††thanks: The first three authors contributed equally to this work.††thanks:
The first three authors contributed equally to this work.††thanks: The first
three authors contributed equally to this work.

# Trainability barriers and opportunities in quantum generative modeling

Manuel S. Rudolph  Institute of Physics, Ecole Polytechnique Fédérale de
Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Sacha Lerch  Institute of
Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland  Supanut Thanasilp  Institute of Physics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Chula Intelligent
and Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn
University, Bangkok, Thailand, 10330  Oriel Kiss  European Organization for
Nuclear Research (CERN), Geneva 1211, Switzerland  Department of Nuclear and
Particle Physics, University of Geneva, Geneva 1211, Switzerland  Sofia
Vallecorsa  European Organization for Nuclear Research (CERN), Geneva 1211,
Switzerland  Michele Grossi  European Organization for Nuclear Research
(CERN), Geneva 1211, Switzerland  Zoë Holmes  Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

(September 1, 2024)

###### Abstract

Quantum generative models, in providing inherently efficient sampling
strategies, show promise for achieving a near-term advantage on quantum
hardware. Nonetheless, important questions remain regarding their scalability.
In this work, we investigate the barriers to the trainability of quantum
generative models posed by barren plateaus and exponential loss concentration.
We explore the interplay between explicit and implicit models and losses, and
show that using implicit generative models (such as quantum circuit-based
models) with explicit losses (such as the KL divergence) leads to a new
flavour of barren plateau. In contrast, the Maximum Mean Discrepancy (MMD),
which is a popular example of an implicit loss, can be viewed as the
expectation value of an observable that is either low-bodied and trainable, or
global and untrainable depending on the choice of kernel. However, in
parallel, we highlight that the low-bodied losses required for trainability
cannot in general distinguish high-order correlations, leading to a
fundamental tension between exponential concentration and the emergence of
spurious minima. We further propose a new local quantum fidelity-type loss
which, by leveraging quantum circuits to estimate the quality of the encoded
distribution, is both faithful and enjoys trainability guarantees. Finally, we
compare the performance of different loss functions for modelling real-world
data from the High-Energy-Physics domain and confirm the trends predicted by
our theoretical results.

##  I Introduction

The advent of quantum computing has opened up new avenues for solving
classically intractable problems Harrow and Montanaro (2017); Biamonte  _et
al._ (2017); Huang  _et al._ (2022); Daley  _et al._ (2022). Naturally,
researchers gravitate towards finding the first high-value applications that
could be tackled with near- and mid-term quantum devices Preskill (2018). This
includes not only speed-ups Harrow  _et al._ (2009); Lloyd  _et al._ (2014);
Huang  _et al._ (2021, 2022), but potentially superior memory efficiency
Anschuetz  _et al._ (2022) or concrete qualitative improvements Alcazar  _et
al._ (2020); Gili  _et al._ (2022a). Quantum machine learning (QML) is one of
the domains that attracts this attention Biamonte  _et al._ (2017). Quantum
systems, in being inherently probabilistic, are particularly well suited to
generative modelling tasks Perdomo-Ortiz  _et al._ (2018). Generative models
aim to learn the underlying distribution of a dataset and thereby provide a
means of generating new data samples that are similar to the original data. As
well as providing a naturally efficient means of generating samples, quantum
generative models can provably encode probability distributions that are out
of reach for classical models Coyle  _et al._ (2020); Sweke  _et al._ (2021);
Gao  _et al._ (2022), and have been proposed for various applications, such as
handwritten digits Rudolph  _et al._ (2022a), finance Coyle  _et al._ (2021)
or High-Energy-Physics Delgado and Hamilton (2022).

Despite the excitement surrounding the potential of generative QML, there
remain substantial questions concerning its scalability. This is non-trivial
to assess since implementations are constrained by hardware limitations to
small-scale proof-of-principle problems Hamilton  _et al._ (2019); Leyton-
Ortega  _et al._ (2021); Coyle  _et al._ (2021); Zhu  _et al._ (2019); Rudolph
_et al._ (2022a). Thus analytic results are essential to guide the successful
development of this field. Of particular concern is the growing body of
literature on cost function concentration and barren plateaus McClean  _et
al._ (2018); Arrasmith  _et al._ (2022); Larocca  _et al._ (2022a); Cerezo and
Coles (2021); Arrasmith  _et al._ (2021); Holmes  _et al._ (2021); Zhao and
Gao (2021); Thanasilp  _et al._ (2022), where loss function values can
exponentially concentrate around a fixed value and loss gradients vanish
exponentially with growing problem size. This phenomenon, which exponentially
increases the resources required for training, originates from different
sources McClean  _et al._ (2018); Holmes  _et al._ (2022); Larocca  _et al._
(2022a); Cerezo  _et al._ (2021); Marrero  _et al._ (2021); Patti  _et al._
(2021); Wang  _et al._ (2021a, b); Thanasilp  _et al._ (2021); Leone  _et al._
(2022); Li  _et al._ (2022), and has been studied in a number of architectures
McClean  _et al._ (2018); Larocca  _et al._ (2022a); Napp (2022); Pesah  _et
al._ (2021); Larocca  _et al._ (2022b); Tangpanitanon  _et al._ (2020); Sharma
_et al._ (2022); Rudolph  _et al._ (2021); Thanasilp  _et al._ (2022) as well
as classes of cost function Cerezo  _et al._ (2021); Napp (2022); Thanasilp
_et al._ (2021). However, its impact on quantum generative modelling has thus
far been largely overlooked.

In this work, we provide a thorough study of trainability barriers and
opportunities in quantum generative modelling. Critical to our analysis is the
distinction between explicit and implicit models and losses. Explicit models
provide efficient access directly to the model probabilities, whereas implicit
models only provide samples drawn from their distribution Mohamed and
Lakshminarayanan (2016). Quantum circuit Born machines (QCBMs) Benedetti  _et
al._ (2019), the focus of this work, encode a probability distribution in an
$n$-qubit pure state and thus are a paradigmatic example of an implicit model.
Mirroring the capabilities of the models, explicit losses are those that are
formulated explicitly in terms of the model and target probabilities, whereas
implicit losses compare samples from the model and the training distribution.
The most commonly used explicit loss for quantum generative models is the
Kullbach-Leibler (KL) divergence Csiszar (1975). Other examples include the
Jensen-Shannon divergence (JSD), the total variation distance (TVD) and the
classical fidelity. The Maximum Mean Discrepancy (MMD) Gretton  _et al._
(2012a) on the other hand is one of the leading examples of an implicit loss.

Circuit depth | Explicit loss (pairwise) |  |  Implicit loss
---|---|---|---
(MMD) |  |  |
| Conventional strategy | Quantum strategy |
Product | No |  |
(Corollary 2) | Yes |  |
(Local Quantum Fidelity Cerezo  _et al._ (2021)) | Yes |  |
($\sigma\in\Theta(n)$, Theorem 2) |  |  |
Shallow |  |  |  Yes
($\sigma\in\Theta(n)$, Conjecture 1) |  |  |
Deep |  |  No McClean  _et al._ (2018); Holmes  _et al._ (2022) |  No McClean  _et al._ (2018); Holmes  _et al._ (2022)

Table 1: Summary of our main results. This table summarizes our key analytical
results on the trainability of different loss functions in quantum generative
modelling tasks. Without a strong inductive bias, pairwise explicit losses are
untrainable for all circuit depths with the conventional sampling strategy. A
quantum strategy could be utilised to efficiently estimate the local quantum
fidelity, Eq. (48), which is trainable for a shallow-depth circuit. The MMD
using a classical Gaussian kernel with a linearly-scaled bandwidth
($\sigma\in\Theta(n)$) is expected to be trainable for a shallow-depth
circuits. Note that ‘Yes’ here indicates the existence of regimes with
trainability guarantees- it does not preclude untrainable regimes including,
for example, the use of global quantum fidelity or the MMD with a fixed
bandwidth.

Here we argue that the tension between using an implicit generative model
(providing only samples) with an explicit loss (requiring access to
probabilities) leads to a new flavour of barren plateau. This result
disqualifies all before-mentioned explicit losses, and crucially the KL
divergence, for efficient training of QCBMs without a strong inductive bias
towards the target distribution. In contrast, the MMD as an implicit loss
exhibits more nuanced behaviour and can be either trainable or untrainable. By
viewing the classical MMD loss as the expectation value of a quantum
observable, we show that varying the bandwidth parameter of a Gaussian kernel
interpolates the MMD loss between a loss composed of predominantly global
terms and one composed of low-bodied terms with either exponentially or
polynomially decaying loss variances in the number of qubits. These results
are summarised in Table I.

In parallel, we provide insights into how the globality of a generative loss
affects the types of correlations in a dataset that can reliably be learned.
In particular, we show that a $k$-bodied loss (see Fig. 4) cannot distinguish
between distributions that agree on all $k$-marginals but disagree about
higher-order correlations. Hence we argue that in the context of quantum
generative modelling it is advantageous to train on full-bodied losses, that
is losses containing both low and high-bodied terms, rather than the purely
local losses advocated elsewhere in quantum machine learning. The MMD is then
a promising candidate choice for the training of QCBMs as its bodyness can be
controlled via the bandwidth parameter.

We additionally expand the pool of viable loss functions by proposing a new
local quantum fidelity-type loss which leverages what we call a quantum
strategy for evaluating losses. This is to be contrasted with the conventional
measurement strategy which simply uses samples from the model distribution in
the computational basis. We provide an efficient training protocol using the
local quantum fidelity loss with provable trainability guarantees.

Finally, we support our analysis with a comparison of the performance of the
KL divergence, MMD and local quantum fidelity losses for modelling High-
Energy-Physics (HEP) data. Specifically, we consider electron energy
depositions in the electromagnetic calorimeter (ECAL) part of detectors
involved in a typical proton-proton collision experiment at the LHC. We learn
to generate hits in the detector as black and white images of various sizes,
with up to 16 qubits. We confirm that the properly-tuned MMD and the local
quantum fidelity losses remain trainable using a restrictive shot budget,
while training with the KL divergence becomes increasingly futile.

Figure 1: The generative modelling framework using quantum circuit Born
machines. Given a training dataset $\tilde{P}$ with distribution
$\tilde{p}(\boldsymbol{x})$ over discrete data samples $\boldsymbol{x}$, the
goal of a QCBM is to learn a distribution
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$ which models the real-world
distribution $p(\boldsymbol{x})$ from which the training data itself was
sampled. This is done by tuning the parameters $\boldsymbol{\theta}$ of a
parametrized quantum circuit such that the QCBM minimizes a loss function that
estimates the distance between the model and the training distribution. The
QCBM is an implicit model and can thus in general not be paired with an
explicit loss function, but it may be trainable using an implicit loss. In
contrast to the conventional loss estimation strategy (solid lines) of
generating a set of samples $\tilde{Q}_{\boldsymbol{\theta}}$ and forming an
empirical distribution $\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x})$,
strategies that are ‘more quantum’ (dashed lines) can be employed with the aim
of allowing QCBMs to be trained with loss functions which conventionally
appear explicit.

##  II Framework

The goal of generative modelling is to use samples from a target distribution
$p(\boldsymbol{x})$ to learn a model of $p(\boldsymbol{x})$ which can be used
to generate new samples. More concretely, as sketched in Fig. 1, a generative
model takes as input a training dataset $\tilde{P}$ consisting of
$M=|\tilde{P}|$ samples drawn from the target distribution
$p(\boldsymbol{x})$. This training set can be used to construct the empirical
probability distribution $\tilde{p}(\boldsymbol{x})$ for all samples
$\boldsymbol{x}\in\tilde{P}$. The training dataset, or the training
distribution, is then used to train the variational parameters
$\boldsymbol{\theta}$ of a parameterized probability distribution
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$. If successful, the output of the
algorithm is a set of optimized parameters $\boldsymbol{\theta_{\rm opt}}$
such that the trained model $q_{\boldsymbol{\theta_{\rm
opt}}}(\boldsymbol{x})$ well-approximates the unknown target distribution
$p(\boldsymbol{x})$. The trained model $q_{\boldsymbol{\theta_{\rm
opt}}}(\boldsymbol{x})$ can then be used to generate new and previously unseen
data. For compactness, we use the notation $p$ and $q_{\boldsymbol{\theta}}$
to denote the target and model distributions respectively.

The process of training requires a loss function
$\mathcal{L}(\boldsymbol{\theta})$ which estimates the distance between the
model distribution $q_{\boldsymbol{\theta}}(\boldsymbol{x})$ and the training
distribution $\tilde{p}(\boldsymbol{x})$. For typical choices in loss function
(detailed further in Section II.2), the loss is minimised when the model
parameters $\boldsymbol{\theta}$ are tuned such that the model distribution
perfectly matches the empirical distribution obtained from the training data.
That is, $\mathcal{L}(\boldsymbol{\theta})=0$ if and only if
$q_{\boldsymbol{\theta}}(\boldsymbol{x})=\tilde{p}(\boldsymbol{x})$ over the
entire data space $\mathcal{X}$. Thus, by perfectly minimizing the loss, one
perfectly learns the empirical distribution $\tilde{p}(\boldsymbol{x})$ but
not the true target distribution $p(\boldsymbol{x})$. This scenario is
commonly called overfitting111In contrast, discriminative machine learning
models can be perfectly minimized on the training data and not be overfitted..
To allow for generalization Gili  _et al._ (2022b), whereby the model can
generate novel data with similar properties to the training data, one seeks to
significantly reduce (but not perfectly minimize) the training loss. While
generalization is the end-all goal of generative models, it is not the focus
of this work. Instead, we focus on the training component of the generative
framework, as failing to train also prohibits generalization.

###  II.1 Quantum circuit models

One prototypical quantum generative model is the quantum circuit Born machine
(QCBM) Cheng  _et al._ (2018); Benedetti  _et al._ (2019); Liu and Wang
(2018); Coyle  _et al._ (2020). Owing its name to the Born rule of quantum
mechanics, a QCBM encodes a probability distribution over discrete data (here
bitstrings) in an $n$-qubit pure quantum state that depends on a parameterized
unitary $U(\boldsymbol{\theta})$,

$\displaystyle q_{\boldsymbol{\theta}}(\boldsymbol{x})$
$\displaystyle=|\langle\boldsymbol{x}|U(\boldsymbol{\theta})|\boldsymbol{0}\rangle|^{2}\,.$
(1)

Here $\ket{\boldsymbol{x}}$ is a computational basis state corresponding to a
bitstring $\boldsymbol{x}$ and, without loss of generality, an initial state
can be chosen as $\ket{\boldsymbol{0}}=\ket{0}^{\otimes n}$. We note that
estimating $q_{\boldsymbol{\theta}}(\boldsymbol{x})$ is equivalent to finding
the expectation value of a global projector
$|\boldsymbol{x}\rangle\langle\boldsymbol{x}|$. More fundamentally, QCBMs
enable the encoded distribution to be efficiently sampled simply by measuring
in a chosen computational basis. That is, every measurement of the quantum
state provides an unbiased sample from the encoded distribution (in an ideal
noise-free setting). This is a very desirable property in generative models
that many (classical) generative models do not share with the QCBM. Sampling
techniques for classical generative models are often unreliable and may break
down for certain distributions, as is the case for restricted Boltzmann
machines (RBMs) Smolensky (1986); Hinton (2012). Born machines represent an
effort to create a powerful, flexible and efficient generative model for
classical discrete data, and as well as numerous ‘standard’ digital quantum
implemenations Hamilton  _et al._ (2019); Leyton-Ortega  _et al._ (2021);
Coyle  _et al._ (2021); Zhu  _et al._ (2019); Rudolph  _et al._ (2022a), they
have been widely implemented using tensor networks Han  _et al._ (2018a);
Cheng  _et al._ (2019); Vieijra  _et al._ (2022); Wall  _et al._ (2021),
continuous variable hardware Čepaitė  _et al._ (2022), in a conditional
setting Kiss  _et al._ (2022); Benedetti  _et al._ (2021), with non-
linearities Gili  _et al._ (2023).

An important, but rather subtle, distinction in generative modelling is that
between explicit and implicit generative models Mohamed and Lakshminarayanan
(2016); Jerbi  _et al._ (2023). Explicit generative models are ones that allow
efficient access to the model probability
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$ for any data sample
$\boldsymbol{x}$. Here, “efficient” means that the probabilities can be
computed in a time and memory that are polynomial in the size of the data
samples, i.e., $\mathcal{O}(\operatorname{poly}(n))$ resources. Explicit
(classical) generative models include for example auto-regressive models Van
Den Oord  _et al._ (2016), RNNs Rumelhart and McClelland (1987), tensor
networks without loops (which includes tensor network Born machines) Han  _et
al._ (2018a); Cheng  _et al._ (2019), and many forms of density estimators. In
contrast, implicit models lack this property and instead offer efficient
access to samples from $q_{\boldsymbol{\theta}}(\boldsymbol{x})$, which some
forms of explicit models may struggle with. A popular example of an implicit
generative model are Generative Adversarial Networks (GANs) Goodfellow  _et
al._ (2014) that leverage an implicit training scheme to learn powerful
generators.

In the case of QCBMs implemented on quantum devices, it becomes evident that
we do not have (efficient) explicit access to
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$, but only to samples of the
distribution in the computational basis. Consequently, QCBMs can be classified
as implicit generative models. In this work, we study the trainability issues
that QCBMs suffer from as a result.

###  II.2 Loss functions

Similarly to the distinction between explicit and implicit generative models,
we draw a distinction between explicit and implicit loss functions. In broad
terms, explicit losses are those that can only be formulated explicitly in
terms of the target and model probabilities, whereas implicit losses are those
that can be formulated in terms of an average over model and training data
samples. This distinction at the level of loss functions thus mirrors the
capabilities and limitations of explicit and implicit generative models.

More concretely, we define an explicit loss as a loss function $\mathcal{L}$
that can be written solely as a function of the probabilities of the target
and model distributions, without any dependence on the data itself. Explicit
losses thus take the general form

$\mathcal{L}_{\rm
expl}(\boldsymbol{\theta}):=\sum_{\boldsymbol{x}_{1}...\boldsymbol{x}_{r}}f\Big{(}p(\boldsymbol{x}_{1}),...,p(\boldsymbol{x}_{r}),q_{\boldsymbol{\theta}}(\boldsymbol{x}_{1}),...,q_{\boldsymbol{\theta}}(\boldsymbol{x}_{r})\Big{)}\,,$
(2)

where $f(\cdot)$ is a function that depends on the target probabilities
$p(\boldsymbol{x}_{i})$ and model probabilities
$q_{\boldsymbol{\theta}}(\boldsymbol{x}_{i})$ for data variables
$\boldsymbol{x}_{i}\in\mathcal{X}$ with $i=1,\,...\,,r$. For this loss to be
useful, the function $f$ should be chosen such that it measures the distance
between the probability distributions $p$ and $q_{\boldsymbol{\theta}}$.
Crucially, the function $f$ does not take the data values $\boldsymbol{x}$
themselves as arguments.

While in full generality explicit losses could compare multiple copies of the
target and model probabilities (i.e., we can have $r>1$), in practice, they
usually take the simpler form

$\displaystyle\mathcal{L}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}f(p(\boldsymbol{x}),q_{\boldsymbol{\theta}}(\boldsymbol{x}))\;.$
(3)

We call such losses pairwise explicit losses since they compare the model and
target probabilities on the same data samples, or in our case, bitstrings. The
pairwise explicit loss covers all so-called $f$-divergences Csiszár (1967),
including the commonly encountered KL divergence (KLD) Kullback and Leibler
(1951),

$\displaystyle\mathcal{L}^{\rm
KLD}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}p(\boldsymbol{x})\log\left(\frac{p(\boldsymbol{x})}{q_{\boldsymbol{\theta}}(\boldsymbol{x})}\right)\;,$
(4)

the reverse-KLD,

$\displaystyle\mathcal{L}^{\rm rev-
KLD}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})\log\left(\frac{q_{\boldsymbol{\theta}}(\boldsymbol{x})}{p(\boldsymbol{x})}\right)\;,$
(5)

the Jensen-Shannon divergence (JSD) Lin (1991),

$\displaystyle\mathcal{L}^{\rm
JSD}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}\Big{[}$
$\displaystyle
p(\boldsymbol{x})\log\left(\frac{p(\boldsymbol{x})}{p(\boldsymbol{x})+q_{\boldsymbol{\theta}}(\boldsymbol{x})}\right)+$
$\displaystyle
q_{\boldsymbol{\theta}}(\boldsymbol{x})\log\left(\frac{q_{\boldsymbol{\theta}}(\boldsymbol{x})}{p(\boldsymbol{x})+q_{\boldsymbol{\theta}}(\boldsymbol{x})}\right)\Big{]}\;,$
(6)

and the total variation distance (TVD),

$\displaystyle\mathcal{L}^{\rm
TVD}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}|p(\boldsymbol{x})-q_{\boldsymbol{\theta}}(\boldsymbol{x})|\;.$
(7)

Another example of loss function that can be written in this form is the
classical fidelity,

$\displaystyle\mathcal{L}^{\rm
CF}(\boldsymbol{\theta})=1-\sum_{\boldsymbol{x}\in\mathcal{X}}\sqrt{p(\boldsymbol{x})q_{\boldsymbol{\theta}}(\boldsymbol{x})}\;.$
(8)

Notably, any non-data dependent post-processing of an explicit loss retains
its explicit character. Thus, any non-data dependent function of an explicit
loss (Eq. (2)) may also be considered an explicit loss. For example, the Rényi
divergence Rényi (1961)

$\displaystyle\mathcal{L}_{R,\alpha}(\boldsymbol{\theta})=\frac{1}{\alpha-1}\log\left(\sum_{\boldsymbol{x}}\frac{p^{\alpha}(\boldsymbol{x})}{q_{\boldsymbol{\theta}}^{\alpha-1}(\boldsymbol{x})}\right)\;,$
(9)

with $0<\alpha<\infty$ and $\alpha\neq 1$ can be classified as an explicit
loss function.

On the other hand, we define an implicit loss as one that can be written as an
average over samples drawn from the target and model distributions. That is,
an implicit loss function can be expressed as

$\mathcal{L}_{\rm
impl}(\boldsymbol{\theta}):=\mathbb{E}_{\boldsymbol{x}_{1},...,\boldsymbol{x}_{r}\sim\\{p,q_{\boldsymbol{\theta}}\\}}\,g(\boldsymbol{x}_{1},...,\boldsymbol{x}_{r})\,,$
(10)

where $g(\boldsymbol{x}_{1},...,\boldsymbol{x}_{r})$ is some function that
depends on the data (but not probabilities), and the expectation is taken over
data variables $\boldsymbol{x}_{1},...,\boldsymbol{x}_{r}$ sampled either from
the data distribution $p$ or the model distribution $q_{\boldsymbol{\theta}}$.

As a key example of an implicit loss, we focus on the commonly used Maximum
Mean Discrepancy (MMD) Gretton  _et al._ (2012a) loss. The MMD takes the form

$\displaystyle\mathcal{L}_{\rm MMD}(\boldsymbol{\theta})=\,$
$\displaystyle\mathbb{E}_{\boldsymbol{x},\boldsymbol{y}\sim
q_{\boldsymbol{\theta}}}[K(\boldsymbol{x},\boldsymbol{y})]-2\mathbb{E}_{\boldsymbol{x}\sim
q_{\boldsymbol{\theta}},\boldsymbol{y}\sim
p}[K(\boldsymbol{x},\boldsymbol{y})]$
$\displaystyle+\mathbb{E}_{\boldsymbol{x},\boldsymbol{y}\sim
p}[K(\boldsymbol{x},\boldsymbol{y})]\,,$ (11)

where $K(\boldsymbol{x},\boldsymbol{y})$ is a freely chosen kernel function.
We consider the popular choice of a classical Gaussian kernel, which is
defined as

$\displaystyle K_{\sigma}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle=e^{-\frac{\|\boldsymbol{x}-\boldsymbol{y}\|^{2}_{2}}{2\sigma}}=\prod_{i=1}^{n}e^{-\frac{(x_{i}-y_{i})^{2}}{2\sigma}}\;.$
(12)

Here, $\|.\|_{2}$ is the 2-norm, $\sigma>0$ is the so-called bandwidth
parameter, and $x_{i},y_{i}$ are the values of bit $i$ in bitstring
$\boldsymbol{x},\boldsymbol{y}$, respectively. This kernel in effect provides
a continuous measure of the distance between target and model bitstrings.

Interestingly, an implicit loss can always additionally be expressed in a form
where it contains the target and model probabilities. Taking the MMD loss in
Eq. (11) as a concrete example, the loss can be re-written as

$\displaystyle\mathcal{L}_{\rm MMD}(\boldsymbol{\theta})=$
$\displaystyle\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})q_{\boldsymbol{\theta}}(\boldsymbol{y})K(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle-2\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})p(\boldsymbol{y})K(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle+\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}p(\boldsymbol{x})p(\boldsymbol{y})K(\boldsymbol{x},\boldsymbol{y})\;.$
(13)

However, we stress that due to the data-dependence in the kernel
$K(\boldsymbol{x},\boldsymbol{y})$, the MMD loss function can in general not
be classified as an explicit loss.

Nonetheless, this brings us to the subtle point that explicitness and
implicitness are in fact not strictly mutually exclusive, i.e., one may be
able to find a loss function that satisfies both Eq. (2) and Eq. (10) in
specific cases. For example, for the MMD this occurs if the kernel is chosen
to be a Kronecker delta function,
$K(\boldsymbol{x},\boldsymbol{y})=\delta_{\boldsymbol{x}\boldsymbol{y}}$.
However, such hybrid losses are very much rare edge cases, and the
overwhelming majority of losses are either explicit or implicit. A more
detailed discussion of the technical nuances of the explicit and implicit loss
distinction is provided in Appendix A.

###  II.3 Loss measurement strategies

Central to the trainability of quantum generative models is the measurement
strategy used to estimate the loss. Here we draw a distinction between
conventional and quantum measurement strategies. For simplicity we now
restrict our discussion to implicit quantum generative models such as the
QCBM.

The conventional measurement strategy, which can be employed by both classical
and quantum implicit models, starts by collecting sample data from the target
and model distributions in the bases in which the data distribution is
modelled, e.g., the computational basis for the case of classical data. For an
implicit loss these samples can then be directly used to evaluate the loss
function in Eq. (10). For an explicit loss, this is not possible, and instead
one needs to use the collected samples to recreate an empirical estimate
$\tilde{q}_{\boldsymbol{\theta}}$ of the true model distributions
$q_{\boldsymbol{\theta}}$.

More formally, as sketched in Fig. 1, consider the set of bitstrings
$\tilde{Q}_{\boldsymbol{\theta}}$ obtained after collecting $N$ samples from
the model and the empirical model distribution
$\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x})$ constructed from these
samples. Then, the statistical estimate of the pairwise explicit loss function
$\tilde{\mathcal{L}}(\boldsymbol{\theta})$ in Eq. (3) can be expressed as

$\displaystyle\tilde{\mathcal{L}}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}f(\tilde{p}(\boldsymbol{x}),\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x}))\;.$
(14)

Crucially, since this proxy is all we have access to, the properties of this
statistical estimate are what determine the trainability of an explicit loss
function when evaluated via the conventional strategy. We note that zero-
estimates of the model probabilities with
$\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x})=0$ are often ‘clipped’ with a
small regularization parameter $\epsilon\ll 1$ in order to avoid numerical
instabilities in the loss computation.

This conventional strategy is somewhat classical in the sense that after
sampling is performed on the quantum model, the post processing required to
compute the cost is entirely classical. However, ‘more quantum’ measurement
strategies are also possible. In this case, a quantum circuit is used to
compute functions of the probabilities, potentially more directly and/or
collectively.

For example, rather than computing the classical fidelity in Eq. (8) by
explicitly computing the probabilities
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$, one could encode the target
distribution in a quantum state
$\ket{\phi}=\sum_{\boldsymbol{x}}\sqrt{\tilde{p}(\boldsymbol{x})}\ket{\boldsymbol{x}}$
and compute the quantum fidelity

$\displaystyle\mathcal{L}_{QF}(\boldsymbol{\theta})$
$\displaystyle:=1-|\bra{\phi}\psi(\boldsymbol{\theta})\rangle|^{2}$ (15)
$\displaystyle\sim
1-\left|\sum_{x}\sqrt{\tilde{p}(\boldsymbol{x})q_{\boldsymbol{\theta}}(\boldsymbol{x})}\right|^{2}\;.$
(16)

Up to arbitrary global phase factors (and a mod-square) this is equivalent to
the classical fidelity. However, it can be computed via coherent strategies -
namely a Loschmidt echo circuit Gibbs  _et al._ (2021, 2022); Caro  _et al._
(2022); Volkoff  _et al._ (2021) or a SWAP test Barenco  _et al._ (1997);
Garcia-Escartin and Chamorro-Posada (2013). We note that in this case quantum
generative modelling is equivalent to a state learning problem. While this
expression seemingly requires the entire training dataset to be loaded into a
wavefunction, we present an approach in Sec. III.3 to estimate this cost using
pairwise Hadamard tests.

More generally, it remains an open question if/when commonly encountered
losses for generative modelling can be computed using quantum strategies and
whether or not this brings any advantages222Beyond QCBMs, Quantum Generative
Adversarial Networks (QGANs) Lloyd and Weedbrook (2018) trained with classical
discriminators Zoufal  _et al._ (2019); Situ  _et al._ (2020); Bravo-Prieto
_et al._ (2022) in effect use a conventional measurement strategy, whereas
their variant with quantum discriminators Niu  _et al._ (2022) use a quantum
strategy.. Nonetheless, we suggest that this is an interesting avenue for
future research.

###  II.4 Exponential concentration and barren plateaus

For a quantum generative model to be trained successfully, the loss landscape
must be sufficiently featured to enable a solution to be found. There is a
growing awareness of the importance of barren plateaus, and its sister
phenomenon exponential concentration, for quantum machine learning McClean
_et al._ (2018); Arrasmith  _et al._ (2022); Larocca  _et al._ (2022a); Cerezo
and Coles (2021); Arrasmith  _et al._ (2021); Holmes  _et al._ (2021); Zhao
and Gao (2021); Thanasilp  _et al._ (2022). A barren plateau (BP) is a loss
landscape where the magnitudes of gradients vanish exponentially with growing
problem size McClean  _et al._ (2018); Cerezo  _et al._ (2021); Larocca  _et
al._ (2022a); Marrero  _et al._ (2021); Patti  _et al._ (2021); Holmes  _et
al._ (2022, 2021); Zhao and Gao (2021); Wang  _et al._ (2021a); Thanasilp  _et
al._ (2021); Cerezo and Coles (2021); Arrasmith  _et al._ (2021); Wang  _et
al._ (2021b). Closely related and equally problematic is exponential
concentration where the loss is shown to concentrate with high probability to
a single fixed value Arrasmith  _et al._ (2022). This, with high probability,
results in poorly trained models using a polynomial number of measurement
shots (regardless of the optimization method employed) Arrasmith  _et al._
(2021). More precisely, exponential concentration can be formally defined as
follows.

######  Definition 1 (Exponential concentration).

Consider a quantity $X(\boldsymbol{\alpha})$ that depends on a set of
variables $\boldsymbol{\alpha}$ and can be measured from a quantum computer as
the expectation of some observable. $X(\boldsymbol{\alpha})$ is said to be
deterministically exponentially concentrated in the number of qubits $n$
towards a certain fixed value $\mu$ if

$\displaystyle|X(\boldsymbol{\alpha})-\mu|\leqslant\beta\in O(1/b^{n})\;,$
(17)

for some $b>1$ and all $\boldsymbol{\alpha}$. Analogously,
$X(\boldsymbol{\alpha})$ is probabilistically exponentially concentrated if

$\displaystyle{\rm
Pr}_{\boldsymbol{\alpha}}[|X(\boldsymbol{\alpha})-\mu|\geqslant\delta]\leqslant\frac{\beta}{\delta^{2}}\;\;,\;\beta\in
O(1/b^{n})\;,$ (18)

for $b>1$. That is, the probability that $X(\boldsymbol{\alpha})$ deviates
from $\mu$ by a small amount $\delta$ is exponentially small for all
$\boldsymbol{\alpha}$.

A number of causes of exponential concentration and barren plateaus have been
identified including using parameterized circuits that are too expressive
McClean  _et al._ (2018); Holmes  _et al._ (2022); Larocca  _et al._ (2022a);
Tangpanitanon  _et al._ (2020) or too entangling Marrero  _et al._ (2021);
Sharma  _et al._ (2022); Patti  _et al._ (2021). Hardware noise Wang  _et al._
(2021a); Stilck França and Garcia-Patron (2021); Wang  _et al._ (2021b) has
also been shown to exponentially flatten the loss landscapes, which strongly
hinders the potential of current noisy quantum devices. The exponential
concentration can also happen due to randomness in the training dataset
Thanasilp  _et al._ (2021); Leone  _et al._ (2022); Li  _et al._ (2022). In
addition, there are studies on the exponential concentration in different QML
models including dissipative parametrized quantum circuits Sharma  _et al._
(2022) as well as quantum kernel-based models Thanasilp  _et al._ (2022).

Finally, the choice of loss function can also induce these phenomena. Thus
far, loss concentration has predominantly been studied in the context of
losses of the form

$C(\boldsymbol{\theta})=\Tr[OU(\boldsymbol{\theta})\rho
U(\boldsymbol{\theta})^{\dagger}]\;,$ (19)

where $\rho$ is an $n$-qubit input state and $O$ is a Hermitian operator. In
particular, it has been shown that ‘global’ Cerezo  _et al._ (2021) losses,
i.e., those where $O$ acts non-trivially on $\mathcal{O}(n)$ qubits, induce
loss concentration even for very shallow random circuits. Conversely, local
losses where $O$ acts non-trivially on at most $log(n)$ adjacent qubits (and
more generally low-body losses where the adjacency constraint is lifted - see
panel a) of Fig. 4) have been shown to enjoy trainability guarantees Cerezo
_et al._ (2021); Napp (2022) with shallow unstructured circuits. Furthermore,
we note that how barren plateaus affect parametrized quantum circuits with a
non-linear loss in the discriminative QML setting has been studied in Ref.
Thanasilp  _et al._ (2021).

Here we study exponential concentration for generative modelling tasks on
classical discrete data using implicit quantum generative models, and use our
insights to establish guidelines of how best to train such models. Crucially,
in this generative modeling context, the fixed points of the model
probabilities tend to be exponentially small and the loss function contains
the sum over exponentially many terms. These two together render previously
used tools not directly applicable for studying the trainability of quantum
generative models.

##  III Trainability analysis on loss functions

In this section, we analyse the trainability of different loss functions used
in quantum generative modelling.

###  III.1 Pairwise explicit losses

Figure 2: The problem with pairwise explicit losses. In the space with $2^{n}$
unique $n$-bit bitstrings, samples $\boldsymbol{x}$ generated from an
uninformed model with high probability do not coincide with any of the
training bitstrings. In other words, the empirical model distribution
$\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x})$ and the training
distribution $\tilde{p}(\boldsymbol{x})$ do not both have non-zero
probabilities for any bitstring $\boldsymbol{x}$. On the other hand, an
implicit loss function such as the MMD provides a continuous measure of
distance between the distributions by use of a Gaussian kernel with bandwidth
$\sigma$.

Part of the power of quantum generative models is that they can be used to
continuously parameterise and express distributions over discrete data with
exponential support. That is, an $n$-qubit model can be used to model
distributions over $2^{n}$ different $n$-bitstrings. However, while the true
target distribution may have exponential support, the amount of training data
$\tilde{P}$ is in practise restricted. More precisely, for large $n$ (e.g.,
$n>50$), it is reasonable to assume that the number of bitstrings in the
training dataset scales at most polynomially in $n$. Similarly, the number of
bitstrings samples obtained from the model must also scale at most
polynomially in $n$. That is,
$|\tilde{P}|,|\tilde{Q}_{\boldsymbol{\theta}}|\in\mathcal{O}(\operatorname{poly}(n))$.

This discrepancy between the polynomial support of the training data and the
exponential support of the model, can make it highly challenging to train
implicit models using pairwise explicit loss functions. In loose terms, the
problem is that the only bitstrings that contribute to the evaluation of a
statistical estimate of an explicit cost are those corresponding to bitstrings
$\tilde{P}$ in the training data. To estimate the loss one thus needs good
estimates of the model distributions over the support of $\tilde{P}$. However,
for an implicit model these estimates are obtained via sampling and the set
$\tilde{P}$ contains an exponentially small proportion of the total number of
bitstrings. As such, for generic models (we will explain what we mean by
generic below), the probability of measuring any bitstring in the training set
will also be exponentially small (as sketched in Fig. 2), leading to a poor
statistical estimate of the loss.

####  III.1.1 Concentration of Pairwise Explicit Losses

To make this line of argument more concrete, the first family of models we
will consider are those where the individual model probabilities
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$ are exponentially concentrated over
different values of $\boldsymbol{\theta}$. This is the case for a large family
of unstructured parameterised quantum circuits. Since estimating
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$ is equivalent to computing the
expectation value of the global projector
$|\boldsymbol{x}\rangle\langle\boldsymbol{x}|$, the concentration of
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$ can be viewed as resulting from the
global-measurement induced barren plateau phenomenon Cerezo  _et al._ (2021).
In this case, concentration is observed even for an ansatz that is comprised
of only a single layer of single-qubit rotations. However, alternative
phenomena (e.g. noise Wang  _et al._ (2021a) or expressibility Holmes  _et
al._ (2022)) can also lead to the exponential concentration of
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$. More formally, the following
proposition holds.

######  Proposition 1 (Concentration of model).

For all possible bitstrings $\boldsymbol{x}\in\mathcal{X}$, the underlying
probability $q_{\boldsymbol{\theta}}(\boldsymbol{x})$ of the quantum model
exponentially concentrates towards some exponentially small fixed point
$\mu\in O(1/b^{n})$ for $b>1$ if the quantum generative model is constructed
with:

  * •

A single layer of random single qubit gates
$U(\boldsymbol{\theta})=\bigotimes_{i=1}^{n}U_{i}(\boldsymbol{\theta}_{i})$.
Or, more precisely, if
$\\{U_{i}(\boldsymbol{\theta}_{i})\\}_{\boldsymbol{\theta}_{i}}$ forms a local
2-design on qubit $i$ Cerezo  _et al._ (2021).

  * •

$L$ layers of random $k$-local 2-designs, i.e.,
$U(\boldsymbol{\theta})=\prod_{l=1}^{L}\bigotimes_{j=1}^{n/k}U_{l,j}(\boldsymbol{\theta}_{l,j})$
with each $U_{l,j}(\boldsymbol{\theta}_{l,j})$ acting on $k$ qubits and
$\\{U_{l,j}(\boldsymbol{\theta}_{l,j})\\}_{\boldsymbol{\theta}_{l,j}}$ forming
a $k$-local 2-design over $\boldsymbol{\theta}_{l,j}$ Cerezo  _et al._ (2021).

  * •

A parameterised quantum circuit $U(\boldsymbol{\theta})$ such that its
ensemble over $\boldsymbol{\theta}$ i.e.,
$\\{U(\boldsymbol{\theta})\\}_{\boldsymbol{\theta}}$ forms an approximate
2-design on $n$ qubits McClean  _et al._ (2018); Holmes  _et al._ (2022). This
holds even for the problem-inspired circuits Larocca  _et al._ (2022a).

  * •

A linear-depth quantum circuit subject to local Pauli noise between each layer
Wang  _et al._ (2021a).

Proposition 1 provides examples of cases where the model probabilities
exponentially concentrate over all bitstrings in $\mathcal{X}$. However, we
find that in fact trainability difficulties arise even if model probabilities
are only exponentially concentrated over the training dataset (but perhaps not
on points outside the dataset). That is, all that is required for
untrainability is that the probability of measuring a sample that is also in
the dataset is practically zero. This is likely to be the case even for highly
structured quantum circuits if the generative model is built without a strong
inductive bias. We formalise this intuition in Appendix B.2.

We now argue that the exponential concentration of probabilities
$q_{\boldsymbol{\theta}}(\boldsymbol{x})$ over the dataset causes
$\tilde{\mathcal{L}}(\boldsymbol{\theta})$ to also exponentially concentrate.
To understand why, let us look at the probability of measuring one specific
bitstring (e.g., $\boldsymbol{x}_{0}$ \- the all-zero bitstring) and assume
that $q_{\boldsymbol{\theta}}(\boldsymbol{x}_{0})$ is exponentially
concentrated towards some exponentially small value $\mu$. Then, for any given
parameter constellation, it is highly likely that
$q_{\boldsymbol{\theta}}(\boldsymbol{x}_{0})$ is exponentially close to $\mu$.
To estimate $q_{\boldsymbol{\theta}}(\boldsymbol{x}_{0})$ on a quantum
computer we sample $N$ bitstrings from the quantum model and record the
observations. The chance that none of the sampled bitstrings are the specific
bitstring that we are interested in is
$(1-q_{\boldsymbol{\theta}}(\boldsymbol{x}_{0}))^{N}\approx 1-N\mu$. However,
the number of circuits $N$ that can be efficiently run is necessarily limited
- here we will assume $N\in\text{poly}(n)$. Thus we have that the probability
of not measuring the bitstring we are interested in is exponentially close to
1. That is, the statistical estimate of
$\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x}_{0})$ is almost always zero.
We can then generalize this intuition for a single bitstring to the estimation
of each of the (polynomially many) target bitstrings and therefore the whole
loss function. The following theorem formalizes this argument.

######  Theorem 1 (Concentration of pairwise explicit loss for concentrated
models).

Consider the loss function of the form in Eq. (3). Assume that for all
bitstrings in the training dataset, $\boldsymbol{x}\in\tilde{P}$, the quantum
generative model $q_{\boldsymbol{\theta}}(\boldsymbol{x})$ exponentially
concentrates towards some exponentially small value (as defined in Definition
1). Suppose that $N\in\mathcal{O}(\operatorname{poly}(n))$ samples are
collected from the quantum model corresponding to the set of sampled
bitstrings $\tilde{Q}_{\boldsymbol{\theta}}$, and that the training dataset
$\tilde{P}$ contains $M\in\mathcal{O}(\operatorname{poly}(n))$ samples. We
define the fixed point of the loss as

$\displaystyle\mathcal{L}_{0}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})=\sum_{\boldsymbol{x}\in\mathcal{P}}f(\tilde{p}(\boldsymbol{x}),0)+\sum_{\boldsymbol{x}\in\mathcal{Q}_{\boldsymbol{\theta}}}f(0,\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x}))\;,$
(20)

with $\mathcal{P}$ (and $\mathcal{Q}_{\boldsymbol{\theta}}$) being a set of
unique bitstrings in $\tilde{P}$ (and $\tilde{Q}_{\boldsymbol{\theta}}$).
Then, the probability that the estimated value
$\tilde{\mathcal{L}}(\boldsymbol{\theta})$ is equal to
$\mathcal{L}_{0}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})$ is exponentially
close to 1, i.e.,

$\displaystyle{\rm
Pr}_{\tilde{Q}_{\boldsymbol{\theta}},\boldsymbol{\theta}}[\tilde{\mathcal{L}}(\boldsymbol{\theta})=\mathcal{L}_{0}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})]\geqslant
1-\delta\;,$ (21)

with $\delta\in\mathcal{O}\left(\frac{\operatorname{poly}(n)}{c^{n}}\right)$
for some $c>1$.

As a direct consequence of Theorem 1, the following corollary gives the
concentration points of some specific explicit loss functions mentioned in
this work.

######  Corollary 1 (Concentration points of common explicit loss functions).

Under the same conditions as in Theorem 1, the following loss functions
concentrate at

  * •

KL-divergence:

$\displaystyle\mathcal{L}_{0}^{{\rm
KLD}}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})=\sum_{\boldsymbol{x}\in\mathcal{P}}\tilde{p}(\boldsymbol{x})\log\left(\frac{\tilde{p}(\boldsymbol{x})}{\epsilon}\right)\;.$
(22)

Here $\epsilon\ll 1$ is a clipping value, which is common practice to avoid
the singularity of the logarithm at
$q_{\boldsymbol{\theta}}(\boldsymbol{x})=0$.

  * •

Classical fidelity:

$\displaystyle\mathcal{L}_{0}^{\rm
CF}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})=1\;.$ (23)

  * •

Reverse KL-divergence:

$\displaystyle\mathcal{L}_{0}^{\rm rev-
KLD}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})=\sum_{\boldsymbol{x}\in\mathcal{Q}_{\boldsymbol{\theta}}}\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x})\log\left(\frac{\tilde{q}_{\boldsymbol{\theta}}(\boldsymbol{x})}{\epsilon}\right)\;.$
(24)

  * •

Total variation distance:

$\displaystyle\mathcal{L}_{0}^{\rm
TVD}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})=2\;.$ (25)

Looking at the expressions for the fixed points given above, in the case of
the KL divergence, classical fidelity and total variational distance, the
fixed point is independent of $\boldsymbol{\theta}$. Thus it is clear that the
costs cannot be used to train the quantum circuit model. In the case of the
reverse KL divergence, the fixed point depends on $\boldsymbol{\theta}$ but is
independent of the training data and thus the reverse KL also cannot be used
to train the model to learn the target distribution.

More generally, for all explicit losses of the form Eq. (3), the concentration
point $\mathcal{L}_{0}(\tilde{P},\tilde{Q}_{\boldsymbol{\theta}})$, Eq. (20),
can be separated into two terms: (i) the term that involves only $\tilde{P}$
and (ii) the other that involves only $\tilde{Q}_{\boldsymbol{\theta}}$. In
other words, the $\boldsymbol{\theta}$ dependence of the estimator of the loss
is independent of the target distribution and thus the estimate of the loss is
worthless for training the generative model. This no-go result is rigorously
established in Corollary 2. Our approach is to show that the loss function at
two arbitrary parameter values $\boldsymbol{\theta}_{1}$ and
$\boldsymbol{\theta}_{2}$, contains no information about the training
distribution.

Figure 3: Variance of the KL divergence with finite shots. Concentration of
the KL divergence loss as a function of the number of measurements and qubits
for random product state circuits. Here we take the target distribution to be
$\tilde{p}(\boldsymbol{0})=1$ and take the cutoff of the KLD to be
$\epsilon=10^{-14}$. Vertical lines indicate where the number of measurements
equal $2^{n}$. Thus, we see that the the KLD estimate is biased upwards with
any finite number of measurements, and the number of measurements required to
achieve a reasonable level of uncertainty increases exponentially with the
number of qubits $n$.

######  Corollary 2 (Untrainability of pairwise explicit loss functions).

Under the same conditions as in Theorem 1, the probability that the difference
between the two statistical estimates of the loss function at
$\boldsymbol{\theta}_{1}$ and $\boldsymbol{\theta}_{2}$ does not contain any
information about the training distribution is exponentially close to 1.
Particularly, we have

$\displaystyle{\rm
Pr}_{\tilde{Q}_{\boldsymbol{\theta}},\boldsymbol{\theta}}[\tilde{\mathcal{L}}(\boldsymbol{\theta}_{1})-\tilde{\mathcal{L}}(\boldsymbol{\theta}_{2})=\Delta\mathcal{L}_{0}(\tilde{Q}_{\boldsymbol{\theta}_{1}},\tilde{Q}_{\boldsymbol{\theta}_{2}})]\geqslant
1-2\delta\;,$ (26)

with $\delta\in\mathcal{O}\left(\frac{\operatorname{poly}(n)}{c^{n}}\right)$
for some $c>1$, $\tilde{Q}_{\boldsymbol{\theta}_{1}}$ (and
$\tilde{Q}_{\boldsymbol{\theta}_{2}}$) is a set of sampling bitstrings
obtained from the quantum generative model at the parameter value
$\boldsymbol{\theta}_{1}$ (and $\boldsymbol{\theta}_{2}$), as well as

$\displaystyle\Delta\mathcal{L}_{0}(\tilde{Q}_{\boldsymbol{\theta}_{1}},\tilde{Q}_{\boldsymbol{\theta}_{2}})=\sum_{\boldsymbol{x}\in\mathcal{Q}_{\boldsymbol{\theta}_{1}}}f(0,\tilde{q}_{\boldsymbol{\theta}_{1}}(\boldsymbol{x}))-\sum_{\boldsymbol{x}\in\mathcal{Q}_{\boldsymbol{\theta}_{2}}}f(0,\tilde{q}_{\boldsymbol{\theta}_{2}}(\boldsymbol{x}))\;,$
(27)

with $\mathcal{Q}_{\boldsymbol{\theta}_{1}}$ (and
$\mathcal{Q}_{\boldsymbol{\theta}_{2}}$) being a set of unique bit-strings in
$\tilde{Q}_{\boldsymbol{\theta}_{1}}$ (and
$\tilde{Q}_{\boldsymbol{\theta}_{2}}$). Crucially,
$\Delta\mathcal{L}_{0}(\tilde{Q}_{\boldsymbol{\theta}_{1}},\tilde{Q}_{\boldsymbol{\theta}_{2}})$
does not depend on any $\tilde{p}(\boldsymbol{x})\in\tilde{P}$.

To support our analytic claims we further conducted a numerical study of the
exponential concentration of pairwise explicit costs. For concreteness, we
here decided to focus on the KL divergence. In Fig. 3, we plot the mean and
variance (over $\boldsymbol{\theta}$) of the KL divergence for the target
distribution $\tilde{p}(\boldsymbol{0})=1$ as a function of the number of
measurement shots and qubits. For simplicity we take our model to be a (Haar)
random product state.

We see in Fig. 3a) that with a polynomial number of measurements, as per Eq.
(22), the empirical estimate of the loss concentrates at
$\log(1/\epsilon)\approx 32.2$ for $\epsilon=10^{-14}$. Correspondingly, with
a polynomial number of measurements the variance in Fig. 3b) is exponentially
close to zero. Using an exponential number of measurements, the estimate of
the KL tends towards its true value and the variance is again small. The
transition between these two regimes is marked by a very high variance
corresponding to the case where the measurement count is high enough for there
to be some overlap between the sampled bits strings and the $\boldsymbol{0}$
bitstring, but not enough overlap to obtain a reliable estimate of
$q_{\boldsymbol{\theta}}(\boldsymbol{0})$. This results in the loss estimate
to sporadically fluctuate between $\log(1/\epsilon)$ and
$\log(1/q_{\boldsymbol{\theta}}(\boldsymbol{x}))$ with
$q_{\boldsymbol{\theta}}(\boldsymbol{x})>0$. While in Fig. 3 the target
dataset consists of a single bitstring, larger datasets only shift the curves
to the left by a polynomial amount.

##### Broader Implications.

While our results above are formulated for training QCBMs with pairwise
explicit costs, we argue that the underlying problem is more general and
immune to simple solutions. One approach, for example, might be to take non-
data-dependent functions of pairwise explicit losses, as in the case of the
Rényi-divergence in Eq. (9). However, such loss functions exponentially
concentrate in the same manner as the explicit losses themselves when
employing the conventional measurement strategy. A more promising but
challenging approach would be to attempt to measure such losses via quantum
strategies. We discuss this further in Section III.3.

More generally, while we provide strict no-go results only for pairwise
explicit losses, we believe that any explicit losses in the general form of
Eq. (2) will suffer from concentration or exponential imprecision due to the
inherent inability of implicit models to accurately estimate the model
probabilities in polynomial time333A possible exception is if a particular
implicit model instead allows for efficient estimation of gradients of an
explicit loss function, as it is the case for RBMs training on the KL
divergence loss function. We are however not aware of any practical explicit
loss function that cannot be brought into the pairwise explicit form.

We further stress that our results hold for unstructured ansätze or ansätze
that lack an appropriate inductive initial bias. Thus, while explicit losses
such as the KLD will not work at scale with implicit models straight out-of-
the-box, our no-go theorems could be side-stepped using clever initialization
strategies in conjunction with specialized ansätze. For example, while we
argue in Appendix B.2 that initializing the quantum circuit model on a subset
of training states will not alleviate the fundamental issue when using a
generic ansatz, this may work if one leverages a quantum circuit that
constrains the model to the symmetry sector of the data. Among other hard
constraints, this is conceivable if the data consists only of samples with a
certain hamming weight or cardinality, as it can be the case in certain
financial applications Chang  _et al._ (2000); Alcazar  _et al._ (2021).
However, many real world datasets may not contain strong symmetries that one
can leverage so straight-forwardly. It is therefore critically important to
study the effect of strong parameter initializations and inductive biases
using explicit losses– both theoretically and experimentally.

###  III.2 Implicit losses: Maximum Mean Discrepancy

In the previous section we saw that an explicit loss function, used in
conjunction with an implicit generative model and the conventional sampling
strategy, exhibits exponential concentration and hence is untrainable. The
root cause was, at least in part, a miss-match between using an explicit loss
function with an implicit model. Thus it is natural to ask whether an implicit
quantum loss would fare better.

Here we focus on analysing the MMD loss function (see Eqs. (11) and (13)),
which is a commonly-used implicit loss. In contrast to the pairwise explicit
losses discussed previously, each bitstring drawn from the model is generally
compared with all training bitstrings, with the kernel function
$K(\boldsymbol{x},\boldsymbol{y})$ controlling the contribution of each
comparison. With a poor choice in kernel it is clear that the MMD will be
susceptible to exponential concentration. For example, the Gaussian kernel
with the bandwidth $\sigma\rightarrow 0$ is equivalent to a delta function
kernel,
$K(\boldsymbol{x},\boldsymbol{y})=\langle\boldsymbol{x},\boldsymbol{y}\rangle=\delta_{\boldsymbol{x}\boldsymbol{y}}$.
In this case the MMD reduces to the pairwise explicit loss
$\sum_{\boldsymbol{x}\in\mathcal{X}}(p(\boldsymbol{x})-q_{\boldsymbol{\theta}}(\boldsymbol{x}))^{2}$
(see Appendix A for details), and consequently is subject to our no-go result
in Theorem 1. This thus prompts the question of how exactly $\sigma$ affects
trainability.

##### Properties of the MMD loss.

To study the properties of the MMD loss, it is helpful to note that each term
in the MMD can be viewed as the expectation value of an observable whose
properties depend on the choice of $\sigma$. This change in perspective allows
us to leverage existing knowledge from the VQA trainability literature. In
particular, prior no-go results on VQAs with observable-type loss functions
are now directly applicable here, including those on cost function induced
Cerezo  _et al._ (2021), expressiblity-induced McClean  _et al._ (2018);
Holmes  _et al._ (2022), and noise-induced Wang  _et al._ (2021a) barren
plateaus.

Specifically, each term in the MMD can be written as

$\displaystyle\mathcal{M}(\rho,\rho^{\prime})=\Tr[O^{(\sigma)}_{\rm
MMD}(\rho\otimes\rho^{\prime})]\;,\;$ (28)

where we have defined the MMD observable

$O^{(\sigma)}_{\rm
MMD}:=\sum_{\boldsymbol{x},\boldsymbol{y}}K_{\sigma}(\boldsymbol{x},\boldsymbol{y})|\boldsymbol{x}\rangle\langle\boldsymbol{x}|\otimes|\boldsymbol{y}\rangle\langle\boldsymbol{y}|\;.$
(29)

This observable acts on $2n$ qubits, namely $n$ qubits corresponding to the
QCBM,
$\rho_{\boldsymbol{\theta}}=\ket{\psi(\boldsymbol{\theta})}\bra{\psi(\boldsymbol{\theta})}$,
and $n$ qubits corresponding to the dataset,
$\rho_{\tilde{p}}=\sum_{\boldsymbol{y}}\tilde{p}(\boldsymbol{y})|\boldsymbol{y}\rangle\langle\boldsymbol{y}|$.
For the first term in the MMD, both $\boldsymbol{x}$ and $\boldsymbol{y}$ are
sampled from the QCBM and we have
$\rho=\rho^{\prime}=\rho_{\boldsymbol{\theta}}$. The cross-term instead has
$\rho=\rho_{\boldsymbol{\theta}}$ and $\rho^{\prime}=\rho_{\tilde{p}}$, and
the final term has $\rho=\rho^{\prime}=\rho_{\tilde{p}}$.

Figure 4: Bodyness of the MMD loss. a) We illustrate the difference between
‘low-body’, ‘local’, ‘global’ and ‘full-body’ operators. An operator $O$ is
low-bodied if it acts non-trivially on at most $\mathcal{O}(\log(n))$ qubits.
If a low-bodied operator acts on qubits that are adjacent to each other, then
$O$ is said to be local. On the other hand, $O$ is global if it acts non-
trivially on $\Theta(n)$ qubits. Lastly, a full-body operator consists of the
sum of several operators that are low-body/local and global. b) We depict the
weight $w_{\sigma}(l)$ for the terms in the MMD operator as a function of
their bodyness $k$ for $n=50$, $100$ and $200$ qubits and the bandwidths
$\sigma=0$, $1$ and $n/4$. The average weight over these three $\sigma$ values
is also shown. For small $\sigma$, the MMD operator is a sum of predominantly
global operators, i.e., with $\sigma\in\mathcal{O}(1)$ the mean bodyness is
$\Theta(n)$. In contrast, $\sigma\in\Theta(n)$ results in predominantly low-
bodied operators. c) Sketch of the expected landscapes for low-body, global
and full-body losses respectively. Because low-body and global operators are
exclusively sensitive to low-body and global features, respectively, their
loss landscapes exhibit spurious minima, which don’t coincide with the minimum
of the true target distribution. A full-body loss on the other hand should
have a single optimal solution solution where all its constituent operator’s
minima align.

In the Pauli basis, the MMD observable $O^{(\sigma)}_{\rm MMD}$ takes the
elegant form

$\displaystyle O^{(\sigma)}_{\rm MMD}=\sum_{l=0}^{n}w_{\sigma}(l)D_{2l}\;,$
(30)

where $D_{2l}$ are normalized $2l$-body diagonal operators (defined explicitly
in Appendix C.1), and

$w_{\sigma}(l)={n\choose l}(1-p_{\sigma})^{n-l}p_{\sigma}^{l}$ (31)

are Bernoulli-distributed weights with effective probability

$p_{\sigma}=(1-e^{-1/2\sigma})/2\;.$ (32)

Thus estimating the MMD loss function in Eq. (11) using a batch of
measurements $\tilde{Q}$ is equivalent to using the same measurements to
estimate a weighted expectation of the observables $D_{2l}$.

The properties of the MMD observable clearly depend on the distribution of the
terms of different bodyness through the $w_{\sigma}(l)$ factor. Fig. 4 shows
how $w_{\sigma}(l)$ are distributed for different $\sigma$. Owing to the
Bernoulli-distributed weights, we can straight-forwardly provide the average
bodyness of $O^{(\sigma)}_{\rm MMD}$, which is given by

$\displaystyle\mathbb{E}_{l\sim\omega_{\sigma}(l)}[2l]=2np_{\sigma}\;,$ (33)

and the variance in the bodyness, which is

$\displaystyle{\rm
Var}_{l\sim\omega_{\sigma}(l)}[2l]=4np_{\sigma}(1-p_{\sigma})\;.$ (34)

From these expressions it follows that the MMD loss is predominantly composed
of global operators when $\sigma\in\mathcal{O}(1)$. More concretely the
following proposition holds.

######  Proposition 2 (MMD consists largely of global terms for
$\sigma\in\mathcal{O}(1)$).

For $\sigma\in\mathcal{O}(1)$, the average bodyness of the MMD operator
containing Pauli terms with weight $w_{\sigma}(l)$ is

$\mathbb{E}_{l\sim w_{\sigma}(l)}[2l]\in\Theta(n)\,.$ (35)

Similarly, the variance in the bodyness is given by

${\rm Var}_{l\sim w_{\sigma}(l)}[2l]\in\Theta(n)\,.$ (36)

This shows that with fixed-size bandwidths $\sigma$, as is commonly done
(e.g., Ref. Liu and Wang (2018)), the MMD suffers from global loss function-
induced barren plateaus Cerezo  _et al._ (2021) and hence is untrainable. This
practice of using constant bandwidths is carried over from classical ML
literature Li  _et al._ (2017); Wang  _et al._ (2018); Li  _et al._ (2018),
but Proposition 2 shows that this is fundamentally incompatible with quantum
generative models using unstructured circuits.

In contrast, we show that if the bandwidth scales linearly in the number of
qubits, $\sigma\in\Theta(n)$, the MMD loss function is approximately low-
bodied. We recall that being low-bodied is more general than being local, the
latter corresponding to the case where an operator is low-bodied and each term
only acts non-trivially on adjacent qubits. The following proposition
formalizes this relation by quantifying the error made when truncating the MMD
observable after a certain bodyness.

######  Proposition 3 (MMD consists largely of low-body terms for
$\sigma\in\Theta(n)$).

Let $\tilde{\mathcal{L}}^{(\sigma,k)}_{\rm MMD}(\boldsymbol{\theta})$ be a
truncated MMD loss with a truncated operator $\tilde{O}^{(\sigma,k)}_{\rm
MMD}$ that contains up to the $2k$-body interactions in $O^{(\sigma)}_{\rm
MMD}$,

$\displaystyle\tilde{O}^{(\sigma,k)}_{\rm
MMD}:=\sum_{l=0}^{k}w_{\sigma}(l)D_{2l}\;,$ (37)

where $w_{\sigma}(l)$ are Bernoulli-distributed weights defined in Eq. (31).
For $\sigma\in\Theta(n)$, the difference between the exact and local
approximation of the loss is bounded as

$\displaystyle|\mathcal{L}^{(\sigma)}_{\rm
MMD}(\boldsymbol{\theta})-\tilde{\mathcal{L}}^{(\sigma,k)}_{\rm
MMD}(\boldsymbol{\theta})|\leqslant\epsilon(k)\;,$ (38)

with

$\displaystyle\epsilon(k)\in\mathcal{O}\left(n(c/k)^{k}\right)\;,$ (39)

for some positive constant $c$.

This implies that one can view the MMD loss with a bandwidth
$\sigma\in\Theta(n)$ as composed almost exclusively of low-body contributions.
We therefore expect, given the results of Refs. Cerezo  _et al._ (2021); Napp
(2022), that the the MMD is trainable for $\sigma\in\Theta(n)$ for quantum
generative models which employ shallow quantum circuits. We note that there
appears to be no merit in increasing $\sigma$ beyond $\Theta(n)$, as that
simply increases the relative weight of the constant $l=0$ term in Eq. (30).
That is, the MMD operator tends towards the trivial identity measurement for
$\sigma\rightarrow\infty$.

To probe this further, and get a better understanding of the effect of
$\sigma$ on the trainability of the MMD loss, we start by considering the case
of QCBM with a product ansatz. This allows us to find a closed-form expression
of the MMD variance as a function of the circuit parameters (Supplemental
Proposition 2) from which we can study the concentration of the MMD for
different $\sigma$ values. Our findings are summarized by the following
Theorem (proven in Appendix C.2.3).

######  Theorem 2 (Product ansatz trainability of MMD, informal).

Consider the MMD loss function $\mathcal{L}^{(\sigma)}_{\rm
MMD}(\boldsymbol{\theta})$ as defined in Eq. (11), which uses the classical
Gaussian kernel as defined in Eq. (12) with the bandwidth $\sigma>0$, and a
quantum circuit generative model that is comprised of a tensor-product ansatz
$U=\bigotimes_{i}^{n}U_{i}(\theta_{i})$ with
$\\{U_{i}(\theta_{i})\\}_{\theta_{i}}$ being single-qubit (Haar) random
unitaries. Given a training dataset $\tilde{P}$, the asymptotic scaling of the
variance of the MMD loss depends on the value of $\sigma$.

For $\sigma\in\mathcal{O}(1)$, we have

$\displaystyle{\rm Var}_{\boldsymbol{\theta}}[\mathcal{L}^{(\sigma)}_{\rm
MMD}(\boldsymbol{\theta})]\in\mathcal{O}(1/b^{n})\;,$ (40)

with some $b>1$.

On the other hand, for $\sigma\in\Theta(n)$, we have

$\displaystyle{\rm Var}_{\boldsymbol{\theta}}[\mathcal{L}^{(\sigma)}_{\rm
MMD}(\boldsymbol{\theta})]\in\Omega(1/n)\;.$ (41)

We numerically verify Theorem 2 in Fig. 5. In panel a) we show that the
analytical predictions for different bandwidths coincide perfectly with the
numerical estimates. The exponentially vanishing loss variances observed for
$\sigma\in O(1)$ are expected to render the loss untrainable. This is
demonstrated in panel b), where we further train a QCBM with $\sigma=n/4$
(which approximately maximizes the variance) and $\sigma=1$. We find that a
QCBM with $\sigma=n/4$ can be successfully trained even for $n=1000$ qubits.
In contrast, the training starts to fail to learn the $|\boldsymbol{0}\rangle$
target state after $n\approx 50$ and is fully untrainable at $n=100$ when
$\sigma=1$ is used.

Figure 5: $\bm{\sigma}$-dependence of the MMD loss function. a) Comparison of
the MMD variance between the analytical prediction in Eq. (182) and empirical
variance using 100 measurements from a product state ansatz. b) Training a
product state ansatz on the $|\boldsymbol{0}\rangle$ target state for
$\sigma=1$ and $\sigma=n/4$ using the CMA-ES Igel  _et al._ (2007) optimizer
and 512 measurements. In both cases the QCBM ansatz consists of a single layer
of Ry rotations on each qubit.

It is interesting to note that the approximately optimal bandwidth
$\sigma\sim\frac{n}{4}$ for the product state ansatz coincides with the so-
called median heuristic Gretton  _et al._ (2012a) from classical ML
literature. For random circuits, the median (hamming) distance between
bitstrings is in fact $\frac{n}{2}$, which we satisfy with the factor of $2$
in our kernel convention.

To go towards more practical generative modelling, we recall that Ref. Napp
(2022) proves that cost functions of the form of Eq. (19) using $2k$-body
observables with $k\in\mathcal{O}(\log(n))$ are trainable using 1D-random
$\log(n)$ depth circuits. Since Proposition 3 implies that the MMD is well
approximated by a $\log(n)$-body cost, it should follow that the MMD is also
trainable at $\log(n)$ depths. There are a few technical caveats associated
with constructing a full proof. For example, the first term of the MMD
requires working with 4-designs instead of 2-designs, and the second term
depends on the target distribution, leading to additional subtleties. However,
there is no strong reason to expect that these technicalities make the MMD
untrainable. Hence, we propose the following conjecture.

######  Conjecture 1.

A QCBM composed of a shallow $\log(n)$ depth unstructured circuit is trainable
with the MMD loss function as defined in Eq. (11) using the Gaussian kernel in
Eq. (12) as long as the bandwidth $\sigma\in\Theta(n)$. That is, we have

$\displaystyle{\rm Var}_{\boldsymbol{\theta}}[\mathcal{L}^{(\sigma)}_{\rm
MMD}(\boldsymbol{\theta})]\in\Omega(1/\operatorname{poly}(n))\;.$ (42)

This conjecture is further supported by our numerical evidence for the
trainability of the MMD for deeper circuits and more realistic datasets shown
in Fig. 6. Here we plot the loss variance as a function of circuit depth $L$
and the number of qubits $n$ for $\sigma=n/4$ on four datasets from four
different target distributions. We observe that the polynomial scaling of the
loss variance does in fact extend beyond product states to shallow circuits,
i.e., $L\in\mathcal{O}(\log(n))$. However, for sufficiently deep circuits,
i.e., $L\in\Omega(n)$, the MMD variance appears to decay exponentially. This
aligns with expressibility-induced barren plateaus observed in other VQA
applications, which occur even for maximally local loss functions, i.e.,
$k=1$.

Figure 6: Study of loss concentration with the MMD loss function. Numerical
evidence that the MMD loss with Gaussian bandwidth parameter $\sigma=n/4$ does
not exhibit global or explicit loss function barren plateaus, but does exhibit
loss concentration with deep quantum circuits. We study the loss concentration
in randomly initialized line-topology circuits for various datasets, and
increasing number of qubits $n$ and circuit depth. The GHZ dataset consists of
the all-0 and all-1 bitstrings ($\mathcal{O}(1)$ support), the
$\mathcal{O}(n)$ and $\mathcal{O}(n^{2})$ datasets consist of $n$ and $n^{2}$
random bitstrings, respectively, and the cardinality dataset contains all
bitstrings with $\frac{n}{2}$ cardinality ($\mathcal{O}(2^{n})$ support).
There does not appear to be a strong data-dependence for the magnitude of the
loss variance.

##### Large gradients are not enough.

Our results so far appear to indicate that picking a single bandwidth
$\sigma\in\Theta(n)$ maximizes the trainability of the MMD loss function with
a Gaussian kernel. While it is true that this choice maximizes the expected
magnitude of initial gradients for a QCBM, non-vanishing gradients are a
necessary condition but not sufficient to guarantee reliable training
performance. And in fact it turns out that while low-body losses exhibit large
gradients they come with other limitations. Particularly, we show that the
bodyness of a generative loss function defines the maximal order of marginals
of the target distribution that can be distinguished. That is, the model only
learns to match the target distribution on subsets of bits, i.e. on its
marginals. This introduces a continuous family of minima which are
indistinguishable from the true minimum when using a low-bodied loss function,
but which are systematically wrong for the purposes of generative modelling.
The worry is that the non-vanishing loss gradients in low-bodied losses are
predominantly due to the presence of such spurious minima and do not point in
the direction of the true global minimum. This is sketched in Fig. 4.

Formally, let $q_{\boldsymbol{\theta}}(\boldsymbol{x}_{A})$ denote the
marginal model distribution on a subset $A\subseteq\\{1,2,...,n\\}$ of qubits,
and $\tilde{p}(\boldsymbol{x}_{A})$ the marginal target distribution on that
same subset. For more details we refer to Eq. (237) and Eq. (239) in Appendix
C.3. The connection between the bodyness of the loss operator and the
marginals of the model and target distributions is then formalized in the
following Proposition.

######  Proposition 4 (The truncated MMD loss is not faithful).

Consider a distribution $q_{\boldsymbol{\theta}}(\boldsymbol{x})$ that agrees
with the training distribution $\tilde{p}(\boldsymbol{x})$ on all the
marginals up to $k$ bits, but disagrees on higher-order marginals. The
distribution $q_{\boldsymbol{\theta}}(\boldsymbol{x})$ minimizes the truncated
MMD loss. That is, suppose

$\displaystyle
q_{\boldsymbol{\theta}}(\boldsymbol{x}_{A})=\tilde{p}(\boldsymbol{x}_{A})\;,$
(43)

for all $A\subseteq\\{1,2,...,n\\}$ with $|A|\leqslant k$, then

$\displaystyle\tilde{\mathcal{L}}^{(\sigma,k)}_{\rm
MMD}(\boldsymbol{\theta})=0\;.$ (44)

Crucially, this is true even if for some $B\subseteq\\{1,2,...,n\\}$ with
$|B|>k$

$\displaystyle
q_{\boldsymbol{\theta}}(\boldsymbol{x}_{B})\neq\tilde{p}(\boldsymbol{x}_{B})\;.$
(45)

In other words, if the MMD operator can be approximated well by a truncated
operator with at most $2k$-body terms, model distributions that match the
target distribution exactly up to $k$-body marginals or higher cannot be
distinguished from ones that match fully. As an example of such distributions,
consider the uniform distribution over the bitstrings $[001,011,101,110]$,
where the third bit is the bit-wise addition of the previous two bits. Using
only second-order marginals, it is not possible to distinguish this correlated
distribution from the uniform distribution over all eight possible outcomes.

Notably, long-range correlations in the data can still be learned by the low-
bodied MMD loss, just not ones that are particularly high-order444Note that
this is in contrast to a loss composed purely of local terms which would be
restricted to learning local/short-range correlations.. Not all distributions
will however exhibit such higher-order correlations and thus some
distributions will be learnable using losses composed of low-body terms.

Proposition 4 thus establishes that to fulfil the promise of quantum
generative models, that is to be able to learn both long-range and many-body
correlations, one cannot use exclusively low-body losses. However, such a
requirement is in immediate tension with the low-bodyness required for the
trainability guarantees (see Theorem 2). In particular, in Proposition 3 we
show that for $\sigma\in\Theta(n)$ the contribution of $k\in\Theta(n)$ terms
are exponentially small in $n$. Thus, although the loss is still strictly
faithful given an infinite shot budget, with a reasonable shot budget we will
not be able to resolve the contribution from the exponentially small high-body
terms. Hence, there can be spurious minima that we cannot resolve from the
true minimum and therefore for all practical purposes the loss is effectively
not truly faithful.

One approach to resolving this tension would be to adapt the initial value of
$\sigma$ from $\Theta(n)$, where the loss exhibits large gradients but
predominantly learns low-order marginals, towards $\mathcal{O}(1)$ to also
learn high-order correlations as the model improves. This is in line with
studies from the classical ML literature showing that bandwidths for optimal
MMD performance are oftentimes smaller than the so-called median heuristic
Gretton  _et al._ (2012b); Sutherland  _et al._ (2016); Garreau  _et al._
(2017), which coincides with our result of $\sigma\in\Theta(n)$. Another
approach, which is also already employed in classical ML literature, is to use
a kernel that averages the effects of several $\sigma$ Li  _et al._ (2017);
Wang  _et al._ (2018); Li  _et al._ (2018). That is, the kernel is taken to be

$K_{\textbf{c}}(\boldsymbol{x},\boldsymbol{y})=\frac{1}{|\textbf{c}|}\sum_{i\in\textbf{c}}K_{\sigma_{i}}(\boldsymbol{x},\boldsymbol{y})\equiv\sum_{l=1}^{k}\langle
w_{\sigma}(l)\rangle_{\textbf{c}}D_{2l}$ (46)

for a set of bandwidths $\textbf{c}=\\{\sigma_{1},\sigma_{2},...\\}$. The
resulting weight of each $2k-$body term of the new MMD observable is an
average of the weightings corresponding to each $\sigma_{i}$ in $\mathbf{c}$
as shown in Fig. 4. Theorem 2 shows that for a QCBM without inductive bias to
not fall prey to exponential concentration, at least one of the $\sigma_{i}$
needs to be $\Theta(n)$. But the results of Proposition 4 suggest that for
data sets exhibiting high-order correlations a small bandwidth
$\sigma_{i}\in\mathcal{O}(1)$ is required for correct convergence. It stands
to reason that the optimal set c contains a spectrum of bandwidths that both
enable trainability and faithful convergence to the target distribution (as
sketched in Fig. 4c)). How successful this strategy is in practice remains to
be determined.

##### Broader Implications.

Our work highlights that one can treat classical machine learning losses as
quantum observables to study their properties. This implies that our results
transfer to other types of quantum generative models beyond the QCBM that will
also be affected by the fundamental limitations described by Proposition 4. In
fact, we show in Appendix C.4 that any generative modelling loss function for
classical data that can be brought into the form
$\mathcal{L}(\boldsymbol{\theta})=\text{Tr}[\mathcal{M}\rho_{\boldsymbol{\theta}}]$,
with a diagonal measurement operator $\mathcal{M}$, faces the same tension
described above. That is, if $\mathcal{M}$ contains at most $k$-body terms in
the Pauli basis representation, then the loss cannot distinguish two
distributions that agree on all $k$-order marginals but disagree on higher-
order marginals. Thus losses composed exclusively of local terms (with the
conventional measurement strategy) cannot be used in generative modelling to
learn complex higher-order correlations.

With a little thought it becomes clear that an exclusively global loss is also
undesirable. Not only do such losses exhibit exponential concentration for
unstructured circuits, they will also in general possess spurious minima in
virtue of only probing global properties of the distribution (i.e. the average
global parity), as shown in Fig 4. Instead we advocate using full-body losses
which contain both low and high-body terms, such as those obtained by
averaging in Fig. 4. Even then, global contributions cannot be vanishingly
small or else they will not be possible to resolve with a realistic shot
budget.

For another example, one may aim to train a quantum generative model using a
QGAN framework, where a Discriminator $D$ provides a score $D(\boldsymbol{x})$
to every sample. The corresponding operator can then be written as
$\mathcal{M}=\sum_{\boldsymbol{x}}D(\boldsymbol{x})|\boldsymbol{x}\rangle\langle\boldsymbol{x}|$.
The Discriminator may have to initially implement an effectively low-bodied
operator to facilitate initial gradients, but later in training become higher-
bodied to learn global features. That is not to say that the Discriminator
should only classify marginals of the bitstring such as in Ref. Leadbeater
_et al._ (2021). Rather, the architecture and initialization should be such
that the operator $\mathcal{M}$ in the Pauli basis initially contains low-body
terms but can include high-body terms during convergence. Interestingly, the
interpolation from trainable to faithful could be naturally full-filled during
training when the Generator and Discriminator are optimized in tandem.

Fine tuning the interplay between the loss function gradients, density of
local minima and the faithfulness of a generative loss is beyond the scope of
this work, but is an important direction for future research. We especially
emphasize the necessity to evaluate the implications of our results on models
and datasets of practical relevance. In Section IV we take steps in this
direction by investigating training a QCBM to model real data from the HEP
domain.

###  III.3 Quantum strategies: quantum fidelity

Figure 7: Study of loss concentration with the local quantum fidelity loss
function. Numerical evidence that the local quantum fidelity loss function
does not exhibit global or explicit loss function barren plateaus. It does
however exhibit expressivity-induced barren plateaus, as it is the case for
all VQA-type loss functions in the form of Eq. (19). In contrast, the global
quantum fidelity variance decays exponentially at all circuit depths. The
numerical setup is the same as for the MMD in Fig. 6.

While the classical fidelity in Eq. (8) is an explicit cost function, the
quantum fidelity, defined in Eq. (15), allows for a simple known quantum
estimation strategy. Key to the quantum fidelity loss is to interpret the
training distribution as a target state
$\ket{\phi}=\sum_{\boldsymbol{x}}\sqrt{\tilde{p}(\boldsymbol{x})}\ket{\boldsymbol{x}}$.
The QCBM model loss can then be rewritten as the expectation of an observable,
e.g. in the form of Eq. (19), with $\rho=|\phi\rangle\langle\phi|$ and
$O=|\boldsymbol{0}\rangle\langle\boldsymbol{0}|$ being the all-zero projective
measurement. Crucially, as $O=|\boldsymbol{0}\rangle\langle\boldsymbol{0}|$ is
a global projector, the quantum fidelity is subject to a globality-induced
barren plateaus Cerezo  _et al._ (2021) and the loss exponentially
concentrates towards one Arrasmith  _et al._ (2022). That is, we have

$\displaystyle{\rm
Var}_{\boldsymbol{\theta}}[\mathcal{L}_{QF}(\boldsymbol{\theta})]\in\mathcal{O}(1/b^{n})\;.$
(47)

This global-measurement-induced barren plateau can however be avoided by
localising $\mathcal{L}_{QF}(\boldsymbol{\theta})$. That is, we replace the
global projective measurement $|0\rangle\langle 0|$ with its local version
$H_{L}=\frac{1}{n}\sum_{i=1}^{n}\ket{0_{i}}\bra{0_{i}}\otimes\mathbb{1}_{\bar{i}}$,
where $\bar{i}$ indicates all qubits except qubit $i$. The new localised
version of the quantum fidelity loss is given by

$\displaystyle\mathcal{L}_{QF}^{(L)}(\boldsymbol{\theta})=1-\bra{\phi}U(\boldsymbol{\theta})H_{L}U^{\dagger}(\boldsymbol{\theta})\ket{\phi}\;.$
(48)

This local loss is faithful to its global variant for product state training
in the sense that it vanishes under the same conditions Khatri  _et al._
(2019), i.e. when the QCBM distribution matches the data distribution exactly.
However, it enjoys trainability guarantees via the results of Ref. Cerezo  _et
al._ (2021). This implies that, unlike the MMD and other classical losses that
utilize the conventional measurement strategy, the local quantum fidelity can
effectively distinguish between high-order marginals even at $k=1$ bodyness.
However, although the local loss function can evade global measurement-induced
BPs, it still suffers under BPs from other sources, such as expressibility or
noise. Additionally, it is not yet explored how practical a fidelity loss is
for the purposes of generalizing from training data.

Fig. 7 depicts numerical variance results for the fidelity loss on a range of
datasets, circuit depths and numbers of qubits. For all datasets, the local
quantum fidelity exhibits only polynomially decaying variance over random
parameters when the quantum circuits are not too deep. As a reference, we
additionally depict the global quantum fidelity which exponentially decays for
all circuit depths.

The challenge now becomes how to estimate
$\mathcal{L}_{QF}^{(L)}(\boldsymbol{\theta})$ using measurements from the
quantum computer. The seemingly straight-forward approach is to prepare the
initial state $\ket{\phi}$, evolve it under
$U^{\dagger}(\boldsymbol{\theta})$, and then evaluate the observable defined
by $H_{L}$ through measurements in the computational basis. However, loading
classical data into a quantum state $\ket{\phi}$ is not expected to be
feasible in general. In Appendix D, we propose an approach that can be used to
estimate $\mathcal{L}_{QF}^{(L)}(\boldsymbol{\theta})$ using a series of
Hadamard tests without needing to prepare $\ket{\phi}$. We note that, while in
theory our approach requires a number of Hadamard tests that scales with the
amount training data, we expect stochastic techniques, such as stochastic
gradient descent Sweke  _et al._ (2020), to be sufficient in practice.

##### Broader Implications.

In this section we have presented one example of a quantum strategy to measure
a fidelity-based loss for quantum generative modelling. While this approach
puts more load on the quantum computer as compared to losses employing the
conventional measurement strategy, it enjoys simultaneous trainability and
faithfulness to the target distribution.

An interesting extension would be to explore other quantum approaches for
efficiently training QML models. One could for example attempt to compute the
KL divergence or other explicit losses directly on the quantum computer.
Although the implementation of non-linear operations on quantum computers has
been demonstrated in Ref. Holmes  _et al._ (2023); Gilyén  _et al._ (2018);
Martyn  _et al._ (2021), we are not yet aware of quantum strategies beyond one
related demonstration for the Rènyi divergence in Ref. Kieferova  _et al._
(2021). One alternative approach would be to attempt to indirectly turn the
QCBM into an explicit generative model by estimating its probabilities using
amplitude amplification or other techniques.

##  IV Training on a HEP dataset

Figure 8: Finite-shot comparison of loss functions. TVD, computed with
infinite statistics, on the training curve of the QCBMs with varying number of
qubits $n=4$, $n=9$ and $n=12$ (rows) and layers (symbols), where the
gradients are computed with different number of shots (columns) for different
loss function (colours). Figure 9: 16 qubit QCBM training. a) TVD, computed
with infinite statistics on the training curve of $n=16$ QCBM with
$\log_{2}{n}=4$ layers trained with 1000 shots. Since the computations are
more expensive, only one training curve is shown for the MMD, while for the
KLD and LQF, ten and six repetitions are performed respectively. b) Histograms
of the trained QCBMs, where only the 15 most-occurring images, shown in panel
c), are displayed. The black lines denote the target probabilities.

In this section, we perform realistic training of QCBMs on a more practical
dataset which is derived from HEP colliders experiments. We compare the
implicit cost functions MMD and local quantum fidelity (LQF) with the explicit
KL divergence for an increasing number of circuit depth $L$ and the number of
qubits $n$, and across several measurement budgets. To summarize our results,
we observe that the presence of shot noise causes the training with KLD to
fail, while the MMD and LQF both hold up significantly better.

##### Dataset.

We consider a dataset consisting of energy depositions in an electromagnetic
calorimeter (ECAL) Pierini and Zhang (2020). The data was generated using a
Monte Carlo approach (theGeant4 toolkit Agostinelli  _et al._ (2003)), which
accurately describes the ECAL detector behaviour under a typically proton-
proton collision at a LHC experiment. The dataset consists of the energy
deposition on a $25\times 25\times 25$ grid, that we downsized to a two
dimensional grid of various sizes. The images are converted to a black and
white scale by considering the pixel ‘hit’ if the energy deposition exceeds a
certain threshold, which is chosen as one tenth of the mean energy deposit. We
map each pixel to a qubit and take the state $|1\rangle$ to represent a hit.
This dataset naturally has a polynomial support and thus is precisely the type
of dataset that we might hope to learn using quantum machine learning.

##### Training.

We use a parametrized quantum circuit of the form

$U(\boldsymbol{\theta})=\left[\prod_{l=1}^{L}\text{R}(\boldsymbol{\theta}_{l})W_{l}(\boldsymbol{\alpha}_{l})\right]\text{R}(\boldsymbol{\theta}_{0})\;,$
(49)

where R$(\boldsymbol{\theta}_{l})$ is a layer of arbitrary single qubit
unitaries that can be parameterised using $3n$ Euler angles,
$W(\boldsymbol{\alpha}_{l}):=\prod_{i=1}^{n-1}\text{CX}_{i,i+1}\text{RY}_{i}(\alpha_{l}^{i})\text{CX}_{i,i+1}$
acts as parametrized entangling gate with CXi,j a CNOT gate between qubits $i$
and $j$ and RY${}_{i}(\alpha_{l}^{i})$ a single qubit rotation of qubit $i$
around the $y$-axis, and the parameters
$\boldsymbol{\theta}=\\{\boldsymbol{\theta}_{l},\boldsymbol{\alpha}_{l}\\}$.
We use the total variation distance (TVD), see Eq. (7), as a common metric to
assess the performance of each loss function. To verify performance
accurately, we compute the TVD using exact simulation. The gradients for each
loss are computed using the parameter shift rule Schuld  _et al._ (2019) which
provides estimates of the analytical gradient, and the parameters are updated
with the ADAM Kingma and Ba (2015) optimizer with a decaying learning rate
lr$(t)$ = $\max{(0.01e^{-\beta t},10^{-5})}$, where $t$ is the optimization
step and $\beta=0.005$ the decaying rate. The computation of the KLD is
stabilised using a regulariser of $\epsilon=10^{-6}$, which has been tuned
through trial and error. To follow best-practices with the MMD, we average the
gradient estimation over several different bandwidths

$\boldsymbol{\sigma}=\begin{pmatrix}0.01,&0.1,&0.25,&0.5,&1,&10\end{pmatrix}n\;,$
(50)

which incurs no additional quantum resources. This makes the loss full-bodied
and thus keeps the model trainable while aiding convergence. We note that
these are likely not the optimal bandwidths to average over but it
demonstrates the best-practice approach.

##### Results.

Fig. 8 shows the TVD, computed with infinite statistics, on the training curve
of the QCBMs with varying number of qubits $n\in\\{4,9,12\\}$ (rows) and
layers (symbols), where the gradients are computed with different number of
shots (columns) for different loss function (colours). The lines denote the
median over ten random parameter initialisation while the shaded area denotes
the $25\%$ to $75\%$ percentile. We observe that the performance of the KLD
quickly deteriorates as the number of shots is reduced while the MMD and local
quantum fidelity (LQF) remain more stable. We further observe that increasing
the expressivity of the QCBM from $\log_{2}{n}$ to $n$ layers does not lead to
a significant increase in performance for a low number of shots.

To demonstrate the scalability to larger systems, we also train a $n=16$ QCBM
with $\log_{2}{n}=4$ layers and 1000 shots per function evaluation. We
performed a manual search over different ansatz structures and used the best
performing one across the losses. Hence, a slightly different ansatz from Eq.
(49) is used, where the difference is in the entanglement map, which is
replaced by $\prod_{i<j}\text{CX}_{i,j}$. In panel a) of Fig. 9, we depict the
median and $25\%$ to $75\%$ percentiles for the KLD and LQF over 10 and 6
random repetitions respectively, whereas we only show a single representative
example for the MMD. In panel b) we show the probability histograms of the 15
most occurring images in the dataset, as well as the final respective model
probabilities. The corresponding $4\times 4$ pixel images are displayed at the
bottom panel c).

In this 16-qubit example, it appears that the LQF is no longer performing on-
par with the MMD, as was the case in Fig. 8 for smaller system sizes. A
possible explanation is that one chooses all relative phases in the data state
$|\phi\rangle=\sum_{\boldsymbol{x}}\sqrt{\tilde{p}(\boldsymbol{x})}|\boldsymbol{x}\rangle$,
which strongly reduces the number of wavefunctions that minimize the LQF loss
even though they may produce the desired measurement distribution. This may
not only produce less solutions, it also enforces that the ansatz needs to be
able to express exactly that state. While this could be leveraged using
specialized real-valued ansätze, this is not attempted here. We conclude that
the practical properties of the LQF loss as compared to implicit losses using
the conventional measurement strategy are still to be studied in more detail.

To emphasise the importance of the size of the support, in Appendix E we also
consider an exponential version of the dataset, by using a negative logarithm
transformation. We find in this case that the KLD does not suffer from
exponential concentration and can be trained. This explains the successes
previously observed for training QCBMs using the KLD for small scale problems.
However, as the amount of classical training data cannot scale exponentially
these successes are not relevant to larger, non-classically simulable,
problems.

##  V Discussion

In this work, we have introduced the notion of explicit and implicit losses,
which broadly reflect the capabilities of explicit and implicit generative
models Mohamed and Lakshminarayanan (2016). We argue that these concepts
provide a useful framework to understand the trainability of quantum
generative models. In particular, we argue that the mismatch between the
indirect access to probabilities provided by implicit models with the explicit
probabilities required by explicit losses renders implicit models untrainable
via explicit losses. More concretely, focusing our attention on quantum
circuit Born machines as a commonly used implicit model, we prove that
pairwise explicit losses exponentially concentrate (Theorem 1). This result
prohibits efficient training using a large class of commonly-used losses
including the KL divergence, JS divergence and the total variation distance.
Such losses may however be usable with explicit “quantum” generative models
such as tensor network Born machines Han  _et al._ (2018b); Cheng  _et al._
(2019); Vieijra  _et al._ (2022).

Crucially, our results assume access to a polynomial (in the number of qubits)
number of training data samples and measurements from the quantum circuit.
With only moderate numbers of qubits, this assumption is unnecessary and
explicit losses such as the KL divergence may appear to be trainable (see e.g.
Refs. Hamilton  _et al._ (2019); Leyton-Ortega  _et al._ (2021); Coyle  _et
al._ (2021); Zhu  _et al._ (2019); Rudolph  _et al._ (2022a)). More generally,
if we restrict the number of qubits used to classically simulable sizes, this
assumption can be lifted and one could use quantum generative models purely
for their efficient sampling capabilities. However, to harness the full
potential of quantum generative modelling one surely wants to push to non-
classically tractable problem sizes, at which point this assumption is
essential. For example, even with only 50 qubits, access to $\sim
2^{50}\approx 10^{15}$ training samples or quantum measurements is
unrealistic.

While formulated initially for random quantum circuits, the intuition
underlying Theorem 1 suggests our no-go result extends to scenarios where the
implicit generative model’s measurement distribution only has polynomial
support (e.g., near-identity initialization of the circuit Grant  _et al._
(2019)), as well as beyond the pairwise explicit form of the explicit loss.
One exception may be if the quantum generative model has a strong inductive
bias. Hence our work further motivates the search for new methods for
constructing parameterised circuits with strong inductive biases (e.g. via
warm starts Rudolph  _et al._ (2022b) or incorporating symmetry constraints
Larocca  _et al._ (2022b); Schatzki  _et al._ (2022); Nguyen  _et al._ (2022);
Ragone  _et al._ (2022); Meyer  _et al._ (2022)).

In contrast to explicit losses, implicit losses are naturally suited to
training implicit models. Within this line of thought, we have identified the
MMD loss with a Gaussian kernel as a promising implicit loss for training
QCBMs. We show that this loss can be interpreted as the expectation value of a
quantum observable, where crucially the properties of the observable depend on
the bandwidth parameter $\sigma$. In the common case where $\sigma$ is
independent of the system size, $\sigma\in\mathcal{O}(1)$, the observable
becomes predominantly global and thus exponentially concentrates. Conversely,
when $\sigma$ scales linearly with the system size, $\sigma\in\Theta(n)$, the
low-body interaction terms in the observable are largely dominant over the
global terms and hence exhibits large gradients. We provide a rigorous
theoretical guarantee for the case of a QCBM with a randomly initialized
tensor product ansatz, showing that the MMD variance scales at least
polynomially with the number of qubits. Based on the low-body operator
interpretation of the MMD loss and numerical evidence up to 20 qubits, we
further argue that the MMD loss with $\sigma\in\Theta(n)$ should remain
trainable for quantum circuits with $\mathcal{O}(\log(n))$ depth.

Our main results for explicit and implicit losses assume the conventional
strategy for estimating a generative loss function from an implicit model,
where the model provides a set of samples in the computational basis, which
are then used to estimate the loss in conjunction with the training data
samples. While this is the standard classical strategy, quantum generative
models can employ alternative quantum strategies by leveraging quantum
computing power. As an example, we propose the local quantum fidelity as a
trainable loss function for generative modelling. Developing alternative
quantum strategies for training quantum generative models is an interesting
avenue for future research. A natural candidate might be, as suggested in Ref.
Coyle  _et al._ (2020), to implement the MMD loss with a quantum kernel, where
the kernel values themselves are estimated using quantum computers. While one
could hope for a potential quantum advantage with this approach (especially
when training on quantum data), there is the additional challenge that quantum
kernels without inductive bias tend to exponentially concentrate Thanasilp
_et al._ (2022).

To put our conclusions to the test, we studied how these loss functions
perform in more practical scenarios with data derived from High Energy Physics
experiments at the LHC. This dataset naturally satisfies our assumptions of a
polynomial number of data samples at all system sizes. Our training results
are found to be consistent with our theoretical predictions in which both the
MMD and quantum fidelity losses significantly outperform the KLD loss when a
strict measurement budget is employed.

Finally, while our work addresses the question of whether a given loss
exhibits non-exponentially vanishing gradients, we stress that this is just
one ingredient among many to ensure the success of quantum generative
modelling. Of particular importance is the observation that models with local
losses will generally struggle to learn global correlations due to the
function’s inability to distinguish high-order features in the data. Hence we
advocate using full-body losses, which contain both low and high-body terms,
for quantum generative modelling. More broadly, the ability of a model to
successfully generalize will also presumably depend on the choice in loss, but
this is beyond the scope of this work. Nonetheless, ensuring non-vanishing
loss gradients and ensuring faithfulness of the loss function are critical
steps since failing here precludes the successful training and generalization
of quantum generative models. Hence, our work constitutes an important first
step to understanding of the barriers that need to be overcome to achieve a
quantum advantage in generative modelling.

###### Acknowledgements.

The authors would like to acknowledge Christa Zoufal and Marco Cerezo for
helpful discussions. ST and ZH acknowledge support from the Sandoz Family
Foundation-Monique de Meuron program for Academic Promotion. OK, SV and MG are
supported by CERN through the CERN Quantum Technology Initiative.

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one´

Appendix

##  Appendix A Technical nuances in explicit and implicit losses

In this work, we have argued that there are two main classes of loss functions
for generative modelling tasks. First are explicit losses which have the form

$\mathcal{L}_{\rm
expl}(\boldsymbol{\theta}):=\sum_{\boldsymbol{x}_{1}...\boldsymbol{x}_{r}}f\Big{(}p(\boldsymbol{x}_{1}),...,p(\boldsymbol{x}_{r}),q_{\boldsymbol{\theta}}(\boldsymbol{x}_{1}),...,q_{\boldsymbol{\theta}}(\boldsymbol{x}_{r})\Big{)}\,,$
(51)

where $f(\cdot)$ is a function that depends on the target probabilities
$p(\boldsymbol{x}_{i})$ and model probabilities
$q_{\boldsymbol{\theta}}(\boldsymbol{x}_{i})$ for data variables
$\boldsymbol{x}_{i}\in\mathcal{X}$ with $i=1,\,...\,,r$ (but not the data
samples themselves). The other are implicit loss functions which are expressed
as

$\mathcal{L}_{\rm
impl}(\boldsymbol{\theta}):=\mathbb{E}_{\boldsymbol{x}_{1},...,\boldsymbol{x}_{r}\sim\\{p,q_{\boldsymbol{\theta}}\\}}\,g(\boldsymbol{x}_{1},...,\boldsymbol{x}_{r})\,,$
(52)

where $g(\boldsymbol{x}_{1},...,\boldsymbol{x}_{r})$ is some function that
depends on the data (but not probabilities), and an expectation is over data
variables $\boldsymbol{x}_{1},...,\boldsymbol{x}_{r}$ sampled either from the
data distribution $p$ or the model distribution $q_{\boldsymbol{\theta}}$.

While almost all practical loss functions for generative models fall either
into the definitions of explicit or implicit losses (see Sec. II.2), there
exists a technical caveat which allows loss functions to be classified both as
explicit and implicit. To provide a rather general example, we consider a
general loss of the form

$\displaystyle\mathcal{L}(\boldsymbol{\theta})$
$\displaystyle=\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})p(\boldsymbol{y})g(\cdot)$
(53) $\displaystyle=\mathbb{E}_{\boldsymbol{x}\sim
q_{\boldsymbol{\theta}},\boldsymbol{y}\sim p}[g(\cdot)]$ (54)

where the function $g$ is for now left arbitrary. By comparing Eq. (53) and
Eq. (51), we can see that for $\mathcal{L}(\boldsymbol{\theta})$ to be an
explicit loss $g$ cannot depend on $\boldsymbol{x},\boldsymbol{y}$.
Conversely, by comparing Eq. (54) and Eq. (52), for
$\mathcal{L}(\boldsymbol{\theta})$ to be implicit, $g$ cannot depend on
$p(\boldsymbol{x})$ or $q_{\boldsymbol{\theta}}(\boldsymbol{y})$ . Thus, it
appears impossible for such a function to be both explicit and implicit.
However, taking $g$ as the Kronecker delta
$\delta_{\boldsymbol{x},\boldsymbol{y}}$ allows one to straddle the
definitions. That is, for $g=\delta_{\boldsymbol{x},\boldsymbol{y}}$,
$\mathcal{L}(\boldsymbol{\theta})$ is implicit as it can still be written as
an average over samples in Eq. (54) but also we can write
$\mathcal{L}(\boldsymbol{\theta})=\sum_{\boldsymbol{x}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})p(\boldsymbol{x})$
which is of the form of an explicit loss given in Eq. (51).

This case is not just purely hypothetical and arises, for example, for the MMD
loss, Eq. (13), with the kernel
$K(\boldsymbol{x},\boldsymbol{y})=\delta_{\boldsymbol{x},\boldsymbol{y}}$.
Here we have

$\displaystyle\mathcal{L}(\boldsymbol{\theta})=$
$\displaystyle\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})q_{\boldsymbol{\theta}}(\boldsymbol{y})\delta_{\boldsymbol{x},\boldsymbol{y}}-2\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})p(\boldsymbol{y})\delta_{\boldsymbol{x},\boldsymbol{y}}+\sum_{\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}}p(\boldsymbol{x})p(\boldsymbol{y})\delta_{\boldsymbol{x},\boldsymbol{y}}$
$\displaystyle=$
$\displaystyle\sum_{\boldsymbol{x}\in\mathcal{X}}q_{\boldsymbol{\theta}}(\boldsymbol{x})^{2}-2q_{\boldsymbol{\theta}}(\boldsymbol{x})p(\boldsymbol{x})+p(\boldsymbol{x})^{2}$
$\displaystyle=$
$\displaystyle\sum_{\boldsymbol{x}\in\mathcal{X}}\left(q_{\boldsymbol{\theta}}(\boldsymbol{x})-p(\boldsymbol{x})\right)^{2}\;,$
(55)

which takes the form of a pairwise explicit loss (see Eq. (3)). Consequently,
the MMD is always an implicit loss function, but with this particular choice
of kernel, it can additionally gain an explicit character. Interestingly, this
loss is now very related to the TVD loss in Eq. (7) for which there exists no
implicit form.