Majorana Zero Modes and Topological Qubits

Combinatorics and more 2025-02-21

This post contains the first item, devoted to Majorana zero modes, from an ambitious planned post on some quantum physics mysteries, related to quantum information and computation. (Some items are also related to noise sensitivity and associated Fourier analysis.)

Majorana zero modes are fascinating quantum objects in their own right whose creation would be an important first step towards topological quantum computers.

I decided to post this item separately because a few days ago, Microsoft announced successful creation of both Majorana zero modes and topological quantum qubits. (See this post over “Shtetl Optimized”, and also here.) This announcement (as earlier announcements in the area) is controversial and the paper should be carefully scrutinized.

Majorana zero modes and topological quantum computing

On the left you can see David DiVincenzo’s famous 7-steps road map to quantum computers (the picture is taken from a paper by Devoret and Schoelkopf). On the right an analogous roadmap for topological quantum computing. My argument about quantum computation would exclude reaching high-quality topological qubits and I am curious if my theory allows the creation of Majorana zero modes.

Question 1: Is it possible to create convincing demonstrations of Majorana zero modes (MZM)?

Majorana zero modes are quantum states whose creation is considered a crucial first step towards topological quantum computing.

A few more details on the physics: a Majorana fermion (proposed by Majorana in 1937) is a fermion that is its own antiparticle. They were not decisively observed in high-energy physics but commonly believed to describe certain neutrinos. Condensed-matter analogs of Majorana fermions were constructed and they represent the Bogoliubov quasiparticle. Majorana zero modes are Majorana fermions with an additional property and they were not (convincingly) constructed experimentally so far. See the review paper from 2015 Majorana zero modes and topological quantum computation‏ by S. Sarma, M. Freedman, & C. Nayak. The area of topological quantum computing was pioneered by Kitaev in the late 90s. Here is a classic 2003 survey article Topological quantum computing by Freedman, Kitaev, Larsen, and Wang.

Where do we stand now? There were a dozen or so experiments claiming to achieve MZM. (And quite a few other experiments claiming nonabelian anyons and related phenomena.) Proponents regard these experiments as good evidence but admit that other explanations not involving MZM are still possible. Opponents claim that in all the experiments it is much more likely that some mundane “quantum dots” were observed. In addition, opponents raised concerns regarding the quality and reliability of MZM experiments and at least one major MZM paper had to be retracted. Proponents disagree with some of these concerns and find them harmful to the field. Researchers from both sides believe that MZM can eventually be experimentally created. We mentioned this debate here, and see also this SO post (and discussion) “Sayorana Majorana”.

There are some attempts to demonstrate MZM on NISQ computers, see papers by Stenger et al., Rančić, and Mi et al. . (Even if successful, these attempts will only show some aspects of MZMs while not others.) Representing MZM over NISQ computers opens the door to a “no go” approach based on the complexity class LDP of Guy Kindler and me.

Question 2: Is it the case that samples from states demonstrating MZM on a NISQ computer goes (for large numbers of qubits) beyond the Kalai-Kindler LDP complexity class?

A positive answer to Question 2 would give some support to the assertion that MZM’s are inherently out of reach (namely to a negative answer to Question 1).

The computational class LDP

LDP is the class of probability distributions P on o-1 vectors $(x_1,x_2,\dots, x_n)$ of length n, that can be approximated by polynomials of bounded degree D. This class is a small subclass of probability distributions that can be approximated by ${\bf AC}^0$ circuits (bounded depth circuits), which is itself a very tiny subclass of the class BPP of probability distributions that can be approximated by classical computation. The class LDP arose as a complexity class for sampling problems but we can adopt the same definitions for real functions (or Boolean functions) of n 0-1 variables.

Functions in LDP are efficiently learnable. (This is in contrast with functions in the much larger class ${\bf AC}^0$ .)

Additional details, comments, links and references

The very recent news from Microsoft

1) About the very recent Microsoft announcement. It is based on a paper that is just being published by “Nature” that was released as an arXiv preprint a year ago. Apparently, it is also based on further experimental work claimed by the Microsoft team in the past year. A curious aspect of the paper is that the editors and referees do not endorse the author’s claim that Majorana zero modes were demonstrated, but see other qualities in the physical devices described in the paper that merit publication. (Achieving a topological qubit is a further difficult step beyond MZMs.)

About the theory

2) There were several ideas about implementing topological quantum computing since the early 2000. It is related to “nonabelian anyons,” remarkable (still) hypothetical states of matter predicted by physicists as early as the 1980s. The 1991 paper by Greg Moore and Nick Read “Nonabelions in the fractional quantum hall effect” plays a special role. See this Wikipedea article about “anyons“. Nonabelian anyons are related to fascinating mathematics.

3) “Kitaev’s wire” were especially simple (they are sometimes referred to as “poor-men nonabelian anyons”), but there were good physics reasons why they are not realistic. (Kitaev’s surface codes are abelian anyons.)

4) In 2010 two groups of researchers offered a physical way to implement “Majorana zero modes.” One group was

Y. Oreg, G.Refael, F. Von Oppen: Helical liquids and Majorana bound states in quantum wires, and the other was

Lutchyn, R. M., Sau, J. D. & Sarma, S. D. Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures.

Experimental progress

5) In 2012 the first experimental majorana zero modes were created in the paper Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices by V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven.

6) A year later the paper Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures by Eduardo J. H. Lee, Xiaocheng Jiang, Manuel Houzet, Ramon Aguado, Charles M. Lieber, and Silvano De Franceschi offered a much more mundane explanation to the states created in item 5 as “Andreev particles”.

More experimental claims and controversy

7) Since then there have been a dozen or so experimental claims about achieving Majorana zero modes. Proponents regard these experiments as good evidence but admit that other explanations not involving MZM are still possible. Opponents claim that in all the experiments it is much more likely that some mundane quantum states were observed.

8) In addition, opponents raised concerns regarding the quality and reliability of MZM experiments and at least one major MZM paper had to be retracted. Proponents disagree with some of these concerns and find them harmful to the field. (See also this article in Quanta Magazine.)

A few more comments

9) The attempts to simulate MZMs on quantum circuits, if successful, will only demonstrate (with poor fidelity) some aspects of physically constructed MZM’s. In the other direction, showing that “MZM states” constructed by quantum circuits are not in LDP would provide an argument against the possibility of physically constructing MZMs.

10) Some researchers thought that topological quantum computing is the only viable avenue to quantum computing since achieving the error rate needed for quantum circuits is infeasible. Mike Freedman gave a lecture at Microsoft “Quantum computing: the minority report,” with this massage, and this sentiment is expressed in the abstract of Freedman, et al.’s 2003 survey article mentioned above.

Late and early conversations

11) In recent years, I had useful conversations on the subject with Adi Stern and with Sergey Frolov. I am thankful to them both.

12) In my very early days of skepticism (2005) Nick Read challenged me to extend my skeptical study to topological quantum computing. (John Preskill made a similar challenge in an email correspondence a year later; see below.). BTW, here is a nice 2012 article Topological phases and quasiparticle braiding‏, by Nick. My general argument about quantum computers excludes good quality topological qubits, and Question 2 above is related to the feasibility of MZM’s.

13) Even earlier, in the late 90, before I got personally interested in quantum computing, I had some nice conversations (sometimes over lunch) about topological quantum computing, physics, and knot invariants, in Microsoft (Mike Freedman, Alexei Kitaev) and Jerusalem (Dorit Aharonov, Dror Bar-Nathan, and Michael Larsen).

My lecture at Microsoft: Good news and bad news

15) In 2018 I was invited to give a lecture at Microsoft about quantum computers. I concluded my abstract with the sentence:

“So the bad news is that Microsoft will fail in its quantum computer endeavor and the good news is that Google and IBM will fail as well.”

The Microsoft people felt uncomfortable about this sentence and asked me to consider modifying the abstract to which I gladly agreed and wrote instead:

“The good news is that understanding the failure of quantum computers promises interesting insights and new connections for quantum physics.”

Three recent posts on quantum computing

From a 2006 email of John Preskill:

“The core of the idea of quantum fault tolerance is that the logical information processed by a quantum computer can be stored in a form that is inaccessible to local observers and therefore robust against local noise. A particularly vivid realization of this idea is the basis of topological quantum computing: for n widely separated anyons in a nonabelian topologically ordered two dimensional medium, there is an exponentially large Hilbert space describing the possible ways for the anyons to be “fused” together. All of the states in this Hilbert space look identical to any observer who can inspect the anyons only one at a time. Furthermore, the information can be processed if the anyon world lines execute a braid in spacetime, even though the anyons never come close to one another.

It might be interesting to think about what plausible noise model might prevent a highly entangled quantum state from being created in a topological quantum computer. Nonabelian anyons have not yet been seen experimentally, but probably will be within the next couple of years (in experiments with fractional quantum Hall systems). It might then be possible to test your hypotheses concerning nonlocal noise in such systems in the next ten years of so.”