1. Thread on topological quantum computing (TQC). TQC is a radical approach to quantum computing where the individual topological qubits (Tqb) are intrinsically protected against decoherence by physics, making quantum error correction unnecessary. #quantum #QuantumComputing
1.1 TQC involves highly sophisticated mathematics and physics that is of intrinsic intellectual interest on its own quite distinct from its potential as a prospective QC platform. This thread highlights TQC, a central research topic in our center
2. Topological immunity in TQC arises from an intrinsic quantum ground state degeneracy in Tqb, protected by an energy gap, which is not connected to any symmetry, and hence cannot be destroyed by noise or decoherence as long as the gap does not close
3. TQC has been covered in various review articles from CMTC over the last 15 years: arXiv:1501.02813 arXiv:0707.1889
physicstoday.scitation.org/doi/10.1063/1.…
physicstoday.scitation.org/doi/10.1063/1.…
4. TQC has been alluded to as ‘the string theory’ of QC, and this was not meant as a compliment. On the other hand, perhaps a compliment Tqb were hailed as the ‘transistor’ of QC in contrast to the non-topological qubits, which are presumably akin to ‘vacuum tubes’
4.1 Given that quantum error correction is incredibly difficult, both as a matter of principle and a matter of practice, TQC could indeed be the disruptive breakthrough the subject needs for progress
5. There is just one problem. Tqb do not exist yet, except in theory. In spite of 15 years of increasingly focused experimental efforts in TQC. The sad dichotomy of QC is that qubits that exist cannot compute well because they are noisy, and qubits that can compute don't exist
6. TQC got an enormous boost when Microsoft decided to make Tqb its QC platform in contrast to Google, IBM, Intel, Amazon who are all betting on regular qubits necessitating quantum error correction
6.1 In spite of extreme hype in the subject, real progress toward building a commercial fault-tolerant QC has been slow although there has been impressive progress in understanding the physics issues
7. TQC started with pioneering papers by Alexei Kitaev in the 1997-2003 period. Early on, it was proven that TQC is equivalent to regular circuit-level QC in spite of their radical apparent different approaches—QC is universal
8. TQC works using braiding operations of non-Abelian topological excitations (‘anyons’) which follow certain well-defined topological quantum field theories, e.g., (SU)_2 level 2 or (even better) level 3. Braiding is the gate for TQC
8.1 The procedure is simple: Find these anyons and braid them around each other in prescribed manner. Computation done. These anyons, which are neither fermions nor bosons, can exist only in two dimensions (2D)
9. A slight problem: We do not know if such topological excitations exist in nature and they have not yet been seen in the laboratory. But candidate TQC platforms have been proposed and are being actively studied in many laboratories, notably at Microsoft
10. The first concrete Tqb was proposed by Das Sarma (CMTC), Freedman, and Nayak in 2005, who built on the earlier work of Read and collaborators on the fractional quantum Hall effect
10.1 They showed that suitable lithographically fabricated high-quality 2D GaAs-based transistors can produce an elementary Tqb in the presence of a high magnetic field and very low temperature
11. Five years of intense experimental efforts, some supported by Microsoft, followed focusing on the low-temperature, high-field 2D GaAs transistors, and there was considerable excitement and hope as some tantalizing evidence, which could not be reproduced, was reported
11.1 Eventually, the intense search for Tqb in 2D GaAs ended when an alternative theoretical proposal from CMTC appeared in 2009-10
12. This alternative TQC proposal from CMTC (Sau et al. 2010 and Lutchyn et al 2010), followed up on earlier works by Kane and collaborators on topological insulators and by Kitaev on hypothetical 1D spinless p-wave superconductors
12.1 It suggests using 2D or 1D superconductor-semiconductor hybrid systems, and utilizes spin-orbit coupling and Zeeman splitting to convert ordinary s-wave superconductivity into effectively spinless topological p-wave superconductivity
12.2 Such a system should manifest topological Majorana bound states (MBS) which are nonlocal zero-energy defect states obeying the non-Abelian braiding statistics suitable for TQC. MBS are the anyons here
13. The idea of 1D semiconductor nanowires with MBS localized at the wire ends, the so-called Majorana nanowire made of semiconductors InAs or InSb, immediately attracted experimental attention
13.1 A similar 1D theoretical proposal was also made almost simultaneously by an international group of collaborators from Israel, Germany, and California. Microsoft decided to adopt the Majorana nanowire as its TQC platform during 2010-12
14. Concrete proposals for MBS braiding using external gate voltages came soon afterward, and the scene seemed set for rapid progress in TQC. Given that the MBS platform is semiconductor-based, there is every reason to hope that scaling up should be possible
14.1 The only fundamental issue was establishing the existence of MBS in these InAs and/or InSb Majorana nanowires. The theory here is essentially free fermion band structure engineering, which is expected to be sound – all that was needed is its laboratory implementation
15. There was great excitement when an experimental paper appeared in 2012 from Delft (Mourik et al 2012) showing tell-tale signs of predicted topological excitations in the form of enhanced conductance at zero voltage corresponding to zero energy MBS
15.1 This experiment was quickly reproduced during 2012-18 in several laboratories with better and better zero-voltage enhanced conductance creating great optimism about TQC being just around the corner
16. Unfortunately, the hype was premature. Extensive theoretical work, starting already in 2012-13, but accelerating considerably after 2016, following the multiple experimental claims of MBS observations, has now decisively established…
16.1..that the experimental observations of enhanced zero-voltage conductance are unlikely to be evidence for topological MBS but are likely to be ordinary nontopological bound states, Andreev bound states (ABS), which are ubiquitous because of the invariable presence of disorder
17. One silver lining: these ABS are, in a precise mathematical sense, formed by overlapping MBS, and so the task now is to somehow to separate the MBS forming the ABS so that they become isolated nonlocal topological excitations for carrying out TQC
17.1 There are various proposals (2016-2020) on how to convert the ABS to MBS in the laboratory. The first step: Eliminate disorder from the system and make them 1000 times cleaner. This requires herculean materials efforts and infinite financial resources
17.2 Fortunately, the organizations interested in TQC do have infinite resource
18. Two necessary requirements: (1) closing and then reopening of a bulk superconducting gap along with the emergence of a quantized conductance peak at zero voltage; (2) clear nonlocal correlations in the measured zero voltage conductance from the wire ends
18.1 This proves the nonlocal nature of MBS. But any definitive evidence supporting the existence of topological MBS requires a successful braiding experiment
19. Bottom line: There has been enormous experimental progress, but much less than the hype in the popular press indicates. We know what to do to localize topological MBS in nanowires, but it is a very expensive and time-consuming process. It can be done, TQC should happen
19.1 Fortunately, for the TQC protagonists, it is not like that regular circuit level qubits have had spectacular progress either—yes, there are tens of working qubits now, but what remains unsaid is that these noisy physical qubits do not make a QC
19.2 One needs error-free logical qubits, which are nowhere to be found in spite of 25 years of experimental efforts on regular qubits. On the other hand, TQC right now has no qubits
Postscript 1: There have been many experimental claims of the observation of MBS in many other systems during the 2013-2020 period following the Delft 2012 experiment, all based on enhanced zero voltage conductance
Postscript 1.1: They are all likely to be connected with various trivial effects arising from disorder —the best and the most consistent data are still in the semiconductor nanowires which remain the leading TQC platform
Postscript 2: It is impossible to put a definite timeline on when Majorana Tqb will be confirmed in the laboratory, but there is no reason to believe it will not happen with improving sample quality since the theory is definitive
Postscript 2.1: What is pretty clear is that once topological MBS braiding is experimentally demonstrated, TQC may become the only game in town as far as quantum computing goes—regular qubits may disappear just as vacuum tubes did
Postscript 3: An important caveat: MBS-based TQC is nonuniversal since not all gate operations are possible, so one needs to supplement with additional gate operations. MBS however do form a perfect quantum memory
