ArXiv catchup

 Deadlines abound so I haven't been following arXiv postings that closely. Some papers of note from the last few weeks:

Dielectric Mie Voids: Confining Light in Air. The authors demonstrate a metasurface analogue of photonic crystal fibers. A neat idea that could enable ultra-low loss flat optics.

Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps. "We prove that a wide class of variational quantum models -- which are shallow, and exhibit no barren plateus -- have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known." End-users beware: this work adds to evidence that variational quantum algorithms may not be scalable up to useful problem sizes.

Persistent homology analysis of a generalized Aubry-André-Harper model. Our own recent work on applying topological data analysis to study localization transitions in photonic lattices. What we found particularly exciting is that this is our first example of TDA discovering an unanticipated effect - the emergence of disorder-free eigenstates for certain model parameters.

Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks. Spoiler: the nonlinear activation functions are implemented digitally. Incorporating useful nonlinear optical response into integrated photonic neural networks while outperforming conventional electronic circuits is a big challenge. One promising potential solution is to employ measurement+based nonlinearities + feedforward.

Fabrication-Robust Silicon Photonic Devices in Standard Sub-Micron Silicon-on-Insulator Processes. With the continuing interest (and hype) in topological photonics it's important to keep in mind conventional non-topological approaches for designing photonic devices. For example, wider waveguides are more robust to fabrication imperfections, at the expense of being multimode. This is a problem, because waveguide bends will then induce coupling between the different guided modes, corrupting signals. Here the authors demonstrate how a clever choice of bending profile can suppress unwanted inter-modal coupling while not increasing the device footprint.

Breakdown of quantization in nonlinear Thouless pumping. Quantized adiabatic pumping of solitons attracted a lot of interest last year. This theoretical analysis shows how quantization can break down for moderate nonlinearity strengths due to the emergence of loops in the adiabatic energy spectrum, leading to dead-ends in the adiabatic path resulting in sudden non-adiabatic transitions of the soliton.

Topological data analysis using noisy intermediate-scale quantum processors

Topological data analysis (TDA) is an interesting potential application of future quantum computers. One step of the topological data analysis pipeline is to find the null space of the combinatorial Laplacian of a simplicial complex, a problem whose complexity grows exponential in the simplex dimension k. It is remarkable that the combinatorial Laplacian has a certain structure that makes it efficient to implement using quantum circuits, which enabled Lloyd and collaborators to propose a quantum algorithm for topological data analysis back in 2016: Quantum algorithms for topological and geometric analysis of data. However, being based on quantum phase estimation, this algorithm is not suitable for the noisy intermediate-scale quantum (NISQ) processors current available, apart from proof-of-concept experiments.

Last year Ubaru and collaborators proposed a NISQ-friendly TDA algorithm. What is very nice about this algorithm is that it has depth and qubit requirements linear in the number of input data points n, it potentially offers an exponential speed-up compared to the best classical algorithm, and it is not variational. In my opinion, the variational quantum algorithms for NISQ will have serious issues with scaling up to useful (i.e. classically-intractable) system sizes. Therefore I find this NISQ-QTDA algorithm quite exciting.

The NISQ-TDA algorithm uses a few neat tricks: the quantum phase estimation algorithm (not suitable for NISQ devices) is replaced with random sampling of the moments of the graph Laplacian to provide an estimate of the dimension of its null space, giving a circuit depth linear in the number of input data points n. Additionally, intermediate measurements and reset of ancilla qubits reduce the number of ancillas from O(n^2) to O(n). In a subsequent work the group has shown that the boundary operator used to construct the graph Laplacian can be implemented exactly using 2(n-1) two-qubit rotations plus a single qubit rotation. 

QTDA is one of the few quantum machine learning algorithms that offers an exponential speedup without requiring QRAM tricks, which suggests its speedup will be robust to dequantization approaches (i.e. analysis that takes the data-encoding bottleneck into account).

Still, there are a few limitations. The original algorithm by Lloyd and collaborators only computes the Betti number of a single simplicial complex. On the other hand, TDA generally requires computation of the persistent Betti numbers, using a range of scales to obtain a family of simplicial complices. Recent studies [arXiv:2111.00433, arXiv:2202.12965] have proposed quantum algorithms for computing persistent Betti numbers, but only suitable for fault-tolerant quantum computers. It will be interesting to see whether these approaches can be generalized into NISQ-friendly algorithms. This is definitely a subject to keep an eye on.

Cuts to ANU Physics

Research funding shortfall triggers 25% cut to ANU physics department

Australian universities were hit particularly hard by covid: A large proportion of their research funding came from international student fees, and international student numbers plummeted due to the border closure.

 

Budget cuts forced the Australian National University's physics department to cut 25% of its staff and merge its ten departments into five. 

 

As a graduate of ANU physics this is sad to see; nowhere else in Australia could one find such a breadth of expertise. Breadth is important because undergraduates often have no idea in which area they will end up specialising. For example, a bad experience with an optics lab in high school made me swear off anything to do with optics and photonics in university; I even skipped the (recommended) 2nd year optics course. My mind was changed by a 3rd year research project which showed me how the subject had plenty of interesting theoretical and numerical problems for a theorist.

It is appalling that funding agencies would rather dole out money for new buildings than pay salaries for those working inside. For context, ANU physics has just finished the construction of a new building. ANU has also carried out extensive redevelopments of the campus since I graduated, with several examples of old but functional or quite new buildings being demolished to make way for redevelopment.

What's in a name?

Giving your model or result a catchy name greatly increases the impact of your research. 

Compare the citations of Two-dimensional massless electrons in an inverted contact and Quantum spin Hall effect.

The title you give your paper is important. Don't rush it.

On a related note, those working on soliton theory will be familiar with the nonlinear wave equation

$$\partial_t^2 \phi - \partial_x^2 \phi  + m^2 \sin \phi = 0,$$

which is called the Sine-Gordon equation "for obvious reasons" in Rubinstein's original analysis of its soliton solutions. A footnote in this paper gives credit for this brilliant name to Professor Martin Kruskal, who has an impressive list of scientific achievements spanning nonlinear waves, surreal numbers, and wormholes.

Resource-efficient quantum machine learning using matrix product states

Practical applications of noisy intermediate-scale quantum (NISQ) processors will require algorithms that combine qubit-efficiency and gate-efficiency with quantum speedups. This is especially challenging because qubit-efficient algorithms are typically based on highly-entangled qubits that require deep circuits to generate and manipulate.

For example, while optimally qubit-efficient encodings require only log(N) qubits, where N is the input data size, the encoding circuit depth will be polynomial in N. This is bad because NISQ processors are practically limited to circuit depths linear in N. At the other extreme, classical data can be encoded as an N-qubit product state using single qubit rotations, but this kind of entanglement-free encoding will not out-perform a classical computer. What is needed for practical applications is a way to control the level of compression so that one can make full use of the available qubit numbers and circuit depth.

A new preprint posted to arXiv shows how matrix product state-based data encodings allow one to control the trade-off between qubit and gate resources: Data compression for quantum machine learning. Specifically, the bond dimension $\chi$ of a matrix product state controls its degree of entanglement; data can be encoded into a matrix product state using a circuit depth of O(poly($\chi)\log N)$. Qubit compression is achieved by segmenting the data into a product state of patches, where each patch is described by a matrix product state. Larger patches enable stronger compression, at the cost of requiring a larger bond dimension to accurately encode the input data.

As an application of the approach, the authors combine their matrix product state data encoding with a matrix product state-based quantum classification algorithm. Using shallow hardware-efficient circuits the authors demonstrate modest accuracy at classifying images from the Fashion-MNIST dataset. The matrix product state classifier does however require variational optimization of the gate parameters, requiring on the order of 100 gradient-free optimizer iterations to converge for the small problem sizes considered. This will increase the cost of running the classifier on real quantum hardware. It would be timely to study the combination of matrix product state-based data encodings with non-variational shallow quantum algorithms, such as quantum kernel methods.