Through browsing a recent edition of Nature, I recently came across the following articles on topological quantum computing, or which at least touched on aspects of condensed matter physics relevant to said research direction.

These are the papers:

- Topological superconductivity in a phase-controlled Josephson junction, https://arxiv.org/abs/1809.03076
- Evidence of topological superconductivity in planar Josephson junctions, https://arxiv.org/abs/1809.03037

and one bonus article from more recently:

- Tuning Topological Superconductivity in Phase-Controlled Josephson Junctions with Rashba and Dresselhaus Spin-Orbit Coupling, https://arxiv.org/abs/1904.08981

The bottom line of this work is that it appears to me that researchers at Microsoft, and various other institutions, are closing in on building a working topological qubit that would lend itself to scaling up to a practical quantum computer.

What is a topological quantum computer? A good question. This jargon essentially means, ‘use a Majorana fermion (a particle that is its own antiparticle) to do something weird with anyons in order to achieve highly stable qubits through topological properties that are hard to deform, then couple these together to make a computer’.

More on majorana fermions can be found here: https://arxiv.org/abs/1712.06156 . The topological ‘hardness’ of these exotic things that arise within certain condensed matter systems essentially rests on something-something braid groups, per a talk I remember Michael Freedman giving at a conference I attended in New Zealand about 13 years ago (January 2006) in Taipa, New Zealand. That was actually a very interesting conference, with a number of luminaries, including John Conway who spoke about his Game of Life. Anyway, enough name dropping; my point was that Michael Freedman spoke briefly about braid groups there as part of his work for a thinktank at Microsoft, “Microsoft Station Q”.

Another good introduction to topological quantum computing can be found here: https://arxiv.org/abs/1802.06176 .

What is a non-Abelian anyon? First of all, an abelian anyon is a quasiparticle that obeys statistics between Fermi-Dirac and Bose-Einstein. State is preserved with Abelian Anyons.

A non-abelian Anyon is a type quasiparticle that does not necessarily preserve state. In the landmark 1988 paper by Jürg Fröhlich, *“Statistics of fields, the Yang-Baxter equation, and the theory of knots and links.” Nonperturbative quantum field theory. Springer US, 1988. 71-100.*, Fröhlich described the properties of such potential quasiparticles (paywall).

In https://arxiv.org/abs/1802.06176 which is in fact a rather excellent introduction to this subject, the usage of non-abelian Anyons in this mix is made clearer. Indeed, on page 3:

There are three fundamental steps in performing a topological quantum computation, illustrated in Fig. 1.

1. Creating qubits from non-Abelian anyons.

2. Moving the anyons around—‘braiding’ them—to perform a computation.

3. Measuring the state of the anyons by fusion

https://arxiv.org/abs/1802.06176

Physical realisation of this system is where things are currently in a state of flux. Indeed, certain types of non-Abelian anyons can be also what are known as Majorana fermions.

A Majorana fermion is a neutral spin-^{1}⁄_{2} particle that can be described by a real wave equation (the Majorana equation (1937)). A property of solutions to this wave equation is that they happen to be their own antiparticle.

As to the papers above, it will take some digging to really get to the heart of where things are in this field at the moment, but it looks totally fascinating.

In https://arxiv.org/abs/1712.06156 , the researchers claim that they were able to produce chiral majorana fermions, a type of non-abelian Anyon, in a “*hybrid device of [a] quantum anomalous Hall insulator and a conventional superconductor*“.

In https://arxiv.org/abs/1809.03076, the researchers were able to exhibit “Topological superconductivity in a phase-controlled Josephson junction” (paper title). From the abstract:

Topological superconductors can support localized Majorana states at their boundaries. These quasi-particle excitations obey non-Abelian statistics that can be used to encode and manipulate quantum information in a topologically protected manner.

https://arxiv.org/abs/1809.03076

So they essentially claim that they were able to implement one of the building blocks of the proposed system in https://arxiv.org/abs/1802.06176 through control of a type of Josephson junction. Interesting. They go on:

While signatures of Majorana bound states have been observed in one-dimensional systems, there is an ongoing effort to find alternative platforms that do not require fine-tuning of parameters and can be easily scalable to large numbers of states. Here we present a novel experimental approach towards a two-dimensional architecture. Using a Josephson junction made of HgTe quantum well coupled to thin-film aluminum, we are able to tune between a trivial and a topological super-conducting state by controlling the phase difference φ across the junction and applying an in-plane magnetic field.

https://arxiv.org/abs/1809.03076

In https://arxiv.org/abs/1809.03037, the researchers seem to have built on the previous result and were able to exhibit “Evidence of topological superconductivity in planar Josephson junctions”. From the abstract (my emphasis):

Majorana zero modes are quasiparticle states localized at the boundaries of topological superconductors

https://arxiv.org/abs/1809.03037that are expected to be ideal building blocks for fault-tolerant quantum computing. Several observations of zero-bias conductance peaks measured in tunneling spectroscopy above a critical magnetic field have been reported as experimental indications of Majorana zero modes in superconductor/semiconductor nanowires. On the other hand, two dimensional systems offer the alternative approach to confine Majorana channels within planar Josephson junctions, in which the phase difference {\phi} between the superconducting leads represents an additional tuning knob predicted to drive the system into the topological phase at lower magnetic fields. Here, we report the observation of phase-dependent zero-bias conductance peaks measured by tunneling spectroscopy at the end of Josephson junctions realized on a InAs/Al heterostructure. Biasing the junction to {\phi} ~ {\pi} significantly reduces the critical field at which the zero-bias peak appears, with respect to {\phi} = 0. The phase and magnetic field dependence of the zero-energy statesis consistent with a model of Majorana zero modes in finite-size Josephson junctions. Besides providing experimental evidence of phase-tuned topological superconductivity,our devices are compatible with superconducting quantum electrodynamics architectures and scalable to complex geometries needed for topological quantum computing.

So it looks like back in September 2018 folks were closing in on what is required for fault tolerant quantum computing.

Finally, in the April 2019 paper (recent! how intriguing…) https://arxiv.org/abs/1904.08981, the authors describe “Tuning Topological Superconductivity in Phase-Controlled Josephson Junctions with Rashba and Dresselhaus Spin-Orbit Coupling”. From the abstract:

Recently, topological superconductors based on Josephson junctions in two-dimensional electron gases with strong Rashba spin-orbit coupling have been proposed as attractive alternatives to wire-based setups. Here, we elucidate how phase-controlled Josephson junctions based on quantum wells with [001] growth direction and an arbitrary combination of Rashba and Dresselhaus spin-orbit coupling can also host Majorana bound states for a wide range of parameters as long as the magnetic field is oriented appropriately. Hence, Majorana bound states based on Josephson junctions can appear in a wide class of two-dimensional electron gases.

https://arxiv.org/abs/1904.08981

So it seems that there has been an alternative architecture proposed. The people involved also do not seem to be Microsoft related, but a different collaboration. Where was this alternative architecture originally proposed? Unclear, nonetheless, searching for references to “Rashba”, we are led to this part of the paper:

[Regarding implementing architectures to manipulate a 2D electron gas], among these proposals, those based on phase-controlled Josephson junctions with Rashba SOC [see Fig. 1(a)] offer an attractive alternative.37,38,42–47. Here, the interplay be-tween an in-plane Zeeman field parallel to the super-conductor/normal (S/N) interfaces, Rashba SOC, and the Andreev bound states formed in the normal re-gion induces topological superconductivity with Majorana bound states at the ends of the junction .

https://arxiv.org/abs/1904.08981

So it appears nothing fundamentally new here, just consolidation in the field as other teams replicate results.

Interesting area to watch, but the key takeaway for me is from https://arxiv.org/abs/1809.03037:

Besides providing experimental evidence of phase-tuned topological superconductivity,

our devices are compatible with superconducting quantum electrodynamics architectures and scalable to complex geometries needed for topological quantum computing.

Together with advances from the new field of twistronics potentially unlocking the potential for room temperature superconductivity, this is looking tremendously interesting indeed.