# Orthogonal polynomials in weighted Bergman spaces

Erwin Miña-Díaz Corresponding author; Email<EMAIL_ADDRESS>The
University of Mississippi, Department of Mathematics, Hume Hall 305,P. O. Box
1848, University, MS 38677-1848, USA.

###### Abstract

Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form

$w(z)=|v(z)|^{2}\prod_{k=1}^{s}\left|\frac{z-a_{k}}{1-z\overline{a}_{k}}\right|^{m_{k}}\,,\quad
m_{k}>-2,\ |a_{k}|<1,$

where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$, and let
$(p_{n})_{n=0}^{\infty}$ be the sequence of polynomials ($p_{n}$ of degree
$n$) orthonormal over $\mathbb{D}$ with respect to $w$. We give an integral
representation for $p_{n}$ from which it is in principle possible to derive
its asymptotic behavior as $n\to\infty$ at every point $z$ of the complex
plane, the asymptotic analysis of the integral being primarily dependent on
the nature of the first singularities encountered by the function
$v(z)^{-1}\prod_{k=1}^{s}(1-z\overline{a}_{k})^{-1}$.

Keywords: Bergman orthogonal polynomials, integral representation, asymptotic
behavior, strong asymptotics.

##  1 Introduction

Let $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ be the unit disk, and let
$\mathbb{T}:=\partial\mathbb{D}$ be the unit circle. We use the letter
$\sigma$ to denote the area measure divided by $\pi$, so that
$\sigma(\mathbb{D})=1$.

A function $w(z)\geq 0$ defined on $\mathbb{D}$ is said to be a weight,
provided that $0<\int_{\mathbb{D}}wd\sigma<\infty$, and that for every disk
$D(z_{0},\epsilon)$ centered at a point $z_{0}\in\mathbb{T}$ of radius
$\epsilon>0$,

$\int_{D(z_{0},\epsilon)\cap\mathbb{D}}w(z)d\sigma(z)>0.$

We write $\mathcal{H}(\mathbb{D})$ for the linear space of holomorphic
functions in $\mathbb{D}$. For a weight $w$ in $\mathbb{D}$, the weighted
Bergman space $A^{2}_{w}$ is the subspace of $\mathcal{H}(\mathbb{D})$ defined
as

$A^{2}_{w}:=\left\\{f\in\mathcal{H}(\mathbb{D}):\int_{\mathbb{D}}|f(z)|^{2}w(z)d\sigma(z)<\infty\right\\}.$

The space $A^{2}_{w}$ is naturally endowed with the inner product

$\displaystyle\langle
f,g\rangle_{w}:=\int_{\mathbb{D}}f(z)\overline{g(z)}w(z)d\sigma(z),$ (1.1)

which, by means of the Gram-Schmidt orthogonalization process applied to the
powers $(z^{n})_{n=0}^{\infty}$, produces a unique sequence of polynomials
$(p_{n})_{n=0}^{\infty}$, $p_{n}$ of degree $n$ and positive leading
coefficient, such that

$\displaystyle\langle p_{n},p_{m}\rangle_{w}=\delta_{n,m},\quad n,m\geq 0.$
(1.2)

These polynomials are often referred to as Bergman polynomials. We will denote
the leading coefficient of $p_{n}$ by $\gamma_{n}$:

$p_{n}(z)=\gamma_{n}z^{n}+\cdots,\quad\gamma_{n}>0.$

In this paper, we study the asymptotic behavior of $p_{n}$ as $n\to\infty$ for
weights of the form

$\displaystyle
w(z)=|v(z)|^{2}\prod_{k=1}^{s}\left|\frac{z-a_{k}}{1-z\overline{a}_{k}}\right|^{m_{k}},\quad
s\geq 0,$ (1.3)

where $v$ is analytic and never zero in $\overline{\mathbb{D}}$, and

$m_{k}\in(-2,\infty)\setminus\\{0\\},\quad|a_{k}|<1,\quad 1\leq k\leq s.$

The condition that $m_{k}>-2$ guarantees the integrability of $w$ over
$\mathbb{D}$. When $s=0$, the product in (1.3), as well as any other product
of the form $\prod_{k=1}^{0}$, is to be interpreted as being constant $1$. We
assume for convenience that $v(0)>0$.

The restrictions $u=w|_{\mathbb{T}}$, with $w$ as in (1.3), generate the class
$\mathcal{P}_{\mathbb{T}}$ of positive analytic weights on the unit circle,
that is, weights $u$ of the form $u(z)=|v(z)|^{2}$, $z\in\mathbb{T}$, with $v$
analytic and free of zeros in $\overline{\mathbb{D}}$. Our investigation is
motivated by the type of results obtained in [4] for the sequence
$(\varphi_{n})_{n=0}^{\infty}$ of Szegő polynomials corresponding to a weight
$u\in\mathcal{P}_{\mathbb{T}}$. These are polynomials satisfying the
orthonormality conditions

$\int_{\mathbb{T}}\varphi_{n}(z)\overline{\varphi_{m}(z)}u(z)|dz|=\delta_{n,m}\,,\quad
n,m\geq 0.$

It has been shown in [4] that for a weight in $\mathcal{P}_{\mathbb{T}}$, the
Szegő polynomial $\varphi_{n}$ admits an asymptotic expansion whose terms can
be generated by the successive application of a couple of Cauchy integral
operators. The dominant term of the expansion is an integral whose behavior
depends on the type of singularities encountered by the analytic continuation
of the function $v^{*}(z)=1/\overline{v(1/\overline{z})}$. When these
singularities are isolated or branch points, techniques involving contour
deformation and residue computations allow for a precise asymptotic
description of $\varphi_{n}$ inside the unit circle.

To a degree, the results that we present in this paper can be viewed as an
extension to the Bergman setting of those obtained in [4]. Our main result
(Theorem 2.1) asserts that for a weight as in (1.3), the Bergman polynomial
$p_{n}$ can be asymptotically represented by a Cauchy-type integral whose
behavior is influenced by the singularities of the function
$v^{*}(z)/\prod_{k=1}^{s}(z-a_{k})$. The elegance and applicability of this
integral resides in its ability to encode the behavior of $p_{n}$ at every
point of the complex plane.

Previously in [6] (see also [7]), we established similar results for the
subfamily of (1.3) consisting of weights $w=|h|^{2}$, $h$ a polynomial without
zeros in $\mathbb{T}$. The proof that we employed at the time was heavily
based on constructing (for an $h$ with all its zeros in $\mathbb{D}$) an
asymptotic expansion for $p_{n}$ similar to the one given in [4] for Szegő
polynomials. For more general weights, however, our attempt at constructing
these expansions has so far been unsuccessful, thus requiring us to look for a
different approach. Our success this time comes from a better understanding of
the reproducing kernel associated with the weights (1.3), and the key idea of
iterating a certain identity (i.e., (6.5)), the details of which are explained
later in Section 6.

We point out that there are some results in the literature that apply (fully
or partially) to the weights under our consideration. In [10], Simanek studied
the asymptotic properties of $L^{p}(\mu)$-extremal polynomials for a broad
class of measures $\mu$ supported on analytic regions. For a weight as in
(1.3), Theorem 1.2 and Theorem 3.1 of [10] yield

$\lim_{n\to\infty}\frac{p_{n}(z)}{\sqrt{n}z^{n}}=\frac{1}{\overline{v(1/\overline{z})}},\quad|z|>1,$
(1.4)

the convergence being uniform for $|z|\geq r>1$.

A theorem by Suetin [11, Thm. 3.3] for continuous, positive weights that
satisfy a Lipschitz condition of order $\alpha<1$, establishes (1.4) with an
estimate of $O((\ln n/n)^{\alpha/2})$ for the speed of convergence, which at
$z=\infty$ [11, Thm. 3.1] improves to $O((\ln n/n)^{\alpha})$. A better, sharp
estimate will be given in Corollary 2.2 below.

We also mention an earlier result by Korovkin [3] that (when orthogonality is
considered over the unit disk) applies to weights whose restriction to an
annulus $\rho<|z|<1$ is of the form $|v|^{2}$, with $v$ analytic and free of
zeros in $\rho<|z|\leq\infty$. Korovkin proved that for these weights, (1.4)
holds in the larger set $|z|>\rho$. We will see that for a weight as in (1.3),
(1.4) also holds in the exterior of some circle of radius $<1$.

##  2 Statement of results

In this section, $w$ always stands for a weight of the form (1.3), and $p_{n}$
is the corresponding $n$th degree Bergman polynomial.

Let

$\displaystyle q(z):=\prod_{k=1}^{s}(z-a_{k}),\quad
q^{*}(z):=\prod_{k=1}^{s}(1-\overline{a}_{k}z),$ (2.1)

and define $\rho_{w}\geq 0$ to be the smallest number such that the function

$\frac{1}{v(z)q^{*}(z)}$

can be analytically continued to the disk $|z|<1/\rho_{w}$. By our assumptions
on the weight $w$, it is the case that $\rho_{w}<1$.

Equivalently, if we define

$\displaystyle v^{*}(z):=\frac{1}{\overline{v(1/\overline{z})}},$ (2.2)

then $\rho_{w}$ is the smallest number such that the function $v^{*}/q$ has an
analytic continuation to $|z|>\rho_{w}$.

The circle $|z|=\rho_{w}$ defines the boundary where $p_{n}$ has a marked
change in behavior. This can be seen already in a weak sense from the fact
that for all but finitely many $z\in\mathbb{D}$,

$\limsup_{n\to\infty}|p_{n}(z)|^{1/n}=\max\\{|z|,\rho_{w}\\}.$

As we explain below in Section 4, this is a consequence of a connection
between the orthonormal polynomials and the reproducing kernel
$K_{h}(z,\zeta)$ of the Bergman space $A^{2}_{h}$ associated with the weight

$h(z):=\prod_{k=1}^{s}\left|\frac{z-a_{k}}{1-\overline{a}_{k}z}\right|^{m_{k}}\,.$

This kernel is a function defined on $\mathbb{D}\times\mathbb{D}$, analytic in
$z$ and $\overline{\zeta}$, uniquely determined by the property that for all
$f\in A^{2}_{h}$,

$\displaystyle
f(z)=\int_{\mathbb{D}}f(\zeta)K_{h}(z,\zeta)h(\zeta)d\sigma(\zeta),\quad
z\in\mathbb{D}.$

A structural formula for $K_{h}$ has been given in [5], and we shall elaborate
on it later in Section 4, but for our immediate purpose, it suffices to note
that

$\displaystyle L(z,\zeta):=\zeta^{-2}K_{h}(z,1/\overline{\zeta})$

is a rational function that can be written as

$\displaystyle
L(z,\zeta)=\frac{1}{(\zeta-z)^{2}}+\frac{V(z,\zeta)}{(\zeta-z)q^{*}(z)q(\zeta)},$
(2.3)

with $V(z,\zeta)$ a polynomial in the variables $z$ and $\zeta$. We can now
state our main result.

######  Theorem 2.1.

There exists a sequence $(H_{n})_{n=0}^{\infty}$ of analytic functions in
$\mathbb{D}$, all of them vanishing at the origin, such that:

  1. i)

$H_{n}(z)=O(1/n)$ uniformly as $n\to\infty$ on compact subsets of
$\mathbb{D}$;

  2. ii)

For every $r$ and $z$ such that $\rho_{w}<r<1$ and $|z|<r$,

$\displaystyle v(0)\gamma_{n}p_{n}(z)={}$ $\displaystyle\frac{1}{2\pi
iv(z)}\int_{|\zeta|=r}(vv^{*})(\zeta)L(z,\zeta)\zeta^{n+1}(1+H_{n}(\zeta))d\zeta\,.$
(2.4)

From (2.3), we see that the integral in (2.4) is well-defined for the stated
values of $z$. Its behavior as $n\to\infty$ changes according to the location
of $z$ relative to the critical circle $|z|=\rho_{w}$. For $z$ outside this
circle, (2.4) yields the following strong asymptotic formula.

######  Corollary 2.2.

For every $r>\rho_{w}$,

$\displaystyle\frac{p_{n}(z)}{\sqrt{n}\,z^{n}}=v^{*}(z)+O(1/n)$ (2.5)

uniformly on $|z|\geq r$ as $n\to\infty$.

If we evaluate (2.5) at $z=\infty$, then we get

$\displaystyle\gamma_{n}=\frac{\sqrt{n}}{v(0)}\left(1+O\left(1/n\right)\right),\quad(n\to\infty).$
(2.6)

But in reality, we will need to have (2.6) already established before we can
prove (2.4).

The $O(1/n)$ estimate for the error term in (2.5) and (2.6) is, generally
speaking, best possible, as this estimate becomes exact for the weight
$w(z)=|z-b|^{2}$, $|b|>1$. This can be seen from the explicit formulas for
$p_{n}$ and $\gamma_{n}$ discussed in [6, Remark 2], [7, Remark 6].

In the next section, we touch on a few examples that illustrate how the
behavior of $p_{n}$ in the closed disk $|z|\leq\rho_{w}$ is affected by the
nature of the singularities that the function $v^{*}/q$ has on the critical
circle $|z|=\rho_{w}$. If these singularities are poles or branch points, then
the asymptotic analysis of the integral in (2.4) is relatively
straightforward, and has been carried out in detail for integrals of a similar
type, for instance, in [4] and [8] ([4] also provides an example with an
essential singularity).

The function $L$ that occurs in (2.4) is defined through the kernel $K_{h}$.
From the formula for this kernel given in [5, Thm. 1.1], we can see that

$\displaystyle L(z,\zeta)={}$
$\displaystyle\frac{1}{(\zeta-z)^{2}}+\sum_{k=1}^{s}\frac{\frac{m_{k}}{2}(1-|a_{k}|^{2})}{(\zeta-z)(\zeta-
a_{k})(1-z\overline{a}_{k})}+\frac{J(z,\zeta)}{q^{*}(z)q(\zeta)},$ (2.7)

where $J(z,\zeta)$ is a polynomial in the variables $z$ and $\zeta$ of degree
at most $2(s-2)$ ($s-2$ in each independent variable). For $0\leq s\leq 2$,
$J(z,\zeta)$ is constant:

$J(z,\zeta)=\begin{cases}0,&\ s=0,\ 1,\\\
K_{h}(0,0)-1-\frac{m_{1}}{2}(1-|a_{1}|^{2})-\frac{m_{2}}{2}(1-|a_{2}|^{2}),&\
s=2.\end{cases}$

For $s>2$, however, $J(z,\zeta)$ has a somewhat complex structure, but it can
be explicitly written in terms of the values of $K_{h}(z,\zeta)$ and its
derivatives at $(0,0)$. These values can in turn be numerically computed by
integration.

##  3 Examples

###  3.1 Case $s=0$

For $s=0$, that is, when $w=|v|^{2}$ with $v$ analytic and never zero in
$\overline{\mathbb{D}}$, (2.4) becomes

$\displaystyle v(0)\gamma_{n}p_{n}(z)={}$ $\displaystyle\frac{1}{2\pi
iv(z)}\int_{|\zeta|=r}\frac{v(\zeta)v^{*}(\zeta)\zeta^{n+1}}{(\zeta-z)^{2}}(1+H_{n}(\zeta))d\zeta\,.$
(3.1)

By the definition of $\rho_{w}$, the function $v^{*}$ is analytic in
$|z|>\rho_{w}$, but it has singularities on the circle $|z|=\rho_{w}$. For $z$
in the disk $|z|\leq\rho_{w}$, the behavior as $n\to\infty$ of the integral in
(3.1) depends on the nature of these first singularities. For instance, if
$v(z)=\prod_{k=1}^{\ell}(1-\overline{b}_{k}z)^{r_{k}}$ is a polynomial with
all its zeros outside the unit circle (i.e., $0<|b_{k}|<1$), then $v^{*}$ is a
rational function whose poles are the points $b_{k}$:

$v^{*}(z)=z^{r}\prod_{k=1}^{\ell}(z-b_{k})^{-r_{k}},\quad
r=\sum_{k=1}^{\ell}r_{k}.$

The integral in (3.1) can then be computed without difficulty by using the
residue theorem, and we recover the results of [7].

To illustrate a possible approach in the presence of branch points, let us
sketch the analysis for the weight $v(z)=(1-\overline{b}z)^{r}$, $|b|<1$,
$r\in\mathbb{R}\setminus\mathbb{Z}$. For this weight, (3.1) takes the form

$\displaystyle\gamma_{n}p_{n}(z)\sim{}$ $\displaystyle\frac{1}{2\pi
iv(z)}\int_{|\zeta|=r}\left(\frac{\zeta-b}{\zeta}\right)^{-r}\frac{v(\zeta)\zeta^{n+1}}{(\zeta-z)^{2}}d\zeta\,.$
(3.2)

For a number $\mu$ such that $0<\mu<|b|$, let $C_{\mu}$ be the positively
oriented contour that results from attaching the two-sided segment $[\mu
b/|b|,b]$ to the circle $|\zeta|=\mu$. For $|z|<\mu$, the integral in (3.2)
can be taken over $C_{\mu}$ without affecting its value, and can further be
split as the sum of an integral over $|\zeta|=\mu$, which is $O(\mu^{n})$,
plus an integral over the two-sided segment $[\mu b/|b|,b]$, which provides
the dominant behavior as $n\to\infty$. In this way, we get

$\displaystyle\gamma_{n}p_{n}(z)\sim{}$ $\displaystyle\frac{\sin(\pi r)}{\pi
v(z)}\int_{[\mu
b/|b|,b]}\left|\frac{\zeta-b}{\zeta}\right|^{-r}\frac{v(\zeta)\zeta^{n+1}}{(\zeta-z)^{2}}d\zeta.$
(3.3)

Noting that

$\displaystyle\frac{v(\zeta)}{(\zeta-z)^{2}}=\frac{v(b)}{(b-z)^{2}}+O(\zeta-b)$

uniformly as $\zeta\to b$ on $|z|\leq R<\mu$, and bearing in mind a few basic
properties of the Gamma and Beta functions, we can conclude from (3.3) that

$\displaystyle\begin{split}\gamma_{n}p_{n}(z)\sim{}&\frac{\sin(\pi
r)(1-|b|^{2})^{r}b^{n+2}}{\pi(1-\overline{b}z)^{r}(b-z)^{2}}\int_{\mu/|b|}^{1}t^{n+1+r}(1-t)^{-r}dt\\\
\sim{}&\frac{(1-|b|^{2})^{r}b^{n+2}}{(1-\overline{b}z)^{r}(b-z)^{2}}\cdot\frac{r(r+1)\cdots(r+n+1)}{(n+2)!}\,.\end{split}$
(3.4)

These computations actually work if $r<1$, which guarantees integrability over
the segment $[\mu b/|b|,b]$. When $r>1$, we first need to integrate by parts
in (3.2) as many times as needed to bring $r$ into the appropriate range, and
then proceed as we just explained (the final outcome is the same as (3.4)).
More detailed explanations can be found, for instance, in the proof of
Proposition 1 of [8].

If $v$ is meromorphic in $\mathbb{C}$ without zeros, then $\rho_{w}=0$,
$v^{*}$ is an analytic functions in $\overline{\mathbb{C}}\setminus\\{0\\}$,
and (2.5) holds for $|z|>0$. For $v(z)=e^{z}$, there is a curious relationship
between the sequences of Szegő and Bergman polynomials.

######  Proposition 3.1.

Let $(\psi_{n})_{n=0}^{\infty}$ be the sequence of monic polynomials that are
orthogonal with respect to $|e^{z}|^{2}$ on the unit circle, and let
$\psi^{*}_{n}(z):=z^{n}\overline{\psi_{n}(1/\overline{z})}$. Then,
$\psi_{n}(0)\not=0$ and

$\displaystyle\gamma_{n}^{-1}p_{n}(z)=\frac{\psi^{*}_{n}(z)+{\psi^{*}_{n}}^{\prime}(z)}{\overline{\psi_{n}(0)}},\quad
n\geq 0.$ (3.5)

###### Proof.

Let $\phi_{n}$ denote the numerator of the fraction that occurs in (3.5). With
the aid of Green’s Formula [2, p. 10], we find that

$\displaystyle\begin{split}\int_{\mathbb{D}}\overline{\phi_{n}(z)}z^{m}|e^{z}|^{2}d\sigma(z)={}&\int_{\mathbb{D}}\overline{\left(e^{z}\psi^{*}_{n}(z)\right)^{\prime}}z^{m}e^{z}d\sigma(z)\\\
={}&\frac{1}{2\pi i}\int_{|z|=1}\overline{e^{z}\psi^{*}_{n}(z)}z^{m}e^{z}dz\\\
={}&\frac{1}{2\pi}\int_{|z|=1}\overline{\psi^{*}_{n}(z)}z^{m+1}|e^{z}|^{2}|dz|\,.\end{split}$
(3.6)

Since the monic polynomial $\psi_{n}$ is characterized by the orthogonality
property that the last integral in (3.6) equals zero for all $0\leq m\leq
n-1$, it follows that for these values of $m$, $\phi_{n}$ is orthogonal to
$z^{m}$ over $\mathbb{D}$ with respect to the weight $|e^{z}|^{2}$. This
implies that $\phi_{n}$ is of degree $n$, unless $\phi_{n}$ is identically
zero. The latter is impossible because $\psi^{*}_{n}$ is not the constant
zero. Since $\phi_{n}$ is then of degree $n$, so is $\psi_{n}^{*}$, so that
$\psi_{n}(0)\not=0$, and thus the right-hand side of (3.5) (which equals
$\phi_{n}/\overline{\psi_{n}(0)}$) is precisely the monic Bergman orthogonal
polynomial $\gamma_{n}^{-1}p_{n}$. ∎

By virtue of (3.5) and the results of [4], $w(z)=|e^{z}|^{2}$ is another
example of a weight on $\mathbb{D}$ for which a full asymptotic expansion for
$p_{n}$ can be derived.

###  3.2 Case $s\geq 1$

We will assume that

$\rho_{a}:=\max_{1\leq k\leq s}|a_{k}|>0,$

for otherwise $s=1$ and $a_{1}=0$, in which case the asymptotic analysis of
the integral in (2.4) is practically the same as for $s=0$.

The weight

$\displaystyle
w(z)=\frac{\prod_{k=1}^{s}|z-a_{k}|^{m_{k}}}{\prod_{k=1}^{s}|1-z\overline{a}_{k}|^{m_{k}+r_{k}}}$
(3.7)

with $r_{k}\in\mathbb{R}$ and $m_{k}\in(-2,\infty)\setminus\\{0\\}$, serves as
a good prototype for illustrating a few situations that might arise. This
weight can be written in the form (1.3) with

$v(z)=\prod_{k=1}^{s}(1-z\overline{a}_{k})^{-r_{k}/2},\quad
v^{*}(z)=\prod_{k=1}^{s}\left(\frac{z-a_{k}}{z}\right)^{r_{k}/2},$

the branches of the powers being chosen so that $v(0)=1$.

If every $r_{k}$ is a positive, even integer, then $v^{*}/q$ is analytic in
$|z|>0$, so that $\rho_{w}=0$. Set

$d:=\frac{1}{2}\sum_{k=1}^{s}r_{k},\quad
p(z):=\prod_{k=1}^{s}\left(z-a_{k}\right)^{r_{k}/2}\,.$

From (2.5) and Hurwitz’s Theorem, we see that as $n\to\infty$, $n-d$ zeros of
$p_{n}$ approach the origin, while each $a_{k}$ attracts $r_{k}/2$ zeros,
multiplicities being counted. But we can be more precise, for in this case the
integral in (2.4) can be computed directly by using the residue theorem. For
$n+1\geq d$, the computation simplifies into

$\displaystyle\begin{split}\gamma_{n}p_{n}(z)={}&(n+1-d)z^{n-d}p(z)(1+H_{n}(z))+z^{n+1-d}p(z)H^{\prime}_{n}(z)\\\
&+z^{n+1-d}(1+H_{n}(z))\frac{p(z)}{2}\sum_{k=1}^{s}\frac{(m_{k}+r_{k})(1-|a_{k}|^{2})}{(z-a_{k})(1-z\overline{a}_{k})}\,.\end{split}$
(3.8)

It follows from (3.8) that for $n\geq d$,

$\gamma_{n}p_{n}(z)=\gamma_{n}^{2}z^{n-d}\left(\prod_{k=1}^{s}(z-a_{k})^{\frac{r_{k}}{2}-1}\right)\prod_{k=1}^{s}(z-z_{n,k})\,,$

with $z_{n,k}\to a_{k}$ as $n\to\infty$.

By substituting this expression for $\gamma_{n}p_{n}$ back into (3.8) and
simplifying, we get (with $q$ as in (2.1) and $q_{k}=q/(z-a_{k})$)

$\displaystyle\begin{split}\gamma_{n}^{2}\prod_{k=1}^{s}(z-z_{n,k})={}&(n+1-d)q(z)(1+H_{n}(z))+zq(z)H^{\prime}_{n}(z)\\\
&+\frac{z(1+H_{n}(z))}{2}\sum_{k=1}^{s}\frac{q_{k}(z)(m_{k}+r_{k})(1-|a_{k}|^{2})}{1-z\overline{a}_{k}}\,.\end{split}$
(3.9)

Evaluating at $z=a_{j}$, we obtain

$\displaystyle\gamma_{n}^{2}(a_{j}-z_{n,j})\underset{k\not=j}{\prod_{1\leq
k\leq s}}(a_{j}-z_{n,k})={}$
$\displaystyle\frac{a_{j}(m_{j}+r_{j})}{2}(1+H_{n}(a_{j}))\underset{k\not=j}{\prod_{1\leq
k\leq s}}(a_{j}-a_{k}).$

This and (2.6) shows that $|a_{j}-z_{n,j}|=O(1/n)$ as $n\to\infty$, which
leads us to conclude that for all $1\leq j\leq s$,

$\displaystyle
z_{n,j}=a_{j}-\frac{a_{j}(m_{j}+r_{j})}{2n}+O(n^{-2})\quad(n\to\infty).$

Since $H_{n}(0)=0$, evaluating (3.9) at $0$ results in

$\displaystyle\begin{split}\gamma_{n}^{2}={}&(n+1-d)\prod_{k=1}^{s}\frac{a_{k}}{z_{n,k}}\\\
={}&(n+1-d)\left(1+\frac{1}{2n}\sum_{k=1}^{s}(m_{k}+r_{k})+O(n^{-2})\right)\,.\end{split}$

Notice the resemblance of $p_{n}$ to the so-called Bernstein-Szegő polynomial
$z^{n-d}\prod_{k=1}^{s}(z-a_{k})^{\frac{r_{k}}{2}}$, $n\geq d$, which is the
$n$th degree monic orthogonal polynomial on the unit circle for the weight
$\prod_{k=1}^{s}|1-z\overline{a}_{k}|^{-r_{k}}$ obtained by restricting (3.7)
to $\mathbb{T}$.

If every $r_{k}$ is an even, non-positive integer, then $v^{*}/q$ has for
singularities a pole at each $a_{k}$, and so $\rho_{w}=\rho_{a}$. These poles
turn out to be simple if we choose $r_{k}=0$ for all $k$, in which case the
integral in (2.4) reduces to

$\displaystyle\gamma_{n}p_{n}(z)={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=r}L(z,\zeta)\zeta^{n+1}(1+H_{n}(\zeta))d\zeta\,.$ (3.10)

By a residue calculation, we find from (3.10) that if $|z|<1/\rho_{a}$ and
$z\not=a_{k}$ for all $1\leq k\leq s$, then

$\displaystyle\gamma_{n}p_{n}(z)={}$
$\displaystyle(n+1)z^{n}(1+H_{n}(z))+z^{n+1}H^{\prime}_{n}(z)$
$\displaystyle+z^{n+1}(1+H_{n}(z))\sum_{k=1}^{s}\frac{\frac{m_{k}}{2}(1-|a_{k}|^{2})}{(z-a_{k})(1-z\overline{a}_{k})}$
$\displaystyle+\sum_{k=1}^{s}a_{k}^{n+1}(1+H_{n}(a_{k}))\left(\frac{\frac{m_{k}}{2}(1-|a_{k}|^{2})}{(a_{k}-z)(1-z\overline{a}_{k})}+\frac{J(z,a_{k})}{q^{*}(z)q^{\prime}(a_{k})}\right).$

This implies that (2.5) holds uniformly on any closed subset of
$|z|\geq\rho_{a}$ that contains no $a_{k}$, while

$\displaystyle p_{n}(z)={}$
$\displaystyle\sum_{k:|a_{k}|=\rho_{a}}a_{k}^{n+1}\left(\frac{\frac{m_{k}}{2}(1-|a_{k}|^{2})}{(a_{k}-z)(1-z\overline{a}_{k})}+\frac{J(z,a_{k})}{q^{*}(z)q^{\prime}(a_{k})}\right)+O(\rho_{a}^{n}/n)$

uniformly as $n\to\infty$ on compact subsets of $|z|<\rho_{a}$.

For $z=a_{j}$ with $|a_{j}|=\rho_{a}$, a similar residue calculation gives

$p_{n}(a_{j})=(1+m_{j}/2)a_{j}^{n}(1+O(1/n))\,.$

If for some $k$, the number $r_{k}$ is not an even integer, then $v^{*}/q$ has
a branch point at $a_{k}$, and the asymptotic analysis can be carried out with
the ideas sketched in Subsection 3.1.

##  4 Reproducing kernel

For the inner product (1.1) corresponding to a weight $w$ on $\mathbb{D}$, let
us set $\|f\|_{w}:=\sqrt{\langle f,f\rangle_{w}}$. The Bergman space
$A^{2}_{w}$ is the linear space of functions $f\in\mathcal{H}(\mathbb{D})$
such that $\|f\|_{w}<\infty$, endowed with the inner product (1.1). We will
suppress the index $w$ and simply write $\|\cdot\|$ and $A^{2}$ for $w\equiv
1$.

From this point onward, $w$ will represent a weight of the form (1.3), and
$(p_{n})$ is the corresponding sequence of orthonormal polynomials satisfying
(1.2).

Since the weight $w$ is bounded in $\mathbb{D}$, it follows that there is a
constant $c_{1}$ such that $\|f\|_{w}\leq c_{1}\|f\|$.

With the help of Cauchy’s integral formula, one can verify that for all
$0\leq\delta<1$, there exists a constant $C_{\delta}$ such that

$\int_{\mathbb{D}}|f(z)|^{2}d\sigma(z)\leq
C_{\delta}\int_{\delta<|z|<1}|f(z)|^{2}d\sigma(z)\,,\quad
f\in\mathcal{H}(\mathbb{D}).$

Since $w$ has a positive minimum on any annulus $\delta\leq|z|\leq 1$ that
does not contain any of the points $a_{k}$, the latter inequality implies that
there exists a constant $c_{2}$ such that $\|f\|\leq c_{2}\|f\|_{w}$ for every
$f\in\mathcal{H}(\mathbb{D})$.

We have shown that the norms $\|\cdot\|$ and $\|\cdot\|_{w}$ are equivalent on
$\mathcal{H}(\mathbb{D})$, and therefore $A^{2}_{w}$ and $A^{2}$ contain the
same members. Moreover, it is now easy to transfer some of the properties of
$A^{2}$ to $A^{2}_{w}$. For instance, since $A^{2}$ is a Hilbert space [1, p.
8], so is $A^{2}_{w}$. We also know that the polynomials are dense in $A^{2}$
[1, p. 11], and convergence in $A^{2}$ implies normal convergence in
$\mathbb{D}$, so that the same is true in $A^{2}_{w}$, and consequently, we
have the (generalized) Fourier expansion identity

$\displaystyle f(z)=\sum_{k=0}^{\infty}\langle
f,p_{n}\rangle_{w}p_{n}(z),\quad f\in A^{2}_{w},$ (4.1)

with the series converging normally in $\mathbb{D}$.

More can actually be said. If $0\leq\varrho(f)\leq 1$ denotes the smallest
number such that $f$ admits an analytic continuation to $|z|<1/\varrho(f)$,
then

$\displaystyle\limsup_{n\to\infty}|\langle
f,p_{n}\rangle_{w}|^{1/n}=\varrho(f)$

and the normal convergence of (4.1) extends to the disk $|z|<1/\varrho(f)$.
This stronger statement was proven by Walsh [12, pp. 130-131] for strictly
positive continuous weights over Jordan domains, but the proof still applies
(with only minor modifications) to continuous weights that are positive near
the boundary of the domain (see also [9, p. 336]).

Because point evaluation functionals are bounded in $A^{2}$ (see, e.g. [1, Ch.
1]), the same is true in $A^{2}_{w}$, and there exists a reproducing kernel
$K_{w}(z,\zeta)$ defined on $\mathbb{D}\times\mathbb{D}$, analytic in $z$ and
$\overline{\zeta}$, such that for all $f\in A^{2}_{w}$ and $z\in\mathbb{D}$,

$\displaystyle
f(z)=\int_{\mathbb{D}}f(\zeta)K_{w}(z,\zeta)w(\zeta)d\sigma(\zeta).$

This kernel has the symmetry property $K(z,\zeta)=\overline{K(\zeta,z)}$, so
that the variables can be interchanged at will.

Similarly, the space $A^{2}_{h}$ corresponding to the weight

$h(z)=\prod_{k=1}^{s}\left|\frac{z-a_{k}}{1-\overline{a}_{k}z}\right|^{m_{k}}$

has a reproducing kernel $K_{h}(z,\zeta)$, and by direct integration one can
verify that

$\displaystyle K_{w}(z,\zeta)=\frac{K_{h}(z,\zeta)}{v(z)\overline{v(\zeta)}}.$
(4.2)

If we now let $\tau_{\zeta}$ denote, for $\zeta\in\mathbb{D}$, the smallest
number such that $K(z,\zeta)$, as a function of $z$, has an analytic
continuation to $|z|<1/\tau_{\zeta}$, then by (4.1) we see that for every
fixed $\zeta\in\mathbb{D}$,

$\displaystyle\frac{K_{h}(z,\zeta)}{v(z)\overline{v(\zeta)}}=\sum_{k=0}^{\infty}p_{k}(z)\overline{p_{k}(\zeta)},\quad|z|<1/\tau_{\zeta},$
(4.3)

with the convergence of the series being uniform on every disk $|z|\leq r$ of
radius $r<1/\tau_{\zeta}$. Moreover,

$\limsup_{n\to\infty}|p_{n}(\zeta)|^{1/n}=\tau_{\zeta},\quad\zeta\in\mathbb{D}.$

######  Proposition 4.1.

For all but finitely many $\zeta\in\mathbb{D}$,

$\tau_{\zeta}=\max\\{|\zeta|,\rho_{w}\\}.$

###### Proof.

To prove this proposition we make use of a result from [5], where it has been
shown that $K_{h}(z,\zeta)$ is a rational function in $z$ and
$\overline{\zeta}$. As a function of $z$ (and for fixed $\zeta\in\mathbb{D}$)
it has a double pole at $1/\overline{\zeta}$ and a simple pole at
$\overline{a}^{-1}_{k}$, $1\leq k\leq s$. More precisely, from the
representation in [5, Thm. 1.1] and (4.2) above, we find that $K_{w}(z,\zeta)$
can be written as

$\displaystyle K_{w}(z,\zeta)={}$
$\displaystyle\frac{\Pi(z,\zeta)}{(1-z\overline{\zeta})^{2}q^{*}(z)v(z)\overline{q^{*}(\zeta)}\overline{v(\zeta)}},$
(4.4)

the numerator having the structure

$\displaystyle\begin{split}\Pi(z,\zeta)={}&\overline{q^{*}(\zeta)}q^{*}(z)+(1-z\overline{\zeta})\sum_{k=1}^{s}\frac{m_{k}}{2}(1-|a_{k}|^{2})\overline{q^{*}_{k}(\zeta)}q_{k}^{*}(z)\\\
&+(1-z\overline{\zeta})^{2}\Pi_{1}(z,\zeta)\,,\end{split}$ (4.5)

where

$q_{k}(z):=\underset{j\not=k}{\prod_{1\leq j\leq s}}(z-a_{j}),\quad
q^{*}_{k}(z):=\underset{j\not=k}{\prod_{1\leq j\leq
s}}(1-\overline{a}_{j}z)\,,$

and $\Pi_{1}(z,\zeta)$ is a polynomial in $z$ and $\overline{\zeta}$.

Suppose first that $\rho_{w}<|\zeta|<1$. Since $\rho_{w}$ is the smallest
number such that the function $(v(z)q^{*}(z))^{-1}$ is analytic in
$|z|<{1/\rho_{w}}$, (4.4) implies that $K_{w}(\cdot,\zeta)$ is analytic in the
punctured disk $\\{z:|z|<1/\rho_{w},\ z\not=1/\overline{\zeta}\\}$, with a
pole at $z=1/\overline{\zeta}$ (and therefore $\tau_{\zeta}=|\zeta|$) except
possibly when $\zeta=a_{k}$ for some $1\leq k\leq s$.

Assume now that $|\zeta|\leq\rho_{w}$, so that $K_{w}(\cdot,\zeta)$ is
analytic in $|z|<1/\rho_{w}$, and consequently $\tau_{\zeta}\leq\rho_{w}$. If
$\tau_{\zeta}<\rho_{w}$, then again by (4.4), $(vq^{*})^{-1}$ has to be a
meromorphic function in $|z|<\tau_{\zeta}$, and therefore it has only finitely
many singularities (all of them poles) on the circle $|z|=1/\rho_{w}$.
Moreover, at any one of these singularities $z_{0}$, we must have
$\Pi(z_{0},\zeta)=0$. Then, to finish the proof it suffices to show that the
polynomial $\Pi(z_{0},\zeta)$ in the variable $\overline{\zeta}$ is not
constant zero, for this will let us conclude that for only finitely many
$\zeta$ it is possible to have $\tau_{\zeta}<\rho_{w}$.

From (4.5), we see that if $z_{0}$ is not a zero of $q^{*}$, then

$\Pi(z_{0},1/\overline{z}_{0})=\overline{q^{*}(1/\overline{z}_{0})}q^{*}(z_{0})\not=0.$

If $q^{*}(z_{0})=0$, say $z_{0}=1/\overline{a}_{k}$, then

$\left.\frac{\partial}{\partial\overline{\zeta}}\Pi(z_{0},\zeta)\right|_{\zeta=1/\overline{z}_{0}}=-\frac{z_{0}m_{k}}{2}(1-|a_{k}|^{2})\overline{q^{*}_{k}(1/\overline{z}_{0})}q_{k}^{*}(z_{0})\not=0.$

∎

##  5 Preliminary results

###  5.1 Estimates for the Faber polynomials

By the definition of $\rho_{w}$, the function $v^{*}/q$ is analytic in
$|z|>\rho_{w}$, and so is therefore $v^{*}$.

The Laurent expansion of $v^{*}$ in $|z|>\rho_{w}$ is of the form

$v^{*}(z)=c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots,\quad
c_{0}=v(0)^{-1},$

so that for every integer $n\geq 0$,

$z^{n}v^{*}(z)=c_{0}z^{n}+c_{1}z^{n-1}+c_{2}z^{n-2}+\cdots+c_{n}+\frac{c_{n+1}}{z}+\frac{c_{n+2}}{z^{2}}+\cdots.$

The polynomial part

$F_{n}(z):=c_{0}z^{n}+c_{1}z^{n-1}+c_{2}z^{n-2}+\cdots+c_{n}$

is known as the Faber polynomial for $z^{n}v^{*}(z)$. Note that

$\displaystyle F_{n}(z)=z^{n}v^{*}(z)+O(1/z)\quad(z\to\infty).$ (5.1)

For every $\rho$ such that $\rho_{w}<\rho<1$, we also have

$F_{n}(z)=z^{n}v^{*}(z)+\frac{1}{2\pi
i}\int_{|\zeta|=\rho}\frac{\zeta^{n}v^{*}(z)}{\zeta-z}d\zeta,\quad|z|>\rho,$

and thus if $\rho_{w}<\rho_{1}<\rho<1$, then as $n\to\infty$

$\displaystyle F_{n}(z)=z^{n}v^{*}(z)+O(\rho_{1}^{n}),\quad|z|\geq\rho.$ (5.2)

######  Lemma 5.1.

Suppose that $S(z)$ is a function analytic in some open set containing the
annulus $r\leq|z|\leq 1$, and that $|S(z)|=1$ when $|z|=1$. Then, there exists
a constant $M$ such that

$|1-|S(z)|^{2}|\leq M(1-|z|),\quad r\leq|z|\leq 1.$

###### Proof.

Let

$M_{1}:=\max_{r\leq|z|\leq 1}|S(z)|,\quad M_{2}:=\max_{r\leq|z|\leq
1}|S^{\prime}(z)|.$

For all $z$ is the annulus $r\leq|z|\leq 1$, the segment $[z,z/|z|]$ is also
contained in that annulus, so that

$\displaystyle\left|1-|S(z)|^{2}\right|={}$
$\displaystyle(1+|S(z)|)\left||S(z/|z|)|-|S(z)|\right|$ $\displaystyle\leq{}$
$\displaystyle(1+M_{1})\left|S(z/|z|)-S(z)\right|$ $\displaystyle\leq{}$
$\displaystyle(1+M_{1})\left|\int_{[z,z/|z|]}S^{\prime}(\zeta)d\zeta\right|\leq(1+M_{1})M_{2}(1-|z|).$

∎

######  Proposition 5.2.

As $n\to\infty$,

$\displaystyle\int_{\mathbb{D}}|F_{n}(z)|^{2}w(z)d\sigma={}$
$\displaystyle\frac{1}{n}\left(1+O\left(n^{-1}\right)\right)\,.$ (5.3)

###### Proof.

For $\rho$ in the range $\rho_{w}<\rho<1$, we can write

$\displaystyle\int_{\mathbb{D}}|F_{n}(z)|^{2}w(z)d\sigma(z)={}$
$\displaystyle\int_{|z|\leq\rho}|F_{n}(z)|^{2}w(z)d\sigma(z)+\int_{\rho\leq|z|\leq
1}|F_{n}(z)|^{2}w(z)d\sigma(z).$

The first of these two summands can be estimated with the help of (5.2) and
the maximum modulus principle:

$\displaystyle\int_{|z|\leq\rho}|F_{n}(z)|^{2}w(z)d\sigma(z)=O(\rho^{2n})\qquad(n\to\infty).$
(5.4)

The function

$S(z):=\frac{v(z)v^{*}(z)}{\prod_{k=1}^{s}(1-z\overline{a}_{k})^{m_{k}/2}}\prod_{k=1}^{s}\left(\frac{z-a_{k}}{z}\right)^{m_{k}/2}$

has a single-valued analytic branch in the annulus $\rho_{w}<|z|<1/\rho_{w}$,
and $|S(z)|=1$ for $|z|=1$. Note also that

$|v^{*}(z)|^{2}w(z)=|z|^{m}|S(z)|^{2},\quad\rho_{w}<|z|<1/\rho_{w}\,,$

where $m:=\sum_{k=1}^{s}m_{k}$. This equality and (5.2) imply that as
$n\to\infty$,

$\displaystyle\int_{\rho\leq|z|\leq
1}|F_{n}(z)|^{2}w(z)d\sigma(z)=\int_{\rho\leq|z|\leq
1}|z|^{2n+m}|S(z)|^{2}d\sigma(z)+O(\rho^{2n})\,.$ (5.5)

By Lemma 5.1, there exists a constant $M$, depending only on $S$ and $\rho$,
such that whenever $2n+2>m$,

$\displaystyle\int_{\rho\leq|z|\leq 1}$
$\displaystyle|z|^{2n+m}|S(z)|^{2}d\sigma(z)$
$\displaystyle\leq\int_{\rho\leq|z|\leq
1}|z|^{2n+m}\left||S(z)|^{2}-1\right|d\sigma(z)+\int_{\rho\leq|z|\leq
1}|z|^{2n+m}d\sigma(z)$ $\displaystyle\leq M\int_{\rho\leq|z|\leq
1}|z|^{2n+m}(1-|z|^{2})d\sigma(z)+\frac{1-\rho^{2n+m+2}}{n+1+m/2}$
$\displaystyle=M\left(\frac{1-\rho^{2n+m+2}}{n+1+m/2}-\frac{1-\rho^{2n+m+4}}{n+2+m/2}\right)+\frac{1-\rho^{2n+m+2}}{n+1+m/2}$
$\displaystyle=\frac{1}{n+1+m/2}+O(n^{-2})\,.$

Thus, we have proven that (5.5) is equal to $n^{-1}(1+O(n^{-1}))$, and that
(5.4) decreases geometrically fast, so that (5.3) takes place. ∎

The following observation is not about the Faber polynomials per se, but it
will be needed soon in the next section.

Suppose that $E_{n}(z)=e_{0}+e_{1}z+\cdots+e_{n}z^{n}$ is a polynomial of
degree $n$, and let $0\leq r<1$. Since

$\int_{|z|=1}|E_{n}(z)|^{2}|dz|=2\pi\sum_{k=0}^{n}|e_{k}|^{2},$

and

$\int_{r\leq|z|\leq
1}|E_{n}(z)|^{2}d\sigma(z)=\sum_{k=0}^{n}\frac{(1-r^{2k+2})|e_{k}|^{2}}{k+1},$

it follows that

$\displaystyle\int_{|z|=1}|E_{n}(z)|^{2}|dz|\leq\frac{2\pi(n+1)}{1-r^{2}}\int_{r\leq|z|\leq
1}|E_{n}(z)|^{2}d\sigma(z)\,.$ (5.6)

###  5.2 Behavior of $\gamma_{n}$ and related quantities

######  Proposition 5.3.

As $n\to\infty$,

$\displaystyle\gamma_{n}=c_{0}\sqrt{n}\left(1+O\left(n^{-1}\right)\right).$
(5.7)

###### Proof.

Among all monic polynomials $P$ of degree $\leq n$, the orthogonal polynomial
$\gamma_{n}^{-1}p_{n}$ is the one with smallest $\|\cdot\|_{w}$ norm. This
follows from the identity

$\|P\|^{2}_{w}=\|P-\gamma_{n}^{-1}p_{n}\|^{2}_{w}+\|\gamma_{n}^{-1}p_{n}\|_{w}^{2}.$

Because of this extremality property, we have

$\displaystyle 1={}$
$\displaystyle\int_{\mathbb{D}}|p_{n}(z)|^{2}w(z)d\sigma(z)\leq\frac{\gamma^{2}_{n}}{c_{0}^{2}}\int_{\mathbb{D}}|F_{n}(z)|^{2}w(z)d\sigma(z),$

which by (5.3) yields

$\displaystyle
1\leq\frac{\gamma^{2}_{n}}{c_{0}^{2}n}\left(1+O(n^{-1})\right)\,.$ (5.8)

Let $2m_{0}$ be an even integer such that $2m_{0}\geq m_{k}$ for all $1\leq
k\leq s$. Since

$\left|\frac{z-a_{k}}{1-\overline{a}_{k}z}\right|<1,\quad|z|<1,$

we have that with $q$ and $q^{*}$ as defined by (2.1) (recall (1.3)),

$w(z)\geq\left|\frac{v(z)q(z)^{m_{0}}}{q^{*}(z)^{m_{0}}}\right|^{2},\quad|z|<1.$

Hence

$\displaystyle 1={}$
$\displaystyle\int_{\mathbb{D}}|p_{n}(z)|^{2}w(z)d\sigma(z)\geq\int_{\mathbb{D}}\left|p_{n}(z)q(z)^{m_{0}}\overline{g(z)}\right|^{2}d\sigma(z),$
(5.9)

with

$g(z)=\frac{v(z)}{q^{*}(z)^{m_{0}}}.$

This function $g$ is analytic in $\overline{\mathbb{D}}$, and from its
Maclaurin expansion

$g(z)=v(0)+\sum_{k=1}^{\infty}g_{k}z^{k}$

we get (recall that $c_{0}=v^{*}(\infty)=1/v(0)>0$)

$\displaystyle\overline{g(re^{i\theta})}=c_{0}^{-1}+\sum_{k=1}^{\infty}\overline{g_{k}}r^{k}e^{-ik\theta}.$
(5.10)

The polynomial $q(z)^{m_{0}}$ is a monic polynomial of degree $sm_{0}$, so
that if we write

$p_{n}(z)q(z)^{m_{0}}=\sum_{k=0}^{n+sm_{0}}b_{n,k}z^{n+sm_{0}-k},$

then

$e^{-i(n+sm_{0})\theta}p_{n}(re^{i\theta})q(re^{i\theta})^{m_{0}}=\gamma_{n}r^{n+sm_{0}}+\sum_{k=1}^{n+sm_{0}}b_{n,k}r^{n+sm_{0}-k}e^{-ik\theta}\,.$

This and (5.10) yields

$\displaystyle\int_{0}^{2\pi}\left|e^{-i(n+sm_{0})\theta}p_{n}(re^{i\theta})q(re^{i\theta})^{m_{0}}\overline{g(re^{i\theta})}\right|^{2}d\theta\geq
2\pi c_{0}^{-2}\gamma_{n}^{2}r^{2n+2sm_{0}},$

and so we obtain from (5.9)

$\displaystyle 1\geq{}$
$\displaystyle\frac{1}{\pi}\int_{0}^{1}rdr\int_{0}^{2\pi}\left|e^{-i(n+sm_{0})\theta}p_{n}(re^{i\theta})q(re^{i\theta})^{m_{0}}\overline{g(re^{i\theta})}\right|^{2}d\theta$
$\displaystyle\geq{}$
$\displaystyle\frac{\gamma_{n}^{2}c_{0}^{-2}}{n+sm_{0}+1}\,.$

This inequality and that of (5.8) combine into (5.7). ∎

Let us now look into the quantities

$\displaystyle\alpha_{n,k}:=\frac{1}{2\pi
i}\int_{|\zeta|=1}v(\zeta)\zeta^{n-1}\overline{p_{k}(\zeta)}d\zeta\,,\quad
n,k\geq 0.$ (5.11)

These numbers $\alpha_{n,k}$ will play an important role in the proof of
Theorem 2.1. Since $\zeta=1/\overline{\zeta}$ when $|\zeta|=1$, and since
$z^{k}\overline{p_{k}(1/\overline{z})}$ is a polynomial whose value at zero is
$\gamma_{k}$, it follows directly from (5.11) by an application of Cauchy’s
integral formula that

$\displaystyle\alpha_{n,k}={}$ $\displaystyle\begin{cases}0,&\quad 0\leq k\leq
n-1,\\\ \gamma_{n}/c_{0},&\quad k=n.\end{cases}$ (5.12)

We shall presently see that the numbers $\alpha_{n,k}$ are uniformly bounded
once $k>n$, but a better estimate will be achieved later in (6.26).

######  Proposition 5.4.

There exists a constant $C$ such that

$\displaystyle|\alpha_{n,k}|\leq C\,,\quad k>n\geq 0\,.$ (5.13)

###### Proof.

From the definition (5.11), we see that

$\displaystyle\overline{\alpha_{n,k}}={}$ $\displaystyle-\frac{1}{2\pi
i}\overline{\int_{|\zeta|=1}v(\zeta)\zeta^{n-1}\overline{p_{k}(\zeta)}d\zeta}$
$\displaystyle={}$ $\displaystyle-\frac{1}{2\pi
i}\int_{|\zeta|=1}\overline{v(\zeta)\zeta^{n-1}}p_{k}(\zeta)\overline{d\zeta}$
$\displaystyle={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=1}\overline{v(\zeta)\zeta^{n+1}}p_{k}(\zeta)d\zeta\,.$

Since $\overline{v(z)}=\overline{v(1/\overline{z})}=1/v^{*}(z)$ for $|z|=1$,
this can also be written as

$\displaystyle\overline{\alpha_{n,k}}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=1}\frac{\zeta^{-n-1}}{v^{*}(\zeta)}p_{k}(\zeta)d\zeta\,.$
(5.14)

To prove (5.13), we first observe that by (5.1), if $k>n\geq 0$, then

$\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=1}\overline{v(\zeta)\zeta^{n+1}}F_{k}(\zeta)d\zeta={}$
$\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=1}\frac{\zeta^{-n-1}}{v^{*}(\zeta)}\left(\zeta^{k}v^{*}(\zeta)+O(1/\zeta)\right)d\zeta$
$\displaystyle={}$ $\displaystyle 0.$

Therefore, for all $k>n\geq 0$, we have

$\displaystyle\overline{\alpha_{n,k}}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=1}\overline{v(\zeta)\zeta^{n+1}}\left(p_{k}(\zeta)-\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)\right)d\zeta,$

and consequently,

$\displaystyle|\alpha_{n,k}|\leq{}$
$\displaystyle\left(\int_{|\zeta|=1}|v(\zeta)|^{2}|d\zeta|\right)^{1/2}\left(\int_{|\zeta|=1}\left|p_{k}(\zeta)-\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)\right|^{2}|d\zeta|\right)^{1/2}.$
(5.15)

We then proceed to estimate the second factor in (5.15). Fix a number $r<1$
such that every $a_{k}$ is contained in the disk $|z|<r$. Noting that

$d_{r}:=\min_{r\leq|z|\leq 1}w(z)>0,$

and aided by (5.6), we obtain

$\displaystyle\begin{split}\int_{|\zeta|=1}\left|p_{k}(\zeta)-\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)\right|^{2}|d\zeta|\leq{}&\frac{2\pi
k}{1-r^{2}}\int_{r\leq|\zeta|\leq
1}\left|p_{k}(\zeta)-\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)\right|^{2}d\sigma(z)\\\
\leq{}&\frac{2\pi
k}{(1-r^{2})d_{r}}\int_{\mathbb{D}}\left|p_{k}(\zeta)-\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)\right|^{2}w(z)d\sigma(z).\end{split}$

The latter integral can be estimated by using (5.3) and (5.7) as follows:

$\displaystyle\begin{split}\int_{\mathbb{D}}&\left|p_{k}(\zeta)-\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)\right|^{2}w(z)d\sigma(z)\\\
={}&1-2\Re\left(\int_{\mathbb{D}}p_{k}(\zeta)\overline{\frac{\gamma_{k}}{c_{0}}F_{k}(\zeta)}w(z)d\sigma(z)\right)+\frac{\gamma^{2}_{k}}{c^{2}_{0}}\int_{\mathbb{D}}\left|F_{k}(\zeta)\right|^{2}w(z)d\sigma(z)\\\
={}&\frac{\gamma^{2}_{k}}{c^{2}_{0}}\int_{\mathbb{D}}\left|F_{k}(\zeta)\right|^{2}w(z)d\sigma(z)-1=O\left(k^{-1}\right)\,.\end{split}$

This estimate and the previous inequality show that the right-hand side of
(5.15) is uniformly bounded in $k$, which proves (5.13). ∎

##  6 Proof of Theorem 2.1

From the structural formula for $K_{h}$ established in [5], it follows that

$\displaystyle K_{h}(z,1/\overline{\zeta})=\zeta^{2}L(z,\zeta),$ (6.1)

where $L(z,\zeta)$ is a rational function of the form (recall (2.1))

$\displaystyle
L(z,\zeta)=\frac{1}{(\zeta-z)^{2}}+\frac{V(z,\zeta)}{(\zeta-z)q^{*}(z)q(\zeta)}$
(6.2)

with $V(z,\zeta)$ a polynomial in the variables $z$ and $\zeta$.

By the symmetry of the kernel, the series in (4.3) converges locally uniformly
for $|\zeta|<1/\tau_{z}$, for each fixed $z\in\mathbb{D}$. Since $\tau_{z}<1$,
we can combine (4.3), (6.1) and (2.2) to obtain

$\displaystyle\frac{(vv^{*})(\zeta)L(z,\zeta)\zeta^{n+1}}{v(z)}=v(\zeta)\zeta^{n-1}\sum_{k=0}^{\infty}\overline{p_{k}(1/\overline{\zeta})}p_{k}(z)\,,\quad
z\in\mathbb{D},\ |\zeta|=1.$ (6.3)

Let us now define, for every $n\geq 0$, a function $Q_{n}$ analytic in
$|z|<1/\rho_{w}$ as follows. For all $\rho_{w}<r<1/\rho_{w}$,

$\displaystyle Q_{n}(z):=\frac{1}{2\pi
iv(z)}\int_{|\zeta|=r}(vv^{*})(\zeta)L(z,\zeta)\zeta^{n+1}d\zeta,\quad|z|<r.$
(6.4)

By Cauchy’s theorem, it is clear that this function is well-defined, meaning
that for $z$ fixed, the integral in (6.4) produces the same value for any
$r>|z|$.

If we make use of (5.11) and (5.12) when integrating (6.3) over the unit
circle with respect to $\zeta$, we find that for all $n\geq 0$,

$\displaystyle\alpha_{n,n}p_{n}(z)={}$ $\displaystyle
Q_{n}(z)-\sum_{k=n+1}^{\infty}\alpha_{n,k}p_{k}(z),\quad|z|<1.$ (6.5)

Our next step is crucial, and consists of iterating (6.5). This is best
carried out in two separate lemmas.

For any triplet of integers $k$, $m$, and $n$ such that $m,n\geq 0$ and $k\geq
n+m+1$, we define corresponding numbers $h(n,m)$ and $g(n,m,k)$ in the
following recursive manner:

$\displaystyle h(n,0)=1,\quad g(n,0,k)=-\alpha_{n,k},\quad k>n\geq 0,$ (6.6)
$\displaystyle h(n,m+1)=\frac{g(n,m,n+m+1)}{\alpha_{n+m+1,n+m+1}}\,,$ (6.7)
$\displaystyle g(n,m+1,k)=g(n,m,k)-h(n,m+1)\alpha_{n+m+1,k},\quad k\geq
n+m+2.$ (6.8)

######  Lemma 6.1.

For all integers $n,m\geq 0$, the equality

$\displaystyle\alpha_{n,n}p_{n}(z)={}$
$\displaystyle\sum_{j=0}^{m}h(n,j)Q_{n+j}(z)+\sum_{k=n+m+1}^{\infty}g(n,m,k)p_{k}(z)$
(6.9)

holds true for $|z|<1$.

###### Proof.

The proof is by induction on $m$. For $m=0$, (6.9) reduces to (6.5). Assuming
(6.9) is true for some $m$, we can use (6.5) (with $n$ replaced by $n+m+1$) to
get

$\displaystyle\alpha_{n,n}p_{n}(z)={}$
$\displaystyle\sum_{j=0}^{m}h(n,j)Q_{n+j}(z)+\sum_{k=n+m+2}^{\infty}g(n,m,k)p_{k}(z)$
$\displaystyle+\frac{g(n,m,n+m+1)}{\alpha_{n+m+1,n+m+1}}\alpha_{n+m+1,n+m+1}p_{n+m+1}(z)$
$\displaystyle={}$
$\displaystyle\sum_{j=0}^{m}h(n,j)Q_{n+j}(z)+\sum_{k=n+m+2}^{\infty}g(n,m,k)p_{k}(z)$
$\displaystyle+\frac{g(n,m,n+m+1)}{\alpha_{n+m+1,n+m+1}}\left(Q_{n+m+1}(z)-\sum_{k=n+m+2}^{\infty}\alpha_{n+m+1,k}p_{k}(z)\right)$
$\displaystyle={}$
$\displaystyle\sum_{j=0}^{m}h(n,j)Q_{n+j}(z)+\frac{g(n,m,n+m+1)}{\alpha_{n+m+1,n+m+1}}Q_{n+m+1}(z)$
$\displaystyle+\sum_{k=n+m+2}^{\infty}\left(g(n,m,k)-\frac{g(n,m,n+m+1)}{\alpha_{n+m+1,n+m+1}}\alpha_{n+m+1,k}\right)p_{k}(z)\,.$

The latter three terms are precisely what we obtain if we replace $m$ by $m+1$
in (6.9) and use the defining relations (6.7) and (6.8). ∎

######  Lemma 6.2.

Suppose that there exist constants $C,C^{\prime},\lambda\geq 0$ such that

$\displaystyle\frac{1}{\alpha_{n,n}}\leq\frac{C^{\prime}}{\sqrt{n}},\quad|\alpha_{n,k}|\leq\frac{C}{k^{\lambda}},\quad
k>n\geq 0.$ (6.10)

Then, we have the series representation

$\displaystyle\alpha_{n,n}p_{n}(z)={}$
$\displaystyle\sum_{j=0}^{\infty}h(n,j)Q_{n+j}(z),\quad|z|<1,$ (6.11)

with $h(n,0)=1$ and the remaining coefficients satisfying the inequality

$\displaystyle|h(n,j)|\leq\frac{CC^{\prime}}{(n+j)^{\lambda+1/2}}\prod_{\ell=1}^{j-1}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right),\quad
j\geq 1\,.$ (6.12)

###### Proof.

By hypothesis, and by the defining relations (6.6)-(6.8), we have

$h(n,0)=1,\quad|g(n,0,k)|\leq\frac{C}{k^{\lambda}},\quad k\geq n+1,$
$\displaystyle|h(n,m+1)|\leq{}$
$\displaystyle\frac{C^{\prime}|g(n,m,n+m+1)|}{\sqrt{n+m+1}}\,,$
$\displaystyle|g(n,m+1,k)|\leq{}$
$\displaystyle|g(n,m,k)|+\frac{C}{k^{\lambda}}|h(n,m+1)|,\quad k\geq n+m+2.$

From these inequalities, we proceed by induction (in the second variable $m$)
to prove (6.12) and that

$\displaystyle|g(n,m,k)|\leq\frac{C}{k^{\lambda}}\prod_{\ell=1}^{m}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right),\quad
m\geq 0,\ k\geq n+m+1.$ (6.13)

For $m=0$ the inequality (6.13) is correct. Then, assuming it is also correct
for $g(n,m,k)$, we have

$\displaystyle|h(n,m+1)|\leq\frac{C^{\prime}}{\sqrt{n+m+1}}\frac{C}{(n+m+1)^{\lambda}}\prod_{\ell=1}^{m}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)$

and

$\displaystyle|g(n,m+1,k)|\leq{}$
$\displaystyle\frac{C}{k^{\lambda}}\prod_{\ell=1}^{m}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)$
$\displaystyle+\frac{C}{k^{\lambda}}\frac{C^{\prime}C}{(n+m+1)^{\lambda+1/2}}\prod_{\ell=1}^{m}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)$
$\displaystyle\leq{}$
$\displaystyle\frac{C}{k^{\lambda}}\left(1+\frac{C^{\prime}C}{(n+m+1)^{\lambda+1/2}}\right)\prod_{\ell=1}^{m}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)\,.$

Let us now think of $z\in\mathbb{D}$ as being fixed. Since
$\tau_{z}=\limsup_{k\to\infty}|p_{k}(z)|^{1/k}<1$, we can find $\epsilon>0$
and a constant $A_{\epsilon}$ such that $\tau_{z}+\epsilon<1$ and
$|p_{k}(z)|\leq A_{\epsilon}(\tau_{z}+\epsilon)^{k}$ for all $k\geq 0$. This
and (6.13) imply that

$\displaystyle\begin{split}&\sum_{k=n+m+1}^{\infty}|g(n,m,k)p_{k}(z)|\\\ &\leq
A_{\epsilon}C\prod_{\ell=1}^{m}\left(1+\frac{CC}{(n+\ell)^{\lambda+1/2}}\right)\sum_{k=n+m+1}^{\infty}\frac{(\tau_{z}+\epsilon)^{k}}{k^{\lambda}}\\\
&{}\leq{}\frac{A_{\epsilon}C(\tau_{z}+\epsilon)^{n+1}}{(n+m+1)^{\lambda}(1-(\tau_{z}+\epsilon))}\prod_{\ell=1}^{m}\left[\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)(\tau_{z}+\epsilon)\right].\end{split}$
(6.14)

Since

$\lim_{\ell\to\infty}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)(\tau_{z}+\epsilon)=(\tau_{z}+\epsilon)<1,$

the last product that occurs in (6.14) converges to zero as $m\to\infty$, and
so we conclude that for $|z|<1$,

$\lim_{m\to\infty}\sum_{k=n+m+1}^{\infty}|g(n,m,k)p_{k}(z)|=0,$

which in view of (6.9) confirms the validity or (6.11). ∎

Let us now define

$H_{n}(z):=\sum_{j=1}^{\infty}h(n,j)z^{j}\,.$

By Lemma 6.2, when constants $C,C^{\prime},\lambda\geq 0$ can be found such
that (6.10) holds, then (see (6.12))

$|H_{n}(z)|\leq\frac{CC^{\prime}}{(n+1)^{\lambda+1/2}}\sum_{j=1}^{\infty}|z|^{j}\prod_{\ell=1}^{j-1}\left(1+\frac{CC^{\prime}}{(n+\ell)^{\lambda+1/2}}\right)\,.$

This would imply that $H_{n}(z)$ is well-defined and analytic in $\mathbb{D}$,
and that

$\displaystyle|H_{n}(z)|=O(n^{-\lambda-1/2})$ (6.15)

uniformly as $n\to\infty$ on every disk $|z|\leq r$ of radius $r<1$. Moreover,
from the definition of $Q_{n}$ and (6.11), we would also have that for every
$r$ in the range $\rho_{w}<r<1$,

$\displaystyle\alpha_{n,n}p_{n}(z)={}$ $\displaystyle\frac{1}{2\pi
iv(z)}\int_{|\zeta|=r}(vv^{*})(\zeta)L(z,\zeta)\zeta^{n+1}(1+H_{n}(\zeta))d\zeta,\quad|z|<r.$
(6.16)

By (5.12) and Proposition 5.3, the first inequality of (6.10) is always
satisfied, and by Proposition 5.4, the second inequality of (6.10) holds true
with $\lambda=0$. Therefore, we can say for sure that $H_{n}(z)=O(n^{-1/2})$
uniformly as $n\to\infty$ on compact subsets of $\mathbb{D}$.

We now proceed to get a better estimate for the coefficients $\alpha_{n,k}$ of
(5.11). Let $\rho_{v}$ be the smallest number such that $v^{*}$ can be
analytically continued without ever vanishing in $|z|>\rho_{v}$, and suppose
that $\mu$ and $\eta$ are such that

$\displaystyle\rho_{w}<\mu<1,\quad\max\\{\mu,\rho_{v}\\}<\eta<1.$ (6.17)

Bearing in mind the representation (6.2), the contour of integration in (6.16)
can be retracted to the circle $|\zeta|=\mu$ if we properly account for
residues. Thus, for $\mu<|z|<1$, we have

$\displaystyle\begin{split}\alpha_{n,n}p_{n}(z)={}&R_{n}(z)+\frac{1}{v(z)}\left[v(z)v^{*}(z)z^{n+1}(1+H_{n}(z))\right]^{\prime}\\\
&+\frac{z^{n+1}v^{*}(z)V(z,z)(1+H_{n}(z))}{q^{*}(z)q(z)},\end{split}$ (6.18)

with

$\displaystyle R_{n}(z)={}$ $\displaystyle\frac{1}{2\pi
iv(z)}\int_{|\zeta|=\mu}(vv^{*})(\zeta)L(z,\zeta)\zeta^{n+1}(1+H_{n}(\zeta))d\zeta$
$\displaystyle={}$ $\displaystyle\frac{1}{2\pi
iv(z)}\int_{|\zeta|=\mu}\frac{v(\zeta)v^{*}(\zeta)\zeta^{n+1}}{(\zeta-z)^{2}}(1+H_{n}(\zeta))d\zeta$
$\displaystyle+\frac{1}{2\pi
iv(z)q^{*}(z)}\int_{|\zeta|=\mu}\frac{(v^{*}/q)(\zeta)v(\zeta)V(z,\zeta)\zeta^{n+1}}{\zeta-z}(1+H_{n}(\zeta))d\zeta\,.$

By Cauchy’s theorem, we can write (5.14) as

$\displaystyle\overline{\alpha_{n,k}}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=\eta}\frac{\zeta^{-n-1}}{v^{*}(\zeta)}p_{k}(\zeta)d\zeta\,.$

In this integral, we replace $p_{k}$ by its representation (6.18) to get

$\displaystyle\alpha_{k,k}\overline{\alpha_{n,k}}={}$ $\displaystyle
I^{1}_{n,k}+I^{2}_{n,k}+I^{3}_{n,k},$ (6.19)

where

$\displaystyle I^{1}_{n,k}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=\eta}\frac{\zeta^{-n-1}}{v^{*}(\zeta)}R_{k}(\zeta)d\zeta\,,$
(6.20) $\displaystyle I^{2}_{n,k}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=\eta}\frac{\zeta^{-n-1}\left[v(\zeta)v^{*}(\zeta)\zeta^{k+1}(1+H_{k}(\zeta))\right]^{\prime}}{v(\zeta)v^{*}(\zeta)}d\zeta\,,$
$\displaystyle I^{3}_{n,k}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=\eta}\frac{\zeta^{k-n}V(\zeta,\zeta)(1+H_{k}(\zeta))}{q^{*}(\zeta)q(\zeta)}d\zeta\,.$
(6.21)

We proceed to estimate $I^{1}_{n,k}$. Note that $R_{k}(z)$ is well-defined and
analytic for values of $z$ in the annulus $\mu<|z|<1/\rho_{w}$, and that
$R_{k}(z)=O(\mu^{k})$ uniformly as $k\to\infty$ on any circle $|z|=r$ with
$\mu<r<1/\rho_{w}$. Then, the value of the integral in (6.20) does not change
if the circle of integration $|\zeta|=\eta$ is moved to the circle
$|\zeta|=1/\mu$, which leads to

$\displaystyle\begin{split}I^{1}_{n,k}=O\left(\mu^{k+n}\right),\quad\rho_{w}<\mu<1,\end{split}$
(6.22)

with the implied constant depending only on the value of $\mu$.

Next, we estimate $I^{2}_{n,k}$. By a direct computation of the derivative, we
find

$\displaystyle\frac{\left[v(\zeta)v^{*}(\zeta)\zeta^{k+1}(1+H_{k}(\zeta))\right]^{\prime}}{v(\zeta)v^{*}(\zeta)}={}$
$\displaystyle(k+1)\zeta^{k}(1+H_{k}(\zeta))+\zeta^{k+1}H_{k}^{\prime}(\zeta)$
$\displaystyle+\frac{v^{\prime}(\zeta)}{v(\zeta)}\zeta^{k+1}(1+H_{k}(\zeta))$
$\displaystyle+\frac{{v^{*}}^{\prime}(\zeta)}{v^{*}(\zeta)}\zeta^{k+1}(1+H_{k}(\zeta))\,.$

Since $H_{k}$ and $v$ are analytic in $\mathbb{D}$, we see that for $k>n$,
$I^{2}_{n,k}$ reduces to

$\displaystyle I^{2}_{n,k}={}$ $\displaystyle\frac{1}{2\pi
i}\int_{|\zeta|=\eta}\frac{{v^{*}}^{\prime}(\zeta)}{v^{*}(\zeta)}\zeta^{k-n}(1+H_{k}(\zeta))d\zeta\,.$
(6.23)

The number $\rho_{v}$ is also the smallest number such that the function
${v^{*}}^{\prime}/v^{*}$ has an analytic continuation to $|z|>\rho_{v}$, and
so we get from (6.23)

$\displaystyle I^{2}_{n,k}={}$ $\displaystyle
O(\eta^{k-n}),\quad\rho_{v}<\eta<1,$ (6.24)

with the implied constant depending only on $\eta$.

By comparing (6.2) and (2.7), we see that

$\displaystyle\frac{V(\zeta,\zeta)}{q^{*}(\zeta)q(\zeta)}={}$
$\displaystyle\sum_{j=1}^{s}\frac{\frac{m_{j}}{2}(1-|a_{j}|^{2})}{(\zeta-
a_{j})(1-\zeta\overline{a}_{j})}.$

We substitute this expression in (6.21) and compute the resulting integral by
means of the residue theorem to obtain

$\displaystyle I^{3}_{n,k}={}$
$\displaystyle\sum_{j=1}^{s}\frac{m_{k}}{2}a_{j}^{k-n}(1+H_{k}(a_{j}))=O(\rho_{a}^{k-n}),$
(6.25)

with $\rho_{a}=\max_{1\leq k\leq s}|a_{k}|$.

Having established (6.22), (6.24), and (6.25), it follows from (6.19) that
with $\mu$ and $\eta$ as specified in (6.17),

$\displaystyle\alpha_{k,k}\overline{\alpha_{n,k}}={}$ $\displaystyle
O\left(\mu^{k+n}\right)+O(\eta^{k-n})+O(\rho_{a}^{k-n}).$

Since $v^{*}/q$ is analytic in $|z|>\rho_{w}$, so is $v^{*}$ alone, and thus
$\rho_{w}\leq\rho_{v}$. With this observation in mind, and recalling (5.12)
and (5.7), we conclude from the latter equality that for every $\eta$ in the
range $\rho_{v}<\eta<1$, there exists a constant $C_{\eta}$ such that

$\displaystyle|\alpha_{n,k}|\leq\frac{C_{\eta}(\eta^{k-n}+\rho_{a}^{k-n})}{\sqrt{k}},\quad
k>n\geq 0.$ (6.26)

This shows that the second inequality of (6.10) is satisfied with
$\lambda=1/2$. It follows from the discussion leading to (6.15) that Formula
(6.16) holds with $H_{n}(z)=O(n^{-1})$, completing the proof of Theorem 2.1.

From (6.18), we see that for every $\mu$ and $r$ such that $\rho_{w}<\mu<r<1$,
the equality

$\displaystyle\frac{\alpha_{n,n}p_{n}(z)}{(n+1)z^{n}}={}$ $\displaystyle
O(n^{-1}(\mu/r)^{n})+v^{*}(z)(1+H_{n}(z))+\frac{zv^{*}(z)}{n+1}H^{\prime}_{n}(z)$
$\displaystyle+\frac{z}{n+1}\left(\frac{(vv^{*})^{\prime}(z)}{v(z)}+\frac{v^{*}(z)V(z,z)}{q(z)q^{*}(z)}\right)(1+H_{n}(z))$

holds uniformly for $|z|=r$ as $n\to\infty$. This quickly leads to (2.5) for
$|z|=r$, and by the maximum modulus principle, for $|z|\geq r$ as well.

## References

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# Quantum circuits generating four-qubit maximally entangled states

Marc Bataille
<EMAIL_ADDRESS>

LITIS laboratory, Université Rouen-Normandie 685 Avenue de l’Université, 76800
Saint-Étienne-du-Rouvray. France.

###### Abstract

We describe quantum circuits generating four-qubit maximally entangled states,
the amount of entanglement being quantified by using the absolute value of the
Cayley hyperdeterminant as an entanglement monotone. More precisely we show
that this type of four-qubit entangled states can be obtained by the action of
a family of CNOT circuits on some special states of the LU orbit of the state
$\ket{0000}$.

##  1 Introduction

The original idea of using the hyperdeterminant to classify multipartite
entanglement goes back to Miyake [9, 8]. The hyperdeterminant (in the sense of
Gelfand et al. [4]) is a generalization of the determinant to higher
dimensions. Let
$\ket{\psi}=\sum_{i_{0},i_{1},\dots,i_{n-1}\in\\{0,1\\}}a_{i_{0}i_{1}\dots
i_{n-1}}\ket{i_{0}i_{1}\dots i_{n-1}}$ be the state vector of an $n$-qubit
system in the Hilbert space $\mathcal{H}^{\otimes n}=(\mathbb{C}^{2})^{\otimes
n}$, then the hyperdeterminant of the format $2^{n}$, denoted in this paper by
$\Delta_{n}$, is an homogenous multivariate polynomial in the $2^{n}$
variables $a_{i_{0}i_{1}\dots i_{n-1}}$, with coefficients in $\mathbb{Z}$. It
is invariant (up to a sign) by permutation of the qubits and also invariant by
the action of the group $SLOCC$, the group of stochastic local operations
assisted by classical communication, assimilated to the cartesian product
$SL(2,\mathbb{C})^{n}$. According to Miyake [9, 8], the more generic
entanglement holds only for the states on which the hyperdeterminant does not
vanish and the absolute value of $\Delta_{n}$ quantifies the amount of generic
entanglement.

In this article, we focus on a four-qubit quantum system. In this case, the
hyperdeterminant $\Delta_{4}$ is of degree 24 and an expression of
$\Delta_{4}$ in terms of fundamental SLOCC invariant polynomials of lower
degree was given by Luque and Thibon [7]. Following Miyake, we consider as
Gour and Wallach [6], that a four-qubit state with the highest amount of
generic entanglement can be defined as a state maximizing the absolute value
of $\Delta_{4}$. In the rest of the paper, we refer to this type of state as a
maximum hyperdeterminant state (sometimes abbreviated as MHS). In a paper from
2012 [6], Gour and Wallach conjectured that the state $\ket{L}$ (see Fig. 1)
is the unique maximum hyperdeterminant state, up to a local unitary operation.
This conjecture was proved in 2013 by Chen and Djokovic [3] and the maximal
value of $|\Delta_{4}|$ is
$\frac{1}{2^{8}3^{9}}=\frac{1}{\numprint{5038848}}\simeq 1.98\times 10^{-7}$.
Moreover, the state $\ket{L}$ has also the property to be the only state (up
to local unitary operations) maximizing the average Tsallis $\alpha$-entropy
of entanglement, for all $\alpha>2$ [5]. Let us also mention two other maximum
hyperdeterminant states, which have the property of having real coordinates
(see Fig. 1): $\ket{\Phi_{5}}$ (reported by Osterloh and Siewert in [11] and
by Alsina in his PhD thesis [1]) and $\ket{M_{2222}}$ (reported by Hamza
Jaffali in his PhD thesis, unpublished) .

In Quantum Information and Computation, entangled states, and in particular
maximally entangled states, play the role of an important physical resource
(see e.g. the introduction of [3]). Despite of that, to our knowledge, there
is no proposal in the academic literature for quantum circuits capable of
producing the state $\ket{L}$ or any other MHS. The goal of this work is
merely to fill this gap by describing a family of quantum circuits that enable
the generation of maximum hyperdeterminant states. We show that a MHS can be
obtained by the action a certain type of CNOT gate circuits on a fully
factorized state, namely a state of the LU orbit of $\ket{0000}$. As a
consequence of this result, one can construct quantum circuits of relatively
small depth generating the three states $\ket{L}$, $\ket{\Phi_{5}}$ and
$\ket{M_{2222}}$.

The paper is structured as follows. Section 2 is a reminder on quantum
circuits of $\mathtt{CNOT}$ gates and $\mathtt{SWAP}$ gates, where we
introduce most of our notations and give some useful conjugation rules between
these gates. In Section 3, we present the methodology and algorithms used in
our numerical approach to find circuits generating maximum hyperdeterminant
states. The two next sections (4 and 5) are dedicated to the description of
these circuits. Finally, in Section 6, we propose three simple quantum
circuits generating the states $\ket{L}$, $\ket{\Phi_{5}}$ and $M_{2222}$, as
well as an implementation of a circuit generating the state $\ket{L}$ into a
quantum computer provided by the IBM quantum experience at https://quantum-
computing.ibm.com/.

This article goes along with a Python module than can be downloaded at
https://github.com/marcbataille/maximum-hyperdeterminant-states. The module
provides an implementation of the different algorithms, quantum gates and
quantum states used in this work. As the proof of some assertions (mostly
numerical equalities) consists only of basic linear algebra and calculus, we
chose to refer the reader to the corresponding function of the module that
does the job.

$\ket{L}=\frac{1}{\sqrt{3}}(\ket{u_{0}}+\omega\ket{u_{1}}+\omega^{*}\ket{u_{2}})$
(1)

$\text{where : }\omega=\mathrm{e}^{\frac{\mathrm{i}\pi}{3}}$

$\phantom{where:}\ket{u_{0}}=\frac{1}{2}(\ket{0000}+\ket{0011}+\ket{1100}+\ket{1111})=\ket{\Phi^{+}}\ket{\Phi^{+}}$

$\phantom{where:}\ket{u_{1}}=\frac{1}{2}(\ket{0000}-\ket{0011}-\ket{1100}+\ket{1111})=\ket{\Phi^{-}}\ket{\Phi^{-}}$

$\phantom{where:}\ket{u_{2}}=\frac{1}{2}(\ket{0101}+\ket{0110}+\ket{1001}+\ket{1010})=\ket{\Psi^{+}}\ket{\Psi^{+}}$

$\phantom{where:}\ket{\Phi^{\pm}}=\frac{1}{\sqrt{2}}(\ket{00}\pm\ket{11}),\quad\ket{\Psi^{\pm}}=\frac{1}{\sqrt{2}}(\ket{01}\pm\ket{10})$

$\ket{\Phi_{5}}=\frac{1}{\sqrt{6}}(\ket{0001}+\ket{0010}+\ket{0100}+\ket{1000}+\sqrt{2}\ket{1111})$
(2)

$\ket{M_{2222}}=\frac{1}{\sqrt{8}}\ket{v_{1}}+\frac{\sqrt{6}}{4}\ket{v_{2}}+\frac{1}{\sqrt{2}}\ket{v_{3}}$
(3)

$\text{where:
}\ket{v_{1}}=\frac{1}{\sqrt{6}}(\ket{0000}+\ket{0101}-\ket{0110}-\ket{1001}+\ket{1010}+\ket{1111})$

$\phantom{where:}\ket{v_{2}}=\frac{1}{\sqrt{2}}(\ket{0011}+\ket{1100})$

$\phantom{where:}\ket{v_{3}}=\frac{1}{\sqrt{8}}(-\ket{0001}+\ket{0010}-\ket{0100}+\ket{0111}+\ket{1000}-\ket{1011}+\ket{1101}-\ket{1110})$

Figure 1: 4-qubits states for which $|\Delta_{4}|$ is maximal.

##  2 Quantum circuits of $\mathtt{CNOT}$ and $\mathtt{SWAP}$ gates

In this section we introduce the main notations and conventions of the paper
and we recall the definition of some classical quantum gates (Table 1) as well
as some properties of the $\mathtt{CNOT}$ gates and $\mathtt{SWAP}$ gates
often used in the rest of the article.

Let $n\geqslant 1$ be the number of qubits of the considered quantum register.
We label each qubit from 0 to $n-1$, thus following the usual convention. For
coherence we also number the lines and columns of a $n\times n$ matrix from 0
to $n-1$ and we consider that a permutation in the symmetric group
$\mathfrak{S}_{n}$ is a bijection of $\\{0,\dots,n-1\\}$.

Name | Symbol | Matrix
---|---|---
Pauli-$\mathtt{X}$ | $X$ | $\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$
Pauli-$\mathtt{Y}$ | $Y$ | $\begin{bmatrix}0&-\mathrm{i}\\\ \mathrm{i}&0\end{bmatrix}$
Pauli-$\mathtt{Z}$ | $Z$ | $\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$
Rotation around the $\hat{x}$ axis | $R_{x}(\theta)$ | $\mathrm{e}^{-\mathrm{i}\theta X/2}=\begin{bmatrix}\cos\frac{\theta}{2}&-\mathrm{i}\sin\frac{\theta}{2}\\\ -\mathrm{i}\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{bmatrix}$
Rotation around the $\hat{y}$ axis | $R_{y}(\theta)$ | $\mathrm{e}^{-\mathrm{i}\theta Y/2}=\begin{bmatrix}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\ \sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{bmatrix}$
Rotation around the $\hat{z}$ axis | $R_{z}(\theta)$ | $\mathrm{e}^{-\mathrm{i}\theta Z/2}=\begin{bmatrix}\mathrm{e}^{-\mathrm{i}\theta/2}&0\\\ 0&\mathrm{e}^{\mathrm{i}\theta/2}\end{bmatrix}$
Phase | $P$ | $\begin{bmatrix}1&0\\\ 0&\mathrm{i}\end{bmatrix}$
T-gate | $T$ | $\begin{bmatrix}1&0\\\ 0&\mathrm{e}^{\mathrm{i}\pi/4}\end{bmatrix}$
Hadamard | $H$ | $\frac{\sqrt{2}}{2}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix}$
Table 1: Classical single qubit unitary gates

If two normalized vectors $\ket{\psi}$ and $\ket{\psi^{\prime}}$ of the
Hilbert space $\mathcal{H}^{\otimes n}$ are equal up to a global phase, then
they represent physically the same state and we write
$\ket{\psi}\simeq\ket{\psi^{\prime}}$. In the same way, we write $U\simeq
U^{\prime}$ for two unitary operators which are equal up to a global phase. In
the design of quantum circuits, we use the following correspondences between
the classical gates :

$\displaystyle R_{z}(\pi)\simeq Z,\ R_{z}(\pi/2)\simeq P,\ R_{z}(-\pi/2)\simeq
P^{{\dagger}},\ R_{z}(\pi/4)\simeq T$ (4) $\displaystyle R_{y}(\pi)\simeq Y,\
R_{y}(\pi/2)=HZ=XH,\ R_{y}(-\pi/2)=ZH=HX$ (5)

When we apply locally a single-qubit gate $U$ to the qubit $i$ of a $n$-qubit
register, the corresponding action on the $n$-qubit system is that of the
unitary operator

$U_{i}=\underbrace{I\otimes\dots\otimes I}_{i\ \mathrm{times}}\otimes
U\otimes\underbrace{I\otimes\dots\otimes I}_{n-i-1\text{ times}}=I^{\otimes
i}\otimes U\otimes I^{\otimes n-i-1},$ (6)

where $\otimes$ is the Kronecker product of matrices and $I$ the identity
matrix in dimension 2. As an example, if $n=4$, $H_{1}=I\otimes H\otimes
I\otimes I$ and $H_{0}H_{3}=H\otimes I\otimes I\otimes H$. We also use vectors
of $\mathbb{F}_{2}^{n}$ as labels to indicate the set of qubits on which the
single-qubit unitary $U$ is applied. Let $v=[v_{0},\dots,v_{n-1}]^{t}$ be a
(column) vector of $\mathbb{F}_{2}^{n}$, we denote by $U_{v}$ the product
$\prod_{i}U_{i}^{v_{i}}$.

A $\mathtt{CNOT}$ gate with target on qubit $i$ and control on qubit $j$ is
denoted by $X_{[ij]}$ (not to be confused with $X_{i}$ which denotes a
Pauli-$\mathtt{X}$ gate applied on qubit $i$). The group generated by the
$\mathtt{CNOT}$ gates acting on an $n$-qubit quantum system is denoted by
${\left\langle\mathtt{CNOT}\right\rangle}_{n}$. Let us denote by
$\mathrm{GL}_{n}(\mathbb{F}_{2})$ the general linear group over
$\mathbb{F}_{2}$ in dimension $n$. A transvection matrix $[ij]$ ($i,j=1\dots
n-1$ and $i\neq j$), is the matrix of $\mathrm{GL}_{n}(\mathbb{F}_{2})$
defined by $[ij]=I_{n}+E_{ij}$, where $I_{n}$ is the identity in dimension $n$
and $E_{ij}$ is the matrix with all entries equal to zero but the entry
$(i,j)$ that is equal to 1. We recall that the transvection matrices generate
the group $\mathrm{GL}_{n}(\mathbb{F}_{2})$ and that multiplying a matrix $M$
to the left by a transvection matrix $[ij]$ is equivalent to adding the row
$j$ to the row $i$ of $M$. From these facts, one can deduce that the group
${\left\langle\mathtt{CNOT}\right\rangle}_{n}$ is isomorphic to
$\mathrm{GL}_{n}(\mathbb{F}_{2})$, a possible isomorphism associating, to any
gate $X_{[ij]}$, the transvection matrix $[ij]$ (see [2] for more details).
The order of ${\left\langle\mathtt{CNOT}\right\rangle}_{n}$ is therefore equal
to the order of $\mathrm{GL}_{n}(\mathbb{F}_{2})$ :

$|{\left\langle\mathtt{CNOT}\right\rangle}_{n}|=2^{\frac{n(n-1)}{2}}\prod_{i=1}^{n}(2^{i}-1).$
(7)

Let $A$ be any matrix in $\mathrm{GL}_{n}(\mathbb{F}_{2})$, we denote by
$X_{A}$ the element of ${\left\langle\mathtt{CNOT}\right\rangle}_{n}$
associated to $A$, i.e. $X_{A}$ is the product of any sequence of
$\mathtt{CNOT}$ gates $X_{[i_{1},j_{i}]},\dots,X_{[i_{p},j_{p}]}$ such that
$A$ can be decomposed in the product of the transvection matrices
$[i_{1},j_{i}],\dots,[i_{p},j_{p}]$.

The $\mathtt{SWAP}$ gate that exchanges qubits $i$ and $j$ is denoted by
$S_{(ij)}$. Let $\sigma$ be a permutation of the symmetric group
$\mathfrak{S}_{n}$. We also denote by $\sigma$ the permutation matrix
associated to the permutation $\sigma$. This matrix is defined as the matrix
$A=(a_{ij})$ in $\mathrm{GL}_{n}(\mathbb{F}_{2})$ such that $a_{ij}=1$ if and
only if $i=\sigma(j)$. We recall that multiplying a matrix $M$ to the left by
$\sigma$ is equivalent to applying the permutation $\sigma$ to the rows of
$M$. In this case, each row $R_{i}$ is replaced by the row
$R_{\sigma^{-1}(i)}$. The group of permutation matrices is a subgroup of
$\mathrm{GL}_{n}(\mathbb{F}_{2})$ which is isomorphic to the group generated
by the $\mathtt{SWAP}$ gates acting on $n$ qubits : to each $\mathtt{SWAP}$
gate $S_{(ij)}$ corresponds the transposition matrix $(ij)$. We denote by
$S_{\sigma}$ the product of any sequence of $\mathtt{SWAP}$ gates
$S_{(i_{1}j_{1})},\dots,S_{(i_{p},j_{p})}$ such that
$\sigma=(i_{1}j_{1})\dots(i_{p}j_{p})$. Let $\tau$ be a transposition of
$\mathfrak{S}_{n}$, it is easy to check that
$S_{\tau}X_{[ij]}S_{\tau}=X_{[\tau(i)\tau(j)]}$, hence by induction

$S_{\sigma}X_{[ij]}S_{\sigma}^{-1}=X_{[\sigma(i)\sigma(j)]},$ (8)

for any permutation $\sigma$. Let $U$ be a single-qubit unitary matrix and
$U_{i}$ the unitary corresponding to the action of $U$ on qubit $i$ (Identity
(6)), one has for any permutation $\sigma$ and any vector $v$ in
$\mathbb{F}_{2}^{n}$ :

$\displaystyle S_{\sigma}U_{i}S_{\sigma}^{-1}=U_{\sigma(i)}$ (9)
$\displaystyle S_{\sigma}U_{v}S_{\sigma}^{-1}=U_{\sigma v}$ (10)

The Pauli group for $n$ qubits is the group generated by the Pauli gates
$X_{i},Y_{i}$ and $Z_{i}$ ($0\leqslant i\leqslant n-1$). Since
$Y=\mathrm{i}XZ$ and $XZ=-ZX$, any element of this group can be written
uniquely in the form

$\mathrm{i}^{\lambda}X_{u}Z_{v},$ (11)

where $u$ and $v$ are two vectors of the space $\mathbb{F}_{2}^{n}$ and
$\lambda\in\\{0,1,2,3\\}$.

It is not difficult to prove the following conjugation rules of a Pauli gate
by a $\mathtt{CNOT}$ gate : $X_{[ij]}Z_{i}X_{[ij]}=Z_{i}Z_{j}$,
$X_{[ij]}Z_{j}X_{[ij]}=Z_{j}$, $X_{[ij]}X_{i}X_{[ij]}=X_{i}$ and
$X_{[ij]}X_{j}X_{[ij]}=X_{i}X_{j}$. These rules can be generalized as

$X_{A}X_{u}Z_{v}X_{A}^{-1}=X_{Au}Z_{A^{-t}v},$ (12)

where $u$ and $v$ are vectors in $\mathbb{F}_{2}^{n}$, $A$ is a matrix in
$\mathrm{GL}_{n}(\mathbb{F}_{2})$ and $A^{-t}$ a shorthand for
$\left(A^{-1}\right)^{t}$.

##  3 Methodology used in the numerical exploration

We address the following problem : is it possible to generate a state
maximizing $\Delta_{4}$ by applying a $\mathtt{CNOT}$ gate circuit on a state
of the LU orbit of $\ket{0000}$ ? We use the classical Z-Y decomposition of a
single qubit unitary operator in the form
$\mathrm{e}^{\mathrm{i}\varphi}R_{z}(\alpha)R_{y}(\beta)R_{z}(\alpha^{\prime})$
(see e.g. [10, Th. 4.1]). Using this decomposition, any fully factorized
unitary operator $U$ depends, up to a global phase, on the 12 real parameters
of the matrix

$\mathcal{P}=\begin{bmatrix}\alpha_{0}&\beta_{0}&\alpha_{0}^{\prime}\\\
\alpha_{1}&\beta_{1}&\alpha_{1}^{\prime}\\\
\alpha_{2}&\beta_{2}&\alpha_{2}^{\prime}\\\
\alpha_{3}&\beta_{3}&\alpha_{3}^{\prime}\end{bmatrix}.$ (13)

We define the unitary $U(\mathcal{P})$ by

$\displaystyle\begin{split}U(\mathcal{P})=R_{z}(\alpha_{0})R_{y}(\beta_{0})R_{z}(\alpha_{0}^{\prime})&\otimes
R_{z}(\alpha_{1})R_{y}(\beta_{1})R_{z}(\alpha_{1}^{\prime})\\\ \otimes
R_{z}(\alpha_{2})&R_{y}(\beta_{2})R_{z}(\alpha_{2}^{\prime})\otimes
R_{z}(\alpha_{3})R_{y}(\beta_{3})R_{z}(\alpha_{3}^{\prime}).\end{split}$ (14)

As a rotation around the $\hat{z}$ axis applied to $\ket{0}$ is just a change
of phase, it is possible to write any state vector of the LU orbit of
$\ket{0000}$ (up to a global phase), by using only two parameters for each
qubit. So, any state in the LU orbit of $\ket{0000}$ is equal (up to a global
phase) to the state $\ket{\mathcal{P}}$ defined by

$\ket{\mathcal{P}}=R_{z}(\alpha_{0})R_{y}(\beta_{0})\otimes
R_{z}(\alpha_{1})R_{y}(\beta_{1})\otimes
R_{z}(\alpha_{2})R_{y}(\beta_{2})\otimes
R_{z}(\alpha_{3})R_{y}(\beta_{3})\ket{0000}.$ (15)

Using the definition of the rotation matrices around the $\hat{z}$ and
$\hat{y}$ axes, one has

$\ket{\mathcal{P}}=(a_{0}\ket{0}+a_{1}\ket{1})\otimes(b_{0}\ket{0}+b_{1}\ket{1})\otimes(c_{0}\ket{0}+c_{1}\ket{1})\otimes(d_{0}\ket{0}+d_{1}\ket{1}),$
(16)

$\text{where :
}(a_{0},a_{1})=(\mathrm{e}^{-\mathrm{i}\alpha_{0}/2}\cos\frac{\beta_{0}}{2},\mathrm{e}^{\mathrm{i}\alpha_{0}/2}\sin\frac{\beta_{0}}{2})$,

$\phantom{\text{where :
}}(b_{0},b_{1})=(\mathrm{e}^{-\mathrm{i}\alpha_{1}/2}\cos\frac{\beta_{1}}{2},\mathrm{e}^{\mathrm{i}\alpha_{1}/2}\sin\frac{\beta_{1}}{2})$,

$\phantom{\text{where :
}}(c_{0},c_{1})=(\mathrm{e}^{-\mathrm{i}\alpha_{2}/2}\cos\frac{\beta_{2}}{2},\mathrm{e}^{\mathrm{i}\alpha_{2}/2}\sin\frac{\beta_{2}}{2})$,

$\phantom{\text{where :
}}(d_{0},d_{1})=(\mathrm{e}^{-\mathrm{i}\alpha_{3}/2}\cos\frac{\beta_{3}}{2},\mathrm{e}^{\mathrm{i}\alpha_{3}/2}\sin\frac{\beta_{3}}{2})$.

Any state resulting from the action of an unitary operator in
${\left\langle\mathtt{CNOT}\right\rangle}_{4}$, on a state of the LU orbit of
$\ket{0000}$ can be written (up to a global phase) in the form
$X_{A}\ket{\mathcal{P}}$, where $A$ is a matrix in
$\mathrm{GL}_{4}(\mathbb{F}_{2})$ and $\mathcal{P}$ a matrix of parameters.

In order to determine the states of type $X_{A}\ket{\mathcal{P}}$ capable of
maximizing $|\Delta_{4}|$, it is sufficient to consider the right cosets of
the subgroup of ${\left\langle\mathtt{CNOT}\right\rangle}_{4}$ generated by
the $\mathtt{SWAP}$ gates (group
${\left\langle\mathtt{SWAP}\right\rangle}_{4}\simeq\mathfrak{S}_{4})$, because
$|\Delta_{4}|$ is invariant under permutation of the qubits, i.e.
$|\Delta_{4}(X_{A}\ket{\mathcal{P}})|=|\Delta_{4}(X_{\sigma
A}\ket{\mathcal{P}})|$ for any permutation matrix $\sigma$. The order of the
group ${\left\langle\mathtt{CNOT}\right\rangle}_{4}$ is 20160 (Identity (7)),
so the number of right cosets of
${\left\langle\mathtt{SWAP}\right\rangle}_{4}$ in
${\left\langle\mathtt{CNOT}\right\rangle}_{4}$ is $20160/24=840$. For each
coset, we compute a representative of minimal length in the generators
$X_{[ij]}$ (function right_cosets_perm_GL4 of the Python module).

The computation of $\Delta_{4}$ for a given state is performed using the
algorithm proposed by Luque and Thibon in [7, Section IV] (function hyper_det
of the Python module). After eliminating all coset representatives $X_{A}$
such that $|\Delta_{4}(X_{A}\ket{\mathcal{P}})|$ vanishes for any
$\mathcal{P}$, we obtain a list of 333 representatives (function
non_zero_HD_strings of the Python module). For each of them, we use a random
walk on the search space defined by the eight parameters of
$\ket{\mathcal{P}}$ in order to maximize the value of
$|\Delta_{4}(X_{A}\ket{\mathcal{P}})|$ (function search_max_HD of the Python
module). We check that it is possible to reach the maximal value of
$\frac{1}{2^{8}3^{9}}$ for $|\Delta_{4}|$ (accuracy $10^{-22}$) for only 12
coset representatives. These cosets are described in Section 5. Finally, from
the approximate values of $\mathcal{P}$ computed by the random walk heuristic,
it is possible to deduce the exact values of $\mathcal{P}$ such that
$|\Delta_{4}(X_{A}\ket{\mathcal{P}})|=\frac{1}{2^{8}3^{9}}$.

##  4 A $\mathtt{CNOT}$ circuit to reach the maximum of $|\Delta_{4}|$

Let $i,j,k,\ell$ be distinct integers in $\\{0,1,2,3\\}$. We define
$M_{k}^{(i,j)}$, a product of $\mathtt{CNOT}$ gates, and $A_{k}^{(i,j)}$, the
bit matrix of $\mathrm{GL}_{4}(\mathbb{F}_{2})$ associated to $M_{k}^{(i,j)}$,
as follows :

$\displaystyle M_{k}^{(i,j)}$
$\displaystyle=X_{[ij]}X_{[jk]}X_{[ki]}X_{[i\ell]}X_{[\ell j]}$ (17)
$\displaystyle A_{k}^{(i,j)}$ $\displaystyle=[ij][jk][ki][i\ell][\ell j]$ (18)

In this section, we show how to reach the maximum of $|\Delta_{4}|$ using the
operator

$M_{2}^{(0,1)}=X_{[01]}X_{[12]}X_{[20]}X_{[03]}X_{[31]}.$ (19)

The results are extended to any operator of type $M_{k}^{(i,j)}$ in the next
section. Since $A_{2}^{(0,1)}=[01][12][20][03][31]=\begin{bmatrix}0&1&1&0\\\
1&0&1&1\\\ 1&1&1&1\\\ 0&1&0&1\end{bmatrix}$ and
$A_{3}^{(0,1)}=[01][13][30][02][21]=\begin{bmatrix}0&1&0&1\\\ 1&0&1&1\\\
0&1&1&0\\\ 1&1&1&1\end{bmatrix}$, we remark that
$A_{3}^{(0,1)}=(023)A_{2}^{(0,1)}$, so

$M_{3}^{(0,1)}=S_{(023)}M_{2}^{(0,1)},$ (20)

which means that $M_{3}^{(0,1)}$ and $M_{2}^{(0,1)}$ represent the same coset.
This coset is denoted by $\overline{(0,1)}$.

######  Proposition 1.

Let $\mathcal{P}_{\mathrm{max}}$ and $\mathcal{P}_{\mathrm{max}}^{\prime}$ be
the two matrices of parameters defined by

$\mathcal{P}_{\mathrm{max}}=\begin{bmatrix}\pi/2&\pi/2&0\\\ \pi/2&\pi/2&0\\\
\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\\\
\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\end{bmatrix},\qquad\mathcal{P}_{\mathrm{max}}^{\prime}=\begin{bmatrix}\pi/2&\pi/2&0\\\
\pi/2&\pi/2&0\\\ 3\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\\\
3\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\end{bmatrix},$ (21)

then the states

$\ket{\psi_{\mathrm{max}}}=M_{2}^{(0,1)}\ket{\mathcal{P}_{\mathrm{max}}}$ (22)

and

$\ket{\psi_{\mathrm{max}}^{\prime}}=M_{2}^{(0,1)}\ket{\mathcal{P}^{\prime}_{\mathrm{max}}}$
(23)

maximize the absolute value of the four-qubit hyperdeterminant. One has

$\Delta_{4}(\ket{\psi_{\mathrm{max}}})=\Delta_{4}(\ket{\psi_{\mathrm{max}}^{\prime}})=-\frac{1}{2^{8}3^{9}},$
(24)
$\ket{\psi_{\mathrm{max}}}=\frac{\sqrt{3}}{3}\ket{w_{1}}+\frac{3+\sqrt{3}}{6}\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}\ket{w_{2}}+\frac{3-\sqrt{3}}{6}\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}\ket{w_{3}}$
(25)

and

$\ket{\psi_{\mathrm{max}}^{\prime}}=\frac{\sqrt{3}}{3}\ket{w_{1}}+\frac{3+\sqrt{3}}{6}\mathrm{e}^{-\mathrm{i}\frac{\pi}{4}}\ket{w_{2}}+\frac{3-\sqrt{3}}{6}\mathrm{e}^{\mathrm{i}\frac{3\pi}{4}}\ket{w_{3}},$
(26)

where

$\displaystyle\ket{w_{1}}$
$\displaystyle=\frac{1}{\sqrt{8}}(\ket{0001}+\mathrm{i}\ket{0011}+\ket{0101}-\mathrm{i}\ket{0111}+\ket{1000}+\mathrm{i}\ket{1010}+\ket{1100}-\mathrm{i}\ket{1110}),$
$\displaystyle\ket{w_{2}}$
$\displaystyle=\frac{1}{2}(-\ket{0000}-\mathrm{i}\ket{0110}-\mathrm{i}\ket{1011}+\ket{1101}),$
$\displaystyle\ket{w_{3}}$
$\displaystyle=\frac{1}{2}(\ket{0010}+\mathrm{i}\ket{0100}-\mathrm{i}\ket{1001}+\ket{1111}).$

###### Proof.

The different assertions can be checked using the function

check_psi_max_is_MHS of the Python module. ∎

In our numerical search for matrices of parameters $\mathcal{P}$ such that
$M_{2}^{(0,1)}\ket{\mathcal{P}}$ maximizes $\Delta_{4}$, it appears that all
values of $\mathcal{P}$ computed by the random walk heuristic are related to
$\mathcal{P}_{\mathrm{max}}$ or to $\mathcal{P}_{\mathrm{max}}^{\prime}$ by
simple operations. These operations are described by the following lemma and
its corollary. Numerical results suggest that these operations applied to the
matrices $\mathcal{P}_{\mathrm{max}}$ or $\mathcal{P}_{\mathrm{max}}^{\prime}$
are sufficient to describe all the possible matrices $\mathcal{P}$ such that
$M_{2}^{(0,1)}\ket{\mathcal{P}}$ is a MHS (Conjecture 7).

######  Lemma 2.

Let $\mathcal{P}$ be a matrix of parameters and, for any $k$ in
$\\{0,1,2,3\\}$, let us denote by :

$\mathcal{P}_{\alpha_{k}+\pi}$ the matrix obtained from $\mathcal{P}$ by
adding $\pi$ to the parameter $\alpha_{k}$,

$\mathcal{P}_{-\beta_{k}}$, the matrix obtained from $\mathcal{P}$ by taking
the opposite of $\beta_{k}$,

$\mathcal{P}_{-\alpha_{k},\ \beta_{k}+\pi}$, the matrix obtained from
$\mathcal{P}$ by taking the opposite of $\alpha_{k}$ and adding $\pi$ to
$\beta_{k}$ . Then :

$\displaystyle\ket{\mathcal{P}_{\alpha_{k}+\pi}}=-\mathrm{i}Z_{k}\ket{\mathcal{P}}$
(27) $\displaystyle\ket{\mathcal{P}_{-\beta_{k}}}=Z_{k}\ket{\mathcal{P}}$ (28)
$\displaystyle\ket{\mathcal{P}_{-\alpha_{k},\
\beta_{k}+\pi}}=-\mathrm{i}Y_{k}\ket{\mathcal{P}}$ (29)

###### Proof.

We prove only Identity (29), the proofs of Identities (27) and (28) being
similar. Without loss of generality, we suppose that the last column of the
matrix $\mathcal{P}$ is null and $k=0$. On the one hand :

$-\mathrm{i}Y_{0}\ket{\mathcal{P}}=(-\mathrm{i}Y(a_{0}\ket{0}+a_{1}\ket{1}))\otimes(b_{0}\ket{0}+b_{1}\ket{1})\otimes(c_{0}\ket{0}+c_{1}\ket{1})\otimes(d_{0}\ket{0}+d_{1}\ket{1})$,
$\phantom{-\mathrm{i}Y_{0}\ket{\mathcal{P}}}=(-a_{1}\ket{0}+a_{0}\ket{1})\otimes(b_{0}\ket{0}+b_{1}\ket{1})\otimes(c_{0}\ket{0}+c_{1}\ket{1})\otimes(d_{0}\ket{0}+d_{1}\ket{1})$,
where
$(a_{0},a_{1})=(\mathrm{e}^{-\mathrm{i}\alpha_{0}/2}\cos\frac{\beta_{0}}{2},\mathrm{e}^{\mathrm{i}\alpha_{0}/2}\sin\frac{\beta_{0}}{2})$,

On the other hand :

$\ket{\mathcal{P}_{-\alpha_{0},\
\beta_{0}+\pi}}=(a_{0}^{\prime}\ket{0}+a_{1}^{\prime}\ket{1})\otimes(b_{0}\ket{0}+b_{1}\ket{1})\otimes(c_{0}\ket{0}+c_{1}\ket{1})\otimes(d_{0}\ket{0}+d_{1}\ket{1})$,
where
$(a_{0}^{\prime},a_{1}^{\prime})=(\mathrm{e}^{-\mathrm{i}(-\alpha_{0}/2)}\cos\frac{\beta_{0}+\pi}{2},\mathrm{e}^{-\mathrm{i}\alpha_{0}/2}\sin\frac{\beta_{0}+\pi}{2})=(-a_{1},a_{0})$.

Hence $-\mathrm{i}Y_{0}\ket{\mathcal{P}}=\ket{\mathcal{P}_{-\alpha_{0},\
\beta_{0}+\pi}}$. ∎

######  Corollary 3.

Let $A$ be a matrix in $\mathrm{GL}_{4}(\mathbb{F}_{2})$ and $\mathcal{P}$ a
matrix of parameters.

If $|\Delta_{4}|$ is maximal for $X_{A}\ket{\mathcal{P}}$, then $|\Delta_{4}|$
is also maximal for $X_{A}\ket{\mathcal{P}_{\alpha_{k}+\pi}}$,
$X_{A}\ket{\mathcal{P}_{-\beta_{k}}}$ and
$X_{A}\ket{\mathcal{P}_{-\alpha_{k},\ \beta_{k}+\pi}}$, for any $k$ in
$\\{0,1,2,3\\}$.

###### Proof.

Suppose that $|\Delta_{4}|$ is maximal for $X_{A}\ket{\mathcal{P}}$. Let
$\mathcal{P}^{\prime}\in\\{\mathcal{P}_{\alpha_{k}+\pi},\mathcal{P}_{-\beta_{k}},\mathcal{P}_{-\alpha_{k},\
\beta_{k}+\pi}\\}$. From Lemma 2 and Identity (11), there exists two vectors
$u$ and $v$ in $\mathbb{F}_{2}^{4}$ such that
$\ket{\mathcal{P}^{\prime}}\simeq X_{u}Z_{v}\ket{\mathcal{P}}$. Hence
$X_{A}\ket{\mathcal{P}^{\prime}}\simeq X_{A}X_{u}Z_{v}\ket{\mathcal{P}}\simeq
X_{A}X_{u}Z_{v}X_{A}^{-1}X_{A}\ket{\mathcal{P}}$. So, using Identity (12), we
deduce that $X_{A}\ket{\mathcal{P}^{\prime}}\simeq
X_{Au}Z_{A^{-t}v}X_{A}\ket{\mathcal{P}}$ and consequently,
$X_{A}\ket{\mathcal{P}^{\prime}}$ is in the LU orbit of
$X_{A}\ket{\mathcal{P}}$, which implies that $|\Delta_{4}|$ is maximal for the
state $X_{A}\ket{\mathcal{P}^{\prime}}$. ∎

######  Example 4.

Let us apply the following sequence of operations on
$\mathcal{P}_{\mathrm{max}}$ : $\alpha_{0}\leftarrow\alpha_{0}+\pi$,
$\beta_{2}\leftarrow-\beta_{2}$, $\alpha_{3}\leftarrow-\alpha_{3}$,
$\beta_{3}\leftarrow\beta_{3}+\pi$. The resulting matrix of parameters is
$\mathcal{P}=\begin{bmatrix}3\pi/2&\pi/2&0\\\ \pi/2&\pi/2&0\\\
\pi/4&-\cos^{-1}\frac{\sqrt{3}}{3}&0\\\
-\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}+\pi&0\end{bmatrix}$ and
$\ket{\mathcal{P}}=(-\mathrm{i}Z_{0})Z_{2}(-\mathrm{i}Y_{3})\ket{\mathcal{P}_{\mathrm{max}}}\simeq
X_{3}Z_{0}Z_{2}Z_{3}\ket{\mathcal{P}_{\mathrm{max}}}$. Let $u=[0,0,0,1]^{t}$
and $v=[1,0,1,1]^{t}$. One has : $A^{(0,1)}_{2}=\begin{bmatrix}0&1&1&0\\\
1&0&1&1\\\ 1&1&1&1\\\ 0&1&0&1\end{bmatrix}$, $A^{(0,1)}_{2}u=[0,1,1,1]^{t}$,
$\left(A^{(0,1)}_{2}\right)^{-t}=\begin{bmatrix}1&0&1&0\\\ 1&1&1&1\\\
0&1&1&1\\\ 1&0&0&1\end{bmatrix}$,
$\left(A^{(0,1)}_{2}\right)^{-t}v=[0,1,0,0]^{t}$. Hence
$M^{(0,1)}_{2}\ket{\mathcal{P}}\simeq
X_{1}X_{2}X_{3}Z_{1}\ket{\psi_{\mathrm{max}}}$.

######  Remark 5.

We observe that the matrices $\mathcal{P}_{\mathrm{max}}$ and
$\mathcal{P}_{\mathrm{max}}^{\prime}$ are not related by the operations on
parameters described in Lemma 2, i.e. there does not exist any gate
$X_{u}Z_{v}$ in the four-qubit Pauli group such that
$\ket{\mathcal{P}_{\mathrm{max}}^{\prime}}\simeq
X_{u}Z_{v}\ket{\mathcal{P}_{\mathrm{max}}}$. This implies that the state
$\ket{\psi_{\mathrm{max}}}$ and the state $\ket{\psi_{\mathrm{max}}^{\prime}}$
define distinct orbits by the action of the four-qubit Pauli group. Actually,
from Identities (21) and (4), one has
$\ket{\mathcal{P}_{\mathrm{max}}^{\prime}}\simeq
P_{2}P_{3}\ket{\mathcal{P}_{\mathrm{max}}}$. Using the method described in
Section 3, we compute a matrix of parameters
$\mathcal{P}_{\psi\rightarrow\psi^{\prime}}=\begin{bmatrix}-\pi/2&-\pi/2&-\pi/2\\\
\pi/2&\pi&\pi\\\ 0&\pi/2&\pi\\\ -\pi/2&-\pi/2&-\pi/2\end{bmatrix}$ and a phase
$\varphi=-\frac{\pi}{3}$ such that
$\ket{\psi_{\mathrm{max}}^{\prime}}=\mathrm{e}^{\mathrm{i}\varphi}U(\mathcal{P}_{\psi\rightarrow\psi^{\prime}})\ket{\psi_{\mathrm{max}}}$.
Then, using Identities (4) and (5), we obtain :

$\ket{\psi_{\mathrm{max}}^{\prime}}\simeq PHP^{{\dagger}}\otimes PX\otimes
H\otimes PHP^{{\dagger}}\ket{\psi_{\mathrm{max}}}$ (30)

This last identity can be checked using the function _check_psi_to_psi_prime_
of the Python module.

######  Remark 6.

Since $M_{3}^{(0,1)}=S_{(023)}M_{2}^{(0,1)}$ , it is easy to see that the set
of all matrices of parameters $\mathcal{P}$ having their last column null such
that $M^{(0,1)}_{3}\ket{\mathcal{P}}$ maximizes $|\Delta_{4}|$ is equal to the
set of all matrices of parameters $\mathcal{P}$ having their last column null
such that $M^{(0,1)}_{2}\ket{\mathcal{P}}$ maximizes $|\Delta_{4}|$. We denote
this set by $\mathrm{PMAX}^{(0,1)}$.

######  Conjecture 7.

Any matrix in $\mathrm{PMAX}^{(0,1)}$ can be obtain from
$\mathcal{P}_{\mathrm{max}}$ or from $\mathcal{P}_{\mathrm{max}}^{\prime}$ by
a sequence of the operations on parameters described by Lemma 2.

##  5 All $\mathtt{CNOT}$ circuits to reach the maximum of $|\Delta_{4}|$

We generalize the results of the previous section by describing all the four-
qubit $\mathtt{CNOT}$ circuits that enable to produce a state maximizing
$|\Delta_{4}|$ when they act on the LU orbit of $\ket{0000}$.

######  Proposition 8.

Let $i,j,k,\ell$ be distinct integers in $\\{0,1,2,3\\}$, then $M_{k}^{(i,j)}$
and $M_{\ell}^{(i,j)}$ define the same right coset of the subgroup
${\left\langle\mathtt{SWAP}\right\rangle}_{4}$ in
${\left\langle\mathtt{CNOT}\right\rangle}_{4}$. This right coset is denoted by
$\overline{(i,j)}$ :

$\overline{(i,j)}=\\{S_{\sigma}X_{[ij]}X_{[jk]}X_{[ki]}X_{[i\ell]}X_{[\ell
j]}\mid\sigma\in\mathfrak{S}_{4}\\}.$ (31)

###### Proof.

Let $\sigma$ be the permutation $\begin{pmatrix}0&1&2&3\\\
i&j&k&\ell\end{pmatrix}$. Using Identity (8), we conjugate each member of the
equality $M_{3}^{(0,1)}=S_{(023)}M_{2}^{(0,1)}$ (Identity (20)) by
$S_{\sigma}$ and obtain $M_{\ell}^{(i,j)}=S_{(ik\ell)}M_{k}^{(i,j)}$. ∎

######  Proposition 9.

Let $i,j,i^{\prime},j^{\prime}$ in $\\{0,1,2,3\\}$ such that $i\neq j$ and
$i^{\prime}\neq j^{\prime}$. If $(i,j)$ and $(i^{\prime},j^{\prime})$ are
distinct couples, then $\overline{(i,j)}$ and
$\overline{(i^{\prime},j^{\prime})}$ are distinct cosets.

###### Proof.

We check that for any $i,j,k,i^{\prime},j^{\prime},k^{\prime}$ in
$\\{0,1,2,3\\}$ ($i,j,k$ distinct and $i^{\prime},j^{\prime},k^{\prime}$
distinct), if $A_{k}^{(i,j)}=\sigma A_{k^{\prime}}^{(i^{\prime},j^{\prime})}$
for some permutation matrix $\sigma$, then $(i,j)=(i^{\prime},j^{\prime})$
(function check_distinct_cosets of the Python module). ∎

######  Proposition 10.

For any permutation $\sigma$, one has :
$S_{\sigma}\overline{(i,j)}S_{\sigma}^{-1}=\overline{(\sigma(i),\sigma(j))}$.

###### Proof.

Using Identity (8) one has
$S_{\sigma}M_{k}^{(i,j)}S_{\sigma}^{-1}=M_{\sigma(k)}^{(\sigma(i),\sigma(j))}$.
The result follows from Proposition 8. ∎

Remark 6 can be generalized to any coset $\overline{(i,j)}$ and one can define
$\mathrm{PMAX}^{(i,j)}$ as being the set of all matrices $\mathcal{P}$ having
their last column such that the state $M_{k}^{(i,j)}\ket{\mathcal{P}}$
maximizes $|\Delta_{4}|$.

######  Proposition 11.

Let $i,j$ be distinct integers in $\\{0,1,2,3\\}$ and $\sigma$ be a
permutation such that $\sigma(0)=i$ and $\sigma(1)=j$, then :

$\mathrm{PMAX}^{(i,j)}=\sigma\mathrm{PMAX}^{(0,1)}.$ (32)

###### Proof.

We prove that $\mathrm{PMAX}^{(i,j)}\subset\sigma\mathrm{PMAX}^{(0,1)}$, the
other inclusion being similar. Let $\mathcal{P}$ be a matrix of parameters in
$\mathrm{PMAX}^{(i,j)}$. Then $M_{k}^{(i,j)}\ket{\mathcal{P}}=\ket{\psi}$ and
$\ket{\psi}$ maximizes $|\Delta_{4}|$, so
$S_{\sigma}^{-1}\ket{\psi}=S_{\sigma}^{-1}M_{k}^{(i,j)}\ket{\mathcal{P}}=S_{\sigma}^{-1}M_{k}^{(i,j)}S_{\sigma}S_{\sigma}^{-1}\ket{\mathcal{P}}=M_{k^{\prime}}^{(0,1)}S_{\sigma}^{-1}\ket{\mathcal{P}}$,
where $k^{\prime}\in\\{2,3\\}$. We observe that
$S_{\sigma}^{-1}\ket{\mathcal{P}}$ can be rewritten as
$\ket{\sigma^{-1}\mathcal{P}}$ by using Identity (9), hence
$S_{\sigma}^{-1}\ket{\psi}=M_{k^{\prime}}^{(0,1)}\ket{\sigma^{-1}\mathcal{P}}$.
As $S_{\sigma}^{-1}\ket{\psi}$ maximizes $|\Delta_{4}|$, there exists
$\mathcal{P}^{\prime}$ in $\mathrm{PMAX}^{(0,1)}$ such that
$\sigma^{-1}\mathcal{P}=\mathcal{P}^{\prime}$, so
$\mathcal{P}=\sigma\mathcal{P}^{\prime}$ and we deduce that
$\mathrm{PMAX}^{(i,j)}\subset\sigma\mathrm{PMAX}^{(0,1)}$. ∎

Our numerical results based on the use of the random walk heuristic suggest
the following conjecture.

######  Conjecture 12.

Any four-qubit $\mathtt{CNOT}$ circuit capable of maximizing $|\Delta_{4}|$ by
acting on the LU orbit of $\ket{0000}$ belongs to one of the 12 cosets
$\overline{(i,j)}$.

######  Example 13.

Let $\sigma=(013)$. The unitary $S_{\sigma}M_{2}^{(0,1)}$ acting on the state
$\ket{\mathcal{P}_{\mathrm{max}}}$ generates the state
$S_{(013)}\ket{\psi_{\mathrm{max}}}$ that maximizes $|\Delta_{4}|$. This state
can be produced by the unitary
$M_{\sigma(2)}^{(\sigma(0),\sigma(1))}=M_{2}^{(1,3)}$ acting on the state
$\ket{\sigma\mathcal{P}_{\mathrm{max}}}$, where
$\sigma\mathcal{P}_{\mathrm{max}}=\sigma\begin{bmatrix}\pi/2&\pi/2&0\\\
\pi/2&\pi/2&0\\\ \pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\\\
\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\end{bmatrix}=\begin{bmatrix}\pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\\\
\pi/2&\pi/2&0\\\ \pi/4&\cos^{-1}\frac{\sqrt{3}}{3}&0\\\
\pi/2&\pi/2&0\end{bmatrix}$.

##  6 Circuits generating the states $\ket{L}$, $\ket{\Phi_{5}}$ and
$\ket{M_{2222}}$

Let $\ket{\psi}$ be a state in the set
$\\{\ket{L},\ket{\Phi_{5}},\ket{M_{2222}}\\}$. Following the Gour-Wallach
conjecture [6] proved by Chen and Djokovic [3], there exists a matrix
$\mathcal{P}$ of parameters and a phase $\varphi$ such that

$\ket{\psi}=\mathrm{e}^{\mathrm{i}\varphi}U(\mathcal{P})\ket{\psi_{\mathrm{max}}}.$
(33)

In practice, one has to solve a 16 equations non linear system but its
resolution seems to be out of reach of current equation solvers (we used Maple
and Python SymPy solvers). However, one can turn the problem of finding a
solution of (33) into an optimization problem thanks to this simple remark :
$\ket{\psi}=\mathrm{e}^{\mathrm{i}\varphi}U(\mathcal{P})\ket{\psi_{\mathrm{max}}}$
if and only if the sum of the absolute values of the 16 coordinates of
$\ket{\psi}-\mathrm{e}^{\mathrm{i}\varphi}U(\mathcal{P})\ket{L}$ vanishes.
Again, we use a random walk on a search space of 13 parameters (the 12
parameters of $\mathcal{P}$ plus the phase $\varphi$) to minimize this sum and
obtain an approximate solution of the system (function
search_LU_from_state1_to_state2 of the Python module). From this approximate
solution, it is possible to deduce an exact solution. The results are
summarized in Figure 2. Finally, by combining these results with those of
Proposition 1, it is possible to propose simple quantum circuits generating
the states $\ket{L}$, $\ket{\Phi_{5}}$ and $\ket{M_{2222}}$ (Figure 3,
function check_circuits of the Python module).

$\ket{L}=\mathrm{e}^{-\mathrm{i}\frac{11\pi}{12}}U(\mathcal{P}_{\psi\rightarrow
L})\ket{\psi_{\mathrm{max}}}$, where $\mathcal{P}_{\psi\rightarrow
L}=\begin{bmatrix}\pi&\pi/2&-\pi/2\\\ \pi/2&-\pi/2&\pi\\\ 0&\pi&\pi\\\
0&-\pi/2&\pi/2\end{bmatrix}$

$\ket{\Phi_{5}}=\mathrm{e}^{-\mathrm{i}\frac{7\pi}{12}}U(\mathcal{P}_{\psi\rightarrow\Phi_{5}})\ket{\psi_{\mathrm{max}}}$,
where
$\mathcal{P}_{\psi\rightarrow\Phi_{5}}=\begin{bmatrix}-\pi/3&\theta-\pi&-3\pi/4\\\
\pi/3&\theta&3\pi/4\\\ \pi&\theta&3\pi/4\\\
2\pi/3&\pi-\theta&\pi/4\end{bmatrix}$

and $\theta=\cos^{-1}\frac{\sqrt{3}}{3}$

$\ket{M_{2222}}=\mathrm{e}^{\mathrm{i}\frac{5\pi}{12}}U(\mathcal{P}_{\psi\rightarrow
M_{2222}})\ket{\psi_{\mathrm{max}}}$, where $\mathcal{P}_{\psi\rightarrow
M_{2222}}=\begin{bmatrix}\pi/2&\pi/4&0\\\ -\pi/2&-\pi/4&\pi/2\\\
0&-\pi/2&\pi/4\\\ \pi/2&3\pi/4&\pi\end{bmatrix}$

Figure 2: LU operators generating the states $\ket{L}$, $\ket{\Phi_{5}}$ and
$\ket{M_{2222}}$, from the state $\ket{\psi_{\mathrm{max}}}$.

$\theta=\cos^{-1}\frac{\sqrt{3}}{3}$

$\mathrm{e}^{-\mathrm{i}\frac{7\pi}{12}}\ket{L}=$
$\ket{0}$$\ket{0}$$\ket{0}$$\ket{0}$$H$$H$$R_{y}(\theta)$$R_{y}(\theta)$$P$$P$$T$$T$$P$$Z$$X$$P$$H$$H$$H$$Z$$P^{{\dagger}}$$Z$

$\mathrm{e}^{-\mathrm{i}\frac{\pi}{6}}\ket{\Phi_{5}}=$
$\ket{0}$$\ket{0}$$\ket{0}$$\ket{0}$$H$$H$$R_{y}(\theta)$$R_{y}(\theta)$$P$$P$$T$$T$$T$$T$$T$$T$$X$$P$$P$$Y$$R_{y}(\theta)$$R_{y}(\theta)$$R_{y}(\theta)$$R_{y}(-\theta)$$R_{z}(-\frac{\pi}{3})$$R_{z}(\frac{\pi}{3})$$Z$$R_{z}(\frac{2\pi}{3})$

$\mathrm{e}^{-\mathrm{i}\frac{13\pi}{24}}\ket{M_{2222}}=$
$\ket{0}$$\ket{0}$$\ket{0}$$\ket{0}$$H$$H$$R_{y}(\theta)$$R_{y}(\theta)$$P$$P$$T$$T$$P$$T$$Z$$R_{y}(\frac{\pi}{4})$$R_{y}(-\frac{\pi}{4})$$H$$R_{y}(\frac{3\pi}{4})$$P$$P^{{\dagger}}$$Z$$P$

Figure 3: Quantum circuits generating the states $\ket{L}$, $\ket{\Phi_{5}}$
and $\ket{M_{2222}}$ up to a global phase. For better readability, most of the
rotations around the $\hat{y}$ and $\hat{z}$ axes defined by the matrices of
parameters are written using the universal single-qubit gates $H,P,T$ (see
Identity (4)).

We implemented the circuit generating the state $\ket{L}$ in one of the
quantum computers publicly available at https://quantum-computing.ibm.com/. In
those computers full connectivity between the qubits is not achieved and the
direct connections allowed between two qubits are given by a graph. Moreover,
due to the noise in the gates, and particularly the 2-qubit gates, it is of
crucial importance to use as few $\mathtt{CNOT}$ gates as possible. We chose
the 5-qubit ibmq_quito computer because its graph is
$\\{\\{1,0\\},\\{1,2\\},\\{1,3\\},\\{3,4\\}\\}$, hence the gates
$X_{[01]},X_{[12]}$ and $X_{[31]}$ of the $\mathtt{CNOT}$ subcircuit that
implements the operator
$M^{(0,1)}_{2}=X_{[01]}X_{[12]}X_{[20]}X_{[03]}X_{[31]}$ are already native
gates. The two other gates of the $\mathtt{CNOT}$ subcircuits, namely
$X_{[20]}$ and $X_{[03]}$ are not native $\mathtt{CNOT}$ gates and can be
simulated thanks to the use of $\mathtt{SWAP}$ gates. Finally the operator
$M^{(0,1)}_{2}$ can be implemented using only 11 native $\mathtt{CNOT}$ gates
:

$M^{(0,1)}_{2}=X_{[01]}X_{[12]}\underbrace{X_{[01]}X_{[10]}X_{[01]}}_{S_{(01)}}X_{[21]}X_{[13]}\underbrace{X_{[01]}X_{[10]}X_{[01]}}_{S_{(01)}}X_{[31]}.$
(34)

After compilation by the IBM algorithm (the process is called transpilation on
the website), the quantum circuit implementing the state $\ket{L}$ uses 22
single-qubit native gates, 11 $\mathtt{CNOT}$ gates and has a total depth of
18 (see Figure 4). However, despite of this moderate length, we observed after
measurement the apparition of a large quantity of scorias (see the bar chart
in Figure 4). Indeed, from Identity (1), one has

$\ket{L}=\textstyle\frac{1}{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{6}}(\ket{0000}+\ket{1111})+\textstyle\frac{\sqrt{3}}{6}\mathrm{e}^{-\mathrm{i}\frac{\pi}{3}}(\ket{0011}+\ket{0101}+\ket{0110}+\ket{1001}+\ket{1010}+\ket{1100}),$
(35)

so the states
$\ket{0001},\ket{0010},\ket{0100},\ket{1000},\ket{0111}\ket{1011},\ket{1101}$
or $\ket{1110}$ should not appear after measurement. The main causes of this
problem are, on one hand the measurement errors (average readout error is
about $3$ percent on this device), on the other hand the noise in the gates
($\mathtt{CNOT}$ gate average error is about $1.3$ percent). This suggests
that there are still significant technological challenges to overcome before
we can implement the state $\ket{L}$ in a reliable fashion.

Original circuit :

Transpiled circuit :

Measurements :

Figure 4: Implementation in the ibmq_quito quantum computer of a circuit
generating the state $\ket{L}$. The bar chart is based on 2000 measurements.

##  7 Conclusion and perspectives

In this work we described how a $\mathtt{CNOT}$ circuit acting on a factorized
state can produce four-qubit maximum hyperdeterminant states and we proposed a
quantum circuit generating the state $\ket{L}$, whose interesting properties
where described by Gour and Wallach [6] and by Chen and Djokovic [3]. It would
be interesting to know whether it is possible to generalize this result when
the number of qubits $n$ is greater than 4. Is it still possible to reach a
MHS by a $\mathtt{CNOT}$ circuit acting on a factorized state ? What would be
in this case the generalization of the unitary $M^{(i,j)}_{k}$ to higher
dimensions ? However, answering these questions seems to be currently out of
reach because an explicit polynomial expression of the hyperdeterminant is
known only up to 4 qubits. A first approach would be to know if a generically
entangled state (i.e. a state $\ket{\psi}$ such that
$\Delta_{n}(\ket{\psi})\neq 0$) can be produced by a $\mathtt{CNOT}$ circuit
acting on a factorized state in the case of any $n$-qubit system. Indeed, the
vanishing of $\Delta_{n}$ can be tested using the following criterion [4, p.
445] : let $A=\displaystyle\sum_{0\leqslant i_{0},\dots,i_{n-1}\leqslant
1}a_{i_{0}\dots i_{n-1}}x^{(0)}_{i_{0}}\dots x^{(n-1)}_{i_{n-1}}$ be the
multilinear form associated to the $n$-qubit state
$\ket{\psi}=\displaystyle\sum_{0\leqslant i_{0},\dots,i_{n-1}\leqslant
1}a_{i_{0}\dots i_{n-1}}\ket{i_{0}\dots i_{n-1}}$, then the condition
$\Delta_{n}(\ket{\psi})=0$ means that the system

$\\{A=\frac{\partial A}{\partial x^{(0)}_{0}}=\frac{\partial A}{\partial
x^{(0)}_{1}}=\dots=\frac{\partial A}{\partial x^{(n-1)}_{0}}=\frac{\partial
A}{\partial x^{(n-1)}_{1}}=0\\}$ (36)

has a solution $(x^{(0)}_{0},x^{(0)}_{1},\dots,x^{(n-1)}_{0},x^{(n-1)}_{1})$
such that $(x^{(i)}_{0},x^{(i)}_{1})\neq(0,0)$ for any $i=0\dots n-1$. Such a
solution is called non trivial. Therefore, to show that a state $\ket{\psi}$
is generically entangled, it is sufficient to prove that the system
corresponding to $\ket{\psi}$ has no solutions apart from the trivial
solutions. We will go back to these questions in future works.

##  8 Acknowledgements

The author acknowledges the use of the IBM Quantum Experience at
https://quantum-computing.ibm.com/. The views expressed are those of the
author and do not reflect the official policy or position of IBM or the IBM
Quantum Experience team.

Part of this work was performed using computing resources of the CRIANN
(Normandy, France).

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