Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The barren plateau phenomenon is a 'vanishing gradient' effect that arises in sufficiently randomly initialized parameterized quantum circuits. Specifically, the norm of the gradient falls exponentially with the number of quantum registers in the circuit. While not a problem for classical neural networks due to efficient gradient estimation procedures, gradients for parameterized quantum circuits are obtained by statistical sampling. Estimating small gradients therefore adds an exponential overhead, eliminating most possible computational advantages.
In recent years there has been an effor to handle this problem by suggesting initializations that are not 'fully random' on the space of circuits. This paper takes the following approach: a (practically reasonable) architecture is chosen and the parameters are initialized from a Gaussian distribution. The main technical message is a proof that if the variance of the normal distributions is chosen as $1/L$ where $L$ is the number of layers, the gradient decays polynomially with the number of qubits $n$ and layers $L$.
Models with the propsoed initialization are evaluated on a variational quantum eigensolver setup to find the ground state of the Heisenberg model and LiH Hamiltonian. On the considered examples, Gaussian initialization appears to outperform the setting where parameters are initialized uniformly.
strengths_and_weaknesses: The main contribution of the paper is the lower bound on the gradient. This proof is very non-trivial and involves techniques and intermediate results, that I think may be of interest when analyzing properties of Gaussian initializations in quantum circuits in general. I have not checked every statement, but I am overall convinced of correctness of the proofs. On a technical level, I think the contribution of the paper is solid. The experimental results also line up with the conclusions drawn, indicating that the phenomenon described may have applicability beyond the theorem setting.

I have two concerns: firstly, due the Gaussian initialization I am not sure that the proof in the paper suffices to say that the gradients are lower bounded *throughout* training since the distribution has morphed from that at initialization. Note that this was not an issue for the original derivation of 'barren plateau' as Haar distributions are invariant under the shifts induced by training. The second is that the restriction to Gaussians with deviation decaying as 1/L, essentially restricts the initialization to a constant neighborhood of the identity. This assumption seems to put more of a bias on the initialization than most existing approaches, and may create the possibility of adversarial problems where convergence is heavily slowed.
Experimentally, the advantage over initializing the parameters to zero seems like an artifact of their being a stationary point at identity, and the addition of some noise to perturb the initial state (in the noisy simulations) seems to remove most observed advantage for the proposed scheme.
questions: - Can there be a general argument about how the results with Gaussian initialization are maintained during the training? Can there be cases where the gradient starts large but reaches a barren region?
- As an associated question, can we reason about the bias induced by keeping the initial unitary close to the identity due to the choice of parameter variance?
- The barren plateau phenomenon was established first for Haar initializations. I did not see a comparison to that setting. Is it similar to the uniformly initialized parameter case.
limitations: Yes
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Variational quantum circuits are parametrized models that can be trained to perform mappings using gradient descent methods. In this paper, the authors propose an initialisation strategy to avoid the problem of vanishing gradients that occur when the number of qubits and the circuit depth grow. 
strengths_and_weaknesses: - Strengths:
	- Well-written
	- New initialisation: Several initialisation strategies have been studied for classical neural networks but few existing work extend these results to the quantum case. In this work, the authors apply the Gaussian initialisation strategy to variational quantum circuits and study how it may affect the training procedure by providing a theoretical and experimental analysis. 
	- Theoretical analysis: The authors start by describing the Gaussian initialisation technique and provide theoretical guarantees in different settings. The first setting corresponds to the case when the circuit architecture is made using trainable 1-qubit gates and the output is projected using local observables. These results are then extended to the global observable case and 2-qubit gates. 
	- Experimental analysis: The authors apply their technique to two quantum machine learning problems where they perform numerical simulations to study experimentally the training behaviour of the parameters. 

- Weaknesses:
	- The zero initialisation strategy seems to be fine for the performed experiments. 
questions: - Theoretical analysis: what happens when the training procedure requires many update steps ? in other words, if the convergence is very slow, is the initialisation technique still useful to avoid vanishing gradients ?
- Experimental analysis: why is the zero initialisation strategy providing competitive results to the Gaussian one ?
limitations: No
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper introduces a new initialization strategy for quantum variational circuits. This gaussian initialization strategy is shown to exponentially increase the upper bound on the gradient, with substantial implications for addressing optimization concerns of medium to large scale quantum machine learning models. Two empirical examples are provided, which show this initialization strategy demonstrates an improvement in performance. 

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Note: after author responses score 6 -> 7
strengths_and_weaknesses: Pros:
- This work provides a very important improvement in gradient bounds. Given the scale of concerns for optimising QVCs, this is a very exciting result. 
- The results for global observables are especially interesting, since previous works have focused on the benefits of local observables to trainability. 
- The paper is generally well written and conveys the point effectively
- Circuit diagrams are well done and add to the understandability
- Code is provided in supplementary material, which greatly improves replicability and experimental verification
- The survey of related work is both useful and extensive

Cons:
- The biggest problem is the empirical results. Although these experiments are just examples and the main result is the theoretical proofs, they don’t add as much as they could. Using shot noise, instead of added measurement noise would improve the realism. It would also be beneficial to add examples with more realistic circuit noise (e.g. depolarizing channels). Additionally, showing the gradient norm in Figure 3 (like in Figure 2) would be beneficial
- Citations could be condensed, e.g. “quantum simulations [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]” -> “quantum simulations [14-23]”
- Empirical comparisons to other initialization strategies would be beneficial (e.g. block initialization)
- The horizontal lines on the graphs don’t aid interpretability

questions: - The heisenberg gamma squared is said to be 1/160 (line 260), however, in the code gamma is set to 1/(8 * n_layer)**0.5. Since n_layer is 10, gamma = 1/sqrt(80) and gamma squared is thus 1/80. There seems to be a factor of two missing, where does this come from?
- Do the optimization improvements from this initialization strategy carry over to gradient free optimizers?
- A variety of gradient based optimizers have been used for QVCs, is there a reason why Adam was chosen over other methods?
- The zero initialization seems to perform very well on LiH (almost exact overlap with gaussian). Is this just a unique aspect of the problem or is zero initialization comparable in performance on large circuits (when the variance becomes very small)?

limitations: The authors sufficiently addressed the potential negative societal impact of the work.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors demonstrate that over a Gaussian prior of appropriate width, the second moment of the derivatives of a class of variational quantum algorithms (VQAs) is only polynomially small in the problem size $n$ and circuit depth $L$ (for constant problem locality $S$), which is exponentially larger than the traditional barren plateau bounds (taken uniformly over parameters). The authors also give a bound when the locality $S$ grows with the problem size, that lower bounds the expected derivative second moment by (a finite fraction of) the initial squared derivative. The authors then demonstrate numerically that their initialization scheme gives much better optimization performance than uniform initialization in a variety of VQA tasks.
strengths_and_weaknesses: The introduced bounds are novel, and though I did not check the proofs in complete detail, the authors' technical methods and proofs seem correct. I also enjoyed that these results are essentially a more rigorous understanding of the intuitive fact that training VQAs while near the identity should be similar to low-depth VQAs, and not experience barren plateaus. I think a couple of weaknesses of the paper, though, are maybe claiming too much from the shown results.

First, in their discussion of Theorem 4.2 (and Corollary 4.3), the authors imply that their results are enough to show that Gaussian initialization completely absolves VQAs with global cost functions from barren plateaus. However, these results only lower bound the second moment of the derivative with (a finite fraction of) the initial square of the derivative. This initial derivative can be very small; see, for instance, the global cost function warm-up example in "Cost function dependent barren plateaus in shallow parametrized quantum circuits" (Cerezo et al. 2021), where this initial derivative is zero, giving a trivial bound. I would have enjoyed more discussion (or examples) arguing that this bound is typically only polynomially (not superpolynomially) small.

Second, the authors' results (including now also Theorem 4.1) still rely on assumptions in the training of the model. Namely, once training is far away from the initial $\vec{\theta}=\vec{0}$, it is no longer well-approximated by a Gaussian of polynomially small variance. In fact, for a number of parameters growing superlogarithmically with $n$, roughly the volume of "allowed" region where these results are expected to hold is superpolynomially small in the volume of parameter space (polynomially small in diameter). This (that is, superlogarithmically large depth) is the regime previous barren plateau results kick in (i.e. when averaged uniformly over parameters), and I suspect they may be related. I recommend the authors make this limitation of their work more clear.

These limitations aside, I still find the work a nice, rigorous interpretation of a common approach to circumventing barren plateaus; when one has a good guess for where in parameter space the optimum is (say, near $\vec{\theta}=\vec{0}$), and expect optimization to stay within this region, barren plateaus may be avoided.
questions: * I suggest making the previously discussed limitations of the work more clear (in particular, that these results require staying near $\vec{\theta}=\vec{0}$ to hold).
* I suggest providing more intuition as to when Theorem 4.2 and Corollary 4.3 give nontrivial bounds.
* I suggest providing more examples of when the authors' results are expected to hold (e.g. in chemistry where the Hartree--Fock state is a good approximation of the true ground state, etc.).
limitations: I previously discussed what limitations of the work I believe should be more explicitly mentioned by the authors; namely, I suggest tempering the claims that Gaussian initialization solves all instances of barren plateaus, and providing more examples or intuition as to situations where one might expect the implicit assumptions on training (i.e. staying near $\vec{\theta}=\vec{0}$) to hold.
review_title: null

Meta-review: metareview: The authors propose a new random initialization of quantum neural networks which could avoid generating vanishing gradients.  Specifically, the new random (Gaussian) initialization scheme will depend on the shape of the ansatz so that the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase.  This finding is also supported by the associated empirical study.  The reviewers consider this an important step toward the understanding of the trainability of variational quantum circuits. However, some limitations of the proposal are also discussed in the reviews, and we hope the authors can make an explicit discussion of these limitations in the final version. 
confidence: Certain
Decision: Accept