On the design of photonic quantum circuits

The paper "On the design of photonic quantum circuits" presents a framework for designing and optimizing photonic quantum circuits involving Gaussian states, unitaries, channels, and non-Gaussian effects.

The main feat is the coherent introduction of a recurrent relation to define a phase space representation of Gaussian objects using Fock representation. The method comes with a description of state-of-the-art optimization techniques of quantum circuits and specifically their photonic implementation, such as Riemann optimization, and high-performance computational techniques such as auto-differentiation. Probably the most important and useful part of this paper is that it comes with an open-source library MrMustard. I have already played with the library (before I was asked to review this paper), and it is clearly of very high quality and very intuitive to use. Importantly, and this is why I believe that the software as well as this paper will have of huge impact: The software is continuously expanded (the last update on GitHub was 3 days ago, and it counts 18 contributors).

The manuscript is an extended documentation which describes the underlying mathematics of the software. As such, it will likely become a standard text for newcomers in the field who want to learn about the details of Gaussian quantum optics.

I think they should certainly be published, but I would like the authors to consider these points before they submit a final version:

1) I see that the authors changed the manuscript title on arxiv between versions 3 and 4 from 'The recursive representation of Gaussian quantum mechanics' to 'On the design of photonic quantum circuits'. I believe that the current title does not well describe the content of the paper. After all, there are by now dozens of papers that perform designs of photonic quantum circuits. I recommend to fix and make it clear what the content is about.

2) Furthermore, while computer-designed quantum photonic circuits are heavily studied since around 2016 (10.1088/1367-2630/18/7/073033, 10.1103/PhysRevLett.116.090405, 10.1088/2058-9565/aaf59e, https://github.com/Budapest-Quantum-Computing-Group/piquasso, https://github.com/xvalcarce/QuantumOpticalCircuits.jl used in 10.1103/PhysRevA.107.062607 and many others), the authors do not cite any other works in this field [except other Xanadu paper in ref 18 and 19, QuTiP which has a different purpose and a very recent 2022 Perceval software, and later some graph-based approaches in ref45]. I recommend comparing and contrasting their methods with currently existing techniques and explaining their applicability. As the authors have the "design" in the title, the comparison should talk about other design approaches (not just simulators).

3) The algorithm performs high-performance continuous optimization. However, is it possible to perform discrete optimization, i.e. starting with a circuit ansatz and not only finding the correct parameters but trying to reduce the circuit to smaller component numbers?

4) [optional] It would be interesting to give an overview of other fields that apply auto-differentiation. For example, what distinguishes the authors' methods from classical optics design methods in the field of photonic material design (e.g. 10.1021/acsphotonics.0c00327). This field is powered by the adjoint method, which is a special case of auto-differentiation.

The manuscript provides a method to optimize quantum circuits based on Gaussian processes. This optimization is rendered possible thanks to a linear recurrence relation connecting the Fock and the Gaussian (Wigner) representations of quantum states, unitaries and channel. Specifically, optimization over Gaussian processes of a cost function in the Fock space is done by the backpropagation of a gradient from the Fock representation to the Gaussian representation and back. At the core of the presented method is a Riemannian optimization ; the authors derived a geodesic updating formula allowing to update symplectic matrices (which are the object representing Gaussian processes) along a geodesic path and according to a Riemannian gradient that can be computed from the cost function.
To display the proposed optimization methods, the authors provide three numerical experiments. They are performed using an open-source library which implements the optimization, thanks to contributions from the authors. Numerical results display convergence in the optimization of non-trivial tasks, therefore emphasising the optimization capability of both the proposed method and its implementation.

The method presented in this manuscript will appeal to readers working with photonic circuits, as I can think of multiple use-cases in which having a performant and efficient Gaussian and non-Gaussian (photodetectors) processes optimization framework would be highly appreciated. Furthermore, the Python package is of high-quality, from its simplicity of use to its documentation.
In my opinion, however, the discussion section is missing some important elements, contrasting the optimization method against existing ones, and mentioning potential limits/drawbacks of the approach (cf. Requested Changes).

Given the quality of this manuscript, the usefulness and novelty of the proposed framework, and provided that the authors answer the points mentioned in the requested changes section, I will support the publication of this manuscript in SciPost.

1. Section "V-F. Discussion"

As a general remark, the quality of this subsection does not match the rest of the manuscript. I would therefore suggest a rewrite of this section.

More specifically, here are some keys points I would like to see included in the discussion:

a. The authors refer to a missing Appendix, containing convergences results. Such results are more than welcome, the new manuscript version should therefore include them.
b. A comparison against different methods for the optimisation of general Gaussian processes would be appreciated. For example, how much faster does this method converge vs a black-box optimisation in the case where the objective function is to find a process matching (fidelity) a random unitary? Similar question if you optimise Gaussian components individually vs in bulk?
c. What is the scalability of this method? For example, given the task of maximizing the fidelity to a randomly generated unitary on n-bosonic modes, how does the convergence-time scale with n?
d. I would like to read more about the potential drawbacks and limits of this method. For example: Are there cases in which the optimisation is unstable/unreliable? What is the main computational bottleneck ? The authors mention some occurring “inaccuracies”, where do they originate from (renormalization?), and can we detect them to make the optimisation more trustworthy?


2 - References and Bibliography

A lot of the references in the bibliography are not with clickable URLs.
On a side note, I also suggest these references to the authors:

A review on Gaussian quantum information - 10.1103/RevModPhys.84.621
A framework in which heralded Gaussian states are simulated using linear combination of Gaussian states - 10.1103/PhysRevA.107.062607
A quick references to common Gaussian states and operations - arXiv:2102.05748
Quantum Optics book in which quantum system in phase space are described - 10.1002/3527608524
A framework with auto-differentiation of optical setup processes - arXiv:2310.08408

3 - Editing

Some suggestions regarding typos or English formulation:

p.5: "k + 1i is like k but the i-th index has been increased by 1" -> "k +1i is the vector k with the i-th element increased by 1.".
p.6: "and then we can write" -> "allowing us to write" .
p.12: "its last two modes:" -> replace the column by a dot.