arXiv highlights

Here are some papers that caught my eye over the past month:

Germain Curvature: The Case for Naming the Mean Curvature of a Surface after Sophie Germain

This essay argues that the intrinsic curvature of a surface, aka the Gaussian curvature, should be named instead the Germain curvature, since Gauss was not the first to study it.

I remember attending a lecture by Sir Michael Berry (of Berry phase fame) where he made a compelling argument against naming new objects or effects after people, on account of the three "Laws of Discovery":

"1. Discoveries are rarely attributed to the correct person

2.Nothing is ever discovered for the first time

3. To come near to a true theory, and to grasp its precise application, are two very different things, as the history of science teaches us. Everything of importance has been said before by someone who did not discover it."

Indeed, versions of the Berry phase had been previously decades before Berry, by Pancharatnam, Rytov, and others. For this reason he prefers the name "geometric phase." Similarly, intrinsic curvature is perhaps a more suitable alternative to Gaussian curvature.

The problem with naming effects after people is that the nature of the effect becomes opaque unless one already knows what it means. The situation becomes even worse when different groups decide to name the same effect after different people. On the other hand, simple yet descriptive names including geometric phase and intrinsic curvature reveal some sense of what is meant to the outsider. The absence of a simple-sounding name may indicate that we don't really understand the effect.

An Aperiodic Monotile

The authors discover a family of shapes that can tile the 2D plane, but only aperiodically. The shapes are non-convex mirror-asymmetric polygons. Tiling the plane involves placing a mixture of the polygon and its reflection, but the two can never be arranged to form a regular pattern. Can this kind of aperiodic tiling lead to novel physical properties of some system or model? For example, tight binding lattices can be obtained from tilings by identifying corners as "sites", with coupling between sites linked by edges of the tiling shapes.

Spectral localizer for line-gapped non-Hermitian systems

The localizer theory I have discussed previously (here and here) is now generalized to non-Hermitian systems! This is relevant to understanding the properties and robustness of certain topological laser models.

A quantum spectral method for simulating stochastic processes, with applications to Monte Carlo

This preprint shows that the quantum Fourier transform can be used to efficiently simulate random processes such as Brownian motion. In contrast to previous "digital" quantum Monte-Carlo approaches, here the authors consider an encoding in which the value of the random variable is encoded in the amplitude of the quantum state, with different basis vectors corresponding to different time steps. Since Prakash's earlier work on quantum machine learning using subspace states was the inspiration of our recent quantum chemistry work I think this paper is well worth a closer read!

Photonic quantum computing with probabilistic single photon sources but
without coherent switches

 If you want to learn more about the photonic approach for building a fault tolerant quantum computer (being pursued by PsiQ), you should read Terry Rudolph's always-entertaining papers. Even though the approaches presented in this manuscript (first written in 2016-2018) are now obsolete this is still well worth a read as a resource on the key ingredients of potentially-scalable methods for linear optical quantum computing.

An Improved Classical Singular Value Transformation for Quantum Machine Learning

The field of quantum machine learning has seen two phases. The first phase was sparked by the discovery of the HHL algorithm. HHL and its descendants promised an exponential speedup for certain linear algebra operations appearing widely-used machine learning techniques, arguably triggering the current boom in quantum technologies. However, running these algorithms on any useful problem will require a full fault-tolerant quantum computer.

Consequently, novel quantum algorithms for machine learning have attracted interest as a possible setting for achieving useful quantum speedups before a large scale fault-tolerant quantum computer can be developed. The power of these newer algorithms is much less certain and still under intense debate. Nevertheless, researchers could find solace in the hope that, even if these NISQ-friendly algorithms do not end up being useful, eventually we will achieve a quantum advantage using HHL-based algorithms.

The dequantization techniques pioneered by Ewin Tang and collaborators are starting to suggest that even a quantum advantage based on fault-tolerant algorithms such as HHL may turn out to be a mirage. This paper presents a new efficient classical sampling-based algorithm that reduces the potential quantum speedup for singular value transformations from exponential to polynomial. This affects a variety of quantum machine learning algorithms, including those for topological data analysis, recommendation systems, and linear regression.

 

 

Fusion-based quantum computation

A brief summary of the great talks by Mercedes Gimeno-Segovia and Terry Rudolph at the 6th Quantum Africa Conference on PsiQ's approach for building a large scale photonic fault-tolerant quantum computer using entangling measurements: fusion-based quantum computation. See also an earlier perspective by Rudolph published in APL Photonics.

Fusion-based quantum computation can be seen as a middle ground between the gate model (employed in most other platforms for quantum computing) and measurement-based quantum computation, which replaced nonlinear operations (hard to do using quantum states of light) with a suitable sequence of single qubit measurements.

The first step in measurement-based quantum computation is to prepare a large scale entangled resource state that is big enough to perform the desired computation. This is hard.

Fusion-based on the other hand employs two-qubit (entangling) measurements as its basic building block. This allows the calculation to be performed using a steady stream of constant-size resource states. These resource states also require multiple photons to be entangled, which is hard, but at least their size is bounded. Creating constant size resource states with sufficient fidelity and speed is the challenge which PsiQ believes they can solve.

What distinguishes PsiQ from their competitors is that their platform is easily scalable to billions of logical qubits and gates using commercial silicon lithography employed to make all our conventional electronic computers. I think the only other platform which can make such a claim are the silicon qubits being pursued by a few groups, including the UNSW spin-off Silicon Quantum Computing. By contrast, the quantum processors currently available on the cloud (superconducting circuits and ions trap) are (relatively) large and clunky devices requiring painstaking optimisation and babysitting but nevertheless might only support a few logical qubits if quantum error correction can be implemented.

An interesting point raised in the talks was that error-corrected devices will be very different from any of the currently-available quantum processors; once you have working error correction many of the details of the physical qubits become unimportant. Thus, PsiQ's approach is to create billions of "good-enough" physical qubits rather than a lower number of perfect qubits.

PsiQuantum's take on quantum chemistry using quantum computers

PsiQuantum takes a contrarian view on quantum computing: forget NISQ, focus on fault-tolerant systems with millions of qubits and error correction.

A few weeks ago PsiQuantum published a paper analyzing the resources required to carry out commercially-relevant simulations of battery chemistry: Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules. They also posted a summary on their website.

So far I only had a chance to skim the paper, but as someone focusing on NISQ algorithms I found the different approaches required for fault-tolerant algorithms quite interesting. For example, I had thought that state preparation might be a bottleneck in applying quantum phase estimation to classically-intractable quantum systems, but this doesn't appear to be a problem for high precision quantum chemistry applications:

Concerns over the viability of preparing such an ansatz efficiently are encapsulated by the “orthogonality catastrophe,” the observation that the overlap between the true ground state and some ansatz wave function decreases exponentially as the system size increases. However, it has been shown that there are sophisticated classical methods for preparing trial wavefunctions with sufficient overlap, for states of up to O(100) orbitals using simple-to-prepare states such as the single Slater determinant obtained from the Hartree-Fock method (alternatively, methods for multideterminant state preparation can be used) [arXiv:1809.05523]

Another part of the paper I found interesting was Appendix C, which presents a logarithmic-depth circuit for state preparation using Givens rotations. From what I understand, this construction is limited to the fault-tolerant regime, since it requires multi-qubit Pauli operations.