From structured light to structured waves

Introductory textbooks on quantum mechanics and electromagnetism typically use plane waves, standing waves formed from a superposition of two counter-propagating plane waves, or twin slit interference to illustrate concepts such as the role of boundary conditions, phase velocity, energy transport, interference, and so on. However, these special cases actually too simple to capture the full breadth of wave physics. A nice example of this is shown in Figure 1 of the review article "Singular Optics: Optical Vortices and Polarization Singularities", which was one of my first introductions to the field:

(a) Twin slit interference results in alternating bright and dark fringes, the latter corresponding to lines of vanishing intensity. (b) In three slit interference the intensity only vanishes at points, which correspond to singularities of the field's phase in (c). Panel (d) shows a close up of a pair of oppositely-charged phase singularities.

In the 2000s and early 2010s, studies of the peculiarities of these kinds of structured wave fields were focused on optics, where the availability of devices such as spatial light modulators made phenomena associated with structured waves conveniently accessible. There is now growing interest in structured waves in other wave systems including electron beams, acoustics, water waves, and condensed matter physics. Recent developments in this direction are summarised in a Journal of Optics article, "Roadmap on structured waves," that was just published today. Thanks to Konstantin Bliokh for the invitation to contribute to this article and the mammoth effort of collating and coordinating the work of the 49 coauthors!

Quantum chemistry with subspace states: the conclusion

 Just over a year ago I wrote about a paper on quantum machine learning using subspace states, which inspired a project we undertook on applications of similar quantum states to variational quantum circuits for quantum chemistry and condensed matter physics. Over the weekend our manuscript was published in Physical Review A!

We were fortunate to have three knowledgeable referees who gave constructive and insightful comments on the original manuscript. We heavily revised the manuscript compared to the original arXiv preprint to not only improve the presentation, but also emphasize the broader applicability of the subspace space approach, specifically the ability to prepare correlated fermionic ansatz states beyond pairwise correlations. Our approach can yield substantially shallower quantum circuits for solving problems where the electron density (number of electrons d / number of orbitals used N) is small, for example when trying to extrapolate finite basis set calculations to the complete basis set limit. This is illustrated in the figure below, taken from the paper:

Estimated two-qubit gate depth per occupied mode d to prepare an N-mode Slater determinant and pairwise-correlated ansatz states using subspace states, compared to existing d-independent and linear in N approaches.

Will there be a useful quantum advantage for topological data analysis?

 

We don't know yet. 

Prominent applications of topological data analysis (TDA) including Mapper-based visualisation are based on fast and interpretable heuristics. While quantum TDA may speed up the calculation of high-dimensional Betti numbers, it doesn't help with understanding when and why high-dimensional topological features might be important, and whether they need to be computed to high precision or classical Monte Carlo methods can give a sufficient accuracy in practice. What high-dimensional Betti numbers might be good for needs to be determined empirically using machine learning benchmark datasets and the best classical algorithms.

Beyond the Betti number problem, for which quantum algorithms have already been proposed, it will be interesting to explore what other TDA methods could be sped up using quantum subroutines. For example, the nonzero eigenvalues of the persistent Laplacian also seem to be useful as features for machine learning algorithms, in contrast to traditional persistent homology methods that focus only on the zero or near-zero eigenvalues of the persistent Laplacian. The speedup for quantum persistent homology comes from being able to construct the persistent Laplacian exponentially faster the best-known classical methods. If there is useful information that can be extracted from the persistent Laplacian without requiring the quantum singular value transformation or rejection sampling, the resource requirements for a quantum advantage would be reduced enormously.

Thanks to the the team at QCWare for inviting me to give a seminar on this topic and the thought-provoking discussions afterwards!

arXiv highlights

Quantum-noise-limited optical neural networks operating at a few quanta per activation

Suitably-trained optical neural networks can still perform classification tasks accurately using low intensity light with a low signal to noise ratio. This suggests that specialized light-based analogue hardware for machine learning may offer a route towards reducing the enormous energy consumption of neural networks!

Dissipative mean-field theory of IBM utility experiment

Another approach towards reproducing the results of IBM's kicked Ising model quantum simulation experiment, this time using mean field theory. The Appendix gives a simple rule of thumb for estimating the quantum volume of specific devices based on their two-qubit gate and readout fidelities and compares some different hardware providers.

Maximally-Localized Exciton Wannier Functions for Solids

Wannier functions - localized states constructed as a superposition of Bloch waves from an energy band of interest - are an important tool of the condensed matter physicists' trade. This work presents a method for constructing maximally-localized Wannier functions for multi-particle states, focusing on applications to excitons (electron-hole pairs).

Tensorized orbitals for computational chemistry

This work presents a tensor network-based compression of the matrix elements that need to be computed and stored when performing quantum chemistry calculations, based on Tensor Cross Interpolation. This is yet another example of how tools from quantum many-body physics can be used to speed up time-consuming computational tasks - no working quantum computer needed!