}^{\ast}=3B_{n}$ and
$B_{n}=\frac{B_{2n+1}^{\ast\ast}-B_{2n}^{\ast\ast}}{2}$ by Theorems 2.3 and
2.9, we deduce that
$B_{n}^{\ast}=\frac{3B_{2n+1}^{\ast\ast}-3B_{2n}^{\ast\ast}}{2}$. The others
can be proved similarly.

Thus we construct a one-to-one correspondence between all almost balancing
numbers of first type and all almost balancing numbers of second type.

##  3 Relationship with Pell Numbers.

In this section, we consider the relationship between all almost balancing
numbers of first and second type and Pell numbers. It is known that the
general terms of all balancing numbers can be given in terms of Pell numbers,
namely

$B_{n}=\frac{P_{2n}}{2},b_{n}=\frac{P_{2n-1}-1}{2},C_{n}=P_{2n}+P_{2n-1},c_{n}=P_{2n-1}+P_{2n-2}$
(18)

for $n\geq 1$. Similarly we can give the following theorem.

######  Theorem 3.1.

The general terms of all almost balancing numbers of first type are

$\displaystyle B_{n}^{\ast}=\frac{3P_{2n}}{2},\
b_{2n}^{\ast}=\frac{4P_{2n}+P_{2n-1}-1}{2},\ C_{n}^{\ast}=3P_{2n}+3P_{2n-1},$
$\displaystyle c_{2n-1}^{\ast}=5P_{2n-1}+P_{2n-2},\
c_{2n}^{\ast}=3P_{2n+1}-P_{2n}$

for $n\geq 1,$ and $b_{2n-1}^{\ast}=\frac{8P_{2n-2}+3P_{2n-3}-1}{2}$ for
$n\geq 2,$ and the general terms of all almost balancing numbers of second
type are

$\displaystyle B_{2n}^{\ast\ast}=\frac{P_{2n}+2P_{2n-1}}{2},\
b_{n}^{\ast\ast}=\frac{3P_{2n-1}-1}{2}$ $\displaystyle
C_{2n}^{\ast\ast}=3P_{2n}-P_{2n-1},\ c_{n}^{\ast\ast}=3P_{2n-1}+3P_{2n-2}$

for $n\geq 1,$ and
$B_{2n-1}^{\ast\ast}=\frac{3P_{2n-2}+2P_{2n-3}}{2},C_{2n-1}^{\ast\ast}=5P_{2n-2}+P_{2n-3}$
for $n\geq 2$.

######  Proof 3.2.

Note that $B_{n}^{\ast}=3B_{n}$ and $B_{n}=\frac{P_{2n}}{2}$. So
$B_{n}^{\ast}=\frac{3P_{2n}}{2}$. Since $B_{2n-1}^{\ast\ast}=B_{n-1}$
$+C_{n-1}$ by Theorem 2.3 and $B_{n}=\frac{P_{2n}}{2},C_{n}=P_{2n}+P_{2n-1}$
by (18), we easily get

$B_{2n-1}^{\ast\ast}=\frac{P_{2n-2}}{2}+P_{2n-2}+P_{2n-3}=\frac{3P_{2n-2}+2P_{2n-3}}{2}$

for $n\geq 2$ as we wanted. The other cases can be proved similarly.

In Theorem 3.1, we can give the general terms of all almost balancing numbers
of first and second type in terms of Pell numbers. Conversely, we can give the
general terms of Pell numbers in terms of almost balancing numbers of first
and second type as follows:

######  Theorem 3.3.

The general terms of Pell numbers are $P_{2n}=\frac{2B_{n}^{\ast}}{3}$ and
$P_{2n-1}=b_{2n-1}^{\ast}-b_{2n-2}^{\ast}$ for $n\geq 1,$ or
$P_{2n}=B_{2n+1}^{\ast\ast}-B_{2n}^{\ast\ast}$ and
$P_{2n-1}=\frac{2b_{n}^{\ast\ast}+1}{3}$ for $n\geq 1$.

######  Proof 3.4.

It can be easily deduced from Theorem 3.1.

Thus we construct a one-to-one correspondence between all almost balancing
numbers of first and second type and Pell numbers.

##  4 Concluding Remarks

For almost balancing and almost Lucas-balancing numbers of first and second
type, in [almost] Panda and Panda proved in Theorem 3.1 that the solutions of
the Diophantine equation $8x^{2}+9=y^{2}$ in positive integers are given by
$x=3B_{n}$ and $y=3C_{n}$ for $n\geq 1$. Similarly they proved in Theorem 3.2
that the solutions of the Diophantine equation $8x^{2}-7=y^{2}$ in positive
integers constitute two classes: the first class is $(x,y)=(B_{n}-2B_{n-1},$
$C_{n}-2C_{n-1})$, and the second class is
$(x,y)=(2B_{n}-B_{n-1},2C_{n}-C_{n-1})$, for $n\geq 1$. Since
$B_{n}=\frac{\alpha^{2n}-\beta^{2n}}{4\sqrt{2}}$ and
$C_{n}=\frac{\alpha^{2n}+\beta^{2n}}{2}$, we easily deduce that

$\displaystyle B_{n}-2B_{n-1}$
$\displaystyle=\frac{\alpha^{2n}-\beta^{2n}}{4\sqrt{2}}-2(\frac{\alpha^{2n-2}-\beta^{2n-2}}{4\sqrt{2}})$
$\displaystyle=\frac{\alpha^{2n-2}(1+2\sqrt{2})+\beta^{2n-2}(-1+2\sqrt{2})}{4\sqrt{2}}$
$\displaystyle=\frac{\alpha^{2n-2}-\beta^{2n-2}}{4\sqrt{2}}+\frac{\alpha^{2n-2}+\beta^{2n-2}}{2}$
$\displaystyle=B_{n-1}+C_{n-1}$ $\displaystyle=B_{2n-1}^{\ast\ast}$

for $n\geq 1$. Similarly it can be shown that:

$\displaystyle C_{n}-2C_{n-1}=8B_{n-1}+C_{n-1}=C_{2n-1}^{\ast\ast},\qquad
2B_{n}-B_{n-1}=-B_{n}+C_{n}=B_{2n}^{\ast\ast},$ $\displaystyle
2C_{n}-C_{n-1}=8B_{n}-C_{n}=C_{2n}^{\ast\ast},$

for $n\geq 1$, that is, we get same result in Theorem 2.7. Similarly for the
almost cobalancing numbers of first and second type in [almosttez], Panda
proved in Theorem 4.3.1 that the values of $x$ satisfying the Diophantine
equation $8x^{2}+8x+9=y^{2}$ in positive integers partition in two classes.
The first class is given by $U_{n}=$ $\frac{3B_{n}+B_{n-1}-1}{2}$ and the
second class is $V_{n}=\frac{3B_{n}+B_{n+1}-1}{2}$ for $n\geq 1$. Here we
notice that

$\displaystyle U_{n}$ $\displaystyle=\frac{3B_{n}+B_{n-1}-1}{2}$
$\displaystyle=\frac{3(\frac{\alpha^{2n}-\beta^{2n}}{4\sqrt{2}})+\frac{\alpha^{2n-2}-\beta^{2n-2}}{4\sqrt{2}}-1}{2}$
$\displaystyle=\frac{6(\frac{\alpha^{2n}-\beta^{2n}}{4\sqrt{2}})-\frac{\alpha^{2n}+\beta^{2n}}{2}-1}{2}$
$\displaystyle=\frac{6B_{n}-C_{n}-1}{2}$ $\displaystyle=4b_{n}-b_{n-1}+1$
$\displaystyle=b_{2n-1}^{\ast}$

and similarly it can be shown that $V_{n}=b_{2n}^{\ast}$ as we proved in
Theorem 2.7. But he did not determine the general terms of almost Lucas-
cobalancing numbers of first and second type. Apart from these in this paper,

  1. 1.

we determined the general terms of almost Lucas-cobalancing numbers of first
and second type in Theorem 2.7.

  2. 2.

we can give the general terms of all balancing numbers in terms of all almost
balancing numbers of first and second type in Theorem 2.9. Thus we construct a
one-to-one correspondence between all balancing numbers and all almost
balancing numbers of first and second type.

  3. 3.

We can give the general terms of all almost balancing numbers of first type in
terms of all almost balancing numbers of second type and conversely give the
general terms of all almost balancing numbers of second type in terms of all
almost balancing numbers of first type in Theorem 2.11. Thus, we construct a
one-to-one correspondence between all almost balancing numbers of first type
and of second type.

  4. 4.

We can give the general terms of all almost balancing numbers of first and
second type in terms of Pell numbers in Theorem 3.1 and conversely give the
general terms of Pell numbers in terms of almost balancing numbers of first
and second type in Theorem 3.3. Thus, we construct a one-to-one correspondence
between all almost balancing numbers of first and second type and Pell
numbers.

## References

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  * [2] PaperB-Pa Behera A. and Panda G.K. On the Square Roots of Triangular Numbers Fibonacci Quart. 37 1999 2 98-105
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February 25, 2019March 16, 2021Attila Bérczes


# Entanglement dynamics in U(1) symmetric hybrid quantum automaton circuits

Yiqiu Han<EMAIL_ADDRESS>Department of Physics, Boston College, Chestnut Hill,
MA 02467, USA  Xiao Chen  Department of Physics, Boston College, Chestnut
Hill, MA 02467, USA

###### Abstract

We study the entanglement dynamics of quantum automaton (QA) circuits in the
presence of U(1) symmetry. We find that the second Rényi entropy grows
diffusively with a logarithmic correction as $\sqrt{t\ln{t}}$, saturating the
bound established by Ref.[1]. Thanks to the special feature of QA circuits, we
understand the entanglement dynamics in terms of a classical bit string model.
Specifically, we argue that the diffusive dynamics stems from the rare slow
modes containing extensively long domains of spin 0s or 1s. Additionally, we
investigate the entanglement dynamics of monitored QA circuits by introducing
a composite measurement that preserves both the U(1) symmetry and properties
of QA circuits. We find that as the measurement rate increases, there is a
transition from a volume-law phase where the second Rényi entropy persists the
diffusive growth (up to a logarithmic correction) to a critical phase where it
grows logarithmically in time. This interesting phenomenon distinguishes QA
circuits from non-automaton circuits such as U(1)-symmetric Haar random
circuits, where a volume-law to an area-law phase transition exists, and any
non-zero rate of projective measurements in the volume-law phase leads to a
ballistic growth of the Rényi entropy.

##  1 Introduction

Entanglement is an important measure of correlations between different degrees
of freedom in many-body quantum systems. In a typical system with local
interactions, quantum information propagates ballistically, resulting in
linear growth of entanglement over time [2]. This physics can be understood
through random circuit models, which offer a minimal model for investigating
entanglement dynamics and information scrambling [3, 4, 5, 6, 7].

However, the above-described picture changes slightly when an additional
continuous symmetry is present in the dynamics. If U(1) symmetry is imposed,
it can lead to diffusive transport of the conserved charges. It has been
demonstrated that although the von-Neumann entanglement entropy continues to
grow linearly, the growth of higher Rényi entropies is limited by the
diffusive transport and therefore exhibits sub-ballistic growth [1, 8, 9, 10].
Mathematically it is rigorously proven that the growth of $S^{(n>1)}$ is at
most diffusive, with a logarithmic correction [1], i.e.,

$S^{(n>1)}\leq\frac{n}{n-1}\mathcal{O}(\sqrt{t\ln t}).$ (1)

Motivated by these findings, this paper investigates the entanglement dynamics
in the U(1)-symmetric quantum automaton (QA) circuits with a focus on the
second Rényi entropy $S^{(n=2)}$. In QA circuits, the quantum state is always
an equal-weight superposition of all the allowed basis states with the phases
carrying the quantum information. Due to this special property, $S^{(n=2)}$
can be mapped to a quantity of a classical bit string model [11, 12, 13]. Such
a mapping enables us to study the entanglement dynamics analytically and also
provides an efficient method for numerical simulation. We show that the growth
of $S^{(n=2)}$ is governed by the presence of the rare bit strings that
contain extensively long domains comprising consecutive spin 0s or 1s,
consistent with the physical picture introduced in Ref. [1, 8]. Additionally,
we present numerical evidence demonstrating that the dynamics of $S^{(n=2)}$
actually saturates the upper bound defined in Eq. (1). This saturation is
caused by the diffusive transport (up to a logarithmic correction) of the
boundary of these long domains. Furthermore, for charge-fixed states, we study
the coefficient in front of the diffusive scaling of $S^{(n=2)}(t)$ and find
that it is linearly dependent on the charge filling factor $\nu$ when $\nu$ is
small.

In addition, we are interested in the impact of U(1) symmetry on the
entanglement dynamics of monitored QA circuits. Notably, recent research has
revealed that monitored quantum dynamics give rise to a measurement-induced
entanglement phase transition (MIPT) [14, 15, 16]. This occurs as a result of
the interplay between random unitary evolution and local non-unitary
measurements, driving the system from a highly-entangled volume-law phase to a
disentangled area-law phase [14, 15, 16, 17, 18, 19, 20], or even to other
quantum phases, such as the critical phase, depending on the symmetry and type
of measurements imposed [21, 22, 23, 24, 25, 11]. When U(1)-symmetry is
introduced in monitored Haar random circuits, it is found that any non-zero
rate of single-qubit projective measurements will eliminate the rare slow
modes containing extensively long domains, and the Rényi entropy grows
linearly in time for $0<p<p_{c}$ and exhibits $z=1$ dynamical scaling at the
critical point $p_{c}$ [26].

With these insights in mind, our paper also investigates the entanglement
dynamics of U(1)-symmetric QA circuits under specific measurements that
preserve U(1) symmetry and keep the wave function as an equal weight
superposition of basis states. Interestingly, different from Haar random
circuits, the measurements leave these extensively long domains untouched and
the second Rényi entropy still exhibits diffusive growth in the volume-law
phase. As the measurement rate $p$ increases, we observe a phase transition to
a critical phase where the entanglement entropy grows logarithmically in time.
The critical phase that we observe is a result of both the unique properties
of QA circuits and the presence of U(1) symmetry. It is worth noting that
similar behavior has also been observed in the monitored $\mathbb{Z}_{2}$
symmetric QA circuits [11].

##  2 U(1)-symmetric hybrid QA circuits and two-species particle model

In this paper, we consider 1+1d U(1)-symmetric hybrid QA circuits. The
dynamics consists of local QA unitary operators and composite measurements,
which are chosen to preserve the total charge

$Q=\sum_{i}^{L}\sigma_{i},\text{ where }\sigma_{i}=(1-Z_{i})/2$ (2)

and $Z_{i}$ is the Pauli Z matrix acting on the $i$th site of a chain with $L$
qubits. A QA unitary operator permutes states in the computational basis up to
a phase, i.e.,

$U|n\rangle=e^{i\theta_{n}}|\pi(n)\rangle,$ (3)

where $\pi\in S_{N}$ is an element of the permutation group on a computational
basis with cardinality $N$. Here we take the initial state

$|\psi_{0}\rangle=\sum_{n}\frac{|n\rangle}{\sqrt{N}}$ (4)

to be an equal-weight superposition of two different sets of basis states: (i)
$\\{|n\rangle=|\sigma_{1}\sigma_{2}\dots\sigma_{L}\rangle:\sigma_{i}=\\{0,1\\}\\}$
is all the allowed Pauli Z basis with cardinality $N=2^{L}$ so that

$|\psi_{0}\rangle=|+x\rangle^{\otimes
L}=\big{[}\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\big{]}^{\otimes L}.$ (5)

(ii)
$\\{|n\rangle=|\sigma_{1}\sigma_{2}\dots\sigma_{L}\rangle:\sigma_{i}=\\{0,1\\},\sum_{i}\sigma_{i}=Q\\}$
is the subset of the Pauli Z basis with a fixed extensive charge filling
$\nu\equiv Q/L$, so that $N={L\choose Q}=L!/[(L-Q)!Q!]$.

Figure 1: The setup of the U(1)-symmetric hybrid QA circuit of one time step.
The dashed box encloses the gates within a single layer. Each time step
involves three layers of Fredkin gates, SWAP gates, and CZ gates, interspersed
with composite measurements. The Fredkin and SWAP gates are applied in each
layer with probability $p_{u}<1$, and the measurement appears in each measured
sublayer with probability $p$. The Fredkin and SWAP gates determine the
dynamics of bit string $|n\rangle$ and can be replaced by other QA gates
preserving U(1) symmetry.

Fig.1 depicts a brickwork-patterned U(1) symmetric hybrid QA circuit. Each
time step consists of three layers of QA unitary operators interspersed with
composite measurements with probability $p$. For the unitary part, we consider
Fredkin gates and SWAP gates, along with CZ gates which assign a $\pi$ phase
to the spin configuration $|11\rangle$. The Fredkin gates are three-qubit
gates that interchange qubits $i-1$ and $i+1$ according to the value of the
middle (control) qubit, i.e.,
$|\sigma_{i}1_{i+1}\sigma_{i+2}\rangle\mapsto|\sigma_{i+2}1_{i+1}\sigma_{i}\rangle$
and
$|\sigma_{i}0_{i+1}\sigma_{i+2}\rangle\mapsto|\sigma_{i}0_{i+1}\sigma_{i+2}\rangle$.
Meanwhile, the SWAP gates interchange two neighboring qubits. Together with CZ
gates, they scramble the quantum information and increase the entanglement
entropy of the state until it saturates to the volume-law scaling. As
illustrated in Fig.1, in the first/ second/ third layer of each time step, the
Fredkin gates are applied on sites
$\\{3j-2,3j-1,3j\\}/\\{3j-1,3j,3j+1\\}/\\{3j,3j+1,3j+2\\}$ for $j\in[1,L/3]$,
while the SWAP and CZ gates are applied on sites
$\\{2j-1,2j\\}/\\{2j,2j+1\\}/\\{2j-1,2j\\}$ for $j\in[1,L/2]$. Specifically,
we set the occurring probability of the Fredkin and SWAP gates to be $p_{u}$
and we take $p_{u}<1$ throughout the paper.

Constructing the measurement gates can be quite tricky. To ensure that
$|\psi_{t}\rangle$ remains an equal-weight superposition of basis states, we
introduce a charge-preserving two-qubit composite measurement $M_{1/2}=R\circ
P_{1/2}$. Here $P_{1}$ and $P_{2}$ are the Kraus operators,

$\displaystyle P_{1}$
$\displaystyle=\begin{pmatrix}\frac{1}{\sqrt{2}}&0&0&0\\\ 0&1&0&0\\\
0&0&0&0\\\ 0&0&0&\frac{1}{\sqrt{2}}\end{pmatrix},$ (6) $\displaystyle P_{2}$
$\displaystyle=\begin{pmatrix}\frac{1}{\sqrt{2}}&0&0&0\\\ 0&0&0&0\\\
0&0&1&0\\\ 0&0&0&\frac{1}{\sqrt{2}}\end{pmatrix},$

followed by a two-site rotation operator,

$R=\begin{pmatrix}1&0&0&0\\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\
0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\\ 0&0&0&1\end{pmatrix},$ (7)

which maps $|01\rangle\mapsto(|01\rangle+|10\rangle)/\sqrt{2}$ and
$|10\rangle\mapsto(|01\rangle-|10\rangle)/\sqrt{2}$ 111 In numerical
simulations, we omit the extra $\pi$ phase in front of $|10\rangle$ obtained
from the rotation of $M_{2}$. This is because it is equivalent to applying
$M_{2}$ without the extra $\pi$ phase followed by a controlled phase gate. As
shown later, such a phase gate will not affect the entanglement phase
transition from the volume-law phase to the critical phase. The location of
the phase transition is only determined by the corresponding bit string
dynamics., and acts trivially on $|00\rangle$ and $|11\rangle$, so as to
rotate the wave function back to equal weight (up to a phase) superposition of
the computational basis. In general, the measurement disentangles the system
by discarding the phase information of a quarter of the basis states. However,
considering it is a two-qubit operator as required by the U(1) symmetry, as we
will see later, the measurements together with phase gates can actually induce
entanglement in certain circumstances. As shown in Fig.1, each measured layer
contains two rows of composite measurements, with each row containing
$M_{1/2}$ randomly distributed with probability $p$ on sites
$\\{2j-1,2j\\}/\\{2j,2j+1\\}$ for $i\in[1,L/2]$.

Throughout the paper, we focus on the entanglement dynamics between subsystems
A and B that the system is bi-partitioned into. Specifically, we consider the
second Rényi entropy of A,

$\displaystyle S^{(2)}_{A}(t)$
$\displaystyle=-\ln{\text{Tr}[\rho_{A}^{2}(t)]},$ (8)
$\displaystyle\rho_{A}(t)$
$\displaystyle=\text{Tr}_{B}|\psi_{t}\rangle\langle\psi_{t}|$

where $|\psi_{t}\rangle=\tilde{U}_{t}|\psi_{0}\rangle$ is the wave function of
the quantum trajectory with $\tilde{U}_{t}$ representing the circuit evolution
up to time $t$. We first consider the initial condition (i), where
$|\psi_{0}\rangle=|+x\rangle^{\otimes L}$. In our earlier work, we discovered
an efficient algorithm to compute $S_{A}^{(2)}(t)$ from this initial state
[13]. Additionally, we presented a classical stochastic model to elucidate the
entanglement dynamics [11]. In the following, we will provide a brief overview
of them and apply these methods to our QA circuit with U(1) symmetry.

The purity can be expressed as the expectation of the $\mathsf{SWAP}_{A}$
operator over double copies of the system [27],

$\text{Tr}[\rho_{A}^{2}(t)]=\langle\psi_{t}|_{2}\otimes\langle\psi_{t}|_{1}\mathsf{SWAP}_{A}|\psi_{t}\rangle_{1}\otimes|\psi_{t}\rangle_{2}.$
(9)

The $\mathsf{SWAP}_{A}$ operator swaps the configurations of the copies within
region $A$. Since QA circuits preserve the computational basis that spans
$|\psi_{t}\rangle$, we can insert into Eq.(9) two sets of complete basis which
are acted upon by the circuit in a time-reversed order,

$\displaystyle\text{Tr}[\rho_{A}^{2}(t)]=\sum_{n_{1},n_{2}}\langle\psi_{t}|_{2}\langle\psi_{t}|_{1}\mathsf{SWAP}_{A}|n_{1}\rangle|n_{2}\rangle$
(10) $\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\langle n_{2}|\langle
n_{1}|\psi_{t}\rangle_{1}|\psi_{t}\rangle_{2}$
$\displaystyle=\frac{1}{4^{L}}\sum_{n_{1},n_{2}}e^{-i\Theta_{n_{1}^{\prime}}(t)}e^{-i\Theta_{n_{2}^{\prime}}(t)}e^{i\Theta_{n_{1}}(t)}e^{i\Theta_{n_{2}}(t)},$

where

$e^{i\Theta_{n_{i}}(t)}=\sqrt{2^{L}}\langle
n_{i}|\tilde{U}_{t}|\psi_{0}\rangle,$ (11)

and

$\displaystyle|n_{1}^{\prime}\rangle|n_{2}^{\prime}\rangle$
$\displaystyle\equiv\mathsf{SWAP}_{A}|n_{1}\rangle|n_{2}\rangle$ (12)
$\displaystyle=\mathsf{SWAP}_{A}|\alpha_{1}\beta_{1}\rangle|\alpha_{2}\beta_{2}\rangle$
$\displaystyle=|\alpha_{2}\beta_{1}\rangle|\alpha_{1}\beta_{2}\rangle,$

where $|\alpha_{i}\rangle$ and $|\beta_{i}\rangle$ are the spin configurations
in subsystems $A$ and $B$ of $|n_{i}\rangle$. Therefore, in the numerical
simulation, instead of evolving the wave function $|\psi_{t}\rangle$, we can
apply the circuit on the bit strings in a time-reversed order. When evaluating
Eq. (11) from left to right, although the composite measurement $M$ is non-
unitary, we can still derive the effective action on the bit string $\langle
n|$. For bit strings that have anti-parallel spins on the sites where $M$ is
applied, they are all forced to be either $\langle\dots 01\dots|$ or
$\langle\dots 10\dots|$ after the measurement.

With a few modifications, the above equations used to compute the purity can
also be applied to the initial condition (ii). When $|\psi_{0}\rangle$ is a
charge-fixed state of filling factor $\nu$, only the bit string pairs that
share the same filling factor both before and after the SWAP, i.e.,
$\\{|n_{1}\rangle,|n_{2}\rangle,|n_{1}^{\prime}\rangle,|n_{2}^{\prime}\rangle\\}$
with the same $\nu$, will have nonzero overlap with $|\psi_{0}\rangle$ and
hence contribute to the purity.

Figure 2: The cartoon of the two-species particle model. We use $\circ$ to
denote $h(x)=0$ and $\bullet$ to denote $h(x)=1$, where the black dots
represent $X$ particles and the red dots represent $Y$ particles, and
$\tilde{X}(\tilde{Y})$ represents the rightmost $X$ (leftmost $Y$) particle.
At $t=0$, the $X$ and $Y$ particles are distributed in region $A$ and $B$
respectively. As time evolves, both species begin to expand and intrude into
each other’s territory.

Eq. (10) not only offers a numerical method but also helps us to understand
the entanglement dynamics through the classical bit string dynamics. It is
worth noting that this equation sums up the accumulated phase
$\Theta_{r}=-\Theta_{n_{1}^{\prime}}-\Theta_{n_{2}^{\prime}}+\Theta_{n_{1}}+\Theta_{n_{2}}$
for each bit string pair $\\{|n_{1}\rangle,|n_{2}\rangle\\}$. Under random
time evolution, $\Theta_{r}$ can become a nonzero random number, and we expect
the sum over these phases to be zero. Therefore only the configurations with
$\Theta_{r}=0$ contribute to the purity. This observation leads to a
stochastic particle model in which there are two particle species $X$ and $Y$
representing the bit string difference

$h(x,t)=|n_{1}(x,t)-n_{2}(x,t)|$ (13)

initially distributed in subregion $A$ and $B$ respectively [11, 12], as
illustrated in Fig. 2. As time evolves, the two species originally located in
subregions $A$ and $B$ gradually expand. It is shown that only the
configurations in which $X$ and $Y$ particles have never met up to time $t$
satisfy $\Theta_{r}(t)=0$ and hence contribute to the purity, i.e.,

$S_{A}^{(2)}(t)=-\ln\text{Tr}\rho_{A}^{2}(t)\approx-\ln\frac{N_{0}(t)}{4^{L}}\equiv-\ln
P(t),$ (14)

where $N_{0}(t)$ is the number of bit string pairs in which the two species
never encounter each other up to time $t$. Such an approximation has been
numerically verified in Ref.[12] (For more details, see Appendix.A).

From the above analysis, the growth of the second Rényi entropy is determined
by the dynamics of the endpoint $\tilde{X}$ and $\tilde{Y}$ particles of each
bit string pair, i.e., the rightmost $X$ particle and the leftmost $Y$
particle. Let us first consider the system without any symmetry. Under unitary
dynamics, the $\tilde{X}$ and $\tilde{Y}$ particles move ballistically toward
each other at roughly the same speed, i.e., the distance that an endpoint
particle travels over time $t$ scales as $\Delta l(t)\propto t$. Therefore,
only the configurations whose initial rightmost $X$ and leftmost $Y$ particles
are situated a distance of at least $2\Delta l(t)$ apart can contribute to the
purity. This leads to $P(t)=[2^{2\Delta l(t)}\times 4^{L-2\Delta
l(t)}]/4^{L}=\mathcal{O}(e^{-t})$, which explains the linear growth of
entanglement entropy in the absence of U(1) symmetry.

##  3 Unitary dynamics

We first study the U(1)-symmetric QA circuit without measurements, i.e.,
$p=0$, and take the unitary rate $p_{u}=0.5$. The classical bit string model
allows for numerical simulations of the second Rényi entropy for relatively
large system sizes. To be more specific, we prepare a large sample of randomly
generated bit strings which can have either unfixed or fixed charge filling,
and estimate $S_{A}^{(2)}$ using Eq.(10). In both cases, we find that the
ensemble-averaged early-time $\overline{S^{(2)}_{A}(t)}$ exhibits a sub-
ballistic power-law growth with the exponent close to $1/2$. We also evaluate
$P(t)$ by calculating the fraction of the bit string configurations whose
corresponding $X$ and $Y$ particles never meet up to time $t$. The numerics
indicates that $\overline{-\ln P(t)}$ exhibits the same scaling as the one
obtained using Eq.(10). Hence we conclude that $-\ln P(t)$ provides a reliable
approximation for evaluating $S_{A}^{(2)}$ in U(1)-symmetric QA circuit. By
studying the dynamics of the classical particle model, we can obtain valuable
insights into the underlying physics.

Figure 3: $\overline{S_{A}^{(2)}(t)}$ and $\overline{-\ln P(t)}$ in the QA
circuit with $p=0$. We consider two different initial conditions: the charge-
mixed state and the charge-fixed state with the filling factor $\nu=1/3$. We
take the unitary rate $p_{u}=0.5$ in each layer with the system size $L=120$
and subsystem size $L_{A}=L/2=60$. Both the early-time
$\overline{S_{A}^{(2)}(t)}$ and $\overline{-\ln P(t)}$ for the two initial
conditions have $\sqrt{t\ln t}$ scaling.

Figure 4: The illustration of the initial bit string configuration of (a) fast
modes and (b) slow modes with the same particle representation. For
convenience, we consider the bit strings
$\\{|n_{1}\rangle,|n_{1}^{\prime}\rangle\\}$ whose difference represents the
particle configuration where there are only $X$ particles located in the
region $A$ initially.

More careful examination of $\overline{-\ln P(t)}$ and
$\overline{S_{A}^{(2)}(t)}$ reveals that the power law growth exponent is
slightly larger than $1/2$, which has also been observed in the previous study
[28]. Since the growth is constrained by Eq.(1), the power law exponent cannot
exceed $1/2$. We propose that the deviation of the exponent from $1/2$
observed in the numerics is due to the logarithmic correction. As shown in
Fig.3, both quantities are linearly proportional to $\sqrt{t\ln t}$.
Therefore, in our QA circuit,

$\displaystyle\overline{S_{A}^{(2)}(t)}=\lambda_{EE}\sqrt{t\ln t}.$ (15)

To explain the above results, we study the dynamics of the two-species
particle model. Since the dynamics of the $\tilde{X}$ and $\tilde{Y}$
particles are analogous, we will focus on the displacement $\Delta l(t)$ of
the $\tilde{X}$ particle. Different from the case without U(1) symmetry where
the particles move ballistically, in the presence of U(1) symmetry, there are
two distinct modes of $\tilde{X}$ particles depending on the corresponding bit
string configuration to the right of $\tilde{X}$ particle. As illustrated in
Fig. 4, in typical random bit strings whose domains are of $\mathcal{O}(1)$
length, the $\tilde{X}$ particle moves ballistically with a constant velocity.
In contrast, in Fig. 4, for the bit string configurations with domains of
$\mathcal{O}(L)$ length, the $\tilde{X}$ particle only moves diffusively (up
to some logarithmic correction). Such configurations are rare and only
comprise $\mathcal{O}(e^{-L})$ of the bit string ensemble. To verify the
existence of the slow modes, in numerical simulations, we consider the extreme
case where the initial bit string configurations in subsystem $B$ are a single
domain of spin 0’s which is called “dead region”, i.e.,

$\displaystyle|n_{1}(t=0)\rangle$
$\displaystyle=|\alpha_{1}\rangle\otimes|0\rangle^{\otimes|B|},$ (16)
$\displaystyle|n_{1}^{\prime}(t=0)\rangle$
$\displaystyle=|\alpha_{2}\rangle\otimes|0\rangle^{\otimes|B|}.$

We also study $\Delta l(t)$ without the dead region for comparison.

Figure 5: The distance that an endpoint particle travels $\Delta l(t)$ over
time $t$ under U(1)-symmetric QA unitaries for the initial configurations (a)
without the dead region and (b) with the dead region. Without loss of
generality, we take the probability of Fredkin and SWAP gates to be
$p_{u}=0.5$ in each layer, the system size $L=600$, and the number of spin 1’s
to be $L/3$ in (a) and $L_{A}/3$ in (b), both of which have $\nu=1/3$. We find
that (a) $\overline{\Delta l}\propto t$ and (b) $\overline{\Delta
l}\propto\sqrt{t\ln t}$ before saturation.

As shown in Fig. 5, for the configurations without the dead region,
$\overline{\Delta l}$ grows linearly in time. This is responsible for the
ballistic information spreading observed in the out-of-time-ordered correlator
(OTOC) in a similar U(1) symmetric QA circuit [21]. On the other hand, for
configurations with the dead region, $\overline{\Delta l}$ exhibits diffusive
growth over time, with a logarithmic correction, that is,

$\overline{\Delta l}=\lambda_{l}\sqrt{t\ln t}.$ (17)

The diffusive motion of the $\tilde{X}$ particle comes from the diffusive
dynamics of the rightmost charge (spin 1) located at the boundary of the dead
region. In the simple symmetric exclusion process, one of the most basic
models with U(1) symmetry, it is analytically proven that the displacement of
the rightmost charge expands as $\sqrt{t\ln{t}}$ [29]. Despite the greater
complexity of our model involving the Fredkin gate, we believe that the
underlying physics remains fundamentally the same. In Appendix.B, we examine a
simple QA circuit with the kinetic constraint set by SWAP gates only, enabling
an exact mapping of the bit string dynamics to the simple symmetric exclusion
processes. In addition, we also investigate another QA circuit involving a
four-qubit gate. For both models, we provide numerical evidence confirming the
existence of logarithmic corrections in both Eq.(17) and Eq.(15).

Based on this analysis, the bit string pairs that contribute to the purity
(See Eq.(14)) can be divided into two parts, $P(t)=P^{F}(t)+P^{S}(t)$, where
$P^{F}(t)$ and $P^{S}(t)$ are the fractions of fast modes and slow modes
respectively whose $X$ and $Y$ particles have not encountered each other up to
time $t$. Since the distance between the endpoint $X$ and $Y$ particles
decreases by $2\Delta l(t)$ over time $t$, only the configurations whose
initial two species are located a distance $2\Delta l(t)$ apart contribute to
$P(t)$. Therefore, both $P^{F}(t)$ and $P^{S}(t)$ decay as
$\exp(-\overline{\Delta l(t)})$, whereas $P^{F}(t)\propto\exp(-t)$ and
$P^{S}(t)\propto\exp(-\sqrt{t\ln{t}})$. In the absence of U(1) symmetry, the
bit string ensemble comprises only fast modes, which account for the linear
growth of $S_{A}^{(2)}(t)$ as explained earlier. In the presence of U(1)
symmetry, slow modes consisting of the bit string pairs with identical long
domains with length $\mathcal{O}(L)$ between $\tilde{X}$ and $\tilde{Y}$
emerge. Therefore it takes $\mathcal{O}(L^{2})$ time for $\tilde{X}$ particle
to reach $\tilde{Y}$ particle. For
$S_{A}^{(2)}(t)\approx-\ln[P^{F}(t)+P^{S}(t)]$, $P^{F}(t)$ vanishes at time
$\mathcal{O}(L)$, leaving a diffusive growth of $S_{A}^{(2)}(t)$ caused by the
slow modes up to time $\mathcal{O}(L^{2})$.

Figure 6: The coefficient $\lambda_{EE}$ of
$\overline{S_{A}^{(2)}}=\lambda_{EE}\sqrt{t\ln t}$ vs the filling factor
$\nu$. In the inset, we present $\lambda_{l}$ of the endpoint displacement
$\overline{\Delta l}=\lambda_{l}\sqrt{t\ln t}$ of the slow modes with dead
region vs the filling factor $\nu_{A}$ in subsystem $A$.

Similar reasoning can also be applied to the charge-fixed state. In
particular, we can also analyze the dependence of the coefficient
$\lambda_{EE}$ in Eq.(15) on the filling factor $\nu$. As shown in Fig. 6,
$\lambda_{EE}\propto\nu$ for $\nu\lessapprox 0.3$. To understand this
behavior, we investigate the dependence of $P^{S}$ on $\nu$, given by

$P^{S}=\left[{L-2\Delta l\choose\nu L}\bigg{/}{L\choose\nu
L}\right]^{2}=\left[\frac{(L-2\Delta l)!(L-\nu L)!}{(L-2\Delta l-\nu
L)!L!}\right]^{2}.$ (18)

If $\nu\ll 1-2\Delta l/L$, we can approximate as follows:

$\displaystyle-\ln{P^{S}}$ $\displaystyle\approx-2[(L-2\Delta l)\ln{(L-2\Delta
l)}+(L-\nu L)\ln{(L-\nu L)}-(L-2\Delta l-\nu L)\ln{(L-2\Delta l-\nu
L)}-L\ln{L}]$ (19) $\displaystyle=-4\Delta l\ln{\left(1-\frac{\nu L}{L-2\Delta
l}\right)}-2\nu L\ln{\left(1-\frac{2\Delta l}{L-\nu
L}\right)}+\mathcal{O}(\ln{\Delta l})+\mathcal{O}(\ln{\nu L})$
$\displaystyle\approx 8\nu\Delta l+\mathcal{O}(\ln{\Delta
l})+\mathcal{O}(\ln{\nu L}).$

As time evolves, the entanglement entropy is dominated by the slow modes and
has the scaling $S_{A}^{(2)}(t)\approx-\ln{P^{S}(t)}\propto\nu\Delta l$. We
further investigate the dependence of $\Delta l$ on $\nu$. We examine the
displacement of $\tilde{X}$ particle of the bit string configurations with the
dead region as illustrated in Fig. 4. Specifically, we define $\nu_{A}$ to be
the filling factor in subsystem $A$. As shown in Fig. 6, $\Delta l$ remains
largely unaffected by changes in $\nu_{A}$.

Combining these findings, we can explain the numerical observation of the
linear dependence of $\lambda_{EE}$ on $\nu$ when $\nu$ is small. In addition,
we also observe similar linear dependence on $1-\nu$ for $\nu\gtrapprox 0.7$
(not presented in the figure), which can be understood in a similar way.

##  4 Hybrid dynamics

Now we introduce the composite measurements into the circuit and examine its
impact on the entanglement dynamics. As shown in Fig.7, our finding indicates
that when $p$ is small, the entanglement still grows diffusively with a
logarithmic correction, as described in Eq.(15). This can be explained by the
fact that our composite measurement only acts non-trivially on anti-parallel
neighboring sites, while leaving the bit strings with extensively long domains
unaffected. We expect that this scaling behavior persists throughout the
entire volume-law phase. This is different from the single-qubit projective
measurement which quickly destroys the slow modes with dead regions and leads
to the linear growth of $S_{A}^{(2)}$ in the volume-law phase of the non-
unitary U(1) symmetric Haar random circuit[26].

Figure 7: (a) $S_{A}^{(2)}$ vs $\sqrt{t\ln{t}}$ for $p=0.1$ and $p=0.3$ with
the unitary rate $p_{u}=0.5$. (b) $S_{A}^{(2)}$ vs $\ln{t}$ for $p=0.6$, 0.8
and 1 with $p_{u}=0.1$. For both phases, we take the system size $L=120$ and
the filling factor $\nu=1/3$, under the periodic boundary condition (PBC).

As $p$ increases, we observe a decrease in the coefficient $\lambda$ for
$S_{A}^{(2)}(t)$. Eventually, this diffusive growth is replaced by logarithmic
growth. To expand the tuning range for the ratio $p/p_{u}$, we fix $p$ to be a
finite constant and reduce $p_{u}$. Surprisingly, we find that even when
$p_{u}$ approaches zero, the logarithmic scaling persists. This observation
suggests that there is an entanglement phase transition from a volume-law
phase to a critical phase in our model.

Figure 8: The finite-size data collapse $\overline{n(t)}t^{\alpha}$ vs
$t/L^{z}$ at: (a) $p_{u}=0$, $p=0.5$, with the dynamical exponent $z=1.95$ and
$\alpha=0.28$. (b) $p_{u}=0.1$, $p=1$, with $z=2$ and $\alpha=0.27$. (c)
$p_{u}=0.1$, $p=0.4$. (d) $p_{u}=0.4$, $p=1$. Both (c) and (d) have the same
critical exponents $z=2.5$ and $\alpha=0.26$. The data are calculated over a
variety of system sizes with the filling factor $\nu=1/3$ under PBC.

Such a transition to a critical phase is a special feature of QA circuits and
a similar transition has also been observed in the hybrid QA circuit with
discrete $\mathbb{Z}_{2}$ symmetry [11]. To understand this phase transition,
we can analyze the dynamics of the particles characterizing the bit string
pairs difference, in particular, the particle density
$n(t)\equiv\sum_{x}h(x,t)/L$ [11, 12, 13]. In the $\mathbb{Z}_{2}$ symmetric
QA circuit, it is shown that the particles perform branching-annihilating
random walks (BAW) with an even number of offspring:

$W\leftrightarrow 3W,W+W\xrightarrow[]{p}\emptyset.$ (20)

The first process arises from unitary dynamics, while the second annihilation
process occurs due to the measurement. The competition between these processes
gives rise to a phase transition at a critical value $p_{c}$, which falls into
the parity-conserving (PC) universality class. In the absorbing phase with
$p>p_{c}$, the dynamics are primarily governed by the random walking
particles, which annihilate in pairs upon meeting. This particular dynamics
leads to an algebraic decay with $n(t)\sim t^{-0.5}$, resulting in a power-law
decay of $P(t)$. This, in turn, leads to a critical quantum phase
characterized by logarithmic entanglement dynamics and a dynamical exponent of
$z=2$.

In our model, the particle dynamics is similar and still preserves parity, but
is more complicated. For example, for the bit string pair
$\\{|011\rangle,|001\rangle\\}$, under the Fredkin gate, it becomes
$\\{|110\rangle,|001\rangle\\}$ and the particle representation
$\circ\bullet\circ\mapsto\bullet\bullet\bullet$, i.e., the particles branch
from 1 to 3. However, for the bit string pair $\\{|111\rangle,|101\rangle\\}$
which has the same particle representation as the previous one, it will remain
invariant under the Fredkin gate. On the other hand, the SWAP gate enables
particles to diffuse $\bullet\circ\leftrightarrow\circ\bullet$ regardless of
the bit string configuration. Similarly, under the composite measurement, the
particles will experience pair annihilation $\bullet\bullet\mapsto\circ\circ$
only when the bit string pair is $\\{|10\rangle,|01\rangle\\}$ instead of
$\\{|00\rangle,|11\rangle\\}$. This leads to a phase transition belonging to a
different universality class.

We first focus on the absorbing phase. In the limit $p_{u}=0$, we observe that
the particle density follows a power law behavior $n(t)\sim t^{-\alpha}$ with
$\alpha=0.28$ for all measurement rates $p>0$, which is significantly smaller
than the exponent of $0.5$. This is because particle annihilation only occurs
upon measurement based on specific bit string pairs, as illustrated earlier.
More detailed data analysis suggests that $\overline{n(t)}$ for different
system sizes can be collapsed onto the function

$\overline{n(t)}=t^{-\alpha}f(t/L^{z}),$ (21)

where $z=1.95$, close to that of the critical phase observed in
$\mathbb{Z}_{2}$ symmetric QA circuits. We also take a small but finite
$p_{u}$ and similar scaling behavior is observed. For instance, in Fig.8, we
present the data collapse for $p_{u}=0.1$ and $p=1$, yielding exponents
$\alpha=0.27$ and $z=2$. This absorbing phase with algebraic decay is
responsible for a power law decay of $P(t)$, which further leads to a quantum
critical phase with a logarithmic entanglement scaling. Notice that in this
phase, $P(t)$ is mainly determined by the fraction $P^{F}(t)$ without
extensively long domains.

We further analyze the transition point. By fixing the unitary rate at
$p_{u}=0.1$ and decreasing the measurement rate $p$, we can numerically
identify the critical point. In Fig.8, it is observed that the critical point
occurs at approximately $p=0.4$, with $\alpha=0.26$ and $z=2.5$. The existence
of the phase transition persists as the unitary rate $p_{u}$ increases, until
it reaches $p_{u}=0.4$. At this point (as illustrated in Fig.8), the critical
point is found at the maximum allowed measurement rate $p=1$. It is worth
emphasizing that this phase transition does not fall into the PC universality
class, where the critical exponents are $z^{PC}=1.744$ and
$\alpha^{PC}=0.286$.

##  5 Discussions and outlook

In this paper, we investigate the entanglement dynamics of U(1) symmetric QA
circuits. We show that the second Rényi entropy saturates the upper bound
introduced in Eq.(1), namely $S_{A}^{(2)}(t)\propto\sqrt{t\ln{t}}$. To
understand this behavior, we map the entanglement dynamics to a classical bit
string model and demonstrate that the diffusive dynamics of $S_{A}^{(2)}(t)$
is caused by bit strings containing extensively long domains.

Additionally, we explore the monitored entanglement dynamics under U(1)
symmetry and identify a phase transition from a volume-law phase to a critical
phase as the measurement rate $p$ increases. Within the volume-law phase,
$S_{A}^{(2)}(t)$ continues to exhibit diffusive growth due to the presence of
bit strings with long domains that remain unaffected by the introduced
measurements. On the other hand, the critical phase is characterized by
logarithmic scaling of the entanglement, and its stability is ensured by both
the U(1) symmetry and the basis-preserving nature of QA circuits.

The analysis of the second Rényi entropy can be extended to higher integer
Rényi indices $n$. By making slight modifications to Eq.(10), it can be shown
straightforwardly that the evolution of $S^{(n)}_{A}$ is mapped to a classical
dynamics that encompasses n copies of bit strings. In particular, the volume-
law phase exhibits diffusive dynamics in the presence of a logarithmic
correction. This behavior is governed by these bit string configurations with
extensive long domains.

Notably, in both the volume-law phase and critical phase, the spin transport
exhibits diffusive dynamics and fails to capture the entanglement phase
transition. This is because this measurement-induced transition is visible
solely in the non-linear observable of the density matrix. We confirm this
diffusive transport by numerically computing the correlation functions and the
detailed results are presented in Appendix.C.

It has been established that the volume-law phase can be alternatively
understood as a quantum error correcting code [18, 30, 31, 32, 33]. Previously
in Ref. [12], we presented an interpretation of the quantum error correction
property of the volume-law phase of a generic QA circuit, relating it to the
dynamics of classical bit strings. Furthermore, we demonstrated a connection
between quantum error-correcting codes and classical linear error-correcting
codes. In the case of the QA circuit with U(1) symmetry, we expect that the
volume-law phase continues to exhibit the characteristics of a quantum error-
correcting code. In particular, the dynamics of the associated classical bit
strings reveal that the difference between bit string pairs still preserves
classical information in a non-local manner, thereby functioning as a
classical error-correcting code. It is worth noting that, unfortunately, this
classical error-correcting code is no longer linear. We leave this for future
study.

## Acknowledgements

We thank Hisanori Oshima and Ethan Lake for the useful discussions. This
research is supported in part by the Google Research Scholar Program and is
supported in part by the National Science Foundation under Grant No.
DMR-2219735. We gratefully acknowledge computing resources from Research
Services at Boston College and the assistance provided by Wei Qiu.

##  Appendix A Two-species particle model

In Ref. [11] and Ref. [12], we proposed a two-species particle model which
maps the entanglement dynamics of hybrid QA circuits under different
symmetries to a classical two-species particle model. Before introducing the
particle model, we will first give an overview of the classical bit string
dynamics.

Since the charge-mixed state $|\psi_{0}\rangle=|+x\rangle^{L}$ has already
been discussed in the main text, in this section we will focus on a charge-
fixed initial state $|\psi_{0}\rangle=\frac{1}{\sqrt{N}}\sum_{n}|n\rangle$,
where
$\\{|n\rangle=|\sigma_{1}\sigma_{2}\dots\sigma_{L}\rangle:\sigma_{i}=\\{0,1\\},\sum_{i}\sigma_{i}=Q\\}$
is the set of basis states with a fixed filling factor $\nu$, and $N={L\choose
Q}=L!/[(L-Q)!Q!]$.

Then, the wavefunction at time $t$ can be represented as
$|\psi_{t}\rangle=\tilde{U}_{t}|\psi_{0}\rangle$, where
$\tilde{U}_{t}=M_{t}U_{t}M_{t-1}U_{t-1}\cdots$ represents the hybrid QA
circuit of depth $t$ as an alternating combination of layers of measurements
and unitary evolution. Recall that the second Rényi entropy
$S_{A}^{(2)}=-\log_{2}{\text{Tr}(\rho_{A}^{2})}$. The purity
$\text{Tr}(\rho_{A}^{2})$ equals the expectation value of the
$\mathsf{SWAP}_{A}$ operator over two copies of the state [27, 34],

$\text{Tr}[\rho_{A}^{2}(t)]=\langle\psi_{t}|_{2}\otimes\langle\psi_{t}|_{1}\mathsf{SWAP}_{A}|\psi_{t}\rangle_{1}\otimes|\psi_{t}\rangle_{2}.$
(22)

The wave function can be partitioned into subregions $A$ and $B$

$\displaystyle|\psi_{t}\rangle=\tilde{U}_{t}|\psi_{0}\rangle=\frac{1}{\sqrt{N}}\sum_{i,j}e^{i\theta_{ij}}|\alpha_{i}\rangle_{A}|\beta_{j}\rangle_{B},$
(23)

The $\mathsf{SWAP}_{A}$ operator exchanges the spin configurations
$|\alpha\rangle$ within subsystem $A$ of the double copies of
$|\psi_{t}\rangle$. Then, we insert two sets of complete basis which we call
“bit strings” [13],

$\displaystyle\text{Tr}[\rho_{A}^{2}(t)]$
$\displaystyle=\sum_{n_{1},n_{2}}\langle\psi_{t}|_{2}\langle\psi_{t}|_{1}\mathsf{SWAP}_{A}|n_{1}\rangle|n_{2}\rangle\langle
n_{2}|\langle n_{1}|\psi_{t}\rangle_{1}|\psi_{t}\rangle_{2}$ (24)
$\displaystyle=\sum_{n_{1},n_{2}}\langle\psi_{0}|_{1}\tilde{U}_{t}^{\dagger}|n_{1}^{\prime}\rangle\langle\psi_{0}|_{2}\tilde{U}_{t}^{\dagger}|n_{2}^{\prime}\rangle\langle
n_{1}|\tilde{U}_{t}|\psi_{0}\rangle_{1}\langle
n_{2}|\tilde{U}_{t}|\psi_{0}\rangle_{2}$

where

$\displaystyle|n_{1}^{\prime}\rangle|n_{2}^{\prime}\rangle$
$\displaystyle=\mathsf{SWAP}_{A}|n_{1}\rangle|n_{2}\rangle$ (25)
$\displaystyle=\mathsf{SWAP}_{A}|\alpha_{1}\beta_{1}\rangle|\alpha_{2}\beta_{2}\rangle$
$\displaystyle=|\alpha_{2}\beta_{1}\rangle|\alpha_{1}\beta_{2}\rangle.$

Since the dynamics preserves the total charge of the state, only the bit
string configurations
$\\{|n_{1}\rangle,|n_{2}\rangle,|n_{1}^{\prime}\rangle,|n_{2}^{\prime}\rangle\\}$
with the same filling factor $\nu$ will have non-zero overlap with
$\langle\psi_{0}|$. We represent this subset of bit strings as

$\\{n_{1},n_{2}\\}^{\nu}=\\{|n_{1}\rangle=|\sigma^{0}_{1}\dots\sigma^{0}_{L}\rangle,|n_{2}\rangle=|\sigma^{1}_{1}\dots\sigma^{1}_{L}\rangle:\sum_{i=1}^{L}\sigma_{i}^{\mu}=\nu
L,\sum_{i=1}^{L_{A}}\sigma_{i}^{\mu}+\sum_{i=L_{A}+1}^{L}\sigma_{i}^{1-\mu}=\nu
L,\forall\mu\in\\{0,1\\}\\}.$ (26)

Therefore, Eq.(24) becomes

$\text{Tr}[\rho_{A}^{2}(t)]=\frac{1}{N^{2}}\sum_{\\{n_{1},n_{2}\\}^{\nu}}e^{-i\Theta_{n_{1}^{\prime}}(t)}e^{-i\Theta_{n_{2}^{\prime}}(t)}e^{i\Theta_{n_{1}}(t)}e^{i\Theta_{n_{2}}(t)}.$
(27)

Strictly speaking, there does not exist $\tilde{U}_{t}^{\dagger}$ since the
projective measurements are nonunitary operators. However, we can still deduce
the effective action of the composite measurement $M_{1/2}$ on sites $i$ and
$i+1$ of the bit string $|n\rangle\in\\{n_{1},n_{2}\\}^{\nu}$,

$\langle n|M_{1/2}|\psi\rangle=\begin{cases}\langle
n|\psi_{0}\rangle=\frac{1}{\sqrt{N}}e^{i\theta_{n}}&\text{if
$\sigma_{i}=\sigma_{i+1}$}\\\ \langle
T_{1/2}(n)|\psi_{0}\rangle=\frac{1}{\sqrt{N}}e^{i\theta_{T_{1/2}(n)}}&\text{if
$\sigma_{i}\neq\sigma_{i+1}$}\end{cases}$ (28)

where $\langle T_{1/2}|$ stands for the bit string $\langle n|$ with the spins
on sites $i$ and $i+1$ forced to be in the $|01\rangle/|10\rangle$ state.
Hence, instead of following the quantum trajectory of $|\psi_{t}\rangle$, we
can study the bit string dynamics in a time-reversed order, i.e., evaluate
$\langle n|\tilde{U}_{t}|\psi_{0}\rangle$ from left to right,

$\displaystyle\langle n|\tilde{U}_{t}|\psi_{0}\rangle$ $\displaystyle=\langle
n(t^{\prime}=0)|M_{t}U_{t}M_{t-1}U_{t-1}\cdots|\psi_{0}\rangle$ (29)
$\displaystyle=\langle
n(t^{\prime}=1)|U_{t}M_{t-1}U_{t-1}\cdots|\psi_{0}\rangle$
$\displaystyle=e^{i\theta_{n(t^{\prime}=1)}}\langle
n(t^{\prime}=1)|M_{t-1}U_{t-1}\cdots|\psi_{0}\rangle$ $\displaystyle\cdots$
$\displaystyle=\frac{1}{\sqrt{N}}e^{i\theta_{n(t^{\prime}=1)}}e^{i\theta_{n(t^{\prime}=2)}}\cdots
e^{i\theta_{n(t^{\prime}=t)}}$
$\displaystyle=\frac{1}{\sqrt{N}}e^{i\Theta_{n}(t)},$

where $e^{i\Theta_{n}(t)}$ is one of the accumulated phase terms under time
evolution that are multiplied and summed up over the ensemble of
$\\{|n_{1}\rangle,|n_{2}\rangle\\}^{\nu}$ in Eq. (27) to evaluate
$\text{Tr}\rho_{A}^{2}$.

Before (after) Fredkin | $n_{1}$ | $|110\rangle$ | $|110\rangle$ | $|110\rangle$ | $|110\rangle$ | $|110\rangle$ | $|110\rangle$ | $|110\rangle$ | $|101\rangle$ | $|101\rangle$ | $|101\rangle$ | $|101\rangle$
---|---|---|---|---|---|---|---|---|---|---|---|---
$n_{2}$ | $|101\rangle$ | $|011\rangle$ | $|100\rangle$ | $|010\rangle$ | $|001\rangle$ | $|111\rangle$ | $|000\rangle$ | $|100\rangle$ | $|010\rangle$ | $|001\rangle$ | $|111\rangle$
$|n_{1}-n_{2}|$ | $\circ\bullet\bullet$ | $\bullet\circ\bullet$ | $\circ\bullet\circ$ | $\bullet\circ\circ$ | $\bullet\bullet\bullet$ | $\circ\circ\bullet$ | $\bullet\bullet\circ$ | $\circ\circ\bullet$ | $\bullet\bullet\bullet$ | $\bullet\circ\circ$ | $\circ\bullet\circ$
After (before) Fredkin | $n_{1}$ | $|011\rangle$ | $|011\rangle$ | $|011\rangle$ | $|011\rangle$ | $|011\rangle$ | $|011\rangle$ | $|011\rangle$ | $|101\rangle$ | $|101\rangle$ | $|101\rangle$ | $|101\rangle$
$n_{2}$ | $|101\rangle$ | $|110\rangle$ | $|100\rangle$ | $|010\rangle$ | $|001\rangle$ | $|111\rangle$ | $|000\rangle$ | $|100\rangle$ | $|010\rangle$ | $|001\rangle$ | $|111\rangle$
$|n_{1}-n_{2}|$ | $\bullet\bullet\circ$ | $\bullet\circ\bullet$ | $\bullet\bullet\bullet$ | $\circ\circ\bullet$ | $\circ\bullet\circ$ | $\bullet\circ\circ$ | $\circ\bullet\bullet$ | $\circ\circ\bullet$ | $\bullet\bullet\bullet$ | $\bullet\circ\circ$ | $\circ\bullet\circ$
Before (after) Fredkin | $n_{1}$ | $|101\rangle$ | $|100\rangle$ | $|100\rangle$ | $|100\rangle$ | $|100\rangle$ | $|010\rangle$ | $|010\rangle$ | $|010\rangle$ | $|001\rangle$ | $|001\rangle$ | $|111\rangle$
$n_{2}$ | $|000\rangle$ | $|010\rangle$ | $|001\rangle$ | $|111\rangle$ | $|000\rangle$ | $|001\rangle$ | $|111\rangle$ | $|000\rangle$ | $|111\rangle$ | $|000\rangle$ | $|000\rangle$
$|n_{1}-n_{2}|$ | $\bullet\circ\bullet$ | $\bullet\bullet\circ$ | $\bullet\circ\bullet$ | $\circ\bullet\bullet$ | $\bullet\circ\circ$ | $\circ\bullet\bullet$ | $\bullet\circ\bullet$ | $\circ\bullet\circ$ | $\bullet\bullet\circ$ | $\circ\circ\bullet$ | $\bullet\bullet\bullet$
After (before) Fredkin | $n_{1}$ | $|101\rangle$ | $|100\rangle$ | $|100\rangle$ | $|100\rangle$ | $|100\rangle$ | $|010\rangle$ | $|010\rangle$ | $|010\rangle$ | $|001\rangle$ | $|001\rangle$ | $|111\rangle$
$n_{2}$ | $|000\rangle$ | $|010\rangle$ | $|001\rangle$ | $|111\rangle$ | $|000\rangle$ | $|001\rangle$ | $|111\rangle$ | $|000\rangle$ | $|111\rangle$ | $|000\rangle$ | $|000\rangle$
$|n_{1}-n_{2}|$ | $\bullet\circ\bullet$ | $\bullet\bullet\circ$ | $\bullet\circ\bullet$ | $\circ\bullet\bullet$ | $\bullet\circ\circ$ | $\circ\bullet\bullet$ | $\bullet\circ\bullet$ | $\circ\bullet\circ$ | $\bullet\bullet\circ$ | $\circ\circ\bullet$ | $\bullet\bullet\bullet$
Table 1: The update rule of the particle configurations under the Fredkin gate $|\sigma_{i-1}1_{i}\sigma_{i+1}\rangle\mapsto|\sigma_{i+1}1_{i}\sigma_{i-1}\rangle$. The Fredkin gate only acts nontrivially on bit strings with $\sigma_{i}=1$ and $\sigma_{i-1}\neq\sigma_{i+1}$. For bit string pairs containing such configurations, the corresponding particles either branch/annihilate in pairs, i.e., $\bullet\bullet\bullet\leftrightarrow\circ\bullet\circ$, or diffuse i.e., $\circ\bullet\bullet\leftrightarrow\bullet\bullet\circ$, or $\bullet\circ\circ\leftrightarrow\circ\circ\bullet$. Before $M_{1/2}$ | $n_{1}$ | $|10\rangle$ | $|10\rangle$ | $|10\rangle$ | $|01\rangle$ | $|01\rangle$ | $|11\rangle$
---|---|---|---|---|---|---|---
$n_{2}$ | $|01\rangle$ | $|11\rangle$ | $|00\rangle$ | $|11\rangle$ | $|00\rangle$ | $|00\rangle$
$|n_{1}-n_{2}|$ | $\bullet\bullet$ | $\circ\bullet$ | $\bullet\circ$ | $\bullet\circ$ | $\circ\bullet$ | $\bullet\bullet$
After $M_{1}$ | $n_{1}$ | $|01\rangle$ | $|01\rangle$ | $|01\rangle$ | $|01\rangle$ | $|01\rangle$ | $|11\rangle$
$n_{2}$ | $|01\rangle$ | $|11\rangle$ | $|00\rangle$ | $|11\rangle$ | $|00\rangle$ | $|00\rangle$
$|n_{1}-n_{2}|$ | $\circ\circ$ | $\bullet\circ$ | $\circ\bullet$ | $\bullet\circ$ | $\circ\bullet$ | $\bullet\bullet$
After $M_{2}$ | $n_{1}$ | $|10\rangle$ | $|10\rangle$ | $|10\rangle$ | $|10\rangle$ | $|10\rangle$ | $|11\rangle$
$n_{2}$ | $|10\rangle$ | $|11\rangle$ | $|00\rangle$ | $|11\rangle$ | $|00\rangle$ | $|00\rangle$
$|n_{1}-n_{2}|$ | $\circ\circ$ | $\circ\bullet$ | $\bullet\circ$ | $\circ\bullet$ | $\bullet\circ$ | $\bullet\bullet$
Table 2: The update rule of the particle configurations under the composite
measurement $M_{1/2}=R\circ P_{1/2}$. Note that the composite measurement only
acts nontrivially on bit string configurations with anti-parallel neighboring
spins. For bit string pairs containing such configurations, under the
measurement, the corresponding particle representation either annihilates in
pairs, i.e., $\bullet\bullet\mapsto\circ\circ$, or diffuses, i.e.,
$\bullet\circ\leftrightarrow\circ\bullet$.

In order to understand the dynamics of the relative phase
$\Theta_{r}=-\Theta_{n_{1}^{\prime}}-\Theta_{n_{2}^{\prime}}+\Theta_{n_{1}}+\Theta_{n_{2}}$,
we consider the evolution of the difference between a bit string pair
$\\{|n_{1}\rangle,|n_{2}\rangle\\}$,

$h(x,t)=|n_{1}(x,t)-n_{2}(x,t)|.$ (30)

It is then natural to use the particle representation where the empty site
symbol $\circ$ denotes $h(x)=0$ and the occupied site symbol $\bullet$ denotes
$h(x)=1$. Specifically, we represent the difference at $t=0$ in subregion $A$
(subregion $B$) by $X$ ($Y$) particles. Let $\tilde{X}/\tilde{Y}$ denote the
rightmost $X$ (leftmost $Y$) particle and $x/y$ denote their positions. As
shown in Fig. 2, under the time evolution, the particles start to evolve
according to the update rule. As mentioned in the main text, to determine the
update rule of the particles under the U(1)-symmetric circuit, one should
always refer to the corresponding bit string dynamics. The detailed update
rule is listed in Table.1 and Table.2. Before the two species encounter each
other, the phase generated by each layer of unitary evolution on $|n\rangle$
is $\theta_{n}=\theta_{n}^{[1,x]}+\theta_{n}^{(x,y)}+\theta_{n}^{[y,L]}$,
i.e., the sum of phases generated within the regimes $[1,x]$, $(x,y)$ and
$[y,L]$. The bit string configurations within $[1,x]$ occupied by $X$
particles always satisfy $n_{1}([1,x])=n_{2}^{\prime}([1,x])$ and
$n_{2}([1,x])=n_{1}^{\prime}([1,x])$. Therefore,
$\theta_{n_{1}}^{[1,x]}=\theta_{n_{2}^{\prime}}^{[1,x]}$ and
$\theta_{n_{2}}^{[1,x]}=\theta_{n_{1}^{\prime}}^{[1,x]}$. Similarly, for the
regime occupied by $Y$ particles, since $n_{1}([y,L])=n_{1}^{\prime}([y,L])$
and $n_{2}([y,L])=n_{2}^{\prime}([y,L])$, we always have
$\theta_{n_{1}}^{[y,L]}=\theta_{n_{1}^{\prime}}^{[y,L]}$ and
$\theta_{n_{2}}^{[y,L]}=\theta_{n_{2}^{\prime}}^{[y,L]}$. At the same time,
since there is no bit string difference within the regime $(x,y)$,
$\theta_{n_{1}}^{(x,y)}=\theta_{n_{2}}^{(x,y)}=\theta_{n_{1}^{\prime}}^{(x,y)}=\theta_{n_{2}^{\prime}}^{(x,y)}$.
Therefore, the phase difference along the lattice vanishes:
$-\theta_{n_{1}^{\prime}}-\theta_{n_{2}^{\prime}}+\theta_{n_{1}}+\theta_{n_{2}}=0$.
If for a bit string pair $\\{|n_{1}\rangle,|n_{2}\rangle\\}$, $X$ and $Y$
particles do not meet each other up to time $t$, then the accumulated relative
phase $\Theta_{r}(t)$ is zero and such pair contributes $1/4^{L}$ to the
purity $\text{Tr}[\rho_{A}^{2}(t)]$.

Once the rightmost $X$ particle comes across the leftmost $Y$ particle, the
two-qubit phase gate acting on sites $x$ and $y$ will generate a nonzero
relative phase. For example, if we apply the CZ gate on
$\bullet\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\bullet}$
with a possible corresponding bit string configuration
$\\{|n_{1}\rangle,|n_{2}\rangle,|n_{1}^{\prime}\rangle,|n_{2}^{\prime}\rangle\\}_{x,y}=\\{|10\rangle,|01\rangle,|00\rangle,|11\rangle\\}$,
a relative phase $0+0-0-\pi=-\pi$ is generated. If we apply the CNOT gate on
sites $x$ and $y$,
$\\{|n_{1}\rangle,|n_{2}\rangle,|n_{1}^{\prime}\rangle,|n_{2}^{\prime}\rangle\\}_{x,y}\to\\{|11\rangle,|01\rangle,|00\rangle,|10\rangle\\}$,
i.e., another type of “particle” different from the two species with bit
string configuration
$|n_{1}\rangle_{y}=|n_{2}\rangle_{y}\neq|n_{1}^{\prime}\rangle_{y}=|n_{2}^{\prime}\rangle_{y}$
appears on site $y$ and will spread along the lattice under further evolution.
As time evolves, the configurations for which the two species have met will
generate random accumulated phases, half of which are composed of odd numbers
of $\pi$, while the other half are composed of even numbers of $\pi$. The
accumulated phase terms $e^{i\Theta_{r}}$ of such configurations will add up
to zero and make no contribution to Eq. (27). Therefore, we have

$\displaystyle\text{Tr}\rho_{A}^{2}(t)\approx P(t),$ (31) $\displaystyle
S_{A}^{(2)}(t)\approx-\ln{P(t)},$

where $P(t)$ is the fraction of particle configurations in which $X$ and $Y$
particles never encounter one another up to time $t$. This quantum-classical
correspondence has been numerically verified in Ref. [11].

##  Appendix B Entanglement dynamics of U(1) symmetric QA circuits with other
kinetic constraints

###  B.1 The SWAP Model

Figure 9: The arrangement of gates of the SWAP model in a time step. The
dashed box encloses the gates within a single layer. Each time step involves
two layers of SWAP gates with probability $p_{u}=0.5$, and CZ gates.

We study the entanglement dynamics under the kinetic constraint determined by
SWAP gates solely, so that our circuit is a U(1) symmetric Clifford QA
circuit, which can be efficiently simulated at large system sizes using
stabilizer formalism [35]. We will only consider the unitary dynamics as the
composite measurements can not be simulated by stabilizer formalism. As
illustrated in Fig.9, we consider a circuit where each time step involves two
layers of SWAP gates and CZ gates, with each applying on odd/even sites. To
achieve enough randomness, we take the SWAP rate to be $p_{u}=0.5<1$. In
addition, we consider $|\psi_{0}\rangle=|+x\rangle^{\otimes L}$ as the initial
state and take $L=600$. As shown in Fig.10, the entanglement entropy grows
diffusively with a logarithmic correction, i.e.,
$\overline{S_{A}^{(2)}}\propto\sqrt{t\ln{t}}$.

Figure 10: (a) $\overline{S_{A}^{(2)}(t)}$ of the SWAP model simulated using
the stabilizer formalism. (b) The $\tilde{X}$ particle displacement
$\overline{\Delta l(t)}$ of the slow modes. It is found that both
$\overline{S_{A}^{(2)}(t)}$ and $\overline{\Delta l(t)}$ scale as
$\sqrt{t\ln{t}}$. We take the system size $L=600$ and
$|\psi_{0}\rangle=|+x\rangle^{\otimes L}$ for (a) and the bit strings with
dead regions with the charge filling $\nu_{A}=1/3$ in subsystem $A$ for (b).

On the other hand, we examine the endpoint displacement $\overline{\Delta
l(t)}$ of the bit strings with the dead region with a charge filling factor of
$\nu_{A}=1/3$. The numerics in Fig.10 indicates that $\overline{\Delta
l(t)}\propto\sqrt{t\ln{t}}$ as well. This scaling can be explained by exactly
mapping the spin dynamics of the SWAP model to the simple symmetric exclusion
processes. In this process, each charge undertakes a symmetric random walk,
while being prohibited from jumping to an already occupied site. It has been
analytically shown in Ref. [29] that the position of the rightmost charge
expands as $\sqrt{t\ln t}$.

It is worth noting that Clifford circuits with U(1) symmetry are highly
restricted. For example, all Rényi entropies are equal for Clifford circuits,
which fails to capture the ballistic growth of von Neumann entropy for generic
U(1) symmetric random circuits. Nevertheless, the SWAP model enables us to
verify at a large system size that the second Rényi entropy indeed saturates
the upper bound in Eq. (1) and is dominated by rare slow modes with
extensively long domains.

###  B.2 The Four-qubit CSWAP model

Figure 11: The arrangement of gates of the Four-qubit CSWAP model in a time
step. The dashed box encloses the gates within a single layer. Each time step
involves four layers of the four-qubit CSWAP gates with probability $p_{u}$,
and CZ gates, interspersed with measurements with probability $p$.

Figure 12: $-\overline{\ln{Q}}$ of the Four-qubit CSWAP model at the
measurement rate (a) $p=0$ and $p=0.1$, and (b) $p=0.7$ and $p=0.9$. We
observe that $-\overline{\ln Q}\propto\sqrt{t\ln{t}}$ for (a) and
$-\overline{\ln Q}\propto\ln{t}$ for (b). We take the system size $L=120$,
$p_{u}=0.5$, and charge filling $\nu=1/2$. Figure 13: The endpoint
displacement $\overline{\Delta l(t)}$ of the bit strings with the dead region
of the Four-qubit CSWAP model vs $\sqrt{t\ln{t}}$ at $p=0$ and $p=0.1$. We
take the system size $L=600$, $p_{u}=0.5$, and $\nu_{A}=1/2$.

Now we consider the entanglement dynamics of the U(1) symmetric QA circuits
with the kinetic constraint determined by a four-qubit gate, which swaps the
spins $\sigma_{2}$ and $\sigma_{3}$ if the first spin $\sigma_{1}=0$ or the
fourth spin $\sigma_{4}=1$. It is also called Fredkin gate in other works [28,
36], to distinguish it from the Fredkin gate in the main text, we call it the
Four-qubit CSWAP gate. As shown in Fig.11, each time step of the circuit
consists of four layers of gates under PBC and in each layer, the Four-qubit
CSWAP gates are applied on sites
$\\{4j-3,4j-2,4j-1,4j\\}/\\{4j-2,4j-1,4j,4j+1\\}/\\{4j-1,4j,4j+1,4j+2\\}/\\{4j,4j+1,4j+2,4j+3\\}$
for $j\in[1,L/4]$ with probability $p_{u}=0.5$, and the CZ gates are applied
on sites $\\{2j-1,2j\\}/\\{2j,2j+1\\}/\\{2j-1,2j\\}/\\{2j,2j+1\\}$,
interspersed with composite measurements with probability $p$ applied on both
odd and even sites.

To simplify the numerical simulation, we can fix the position of $\tilde{Y}$
particle to be the boundary between subsystems $A$ and $B$. This is equivalent
to focusing on the single species picture where there are only $X$ particles
and considering a subset of the phase difference in Eq. (27), i.e., the phase
difference of $|n_{1}\rangle$ and $|n_{1}^{\prime}\rangle$ restricted in
regime $B$, denoted as

$Q\equiv\frac{1}{M}\sum_{n_{1},n_{1}^{\prime}}e^{-i\Theta_{n_{1}^{\prime}}^{B}+i\Theta_{n_{1}}^{B}},$
(32)

where $M$ is the number of bit string pairs
$\\{|n_{1}\rangle,|n_{1}^{\prime}\rangle\\}$. With this approximation, the
configurations which contribute to $Q$ are those whose $\tilde{X}$ particles
never reach the middle cut.

We numerically simulate $-\ln{Q(t)}$ for an ensemble of bit strings
$\\{|n_{1}\rangle,|n_{1}^{\prime}\rangle\\}$ with system size $L=120$ and
charge filling $\nu=1/2$. As shown in Fig.12, there exists a similar phase
transition from a volume-law phase to a critical phase: when $p<p_{c}$,
$-\overline{\ln Q}\propto\sqrt{t\ln{t}}$, and when $p>p_{c}$, $-\overline{\ln
Q}\propto\ln{t}$. We believe that this applies to $\overline{S_{A}^{(2)}(t)}$
as well.

Finally, we study the endpoint displacement $\Delta l(t)$ of the bit strings
with the dead region evolved under the Four-qubit CSWAP gates. The numerics in
Fig.13 confirms that $\overline{\Delta l}\propto\sqrt{t\ln{t}}$ for pure
unitary dynamics and small measurement rate $p=0.1$. We believe that this
diffusive growth persists in the whole volume-law phase.

##  Appendix C Spin transport of hybrid QA circuits with U(1) symmetry

Figure 14: The correlation function $C(0,t)$ of (a) the Fredkin-SWAP model and
(b) the Four-qubit CSWAP model. We take the system size $L=300$ and
$p_{u}=0.1$, $\nu=1/3$ for (a), and $p_{u}=0.5$, $\nu=1/2$ for (b). It is
found that for (a), $C(0,t)\propto t^{-0.5}$ for all $p$, and for (b),
$C(0,t)\propto t^{-0.41}$ for $p=0$ and $C(0,t)\propto t^{-0.5}$ for $p>0$.

In this section, we consider the transport properties of the conserved charges
which can be characterized by the spin correlation function

$C(x,t)=\langle Z_{x}(t)Z_{0}(0)\rangle-\langle Z_{x}(t)\rangle\langle
Z_{0}(0)\rangle,$ (33)

where site 0 is in the middle of the system. If we consider a charge-fixed
state $|\psi_{0}\rangle=\frac{1}{\sqrt{N}}\sum_{n}|n\rangle$ with the filling
factor $\nu$, then the correlator of our QA circuit can be sampled using the
classical bit strings via

$\displaystyle C(x,t)$ $\displaystyle=\frac{1}{N}\sum_{n}\langle
n|\tilde{U}_{t}^{\dagger}Z_{x}(0)\tilde{U}_{t}Z_{0}(0)|n\rangle-\frac{1}{N}\sum_{n}\langle
n|\tilde{U}_{t}^{\dagger}Z_{x}(0)\tilde{U}_{t}|n\rangle\frac{1}{N}\sum_{m}\langle
m|Z_{0}(0)|m\rangle$ (34)
$\displaystyle=\frac{1}{N}\sum_{n}Z_{n(t),x}Z_{n,0}-\frac{1}{N^{2}}\sum_{n}Z_{n(t),x}\sum_{m}Z_{m,0},$

where $Z_{n(t),x}$ is the spin value at site $x$ of the bit string
$|n(t)\rangle$ at time $t$. The correlation function for different system
sizes can be collapsed onto the scaling form

$C(x,t)=t^{-1/z}f(x/t^{1/z}).$ (35)

Here we will only focus on the correlation in the time direction $C(0,t)$ for
the Fredkin-SWAP model and the Four-qubit CSWAP model, and observe the
dynamical exponents as we vary the measurement rate $p$.

As shown in Fig.14, for the Fredkin-SWAP model, $C(0,t)\propto t^{-1/z}$ with
$z=2$ for all $p$. The spin transport is diffusive with and without the
measurements and hence fails to reflect the measurement-induced entanglement
phase transition. This is because MIPTs are only visible in observables that
are nonlinear in the density matrix. Similar diffusive dynamics is observed
for all $p>0$ in the Four-qubit CSWAP model, as shown in Fig.14.
Interestingly, when $p=0$, the dynamical exponent $z=2.44>2$, consistent with
$z=8/3$ as discovered in previous literature [28, 36]. Nevertheless, the spin
at the boundary of the extensively long domain still exhibits diffusive
dynamics with a logarithmic correction.

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