The first quantum processor powered by topological qubits

Microsoft has unveiled Majorana 1, claimed to be the world’s first quantum processor powered by topological qubits (see this)

1. How could one stabilize the computations and qubits?

The basic problem is how to realize computations in a stable way and how to make stable enough qubits? Concerning computation, topology comes to rescue.

1. Topological quantum computations (see this can be represented as braidings which are topologically stable under small deformations. Each braid strand represent a unitary evolution of a particle representing a unitary evolution if a qubit and the braiding operation would represent the computation. Braiding can be either time-like dynamical operation for point-like particles in plane or space-like for a braid connects two planes.
2. Since the 2-D plane containing particles as punctures, the homotopy group is non-abelian. This means that the rotation of a puncture around a second puncture of say bound state can transform the state such that transformation is not a mere phase factor but is a rotation which change the directions of the spins of the particles involved. Therefore the exchange of particles which can be seen as basic braiding operation changing the braid strands can induce an operation, which can be used as a basic building brick for a topological quantum computation.

How could one obtain stable qubits? Qubit represented as a spin is not thermodynamically stable and extremely low temperatures are required. This is the case also for the proposed topological quantum computation: the reason is now that superconductivity is required and this is possible only at temperatures of order milli Kelvins. In any case, the notion of qubit should be topologized. How to achieve this? Here Majorana bound bound states have been proposed as an answer (see this).

1. Non-Abelian braid statistics, which means that their exchange realized as a 2-D rotational flow generated by braiding induces, instead of change of a sign in Fermi statistics, a non-Abelian unitary transformation of the state. It could be used to change the directions of their spins and affect the anyons. 2π rotation would induce a non-Abelian rotation instead of a mere sign change or phase factor in brain statistics. This is only possible in dimension 2 where the homotopy group can be non-abelian if there are punctures in the plane that the braids would represent. Similarly, swapping two Majorana fermions in braid produces a SU(2) rotation and can flip the spins and thus the qubits. This swap would be an essential operation in quantum computing. In order to have non-trivial topological quantum computation, one must have non-Abelian braid statistics characterized by a Lie group. Rotation group SO(2) or its covering SU(2) are the minimal options
2. The bound state of two Majorana fermions associated with planar punctures, anyons, would thus obey non-Abelian braid stastistics. It is also possible to affect the second fermion of Majorana bound state by rotating a puncture containing a fermion around the second fermion. Braidings could therefore represent unitary transformations having an interpretation as topological quantum computations. Wikipedia article mentions several realizations of Majorana bound states in superconductors. Quantum vortices in super conductors can provide this kind of states. The ends of the super-conducting wire or of line defects can contain the Majorana fermions. Also fractional Hall effect can provide this kind of states. The realization studied by Microsoft has the fermions of the Majorana fermion at the ends of a superconducting wire.
3. As I understand it, a condensed matter Majorana fermion would correspond formally to a superposition of an electron and a hole. The statistics would no longer be normal but non-Abelian Fermi statistic but would be that of a non-abelian anion. The weird sounding property of this statistics is that the the creation operator is equal to annihilation operator. One obtains two creation operators corresponding to two spin statse and square the creation operator of is unit operator: for fermions it vanishes. This implies that Majorana fermion number is defined only modulo 2 and only the number of fermions modulo 2 matters. Also the anticommutator of two creation operators at different points is equal to unit operator so that the system is highly nonlocal.
4. How the braiding could be realized? One can consider two options. Dance metaphor allows to understand the situation. Imagine that particles are dancers at the parquette. The dance would give rise to a time like braiding. If the feet of the dancers are tied to a wall of the dancing house by threads, also a space-like braiding is induced since the threads get tangled.
5. In the TGD framework, dancers would correspond to particle-like 3-surfaces moving in the plane and the dance would define the dancing pattern as a time-like braiding. This classical view is actually exact in the TGD framework since classical physics is an exact part of quantum physics in TGD. If thee particles are connected to the wall by threads realized as monopole flux tubes, a space-like braiding is induced.
6. These threads bring in mind the wires connecting superconductor and another object and containing Majorana fermions at its ends. Now the second end would be fixed and second would correspond to a moving particle. Majorana bound states would correspond to the ends of the thread and the superconducting flow of the second end would correspond to the dynamical braiding.

2. Algebraic description of Majorana fermions

The dissertation of Aran Sivagure contains a nice description of Majorana fermions (see this). Majorana fermions would be quasiparticles possible in a many-fermion state. They would create from a fermion state with N fermions a superposition of states with fermion numbers N+1 and N-1. They would be created by hermitian operators γ_i^+/-= a^†_i+/- a_i formed from the fermionic oscillator operators satisfying the standard anticommutation relations {a^†_i,a_j}= δ_i,j. Note that one consider also more general Hermitian operators γ_i^+/-= exp(iφ)a^†_i+/- exp(-iφ)a_i.

The anticommutations would be {γ_i^ε₁, γ_i^ε₂}= 2× Id, where one has ε₁=+1 and ε₂=-1 and Id denotes the unit operator. Therefore the statistics is not the ordinary Bose or Fermi statistics and non-Abelian statistics.

What is so remarkable is that that also the anticommutators {γ_i,γ_j} satisfy {γ_i^+/-,γ_j^+/-}= 2Id even when i and j label different points. Therefore these operators are highly non-local meaning long range quantum coherence.

3. Could many-sheeted spacetime allow a more fundamental description of Majorana like states?

The problematic aspect of the notion of Majorana fermion is that the manyfermion states in this kind of situation do not in general have a well-defined fermion number. Physically, fermion conservation is a superselection rule so that the superposition of fermion and hole must physically correspond to a superposition of fermion states, where the hole corresponds to a fermion which is outside the system. TGD suggests an elegant solution of the problem.

1. In condensed matter physics Majorana fermions could be assigned with the vortices of superconductors. In the TGD Universe, these vortices could correspond to monopole flux tubes as body parts of field body. The states created by γ_i would be superpositions of states in which the fermion is at the monopole flux tube or at the normal space-time sheet representing the part of the condensed matter system that we see. The Majorana description would be effective description.
2. The Majorana creation operators γ_i would be replaced with operators which shift the fermion from ordinary space-time sheet to the monopole flux tube and vice versa. From the geometric interpretation it is clear that this operation must be idempotent. This operation must be representatable in terms of annihilation and creation operators. The operators γ_i would be expressible products of creation and annihilation operators acting at the space-time sheets 1 and 2 and one would have

γ_i^ε= a^†₁a₂+ε a^†₂a₁.

One can consider either commutation or anticommutation relations for these operators. Since the operation does not change the total fermion number, the interpretations as a bosonic operator is natural and therefore commutations relations look more plausible.

Neglecting for a moment the indices labelling positions and spins, a rather general expression for the operator γ^ε would be

γ^ε= a₁^†a₂+ ε a₂^†a₁ .

This operator is hermitian. If fermionic anticommutations are true, one has for anticommutator

(γ^ε)²= X+Y ,

X=a₁^†a₂a₁^†a₂+ ε² a₂^†a₁a₂^†a₁ ,

Y=ε₁ε₂ (a₁^†a₂a₂^†a₁+ a₂^†a₁a₁^†a₂) .

If one can assume anticommutativity for the oscillator operators associated with flux tube and ordinary space-time sheet one has X=0. For Y one obtains

Y= ε₁ε₂ (N₁a₂a₂^†+N₂ a₁a₁^†)= ε₁ε₂(-N₁-N₂+2N₁N₂) .

The eigen values of (-N₁-N₂+2N₁N₂) vanish for (N₁,N₂) ∈ {(1,1),(0,0)} and are equal to -1 for (N₁,N₂) ∈ {(1,0),(0,1)} so that the eigenvalues are equal to -ε₁ε₂.

One can consider also the commutator, which is perhaps more natural on the basis of the physical interpretation. The commutator obviously vanishes for ε₁=ε₂. For (ε₁,ε₂)=(1,-1) one has

[γ¹,γ^-1]= ε₁ε₂ (a₁^†a₂a₂^†a₁- a₂^†a₁a₁^†a₂) =-(N₁a₂a₂^†-N₂a₁a₁^†)= N₂-N₁ .

For ε₁=-ε₂, the eigen values of ε₁ε₂(N₂-N₁) vanish for (N₁,N₂) ∈ {(1,1),(0,0)} and belong to {1 ,-1} for (N₁,N₂) ∈ {(1,0),(0,1)}.

Both the anticommutator and commutator resemble that in the Majorana case but are not identical to a unit operator since the eignvalues belong to the set {0,1} for the anticommutator and to the set {0,-1,1} for the commutator.

4. OH-OH⁺+p as topological qubits?

While writing this, I noticed that the OH-OH⁺+p qubits, where p is a dark proton ag monopole flux tubes, that I proposed earlier to play fundamental role in biology and perhaps even make quantum counterparts of ordinary computes possible, are to some degree analogous to Majorana fermions. The extremely nice feature of these qubits would be that superconductivity, in particular biosuperconductivity. would be possible at room temperatures. This is would be possible by the new physics predicted by TGD both at the space-time level and at the level of quantum theory.

1. In TGD space-times are surfaces in H=M^{× CP₂ and many-sheetedness is the basic prediction. Another related prediction is the notion of field body (magnetic/electric) body. Number theoretica view of TGD predicts a hierarchy of effective Planck constants making possible quantum coherence in arbitrarily long length scales. Second new element is zero energy ontology modifying profoundly quantum measurement theory and solving its basic problem.}
2. OH-OH⁺+p qubit means that one considers protons but also electrons can be considered. Now the proton is either in the OH group associated with water molecule in the simplest situation in which Pollack effect occurs or the proton is a dark proton at a monopole flux tube. A proton in OH would be analog of non-hole state and the dark proton in the flux tube be the anaog of hole state.
3. What is new is that the proton being on/off the spacetime surface would represent  a bit.  For Majorana fermions,  the situation is rather similar: the hole  corresponds to the electron being “somewhere else”, which could also correspond to being on a monopole flux tube as I have suggested. In standard quantum computation, a qubit would correspond to a spin. The analog of  the Majorana qubit would be a quasiparticle which is superposition of transitions OH ↔ O^-+p transitions and transitions in which nothing happens: OH goes to itself and O^-+p goes to itself. The Majorana property would correspond to the fact that the transfer between two space-time sheets repeated twice is trivial.
4. If the energies for OH and OH⁺+p are close to each other, the situation is quantum critical and the qubits can be flipped and a process similar to quantum computation becomes possible. Also superconductivity becomes possible at the magnetic flux tubes analogous to magnetic vortices appearing in superconductivity and in fractional Quantum Hall effect. These are truly topological qubits also because the topologies of the spacetime surface for different bit values ​​are different. However, the energy difference must be larger than the thermal energy, otherwise the qubits become unstable. With the help of electric fields, qubits can be sensitized to quantum criticality and their inversion becomes possible.
5. The above argument suggests that a non-abelian statistics could be understood for OH-OH⁺+p qubits. The anticommutation/commutation relations for the operators trasferring protons to the magnetic body would not be identical to those for Majorana oscillator operators the squares of these operators would be proportional to unit operator which is essentially the Majorana property. I have proposed a possible realization for this in a more general case. The exchange of dark protons/qubits would be induced by reconnection of monopole flux tubes:    it would therefore be a purely topological process. Nothing would be done to the dark electrons, but the flux tubes would be reconnected. Strands AB and CD would become strands AD and BC. At the same time, the unilluminated protons would become associated with different O^-. In this exchange, could the final result be represented as an SU(2) rotation for the entire space.

See the article The first quantum processor powered by topological qubits or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/02/the-first-quantum-processor-powered-by.html