# Adversarial Parametric Pose Prior

Andrey Davydov
<EMAIL_ADDRESS>Anastasia Remizova
<EMAIL_ADDRESS>Victor Constantin
<EMAIL_ADDRESS>Sina Honari
<EMAIL_ADDRESS>Mathieu Salzmann
<EMAIL_ADDRESS>

CVLab, EPFL  Pascal Fua
<EMAIL_ADDRESS>
###### Abstract

The Skinned Multi-Person Linear (SMPL) model can represent a human body by
mapping pose and shape parameters to body meshes. This has been shown to
facilitate inferring 3D human pose and shape from images via different
learning models. However, not all pose and shape parameter values yield
physically-plausible or even realistic body meshes. In other words, SMPL is
under-constrained and may thus lead to invalid results when used to
reconstruct humans from images, either by directly optimizing its parameters,
or by learning a mapping from the image to these parameters.

In this paper, we therefore learn a prior that restricts the SMPL parameters
to values that produce realistic poses via adversarial training. We show that
our learned prior covers the diversity of the real-data distribution,
facilitates optimization for 3D reconstruction from 2D keypoints, and yields
better pose estimates when used for regression from images. We found that the
prior based on spherical distribution gets the best results. Furthermore, in
all these tasks, it outperforms the state-of-the-art VAE-based approach to
constraining the SMPL parameters.

##  1 Introduction

The SMPL model [16] is now widely used to parameterize body poses and shapes
[20, 14]. However, it offers no guarantee to produce realistic human bodies
when random values are passed as its inputs. This complicates its usage within
an optimization, regression, or generative frameworks, where it is desirable
that any sample drawn be plausible.

To mitigate this issue, several approaches have been used. In [2], this is
addressed by introducing a Gaussian Mixture Model (GMM) learned on the SMPL
pose. Unfortunately, due to its unbounded nature, it still allows poses far
away from any training example and potentially unrealistic. In SMPL-X [21], a
Variational Autoencoder (VAE) is used instead to learn a low-dimensional
representation of the SMPL parameters. This choice was motivated by the
ability of VAEs to model the distribution of valid data samples in the latent
space as a multivariate Gaussian, which was shown to better approximate the
data distribution than classical models, such as GMMs, while also facilitating
sampling at test time. In both approaches, the learned prior is then used
together with other losses in an optimization-based framework that aims at
finding plausible human meshes. Unfortunately, VAEs have drawbacks. First,
their learned prior tends to be mean-centered and to discard part of the
original data distribution that are far away from it. Furthermore, its
Gaussian prior is unbounded, like that of GMMs. Hence, one can also sample
latent values far away from any in the training set and produce unrealistic
bodies. Adversarial training has been used to bound the parameter prediction
of SMPL in regression-based frameworks [10, 4]. However, this requires
balancing the adversarial loss with other losses. More importantly, no
explicit prior has been learned in such cases, as this training needs to be
repeated for each new task.

In short, these approaches make it necessary to balance different losses and
do not bound the inputs of the SMPL model. In contrast, we aim at learning a
prior, that once learned can be used in an optimization or learning-based
frameworks, without the requirement of enforcing constraints on it. In other
words, the learned prior should be integrated as part of the SMPL model and
the model can be optimized only on the target loss, where the learned prior is
not added as an extra constraint. To this end, we learn an explicit prior,
that constrains the input of the SMPL model to be realistic poses via
adversarial training. This has to be done only once and independently of the
target application so that no further adversarial training is needed. Hence,
it does not require balancing multiple losses in the downstream tasks.

Furthermore, one can use a bounded distribution, such as uniform or spherical
ones, in the input space of the learned prior, which facilitates plausible
sample generation and also its integration in regression frameworks, as by
limiting the output of the preceding component that is passed as input to the
learned prior, one is always guaranteed to have a plausible human
representation. Once trained, our model can be used in many different settings
without further retraining as shown in Fig. 1. We introduce GAN-based pose
prior learning technique that consistently outperforms the VAE-based state-of-
the-art approach for both optimization- and regression-based approaches to
human body pose recovery [21]. Also, we make a comparision between different
choices of latent spaces, out of which the spherical one brings the most
benefit.

##  2 Related Work

###  2.1 Body Representation

There have been many attempts at modeling the human body. The earliest ones
split the body into several simpler shapes and combine them into a unified
model. The introduction of several datasets consisting of diverse body scans
[24] has ushered the age of learnable body models. The SMPL body model [16]
constitutes one of the most successful and easy-to-use models. It uses a
combination of PCA coefficients to model the shape and a regressor that poses
the body from the joints angles. Several extensions have since then been
proposed. SMPL-H [25] includes a more detailed hand model, thus removing one
of the limitations of the original model. More recently, SMPL-X [21], adds
facial expressions to the previous models. Instead of using a mesh
representation, NASA [3] encodes the human body as a sign distance function.
Here, we focus on the SMPL model as we are interested in modeling the human
body itself, and favor a mesh representation which inherently provides
correspondences across, e.g., video frames.

###  2.2 SMPL Parameter Estimation

Since the introduction of the SMPL body model, many approaches have aimed to
estimate the SMPL parameters given either an image [10, 22, 4, 8], some
labels, such as 2D or 3D pose [21, 2, 1], or body silhouettes [13]. Depending
on whether they are optimization- or regression-based, they can be divided
into three categories.

#### Optimization Models.

The first category consists of optimizing the SMPL parameters so as to
minimize an objective function defined in terms of different pose or image
descriptors. Such descriptors can be 2D and/or 3D joint locations [2, 1, 21],
silhouettes [13], or dense correspondences [6]. SMPLify [2] constitutes one of
the first such methods. It uses a GMM to model the pose space and optimizes
the SMPL parameters so as to match 2D joint locations. The unboundedness of
the GMM prior may result in the optimization producing unrealistic poses.
Therefore, VPoser [21] proposed to replace the GMM with a VAE to model the
pose space distribution. While a VAE can model more complicated distributions
than a GMM, it remains unbounded. Furthermore, the mean-centered nature of
VAEs makes it cover the original data distribution only partially, because it
poorly represents data samples away from its distribution’s means. As we will
show later, our approach learns a better and smoother coverage of the data
while addressing the unbounded nature of these approaches.

#### Regression Models.

The second category consists of directly regressing the SMPL parameters given
an input image. Human Mesh Recovery (HMR) [10] is one of the initial methods
that applies such a technique using deep neural networks. Since then it has
been used in several other works, such as [4, 8, 22]. These methods minimize
an adversarial prior together with other target losses. Therefore, the
resulting representation is only usable within the learned model, since no
explicit prior is learned. By contrast, we learn an explicit bounded prior,
which needs to be trained only once. Then, the weights of this learned prior
can be frozen and it can be used in any regression approach by mapping a
feature space to the learned prior latent space.

#### Combined Models.

The two previous categories are compatible with each other and can be used
together. SPIN [12] mixes the two by fine-tuning the regression estimate with
an optimization procedure. EFT [9] takes the pretrained regression network of
[12] and uses its weights as an implicit body prior. It fine-tunes the weights
of the network for every sample in a weakly-annotated dataset to obtain the
body parameters. Although we demonstrate our method separately on optimization
and regression-based tasks, it can be used in the combined approach, as these
models merge the individual components from optimization and regression-based
approaches.

##  3 Method

Figure 1:  Multiple uses of the proposed approach. $(A)$ The adversarial game
between generator $\mathcal{G}$ and discriminator $\mathcal{D}$ guarantees
that the former provides realistic body poses. $(B)$ Given “start” and “end”
latent vectors, one can render the whole sequences of plausible and smooth
body interpolations. $(C)$ To optimize for corresponding poses given target
keypoints $\mathcal{Y}$, akin to VPoser [21], the pretrained generator
$\mathcal{G}$ can be used as an implicit pose prior. $(D)$ The pretrained
$\mathcal{G}$ can be dropped in as a pose prior in a pretrained off-the-shelf
human mesh regressor. It improves the prediction quality of the regressor.

To constrain the SMPL poses we rely on a GAN approach [5]. It involves two
competing networks, a generator ${\mathcal{G}}$ and a discriminator
${\mathcal{D}}$. The generator samples vectors ${\bf z}$, known as latent
vectors, from a set $\mathbb{P}_{z}\subseteq\mathbb{R}^{d}$ and generates a
SMPL pose vector $\hat{\Theta}=\mathcal{G}({\bf z})$, which can be passed to
the SMPL decoder $\phi$ to generate a body mesh ${\cal B}=\phi(({\bf
z}),\beta)$, where $\beta$ denotes the SMPL body shape parameters. The task of
the discriminator is to distinguish poses generated in this manner from those
of a large dataset of poses known to be realistic. By contrast, the generator
is trained to produce poses that fool the discriminator. This process is shown
in Fig. 1 $(A)$.

#### Constraining shape and pose

Training our models, we leave the SMPL shape parameters $\beta$ untouched to
be able to compare with other models, i.e. VPoser [21], as they only learn a
prior for pose. Moreover, the shape part of the model is already data-driven
(with PCA). However, the PCA weights for the shape are also unbounded by the
model and eliminating this problem is also worth further research. We trained
a model in such combined fashion, more information on this can be found in the
supplementary material.

###  3.1 Distribution over Latent Vectors

GAN-based approaches [27, 19, 11] have used several types of distributions
from which to draw their latent vectors, including Gaussian, Uniform, and
Spherical distributions. To test all three, we learn three different sets of
latent vectors:

  * •

GAN-N: ${\bf z}_{N}\sim\mathbb{P}_{z}=\mathcal{N}(0,\mathcal{I})$ (Normal)

  * •

GAN-U: ${\bf z}_{U}\sim\mathbb{P}_{z}=\mathcal{U}(-1,1)$ (Uniform)

  * •

GAN-S: ${\bf z}_{S}\sim\mathbb{P}_{z}=\mathcal{S}\subset\mathbb{R}^{d}$
(Spherical)

where the spherical vectors are sampled by drawing vectors ${\bf z}_{N}$ from
a normal distribution and computing ${\bf z}_{S}:=\frac{{\bf z}_{N}}{\|{\bf
z}_{N}\|_{2}}$.

The unbounded nature of the Gaussian distribution $\mathcal{N}(0,\mathcal{I})$
prevents sampling from rare modes and may make the resulting prior suffer from
the same drawbacks as GMMs and VAEs when used in regression tasks. While the
Uniform distribution does not have such a limitation, it imposes artificial
bounds $[-1,1]^{d}$ that do not have a clear meaning in the output pose space.
Intuitively, because one can smoothly move from one pose to another, we would
rather expect a latent pose space to be continuous, without strict boundaries
as the uniform space. The desirable properties of the latent space, such as
continuity and boundedness are all inherent to the Spherical distribution. Our
experiments show that, in practice, it does indeed tend to perform better than
the others.

###  3.2 Training

We define the generator $\mathcal{G}$ in our GAN architecture to have the same
structure as the decoder of the VAE in VPoser [21]. As for our discriminator
$\mathcal{D}$, we base our structure on that of the HMR approach [10], using
$K+1$ discriminators, one for each joint angle and one for the whole set of
pose parameters.

As can be seen in Fig. 1 $(A)$, we draw samples from the latent space
$\mathbb{P}_{\bf z}$ and train the generator to map them to the SMPL pose
space. The discriminators are trained to distinguish the SMPL pose vectors
$\Theta$, obtained by a real pose dataset, from the ones produced by the
generator $\hat{\Theta}$. The training loss function, aiming to balance the
two opposing goals of the generator and discriminator, can thus be expressed
as

$\displaystyle\min_{\mathcal{G}}\max_{\mathcal{D}}\mathcal{L}(\mathcal{G},\mathcal{D})$
$\displaystyle=\mathbb{E}_{\Theta}[\log\mathcal{D}(\Theta)]+$ (1)
$\displaystyle+\mathbb{E}_{z\sim\mathbb{P}_{z}}[\log(1-\mathcal{D}(\mathcal{G}(z))]\;.$

In our experiments, we update the discriminator weights $10$ times for every
update of the generator. As we found, it leads to the best performance.

We train this model using the common splits of the AMASS dataset [18],
following the procedure for VPoser in SMPL-X [21]. As AMASS provides SMPL-H
parameters [25], the body pose is a bit different from the original SMPL [16,
2] one; it only contains $K=21$ joint angles, with 2 angles from SMPL having
been moved to SMPL-H “hands” articulations.

###  3.3 Using the Generator as a Universal Prior

Being trained once, our generative model can be used in many applications. We
introduce some of them here and present the results in the following section.

#### Interpolation in Latent Space.

One way to gauge the quality of a latent representation is to check how smooth
the interpolation from one latent vector to another is. Ideally, the
transition should vary equally in each step from the source to the target
samples, rather than most of the transformation occurring in only a few steps.
To check this, we randomly select $N$ samples $\\{{\cal
B}{}^{t}_{1},\ldots,{\cal B}{}^{t}_{N}\\}$ from the test set and optimize
(similar to the paragraph below) for every body ${\cal B}{}^{t}_{i}$ the
corresponding latent vector ${\bf z}^{r}_{i}$ that yields the closest mesh
${\cal B}{}^{r}_{i}=\phi(G({\bf z}^{r}_{i}),\beta)$. For each pair of such
latent vectors, we construct an interpolation sequence $\\{{\bf
z}^{r}_{0},{\bf z}^{r}_{1},\ldots,{\bf z}^{r}_{T}\\}$ and $\\{{\cal
B}_{0},{\cal B}_{1},\ldots,{\cal B}_{T}\\}$ using either linear interpolation

$\displaystyle{\bf z}^{r}_{t}$ $\displaystyle=\big{(}1-\frac{t}{T}\big{)}{\bf
z}^{r}_{0}+\frac{t}{T}{\bf z}^{r}_{T}$ (2)

for GAN-N and GAN-U, or spherical interpolation

$\displaystyle{\bf z}^{r}_{t}$
$\displaystyle=\frac{\sin{\big{(}1-\frac{t}{T}\big{)}\theta}}{\sin{\theta}}{\bf
z}^{r}_{0}+\frac{\sin{\frac{t}{T}\theta}}{\sin{\theta}}{\bf z}^{r}_{T}$ (3)

for GAN-S, with ${\cal B}{}^{r}_{t}=\phi(G({\bf z}^{r}_{t}),\beta)$, and
$\theta$ representing the angular distance between two points ${\bf
z}^{r}_{0}$ and ${\bf z}^{r}_{T}$ on a sphere. We discuss spherical
interpolation (SLERP) in more detail, including the proof of Eq. 3 for high
dimensions, in the supplementary material. Ideally, the samples ${\cal
B}_{t-1}$ and ${\cal B}_{t+1}$ should be roughly equidistant from ${\cal
B}_{t}$, indicating smooth transitions. The per-vertex mesh distance is
computed using Eq. 5. The sampling process is depicted in Fig. 1 $(B)$.

#### Optimization from Keypoints.

Given the 2D joint targets $\mathcal{Y}$ obtained from a monocular observation
and assuming neutral SMPL shape parameters $\beta=0$, our goal is to find the
SMPL pose parameters $\hat{\Theta}$ that produce the target $\mathcal{Y}$
using the SMPL model $\phi$, which translates from SMPL space
$(\hat{\Theta},\beta)$ to the space of body meshes ${\cal B}$. Fig. 1 $(C)$
describes the idea. The recovered mesh can be projected to 2D joints using
camera parameters, i.e., $\Pi(\phi(\hat{\Theta},\beta))=\mathcal{Y}$, where
$\Pi$ is the camera projection function. To find the optimal SMPL parameters,
one can minimize $L(\Pi(\phi(\hat{\Theta},\beta)),\mathcal{Y})$, where $L$ is
a loss function such as the $L2$ distance between the 2D mesh joints and the
corresponding target joints.

To better constrain the pose output by SMPL, we make use of our pose prior.
That is, instead of directly optimizing $\hat{\Theta}$, we optimize a vector
${\bf z}$ in the GAN’s latent space and obtain the corresponding
$\hat{\Theta}$ by feeding ${\bf z}$ to the generator $\mathcal{G}$.
Altogether, we therefore solve the optimization problem

$\displaystyle\min_{{\bf z}}\|\Pi(\phi(\mathcal{G}({\bf
z}),\beta)-\mathcal{Y}\|^{2}_{2}\;.$ (4)

#### Image-to-Mesh Regression.

Our GAN models can also be used as drop-in priors to improve existing
pretrained image-to-mesh algorithms [10, 12, 9]. To demonstrate this, we start
from the model of [9], whose architecture is a Resnet50 model based on the one
of [10]. It is pretrained on pseudo ground-truth COCO [15] dataset obtained by
[9]. We then inject our model into it as shown in Fig. 1 $(D)$. More
specifically, we introduce an additional MLP $\mathcal{F}$ that maps
intermediate features of Resnet50 to a latent vector ${\bf z}$ of the pre-
trained SMPL prior, which then can be mapped by $\mathcal{G}({\bf z})$ into
the pose vector $\Theta$ of SMPL. One can then decode pose parameters $\Theta$
into a human mesh ${\cal B}$ using the SMPL model $\phi$. In turn,
$\mathcal{F}$ can be used in conjunction with the pre-trained SMPL prior
$\mathcal{G}$ and the SMPL decoder $\phi$ to reconstruct a complete body mesh,
which can then be compared to the ground-truth targets. We used this process
to train only the $\mathcal{F}$ in an end-to-end setup and obtain the
corresponding body mesh ${\cal B}$.

##  4 Experiments

We now compare the three versions of our approach in sampling the latent
vectors, GAN-N, GAN-U, and GAN-S, with the VPoser VAE-based approach of SMPL-X
[21].

###  4.1 Dataset Coverage

Table 1: Statistics for Recall experiment on Train/Test splits of AMASS dataset [18]. The values are the mean, variance and medians of distances between real samples and closest neighbors among generated samples for every model. GAN models demonstrate indistinguishable behaviour, while VPoser [21] provides consistently larger discrepancy with real set. |  Train set |  Test set
---|---|---
|  $\mu\pm\sigma$ ($\downarrow$) |  median ($\downarrow$) | $\mu\pm\sigma$ | median
GAN-S (Ours) |  4.0$\pm$1.9 | 5.5 |  6.3$\pm$2.6 | 5.5
GAN-U (Ours) | 3.9$\pm$1.9 | 5.4 | 6.2$\pm$2.5 | 5.4
GAN-N (Ours) |  4.0$\pm$1.9 | 3.6 | 6.2$\pm$2.5 | 5.6
VPoser [21] |  5.2$\pm$3.2 | 4.3 |  6.3$\pm$4.0 | 7.3

(a) Recall. “Do real samples live in latent spaces?”

(b) Precision. “How close are fake samples to real?”

Figure 2: Empirical estimation of data coverage of generative models for both
Recall (a) and Precision (b). Experiments with data from the Train set are
drawn with solid lines, and from the Test set with dashed lines. Higher means
better in all charts.

(a)

(b)

(c)

(d)

Figure 3: t-SNE projections of “real” samples from the training set and of
“fakes” generated by GAN-S (a), GAN-U (b), GAN-N (c) and VPoser [21] (d)
models.

GAN-S

sample

GAN-S

1NN

GAN-U sample

GAN-U

1NN

GAN-N sample

GAN-N

1NN

Figure 4: “Worst” samples according to the precision metric In Fig. 2(b). For
each GAN model we show 10 samples with the largest distance to the first
nearest neighbor (1NN) in the training set, ordered from the worst sample on
the right. Generated samples themselves are absolutely plausible human bodies,
despite being away from training samples. Note that in GAN-S and GAN-U these
samples are further away from 1NN compared to GAN-N.

An ideal latent representation should cover the whole space of realistic human
poses and nothing else. In other words, it should have good Recall and
Precision. By recall, we mean that all samples in the training set should be
well approximated by poses our model generates. By precision, we mean that
these generated poses should never deviate too far from the training set.
While recall indicates how well the generated samples cover the dataset
distribution, precision indicates how realistic the generated samples are. We
define these metrics as follows.

#### Recall.

To evaluate recall, we use our pose generator to produce SMPL poses and take
the shape parameters to be a zero vector, which yields a neutral body shape.
Hence, for all models we produce a body mesh ${\cal B}=\phi(\mathcal{G}({\bf
z}),\beta)$ given a sampled latent vector ${\bf z}$ and a fixed $\beta$. We
first generate 6M samples from the pose generator of each model. Then, given a
ground-truth body ${\cal B}^{t}$ from either the training or test set, we
select the generated body ${\cal B}^{r}=\phi(\mathcal{G}({\bf z}^{r}),\beta)$
with minimum vertex-to-vertex distance

$\displaystyle d({\cal B}{}^{r},{\cal B}{}^{t})$
$\displaystyle=\frac{1}{N_{verts}}\sum_{v=1}^{N_{verts}}\|{\cal
B}{}^{r}_{v}-{\cal B}{}^{t}_{v}\|_{2}\;,$ (5)

where $v$ sums over the vertices.

We then repeat this operation for 10k bodies randomly sampled from either the
training or test set. We report the mean, variance and median of the resulting
distances in Table 1. In Fig. 2(a), we plot the cumulative distribution
$\mathbb{P}(d<\epsilon)$ given the values $d({\cal B}{}^{r},{\cal B}{}^{t})$
for each training sample. Note that all versions of our approach deliver
consistently higher values than the VPoser [21], indicating that our models
better cover the entire distribution.

In Fig. 3, we show the t-SNE projection [17] of the resulting SMPL
$\Theta^{r}$ pose vectors superposed on the $\Theta^{t}$ vectors that were
used to generate the training examples. All GANs cover the space spanned by
the training examples more completely than VPoser, which is consistent with
the previous result. In other words, our learned prior can represent more
diverse poses than the other ones.

#### Precision.

Our approach to computing precision mirrors the one we used for recall. We
randomly generate 10k latent points ${\bf z}^{r}$ from every model, and for
each sample ${\cal B}{}^{r}=\phi(G({\bf z}),\beta)$, with a fixed $\beta$,
look in the training or test datasets for the nearest neighbor in terms of the
distance given in Eq 5. If the latent representation only produces poses
similar to those seen in training, this distance should be consistently small.

As shown in Fig. 2(b), GAN models tend to produce meshes that are further away
from the training distribution than the VAE model. This could be interpreted
as a failure to produce realistic poses. However, these unseen samples
correspond to plausible bodies. They are nothing but the result of semantic
interpolation that GANs implicitly learn from the data. In Fig. 4, we show the
worst 10 samples based on the distance metric of Eq. 5 and their closest
nearest neighbors from the training set. Note that all of these samples look
realistic even though they are far from the closest neighbor in the dataset.
This indicates that our generators are able to produce novel samples that were
not observed in the training set, however, this is more observed in GAN-S and
GAN-U compared to GAN-N, as GAN-N generates samples closer to its mean, hence
deviating less to more diverse poses.

###  4.2 Interpolation in Latent Space

To evaluate our model on the first application described in Section 3.3 and in
Fig. 1 $(B)$, we randomly select $N=128$ samples $\\{{\cal
B}{}^{t}_{1},\ldots,{\cal B}{}^{t}_{N}\\}$ from the test set, and, for each
pair, we construct the corresponding interpolation sequence $\\{{\bf
z}^{r}_{0},{\bf z}^{r}_{1},\ldots,{\bf z}^{r}_{T}\\}$ and $\\{{\cal
B}_{0},{\cal B}_{1},\ldots,{\cal B}_{T}\\}$. We use the mean per-vertex
position error (Eq. 5) between body meshes ${\cal B}_{i}$ and ${\cal B}_{j}$
and compute pairwise distances between their body meshes by $\Delta({\cal
B}_{i},{\cal B}_{j})$, which we represent by $\Delta_{ij}$.

The minimal transformation $\Delta_{ij}$ between every consecutive bodies is
equal to $\Delta_{0T}=\frac{\Delta({\cal B}^{i}_{0},{\cal B}^{i}_{T})}{T}$.
For different initial pairs, this value can be drastically different, as the
corresponding bodies might be very close to or very far from each other.
Hence, we normalize the transformation $\Delta_{ij}$ of every sequence by the
expected average transformation $\Delta_{0T}$, yielding $\bar{\Delta}_{ij}$,
which should be $1$ in the minimal case. Note, however, that such an ideal
case can typically only be achieved by going through physically-impossible
poses, for instance by shrinking the arms to go from a body with arms up to
one with arms down. Hence, actual transformations will typically obtain values
higher than $1$, but a good latent space should nonetheless yield values as
constant as possible throughout the entire interpolation steps, indicating a
smooth gradual transition.

Figure 5: Average normalized mesh transformation of consecutive generated samples in interpolations between source and target examples. $x$ axis depicts the number of iterations and the $y$ axis shows the normalized mesh deviation in log-scale. The ideal transformation should be represented as a flat line. VPoser applies most of transformations at the beginning or at the end of the interpolations. GAN-N reduced this affect, while GAN-S obtains the smoothest transitions. Table 2: Comparison of interpolation smoothness between different models. $max$ to $min$ ratios of consecutive deformations for every pair of interpolation is measured according to Eq. 6. Mean, median, and standard deviation of $R_{ij}$ over a set of interpolations is then reported for each model.  |  $\mu\pm\sigma$ ($\downarrow$) |  median ($\downarrow$)
---|---|---
GAN-S (Ours) | 4.3$\pm$3.0 | 3.5
GAN-U (Ours) |  7.1$\pm$7.7 | 5.1
GAN-N (Ours) |  5.7$\times 10^{2}\pm$5.3$\times 10^{2}$ | 8.6
VPoser [21] |  1.8$\times 10^{5}\pm$6.9$\times 10^{5}$ | 9.1

GAN-S

GAN-U

GAN-N

VPoser [21]

GAN-S

GAN-U

GAN-N

VPoser [21]

Figure 6: Examples of interpolations for different generative models. All GAN
models provide smooth (yet semantically very different) interpolations, while
VPoser [21] sticks at one pose for most of the path and “jumps” into the
ending pose. More examples can be found in the supplementary material.

We illustrate the behavior of different models in Fig. 5, where we average the
consecutive interpolation distances $\bar{\Delta}_{ij}$ across all pairs. The
closer the curve is to being horizontal, the smoother is the interpolation.
The VPoser [21] curve indicates that interpolation with this model is subject
to jumping from ${\cal B}^{i}_{0}$ to ${\cal B}^{i}_{T}$ in very few steps,
either at the beginning or the end of the sequence, and the remaining steps
are spent performing small pose adjustments. This effect can also be seen in
GAN-N but to a lesser degree. In contrast, GAN-S and GAN-U both produce smooth
interpolations, with a slight advantage to the spherical distribution. More
plots with interpolation distances for particular pairs can be found in the
supplementary material.

To measure the smoothness of an interpolation, we compute the ratio between
the maximal and minimal transitions in a sequence $\Delta_{ij}$, i.e.,

$\displaystyle R_{ij}=\frac{\max(\Delta_{ij})}{\min(\Delta_{ij})}\;.$ (6)

We report the mean, variance and median of the ratios $R_{ij}$ in Table 2.
These results confirm our previous conclusions: GAN-S yields the smoothest
interpolations, closely followed by GAN-U. In Fig. 6 we show interpolation
between two pairs of samples for different models.

###  4.3 Mesh Optimization from 2D Joints

Let us now turn to the optimization application discussed in Section 3.3 and
in Fig. 1 $(C)$. Given the 2D joint locations, the optimal latent vector can
be found using an iterative optimization algorithms. In our experiments, we
use L-BFGS-B for all models. In the case of GAN-S and GAN-U, we renormalize
the estimated ${\bf z}$ given their input bound at each optimization step.
Note that projecting the 3D body joints to the observed 2D joints on the image
via the camera project $\Pi$ requires access to the camera parameters and the
body orientations. We obtain them in the same way as in [2].

Table 3: Reporting 3D pose error for the bodies recovered through optimization from 2D joint targets. (numbers in mm). |  P-MPJPE ($\downarrow$)
---|---
GAN-S (Ours) | 84.3
GAN-U (Ours) | 90.7
GAN-N (Ours) | 85.4
VPoser [21] | 90.1
GMM [2] | 92.3

In Table 3, we report the 3D pose errors (after rigid alignment) obtained by
recovering the SMPL parameters $\hat{\Theta}$ from 2D joints for the H3.6M
dataset [7], following Protocol 2. Note that GAN-S again yields the best
results for this application, this time closely followed by GAN-N. By
contrast, GAN-U yields a higher error, indicating its input bounds makes it
less suitable for the optimization-based tasks.

###  4.4 Image to Mesh Regression

Table 4: Reporting 3D pose error for the bodies recovered through regression from input images. (numbers in mm). |  MPJPE ($\downarrow$) |  P-MPJPE ($\downarrow$)
---|---|---
$\text{HMR}_{\text{[COCO EFT]}}$ [9] | 75.91 | 68.02
GAN-S (Ours) | 63.24 | 56.32
GAN-U (Ours) | 64.28 | 57.30
GAN-N (Ours) | 67.19 | 59.92
VPoser [21] | 69.18 | 61.71

We train the approach to body regression from an image introduced in Section
3.3 and in Fig. 1 $(D)$ for the three versions of our approach and for the
VPoser [21]. We report accuracy results on test data in Table 4 in terms of 3D
pose error of the recovered bodies after Procrustes alignment (P-MPJPE),
according to Protocol 2 of [7]. Once again GAN-S performs the best, with GAN-U
and GAN-N outperforming VPoser. Our models also report better accuracy
compared to [9], for which the accuracy of the pre-trained model, provided by
the authors, is reported.

##  5 Limitations

Using our GAN priors, the diversity of their learned distributions is limited
by their training sets, which might not be diverse enough for downstream
tasks. This is, however, similar to any other model that learns a distribution
such as VPoser or HMR, which is also limited by its training distribution.

##  6 Conclusion

In this paper we proposed a simple yet effective prior for SMPL model to bound
it to realistic human poses. We show that the learned prior can cover the
diversity of the training distribution, while also being capable of generating
novel unseen samples. Further, we demonstrate the advantage of learning such a
prior in generation, optimization, and regression based frameworks, where the
learned prior can be trained once and for all, then used in any downstream
task without requiring to balance different losses. Our results show that
using a spherical distribution for the learned prior leads to smoother
transition in the generated samples from the latent space, while also yielding
more accurate results for optimization- and regression-based tasks, indicating
this prior is better suited for learning human poses.

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Adversarial Parametric Pose Prior
Supplementary Material

##  Appendix A Proof of SLERP formula

Let us denote two vectors of unit length:

$\displaystyle\begin{cases}&{\bf x},{\bf y}\in\mathbb{S}^{d}\\\ &\|{\bf
x}\|_{2}=\|{\bf y}\|_{2}=1\\\ &<{\bf x}\cdot{\bf
y}>=\cos\theta,\quad\theta\in(0,\pi)\end{cases}$ (7)

where $<{\bf x}\cdot{\bf y}>$ is the inner product between vector ${\bf x}$
and ${\bf y}$, with $\theta$ as the angle between them on a $d$-dimensional
unit sphere $\mathbb{S}^{d}$. With slight abuse of notations, we use ${\bf x}$
to refer to both a point on a sphere, when talking about the sampled
$d$-dimensional point, and also a vector from the origin of this sphere, when
talking about trigonometric operations.

The point ${\bf z}\in\mathbb{S}^{d}$ that lies on the interpolation path from
${\bf x}$ to ${\bf y}$ can be found via the Spherical Linear Interpolation
(SLERP):

$\displaystyle{\bf z}=\frac{\sin(\theta-\phi)}{\sin{\theta}}{\bf
x}+\frac{\sin\phi}{\sin{\theta}}{\bf y}$ (8)

where

$\displaystyle\begin{cases}&\phi=\arccos(<{\bf z}\cdot{\bf
x}>),\quad\phi\in[0,\theta]\\\ &\|{\bf z}\|_{2}=1\end{cases}$ (9)

Initially, SLERP arose in the task of 3D rotations for solid objects [26] and
Eq. 8 can be naturally derived from the 4D rotation intuition via quaternions.
However, Eq. 8 is still valid for $d$-dimensional vectors, even though
quaternion intuition is not applicable anymore. Next, we prove interpretation
used in this equation.

Proof

We want to find the point ${\bf z}$ on the sphere $\mathbb{S}^{d}$ that is a
linear combination of ${\bf x}$ and ${\bf y}$:

$\displaystyle{\bf z}=\alpha{\bf x}+\beta{\bf y}$ (10)

First, we derive the connection between coefficients $\alpha$ and $\beta$ in
Eq. 10. Knowing each point should lie on the unit-sphere we have:

$\displaystyle 1$ $\displaystyle=\|{\bf z}\|^{2}$ $\displaystyle=\|\alpha{\bf
x}+\beta{\bf y}\|^{2}$
$\displaystyle=\alpha^{2}+\beta^{2}+2\alpha\beta\cos\theta$
$\displaystyle=(\alpha^{2}+2\alpha\beta\cos\theta+\beta^{2}\cos^{2}\theta)+\beta^{2}-\beta^{2}\cos^{2}\theta$
$\displaystyle=(\alpha+\beta\cos\theta)^{2}+(\beta\sin\theta)^{2}$
$\displaystyle=\tilde{\alpha}^{2}+\tilde{\beta}^{2}\,$ (11)

where

$\displaystyle\begin{cases}&\tilde{\alpha}=\alpha+\beta\cos\theta\\\
&\tilde{\beta}=\beta\sin\theta\end{cases}$ (12)

Then, we rewrite the denotion for $\phi$ from Eq. 9 and obtain the value of
$\tilde{\alpha}$:

$\displaystyle\cos\phi$ $\displaystyle=<{\bf z}\cdot{\bf x}>$
$\displaystyle=<(\alpha{\bf x}+\beta{\bf y})\cdot{\bf x}>$
$\displaystyle=\alpha+\beta\cos\theta$ $\displaystyle=\tilde{\alpha}$ (13)

Given $\tilde{\alpha}$ and $\tilde{\beta}$ formulations in Eqs. 11 and 13, one
can derive $\tilde{\beta}=\sin{\phi}$. Finally, to obtain the coefficients
$\alpha$ and $\beta$ in Eq. 10, we use Eqs. 12 and 13 as follows:

$\displaystyle\begin{cases}&\beta=\frac{\tilde{\beta}}{\sin{\theta}}=\frac{\sin{\phi}}{\sin{\theta}}\\\
&\alpha=\tilde{\alpha}-\beta\cos\theta=\cos\phi-\frac{\sin{\phi}}{\sin{\theta}}\cos\theta=\frac{\sin{(\theta-\phi)}}{\sin{\theta}},\end{cases}$
(14)

using which we can derive Eq. 8. Choosing $\phi=\frac{t\theta}{T}$, where
$t\in[0,T]$, one gets the desired interpolation between two points ${\bf x}$
and ${\bf y}$, as presented in Eq. 3, which is linear in parameter $t$.

##  Appendix B Interpolation sequences

GAN-S

GAN-U

GAN-N

VPoser [21]

GAN-S

GAN-U

GAN-N

VPoser [21]

GAN-S

GAN-U

GAN-N

VPoser [21]

Figure 7: Examples of interpolations for different generative models, 10
intermediate interpolants each. Corresponding transition distances (100-step
interpolations) are provided in Fig. 8.

(a)

(b)

(c)

Figure 8: Mean per-vertex transition distances of different generative models
for the samples presented in Fig. 7. Each interpolation path takes 100 steps
from start to end, while the corresponding images in Fig. 7 show only each
10-th step. It is clear that VAE-based VPoser [21] provides abrupt
transitions, while GAN-based models provide smoother transitions in the output
space.

In Fig. 7 we show more interpolation sequences for different generative
models: GAN-based GAN-S, GAN-U, GAN-N and VAE-based VPoser [21]. The procedure
of sampling and interpolation is the same as in the manuscript. Also, in Fig.
8 we provide the transition distances that correspond to the interpolations in
Fig. 7. For GAN-based models transition distances for the sampled pairs lie in
the range [$10^{-3}$, $10^{-2}$] mm (per-vertex distance), which is in much
smaller range than VPoser. Note that the average result for all samples is
provided in the main paper (Fig. 5). It is clear that VAE-based VPoser [21]
applies most of the transition either at the beginning or at the end of
interpolations, while GAN-based models, especially GAN-S and GAN-U, provide
smooth interpolations.

##  Appendix C Image-to-Mesh regression

Figure 9:  Mesh predictions in Section 4.4 experiments of the main paper with
the highest (“worst”) P-MPJPE on the subset of H3.6M [7] validation set
(according to H3.6M Protocol 2). Examples are split in triplets: left \- the
input image, middle \- the predicted mesh (with predicted rotation), right \-
the same mesh rotated by $90^{\circ}$ along vertical axis. The error decreases
from left to right and top to bottom (top-left sample obtains the highest
P-MPJPE value).

We demonstrate qualitative results of injecting our module GAN-S into the
pretrained HMR architecture and fine-tuning it on recent in-the-wild pseudo
ground-truth dataset $\text{COCO}_{\text{EFT}}$ [9] (see Sections 3.3 and 4.4
in the main paper for details). We show the worst predictions of our model on
H3.6M [7] validation subset, chosen according to Protocol 2 (corresponding to
the results in Table 4 of the main paper).

##  Appendix D GANs for shape parameters

Figure 10: Examples of interpolating between random points in the latent space
of GAN-$\beta\Theta$ mixed model. Pose and shape are entangled with each
other, which complexifies the usage of such a mixed model in applications.
Figure 11: Example of generating bodies via two independent models,
GAN-$\Theta$ and GAN-$\beta$. Pose varies along axis $x$, Shape varies along
axis $y$. Having indepedent latent models allows to disentangle shape and pose
for generating body. However, as pose and shape are in fact dependent, it
might lead to generating implausible bodies (see last column, bottom rows).

(a)

(b)

(c)

(d)

Figure 12: t-SNE of samples from the entangled GAN-$\beta\Theta$ (c)
identifies a cluster of generated points that are not consistent with real
samples. This cluster consists of outliers, shown in (d), that correspond to
unrealistic bodies. t-SNE plots for independent models, GAN-$\Theta$ (a) and
GAN-$\beta$ (b), demonstrate consistent data coverage.

The SMPL model [16] is parametrized by two sets of parameters: pose $\Theta$
and shape $\beta$. In the main paper, we explore the properties of the former
for a fair comparison with VPoser [21] that only explores the pose prior,
while the latter stays out of context. Despite having relatively constrained
shape parameters in SMPL, by using a data-driven PCA model, the unbounded
nature of PCA does not restrict the SMPL output to always be plausible.

Luckily, the GAN-based approach that we used for poses can easily be applied
for shapes as well. The only difference is the choice of architecture for the
shape discriminator.

To provide a joint model for pose and shape, we explore two GAN variants
(using spherical input space prior):

  * •

Disentangled “shape-only” (“GAN-$\beta$”) and “pose-only” (“GAN-$\Theta$”)
models.

  * •

Entangled “shape+pose” (${\bf z}\mapsto\beta,\Theta$), denoted as
“GAN-$\beta\Theta$”.

In the first case, we train an independent shape model GAN-$\beta$ and use it
together with GAN-$\Theta$, while in the second case, we train both jointly,
using a shared ${\bf z}$ space. Note that in the main paper all GAN-based
models are of a kind “GAN-$\Theta$”, as they map input points to the SMPL-pose
space.

It is important to note that for training GAN-{S,U,N} models we used AMASS
[18] dataset, which contains an abundant number of body poses. However, the
number of subjects (different body shapes) is very few ($346$ in total). To
overcome this issue, for training shape-oriented models we choose another
dataset, SURREAL [28], which is composed of synthetic SMPL bodies. In SURREAL,
the shapes $\beta$ are sampled from CAESAR dataset [23], which contains about
$4k$ different shapes. This number is still very limited, compared to tens of
millions of $\Theta$-poses in AMASS, however, we are still able to train
generative models as a proof of concept.

###  D.1 Disentangled GAN-$\beta$ and GAN-$\Theta$

Trained GAN-$\beta$ (${\bf z}_{1}\mapsto\beta$) coupled with trained
GAN-$\Theta$ model (${\bf z}_{2}\mapsto\Theta$, from the main paper) might
serve as a full prior of the original SMPL [16] model. This approach learns
disentangled priors for pose and shape, in the same spirit as SMPL that
represents pose and shape parameters independently. However, the realistic
pose and shape are not completely disentangled. It means that pose and shape
independently may account for plausible humans but combined together give a
body with self-interpenetrations, as illustrated in Fig. 11.

As for the architecture of the shape discriminator, we follow HMR [10] and use
a 2-layer MLP.

###  D.2 Entangled GAN-$\beta\Theta$

To train a GAN for pose and shape together, we use the discriminators of
GAN-$\Theta$ (GAN-S in the main paper) and GAN-$\beta$ together, and train a
generator with a shared input space ${\bf z}$. In this model, we also use a
“$\beta+\Theta$” discriminator that penalizes the full generated SMPL vector.
In total we have $K+1(\text{pose})+1(\text{shape})+1(\text{pose+shape})$
discriminators.

The examples of random interpolations in the input space of ${\bf z}$ can be
found in Fig. 10. As pose and shape are entangled together, it prevents
utilizing such a mixed model in applications where one of these
characteristics needs to remain fixed.

At the same time, not every sample corresponds to a plausible body (even with
self-interpenetrations allowed). Our experiments show that generated samples
might not look as humans at all. We demonstrate it in Fig. 12 with
corresponding t-SNE visualizations for each GAN model.

##  Appendix E Future work

In most situations, independently sampling pose and shape parameters will
result in realistic bodies. However, this is not always the case.
Consequently, the task of generating plausible bodies when mixing shape and
pose together needs to be further investigated. It might be resolved, for
example, by using the formulation of a conditional GAN, where generating pose
depends on some shape features or vice versa. We explore these aspects in our
further work.


# Measuring Magic via Multifractal Flatness

Xhek Turkeshi  JEIP, USR 3573 CNRS, Collège de France, PSL Research
University, 11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France  Marco
Schirò  JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11
Place Marcelin Berthelot, 75321 Paris Cedex 05, France  Piotr Sierant  ICFO-
Institut de Ciències Fotòniques, The Barcelona Institute of Science and
Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain

(August 31, 2024)

###### Abstract

Universal quantum computing requires non-stabilizer (magic) quantum states.
Quantifying the nonstabilizerness and relating it to other quantum resources
is vital for characterizing the complexity of quantum many-body systems. In
this work, we prove that a quantum state is a stabilizer if and only if all
states belonging to its Clifford orbit have a flat probability distribution on
the computational basis. This implies, in particular, that multifractal states
are magic. We introduce multifractal flatness, a measure based on the
participation entropy that quantifies the wave function distribution flatness.
We demonstrate that this quantity is analytically related to the stabilizer
entropy of the state and present several examples elucidating the relationship
between multifractality and nonstabilizerness. In particular, we show that the
multifractal flatness provides an experimentally and computationally viable
nonstabilizerness certification. Our work unravels a direct relation between
the nonstabilizerness of a quantum state and its wave function structure.

##  I Introduction

Originally developed in the context of fault-tolerant quantum computation and
error-correction [1, 2], stabilizer states and operations play an essential
role in quantum information theory [3], in our understanding of thermalization
[4, 5] and entanglement propagation in many-body systems [6, 7, 8], or
providing insights on the holography [9]. Yet, their classical simulability
[10] implies that additional ingredients are required to attain computational
quantum speed-up [11, 12, 13]. Quantifying the nonstabilizerness (or magic) of
a quantum state is, therefore, a key task in the resource theory of quantum
computation [14, 15, 16, 17, 18, 19], related to the classical computational
cost of simulation quantum physics [20, 18, 21, 22] and to onset of quantum
chaos [23, 24]. While various nonstabilizerness monotones have been proposed
[25, 17, 26, 27, 28, 29, 30, 31], their geometrical nature requires
minimization over large spaces resulting in prohibitive computational costs
already for few-qubit systems. In fact, quantifying nonstabilizerness in many
qubit systems remains a major challenge. Stabilizer entropy provides a
(generally non-monotone [32]) measure of magic that is calculable for few-
qubit states [33, 34, 35, 36] and that is related to the flatness of the
entanglement spectrum [37]. Recent works demonstrated that the stabilizer
entropy is efficiently computable for many-body matrix product states [38, 32,
39] while being experimentally measurable in present quantum devices via
randomized measurement protocols or Bell measurements [40, 41]. These results
allow for studying static and dynamical properties of nonstabilizerness in
one-dimensional many-body systems.

A seemingly unrelated quantity is the inverse participation ratio and the
participation entropy of many-body wave functions. Originally developed in the
context of Anderson localization [42, 43, 44, 45, 46, 47, 48], these
quantities measure the spread of the many-body states on a certain basis of
the Hilbert space. Studies in ground states of many-body systems [49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59], in the problem of many-body localization [60,
61, 62, 63, 64, 45, 65, 66, 67, 68, 69, 70, 71, 72], and monitored unitary
dynamics [73], demonstrated that most quantum states have multifractal
features [74], i.e., they are described by different fractal properties in
different geometrical regions.

This work demonstrates that nonstabilizerness is directly encoded in the
structure of wave functions. Employing methods inspired by recent works [37],
we show that a state has a flat participation distribution along its Clifford
orbit if and only if it is a stabilizer state. Conversely, a state with non-
flat participation distribution, for instance, a multifractal state, is
nonstabilizer, i.e., it has magic. After reviewing the key concepts of
interest, we introduce a multifractal flatness, a measure of the flatness of
participation distributions that we prove is a simple function of the
stabilizer entropy. This allows us to demonstrate that the participation
entropy in the computational basis provides a useful magic witness that is
amenable to computational methods. We illustrate the relation between magic
and participation entropy on examples of a single qubit, many-body product
states, and Haar random states. Finally, we demonstrate that the magic
quantifier introduced in this manuscript is measurable in noisy intermediate-
scale quantum devices [75, 76, 77].

##  II Stabilizer and participation entropies

We consider a system of $N$ qubits with Hilbert space dimension $d=2^{N}$, and
denote $\\{\sigma^{\alpha}\\}_{\alpha=0,1,2,3}$ the Pauli matrices
($\sigma^{0}=\openone$), $|0\rangle,|1\rangle$ the local computational basis
of $\sigma^{3}$ and $\mathcal{P}_{N}$ the set of all $N$-qubit Pauli strings.
The Clifford group $\mathcal{C}_{N}$ is the subset of unitary operations that
map a Pauli string into a _single_ Pauli string. The stabilizer entropy is
defined for a pure normalized state $|\Psi\rangle$ as [33]

$M_{q}(|\Psi\rangle)=\frac{1}{1-q}\log_{2}\sum_{P\in\mathcal{P}_{N}}\frac{(\langle\Psi|P|\Psi\rangle)^{2q}}{d}.$
(1)

It is a measure of nonstabilizerness as: (i) $M_{q}(|\Psi\rangle)\geq 0$, with
the equality holding if and only if the state is a stabilizer
$|\Psi\rangle\in\mathrm{STAB}$, (ii) it is invariant under Clifford
conjugation $M_{q}(C|\Psi\rangle)=M_{q}(|\Psi\rangle)$ for any
$C\in\mathcal{C}_{N}$, (iii) it is additive
$M_{q}(|\Psi\rangle\otimes|\Phi\rangle)=M_{q}(|\Psi\rangle)+M_{q}(|\Phi\rangle)$
[33]. Despite the fact that the stabilizer entropy is not a magic monotone for
generic Rényi index $q$ [32], it is computationally amenable compared to the
magic robustness and stabilizer fidelity [20]. We aim to show that the
stabilizer entropy $M_{2}(|\Psi\rangle)$ is related to the structure of the
wave function quantified by the participation entropy.

Given a pure state $|\Psi\rangle$, the participation entropy is

$S_{q}(\ket{\Psi})=(1-q)^{-1}\log_{2}I_{q}(|\Psi\rangle),$ (2)

where
${I_{q}(|\Psi\rangle)=\sum_{\vec{\sigma}\in\mathcal{B}}|\langle\vec{\sigma}|\Psi\rangle|^{2q}}$
is the inverse participation ratio that involves a sum over the computational
basis $\mathcal{B}=\\{|\vec{\sigma}\rangle,\ \sigma_{i}=0,1,\ i=1,\dots,N\\}$.
The participation entropy (2) quantifies the spreading of the state
$|\Psi\rangle$ over the basis $\mathcal{B}$. Conventionally, in condensed
matter settings, the system size dependence of the participation entropy is
parametrized as [78, 61, 48]

$S_{q}=D_{q}N+c_{q},$ (3)

where $D_{q}$ is the multifractal dimension and $c_{q}$ is a sub-leading term.
In the field of Anderson and many-body localization transition, it is
customary to denote the state $|\Psi\rangle$ localized (fully extended) when
$D_{q}=0$ ($D_{q}=1$). The intermediate regimes, for which $0<D_{q}<1$ and
$D_{q}$ depend non-trivially on the Rényi index $q$ are said to be
multifractal. Here we are concerned with the participation flatness, occurring
when $S_{q_{1}}(\ket{\Psi})=S_{q_{2}}(\ket{\Psi})$ for all $q_{1},q_{2}>0$.
(This condition is equivalent to
$p(\vec{\sigma})\equiv|\langle\vec{\sigma}|\Psi\rangle|^{2}$ being uniform in
its domain, hence justifying the name). Participation flatness implies the
absence of wave function multifractality. However, we remark that the two
notions are not equivalent: participation flatness is defined for a given
state at a fixed system size $N$. Instead, the multifractality is
intrinsically related to the $N$ dependence of the participation entropy (3).
In particular, as we will discuss in the following, a state can be fully-
extended ($D_{q}=1$) or localized ($D_{q}=0$) and still not have a flat
participation distribution due to the non-trivial $q$-dependence of $c_{q}$.

Finally, let us note that the stabilizer entropy (1) has an intrinsic basis
dependence in the choice of Pauli strings as generators of the operator space.
This basis dependence is a feature of the nonstabilizerness and different
frames yield different results [26]. Similarly, the participation entropy (2),
and hence also the participation flatness, depends explicitly on the choice of
many-body basis $\mathcal{B}$. Therefore, we choose the Pauli strings to
quantify the degree of nonstabilizerness and consider the computational basis
to probe the many-body wave function structure.

##  III Nonstabilizerness and multifractality

Stabilizer states are not multifractal and always possess a flat participation
distribution, as shown in [73], which also provides an efficient way to
compute participation entropy of stabilizer state using its tableaux
representation [10, 79]. As we argue in the following, the converse statement
is also true, leading to a novel characterization of magic in terms of the
participation entropies of the many-body wave function.

Figure 1: Pictorial representation of the replica picture. Each _page_ is a
folding of $C|\Psi\rangle$ and its adjoint. Applying $\Lambda^{(q)}_{k}$ on
each site imply a sum over the same physical index and results in _book_
boundary conditions.

Before stating the main results of this work, we introduce the _multifractal
flatness_
$\mathcal{F}(|\Psi\rangle)=I_{3}(|\Psi\rangle)-I_{2}^{2}(|\Psi\rangle)$, which
is a measure of the participation flatness. By concavity of the participation
entropy and using Jensen’s inequality, it is easy to see that
$\mathcal{F}(|\Psi\rangle)\geq 0$, with the equality holding if and only if
the participation distribution
$p_{\vec{\sigma}}\equiv|\langle\vec{\sigma}|\Psi\rangle|^{2}$ is flat or,
equivalently, when $S_{q_{1}}(\ket{\Psi})=S_{q_{2}}(\ket{\Psi})$ for all
$q_{1},q_{2}>0$.

######  Theorem 1.

The average of the multifractal flatness $\mathcal{F}$ over the Clifford orbit
$\mathcal{C}_{\Psi}\equiv\\{C|\Psi\rangle\ |C\in\mathcal{C}_{N}\\}$ is a
measure of nonstabilizerness, with

$\overline{\mathcal{F}}(|\Psi\rangle)\equiv\mathbb{E}_{C\in\mathcal{C}_{N}}[\mathcal{F}(C|\Psi\rangle)]=\frac{2(1-2^{-M_{2}(|\Psi\rangle)})}{(d+1)(d+2)}.$
(4)

Let us highlight an immediate consequence that conceptually bridges the
participation flatness to that of nonstabilizerness.

######  Corollary 1.

A state is a stabilizer if and only if every element of its Clifford orbit is
participation flat. Vice versa, a state has magic if and only if it exists
$C\in\mathcal{C}_{N}$ for which $C|\Psi\rangle$ is not participation flat.

As already mentioned, a stabilizer state has flat participation distribution
[73]. In this case, Eq. (4) is trivially satisfied. Conversely, we must prove
the flatness of the participation distribution for all $\mathcal{C}\ket{\psi}$
states, or in other words, that there is no $C\in\mathcal{C}_{N}$ for which
$S_{q}(C|\Psi\rangle)$ is $q$ dependent. From the positive monotonicity of
$\mathcal{F}$, this fact is equivalent in proving Eq. (4) holds for generic
states. To this end, let us rephrase $I_{q}^{r}$ in the replica formalism as

$I_{q}^{r}=\mathrm{tr}\left[|\Phi^{(rq)}\rangle\langle\Phi^{(rq)}|\left(\Lambda_{1}^{(q)}\right)^{\otimes
r}\cdots\left(\Lambda_{N}^{(q)}\right)^{\otimes r}\right],$ (5)

where $|\Phi^{(rq)}\rangle=|\Psi\rangle^{\otimes rq}$ is the replica state and
$\Lambda^{(q)}_{k}=(|0\rangle\langle 0|)^{q}+(|1\rangle\langle 1|)^{q}$ are
the operators enforcing the book replica boundary condition [50, 73], cf. Fig.
1. From Eq. (5) we see that $I_{q}^{r}$ has a permutation invariance
$\mathcal{S}_{q}^{\otimes r}$ over the replica space. This fact will allow for
simplifications in evaluating the orbit average in Eq. (4). First, let us
compute the $q=3,r=1$ term. Using the 3-design property of the Clifford group
[80, 81] we have

$\mathbb{E}_{\mathcal{C}_{N}}[C^{\otimes
3}|\Phi^{(3)}\rangle\langle\Phi^{(3)}|(C^{\dagger})^{\otimes
3}]=\frac{6}{d(d+1)(d+2)}\Pi_{[3]},$ (6)

with $\Pi_{[k]}=\sum_{\pi\in\mathcal{S}_{k}}U_{\pi}/k!$ the projector onto the
symmetric permutation of $k$ elements. It follows that
$\mathbb{E}_{C\in\mathcal{C}_{N}}[I_{3}(C|\Psi\rangle)]={6}/{((d+1)(d+2))}$.
Instead, the $q=2,r=2$ require the average of the $|\Phi^{(4)}\rangle$. This
object is less trivial and needs insights into the commutant of the Clifford
group, cf. Ref. [82, 83]. We have

$\mathbb{E}_{\mathcal{C}_{N}}[C^{\otimes
4}|\Phi^{(4)}\rangle\langle\Phi^{(\otimes
4)}|(C^{\dagger})^{4}]=\beta_{+}(|\Psi\rangle)\Pi_{+}+\beta_{-}(|\Psi\rangle)\Pi_{-},$
(7)

with $\Pi_{+}=\Pi_{N,4}\Pi_{[4]}$, $\Pi_{-}=(1-\Pi_{N,4})\Pi_{[4]}$ two
projectors and $\Pi_{N,4}=\sum_{P\in\mathcal{P}_{N}}P^{\otimes 4}/d^{2}$,
while the coefficients are defined as
$\beta_{+}=6||\Xi(|\Psi\rangle)||^{2}_{2}/((d+1)(d+2))$ and
$\beta_{-}=24(1-||\Xi(|\Psi\rangle)||^{2}_{2})/((d^{2}-1)(d+2)(d+4))$, with
$||\Xi(|\Psi\rangle)||^{2}_{2}=\sum_{P\in\mathcal{P}_{N}}\bra{\Psi}P\ket{\Psi}^{4}/d^{2}$.
A simple computation imposing the book boundary condition in the replica space
[73] gives

$\mathbb{E}_{C\in\mathcal{C}_{N}}[I_{2}^{2}(C|\Psi\rangle)]=\frac{4-2d||\Xi(|\Psi\rangle)||^{2}_{2}}{(d+1)(d+2)}.$
(8)

Combining these results, it follows Eq. (4) and the proof’s conclusion for
$N\geq 3$. The case $N=1,2$ have a rank deficient Clifford group commutant
[84, 85] and must be evaluated separately. Nevertheless, one can show that the
final result Eq. (4) holds, recalling $d=2^{N}$. $\square$

A few remarks are in order here. At a practical level, the right-hand side of
Eq. (4) can be evaluated via Monte Carlo sampling over the Clifford group
$\mathcal{C}_{N}$, i.e.,
$\overline{\mathcal{F}}_{\mathrm{mc}}\equiv\sum_{C}\mathcal{F}(C|\Psi\rangle)/\mathcal{N}_{\mathrm{real}}$
with $\mathcal{N}_{\mathrm{real}}$ the number of random choices of Clifford
unitaries $C$. We recall that random drawing from the Clifford group is
efficiently implementable [86], and we present an explicit example of the
Monte Carlo estimation in the next sections.

While our theorem (4) links the multifractal flatness averaged over the whole
Clifford orbit $\mathcal{C}_{\Psi}$ to the stabilizer entropy $M_{2}$, we
remark that $\mathcal{F}(C^{*}|\Psi\rangle)$ for a fixed
$C^{*}\in\mathcal{C}_{N}$ is a convenient witness of magic, being
computationally cheap compared to (1). Indeed, $\mathcal{F}(C|\Psi\rangle)\geq
0$ for any $C$ and $\ket{\Psi}$ and is non-zero only if the state has magic.
Hence, if $\mathcal{F}(C^{*}|\Psi\rangle)>0$ we certify that the state has
some amount of magic.

Additionally, the multifractal flatness $\mathcal{F}$ can be extended to the
class
$\mathcal{F}(|\Psi\rangle;q,m)=I_{q}(|\Psi\rangle)-I_{(k-1+q)/m}^{m}(|\Psi\rangle)$,
with $m,q>0$. While we conjecture a result similar to (4) holds, the rapidly
growing commutant dimension of the Clifford group hinders analytical insights
already for $q,k=5$ [87]. Nevertheless, these terms would scale $O(d^{-3})$,
therefore being more difficult to resolve than (4) for practical purposes.
Lastly, we note an expression similar to (4) resembles analogous results for
the entanglement spectrum [37], where an additional dependence on the
entanglement bipartition is present.

In the remaining, we illustrate the relation between magic and the
multifractal flatness $\mathcal{F}$ for several examples of quantum states. In
particular, we show that resolving $\overline{\mathcal{F}}$ generally requires
exponential resources in system size for generic systems, as shown in a
numerical example below. Yet, we show that $\overline{\mathcal{F}}$ can be
estimated in current noisy intermediate-scale quantum devices, provided the
fidelity is preserved.

##  IV Multifractality and magic: examples

###  IV.1 Single qubit

Let us start with the intuitive example of a single qubit ($N=1$) and
highlight the relationship between stabilizer and participation entropies of
the quantum state

$|\Psi_{1}\rangle=\cos\left(\frac{\theta}{2}\right)|0\rangle+e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle,$
(9)

where $\theta,\phi$ are real parameters. The stabilizer entropy (1) of this
state is given by

$\begin{split}M_{q}(|\Psi_{1}\rangle)&=\frac{1}{1-q}\log_{2}\left(1+\cos^{2q}(\theta)+\Omega(\theta,\phi)\right)\\\
\Omega(\theta,\phi)&=(\sin(\theta)\sin(\phi))^{2q}+(\sin(\theta)\cos(\phi))^{2q}\end{split}$
(10)

which is non-vanishing for a generic choice of $\theta$ and $\phi$.
Calculation of the participation entropy (2) yields
$S_{q}(\ket{\Psi_{1}})=\log_{2}[\cos^{2q}(\theta/2)+\sin^{2q}(\theta/2)]/(1-q)$,
which depends non-trivially on $q$ for a generic parameter choice, providing
an example of a typical scenario in which a non-stabilizer state does not have
a flat participation distribution. However, for a fine-tuned choice
$\theta=\pi/2$, the participation entropy is equal to unity, independently of
the value of $q$, hence $\mathcal{F}(\ket{\Psi_{1}})=0$. Still, the state
$\ket{\Psi_{1}}$ has a nontrivial magic for $\theta=\pi/2$ and a generic value
of $\phi$. This shows that flatness of participation distribution in a
selected point of the Clifford orbit $\mathcal{C}_{\Psi_{1}}$ does not imply
that the state is a stabilizer state, illustrating the importance of the
average over the whole Clifford orbit in (4). Indeed, for a single choice of
$C\in\mathcal{C}_{N}$, $\mathcal{F}$ is only a witness of magic: if it is non-
zero, we know that the state has magic, but the converse is not true. If we
act with a Hadamard gate $H\in\mathcal{C}_{1}$ on the state $\ket{\Psi_{1}}$,
we discover a nontrivial $q$ dependency
$S_{q}(H\ket{\Psi_{1}})=(\log_{2}[((1+\sin\theta\cos\phi)/2)^{2q}+((1-\sin\theta\cos\phi)/2)^{2q}])/(1-q)$
for generic values of $\phi$ even at $\theta=\pi/2$, consistently with the
nonstabilizerness of the state $\ket{\Psi_{1}}$ revealed by the non-vanishing
value of $M_{q}(\Psi)$ (10).

###  IV.2 Many-qubit product states

We consider now a state $\ket{\Psi_{N}}$ of $N$ qubits which is a product
state of the single qubit states $\ket{\Psi_{1}}$ (9), i.e.,
$\ket{\Psi_{N}}=\ket{\Psi_{1}}^{\otimes N}$. This state is not entangled but
has an extensive stabilizer entropy
$M_{q}(\ket{\Psi_{N}})=NM_{q}(\ket{\Psi_{1}})$, where $M_{q}(\ket{\Psi_{1}})$
is given by (10), i.e. $\ket{\Psi_{N}}$ is a magic state. Moreover, it is easy
to verify that $S_{q}(\ket{\Psi_{N}})=NS_{q}(\ket{\Psi_{1}})$, i.e. for a
generic choice of $\theta$ and $\phi$, the state $M_{q}(\ket{\Psi_{N}})$ is
multifractal with the multifractal dimension $D_{q}=S_{q}(\ket{\Psi_{1}})$ and
the sub-leading term $c_{q}=0$. This is an example of a situation in which the
presence of multifractality at a given point of the stabilizer orbit of
$\ket{\Psi_{N}}$ implies that the state is magic $\ket{\Psi_{N}}$. However,
this is not the case for the fine-tuned choice $\theta=\pi/2$, for which the
multifractality flatness vanishes, $\mathcal{F}(\ket{\Psi_{N}})=0$, and the
state $\ket{\Psi_{N}}$ is fully extended, $D_{q}=1$, while still being magic
for a generic value of $\phi$. Similarly as for the single qubit, the action
of the Hadamard gates on $\ket{\Psi_{N}}$ yields a state with participation
distribution that is not flat, revealing the nonstabilizerness of the state.

###  IV.3 Random Haar states

We consider a random state $|\Psi\rangle$ and quantify its magic by
calculating the stabilizer entropy via (4). Any such state is obtainable for a
(random) Haar unitary $U\in\mathcal{U}(d)$ acting on a reference state
$|\Psi_{0}\rangle$. For convenience and without loss of generality,
$|\Psi_{0}\rangle=|0\rangle^{\otimes N}$. We begin by computing the Haar
average
$\mathbb{E}_{U\in\mathcal{U}(d)}\left[\overline{\mathcal{F}[U|\Psi\rangle]}\right]$.
As we shall argue, in the limit of large systems $N\gg 1$,
$\overline{\mathcal{F}}$ is self-averaging for any fixed realization
$|\Psi\rangle=U|\Psi_{0}\rangle$.

First, let us note that
$\mathbb{E}_{U\in\mathcal{U}(d)}[\overline{\mathcal{F}}[U|\Psi_{0}\rangle]]=\mathbb{E}_{U\in\mathcal{U}(d)}\mathcal{F}[U|\Psi_{0}\rangle]$.
This fact follows from the Clifford group being a subgroup of the unitary
ensemble $\mathcal{C}_{N}\subset\mathcal{U}(d)$ and by the unitary invariance
of the Haar measure. Recalling that
$\mathbb{E}_{U\in\mathcal{U}(d)}\left[\left(U|\Psi_{0}\rangle\langle\Psi_{0}|U^{\dagger}\right)^{\otimes
k}\right]=k!\Pi_{[k]}/(d(d+1)\dots(d+k-1))$ and using the book boundary
conditions in replica space (cf. Fig. 1), it follows that

$\begin{split}\mathbb{E}_{U\in\mathcal{U}(d)}&\left[I_{q}^{r}(U|\Psi_{0}\rangle)\right]=\\\
&\frac{\sum_{\lambda\vdash
r}d(d-1)\dots(d-K_{\lambda}+1)|\mathcal{S}^{(q)}_{\lambda}|a_{\lambda}}{d(d+1)\dots(d+rq-1)}.\end{split}$
(11)

In the above formula Eq. (11),
$\lambda=\\{(\lambda_{k},n_{k})\\}_{k=1,\dots,K_{\lambda}}$ runs over the
partitions of $r=\sum_{k=1}^{K_{\lambda}}n_{k}\lambda_{k}$, $|A|$ counts the
number of elements in $A$,
${\mathcal{S}^{(q)}_{\lambda}=\bigotimes_{k=1}^{K_{\lambda}}\left(\mathcal{S}_{\lambda_{k}q}^{\otimes
n_{k}}\right)}$ is the symmetry group of the given partition, and the
remaining coefficients are $a_{\lambda}$ are given by

$a_{\lambda}=\frac{r!}{\left[\prod_{k=1}^{K_{\lambda}}\left(\lambda_{k}!\right)^{n_{k}}\right]\left[\prod_{k=1}^{K_{\lambda}}(n_{k}!)\right]}.$
(12)

Specializing to $q=3,r=1$ and $q=r=2$ we have

$\overline{\mathcal{F}}^{U}\equiv\mathbb{E}_{U\in\mathcal{U}(d)}[\overline{\mathcal{F}}[U|\Psi_{0}\rangle]]=\frac{2(d-1)}{(d+1)(d+2)(d+3)}.$
(13)

In particular, we recover the stabilizer entropy computation in Ref. [33],
with
$\mathbb{E}_{U\in\mathcal{U}(d)}\left[2^{-M_{2}[U|\Psi_{0}\rangle]}\right]=4/(d+3)$.
For a realization $U\in\mathcal{U}(d)$, the state
$|\Psi\rangle=U|\Psi_{0}\rangle$ has
$\overline{\mathcal{F}}\simeq\overline{\mathcal{F}}^{U}$, with an
exponentially small error in system size $N$ as a consequence of quantum
typicality [88, 89]. Indeed, the standard deviation of
$\overline{\mathcal{F}}$ over the Haar ensemble is at leading order in $1/d$
is given by
$\mathrm{std}_{U\in\mathcal{U}(d)}(\overline{\mathcal{F}}[U|\Psi_{0}\rangle])\simeq
2\sqrt{34}/d^{5/2}+O\left(d^{-6}\right)$ 111The exact expression is
$\displaystyle\mathrm{std}_{U\in\mathcal{U}(d)}(\overline{F}[U|\Psi_{0}\rangle])^{2}=$
$\displaystyle\frac{8\left(17d^{5}+42d^{4}-106d^{3}-72d^{2}+449d-330\right)}{(d+1)^{2}(d+2)^{2}(d+3)^{2}(d+4)(d+5)(d+6)(d+7)}.$
This formula requires evaluating generic correlators
$\mathbb{E}_{U\in\mathcal{U}(d)}I_{q_{1}}^{r_{1}}[U|\Psi_{0}\rangle]I_{q_{2}}^{r_{2}}[U|\Psi_{0}\rangle]$
and result in expression similar to Eq. (11) but more involved. .

Figure 2: Determining stabilizer entropy with multifractal flatness
$\mathcal{F}$. Upper panel: statistical error $\sigma(M_{2})$ of stabilizer
entropy determined from $\mathcal{N}_{\mathrm{real}}$ measurements of
$\mathcal{F}$ for a random Haar state; the asymptotic
$\mathcal{N}^{-1/2}_{\mathrm{real}}$ scaling is denoted by a dashed line; data
(color online) shown for number of qubits $N=2,3,\ldots,10$; the inset shows
the number of measurements $\mathcal{N}_{0.1}$ needed to achieve error
$\sigma(M_{2})=0.1$ as function of $N$. Bottom panel: the same, but for the
product state $\ket{\Psi_{N}}$ with $\phi=\pi/2$ and $\theta=\pi/2$; the
observed scaling $\mathcal{N}_{0.1}\sim 2^{cN}$ is slower than for random Haar
state (since $c\approx 0.7$), but still exponential in $N$.

The random Haar states constitute an ensemble of magic states with stabilizer
entropy $M_{2}[U|\Psi_{0}\rangle]=N-2+O(1/d)$, which is close to the maximal
one $M_{2}=N$. Their wave function is fully extended over the many-body basis,
as shown by the participation entropy
$S_{q}(U|\Psi_{0}\rangle)=N+(1-q)^{-1}\log_{2}\Gamma(1+q)$, i.e., the
multifractal dimension is $D_{q}=1$. Hence, according to our definition in
this work, the random Haar states are not multifractal. At the same time, the
wave function of a random Haar state does not have a flat participation
distribution. The multifractality flatness is non-vanishing,
$\mathcal{F}(U|\Psi_{0}\rangle)>0$, and $S_{q}(U|\Psi_{0}\rangle)$ depends
non-trivially on the index $q$ via the sub-leading term $c_{q}$ in its system
size dependence, consistently with their nonstabilizerness.

Figure 3: Digital quantum simulation of the magic measure
$\overline{\mathcal{F}}$ using the IBM ibmq_oslo transmon quantum device. For
the proof-of-principle setup considered,
$\overline{\mathcal{F}}^{\mathrm{ex}}$ is the exact nonstabilizerness measure,
while $\overline{\mathcal{F}}^{\mathrm{dig}}$ is the Monte Carlo estimate over
$\overline{\mathcal{F}}_{\mathrm{real}}=80$ random Clifford gates $C$. While
the features in $\theta$ are qualitatively captured, we notice a systematic
shift due to the decoherence and leakage effects of the IBM machine. This is
shown by $\overline{\mathcal{F}}^{\mathrm{sim}}$ obtained using the simulation
model for the machine by the provider. The latter partially accounts for the
noise sources and captures the digital simulation data within one error bar.

##  V Probing magic via multifractal flatness

###  V.1 Numerical example

Here, we show that sampling over the Clifford group in Eq. (4) can be
performed via action of a circuit consisting of local Clifford gates acting on
the state of interest $\ket{\Psi}$. We consider a system of $N\in[2,10]$
qubits, and, as the initial state $\ket{\Psi}$, we take either the random Haar
state $U\ket{\Psi_{0}}$, or a product state $\ket{\Psi_{1}}^{\otimes N}$ of
$N$ single qubit states (9) with $\phi=\pi/4$ and $\theta=\pi/2$. We assume
that the qubits form a $1$-dimensional lattice, select random site $i$, and
act with a random $2$-qubit Clifford gate $U_{2}$ [86, 79] on sites $i$ and
$i+1$, imposing periodic boundary conditions. After each action of the
Clifford gate, we compute the multifractal flatness $\mathcal{F}$ of the
obtained state. By repeating this process, we obtain results
$\mathcal{F}_{1},\mathcal{F}_{2},\dots,\mathcal{F}_{\mathcal{N}_{\mathrm{real}}}$
which we average to calculate an estimator
$\hat{\mathcal{F}}=\sum_{i}\mathcal{F}_{1}/\mathcal{N}_{\mathrm{real}}$ of the
mean multifractal flatness $\overline{\mathcal{F}}(\ket{\Psi})$ over the
Clifford orbit $\mathcal{C}_{\Psi}$.

Reverting Eq. (4), we find that

$M_{2}(\ket{\Psi})=-\log_{2}\left[1-\frac{(d+1)(d+2)}{2}\hat{\mathcal{F}}\right].$
(14)

This allows us to calculate the stabilizer entropy $M_{2}(\ket{\Psi})$ using
$\hat{\mathcal{F}}$, but also shows that the statistical uncertainty
$\sigma(\mathcal{F})$ of the multifractal flatness has to be multiplied by a
factor exponential in the number of qubits $N$ to yield the statistical
uncertainty $\sigma(M_{2})$ of $M_{2}$. For instance, the results from the
preceding Section for random Haar states imply that $\sigma(\mathcal{F})\sim
d^{5/2}/\sqrt{\mathcal{N}}$, which leads to
$\sigma(M_{2})\sim\sqrt{d/\mathcal{N}}$. This, in turn, indicates that in
order to keep the same uncertainty of $M_{2}$ with increasing system size, we
need to increase the number $\mathcal{N}$ of the samples proportionally to the
Hilbert space dimension $d$. Numerical results shown in the upper panel of
Fig. 2 confirm this expectation. Keeping the same uncertainty of the
stabilizer entropy with $\ket{\Psi_{1}}^{\otimes N}$ as the initial state
requires number of samples that is smaller but still exponential in the number
of qubits: $\mathcal{N}\sim 2^{cN}$ (since $c\approx 0.7$).

Results of this numerical experiment provide a quantitative confirmation of
our analytical calculations that relate the stabilizer entropy to the
multifractal flatness of the many-body wave function. Moreover, the action of
local $2$-body Clifford gates is sufficient to probe the Clifford orbit
$\mathcal{C}_{\Psi}$ and obtain an accurate estimation of the multifractal
flatness $\overline{\mathcal{F}}(\ket{\Psi})$. Nevertheless, the sensitivity
of the stabilizer entropy to the statistical uncertainty of
$\overline{\mathcal{F}}(\ket{\Psi})$ makes this method of calculation of
$M_{2}(\ket{\Psi})$ impractical – the computational resources needed to
compute $M_{2}(\ket{\Psi})$ for larger numbers of qubits with a given accuracy
are comparable or larger than a direct calculation of $M_{2}(\ket{\Psi})$
according to the definition (1).

###  V.2 Two-qubit system on quantum device

Lastly, we demonstrate that the participation entropy and
$\overline{\mathcal{F}}$ are observable in current quantum devices. As a
simple proof of principle, we consider a $N=2$ qubit system in
$|\Psi_{0}\rangle=|00\rangle$ with the rotation
$R_{\mathrm{XX}}(\theta)=\exp[-i\theta/2\sigma^{1}\otimes\sigma^{1}]$, and
simulate the system in the IBM transmon quantum device [91]. For specific
angles $\theta$ the rotated state $|\Psi_{\theta}\rangle\equiv
R_{\mathrm{XX}}(\theta)|\Psi_{0}\rangle$ is a stabilizer, otherwise it
contains magic. We estimate
$\overline{\mathcal{F}}(|\Psi_{\theta}\rangle)=\sum_{C_{i}}\mathcal{F}(C_{i}|\Psi_{\theta}\rangle)/\mathcal{N}_{\mathrm{real}}$
in the ibmq_oslo device using $\mathcal{N}_{\mathrm{real}}=80$ realizations of
random Clifford gates $C_{i}$. Our results are shown in Fig. 3.

Compared to the exact value, given by
$\mathcal{F}^{\mathrm{ex}}=\sin^{2}(2\theta)/60$ (orange line), the digital
simulations (red dots) present a systematic shift in values, while
qualitatively describing key features of the system. The discrepancy is due to
decoherence and leakage. Indeed, including partially including these
contributions in the model emulating the device (blue line), the digital
simulation data agrees with the numerical prediction within one error bar.
With higher fidelity gates and more controlled quantum gates, we expect the
decoherence effect to be reduced and $\overline{\mathcal{F}}^{\mathrm{dig}}$
approaching the exact values $\overline{\mathcal{F}}^{\mathrm{ex}}$.

##  VI Conclusion

This work puts forward a correspondence between the concept of wave function
multifractality and nonstabilizerness. Specifically, we introduce multifractal
flatness, a combination of inverse participation ratios that quantifies the
flatness of the wave function probability distribution. We show that
multifractal flatness is a witness of nonstabilizerness and translates to a
measure of magic when the average over the Clifford orbit is considered. We
illustrate the relationship between multifractality and nonstabilizerness on a
few examples of quantum states, for instance recovering known results [33]
about random Haar states.

The connection put forward in this work has practical implications, as we show
by computing the stabilizer entropies by estimating the multifractal flatness
with a repeated action of a local Clifford circuit. As a proof of principle,
we demonstrate that magic, as estimated by the multifractal flatness
$\overline{\mathcal{F}}$, is detectable in current quantum devices.

Recent works [24, 92] revealed a tight relationship between quantum chaos and
nonstabilizerness. It would be interesting to investigate this link through
the lens of multifractality. For instance, in semiclassical quantum systems,
the generalized Lyapunov exponents present multifractal features [93, 94].
Furthermore, models of maximally chaotic quantum many-body systems have been
shown to display rich multifractal behavior [95, 96, 97]. Additionally, it
would be interesting to reveal the equilibrium and out-of-equilibrium
nonstabilizerness of archetypal condensed matter models [98, 39, 35].

Acknowledgments.— We thank L. Piroli, G. Fux, and R. Fazio for enlightening
discussions. X.T. is indebted to S. Pappalardi for consultations on magic,
multifractal, and to G. M. Andolina for discussions. We acknowledge the
workshop ”Dynamical Foundation of Many-Body Quantum Chaos” at Institute Pascal
(Orsay, France) for hosting us while finalizing this manuscript’s writing.
X.T. and M. S. acknowledge support from the ANR grant “NonEQuMat”
(ANR-19-CE47-0001). P.S. acknowledges support from: ERC AdG NOQIA; Ministerio
de Ciencia y Innovation Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I “Retos Colaboración” QUSPIN
RTC2019-007196-7); MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya; Fundació
Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund
FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-1-0013); EU Horizon 2020
FET-OPEN OPTOlogic (Grant No 899794); EU Horizon Europe Program (Grant
Agreement 101080086 — NeQST), National Science Centre, Poland (Symfonia Grant
No. 2016/20/W/ST4/00314); ICFO Internal “QuantumGaudi” project; European
Union’s Horizon 2020 research and innovation program under the Marie-
Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No 847648 (“La
Caixa” Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013,
LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). Views and
opinions expressed in this work are, however, those of the author(s) only and
do not necessarily reflect those of the European Union, European Climate,
Infrastructure and Environment Executive Agency (CINEA), nor any other
granting authority. Neither the European Union nor any granting authority can
be held responsible for them.

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