Exploring channel distinguishability in local neighborhoods of the model space in quantum neural networks

## Abstract

Abstract With the increasing interest in Quantum Machine Learning, Quantum Neural Networks (QNNs) have emerged and gained significant attention. These models have, however, been shown to be notoriously difficult to train, which we hypothesize is partially due to the architectures, called ansatzes, that are hardly studied at this point. Therefore, in this paper, we take a step back and analyze ansatzes. We initially consider their expressivity, i.e., the space of operations they are able to express, and show that the closeness to being a 2-design, the primarily used measure, fails at capturing this property. Hence, we look for alternative ways to characterize ansatzes by considering the local neighborhood of the model space, in particular, analyzing model distinguishability upon small perturbation of parameters. We derive an upper bound on their distinguishability, showcasing that QNNs with few parameters are hardly discriminable upon update. Our numerical experiments support our bounds and further indicate that there is a significant degree of variability, which stresses the need for warm-starting or clever initialization. Altogether, our work provides an ansatz-centric perspective on training dynamics and difficulties in QNNs, ultimately suggesting that iterative training of small quantum models may not be effective, which contrasts their initial motivation.

## 1 Introduction

With the increasing computational requirements of state-of-the-art ML models, the limitations of classical hardware, as theorized by Moore’s law (Shalf, ), are becoming increasingly prevalent (Schulz et al., ; Reed et al., ). Therefore, alternative computing paradigms are heavily studied and the immense potential of Quantum Computing, combined with recent advances in quantum hardware, have made Quantum Machine Learning (QML) a promising candidate.

Within that framework, Quantum Neural Networks (QNNs) are heavily studied, as they are suitable for near-term hardware. Besides initial successes, however, the Barren Plateau phenomenon has been proven under various circumstances, meaning that gradients vanish exponentially in the number of qubits, leading to essentially flat loss landscapes that affect trainability (Cerezo et al., ).

In light of these issues, we hypothesize that the architectures, called ansatzes, are part of the problem. QNN ansatzes consist of fixed and trainable gates. However, due to hardware limitations, mainly so-called Hardware Efficient Ansatzes (HEAs) (Kandala et al., ) are used, that are built primarily considering hardware constraints.
Therefore, we hypothesize that executing the circuit does not lead to a meaningful feature representation.

Currently, it is unclear how individual operations affect the model space, or, when including data, the data representation. There are no standardized ansatzes for Machine Learning (ML) and, due to the many degrees of freedom, ansatz tuning poses an optimization problem with intractable search space. Further, there is little research on the ansatzes themselves, i.e., independently of data and loss, and it is non-obvious how feature extraction works in a QNN.

In particular, in classical ML, a vanilla CNN is ultimately designed to consider local features in lower layers, before considering global features in deeper layers, independently of the dataset. Transformers (Vaswani, ) work similarly, however, the trainable one-qubit and fixed two-qubit gates that are used in QNNs do not allow drawing intuitive conclusions about how features are extracted.

Therefore, the goal of this paper is to analyze and characterize QNN architectures. We aim to take a first step at exploring ansatzes and the model space, initially by considering their expressivity. Using tools from Quantum Information Theory and extensive numerics, we later analyze how the quantum circuit, i.e., the applied operation, changes upon parameter update, thus, providing insights into the model space. Our contributions are as follows.

- •
Based on literature, we argue that closeness to a 2-design, a widely used measure for expressivity, does not adequately represent the possible set of operations in the unitary group.
- •
To investigate the model space, we study how an ansatz changes upon parameter update for different HEAs. In particular, we introduce a measure for the distinguishability of an ansatz upon parameter update and provide an upper bound. It shows that many parameters are required to be able to distinguish the two operations, contrasting with the initial framework.
- •
We provide numerical evidence that each ansatz is hardly distinguishable from itself upon parameter update during training, even in early stages, when larger updates occur.
- •
The observed behavior has remarkable similarities to Barren Plateaus, thus, insinuating that the architectures, independently of the data or loss used, may be flawed and play a fundamental role in the trainability problems observed today.

The paper is structured as follows. In Section 2, we provide background information on QNNs, existing expressivity measures and related work(^1^11For those unfamiliar with quantum computing, we refer to Appendix A.1 for a brief overview.). In Section 3, we discuss the closeness to being a 2-design as a measure of expressivity and Section 4 introduces a measure of distinguishability of quantum channels. We elaborate on our experimental setup in Section 5, and present our results in Section 6. Finally, Section 7 discusses implications of our results and Section 8 concludes the paper.

## 2 Background

### 2.1 Quantum Neural Networks

The small system sizes and the noise in today’s Noisy Intermediate-Scale Quantum (NISQ) technology (Preskill, ) prevents execution of large-scale circuits. Consequently, so-called Variational Quantum Circuits (VQC) are applied, which are hybrid models composed of classical and quantum parts. For ML, they are now being referred to as QNNs due to similar optimization than classical ones, despite being rather related to kernel methods (Schuld, ).

Simply put, a QNN consists of (1) an input state $\ket{\psi_{\bm{X}}}$, (2) an ansatz $U(\bm{\vartheta})$, (3) a loss function, and (4) an optimizer. The features are encoded onto the quantum state by means of a unitary feature encoding method $V(\bm{X})$, and the initial state is obtained as $\ket{\psi_{\bm{X}}}=V(\bm{X})\ket{0}^{\otimes n}$, with $n$ as the number of qubits.
HEAs consists of $L$ layers of trainable one-qubit rotation gates $V_{i}(\vartheta_{i})$ and fixed two-qubit entangling gates $W_{i}$, with $i\in[1,L]$, expressed as Equation 1. A measurement is taken in the end, which is done through a Hamiltonian observable $\hat{O}$, and the result is expressed as the expectation value of the ansatz with respect to $\ket{\psi_{\bm{X}}}$ and $\hat{O}$, as shown in Equation 2.

$$ $U(\bm{\vartheta})=\prod_{i=1}^{L}V_{i}(\vartheta_{i})W_{i}=V_{L}(\vartheta_{L} )W_{L}\dots V_{1}(\vartheta_{1})W_{1}$ (1) $$

$$ $f(\bm{X},\bm{\vartheta})=\bra{\psi_{\bm{X}}}U^{\dagger}(\bm{\vartheta})\hat{O} U(\bm{\vartheta})\ket{\psi_{\bm{X}}}$ (2) $$

The loss is calculated with a classical loss function (e.g., the mean-squared error) on the predicted value (the expectation value of the QNN) and the target value, and passed to the classical optimizer. The optimizer computes the parameter updates and the circuit is again executed on the quantum computer, until some stopping criterion is fulfilled. As the optimization is done classically, this approach has the advantage of allowing a shallow quantum circuit, which limits the propagation of errors during execution in light of the significant amount of noise in today’s hardware.

While HEAs are shallow, they are highly expressive (Holmes et al., ), which is a main promise of QML stemming from the significantly larger Hilbert space. Recent works, however, uncovered that loss landscapes of today’s QNNs are plagued by Barren Plateaus (BPs), meaning exponentially vanishing gradients in problem size, leading to untrainable models (McClean et al., ).

### 2.2 Expressivity of QNNs

The expressivity of a QNN is mainly defined as its ability to represent a wide range of functions (Tao et al., ). While several measures for expressivity have been proposed (Du et al., ; Schuld et al., ; Abbas et al., ), the most common method is to evaluate the similarity between the probability distribution of the unitaries generated by the ansatz and the Haar measure(^2^22We refer to Appendix A.3 for a short introduction to unitary groups and the Haar measure.), namely the uniform distribution over all unitaries $U(d)$ (Cerezo et al., ). When the probability distribution of the unitaries follows the Haar measure up to the $t$-th moment, an ansatz is considered a t-design. Constructing such an ansatz requires exponential time (Harrow and Low, ), however, approximate t-designs can be built efficiently (Dankert et al., ).

Closeness to a 2-design as a measure of expressivity for ansatzes was first defined for the $\ket{0}$ input state by Sim et al. (). In Holmes et al. (), it is expanded to a specific input state (i.e., with encoded data) and the measurement operator, as shown in Equation 3. There, an ansatz is viewed as an ensemble of unitary transformations $\mathbb{U}=\{U(\bm{\vartheta}),\forall\bm{\vartheta}\in\bm{\Theta}\}$ over all possible parameters $\bm{\Theta}$.

$$ $\displaystyle A_{\mathbb{U}}^{(t)}(\cdot)$ $\displaystyle=\int_{U(d)}d\mu(V)V^{\otimes t}(\cdot)(V^{\dagger})^{\otimes t}$ (3) $\displaystyle-\int_{\mathbb{U}}dUU^{\otimes t}(\cdot)(U^{\dagger})^{\otimes t} \quad\forall V\in U(d)$ $$

### 2.3 Related Work

BPs have been proven for deep (McClean et al., ) and shallow circuits with global measurement (Cerezo et al., ), expressivity (Holmes et al., ; McClean et al., ), noise (Wang et al., ), entanglement (Ortiz Marrero et al., ) and it has been linked to cost concentration and narrow gorges (Arrasmith et al., ). Further, even shallow models without BPs have only a small fraction of local minima close to the global minimum (Anschuetz and Kiani, )(^3^33For further information, we refer to Larocca et al. (), who published an excellent review on the topic.).

The problem has been linked to the curse of dimensionality in Cerezo et al. (), therefore, limiting the accessible Hilbert space was proposed. However, they showed that such models can be simulated classically, given an initial data acquisition phase on a quantum computer. This was extended to a trade-off between trainability and dequantizability, i.e., the potential to match the performance using classical models, in Gil-Fuster et al. ().

Besides that, parameter initialization strategies have been proposed, with some even providing better bounds on gradient magnitudes (Rad et al., ; Wang et al., ; Kulshrestha and Safro, ; Grant et al., ), however, the results of Herbst et al. () imply that it is not the type of statistical distribution, but rather the used parameter ranges that could lead to better starting points.

Further, expressivity is commonly discussed to prove a quantum advantage with respect to classical models. The closeness to being a 2-design, as discussed in Section 2.2, is the most widely employed measure, however, others have been proposed. In particular, Equation 3 is directly related to the Frame Potential (Gross et al., ; Roberts and Yoshida, ; Nakaji and Yamamoto, ). Another approach is to analyze the entangling capability as in Sim et al. (), i.e., looking at the entanglement of states produced by random unitaries in the ensemble.
Other measures are the Covering Number from  Du et al. (), analyzing Fourier coefficients (Schuld et al., ; Caro et al., ), or the Effective Dimension from Abbas et al. (). Another approach is to look at unitary t-designs, i.e., only consider the operations without the input state(^4^44Unitary designs and the difference between unitary and state t-designs are discussed in Appendix A.3.). We employ the closeness to being a 2-design in our study, as it is frequently used, due to its intuitive interpretation. While there is a lot of research on measuring expressivity of a whole model, i.e., including data, measurement and loss, there is very little research on the architectures themselves.

Moreover, the field of QML has seen a rise in works on efficient design of ansatzes. Recent works, such as Leone et al. (), have actively highlighted that well-defined ansatzes are a crucial step for large-scale deployment. In the field of ML, such approaches particularly include the field of Geometric Quantum Machine Learning (GQML) (Ragone et al., ; Larocca et al., ; Tüysüz et al., ; Wiersema et al., ), or adaptive ansatzes (Bilkis et al., ).

Further, the practical utility of the HEA has already been questioned and studied by Leone et al. ().
They find that problems with product input states and data that satisfies a volume law of entanglement should avoid using such ansatzes, whereas area law of entanglement data leads to optimization landscapes without BPs. Moreover, the necessity of understanding and characterizing ansatzes has been discussed in Zhang et al. (), as well as in Larocca et al. ().

Changes in quantum channel or loss landscape upon perturbing parameters has not yet been studied for QNNs. However, perturbation analysis and sensitivity analysis have emerged as tools in classical ML. Examples are perturbing the input for explainability (Ivanovs et al., ; Pizarroso et al., ; Fel et al., ), or checking for stability (Testa et al., ). Similarly, weights can be perturbed for trainability (Wen et al., ), or to increase robustness and generalization (Wu et al., ; He et al., ; Dumford and Scheirer, ). To the best of our knowledge, there is no research on perturbing parameters to check how the underlying function changes, as, due to the stochasticity of quantum computing, this research question seems to be much more relevant in the quantum realm.

## 3 2-Design as a Measure of Expressivity

While 2-designs have turned out to be sufficient for many quantum information processing protocols (Holmes et al., ; Harrow and Low, ), in this section, we summarize and connect results from the literature to argue that closeness to a 2-design is an inadequate measure for model expressivity (i.e., the ability to sufficiently approximate the unitary group to a high degree).

For the argument we will use the so-called Welch Bounds which, loosely put, are inequalities related to evenly distributing a finite number of unit vectors in a vector space. This inequality was addressed in Welch (), where a lower bound on the inner-product between unit vectors was provided. Intuitively, a smaller lower bound corresponds to more evenly spread vectors in the vector space, i.e., a more uniform distribution.

Quantum state t-designs reduce integrals of polynomials over all quantum states to averages over a discrete set (cf. Equation 20). These are probability distributions over pure quantum states that replicate properties of the uniform (Haar) measure on the quantum states up to the $t$-th moment.

We measure expressivity of 2-design based architectures by comparing (generalized) Welch bound (Scott, ) values for state 2-designs against its t-design counterparts, with $t>2$. We state the inequality for a state t-design based architecture:

*State t-design Welch bound: *
The choice of the polynomial function for state t-designs is presented in Appendix A.4. We state the following inequality (cf. Appendix B.2 for the derivation):

$$ $\displaystyle\sum_{1\leq j,k\leq n}|\langle\bm{\psi}_{j}|\bm{\psi}_{k}\rangle| ^{2t}$ $\displaystyle\geq\frac{n^{2}}{\binom{d+t-1}{t}}$ (4) $\displaystyle=\frac{n^{2}}{\frac{(d+t-1)(d+t-2).....d}{t!}}\ \ \forall t\in \mathbb{N}.$ $$

It is clear that for an increasing $t$, the term in the denominator $\frac{(d+t-1)(d+t-2).....d}{t!}$ increases rapidly (as compared to Equation 26 for 2-designs), yielding a vanishing overlap between the state vector summands. This makes the inequality in Equation 4 more stringent for increasing $t$ values. This indicates that higher-order designs ($t>2$) approximating the Haar measure up to the $t$-th order impose stronger constraints, requiring greater separation between a sequence of state vectors $\{\bm{\psi}_{1},\bm{\psi}_{2},....,\bm{\psi}_{d}\}$ to maintain equality in the bounds. Intuitively, states forming $t$-designs for $t>2$ are more “sparsely distributed” compared to their 2-design counterparts. This means that 2-design models have far fewer *degrees of freedom* in building complex quantum circuits. On the contrary, t-designs ($t>2$) can efficiently construct higher-order representations capturing more complex interactions between quantum states. Thus, evaluating Welch bounds for $2$-designs corroborates its inadequacy for constituting an expressive architecture.

Currently, there is no consensus in the community about a good measure of expressivity; indeed, it seems to be an open issue to define what even constitutes a highly expressive ansatz. Further, considering that the unitaries grow exponentially in system size, concepts in the Quantum Information Theory literature are often not efficiently computable, adding another layer of complexity to defining a useful measure. Additionally, there is evidence that trainability issues go hand-in-hand with architectures that span large parts of the $SU(N)$, i.e., are expressive with respect to the set of unitaries. Hence, it is questionable whether finding an alternative measure would help in practice in finding trainable ansatzes. Therefore, we adopt a different methodology and study how the QNN changes upon perturbation of parameters, which allows analyzing training dynamics and tracking the extent to which the QNN changes during training.

## 4 Channel Sensitivity

Comparison between quantum channels is an important topic in Quantum Information Theory.
For consistency with respect to, e.g., highly entangling operations, however, operator norms need to be stable under tensor product. Therefore, the diamond norm was proposed in Kitaev () as follows.

###### Definition 1

The diamond norm of a superoperator $T$ is defined as (Harrow and Low, ; Kitaev et al., ).

$$ $\|T\|_{\Diamond}=\sup_{d}\|T\otimes{id}_{d}\|_{\infty}=\sup_{d}\sup_{X\neq 0} \frac{\|(T\otimes id_{d})X\|_{1}}{\|X\|_{1}}$ (5) $$

Here, $\otimes$ denotes the tensor product and $id_{d}$ the identity channel on $d$ dimensions, with $d\leq 2^{n}$, and $n$ as the number of qubits. The $\|X\|_{1}$ in this context is the Trace norm or Schatten 1-norm, which is defined as $\|X\|_{1}=Tr\sqrt{X^{\dagger}X}$. The diamond norm is defined for any superoperator $T$, wherein, superoperators are defined as the set of linear maps acting on a vector space of linear operators.
For a valid quantum channel $\mathcal{E}_{1}$ (a completely positive and trace preserving (CPTP) matrix), the diamond norm is strictly upper bounded by $1$ and lower bounded by 0 0. When using the diamond norm to measure the distance between two CPTP quantum channels $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$, the channels are subtracted. The resulting matrix is not necessarily CPTP, resulting in a revised upper bound of $2$.

Operationally, the diamond norm measures the maximum distinguishability between the output states of the two maps under any input state, i.e., for any input state we will be less or equally able to distinguish the output. A value of 0 0, means that they are indistinguishable, whereas a value of $2$ means they are perfectly distinguishable. Computing the diamond norm is non-trivial, however, can be reduced to a semi-definite program (Watrous, , ).
We apply the diamond norm to evaluate the distinguishability of a parametrized unitary $U(\bm{\vartheta})$, representing a quantum channel, from the unitary $U(\bm{\vartheta}+\bm{\delta})$, where $\bm{\delta}$ is a perturbation of $\bm{\vartheta}$, as follows.

###### Definition 2

We define the channel sensitivity as the distinguishability of the quantum channel from itself upon small perturbation of $\bm{\vartheta}$ by $\bm{\delta}$.

$$ $cs_{U}(\bm{\vartheta},\bm{\delta})=\|U(\bm{\vartheta})-U(\bm{\vartheta}+\bm{ \delta})\|_{\Diamond}$ (6) $$

###### Definition 1

###### Definition 2

## 5 Experimental Setup

**Table 1: Employed ansatzes**
| COMPONENT | CONFIGURATIONS |
| --- | --- |
| Parameterized | [$\text{R}_{X}$, $\text{R}_{Y}$, $\text{R}_{Z}$, $\text{R}_{X}\text{R}_{Y}$, $\text{R}_{Y}\text{R}_{Z}$, $\text{R}_{X}\text{R}_{Z}$, $\text{R}_{X}\text{R}_{Y}\text{R}_{Z}$] |
| Entangling | [CNOT, CZ, CNOT CZ] |

##### Ansatzes

We consider layered architectures in our experiments, each consisting of trainable parameterized and fixed entangling components. For the first, we use rotations around the x- ($\text{R}_{X}$), y- ($\text{R}_{Y}$), and z-axis ($\text{R}_{Z}$) of the Bloch sphere, and the controlled-NOT (CNOT) and controlled-Z (CZ) gates for the latter(^5^55We refer to Appendix A.1 for an overview of the operations.), constituting a widely-used gate set for HEAs. For a list of parameterized and entangling components, we refer to Table 1, and we run our experiments using all combinations.

We do not permute the operators, due to considerations on the expressivity per qubit. That is, if all rotations are applied, all permutations of $R_{X}(\alpha)R_{Y}(\beta)R_{Z}(\gamma)$, with $\alpha,\beta,\gamma\in[0,2\pi]$ span the $SU(2)$, the set of unitary operations on one qubit. Therefore, permuting the operations does not affect expressivity per layer. Similarly, two rotation operations per layer explore a subspace of the $SU(2)$ owing to the Lie-algebraic closure relations, e.g., $R_{X}(\alpha)R_{Y}(\beta)$ spans the upper half of the Bloch sphere, as does $R_{Y}(\alpha)R_{X}(\beta)$. Any permutation allows the same operations per qubit, which is why we deem all permutations as equal.

##### Perturbation

For the experiments, we uniformly sample parameters in range $[0,2\pi]$ and choose a random $95$% of them to perturb their values by $\pm t$% (in our case $[0.1,0.5,1]$). These values were in no way chosen randomly, rather, by training QNNs and collecting summary statistics on parameter updates, we found that these are the ranges where updates are performed. Then, we evaluate the channel sensitivity with the two obtained unitaries. We take 100 samples per parameter involved in the circuit and consider the distribution of the channel sensitivity we obtain for further analysis.

##### Training

Moreover, we want to compare the channel sensitivity for random perturbations and training runs. Therefore, we train binary classification models on the two largest classes of the wine and breast cancer datasets and use PCA to reduce the features to $2^{n}$, with $n$ as the number of qubits. Then, the features are normalized and encoded into the amplitudes of the input state. The measurement is taken in the z-basis of the first qubit, and we use the mean squared error as the loss.

To account for favorable parameter initializations, we train each architecture 50 times. We use 150 iterations, and monitor convergence and parameter changes. Further, we calculate the channel sensitivity between the quantum channels in every iteration, to get an operationally meaningful analysis of how much the model changes. This also allows evaluating whether the model changes more when updated based on the loss function, rather than by just randomly permuting parameters.

##### Qubits

Due to the computational complexity involved in solving the diamond norm, we run our experiments from one to four qubits and one to five layers. Unfortunately, due to matrix dimensions, it is challenging to scale the diamond norm any further. Nonetheless, the results are relevant, considering that they (1) provide a first step at analyzing small scale architectures, and (2) have direct implications for training dynamics that can be expected in the NISQ era.

Today’s QNNs use shallow circuits with a depth in $O(\text{log}\ n)$ and a local cost function, as they can be shown to be trainable (Cerezo et al., ). Even if favorable loss landscapes were found for deeper circuits, adding layers results in noisier loss estimates, prohibiting effective learning. Moreover, scaling in number of qubits is limited due to quantum hardware limitations, or, when the models are simulated classically, to the curse of dimensionality. Therefore, many recent works proposing QML to solve a particular problem use few qubits, e.g., Blance and Spannowsky (), with two qubits, or Yano et al. () with three and four qubits.

Further, our main goal is to take initial steps in quantifying and characterizing QNN model behavior. As has been mentioned before, due to the power of classical ML, we are unlikely to find an advantage in QML in the near term (Schuld and Killoran, ). Therefore, it would be important to focus on more fundamental questions, such as “how can we exploit quantum mechanics for ML purposes?”, to explore the potential of quantum computing for data processing. In that sense, understanding how quantum channels change upon parameter updates is a foundational topic in ansatz design.

##### Framework

Our experiments are implemented using the Python programming language. We use Pennylane (Bergholm et al., ) for obtaining the unitary transformations and QuTIP (Johansson et al., , ) for calculating the diamond norm. Unfortunately, a diamond norm computation between two operators $\|A-B\|_{\Diamond}$ with a finitely-large overlap $A^{\dagger}B\approx\mathbb{I}$, yields numerical instabilities with the QuTIP package. To account for these numerical issues, the operator can be multiplied with a global phase, which fixes a gauge (cf. *Gauge-Fixing*, in Appendix D).

## 6 Results

In the following, we first provide an upper bound on the channel sensitivity based on the applied perturbation, then we discuss our numerical results.

### 6.1 A Bound on the Channel Sensitivity

We can prove an upper bound on the channel sensitivity through Taylor expansion, which is shown in Equation 7. Therefore, we establish a direct dependence of the distinguishability of the channels to the sum of changes applied(^6^66For the precise mathematical derivation, we refer to Appendix C.). The bound holds if the Hermitian generators of the trainable gates are unitary as well, which is relevant for HEAs, as the trainable gates are exponentiations of the $X$, $Y$, and $Z$ Pauli gates. The precision of the bound depends on the magnitude of $\bm{\delta}$, however, considering that the models are trained iteratively in small steps, we expect the bound to hold.

$$ $\begin{split}\|U(\bm{\vartheta})-U(\bm{\vartheta}+\bm{\delta})\|_{\Diamond}&= \|U(\bm{\vartheta})-(U(\bm{\vartheta})+\\ &\sum_{j=1}^{dim(\bm{\delta})}\delta_{j}\frac{\partial U(\bm{\vartheta})}{ \partial\vartheta_{j}}+O(\bm{\delta}^{2}))\|_{\Diamond}\\ &\approx\|-\sum_{j=1}^{dim(\bm{\delta})}\delta_{j}\frac{\partial U(\bm{ \vartheta})}{\partial\vartheta_{j}}\|_{\Diamond}\\ &\leq\frac{\sum_{j=1}^{dim(\bm{\delta})}|\delta_{j}|}{2}\end{split}$ (7) $$

Unless the feature encoding is trained as well, the upper bound on distinguishability cannot be improved by including data. Intuitively, due to the calculation of the diamond being based on the maximum distinguishability from any input state, it includes any possible unitary data encoding as well. Mathematically, this property follows from the unitary invariance of the Schatten 1-norm.

### 6.2 Experiments

Figure: (a) Mean absolute change
Refer to caption: extracted/5918925/figures/mean_change.png

We run extensive numerical experiments to validate our bounds. First, we randomly perturb parameters, then we compare them to the channel sensitivity we observe during actual training runs.

Initially, we verify the assumptions about the magnitude of parameter changes. In training runs, the mean parameter change per update in the first 10 iterations is maximum $1.4$%, with a mean and standard deviation of $0.8\%\pm 0.4\%$, but it quickly decreases. We visualize this in Figure 1(a), with the iterations on the x-axis, and the mean change in parameters on the y-axis. Architectures with the same number of qubits are aggregated and show very narrow confidence intervals. Further, it is also visible that changes in multi-qubit systems tend to be even smaller (less than two orders of magnitude). Moreover, we observe that almost all parameters are updated in every iteration, which is shown in Figure 1(b), where the y-axis shows the percentage of parameters changed.

#### 6.2.1 Random Perturbation

Figure: (a) Change 95% of parameters by 1%
Refer to caption: extracted/5918925/figures/0.95_0.01.png

Based on these observations, we run the random perturbation experiments by changing 95% of parameters by 1%, 0.5%, and 0.1%. We visualize the results in Figure 2, showing the depth of the circuit on the x-axis and the obtained channel sensitivity on the y-axis. Per our bounds, the channel sensitivity strongly depends on the number of parameters in the circuit, hence, we visualize the results based on the number of parameters per layer.

The experiments show the same patterns as the bounds predict. In particular, fixing the qubits and parameters per layer while increasing the depth of the circuit, results in a bigger channel sensitivity. The same can be observed when fixing the number of layers and increasing the qubits. We observe that increasing channel sensitivity is more prevalent for larger systems than for deeper circuits, e.g, architectures with 2 qubits, 5 layers and 3 parameters have a smaller channel sensitivity than architectures with 3 qubits, 5 layers and 2 parameters, despite having the same number of parameters.

The behaviour is expected for ansatzes covering large parts of the search space, as the Hilbert space of a larger system is significantly bigger, scaling as $2^{n}$. This allows more indepedent search directions, therefore, potentially more changes in channel upon update. Deeper circuits can explore larger parts of the Hilbert space too, however, their expressivity stagnates when achieving overparameterization (Larocca et al., ). In this regime, more parameters will not result in more independent search directions, hence, the difference in channel sensitivity is expected to be smaller.

Further, it can be seen that there is a substantial degree of variability in the channel sensitivity. In the majority of experiments, the channels are hardly distinguishable, however, many outliers can be observed. This property could be particularly relevant for warm-starting (Puig et al., ), i.e., starting training from regions with high channel sensitivity might lead to smoother optimization.

#### 6.2.2 Training Updates

Figure: Figure 3: Comparison $\frac{\sum_{j}|\delta_{j}|}{2}$ and channel sensitivity
Refer to caption: extracted/5918925/figures/diamond-norm-vs-bound.png

Additionally, we want to consider how different the quantum channels are when updated based on loss and optimizer. In particular, it could be possible that changing the parameters during training, changes the underlying channel a lot more than perturbing in random directions. Upon analysis of our results, we can confirm the validity of our Taylor expansion even in early stages of iteration, i.e., in the $45,500$ models we trained, the bound holds for every single parameter update taken.

We plot the bounds and channel sensitivity against the iteration in Figure 3 for different layers, qubits and parameters per layer. We mainly observe three things; firstly, the confidence intervals are very narrow, i.e., even though we collect data from 50 different runs for every model, the observable channel sensitivity hardly varies. Further, there is a large discrepancy between our bound and the channel sensitivity, which grows in the number of qubits. Together with our third observation, that the channel sensitivity does not scale as clearly as the bound in the number of parameters, this provides another perspective on training difficulties in today’s QNNs.

While it is not possible to observe the magnitude of channel sensitivity that the bound predicts, the pattern can be observed when looking closely, which is visualized in Figure 4(a), for early training phases, and in Figure 4(b) for later ones. We see that the pattern exists for larger parameter changes in the beginning, however, in fine-tuning or convergence phases, the pattern is not as clear anymore.

Figure: (a) Iterations $<$ 15
Refer to caption: extracted/5918925/figures/dnorm_15.png

## 7 Discussion

Our bound establishes a direct dependence of the distinguishability of the channels during QNN training to the number of parameters. That is, the maximum distinguishability of two output states of the quantum channels, scales with (1) the magnitude of changes and (2) the number of parameters. As iterative updates with a small learning rates are applied, which is independent of the number of parameters in the circuit, the dependence is largely dominated by the number of parameters.

This could significantly contribute to the trainability issues of today’s QNNs. In particular, for reasonably sized models, the channels may be distinguishable in early training stages, but hardly distinguishable when fine-tuning the ansatz later. Comparison of our bound to the channel sensitivity obtained
in training runs reveals that during training, the channels are even less distinguishable, i.e., our bound seems to be quite loose. While the channel sensitivity for random permutation is still considerably lower than our bounds would predict, a lot more variation can be observed.

This confirms our initial motivation of studying the model space independently of the loss and data, as these are global properties of ansatzes. In particular, we want to draw attention to the remarkable similarities to the BP phenomenon. It seems that the architectures significantly contribute to the untrainable models we observe today, independently of data and loss. Our findings have implications to start a more thorough investigation of the model space of the ansatzes that are currently used, and, ultimately, stress the need for good initialization and warm-starting. It is, however, non-obvious how to find such starting points without trial-and-error.

## 8 Conclusion

Altogether, our work provides a first step at exploring the model space of QNNs. In particular, we provide a picture of the local neighborhood of the model space of an ansatz, which suggest that the channels, at least for small scale circuits, do not differ significantly upon parameter update. Our results for random perturbations reveal that there is a substantial degree of variability of the channel sensitivity depending on the neighborhood, hence, stress the need to study the model space more thoroughly. This could reveal valuable information for parameter initialization or warm-starting.

Considering that QNNs were designed for NISQ technology, thus limited to shallow, small-scale circuits due to hardware limitations, the community will be limited to such small instances in the near-term. Therefore, our findings have direct implications on the meaningfulness of iteratively optimizing such circuits. While it is true that the models can be scaled to work for larger circuits, we have overwhelming evidence through works on BPs, that this is inherently difficult. Together with results on classical simulability and dequantizability in the absence of BPs, this work extends the evidence that we need a paradigm shift in Variational Quantum Computing or even QML altogether.

## Reproducibility Statement

We provide the source code for our experiments in a GitHub repository(^7^77[https://github.com/sabrinaherbst/qnn_local_neighborhood](https://github.com/sabrinaherbst/qnn_local_neighborhood)) with extensive documentation and listed dependencies.

## Acknowledgments

The authors would like to thank Adrián Pérez-Salinas for feedback on earlier versions of this work. This work has been partially funded through the Themis project (Trustworthy and Sustainable Code Offloading), Austrian Science Fund (FWF): Grant-DOI 10.55776/PAT1668223, the Standalone Project Transprecise Edge Computing (Triton), Austrian Science Fund (FWF): P 36870-N, and by the Flagship Project HPQC (High Performance Integrated Quantum Computing) # 897481 Austrian Research Promotion Agency (FFG).

## Appendix A Additional Background

### A.1 Quantum Computing

Quantum Computing (QC) works with quantum bits (qubits). While classical bits take either the value 0 or 1, qubits are in a so-called superposition of the two states, meaning they are in both states at once, which is shown mathematically in Equation 8, where $\alpha$ and $\beta$ are called probability amplitudes. We can do calculations with the superposed qubit, however, once we read out results, we have to take a measurement, where the qubit will collapse into state $\ket{0}$ with probability $|\alpha|^{2}$, and into state $\ket{1}$ with probability $|\beta|^{2}$. Due to this inherent non-determinism in QC, executions are always done multiple times.

$$ $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}\quad\text{s.t. }|\alpha|^{2}+|\beta|^{2} =1\quad\alpha,\beta\in\mathbb{C}$ (8) $$

A qubit is characterized by 4 numbers (the real and imaginary parts of $\alpha$ and $\beta$), which, when translated into polar coordinates, gives a convenient geometrical representation on the so-called Bloch sphere. This plays a significant part in today’s QNNs, as the trainable gates are rotations of the individual qubits around the x-, y- and z-axis of this sphere.

Multiple qubits form a quantum register, and its associated quantum state represents $2^{n}$ states at once, where $n$ is the number of qubits, which can make QC very powerful. A quantum state can be entangled, meaning that individual qubits can be correlated, such that an action on one qubit affects the ones it is entangled with as well.

Quantum states are manipulated through unitary transformations or quantum gates ($U^{\dagger}U=\mathbb{I}$, where $U^{\dagger}$ is the adjoint of $U$). Unitary transformations are linear and preserve vector lengths, i.e., applying a unitary to a quantum state ensures that the squared magnitudes of the probability amplitudes still sum up to one. Every unitary is generated by a Hermitian generator ($H=H^{\dagger}$), i.e., $U=e^{-iH}$. We describe the action on an initial quantum state as a so-called unitary evolution, which may involve one or more unitary operations (unitary matrices form a group, hence, are closed under multiplication). We also refer to this as a quantum channel in the following. While the term quantum channel encompasses unitary evolutions, it is a broader term, meaning it can also describe more general dynamics of quantum systems, such as decoherence or noise. It is defined as a linear map that is completely positive and trace preserving (Watrous, ).

Among the most fundamental operations in QC, are the so-called Pauli matrices, which are Hermitian matrices shown in Equation 9. $\sigma_{x}$ applies a NOT (bit-flip) operation ($\ket{0}\rightarrow\ket{1}$, $\ket{1}\rightarrow\ket{0}$), $\sigma_{y}$ changes the phase and bit ($\ket{0}\rightarrow i\ket{1}$, $\ket{1}\rightarrow-i\ket{0}$), and $\sigma_{z}$ applies a phase-flip ($\ket{0}\rightarrow\ket{0}$, $\ket{1}\rightarrow-1\ket{1}$).

$$ $\sigma_{x}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}\quad\sigma_{y}=\begin{bmatrix}0&-i\\ i&0\end{bmatrix}\quad\sigma_{z}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}$ (9) $$

Taking the Pauli matrices as generators, we can rotate a qubit around the x-, y- and z-axis around the Bloch sphere, arriving at the definition of the $R_{X}$, $R_{Y}$ and $R_{Z}$ rotation in Equation 10.

$$ $\text{R}_{\text{X}}(\theta)=e^{-i\frac{\theta}{2}\sigma_{x}}\quad R_{Y}(\theta )=e^{-i\frac{\theta}{2}\sigma_{y}}\quad R_{Z}(\theta)=e^{-i\frac{\theta}{2} \sigma_{z}}$ (10) $$

Further, we define the two entangling operations we use, CNOT (or CX) and CZ, in Equation 11. The operators entangle two qubits $i$ and $j$, where $i$ is referred to as the control qubit and $j$ is the target qubit. In general terms, the entangling gates apply $\sigma_{x}$ or $\sigma_{z}$ to the target qubit if the control is in state $\ket{1}$.

$$ $\text{CNOT}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\\ \end{bmatrix}\quad\text{CZ}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1\\ \end{bmatrix}$ (11) $$

### A.2 Inner-product & overlap between vectors

An inner-product space is defined as a vector space $V$ over a field $\mathbb{F}$ (typically $\mathbb{R}$ or $\mathbb{C}$) endowed with an inner-product map $\langle.|.\rangle$:

$$ $\langle.|.\rangle:V\times V\rightarrow\mathbb{F}$ (12) $$

The inner-product is a generalization of the dot-product between vectors and measures the overlap or coherence between vectors sampled from a vector space.

### A.3 Unitary groups and the Haar Measure

We will provide key information on unitary groups and the Haar measure in this section. For an excellent tutorial on the topic, we refer to Mele (), which we also base this section on. We begin by defining the unitary group $\mathcal{U}(d)$ and the special unitary group $S\mathcal{U}(d)$, which are key concepts when studying quantum computing, as follows.

###### Definition 3

For $d\in\mathbb{N}$, the unitary group $\mathcal{U}(d)$ is the group of isometries of $d$-dimensional complex Hilbert space $\mathbb{C}^{d}$. These are canonically identified with the $d\times d$ unitary matrices ($\mathbb{C}^{d\times d}$):

$$ $\displaystyle\mathcal{U}(d)=\big{\{}\mathcal{V}\in\mathbb{C}^{d\times d}| \mathcal{V}\cdot\mathcal{V}^{\dagger}=\mathcal{V}^{\dagger}\cdot\mathcal{V}= \mathbb{I}_{d}\big{\}}$ (13) $$

The unitary group can be decomposed as $\mathcal{U}(d)=S\mathcal{U}(d)\times U(1)$. Here, the subgroup $S\mathcal{U}(d)$ is called the *special unitary* group, while $z=e^{i\theta}\ \forall z\in U(1)$ is a phase restricted on a unit-circle.

###### Definition 4

For $d\in\mathbb{N}$, the special unitary group $S\mathcal{U}(d)\subset\mathcal{U}(d)$ are a group of $d\times d$ unitary matrices ($\mathbb{C}^{d\times d}$) that have a unit determinant:

$$ $\displaystyle S\mathcal{U}(d)=\big{\{}\mathcal{G}\in\mathcal{U}(d)|\ \text{det }(G)=1\big{\}}$ (14) $$

The Haar measure forms a uniform probability distribution over sets of unitary matrices, in fact, it is unique for compact groups, such as the $\mathcal{U}(d)$.

###### Definition 5

(Mele, ) We define the Haar Measure on the $\mathcal{U}(d)$, as the left and right invariant probability measure $\mu_{H}$ over the group. That is, for all integrable functions $f$ and $\forall V\in\mathcal{U}(d)$:

$$ $\displaystyle\int_{\mathcal{U}(d)}f(U)d\mu_{H}(U)$ $\displaystyle=\int_{\mathcal{U}(d)}f(UV)d\mu_{H}(U)$ (15) $\displaystyle=\int_{\mathcal{U}(d)}f(VU)d\mu_{H}(U)$ (16) $$

Due to properties of a probability measure, it holds that $\int_{S}d\mu_{H}(U)\geq 0$, $\forall S\subseteq\mathcal{U}(d)$ and $\int_{\mathcal{U}(d)}1d\mu_{H}(U)=1$. It is therefore possible to set the expectation value with respect to the probability measure equal to the integral over the Haar measure as $\mathbb{E}_{U\sim\mu_{H}}[f(U)]=\int_{\mathcal{U}(d)}f(U)d\mu_{H}(U)$. This leads to the definition of the moment operator of the Haar measure.

###### Definition 6

(Mele, ) For all operators $O$, the t-th moment operator based on probability measure $\mu_{H}$ is defined as follows.

$$ $M_{\mu_{H}}^{(t)}(O)=\mathbb{E}_{U\sim\mu_{H}}[U^{\otimes t}OU^{\dagger\otimes t}]$ (17) $$

###### Definition 7

(Mele, ) A distribution $v$ over a set of unitaries $S\subseteq\mathcal{U}(d)$ is a unitary t-design if and only if the following holds for all operators $O$.

$$ $\mathbb{E}_{V\sim v}[V^{\otimes t}OV^{\dagger\otimes t}]=\mathbb{E}_{U\sim\mu_ {H}}[U^{\otimes t}OU^{\dagger\otimes t}]$ (18) $$

State t-designs are weaker versions of unitary t-designs in the sense that unitary t-designs induce state t-designs when they act on a fixed input state. These are defined as follows.

###### Definition 8

(Mele, ) A state t-design is a probability distribution $\eta$ over a set of states $S$ if and only if the following holds.

$$ $\mathbb{E}_{\ket{\psi}\sim\eta}[\ket{\psi}\bra{\psi}^{\otimes k}]=\mathbb{E}_{ \ket{\psi}\sim\mu_{H}}[\ket{\psi}\bra{\psi}^{\otimes k}]$ (19) $$

Further, we define an equivalent definition of state t-designs as follows, which will be used as well.

###### Definition 9

(Hoggar, )
Let $\mathbb{C}^{d}$ denote a d-dimensional Hilbert space with orthonormal basis set $\{\ket{n_{k}}\}_{k=1}^{d}$. Owing to their redundancy under global phase transformations and normalization, the quantum states in this space correspond to points in the complex-projective space $\mathbb{CP}^{d-1}$ (Bengtsson and Zyczkowski, ).
Thus, we define the nontrivial complex-projective t-design as a set of states $S\subsetneq\mathbb{CP}^{d-1}$, sampled according to some probability measure $\mu$, satisfying,

$$ $\mathbb{E}_{\ket{\psi}\in S}f(\ket{\psi})=\int_{\mathbb{CP}^{d-1}}f(\ket{\psi} )d\psi.$ (20) $$

Where, $f(\ket{\psi})$ is a polynomial of at most degree t in its amplitudes and complex-amplitudes of $\ket{\psi}$ respectively. The canonical measure $d\psi$ defined on the set of such quantum states, is the unique unit-normalized volume measure invariant under unitary group action $\mathcal{U}(d)$.

Often it is not strictly required to have an exact t-design, but rather a distribution close to a t-design, which is termed $\epsilon$-approximate t-design.

###### Definition 3

###### Definition 4

###### Definition 5

###### Definition 6

###### Definition 7

###### Definition 8

###### Definition 9

### A.4 Choice of polynomial function for state t-designs:

The function $f(\ket{\psi})$ is a polynomial of at most degree $t$. For example, 2-designs, correspond to the average value of a quadratic (second-degree polynomial) function. Hence, a common choice of state 2-design polynomial functions is the overlap (cf. Appendix A.2 for definition) between a collection of states $\{\bm{\psi_{i}}\}_{1\leq i\leq d}$ in the Hilbert space $|\langle\bm{\psi_{j}}|\bm{\psi_{k}}\rangle|^{2}$. Similarly, for an even $t>2$, a canonical choice of the polynomial function $f$ is $|\langle\bm{\psi_{j}}|\bm{\psi_{k}}\rangle|^{2t}$.

## Appendix B Welch bounds and t-designs

##### Theorem 1:

(Welch bounds) Let $n\geq d$. If $\big{\{}\bm{v_{i}}\big{\}}_{1\leq i\leq n}$ is a sequence of unit vectors in $\in\mathbb{C}^{d}$, then,

$$ $\displaystyle\max_{1\leq j,k\leq n,j\neq k}|\langle\bm{v}_{j}|\bm{v}_{k} \rangle|^{2t}$ $\displaystyle\geq\frac{1}{n-1}\bigg{[}\frac{n}{\binom{d+t-1}{t}}-1\bigg{]},\ \forall t\in\mathbb{N},$ (21a) $\displaystyle\text{implies},$ $\displaystyle\sum_{1\leq j,k\leq n}|\langle\bm{v}_{j}|\bm{v}_{k}\rangle|^{2t}$ $\displaystyle\geq\frac{n^{2}}{\binom{d+t-1}{t}}\forall t\in\mathbb{N}.$ (21b) $$

The combinatorial (binomial) factor featuring in the denominator of Equation 21a and Equation 21b can be expanded in terms of the factorials:

$$ $\binom{d+t-1}{t}=\frac{(d+t-1)!}{(d-1)!\ t!}$ (22) $$

Here $!$ corresponds to the factorial symbol, i.e. $n!=n(n-1)(n-2)...1$.

### B.1 Simplification of Equation 21a :

Substituting Equation 22 into Equation 21a yields the following.

$$ $\displaystyle\max_{1\leq j,k\leq n,j\neq k}|\langle\bm{v}_{j}|\bm{v}_{k} \rangle|^{2t}$ $\displaystyle\geq\frac{1}{n-1}\big{[}\frac{n}{\frac{(d+t-1)!}{(d-1)!\ t!}}-1 \big{]}$ (23) $\displaystyle=\frac{1}{n-1}\big{[}\frac{n}{\frac{(d+t-1)(d+t-2)....d}{t!}}-1 \big{]}$ (24) $$

Here in the denominator, we have used the factorial property, $(d+t-1)!/(d-1)!=(d+t-1)(d+t-2)...d(d-1)!/(d-1)!=(d+t-1)(d+t-2)~{}\ldots~{}d$.

### B.2 Simplification of Equation 21b :

Using Equation 22 it follows that:

$$ $\sum_{1\leq j,k\leq n}|\langle\bm{v}_{j}|\bm{v}_{k}\rangle|^{2t}\geq\frac{n^{2 }}{\frac{(d+t-1)(d+t-2)...d}{t!}}.$ (25) $$

For the special case, $t=2$, we get the inequality for 2-designs,

$$ $\sum_{1\leq j,k\leq n}|\langle\bm{v}_{j}|\bm{v}_{k}\rangle|^{4}\geq\frac{n^{2} }{\binom{d+1}{2}}=\frac{n^{2}}{(d+1)d/2}.$ (26) $$

## Appendix C Channel Sensitivity Bound

### C.1 Preliminaries

Before deriving the bound, we want to state some preliminaries for partial derivatives of parameterized unitaries. Equation 27 shows the partial derivative of a circuit (Holmes et al., ) which utilizes the convenient representation of a unitary in terms of its generator. The partial derivative essentially splits the original unitary in two parts, and we define the fractions in Equation 28.

$$ $\partial_{j}U(\bm{\vartheta})=-iU_{j+1\rightarrow dim(\bm{\vartheta})}H_{j}U_{ 1\rightarrow j}$ (27) $$

$$ $\begin{split}U_{l\rightarrow m}(\bm{\vartheta})&=\prod_{j=l}^{m}V_{j}( \vartheta_{j})W_{j}\\ &=\prod_{j=l}^{m}e^{-i\vartheta_{j}H_{j}}W_{j}\end{split}$ (28) $$

### C.2 Derivation of the Bound

We can now analyze the difference in diamond norm upon slight perturbation of parameters in Equation 29. We use a Taylor expansion and truncate after the first order. We arrive at Equation 31 after applying the partial derivative from Equation 27. We use the triangle inequality and homogeneity, which are both necessary conditions for matrix norms (Horn and Johnson, ), to arrive at Equation 32 and Equation 33 respectively. Arriving at Equation 34 is non-trivial, as, in general, it is not guaranteed that multiplying hermitian and unitary matrices results in a unitary. However, HEAs use $R_{X}$, $R_{Y}$ and $R_{Z}$ rotations with generators $X$, $Y$ and $Z$ respectively, which are known to be hermitian and unitary. Since unitaries form a group that is closed under multiplication, it forms a valid quantum channel, which evaluates to $1$(Watrous, , Proposition 3.44).

$$ $\displaystyle\|U(\bm{\vartheta})$ $\displaystyle-U(\bm{\vartheta}+\bm{\delta})\|_{\Diamond}$ (29) $\displaystyle=\|U(\bm{\vartheta})-\bigg{(}U(\bm{\vartheta})+\sum_{j=1}^{dim( \bm{\delta})}\delta_{j}\frac{\partial U(\bm{\vartheta})}{\partial\vartheta_{j} }+O(\bm{\delta}^{2})\bigg{)}\|_{\Diamond}$ (30) $\displaystyle\approx\|U(\bm{\vartheta})-(U(\bm{\vartheta})+\sum_{j=1}^{dim(\bm {\delta})}\frac{\partial U(\bm{\vartheta})}{\partial\theta_{j}}\delta_{j})\|_{\Diamond}$ (31) $\displaystyle=\|-\sum_{j=1}^{dim(\bm{\delta})}\frac{\partial U(\bm{\vartheta}) }{\partial\theta_{j}}\delta_{j}\|_{\Diamond}$ (32) $\displaystyle=\|-\sum_{j=1}^{dim(\bm{\delta})}-\frac{i}{2}U_{j+1\rightarrow dim (\bm{\delta})}H_{j}U_{1\rightarrow j}\delta_{j}\|_{\Diamond}$ (33) $\displaystyle\leq\sum_{j=1}^{dim(\bm{\delta})}\|\frac{i}{2}U_{j+1\rightarrow dim (\bm{\delta})}H_{j}U_{1\rightarrow j}\delta_{j}\|_{\Diamond}$ (34) $\displaystyle=\sum_{j=1}^{dim(\bm{\delta})}|\frac{i}{2}\delta_{j}|\|U_{j+1 \rightarrow dim(\bm{\delta})}H_{j}U_{1\rightarrow j}\|_{\Diamond}$ (35) $\displaystyle=\frac{\sum_{j=1}^{dim(\bm{\delta})}|\delta_{j}|}{2}$ (36) $$

## Appendix D Gauge Fixing

Considering the numerical instabilities whenever $A^{\dagger}B\approx\mathbb{I}$, we perform Gauge-Fixing by adjusting the global phases of the two unitaries, which ensures numerical stability without affecting the diamond norm.

Figure: Algorithm 1 Gauge-fixed diamond norm between difference of quantum channels

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: Do the authors envision that researchers in the field would be able to leverage their findings to build better application-specific models? Could standard architecture search approaches be coupled with suitable distinguishability metrics to build better ansatzes?

What proportion of the expressivity issues observed with QML models are related to the ansatz vs the data encoding?

As all experiments seem to be classical simulations, do the authors have an idea how much will noise when executing the model on a device contribute to expressivity issues?
soundness: 3
strengths: The paper looks from a different angle to the well-known problem of training QNNs and presents new approaches to assessment of quantum model expressivity while discussing the need to develop architectures suitable for QML. This seems like an important work for the QML community.
summary: The study presents the authors' work on evaluating the expressivity of different ansatzes, while trying to propose novel measures and metrics to capture the behaviour of QNNs during training.
weaknesses: The paper did not explain what particular elements or spatial organisation of gates of ansatzes may be the most contributing to expressivity or trainability issues observed in QNNs.

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 3
questions: Question:

1. My understanding is that the diamond norm is derived to provide a channel distinguishability based on a single measurement of the state. Does this translate to distinguishability of a QNN model, where i) we might make use of several measurements and ii) one might make use of a specific measureable (e.g. single qubit magnetisation) rather than the whole state?
2. What can be done with the proposed channel distinguishability metric? The fact that it can be bounded from above for small perturbations does not seem useful. All neural networks (classical or quantum) typically won't change too much in a single training step---why is upper bounding this change in terms of some norm useful to a practitioner?

Suggestions:

2. I think section 3 and 4 can be elaborated on more. In particular, the criticism of 2-designs as a measure of expressivity is hard to read without going back and forth between the main text and the appendix several times.
3. Elaborating on section 3 and 4 would probably need more space. Some of the figures can be moved to the appendix, e.g. the three rows of figure 2 are conveying a similar message, so perhaps two of the rows can be placed in the appendix? Same with figure 3. The conclusions and discussion could be shortened as well.
4.  Ensure that all figures are included in a vector format or lossless format. Maybe I'm not seeing things right, but the quality of the figures seems a little bit off.
5. A minor point unrelated to the core contributions of the paper. But amplitude encoding of features into a QNN feels like a non-standard way of conducting the numerical experiments. Apart from amplitude encoding not being near-term friendly, the resulting QNN will not be particularly expressive. I.e. one will end up with something like `<PCA|Hermitian|PCA>`, which, since the PCA is a linear combination of features, will result in a sum of pairwise interactions with no zeroth, first or cubic and higher order terms. Encoding PCA components into single-qubit rotations would be a more standard approach, to my knowledge.
soundness: 4
strengths: 1. The introduction and background material are particularly well-written and offer a broader ICLR readership a succinct way of learning about quantum neural networks. However, it is possible that too much space has been allocated within this area at the expense of a fuller exposition of novel aspects of the work.
2. The work takes aim at foundational issues in variational quantum circuits and the points made are widely applicable to QNNs as a whole.
3. The proposed channel sensitivity metric based on diamond norms seems to be aiming at something principled, in that it seeks to characterise the action of a unitary on any state. Its application to variational quantum circuits, to my knowledge, is novel.
summary: The paper considers methods of characterising the variational ansätze represented within quantum neural networks. The paper argues against the use of so-called 2-designs as a measure for expressivity that have been used in previous works. The work goes on to introduce a notion of channel sensitivity, which considers how much (in terms of a so-called diamond norm, a norm used more in Quantum Information theory) a unitary changes under perturbation of variational parameters. Bounds for the channel sensitivity are derived by consideration of Taylor series expansions and used to argue against the distinguishability of ansätze with few variational parameters on parameter perturbation. Numerical studies then investigate the notions introduced above in terms of both random perturbations of variatonal parameters, and also perturbations introduced during a training loop.
weaknesses: 1. Sections 3 and 4 feel like they should be the main contributions of the paper, but they squeezed into about a single page. For example, precise mathematical definitions of the how the quantum states in Eq (4) are missing from the main text, which makes it hard to follow.
2. The work as a whole considers expressivity of a variational quantum circuit as a problem of the ansatz and does not give any weight to the importance of feature encoding into the quantum circuit. It is known that the spectral representation of a QNN is directly given by the feature encodings ([Schuld, 2021](https://arxiv.org/abs/2008.08605)).
3. The diamond norm under parameter change seems like an interesting metric, but it remains unclear what can be done with it.
4. Similarly, the bound on the channel distinguishability under small perturbations seems not to be useful. It seems natural that the QNNs will only change gradually under small parameter perturbations such as those encountered during training, so what use is an upper bound?
5. The main points of the work seem somewhat incongruous: the criticism of the 2-design as an expressivity metric seems like it should give way to an alternative expressivity metric. However, the channel distinguishability metric proposed does not provide an expressivity metric.

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 4
questions: 1.	Which optimizer (and optimizer hyperparameters) did you use in the experimental results? 
2.	How do the empirical results depend on the choice of optimizer and its hyperparameters?
3.	Where does the factor $\frac{1}{2}$ in (30) come from? Inserting (25) into (29) would give (30), but without additional factor $\frac{1}{2}$.
4.	Why do you use mean-squared loss, even though you are solving a classification problem?
5.	From its definition, the diamond norm seems to depend on number of qubits. How does an increase in number of qubits affect the diamond norm? 
6.	Can you provide an additional figure, which is the same as Figure 3 but without the upper bounds? 
7.	The reported results are averaged over architectures with the same number of parameters. Can you be more specific about what type of "confidence intervals" you show in the plots?
8.	For comparison reasons, it would be very insightful to analyse channel sensitivity of a non-entangled circuit. For example, for a circuit applying an $R_X$ gate to each qubit independently, what channel sensitivity do you get for $1,2,3,4$ qubits?
9. How do the results depend on your choice of input-encoding?
soundness: 3
strengths: Overall, the proposed research question appears relevant, as a theoretical understanding of quantum neural networks is, to a large extent, still missing. The paper is mostly clearly structured and well-presented. Although limited to small architectures, the considered setting is still of interest, given the hardware limitations. Numerical study considers several important combinations of architectures.
summary: The paper studies quantum neural network expressivity and model distinguishability. The aim is to better understand, at least for small architectures, the model space of so-called Hardware Efficient Ansatzes (HEAs) for quantum neural networks. To do this, the authors first argue why 2-design is an inadequate measure for model expressivity.  Then they define channel sensitivity to measure the effect of parameter changes on quantum neural network output using the diamond-norm from the literature. A Taylor-expansion-based argument allows to provide a natural upper bound on channel sensitivity. Moreover, the paper conducts an extensive numerical study to experimentally measure channel sensitivity for standard HAE quantum neural networks. As its main message, the paper states that its results ultimately suggest that iterative training of small quantum models may not be effective.
weaknesses: Some of the key conclusions drawn from the numerical results are only partially justified by what is presented in the paper, weakening the overall significance of the reported results. More specifically:

-	Some important details regarding the setup of numerical experiments (employed optimizer, optimizer hyperparameters) and results (confidence level for confidence intervals) are currently missing in the paper. Also, it appears that robustness with respect to choice of optimizer / hyperparameters has not been studied. 
-	The upper bound on channel sensitivity is an upper bound, not an equality. Some of the conclusions drawn in the discussion/abstract/introduction would only be valid in case of equality. For example, in the Discussion section, the paper states that channel sensitivity "scales with (1) the magnitude of changes and (2) the number of parameters." This statement is true for the upper bound, but not necessarily for the channel sensitivity, as the upper bound is very loose (as the experiments show) and no lower bound seems to be available. Similarly, the introduction the authors write as one of their four main contributions: "we introduce a measure for the distinguishability of an ansatz upon parameter update and provide an upper bound. It shows that many parameters are required to be able to distinguish the two operations, contrasting with the initial framework." This would be true in case of an equality, but an upper bound (which is loose, as the experiments show) does not seem sufficient to support such a claim.
-	As one of their four main contributions, the authors state "We provide numerical evidence that each ansatz is hardly distinguishable from itself". This seems to be based on Figure 3. Since Figure 3 also shows the (loose) upper bounds, the different scale of bounds and channel sensitivities in Figure 3 makes it impossible to see this solely from the figure.
-	Terminology is slightly confusing:  in Holmes et al (2021) the term expressibility was used. For classical neural networks, expressivity refers to their capability of approximating arbitrary given unknown functions. This is a fundamental property for choosing which neural network architecture to use, but does not seem to be discussed in the paper.

---

confidence: 5
contribution: 3
initial_rating: 6
presentation: 4
questions: Is there a natural reduction of certain, previously-explored measures of expressivity to the authors' introduced measure of expressivity? For instance, I believe demonstrating that having large expressivity based on a measure of closeness to the Haar measure, and showing this implies a large expressivity according to the authors' definition (averaged over parameter space), would strengthen this work.

Furthermore, what relative advantages do the authors' introduced expressivity measure have over expressivity measures based on closeness to the full (i.e., not $t$-design) Haar measure? This impacts the broader reach of this work, as to me it is currently unclear what the relative merits are of the introduced measure to this previously-studied measure.
soundness: 4
strengths: Understanding measures of expressivity for QNNs is important for understanding the to-date loosely-defined "expressivity-trainability trade-off" known to exist in quantum networks (see, e.g., arXiv:2312.09121 and arXiv:2406.07072). The authors, in a very clear fashion, describe shortcomings of certain previously-introduced measures of quantum network expressivity; given these shortcomings, the authors then introduce a novel, natural notion of expressivity that circumvents these shortcomings.
summary: The authors introduce an "expressivity" measure of quantum neural network (QNN) architectures that they argue is more natural than those previously explored in the literature. One example of the latter is a difference measure between the given QNN architecture when randomly initialized and a $t$-design (for some small $t$); the authors show that having a large expressivity according to such a measure does not necessarily indicate that the model explores a large subspace of Hilbert space. This motivates their introduced definition, which is a sensitivity measure in diamond norm of the quantum network upon perturbing the network parameters. The authors then perform some numerical experiments indeed demonstrating that this measure tracks with how one would intuitively expect "expressivity" to behave (it grows with the number of layers and trainable parameters per layer, and so on).
weaknesses: Though not the fault of the authors, the introduced measure of expressivity has the unfortunate side-effect of being difficult to estimate; this limits the scope of their numerical experiments. It also makes the relative advantage of the authors' introduced expressivity measure over an expressivity measure based on closeness to the Haar measure (rather than a small-$t$ $t$-design) unclear, given both are difficult to estimate.

Meta-review: The paper tackles a critical question in quantum neural networks (QNNs), specifically the evaluation of ansatz properties and the challenges associated with training. To this end, the authors introduced the concept of channel sensitivity as a metric for assessing ansatz characteristics. The results present a novel and theoretically robust contribution to the field. While limitations in scalability and practical applicability were acknowledged, the authors' responses successfully addressed most concerns, particularly those related to clarity and experimental rigor. Overall, the paper makes a valuable contribution to the quantum machine learning community.

Decision: Accept (Poster)