# Characterizing randomness in parameterized quantum circuits through
expressibility and average entanglement

Guilherme Ilário Correr<EMAIL_ADDRESS>Instituto de Física de
São Carlos, Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP,
Brasil.  Ivan Medina  Instituto de Física de São Carlos, Universidade de São
Paulo, CP 369, 13560-970, São Carlos, SP, Brasil.  School of Physics, Trinity
College Dublin, Dublin 2, Ireland  Pedro C. Azado  Instituto de Física de São
Carlos, Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP, Brasil.
Alexandre Drinko  Instituto de Física de São Carlos, Universidade de São
Paulo, CP 369, 13560-970, São Carlos, SP, Brasil.  Diogo O. Soares-Pinto
<EMAIL_ADDRESS>Instituto de Física de São Carlos, Universidade de São Paulo,
CP 369, 13560-970, São Carlos, SP, Brasil.

###### Abstract

While scalable error correction schemes and fault tolerant quantum computing
seem not to be universally accessible in the near sight, the efforts of many
researchers have been directed to the exploration of the contemporary
available quantum hardware. Due to these limitations, the depth and dimension
of the possible quantum circuits are restricted. This motivates the study of
circuits with parameterized operations that can be classically optimized in
hybrid methods as variational quantum algorithms (VQAs), enabling the
reduction of circuit depth and size. The characteristics of these
Parameterized Quantum Circuits (PQCs) are still not fully understood outside
the scope of their principal application, motivating the study of their
intrinsic properties. In this work, we analyse the generation of random states
in PQCs under restrictions on the qubits connectivities, justified by
different quantum computer architectures. We apply the expressibility
quantifier and the average entanglement as diagnostics for the characteristics
of the generated states and classify the circuits depending on the topology of
the quantum computer where they can be implemented. As a function of the
number of layers and qubits, circuits following a Ring topology will have the
highest entanglement and expressibility values, followed by Linear/All-to-all
almost together and the Star topology. In addition to the characterization of
the differences between the entanglement and expressibility of these circuits,
we also place a connection between how steep is the increase on the uniformity
of the distribution of the generated states and the generation of
entanglement. Circuits generating average and standard deviation for
entanglement closer to values obtained with the truly uniformly random
ensemble of unitaries present a steeper evolution when compared to others.

##  I Introduction

In recent times, discussions surrounding quantum advantage have become
increasingly prevalent, accompanied by ongoing debates and controversies [1,
2, 3]. Together with the expressive increase in the investments on Quantum
Technologies during the last $12$ years [4], came the pressure of providing
quantum computing solutions to problems of interest. It is known that the
quantum algorithms and protocols that launched the Second Quantum Revolution
as a great promise cannot be implemented in contemporary quantum computers [5,
6, 7], due to the current limitations that avoid the near-term possibility of
fault-tolerant quantum computing. Despite the persistent challenges in
achieving reliable high-quality quantum error correction, the concept of noisy
intermediate-scale quantum (NISQ) computers has emerged as a promising avenue
[8]. These quantum devices exhibit potential in tackling intricate problems
that may surpass the capabilities of classical computers. One possible
approach in unlocking the power of NISQ computers involves the implementation
of parameterized quantum circuits (PQCs). Designed with adjustable parameters,
PQCs introduce flexibility into the quantum computation, which, in turn,
enable an extensive exploration of the Hilbert space with a fixed
architecture. When combined with classical optimizers and by defining a cost
function, they give rise to variational quantum algorithms (VQAs) [9, 10, 11].
The cost function is a trainable function used in hybrid quantum-classical
algorithms to perform a specific task and it might, for instance, quantify the
expected value of an observable with respect to the quantum state generated by
the parameterized quantum circuit, whose value is minimized or maximized to
solve the problem at hand [12, 13]. Together with a classical optimizer, it is
possible to minimize this cost function, thereby bringing the quantum state
closer to the desired solution.

Before the actual training, however, there is the task of constructing the PQC
architecture, which is also called ansatz. In many of the existing works, the
PQC construction has been grounded in heuristic proposals, often drawing
inspiration from established methods, such as adiabatic quantum computing and
chemistry Hartree-Fock methods [10, 14, 15, 16, 17, 18, 19], or imposing
specific restrictions to the resulted circuit [13, 20]. In this work, rather
than relying on heuristics and problem-driven inspiration for the circuit
structure, we have opted to explore the potential topology of connections
feasible within presently available quantum platforms. Research about the
restriction on the connectivity of qubits has found relevance in various
domains, being explored in the context of VQA optimization for thermodynamics
protocols [21], ground state search [20], and to understand the role of the
entanglement [22]. Outside the scope of VQA, it has relevance in fault
tolerant quantum algorithms [23], analysis of superconducting quantum hardware
[24], and pseudorandom quantum circuits [25].

Still, PQCs have characteristics attractive even outside the borders of VQAs
applications. Recent results have shown that these circuit structures can
present characteristics of pseudorandom quantum circuits [26, 27, 28, 29],
such as saturated entanglement, following an area or volume laws, and
distribution of states that resemble an uniformly distributed measure over
state space. Pseudorandom quantum circuits are quantum circuits related with
unitary operators that are randomly distributed over the group manifold [30,
25, 31], many times compared with the uniformly distributed Haar measure.
These circuits can be applied in many important quantum computing tasks, and
find interesting characteristics to the generation of uniformly random
unitaries [30], to the simulation of many-body systems dynamics [32],
complexity theory [33] and scrambling [34, 28]. In this sense, to understand
how this different circuit structure, the PQCs, can replicate random circuits
to some extent is of fundamental and practical concern.

The enhanced exploration of the Hilbert space, facilitated by PQCs, is
quantified through the concept of expressibility [26]. It measures how
uniformly distributed are the output states within the Hilbert space and it
can be correlated with trainability issues in variational quantum algorithms,
such as the existence of barren plateaus [35, 36]. This relation can be best
seen in Ref. [37] and stems from the fact that if our circuit can reach
uniformly distributed states in the Hilbert space, it may be harder to reach
the specific solution state we are looking for. However, a low expressible set
of output states might not even reach the desired solution, so a careful
analysis of this figure of merit is relevant. Also, this quantifier provides
useful information from the viewpoint of the capability of PQCs to achieve
pseudorandom quantum circuits, as it is measuring how close the circuit states
are to a uniform random distribution over state space.

Another crucial consideration is the evaluation of how our circuit generates
entanglement [38, 39, 40, 41]. Highly entangling PQCs are undesirable in the
context of VQAs, as they can lead to barren plateaus many times associated
with the global distribution of local information [42, 27, 28], harming the
obtainment of information through the cost function. The reason why this is
happening is also associated with the convergence of the PQC to a pseudorandom
circuit and $t-$designs, which will present saturated and high entanglement
values [43, 34]. Therefore, in the realm of quantum algorithms and their
applications is of fundamental interest to understand the entanglement
generation in quantum circuits. Referred as entangling capability [26], the
average entanglement can be computed for the output states of random unitaries
and provides information about what is the distribution of entanglement. Many
different quantifiers can be chosen to address this characteristic, but an
interesting one is the order $m$ Scott multipartite entanglement quantifier
[44]. This measure reduces to the well known Meyer-Wallach quantifier when
$m=1$ [45, 46, 25] and has an interpretation in terms of the linear entropy
[47]. Both the average and standard deviation values of this function for the
output states of PQCs are explored here.

In this work, we discuss the expressibility and average multipartite
entanglement of parameterized quantum circuits with restrictions on the qubits
connectivities. Applying the expressibility [26] and the average entanglement
for the Scott quantifier of order $1$ and $2$ [44], we discuss the
consequences of the restrictions on the generation of quantum states. We also
show that, in fact, these quantities are connected and the topology of the
quantum architecture has consequences for the observed features. PQCs
generating random states with average and standard deviation of entanglement
closer to the observed for the ensemble of uniformly distributed random
unitaries have a steeper increase in expressibility. The paper is structured
as follows. In Section II we introduce the PQCs investigated in this work and
the motivation for the choices. In Section III we define the quantifiers
applied for the investigation, i.e., the expressibility and the average Scott
quantifier. In Section IV, we discuss the results for the proposed PQCs.
Finally, the conclusions are presented in Sec. V.

##  II Parameterized Quantum Circuits

###  II.1 Circuit Architectures and Ansätze

The PQCs are one of the building blocks of a VQA and their study is central in
the analysis of possible advantage and to guarantee trainability, while being
constrained by the experimental reality [13, 29, 26, 48, 49]. From the
perspective of random circuits applications, their implementability in real
world experimental setups is also of fundamental importance [25, 50]. Taking
this into account, in this work we consider qubits connectivities (i.e.,
quantum hardware topologies) that appear in contemporary and commercially
available quantum hardware. Considering these different structures related
with quantum computer architectures, we proposed different PQCs to understand
which are the consequences of connections restrictions to the average
entanglement generation and expressibility. We divided the PQCs choices in two
classes, named Ansatz $1$ and Ansatz $2$, whose structure depend on the
different topologies that appear in quantum computers. These different
topologies are presented in Fig. 1, while the two ansätze are shown in Fig. 2.

In Fig. 1, the connectivities for $4$ qubits and their quantum circuit
representations are presented. The use of CNOTs for two qubits gates is due to
their nativity in many quantum computers platforms and to the fact that they
are not parameterized gates. Using this strategy, we can isolate the role of
the connection types, appearing in the CNOT gates, from the role of the
parameterized gates, appearing in the ansätze only as local gates. We chose
the $4$ qubits case to exemplify the graphs because it is the smallest
situation where the connectivities are not degenerated. The simplest case in
Fig. 1 is the No Connections topology (NC), as it is composed by only local
qubit operations and can be implemented in any quantum computer. The
availability of the other connectivities will depend on the hardware. The
Linear topology (LIN) can be implemented in most of the quantum platforms, as
IBM (Vigo [51], Tokyo [52] and Ithaca [53]) or Rigetti [54] quantum computers,
and is composed of two qubits gates between nearest neighbours. The Ring (RIN)
and Star (ST) are also available in the same platforms, however their
possibility will depend on the number of qubits in the circuits. The Ring
connects first neighbours and the first to the last qubit in a circuit, while
Star has connections between one central qubit and every other qubit. The most
complex topology, All-to-all (ATA), will have connections of every qubit with
every qubit (a complete graph) and can be implemented in ion traps quantum
computers with a high number of qubits [55] without the need of additional
SWAP gates. When considering $3$ qubits, the smallest dimension studied here,
the Linear/Star and Ring/All-to-all topologies will be degenerated, therefore
we are going to dub the two equivalent sets as Linear and Ring.

Figure 1: Graphs of the topologies listed and observed in different quantum
computer architectures, together with their quantum circuit representations
consisting of CNOT gates couplings.

Now, these topologies will be part of the circuit structure presented by
ansätze $1$ and $2$, Fig. 2, applied to an initial state $\ket{0}^{\otimes
n}$. The local parameterized rotations used in the ansätze are given by ${\rm
Rj}(\theta)=e^{-i\theta\sigma_{j}}$, where $j=$X,Y, $\sigma_{j}$ are the usual
Pauli matrices and $\theta$ is an arbitrary parameter representing the
rotations angles. The Ansatz 1 is composed of sequential applications of
parameterized rotations RX and RY on each qubit, followed by the connections
according to the chosen topology. The Ansatz 2 shares this same structure with
an additional application of parameterized rotations RX and RY after the
connections step. Both circuits considering only $1$ layer will share the same
average entanglement, as the operations after the connections in Ansatz 2 are
only local unitary operations that preserve entanglement values. We will show
that, in fact, this will be also true for more than $1$ layer. From this
characteristic, we can affirm that Ansatz 2 will have the same entanglement as
Ansatz 1, however with more freedom on which are the local coherences the
different states can have. It is worth remarking that the parameters are
encoded only in the local unitary rotations. Therefore, we can separate the
roles played by the topology, defined only by the connective part of the
circuits (CNOTS gates), and by the local operations, which imprint the
parameters in the system state, leading to local superposition. This will be
of fundamental importance to understand our results and how these circuits can
be useful in different protocols.

Figure 2: One layer of the circuits Ansatz $1$ and Ansatz $2$. The parameters
represented in the figure were randomly sampled from a uniform distribution in
the interval $[-\pi,\pi]$.

We chose local operations RX and RY due to the fact that they are going to
generate random $1$ qubit states over the complete surface of the Bloch sphere
sampling the circuit parameters uniformly, when starting with a $\ket{0}$
state. One of the drawbacks of this choice is that the random $1$ qubit states
will present a bias towards two of the axis of the sphere, however the states
generated effectively cover the space [26]. Table 1 presents the number of
CNOT gates and total number of gates in both circuits Ansatz $1$ and $2$.
Comparing the different topologies, as a function of the number of qubits,
$n$, and number of layers, $l$. The total number of parameters appears
isolated inside brackets in the equations of the total number of gates, so
this number can be easily known looking at the second term inside brackets.
For example, for the Linear topology and Ansatz $2$, there are
$\left[(n-1)+4n\right]l$ gates, being $(n-1)l$ CNOTs and $(4n)l$ parameterized
gates.

Table 1: Number of CNOTs and total number of gates comparing topologies for Ansatz $1$ and $2$ as a function of the number of qubits $n$ and number of layers $l$. The terms inside brackets are separated so the right term in each of them is exactly the number of parameters (total number of gates RX and RY) in each circuit. Topology | Number of CNOTs | Total number Ansatz $1$ | Total number Ansatz $2$
---|---|---|---
No Connections | $0$ | $(2n)l$ | $(4n)l$
Linear | $(n-1)l$ | $\left[(n-1)+2n\right]l$ | $\left[(n-1)+4n\right]l$
Ring | $nl$ | $\left[n+2n\right]l$ | $\left[n+4n\right]l$
Star | $(n-1)l$ | $\left[(n-1)+2n\right]l$ | $\left[(n-1)+4n\right]l$
All-to-all | $\frac{n(n-1)}{2}l$ | $\left[\frac{n(n-1)}{2}+2n\right]l$ | $\left[\frac{n(n-1)}{2}+4n\right]l$

##  III Figures of Merit

###  III.1 Expressibility

In this work we are concerned about the capability of PQCs to generate
uniformly distributed random states over the Hilbert space of interest. When
considering random pure quantum states in a Hilbert space of any finite
dimension $d=2^{n}$, being $n$ the number of qubits, a uniform probability
measure $d\mu(\psi)$ can be obtained using the Fubini-Study (FS) metric [56].
This same measure can be also obtained by a different method. Considering the
uniform probability measure over the unitary group of dimension $d$,
$\mathcal{U}(d)$, sample unitaries and act with them over a fiducial state,
here defined as the tensor product $\ket{0}^{\bigotimes n}$. The obtained
ensemble of states will have the same distribution as the one sampled using
the FS-metrics over the space of states [57, 58].

This measure over the unitary group is called Haar measure and present
interesting properties as invariance considering elements of the group [59].
However, build a faithful replication of the Haar measure can be hard and
costly, requiring circuits with gate complexity that scales exponentially with
the dimension of the system [30]. Also, only part of the properties of the
Haar measure are needed to determine integrals of polynomials of restricted
order or mean values of operators, motivating the definition of $t-$designs
[60, 61, 62]. The $t-$designs are ensembles of unitaries that can replicate
parts of the characteristics of the Haar measure over the unitary group, with
different plausible equivalent definitions [59]. One possible definition is in
terms of the polynomials one might want to calculate integrals over the group
manifold.

######  Definition 1.

(Unitary $t-$design via polynomials)[61, 62] Let
$\\{U(x)\\}_{x=1}^{K}\subset\mathcal{U}(d)$ be a finite set of unitary
operators acting on the space $\mathbb{C}^{d}$. If the set is such that for
every polynomial $P_{(t,t)}(U)$ of degree at most $t$ in the elements and
complex conjugate elements of the matrix $U$,

$\frac{1}{K}\sum_{x=1}^{K}P_{(t,t)}(U(x))=\int_{\mathcal{U}(d)}d\mu(x)P_{(t,t)}(U(x)),$
(1)

is satisfied, we say the set is a unitary $t-$design. The $d\mu(x)$ is the
Haar measure over $\mathcal{U}(d)$.

A measure to compare true $t-$designs with a distribution of states induced
considering a parameterized circuit structure sampling the parameters was
introduced in Ref. [26]. It is defined in terms of the square of the Hilbert-
Schmidt norm of the quantity

$\mathcal{A}^{t}:=\int_{\text{Haar}}(\ket{\psi}\bra{\psi})^{\otimes
t}d\ket{\psi}-\int_{\Theta}(\ket{\phi_{\theta}}\bra{\phi_{\theta}})^{\otimes
t}d\theta,$ (2)

that compares the induced Haar state a $t-$design would construct (first
integral) [33] with the one induced when considering the circuit and sampling
over the parameter space (second integral). The sampling in this work is
performed with a uniform probability measure between $0$ and $2\pi$ for each
individual parameter. This norm can be translated in terms of the statistical
moments of fidelities comparing states generated considering the Haar ensemble
and the circuit ensemble [26],

$||\mathcal{A}^{t}||^{2}_{\text{HS}}:=\Tr\left[(\mathcal{A}^{t})^{\dagger}(\mathcal{A}^{t})\right]=\mathbb{E}\left[F(\bm{\theta},\bm{\phi})^{t}\right]-\mathbb{E}_{\text{Haar}}(F^{t}),$
(3)

being
$F(\bm{\theta},\bm{\phi}):=|\innerproduct{\psi(\bm{\theta})}{\psi(\bm{\phi})}|^{2}$
the fidelity [63] for two circuit generated pure states and the second
expected value obtained by the calculation of fidelities comparing two Haar
random states. This will be greater or equal to zero and the closer it is to
zero, the closer the circuit structure is to generate a $t-$design.

This quantifier considers a particular $t-$design order. To compare with the
Haar measure in a general form, the expressibility quantifier was introduced
[26] based on the Kullback-Leibler divergence or relative entropy [64] between
the circuit distribution of fidelities and the Haar one, whose analytical
values are known for pure states in finite Hilbert spaces [65]. The
expressibility is then defined as [26]

$\text{Expr}:=D_{KL}\left(P_{PQC}(F)||P_{\text{Haar}}(F)\right),$ (4)

where $P_{PQC}$ is the circuit fidelity histogram and $P_{\text{Haar}}$ the
Haar one. The closer this quantity is to zero, the closer the circuit induced
ensemble is to generate uniformly distributed states in the space of states
and we say the more expressible is the circuit.

To compute this quantifier, the parameter vectors are sampled and given as
input for a particular circuit structure. A output state is built for a
sampled parameter vector and the input $\ket{0}^{\otimes n}$ and with two of
these states, is possible to sample a fidelity value. We consider $10^{4}$
parameter vectors according to the Chebyshev inequality [66], giving rise to
$5\cdot 10^{3}$ fidelities that will compose the circuit histogram to be
compared with the Haar histogram that can be analytically built.

###  III.2 Scott entanglement measure and average entanglement

The Scott entanglement measure was proposed to quantify multipartite
entanglement considering the mean bipartite entanglement over every possible
bipartition of a particular size [44, 67]. It was introduced based on the
decomposition of the Meyer-Wallach entanglement measure (MW) [45, 46] in terms
of the linear entropy. The MW computes multipartite entanglement based on the
mean linear entropy [47] over every possible bipartition with size $1$ qubit-
rest of the system. Scott generalized this to any bipartition of size $m$
qubits and the rest of the system. Given an $n$ qubits pure state,
$\rho(n)\equiv\outerproduct{\psi}{\psi}$, the Scott entanglement measure of
order $m$ [44], that we are going to refer either as $Q_{m}$ or $S_{m}$, reads

$\displaystyle Q_{m}\left(\ket{\psi}\right)$ $\displaystyle:=$
$\displaystyle\frac{2^{m}}{2^{m}-1}\left[1-\binom{n}{m}^{-1}\sum_{|S|=m}\Tr\left(\rho_{S}^{2}(n)\right)\right],$
$\displaystyle m$ $\displaystyle\leq$ $\displaystyle\lfloor n/2\rfloor,$ (5)

where $S$ is a size $m$ subset of $\\{1,2,\cdots,n\\}$ indexing the qubits in
the system, $\rho_{S}(n)$ is the reduced state discarding the complementary
set $S^{{}^{\prime}}$ such that $S\cup S^{{}^{\prime}}=\\{1,2,\cdots,n\\}$ and
$\lfloor\cdot\rfloor$ is the floor function. This way, the order $m$ defines
the sizes of the bipartitions considered. The constraint over $m$ defines the
normalization to be always valid and avoid redundancies for different orders.
For example, if $n=3$, $m=1$ and $m=2$ give the same information about the
multipartite entanglement. When $m=1$, the bipartition size considered will be
$1$ qubit and the rest of the system, that is composed of $2$ qubits. The
linear entropies of reduced states with this bipartition are completely
equivalent to the ones considering $m=2$, where the parts are $2$ qubits and
the rest of the system, i.e., $1$ qubit. Therefore, both $m=1$ or $m=2$
represent the same entanglement measure.

The different values of $m$ in the Scott entanglement measure will explore
different characteristics of multipartite entanglement and will define
different entanglement measures. For example, there are states for which the
MW, $m=1$, will present the same entanglement values, even though they have
very different characteristics. We can cite the states of $6$ qubits,
$\ket{GHZ_{6}}=(\ket{0}^{\otimes 6}+\ket{1}^{\otimes 6})/2$ and
$\ket{EPR_{6}}=\ket{\Phi^{+}}^{\otimes 3}$, being
$\ket{\Phi^{+}}=(\ket{00}+\ket{11})/\sqrt{2}$, that will have
$Q_{1}(\ket{GHZ_{6}})=Q_{1}(\ket{EPR_{6}})=1$. These states have very
different entanglement characteristics, being the $\ket{GHZ_{6}}$ sometimes
considered a state with maximum multipartite entanglement, while the
$\ket{EPR_{6}}$ is a type of biseparable state [68, 69]. Choosing $m=2$, this
degeneracy is broken and we have $Q_{2}(\ket{GHZ_{6}})=2/3$ and
$Q_{2}(\ket{EPR_{6}})=4/5$. Therefore, in this work we compute the average
entanglement considering the measures $S_{1}$ and $S_{2}$, so we can
understand if the results can be replicated with both entanglement measures.

To quantify the behavior of the generation of entanglement in the circuits, we
applied a method very similar to the expressibility computation one [26]. We
generate random parameter vectors considering the uniform distribution and,
given the input $\ket{0}^{\otimes n}$ state, calculate the entanglement for
the output states and obtain the average and standard deviation over the
different parameters,

$\displaystyle\langle Q_{m}\rangle_{\Theta}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}}\sum_{i=1}^{\mathcal{N}}Q_{m}(\ket{\psi(\bm{\theta}_{i})}),$
$\displaystyle\sigma_{\Theta}(Q_{m})$ $\displaystyle=$
$\displaystyle\sqrt{\langle Q_{m}^{2}\rangle_{\Theta}-\langle
Q_{m}\rangle_{\Theta}^{2}}.$ (6)

In previous works, only the average entanglement was considered [26, 29] for a
thorough analysis. We can compare these numerical calculations with the
analytical values considering uniformly distributed random states, obtained
with the Haar induced distribution, named Circular Unitary Ensemble (CUE). The
values for the averaged $S_{m}$ and standard deviation for $S_{1}$ are [44,
43]

$\displaystyle\langle Q_{m}\rangle_{CUE}$ $\displaystyle=$
$\displaystyle\frac{2^{n}-2^{m}}{2^{n}+1},$ $\displaystyle\sigma_{CUE}(Q_{1})$
$\displaystyle=$
$\displaystyle\sqrt{\frac{6(2^{n}-4)}{(2^{n}+3)(2^{n}+2)(2^{n}+1)n}+\frac{18\cdot
2^{n}}{(2^{n}+3)(2^{n}+2)(2^{n}+1)^{2}}}.$ (7)

This comparison will be interesting to understand if and when it is possible
to replicate the uniform random entanglement applying PQCs. From the
perspective of $t-$designs, the Scott measures are polynomials of order
$(2,2)$ [25], so $2-$designs are enough to replicate the exact analytical
average values in Eq. (III.2). When discussing the relation between the
entanglement generation and expressibility changes as a function of the number
of layers, this characteristic will be important.

##  IV Results

In this section we are going to discuss the values and relations between the
expressibility and entanglement quantifiers. They are going to be compared
considering the two parameterized structures, ansätze $1$ and $2$, with the
different connectivities, for different numbers of layers and qubits. We
varied the number of circuit layers between $1$ and $5$ and the number of
qubits from $3$ to $8$. Exploring different number of circuit layers is
interesting from the viewpoint of VQAs, where this is sometimes applied to
verify performance changes and supply additional parameters for optimization
[9, 19]. It is also interesting from the perspective of pseudorandom quantum
circuits, where the sequential application of a particular structure evolves
to a simulation of a Haar random unitary [30, 25, 32]. To treat different
dimensions is also of fundamental importance to understand how the size of the
system considered will affect the results and how well the circuit will
perform in these different settings. We start with the expressibility for
Ansatz $1$, followed by the average entanglement. After that, we show that the
entanglement is the same for both ansätze and connect with the expressibility
of Ansatz $2$. The calculations were performed using the quantum circuits
simulation Pennylane Python library [70] and the codes are available on github
111Available on the profile https://github.com/GICorrer..

###  IV.1 Ansatz 1

####  IV.1.1 Expressibility

In Fig. 3, the relative entropy as a function of the number of layers for
Ansatz $1$ is shown for every connectivity with $4$ and $8$ qubits. The first
observation we can make is that the No Connections circuit saturates between
$0.20$ and $0.25$ nats for every studied dimension and cannot evolve to values
closer to $0$. This is expected, as the input state is separable,
$\ket{0}^{\otimes n}$, and the circuit does not generate entanglement. Random
uniformly distributed states should have nonzero entanglement (Eq. (III.2))
and if it is not generated, the circuit expressibility will saturate at a
value not that close to this Haar case. Conversely, the connected topologies
will evolve to values very close to $0$, indicating the increase in how
expressible the circuits are. This aspect highlights that PQCs can replicate
the behavior of pseudorandom quantum circuits up to some order when a sampling
of the parameters is performed. The interesting feature is that PQCs will have
a fixed structure that is repeated, contrasting with many proposals of random
circuits whose positions of gates in the circuits are also random [72, 73].
Other works studied how the uniformly distributed randomness of the states
generated with the circuit will affect optimization and performance in a VQA
scenario [26, 27, 28], however not much attention is given to the capability
of these circuits to generate random states. This behavior is observed for
every dimension studied. A peculiar result is presented for only $1$ layer.
Comparing the different circuit structures there is no established advantage
of generating entanglement, i.e., the connected topologies present relative
entropy values that are very close to the observed for the No Connections
case, even though this circuit is not generating entanglement.

Figure 3: Ansatz 1: relative entropy as a function of the number of layers for
all connectivities, considering (a) 4 qubits and (b) 8 qubits. NC: No
Connections, LIN: Linear, RIN: Ring, ST: Star and ATA: All-to-all.

To understand how the connected topologies are different in terms of the
relative entropy as a function of the number of layers, in Fig. 4 a plot with
logarithmic scale in the $y$-axis is shown. For the smaller dimensions of $3$
and $4$ qubits, Fig. 4 (a) and (b), there is not much difference in the
evolution and they follow a very similar behavior. A saturation is also
observed at $5\cdot 10^{-3}$ nats for both cases. We associate this with the
fact that the rotations $RX$ and $RY$ together do not define a universal set
of unitaries for one qubit, a requirement for the evolution of a circuit
structure to a pseudorandom circuit [30, 25]. A circuit involving universal
one qubit unitaries and CNOTs should converge to the generation of
pseudorandom states with sufficient number of concatenations of the circuit
structure [30]. However, another plausible explanation is that the rotations
$RX$ followed by $RY$ with parameters uniformly sampled are generating biased
states of one qubit, with concentrations around the poles of the Bloch sphere
[26]. A more interesting approach would be to consider a sampling that
replicates the uniform distribution of points in the shell of a $2-$sphere,
where the angles take into account the integration measure
$\sin(\theta)d\theta d\phi$, being $\theta$ the polar angle and $\phi$ the
azimuthal angle. This way, the sampling of states would replicate the action
of a uniformly distributed unitary.

When the dimension increases, from panels (c) to (f) of Fig. 4, it is possible
to notice that the evolutions start to branch, establishing a hierarchy
between the topologies. This difference grows with the dimension and presents
the biggest differences at $8$ qubits, as it is shown in panel (f). The Star
circuit presents the slowest decay to the Haar random case, followed by
Linear/All-to-all almost together and finally the Ring, with the steepest
decrease. Still, a saturation is observed also for $5$ qubits and apparently
has begun for $6$ qubits. In smaller dimensions, the differences in the
connectivities are not very influential, however in higher dimensions we can
clearly observe that the way qubits can be connected affects the
expressibility. In the circuits of this work, the topology only enters the
problem in the CNOT gates part of the circuits, which is also responsible for
the entanglement generation. In this sense, it is expected that the generation
of entanglement and the expressibility of the circuits might be related since
both quantities strongly depends on the topology.

Figure 4: Ansatz 1: relative entropy as a function of the number of layers for
the connected topologies, considering (a) 3 qubits, (b) 4 qubits, (c) 5
qubits, (d) 6 qubits, (e) 7 qubits and (f) 8 qubits. LIN: Linear, RIN: Ring,
ST: Star and ATA: All-to-all.

Fig. 5 highlights the differences discussed in the paragraphs above using
plots with fixed number of layers as a function of the number of qubits. In
Fig. 5 (a), it is possible to observe that all the topologies with the circuit
structure of Ansatz 1 present close values for the expressibility if we
consider only $1$ layer. This is observed for every dimension. Increasing the
number of layers to $3$, Fig. 5 (b), the connected topologies relative
entropies decay to values closer to zero, while the not connected saturates in
higher values. Increasing the dimension, we can observe the hierarchy
establishment. At $5$ layers, Fig. 5 (c), the connected topologies get very
close to zero, despite their differences.

Figure 5: Ansatz 1: relative entropy as a function of the number of qubits for
the connected topologies, considering (a) 1 layer, (b) 3 layers and (c) 5
layers. NC: No Connections, LIN: Linear, RIN: Ring, ST: Star and ATA: All-to-
all.

####  IV.1.2 Average entanglement

We begin the average entanglement results presentation by discussing the
absolute values observed for the circuits. To do so, in Fig. 6 it is shown the
mean entanglement for a fixed number of layers as a function of the number of
qubits. The dark blue line indicates the analytical behavior observed for
uniformly distributed states, which increases to $1$ as a function of the
dimension. This way, for high dimensional systems, the average entanglement of
uniformly distributed states will be very close to maximum [74, 43, 44]. We
can see that, overall, the Ring circuit generates the highest entanglement
values, followed by Linear/All-to-all exactly together and by the Star with
the least values. Increasing the number of layers, the circuits entanglement
gets closer to the analytical values, an expected result as they generate
states more uniformly distributed. In fact, we can say they are approximating
a $2-$design (Def. 1) when the analytical values are achieved 222This
affirmation has to be made with much caution, because not every circuit that
generates entanglement values close to the analytical CUE values approximates
a $2-$design. Still, in this case this is true as the circuits are converging
to the analytical case and a relative entropy decrease was observed.. The Ring
is the only one that can achieve this reproduction with $5$ layers (Fig. 6
(c)), while the others would need more layers to faithfully reproduce it.

If we look at the dependency of the entanglement as a function of the
dimension for only $1$ layer, it is possible to notice a slight increase for
the Ring, All-to-all and Linear topologies, while the Star circuit preserve
the same entanglement values. This can be explained by the fact that the
connections structure for the Star circuit prepares entangled states that have
a strong dependence on the state built with the RX-RY gates step for the
central qubit. Every other qubit is connected to this one and not connected to
any other, so if the state prepared for it does not favor entanglement
generation with the CNOT gates step, increasing the number of qubits will not
influence the entanglement generation. Therefore, the generation of
entanglement is highly connected to this central qubit and changing the number
of qubits, consequently increasing the number of CNOTs, will not affect
entanglement values. In contrast, the generation of entangled states and
distribution of CNOT gates is not confined in the other connected topologies:
The connections are distributed among the qubits that compose the circuit.
Therefore, we can say that increasing the number of qubits will imply in more
possibilities of connections between qubits and favor the average entanglement
increase with the dimension.

Figure 6: Ansatz 1: average Meyer-Wallach/Scott $1$ for all connected
topologies as a function of the number of qubits for (a) 1 layer, (b) 3 layers
and (c) 5 layers. The blue line indicates the analytical values of the CUE
given in Eq. (III.2). LIN: Linear, RIN: Ring, ST: Star and ATA: All-to-all.

An unexpected feature of the chosen connectivities is the coincidence in the
entanglement of the Linear and All-to-all topologies, even though they have
very different number of CNOT gates and complexity of connections, with the
All-to-all presenting a quadratic scale of number of gates with the number of
qubits (see Table 1). To understand why, a specific example is considered. If
we take one layer of a Ansatz $1$ All-to-all circuit with $4$ qubits and
perform the sequence of operations defined in Fig. 7, the result will be
equivalent to a Hadamard gate in the first qubit, nothing in the others and
the steps of connections after that. Using these gates, after the Step $1$ in
Fig. 7, the resulting state will be a $4$ qubits GHZ state, that has maximum
entanglement quantified by the Meyer-Wallach. However, performing Step $2$,
the output state will be
$\ket{\psi_{\text{out}}}=(\ket{00}+\ket{11})/\sqrt{2}\otimes\ket{0}\otimes\ket{0}$,
whose entanglement quantified by the Meyer-Wallach/Scott $1$ is equal to
$1/2$. This way, the step of CNOT gates in the All-to-all topology is creating
and sometimes destroying entanglement at the same time, influencing the
average entanglement. We can then say that different parameter vectors will be
able to generate highly and slightly entangled states due to the connections
choice, resulting in an average that replicates the entanglement generated by
the Linear connected circuit with a way more complex structure. Therefore, the
choice of performing all the CNOT gates in a row is not optimized for
generating high average entanglement and other choices could be more
interesting. One plausible option could be to interleave the local operations
with the CNOT gates, avoiding the sequential CNOTs with the same local
coherences.

Figure 7: Specific example of a sequence of gates applied considering a $4$
qubits All-to-all circuit of Ansatz $1$. In this set, a Hadamard is applied to
the first qubit and nothing is done to the other qubits in the parameterized
operations step. After that, the connections are applied divided into two
intermediate steps, Step $1$ (blue) and Step $2$ (red).

As said before and demonstrated now with data, the generation of entanglement
and the expressibility of the circuits is closely related with the topologies.
However, the connection between these two quantities was not established
hitherto. To try to understand the behavior of the average entanglement in
these circuits and compare with the generation of uniformly distributed
states, Fig. 8 presents a comparison between the circuits mean entanglement
and the CUE analytically obtained values, normalized due to the dependency
with the dimension. Excluding the case of $3$ qubits that already for $2$
layers presents values very close to zero, it is possible to observe a
hierarchy between the topologies where some of them generate entanglement
closer to the random values than others. This hierarchy is the same as the
observed for the expressibility evolutions at higher dimensions, i.e.,
circuits generating average entanglement closer to the CUE will also present a
steeper evolution of the expressibility at higher dimensions. This is an
indicative of a connection, however it does not explain why the same hierarchy
is not observed in the expressibility for smaller dimensions, whose evolutions
for the connected topologies is practically the same.

Figure 8: Ansatz 1: subtraction of the CUE ensemble mean and the circuit
Meyer-Wallach/Scott 1 average, normalized with the CUE value for all connected
topologies as a function of the number of layers for (a) 3 qubits, (b) 4
qubits, (c) 5 qubits, (d) 6 qubits, (e) 7 qubits and (f) 8 qubits. LIN:
Linear, RIN: Ring, ST: Star and ATA: All-to-all.

To discuss this behavior in terms of another quantity, Fig. 9 presents the
standard deviation of the Meyer-Wallach. We can observe that, apart the $3$
qubits case, the circuits standard deviations follow the same hierarchy for
the distance to the analytical values obtained from Eq.(III.2) and indicated
by the traced line. The Ring always generates the closest values, followed by
Linear/All-to-all and the Star. However, looking at the analytical values, a
characteristic of the standard deviation of slowly varying functions sampled
considering uniformly random distributed states stands out. Increasing the
dimension of the system, the values decay exponentially to zero, therefore for
high dimensional systems the random states will present high entanglement (see
Fig. 10) that is concentrated around the mean. This phenomenon is well known
and characterized by Levy’s lemma [74, 76], being also called concentration of
measure. Due to this fact, there is a lot more freedom on the entanglement
values a random state in low dimensional systems can have. However, when
increasing the dimension, the random state will have very specific values for
the entanglement, being these values very close to a maximally entangled
state. From this, we can explain why the dimension is very influential for the
evolution of the expressibility. Even if the circuit can generate entanglement
values that are high and close to the analytical uniformly random values, it
will only affect the evolution of the expressibility when the freedom is
restricted, therefore generating the right amount and distribution of
entanglement is a necessary condition for a steep evolution. This way, we can
affirm that the entanglement generation for only $1$ layer, whose values have
strong dependency on the topology, is influential for the evolution of the
expressibility. Here, we have to take into account that the circuit analysed
is Ansatz $1$, that presents many limitations in the parameterization and
differences from Ansatz $2$ that will be discussed in next section.

Figure 9: Ansatz 1: standard deviation of the Meyer-Wallach/Scott 1 for all
connected topologies as a function of the number of layers for (a) 3 qubits,
(b) 4 qubits, (c) 5 qubits, (d) 6 qubits, (e) 7 qubits and (f) 8 qubits. The
traced line indicates the analytical values as obtained in Eq. (III.2). LIN:
Linear, RIN: Ring, ST: Star and ATA: All-to-all.

One possible argument that could in principle explain this conclusion is that
the circuits are generating average and standard deviation of entanglement
closer to the analytical uniformly distributed states because they are already
generating states more uniformly distributed for only $1$ layer. Then, the
expressibility and statistical moments of entanglement are trivially
connected. However, we can provide two counterarguments against this argument.
The first one is the observation that in Fig. 5 (a), for only $1$ layer, there
is no hierarchy between the circuits and this includes even the No Connections
case. Therefore, there is no circuit that is explicitly and consistently
closer to the Haar case in only $1$ layer of Ansatz $1$ and the entanglement
generation is a fruit of the topology of connections. From another
perspective, to obtain the average value for the uniform case of the Meyer-
Wallach the complete Haar measure is not needed and a $2-$design is already
sufficient, as said before. This way, to quantify how close the circuit is to
a $2-$design for $1$ layer can reveal if the entanglement generation is due to
the structure of the connections or due to the fact that the circuit generated
states are more uniformly distributed. A quantifier for how close a
parameterized quantum circuit is to a $t-$design was presented in Eq. (3).
This quantity was numerically estimated for $20$ independent samples,
considering $t=2$ and Ansatz $1$. The results are presented in Table 2. It is
possible to observe that, within the standard deviation of the calculations,
all the topologies present the same behavior and are all equivalently close to
a $2-$design in only $1$ layer, independent of the dimension. Therefore, we
can argue that the generation of entanglement is due to the structure of the
connections and to the parameterization and not because the random unitaries
related with the circuits generate a $2-$design in $1$ layer.

Table 2: Values for the quantity $||\mathcal{A}^{t=2}||^{2}_{2}$ defined in Eq. (3), that compares the circuit with a $2-$design, obtained for the Ansatz $1$ for different topologies and number of qubits, $n$. The mean values and standard deviations were obtained over $20$ independent calculations. Topology | $n=3$ | $n=4$ | $n=5$ | $n=6$ | $n=7$ | $n=8$
---|---|---|---|---|---|---
No Connections | $(1.3\pm 0.1)\cdot 10^{-2}$ | $(6.5\pm 0.5)\cdot 10^{-3}$ | $(3.0\pm 0.3)\cdot 10^{-3}$ | $(1.2\pm 0.1)\cdot 10^{-3}$ | $(4.6\pm 0.7)\cdot 10^{-4}$ | $(1.6\pm 0.3)\cdot 10^{-4}$
Linear | $(1.3\pm 0.1)\cdot 10^{-2}$ | $(6.6\pm 0.7)\cdot 10^{-3}$ | $(3.0\pm 0.3)\cdot 10^{-3}$ | $(1.2\pm 0.1)\cdot 10^{-3}$ | $(4.5\pm 0.6)\cdot 10^{-4}$ | $(1.7\pm 0.3)\cdot 10^{-4}$
Ring | $(1.3\pm 0.1)\cdot 10^{-2}$ | $(6.7\pm 0.6)\cdot 10^{-3}$ | $(3.0\pm 0.3)\cdot 10^{-3}$ | $(1.2\pm 0.1)\cdot 10^{-3}$ | $(4.3\pm 0.6)\cdot 10^{-4}$ | $(1.7\pm 0.3)\cdot 10^{-4}$
Star | $-$ | $(6.8\pm 0.5)\cdot 10^{-3}$ | $(3.0\pm 0.3)\cdot 10^{-3}$ | $(1.2\pm 0.1)\cdot 10^{-3}$ | $(4.4\pm 0.6)\cdot 10^{-4}$ | $(1.6\pm 0.3)\cdot 10^{-4}$
All-to-all | $-$ | $(6.6\pm 0.6)\cdot 10^{-3}$ | $(3.0\pm 0.3)\cdot 10^{-3}$ | $(1.2\pm 0.2)\cdot 10^{-3}$ | $(4.5\pm 0.8)\cdot 10^{-4}$ | $(1.6\pm 0.3)\cdot 10^{-4}$

These entanglement results are replicated considering the Scott $2$. Fig. 10
presents a comparison for the dimensions of $4$ and $8$ qubits. It is possible
to notice that the values are different, however the results are the same: The
same hierarchy is observed, a very similar evolution and saturation as a
function of the number of layers happen and the same conclusions can be drawn.

Figure 10: Comparison of the average entanglement generated by the different
topologies considering the two entanglement quantifiers S1 and S2 presented in
this work, for the dimensions of (a) 4 qubits - S2, (b) 4 qubits - MW/S1 and
(c) 8 qubits - S2, (d) 8 qubits - MW/S1. LIN: Linear, RIN: Ring, ST: Star and
ATA: All-to-all.

###  IV.2 Ansatz 2

We can now move to the discussion of Ansatz $2$, starting with a comparison
with Ansatz $1$. In Fig. 11 an important result is shown: Both circuit
structures present exactly the same average entanglement. This is also
observed for the Scott $2$ quantifier. This way, their differences are only
related with the expressibility and comes from the additional step of
parameterized local operations in Ansatz $2$. In this sense, with the two
structures we can isolate the role of entanglement generation and the
additional freedom on the local parameters.

Figure 11: Comparison of the average entanglement generated by the different
topologies considering the two circuit structures Ansätze $1$ (A1) and $2$
(A2) presented in this work, for the dimensions of (a) 4 qubits - A1, (b) 4
qubits - A2 and (c) 8 qubits - A1, (d) 8 qubits - A2. LIN: Linear, RIN: Ring,
ST: Star and ATA: All-to-all.

The peculiarities of Ansatz $2$ start to appear at Fig. 12 when comparing the
different topologies together with the No Connections case for $4$ qubits
(Fig. 12 (a)) and $8$ qubits (Fig. 12 (b)). We can see a way quicker evolution
of the relative entropy for all topologies when considering Ansatz $2$
structure. The No Connections case presents the saturation values observed for
Ansatz $1$ already at $1$ layer, while the connected topologies are closer to
zero when compared with the No Connections. By the introduction of additional
local parameters, it was possible to increase the expressibility of the
circuits. Here, the connected and not connected topologies do not have the
same expressibilities in only $1$, in contrast to what happens to Ansatz 1 as
we have discussed in last section. This is explained by the characteristics of
Ansätze $1$ and $2$. The entangled states generated with Ansatz $1$ in $1$
layer have a direct connection between entanglement and local coherences
created by the RX-RY sequence of gates. This is due to the fact that the
generation of entanglement in the CNOTs part of the circuits is only possible
because of the different local coherences created by the parameterized gates.
After that, entanglement and local superpositions are intrinsically connected.
For Ansatz $2$, the additional step of local parameterized operations permits
the manipulation of local superpositions withouth changing entanglement,
therefore for every entanglement value generated after the connections part,
many possible different states can be built using this different circuit
structure.

Figure 12: Ansatz 2: relative entropy as a function of the number of layers
for all connectivities, considering (a) $4$ qubits and (b) $8$ qubits. NC: No
Connections, LIN: Linear, RIN: Ring, ST: Star and ATA: All-to-all.

Now, a detailed study of the characteristics of the connected topologies is
made looking at Fig. 13. The relative entropy evolves in a way similar to the
values observed for Ansatz $1$, however requiring less layers to achieve the
same results. For example, in the case of $4$ qubits (Fig. 13 (b)), the
saturation value is observed already for $2$ layers, while for Ansatz $1$ it
is only observed after $4$ layers. Also, a saturation is now observed in the
$6$ qubits (Fig. 13 (d)) around $1\cdot 10^{-3}$ nats, while the same case for
Ansatz $1$ have a decreasing relative entropy in $5$ layers and values above
$1\cdot 10^{-3}$. In this sense, we again notice that Ansatz $2$ is better at
achieving high expressibilities when compared to Ansatz $1$, as it has a
steeper relative entropy decrease and it applies less entangling gates at the
expense of additional local parameterized gates. Up to $3$ layers, the Ansatz
$2$ overperforms Ansatz $1$. However, with increased number of layers, the
evolutions can get erratic and there are cases where the relative entropy
grows even increasing the number of layers, as Fig. 13 (d) and (f) show, for
example.

In the discussion about Ansatz $1$, it was said that the effects of generating
average and standard deviation of entanglement are influential only for higher
dimensions, being almost imperceptible for $4$ and $5$ qubits. Here, for
Ansatz $2$, we can see the hierarchy for smaller dimensions in $1$ layer and
it is well distinctive. This result may seem to contradict the previous
discussion, however we have to take into account the elements present in
Ansatz $2$ and not in $1$. The topologies enjoying the entanglement
characteristics close to the CUE will have the additional freedom for
generating entangled states with random coherences in Ansatz $2$ and we can
see that this is important for the high expressibility of the circuits. Still,
we can observe that this hierarchy gets more pronounced for higher dimensions
and the conclusions are the same as observed for Ansatz $1$.

The results in this work find agreements with previews works analysing the
expressibility and “entangling capability” of parameterized quantum circuits
[26]. In [26], there are circuits that generate average entanglement very
close to the CUE mean in $1$ layer, although with relatively small
expressibility when compared to circuits generating entanglement values
farther from the CUE mean. There, they say that generating entanglement close
to the uniformly distributed states case is a possible diagnosis for proximity
to the Haar case, however it does not necessarily imply that. This is indeed
true, however, they do not compare circuits with the same characteristics in
terms of the gates applied, as some of them apply parameterized two qubits
gates while others are only composed of not parameterized two qubits gates (CZ
and CNOT, for example), which can result in very different results for
expressibility and, in the context of our results, these are different
analysis. When looking at the evolution as a function of the number of layers
presented in their supplementary material, it is possible to notice that a
circuit generating average entanglement closer to the CUE case, dubbed
“Circuit 2” there, have a steeper expressibility increase as a function of the
number of layers when compared to the others. A result also observed and
systematized in our work.

Figure 13: Ansatz 2: relative entropy as a function of the number of layers
for the connected topologies, considering (a) $4$ qubits and (b) $8$ qubits.
LIN: Linear, RIN: Ring, ST: Star and ATA: All-to-all.

##  V Conclusions

In this work, we studied the characteristics of the generation of states in
different parameterized quantum circuits whose structures were chosen based in
contemporary quantum computer architectures. Five different connectivities
between qubits were analysed, together with two different parameterizations,
generating ten different circuits consisting of parameterized one qubit X and
Y rotations and CNOT gates. To characterize how uniformly distributed are the
random states and what are the average and standard deviation of entanglement
values, we applied the expressibility quantifier together with the averages of
Scott entanglement quantifier of order $1$ and $2$. The results showed that
circuits with a Ring topology will have the steepest increase of the
expressibility as a function of the number of layers and highest entanglement
values, followed by Linear/All-to-all practically together and finally the
Star circuit. The No Connections circuits do not generate entanglement and
will have saturated expressibility values.

These results are influenced by the dimension of the system, however the same
trend is observed. An important correlation is noticed: Circuits generating
entanglement averages and standard deviation deviation closer to the circular
unitary ensemble, that is uniformly distributed over the unitary manifold,
have a steeper increase of the expressibility. This is justified in terms of
the freedom of the random states that can be produced depending on the
dimension of the system, and generating states with closer average and
standard deviation of entanglement favors the increase of expressibility.

The proper entanglement generation and its advantage is explored with the two
possible parameterizations. In Ansatz $1$, only one step of local
parameterized operations is applied before the connections between qubits,
while in Ansatz $2$ the connections between qubits is interleaved with local
parameterized operations. This leads to a quicker increase of the
expressibility as a function of the number of layers, maintaining the
entanglement constant. Therefore, by applying more local unitary operations,
that are less complex and less costly when compared to two qubit operations,
it was possible to obtain a more uniform distribution of states. The results
were replicated with both entanglement measures.

The Meyer-Wallach/Scott quantifiers have been applied to characterize random
circuits and chaotic systems for decades now [25, 43, 44]. For example, in
Ref. [25], the Meyer-Wallach is applied as a possible diagnosis of the
convergence of circuits with different coupling topologies and local gates to
a pseudorandom circuit. Here, we apply the same entanglement measure together
with Scott$-2$ and the entanglement generation is not a diagnosis of
convergence as in these previous works, but is applied as a way to understand
how the different entanglement generation of the circuits influences the
convergence to a uniformly distributed case, quantified by expressibility.

This work provides results that can be applied for choosing quantum
architectures depending on the characteristics expected from the circuits and
places perspectives on the possibility of application of parameterized quantum
circuits for the generation of random states. For example, if the circuit
requires high entanglement values and/or distribution of states close to the
uniform, an interesting candidate would be circuits with Ring topology.
Conversely, if the intention is to have circuits with less entanglement and/or
less uniformly distributed states, the Star would be the best choice between
the options we studied. The circuits Linear/All-to-all are the middle point
between the possibilities. Still, the expressibility quantifier is not the
unique choice and other circuit complexity quantifiers would be interesting to
study these characteristics and provide deeper insights in the behavior of
parameterized quantum circuits and their generation of random states.

###### Acknowledgements.

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior – Brazil (CAPES) – Finance Code 001. I.M.
acknowledges financial support from São Paulo Research Foundation - FAPESP
(Grant No. 2022/08786-2 and No. 2023/14488-7). P.C.A. acknowledges financial
support from Conselho Nacional de Desenvolvimento Científico e Tecnológico -
Brazil (CNPq - Grant No. 160851/2021-1). D.O.S.P acknowledges the support by
the Brazilian funding agencies CNPq (Grant No. 304891/2022-3), FAPESP (Grant
No. 2017/03727-0) and the Brazilian National Institute of Science and
Technology of Quantum Information (INCT/IQ).

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