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Pitkänen, M., Quantum Gravitation & Topological Quantum Computation

Exploration

Quantum Gravitation & Topological Quantum Computation

Matti Pitkänen 1

Abstract

In this article the connection of quantum gravitation, as it is understood in the TGD framework,
with topological quantum computation (TQC) is considered. I sketched the first TGD based vision
about DNA as a TQCer for about 13 years ago. In particular, a model of the system consisting of
DNA and nuclear/cell membrane system acting as a TQCer was discussed. TGD has evolved a lot
after this and there are several motivations for seeing what comes out from combining the recent view
about quantum TGD and TGD inspired quantum biology with this model.

1. There is a rather detailed view about the role of dark matter as phases of ordinary matter with
the effective Planck constant heff = nh0. Large values of heff allow to overcome the problems
due to the loss of quantum coherence.

This leads to the notion of the dark DNA (DDNA), whose codons are realized as dark proton
triplets and proposed to accompany the ordinary DNA. Also dark photon triplets are predicted
and one ends up to a model of communications and control based on dark cyclotron resonance
in which codons serve as addresses and modulation of the signal frequency scale codes the signal
to a sequence of pulses. Nerve pulses could be one application.

2. Quite recently, also the understanding of the possible role of quantum gravitation in biochem-
istry, metabolism, bio-catalysis, and in the function of DNA has considerably increased. The
gravitational variants of hydrogen bonds and valence bonds between metal ions having very
large value of heff = hgr, where hgr = GMm/v0 is the gravitational Planck constant originally
introduced by Nottale, are in a key role in the model and explain metabolic energy quantum
as gravitational energy liberated when dark protons ”drops” from a very long gravitational flux
tube in the transition hgr → h. Also electronic metabolic energy quantum is predicted and
there is empirical support for this.

3. A further motivation comes from the number theoretic vision of quantum TGD. Galois groups
as symmetry groups represent new physics and the natural questions are whether Galois groups
could give rise to number theoretic variants of anyons and what could the TGD counterparts of
the condensed matter (effective) Majorana electrons proposed by Kitaev as anyon like states?

The answer is that quantum superpositions of symmetric hydrogen bonded structures of form
X..H-H+X-H...X are excellent candidates for the seats of dark (heff > nh0 > h) bi-localized
electrons defining TGD analogs of condensed matter Majorana electrons.

The Galois groups permute the roots of a polynomial, which determines a space-time region by
M8 − H duality. The roots correspond to mass squared values, in general algebraic numbers,
and thus to mass hyperboloids in M4

c ⊂M8
c . The H images correspond to 3-hyperboloids with

a constant value a = an of light-cone proper time. Therefore the Galois group can permute
points with time-like separation. Note however that the real or rational parts of two values of a
can be same.

This looks very strange at first but actually confirms with the fact that time-like braidings
defining TQC correspond in TGD time-like braidings (involving also reconnections) of string
like objects defining string world sheets, which are not now time evolutions of space-like entities
as physical state but correspond to time-like entities defining boundary data necessary for fixing
holography completely. Their presence is forced by the small failure of the determinism of the
action principle involved and is completely analogous to the non-determinism for soap films with
frames serving as seats for the failure of determinism.

1Correspondence: Matti Pitkänen http://tgdtheory.com/. Address: Rinnekatu 2-4 A8, 03620, Karkkila, Finland. Email:
matpitka6@gamail.com.

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Pitkänen, M., Quantum Gravitation & Topological Quantum Computation

4. Braidings appear therefore at the level of fundamental TGD and correspond to string world
sheets. They are possible only in 4-D space-time but not in string models.

Also TQC-like processes appear automatically at the level of fundamental physics. In particular,
the number theoretical state function reduction cascade for the Galois group following the time
evolution induced by braiding can be regarded as a generalization of a decomposition of integers
to primes: now primes are replaced by simple groups defining primes for finite groups. Nature
is doing number theory!

5. Also zero energy ontology (ZEO) brings in new elements. The change of the arrow of time in
”big” state function reductions (BSFRs) implies that dissipation with a reversed arrow of time
provides an automatic error correction procedure. Also TQC in which the arrow of time varies
for sub-modules, can be considered.

1 Introduction

In this article the connection of quantum gravitation, as it is understood in the TGD framework, with
topological quantum computation (TQC) is considered. I sketched the first TGD based vision about
DNA as a TQCer for about 13 years ago. In particular, a model of the system consisting of DNA and
nuclear/cell membrane system acting as a TQCer was discussed [19, 18, 31].

TGD has evolved a lot after this and there are several motivations for seeing what comes out from
combining the recent view about quantum TGD and TGD inspired quantum biology with this model.

1. There is a rather detailed view about the role of dark matter as phases of ordinary matter with the
effective Planck constant heff = nh0. Large values of heff allow to overcome the problems due to
the loss of quantum coherence.

This leads to the notion of the dark DNA (DDNA), whose codons are realized as dark proton triplets
and proposed to accompany the ordinary DNA [35, 61]. Also dark photon triplets are predicted
[20] [45, 52] and one ends up to a model of communications and control based on dark cyclotron
resonance in which codons serve as addresses and modulation of the signal frequency scale codes
the signal to a sequence of pulses. Nerve pulses could be one application.

2. Quite recently, also the understanding of the possible role of quantum gravitation in biochemistry,
metabolism, bio-catalysis, and in the function of DNA [60] has considerably increased. The gravi-
tational variants of hydrogen bonds and valence bonds between metal ions having very large value
of heff = hgr, where hgr = GMm/v0 is the gravitational Planck constant [39] [28, 25, 26] originally
introduced by Nottale [7], are in a key role in the model and explain metabolic energy quantum
as gravitational energy liberated when dark protons ”drops” from a very long gravitational flux
tube in the transition hgr → h. Also electronic metabolic energy quantum is predicted and there is
empirical support for this.

3. A further motivation comes from the number theoretic vision of quantum TGD. Galois groups as
symmetry groups represent new physics [51, 49, 50] and the natural questions are whether Galois
groups could give rise to number theoretic variants of anyons and what could the TGD counterparts
of the condensed matter (effective) Majorana electrons proposed by Kitaev [6] as anyon like states?

The answer is that quantum superpositions of symmetric hydrogen bonded structures of form X..H-
H+X-H...X are excellent candidates for the seats of dark (heff > nh0 > h) bi-localized electrons
defining TGD analogs of condensed matter Majorana electrons.

The Galois groups permute the roots of a polynomial, which determines a space-time region by
M8 −H duality. The roots correspond to mass squared values, in general algebraic numbers, and
thus to mass hyperboloids in M4

c ⊂ M8
c . The H images correspond to 3-hyperboloids with a

constant value of light-cone proper time. Therefore the Galois group permutes points with time-like
separation.

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Pitkänen, M., Quantum Gravitation & Topological Quantum Computation

This looks very strange at first but actually confirms with the fact that time-like braidings defining
TQC correspond in TGD time-like braidings (involving also reconnections) of string like objects
defining string world sheets, which are not now time evolutions of space-like entities as physical
state but correspond to time-like entities defining boundary data necessary for fixing holography
completely. Their presence is forced by the small failure of the determinism of the action principle
involved and is completely analogous to the non-determinism for soap films with frames serving as
seats for the failure of determinism.

4. Braidings appear therefore at the level of fundamental TGD and correspond to string world sheets.
They are possible only in 4-D space-time but not in string models.

Also TQC-like processes appear automatically at the level of fundamental physics. In particular,
the number theoretical state function reduction cascade for the Galois group [47] following the time
evolution induced by braiding can be regarded as a generalization of a decomposition of integers to
primes: now primes are replaced by simple groups defining primes for finite groups. Nature is doing
number theory!

5. Also zero energy ontology (ZEO) [41, 55] brings in new elements. The change of the arrow of time
in ”big” state function reductions (BSFRs) implies that dissipation with a reversed arrow of time
provides an automatic error correction procedure. Also TQC in which the arrow of time varies for
sub-modules, can be considered.

1.1 Two visions about physics in TGD framework

TGD leads to two visions about physics discussed in [46, 56]. In the first vision [14, 13, 17] physics is
seen as geometry of space-time identified as 4-surface in H = M4 × CP2, and at a more abstract level,
geometry of the ”world of classical worlds” (WCW) consisting of space of preferred extremals (PEs) of
the basic action principle defining analogs of Bohr orbits as minimal surfaces with singularities.

In the second vision [29] physics is reduced to number theoretic concepts and 4-surfaces in M8 anal-
ogous to momentum space define the basic objects. M8 −H duality [42, 43], analogous to momentum-
position duality, relates the two visions. The 4-surfaces in M8

c (complexified M8), which has interpretation
as complexified octonions, are required to be associative in the sense that their normal space is quater-
nionic.

For given space-time region, they are determined by the roots of polynomial P of real argument
continued to polynomials in M8

c . The roots define a collection of mass shells of M4
c ⊂ M8

c and by
holography they define a 4-D surface of H.

The action principle at the level of H is determined by the twistor lift of TGD and is the sum of
4-D Kähler action and volume term (cosmological constant). It is not fully deterministic and space-time
surfaces in H as PEs analogous to Bohr orbits can be regarded as analogs of soap films with frames,
which correspond to singularities at which determinism fails.

The frames provide additional holographic data besides the hyperbolic 3-surfaces corresponding to
light-bone proper times a = an which are determined by the roots of P . Frames include light-like orbits
of partonic 2-surfaces and string world sheets connecting them. What is new, and consistent with zero
energy ontology (ZEO) [33], is that space-like data are not enough for holography, also time-like data is
required and the string world sheets turn out to be absolutely essential for braiding and TQC.

1.1.1 Physics as geometry

The basic elements of physics as geometry are following.

1. Space-time is identified as minimal 4-surface [57] in H = M4 × CP2. Holography follows from
general coordinate invariance and implies what might be called Bohr orbitology. It turns out that
holography is not quite strict.

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Pitkänen, M., Quantum Gravitation & Topological Quantum Computation

2. Twistor lift of TGD [27] [58, 59] replaces space-time surface with what can be regarded as a counter-
part of its twistor space having X4 as a base space and sphere CP1 as a fiber. The twistor structure
is induced from the product of T (M4) × T (CP2) of twistor spaces T (M4) TC(P2), which are the
only twistor spaces allowing Kähler structure. The induced twistor structure and determined by an
action principle with is 6-D Kähler action existing only for M4 and CP2. Twistor structure requires
dimensional reduction so that one bundle structure and the action reduces to a sum of a volume
term having interpretation in terms of cosmological constant and of 4-D Kähler action as analog of
Maxwell action.

PEs realizing the holography are identified as minimal surfaces [57], which, apart for lower-dimensional
singularities, are also locally extremals of the 4-D Kähler action and possess a holomorphic structure
reducing the field equations to algebraic conditions analogous to Cauchy-Riemann conditions. One
can regard the space-time surface as an analog of soap film spanned by frames assignable to the
singularities at which minimal surface property fails but extremal property for the entire action
remains true so that conservation laws are not lost. As in the case of ordinary soap films, frames are
seats of finite non-determinism interpreted as space-time correlates of quantum non-determinism.

3. The concrete study of the extremals of the action principle leads to the identification of the basic
candidates for the basic PEs. From the point of view of TQC, magnetic flux tubes are the most
interesting objects and define counterparts of the braid strands. The notion of magnetic body (MB)
is central. Its detailed identification is still far from complete: for the latest view about gravitational
MB see [60].

1.1.2 Physics as a generalized number theory and M8 −H duality

Physics as (a generalized) number theory is the dual vision of TGD.

1. p-Adic physics emerged originally from a model for the particle massivation based on p-adic ther-
modynamics for the mass squared of the particle [15, 12]. From the beginning it was clear that
various p-adic physics had to be fused with the real number based physics to a larger framework,
which could be called adelic physics. For mathematical reasons, the natural interpretation of vari-
ous p-adic physics would be in terms of physical and mathematical correlates of cognition. Number
theoretical universality stating that the basic equations of TGD are number-theoretically universal
and make sense in all number fields is a natural constraint on the theory.

2. M8−H duality [42, 43] realizes the number theoretical vision about TGD and also holography. M8
c

identified as complexified M8 and interpreted as complexified octonions, is analogous to momentum
space and 4-surfaces define the basic objects at the level of M8.

The 4-surfaces in M8
c (complexified M8), which have an interpretation as complexified octonions,

are required to be associative in the sense that their normal space is quaternionic. These 4-surfaces
are determined by the roots of polynomials of real argument continued to polynomials in M8

c . The
roots define a collection of 3-D mass shells of M4

c ⊂M8
c and by holography they define a 4-D surface

of M8
c . Physical states correspond to 4-momenta at these mass shells analogous to Fermi balls.

M8 − H duality, analogous to momentum-position duality, relates the two visions by mapping
the 4-surfaces in M8 to those in H. M8 − H duality generalizes to the level of twistor space
[42, 43, 56, 58, 59].

3. One can assign to a given polynomial an algebraic extension of rationals. The collection of points
of the 4-surface of M8

c defines a cognitive representation. The mass shells as sources of holographic
data are however number theoretically exceptional in that the number of points with algebraic M8

c

coordinates is infinite: cognitive explosion takes place both at the level of M8 and H: these values
of the light-one proper time a correspond to very special moments in the life of self, kind of moments
of enlightenment.

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In M8 the points of mass shells are identifiable as quark momenta assumed to be algebraic integers
just as ordinary momenta for particles in a box are integers with suitable choice of momentum unit.
These momenta can also be interpreted as points in extension of p-adic numbers so that number
theoretical universality follows. The p-adic prime in question is identified as the largest ramified
prime of the extension considered.

This gives rise to a hierarchy of algebraic extensions and cognitive representations as unique dis-
cretizations of the 4-surface in M8 and space-time surface and suggests a generalization of compu-
tationalism replacing integers with the hierarchy of algebraic integers for extensions of rationals.

4. The dimension n of algebraic extension is identified as an effective Planck constant heff = nh0
where h0 < h is true. The identification of the value of n0 in h = n0h0 has been proposed [54].
The phases of ordinary matter labelled by the value of n behave in many respects as dark matter
and the identification as dark matter has been proposed. A particularly important class of phases
corresponds to heff = nh0. These phases would play a central role in living matter. The relationship
with galactic dark matter is however somewhat unclear.

What makes these phases so important is the scale of quantum coherence is expected to scale like
heff . Dark phases are also expected to have very weak interaction with ordinary matter and the
proposal is that living matter is controlled by this kind of phases located at MB and approaching
only slowly thermal equilibrium with it: this would have interpretation as aging [62]. The small
value of h and thermal fluctuations spoiling quantum coherence and entanglement belong to the
key problems of QC and dark matter could solve these problems.

5. Galois confinement [49] states that physical states have total momenta, whose components are
ordinary integers. Galois confinement provides a universal mechanism for the formation of bound
states. Galois confinement also applies in spin degrees of freedom and provides spin representations
for the covering of the Galois group. The number theoretic degrees of freedom are of special interest
in QC and suggest that number theoretic quantum computation (NQC) as a counterpart of TQC,
which would involve what might be called Galois anyons. The Galois group could allow identification
as a subgroup of the braid group. This would mean strong restrictions on TQC.

6. M8 −H duality leads to a view about the construction of the counterpart of S-matrix in the TGD
framework [58, 59]. S-matrix would be replaced by the analog of Kähler metric in fermionic degrees
of freedom [48], which by the infinite dimension of Fock space is expected to be highly unique as
also the Kähler metric of WCW [14, 13, 17].

Incoming and outgoing states of particle scattering would be Galois singlets constructed from lower
level states which need not be Galois singlets. Quarks, whose momenta at mass shells are algebraic
integers are free and the scattering would be mere reorganization of Galois singlets to new ones.

Scattering could be also seen as analog of QC and computation in an extension of rationals: both
the input and output would consist of a set of rational integer valued momenta and scattering would
map them to each other.

This applies in the twistor picture also to spins having a representation as points of the twistor
sphere S2 known as Bloch sphere. In this case number theoretic constraints suggest that the set of
quantization axes corresponds to a finite discrete subgroup of SO3) assignable to regular polygons
and Platonic solids.

The quark momenta belonging to the extensions of rationals are invisible, which implies invisible
algebraic complexity of cognition and brings in mind unconscious information processing. Quantum
physics and psychoanalysis would meet!

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1.2 Zero energy ontology (ZEO) and QC

The first basic motivation for the introduction of ZEO was that by the general coordinate invariance
space-time surface as a preferred extremal is a more natural notion than 3-surface. For exact holography,
these notions are equivalent but the identification of space-time surface as minimal surface predicts a small
violation of the strict holography identifiable as a correlate for quantum non-determinism associated with
the physics of cognition or possibly quite generally. This non-determinism would be essential for the
possibility of TQC in TGD.

Second motivation was the basic problem of quantum measurement theory to which ZEO provides an
elegant solution if one assumes that the arrow of time changes in ”big” state functions reductions (BSFRs)
as analogs of ordinary SFRs. In ”small” SFRs, which are analogs of ”weak” measurements introduced in
quantum optics, the arrow is not changed [33] [41, 55].

In the TGD framework, quantum measurement theory generalizes to a quantum theory of conscious
experience in which SSFR defines the basic element of conscious experience. BSFR has an interpretation
as a counterpart of death/sleep. The change of the arrow of time in BSFRs has profound implications
in quantum biology. Since the dissipation with a reversed arrow of time for a subsystem looks like self-
organization from the point of view of a system with an opposite arrow of time [40]. The arrow of time
can change for macroscopic time periods at the MBs with large heff and since MB controls the ordinary
matter, it induces not only effective quantum coherence but also an effective reversal of time also at this
level.

The basic ideas of ZEO [41, 55] are following.

1. In zero energy ontology (ZEO) [41, 55], the pair of incoming and outgoing states of particle scat-
tering are replaced with zero energy state and zero energy states define scattering amplitudes as
entanglement coefficients.

2. At the level of H, positive and negative energy parts of zero energy states are located at boundaries
of causal diamonds (CD), which form a fractal hierarchy. At the level of M8, they reside at the
boundaries of mass shells, which corresponds to the roots of the polynomial defining the space-
time region. M8 −H duality maps these points to the boundary of CD. One can also consider an
alternative for which mass shells as hyperbolic spaces H3 ⊂ M8 are mapped to their counterparts
in H by a map which is essentially inversion (Uncertainty Principle).

3. Scattering events [58, 59] are QC like events. Input (output) data correspond to incoming (outgoing)
quark momenta identified as algebraic integers in an extension of rationals and to spins. Since
fermionic Fock state basis defines a Boolean algebra, the fermionic states define quantum analog
of Boolean algebra, and the scattering amplitudes could be also seen as a quantum generalization
of Boolean maps and realizing statements which are true that is consistent with laws of physics.
These transitions could be interpreted in terms of Boolean cognition.

The replacement of the S-matrix with Kähler metric in fermionic Hilbert space degrees of freedom
represents a new element. The analog of unitary transformation is assigned with CD and from the
point of view of QC, CD could be interpreted as an embedding space analog of gate. Since gates
allow control bits not affected by the unitary transformation, also the Boolean functions, which are
not 1-1, can be realized as unitaries. Same is expected to be true also now.

4. The scattering amplitudes correspond a tensor net-like structure. Physical states are Galois singlets
consisting basically of free quarks. At the number theoretical level, scattering can be seen as a
recombination of Galois singlets to new ones.

ZEO could have a profound impact on QC.

1. Negentropy Maximization Principle (NMP) [16] [53] is the variational principle of TGD inspired
theory of consciousness. Negentropy can correspond to the sum of p-adic negentropies or to the sum

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or p-adic and real negentropies, which can be possible and tends to be so by NMP. For both options,
NMP guarantees that the p-adic entanglement negentropy increases and is positive. It however also
forces the real entanglement entropy to grow. NMP therefore implies cognitive evolution but also
second law.

From the point of view of QC, this picture is very promising since the laws of physics would take
care that the entanglement negentropy grows and also that negentropic entanglement tends to be
stable. This is quite contrary to what standard physics predicts. This leads to evolution [37, 38] in
the sense that the dimension n = heff/h0 of the extension of rationals as a measure of algebraic
complexity tends to increase since this provides larger negentropic resources. This evolution takes
place at MB in human length and time scales and the challenge is to learn to manipulate dark
matter.

2. BSFR could take care of error correction automatically since for the reversed arrow of time dissipa-
tion looks like error correction by self-organization. This error correction is a key feature of living
matter but has remained poorly understood. One can also ask whether BSFRs could make possible
QCs involving sub-QCs in both time directions. Could the use of sub-programs with opposite time
direction allow a faster QC.

1.3 Finite field approximation and QC

Number theoretic vision about QC leads to new ideas about QC itself.

1. The momenta in the extension of rationals as algebraic integers can be interpreted as p-adic integers
in the induced extension of p-adic numbers. The p-adic number field corresponds to prime p, which
is the maximal ramified prime for the polynomial in quesetion.

In the approximation O(p) = 0 they define a finite field F (p, n) having dimension is is not larger
than the dimension of extension but can be smaller. The number of elements is pn and the situation
corresponds to n pinary digits, qupits, instead of qubits. TQC using elements of F (p, n) is an
attractive possibility. Besides this one has also spin degrees of freedom.

2. The elements of F (p, n) can be regarded as roots of some, in general non-unique, polynomial with
degree pn. This polynomial is in general not the polynomial inducing the extension of p-adic
numbers.

3. The Galois group for the finite field should transform to each other the roots of the originalpolyno-
mial interpreted as a polynomial in F (p, n) and is a subgroup of the Galois group for the polynomial
having all points of F (p, n) as its roots.

The automorphism group of quaternions is analogous to Galois group and in the TGD framework
with discretization it looks like a natural notion.

1. In the continuous case, the automorphism group of quaternions is the rotation group SO(3) having
SU(2) as covering group. In the discrete situation, one expects it to be a finite group and would
correspond to symmetries of Platonic solid in non-abelian case and to the symmetries of a regular
polygon in abelian ase. Icosahedron, tetrahedron, and octahedron have triangles as faces and the
proposal is that genetic code realized in terms of bioharmony [20] [45] corresponds to so called
icosa-tetrahedral tessellation of H3 [52].

2. Therefore genetic code and bioharmony could closely relate to the quaternionic aspects of number
theoretic physics and perhaps also to TQC for quantum variant SU(2)q of quaternionic automor-
phisms acting in the normal space of the space-time surface. A natural proposal is that the points
of the icosahedron and tetrahedron correspond to points for the discretized unit sphere known as
Bloch sphere defining possible directions of the quantization axis of spin in TQC.

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Pitkänen, M., Quantum Gravitation & Topological Quantum Computation

3. The finite subgroups of SU(2) are associated with the hierarchy of inclusions of hyper-finite factors
of type II1 and the proposal is that the inclusion of these factors define finite measurement resolution
such that the included factor defines the resolution [32, 23].

1.4 TQC and the new view about space-time

The new view about space-time is highly relevant for the TGD view of TQC.

1.4.1 Galois anyons

The basic problem of the TGD inspired model of TQC is the identification of the topological qubit
identified as an anyon-like state in standard TQC. One could say that topological qubit or its analog
does not correspond to quantum state but representation of braid group or quantum group assignable to
Chern-Simons action. Topological qubits also satisfy a nice algebra defined by the decomposition rules
of the representations of the braid group.

The motivation for this identification is that topological qubits are expected to be highly stable since
the change of the representation is not expected to be probable unlike the change of spin direction.
The non-local character is also an important aspect. The braids defining TQC program as unitary
representation of the braid group allows to identify the gates, which are universal in the sense that they
have finite computational accuracy.

The increase of the order of the covering group as a finite covering of the permutation group SN for N
braid strands allows to improve the accuracy. Kitaev [5, 4] has proposed [6] that anyon-like bi-localized
states of condensed matter Majorana fermions could define stable qubits. Majorana electrons would be
superpositions of electron states localized at the ends of a superconducting wire and would have parity
+/− 1 under permutations of ends of the wire.

In TGD framework the electrons defining analogs could be bi-localized states with localization to the
ends of a monopole flux tube or pair of them. Galois degrees of freedom are a new element and anyons
could correspond to multi-localized states defining representations of Galois group at its orbits consisting
of points of the cognitive representation at mass shell H3. Also spin degrees of freedom would define
Galois representation. If the braidings correspond to lifts of number theoretic symmetries, Galois group
corresponds to a subgroup of the braid group.

In the standard picture of TQC, a computationally interesting situations corresponds to non-Abelian
anyons to guarantee that the states defining topological qubits form a higher-D space. This means that
the swap ab ↔ ba is not a commutative operation inducing a mere phase anymore. Since the status of
Majoran fermions is unclear, it is still unclear whether any anyonic system satisfies this constraint. Galois
groups are in general non-commutative so that this problem disappears.

Physical states would be Galois singlets and anyon-like states would be their building bricks just as
quarks would define building bricks of general Galois singlets including also leptons and various bosons.
Since Galois non-singlet cannot appear as a free particle, one could also understand topological entropy
associated with anyons as relating to the entanglement with environment forced by Galois singletness in
spin degrees of freedom.

1.4.2 Braidings and reconnections as basic elements of TQC

TQC in the TGD Universe involves also other new elements besides Galois groups.

1. The flux tubes connecting the nodes of a tensor net-like structure define natural candidates for
braid strands. Both space-like and time-like braiding are possible.

Time-like braiding defining TQC of the moving nodes connected by flux tubes induces a space-like
braiding so that the TQC is recorded to memory as a kind of log file. Dance metaphor expresses this
neatly: dancers at the parquette are connected by threads, which get braided and form a memory

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representation about the dance. This mechanism could define quite a general representation of
memories based on space-time topology.

2. The fusion defined by the tensor product for the representation of the braid group or associated
quantum groups is a key operation in standard quantum computation. The decomposition of the
tensor products gives a superposition of topological qubits or more general qubit-like entities An
interesting question is whether the fusion could have a more concrete topological meaning. Could
the fusion of flux tubes correspond to a formation of a bound state of flux tubes inside a flux tube?

3. TQC as a braid generalizes to tensor-net (for tensor nets in TGD sense see [24] [36]). The nodes can
have M incoming qubits and N outgoing qubits. The node corresponds to a quantum computation
defined as a map between the incoming and outgoing qubits. In the framework, the nodes would
correspond to CDs For M 6= N is not a unitary transformation 1-1 transformation but can be an
injection so that it is still an isometry at the level of the state space.

4. Besides swap as the basic braiding operation, also reconnection, having the same effect as far as
initial and final states are considered, appears as a basic operation. When the incoming and outgoing
qubits cannot move, reconnection could take the same role as swap and make TQC possible.

5. One can wonder whether this more general view about TQC could be realized in quantum biology.
Could biochemical reactions correspond to fusions of braids of a tensor net, could reconnections and
braidings make it possible to have a larger repertoire of TQCs. Could ZEO-based error correction
requiring only time reversal play a key role in TQC.

1.4.3 Different TGD based views of TQC

TGD suggests several different perspectives of TQC.

1. In the flux tube picture, the basic elements are braiding, reconnections and fusions in which flux
tubes could even form a bound state inside a larger flux tube so that the fusion could have a
geometric meaning. At the level of H, fusion could correspond to a process in which the incoming
particles arriving into the CD form a tensor product. Inside CD fusion occurs and gives rise to a
decomposition of irreps. Measurement selects one irrep first and outgoing states are obtained by an
SFR cascade reducing the total Galois group to the factors defined by relative Galois groups by a
cascade of SSFRs defining cognitive measurements.

Dance metaphor implies a mechanism of memory with spatial braidings representing spatial braid-
ings. This mechanism would be realized in all scales and define kinds of topological Akashic records.
If reconnection is equivalent with swap operation, then TQC is also possible without braiding in-
duced by the motions of braid ends.

2. CDs are counterparts of gates at the level of H and define a fractal hierarchy of gates with sub-CDs
defining sub-modules.

Space-time surface in H can be also seen as a 4-D soap film with frames as seats of non-determinism
and one could assign mental images with this non-determinism. This suggests that the gates at
space-time level correspond to the frames whereas CDs would correspond to entire TQCs at the
level of H. This also suggests that TQC in the TGD sense must allow intermediate SSFRs at the
frames. The situation is far from obvious since fractality is also present and involves a hierarchy of
CDs.

The M8 − H-duality provides a further view about TQC. A highly attractive idea is that TQC
programs can be constructed as functional composites of polynomials giving rise to extensions of
extensions of .... and inclusion hierarchies of corresponding Galois groups, each defining a normal
subgroup of its sup-group.

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The normal subgroup hierarchy makes it possible to understand cognitive measurements as SSFR
cascades reducing the representation of the Galois group to a product of representations for the
subgroup and normal subgroup associated with it. This decomposition could generalize the de-
composition of the anyonic representations. This would also suggest a deep connection with the
paradigm in which computations are functions.

2 What could the replacement of the braid group with the Ga-
lois group mean?

The replacement of the braid group acting on anyons with the Galois group looks a rather innocent
proposal first but has profound implications. The reason is that the Galois group permutes the roots
of the polynomial P , which correspond to different mass shells in M8 and therefore different values of
light-cone proper time in H.

2.1 Functional composition of the polynomials and many-particle states

Functional composition of the polynomials is proposed to give rise to many-particle states.

1. The roots of P correspond to mass shells. Quarks have momenta at these complex mass shells.
Roots and corresponding momenta are in general complex algebraic numbers and total momenta
and mass squared values are real by Galois confinement.

2. Functional composite P = Pn ◦ ... ◦ P1 of polynomials defines the interactions of particles in the
number-theoretical picture. Functional composites are proposed to define particles as many-quark
states and further functional compositions make it possible to engineer many particle states formed
from these.

3. One can also consider iterates of a polynomial as analogs of many particle states involving only a
single kind of particle. Functional decomposition gives as roots inverse iterates of the roots of the
polynomial Q in P = Q◦Q...◦Q [44, 58, 59]. Asymptotically they give rise to an analog of the true
Julia set (https://mathworld.wolfram.com/JuliaSet.html) as a boundary of the filled Julia set.
The inverse iterates near the boundary of the Julia set would correspond to very nearly the same
mass squared values and thus proper time constant hyperboloids.

4. One can regard the roots of Pi as roots with respect to the variable y = Pi−1 ◦ ...P1(x) if y =
Pi−1 ◦ ...P1(x) defines the ground state coordinate. heff = n0h0 would define a natural ground
state for which heff = nh would hold true.

5. If the polynomials appearing in the composite satisfy Pi(0) = 0, one has ”inheritance of roots”.
The roots yi of Pi are mapped to their inverse images (Pi−1 ◦ ...P1)−1(yi) = P−1

1 ◦P−1
2 ... ◦P−1

i−1(x).
This inheritance brings in mind conserved genes. A weaker form of ”inheritance” would be that
some polynomials, say P1, P2, ..., Pk at the lowest level have Pk(0) 6= 0. For P = Q ◦ PF , where
PF = x2 − x− 1 is ”Fibonacci polynomial”, the roots would be of form (−1±

√
5 + 4yn)/2, where

yn is a root of Q. Note that one has P1(0) 6= 0. If one has Pk(0) 6= 0 for k > 1, the roots of
P1 are roots of any P and therefore universal. This suggests the possibility that the ground state
polynomial corresponding to heff = h = n0h0 is non-vanishing at origin.

2.1.1 Ground state polynomial

The ground state polynomial Pg corresponding to heff = h = nh0 is of special interest physically.

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1. The arguments allowing to deduce the value of n0 in h = nh0 lead to a conclusion that the ground
state polynomial Pg [54] corresponding to heff = h = n0h0 corresponds to a Galois group with 7!2

elements.

2. This allows several options. For instance, the semidirect product S7 o S7 could act as a Galois
group. S7 decomposes to a semidirect product of the simple alternating group A7 and Z2 acting as
a normal group. S7 can appear as a maximal Galois group for a polynomial of order 7. In this case
S7 could correspond to Qa = P7 ◦P2 or Qb = P2 ◦P7 and one would have four options P = Qi ◦Qj .
Also P7 ◦ P7 ◦ P2,a ◦ P2,b P2,a ◦ P2,b ◦ P7 ◦ P7 are possible.

3. Second roots appear in all basic formulas of quantum mechanics. Therefore one can argue that
P2 should appear at the bottom of the composite polynomial defining the ground state. Fibonacci
quantum computation involves Golden Mean and the roots x± = (−1 ±

√
5)/2 of Fibonacci poly-

nomial PF (x) = x2 − x − 1. All roots would appear as pairs with members related by the Galois
group of PF . For P1 = PF and Pk(0) = 0 for k > 1 (inheritance), the roots of PF are roots of any
P and Golden Mean would play a key role in fundamental physics.

2.1.2 Mass squared formula and inheritance hypothesis

For Galois singlets, the total momentum has components, which are ordinary integers. Also mass squared
is integer.

1. If the stringy mass formula m2 = n holds true for the quark mass squared values as roots of a
polynomial, one must have m2 =

∑
m2

i = n. This requires that the sum of the inner products of
quark momenta vanishes. The interpretation would be as an additivity of conformal weights. If
every root is realized as quark momentum, m2 =

∑
m2

i equals the constant coefficient of the total
polynomial P giving m2 = P (0).

2. If the strong form of inheritance holds true, one has
∑
m2

i = 0 so that the total conformal weight
vanishes. Could the interpretation be in terms of conformal invariance? Could one say that the
tachyonic total mass squared assignable to the space-like states defined by braid strands compensates
for the non-tachyonic total mass squared?

Total momentum would be light-like and the M8 −H duality should be defined as the map pk →
mk = ~effpk/(p0)2 wheremk belongs to the light-like boundary of CD containing the CDs assignable
to the mass squared values as sub-CDs.

3. In p-adic mass calculations the total conformal weights are however non-vanishing and real. What
could this mean?

(a) The thermal excitations should be excitations of the m2 = 0 state due to interaction with
the environment, which extends the system. The thermal excitations would be described by
polynomials Qex = Pex ◦ P . The roots of Qex would include, besides roots of P (inheritance),
also the roots yn of Pex and these correspond to non-vanishing values of P (yn). yn 6= 0 would
give non-vanishing mass for the thermalized subsystem defined by P .

(b) If one gives up the ”inheritance” hypothesis and allows Pi 6= 0, one has m2 =
∑
m2

i = Pn(0).
Monic polynomials P (x) = xn + an−1x

n−1 + ... + a0 are good candidates for the allowed
polynomials. The coefficients ak are integers so that the mass squared as a conformal weight∑
m2

i = a0 is an integer.

2.1.3 Decomposition of Galois group to a product of relative Galois groups

The Galois group Gal for an extension of extension.... decomposes to a product of the relative Galois
groups Galk/Galk−1.

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1. One can speak of the ground state characterized by some Galois group Gal0. Ordinary Planck
constant h would correspond to Gal0 and in [54] it was proposed to be a product of permutation
groups S7 giving n0 = (7!)2. This allows to interpret CP2 length scale squared as n0lP 2, lP Planck
length. Galois group can be identified as a relative Galois group: as Galois group for extension of
the extension defining the ground state.

2. The structure of the Galois group reflects the functional composition involving a large number of
identical polynomials with the same mass spectrum as free particles. In the functional composite
P ◦Q the mass spectrum S of P is mapped to Q−1(S). Large number of iterations of P produces
Julia set as a fractal. One can speak of an asymptotic mass spectrum.

3. The orbit of Galois group consists of mass shells and its cognitive representation can contain mo-
menta at these mass shells.

Galois symmetry would be a discrete symmetry connecting quarks with different mass values (which
are counterparts of virtual masses rather than real masses). Galois symmetry would be analogous
to a dynamical symmetry and would not commute with Poincare and Lorentz symmetries.

Physical states are Galois singlets and have well defined real mass squared. Galois singlet property of
physical states would imply that these symmetries would be respected. Physical states correspond
to a CD containing sub-CDs... and at the lowest level there would be quarks. Essentially 4-D
objects would be in question.

2.2 M8 −H duality at the level of M4

M8 − H duality maps the algebraic physics at the level of M8 formulated using polynomials to the
geometric physics at the level of H = M4 × CP2 formulated using variational principle and partial
differential equations. The preferred extremal property required by general coordinate invariance reduces
the number of solutions of field equations so that they can correspond to a much smaller set of solutions
of algebraic equations. The holographic aspects of M8 −H duality have been already considered and in
the following only the map M8 ⊃M4 → H ⊃M4 is discussed.

1. M8−H duality maps the surfaces of M8 to minimal surfaces in H having singularities at which only
the field equations for the full action containing also Kähler action besides the volume term hold
true. M8 −H realizes holography: the mass shells determined by the roots of P can be continued
to 4-surfaces containing them.

2. The precise form of M8−H duality is not quite clear. The first question is whether one should allow
complexification of M4 as at the H side. One could define the H image as Mk = ~effRe[pk/m2],
where pk is the quark momentum and at mass shell m2. Mk would define some geometric objects
in H. For physical states m2 is integer and corresponds to a finite value of a = ~eff/m. If the
stringy mass formula m2 =

∑
m2

i = 0 is true, the image belongs to the light-cone boundary.

The image could be a geodesic line of H parallel to mk, which could start from the origin from the
common center of CDs forming a fractal Russian doll hierarchy or from the tip of a given sub-CD.

The image could also be identified as a point or a set of points. The point could be identified as
the intersection of these lines with the boundary of the sub-CD defined by the mass value or its
real part. Also the intersections with boundaries of all sub-CDs involved can be considered. Also
the map of mass shells to M8 to hyperboloids a = an, where a is light-cone proper time and an is
inversely proportional to mass to realize Uncertainty Principle, makes sense.

3. The image of the orbit of the Galois group would correspond to a geodesic line starting at the
centers or tips of various CDs defined by the mass shells. If the CDs are inside each other like a
Russian doll, the geodesics intersect the a = an hyperboloids and the boundaries of corresponding

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sub-CDs corresponding to different values of the light-cone proper time a and are time ordered.
What is highly non-trivial is that the points at the orbit have time- like distances.

2.3 The orbits of the Galois group in H transform hyperboloids to each other

Mass squared values correspond to the roots an of a polynomial and are in general complex algebraic
numbers. Their real projections can be negative and therefore tachyonic. The big surprise during writing
of this article was the trivial observation that the Galois group permutes the mass shells defined by the
roots of P .

If the real projections of mass shells to M4 are mapped to H, Galois group can connect points with
different values of complex ”cosmic time” a = an. This does not conform with the idea that the particles
of the physical state always have space-like distance but could conform with ZEO and non-determinism
inspiring the view that time-like braiding is a physical state rather than its time evolution.

Note however that the spatial distance (M1 −M2)2 in H is space-like for (E1 ≥ m2
1 +m2

2)/m2 in the
coordinate system in which M2 and p2 have a vanishing spatial part. This holds true also for the M4

images.

2.3.1 Orbits of the Galois group as braidings?

Could the orbits of the Galois group for off-mass shell states be identified as braidings?

1. If the braiding is time-like, the value of the real part of the proper time parameter corresponding
to the mass shells or CD sizes increases along the orbit.

This would conform with the idea that the orbit of the Galois group consists of images of mass
shells at the quark level. It also conforms with the breaking of Lorentz and Poincare symmetries
at the level of the Galois group. This finding also justifies the Galois confinement: physical states
correspond to a single value of a.

2. What about number theoretic anyons? These anyons must have non-trivial Galois quantum numbers
and algebraic momenta. Here the relative Galois group is a convenient concept. Galois non-singlet
property is with respect to the relative Galois group and one can forget the huge complexity of the
Galois singlet ground state altogether.

2.3.2 Do Galois anyons require tachyonic states?

The momenta of quarks define the basic representation of the Galois group. One can also imagine
representations in spin degrees of freedom. If only the spin degrees of freedom carry Galois quantum
numbers, the space-time action of the Galois group is trivial. This does not look attractive and does not
conform with time-like braiding. Anyon property therefore suggests the presence of tachyonic momenta.

1. I have played with the idea that quarks and also weak bosons appear in the scale of cells in living
matter as dark quarks or even scaled variants with very small mass. How could the dark quarks
manifest themselves?

I have proposed that the protons of dark nucleon triplets representing codons are connected by
meson-like bonds, which could be colored and confine codons to genes. This could the case also
for the bonds connecting nucleons in the ordinary nuclei. Strong interaction would also make it
possible to have dark neutrons.

I have assigned the Z3 Galois group with the dark nucleon triplets defining dark codons: this is
required by the correct statistics in the model of the genetic code. Could Galois group Z3 correspond
to the center Z3 of the color group SU(3)?

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2. In the original proposal for DNA TQC [18], quark triplets were indeed considered instead of dark
nucleon triplets. Dark tachyonic electrons assignable to symmetric hydrogen bonded structures looks
like a more realistic option. One can also consider mesons with quark and antiquark ends associated
with the ends of the space-like braid strands. Dark tachyonic electrons could be associated with
the ends of string world sheets for which the time dimension corresponds to a space-like normal
dimension.

Could one assign a colored quark pair to anyon-like electron? Leptohadrons [30] are a basic pre-
diction of TGD and there is empirical evidence for them. The predicted mass of the lepto-electron
is very nearly the same as electron mass and evidence for its existence was found already in the
seventies. Lepto-electron would be a color octet: this is allowed in the TGD framework.

Lepto-hadron is associated with the breaking of parity symmetry in nuclear collisions involving
strong electric and magnetic fields not orthogonal to each other. Its description involves Chern-
Simons Kähler action associated also with anyons. The notion of induced gauge field allows its
interpretation as SU(3) Chern-Simons action. A possible identification of lepto-electron would be
as an anyon for which electron would be accompanied by a color octet quark pair formed by the
quarks at the ends of the flux tube.

3. Polynomials can also have roots corresponding to space-like mass squared values. Could dark quarks
be tachyonic in the sense that they have a negative real part of mass squared so that time direction
as a normal direction for this object would be naturally space-like?

4. Could one see time-like braids structures as genuinely 4-D objects predicted by ZEO and the failure
of the strict determinism of the action principle? Singularities as frames span 4-D soap films serve
as a source of non-determinism.

2.3.3 How could dark DNA correspond to time-like braids strands for dark DNA?

The following represents a long list of cautious proposals represented as questions.

1. Can one Galois symmetries acting in time direction have projections acting effectively as 3-D sym-
metries of ordinary matter at time=constant surface.

The Galois group at the level of (presumably gravitational) MB does not act at the level of ordinary
matter. Could the time-like braids at the level of the dark DNA correspond to the ordinary DNA
strands in the sense that the temporal sequences would be mapped to spatial sequences by some
simple rules?

2. Could genes have a representation as time-like braids? Could one imagine a pile of or ordinary
DNA strands and their dark counterparts at different values of a = an such that time like braid
strands would have the same DNA content as the DNA in a = constant or t = constant plane. For
instance, could the intersections of the points of cognitive representation at a = an hyperboloids
with t = constant hyperplane define the DNA strand.

The codons of dark DNA as a temporal sequence would correspond to codons of the ordinary DNA
unless one assumes that only identical codons correspond to the orbits of the Galois group. This
looks like a more reasonable option. Codons themselves would correspond to orbits of the discrete
and finite subgroups of automorphisms of quaternions acting as symmetries of Platonic solids and
regular polygons. Therefore two kinds of Galois groups would be involved.

3. Could the physical DNA correspond to the space-like braidings assignable to the time-like braidings
of dark DNA? Could one realize the representations of the Galois group by using these projections
at the level of ordinary DNA.

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4. Could identical codons of a gene correspond to projections of points related by the Galois group?
If so, the collections of identical codons (64 of them) would correspond to 64 orbits and the anyons
would be realized at these collections as wave functions. Different representations would correspond
to different anyons serving as number theoretical qupits.

2.3.4 String world sheet interpretation of time-like braidings at the level of H

M8 − H duality implies time-like braids correspond to physical states rather than time evolutions of
an ordinary physical state localizable to time= constant hyperplane. The time-like character of states
conforms with ZEO and is implied by the predicted non-determinism in which the singularities of the
minimal surface correspond to loci for the failure of strict determinism. These singularities define analogs
of frames for the space-time surface as an analog of a 4-D soap film. They are a necessary part of the
data allowing to realize holography.

M8−H duality [42, 43] predicts candidates for the singularities as loci of non-determinism. The follow-
ing argument suggests that the 2-D orbits of braid strands defined by string world sheets as fundamental
objects of the TGD Universe giving rise to braidings could characterize the non-determinism.

1. 3-D light-like surfaces defining orbits of partonic 2-surfaces starting at the boundaries of CD and 2-
D string world sheets connecting two light-like 3-surfaces. Strong form of holography, whose status
is uncertain, states that only the partonic 2-surfaces at the boundaries of CD are needed.

2. String world sheets would provide additional data to fix the preferred extremal and the failure of
4-D determinism manifested as the failure of the minimal surface property would be localizable to
the string world sheets. According to the dance metaphor, the ends of the strings would represent
dancers and strings would represent the threads connecting their feet.

String world sheets would be necessary for fixing the space-time surface. This is a profound deviation
from string models, where data at time=constant section would fix the time evolution.

In fully deterministic physics, the direction of time coordinate is normal to t = constant slice. The
normal directions of the string world sheet are analogous to time direction: that they are space-like
conforms with tachyonicity. String world sheet would represent a tachyonic virtual particle exchange
between particles with time-like momenta.

3. Also strings are minimal surfaces apart from singularities. Reconnection is a singularity at which
the string world sheets intersect at a single point and involves failure of determinism. The effect
of reconnection is the same as that of braiding (SWAP). Reconnection therefore corresponds to the
SWAP gate in TQC.

4. The 4-D character of the space-time surface implies that the strings develop spatial braiding during
the dance and can also reconnect. This does not happen in super string models with 10-D embedding
space for strings.

The braiding and reconnection patterns would represent the time evolution of string-like entries
in 4-D space-time so that TQC would reduce to a string model-like theory with one important
exception: braiding and reconnections are not possible in string models.

Gravitational flux tubes would be one particular case of flux tubes. They seem to be key players in
biology and provide a quantum gravitational view about metabolism, biocatalysis, and DNA [60].
TQC involves braiding and flux tubes with strings attached with them: TQC would have a direct
connection with string model type description of quantum gravitation and other interactions.

Tachyonicity of the time-like braids as physical states could be therefore understood. One can look at
the situation also from the point of M8 −H duality to gain additional perspective.

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1. Virtual particles of QFT picture would in TGD framework have a discrete mass squared spectrum
give by the roots of a polynomial and thus algebraic, in general complex, numbers [58, 59]. Their
finite number in zero energy state would resolve the divergence problem of QFTs.

Only quarks appear as fundamental fermions. Mass squared values and momenta of many quark
states constructed are in an extension of rationals without the condition of Galois confinement
implying stringy mass squared spectrum and integer valued momentum components using the scale
of CD as unit.

2. Quarks at mass shells of M4 ⊂ M8 are mapped to geodesic lines of H by M8 −H duality. They
can be also space-like unless one assumes that the real parts of the roots of P are non-negative. For
negative real parts, the momenta would be space-like and define points outside the sub-CD but a
larger CD could contain them.

Could the total momentum of say 3-quark state possibly associated with codon (3N quark state
associated with a gene) be tachyonic? Could the tachyonic quark triplets be located along the
time-like braid strand associated with the codon and define a tachyonic many-quark states?

3. For anyons as tachyons Galois confinement must fail and they should correspond to virtual states
made from quarks. Could the strands of a space-like braid as a string with quark and antiquark at
its ends define an entity analogous to a virtual meson? Could this meson-like entity have non-trivial
color quantum numbers?

How do Galois confinement and color confinement relate? At the level of ”world of classical worlds”
(WCW) quark color corresponds to partial waves in CP2 for cm degrees of freedom for the partonic 2-
surfaces associated with quark. At the level of the space-time surface there are no color partial waves
since fermions do not have color as a spin-like quantum number. I have proposed a Z3 subgroup of
the Galois group as a counterpart for Z3 ⊂ SU(3). Correct statistics requires antisymmetry with
respect to Galois Z3.

One must take this with caution: maybe the braid statistics of anyons could solve the statistics
problem. Note however that braid statistics is analogous to Fermi statistics in that two particles
are not possible in the same state.

The original proposal for DNA as a TQCer, was that DNA and nuclear membrane are connected by
flux tubes having quark and antiquark at their ends. Also DNA strands would be connected by this kind
of strands. The proposal was motivated by the observations and the classical counterpart of color gauge
field is proportional to the induced Kähler form, and can define a coherent field in arbitrarily long scales.

I gave up this proposal a long time ago but it seems that this proposal had some seed of truth in it.
Anyonic electrons replace quarks and antiquarks.

1. What comes in mind first is that the DNA strand and its conjugate involve, besides dark nucleon
triplets, also dark quark/antiquark triplets forced by the time-likeness of the braiding regarded as
a physical state in ZEO. This however leads to problems since dark nucleons are strongly favored.
Doubling of the genetic code without need for it looks ugly. The mere quantum gravitational
modification of the standard chemistry should be enough.

Most importantly, tachyonicity does not require single quark states. Also the dark anyonic electrons
could be virtual particles carrying tachyonic momenta. The 3+3 dark electrons assignable to the
asymmetric HBs of form O..H-N would provide electronic realization of the genetic code. The dark
codons would serve as names, addresses in the symbolic dynamics of TQC involving the resonance
mechanism of communications requiring addresses.

The dark anyonic electrons assignable with G-C bonds would carry tachyonic momenta and make
the braiding possible. The tachyonic electronic momenta assignable to bonds symmetric O...H-O
type bonds connecting water molecules and phosphates would be realized in the same way.

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2. It is good to bring in mind the possible weak points of the scenario once again. Dark protons are
strongly suggested by the Pollack effect and the proposed picture about dark gravitational HBs
with delocalized dark protons [60]. In the original view, dark protons screened the negative charge
of phosphates. In the new picture the negative charge of phosphate is assignable with bi-localized
(anyonic/dark/virtual) electrons of O...H-O +O-H...H: at the level of ordinary matter, DNA is not
negatively charged. In QFT language, one might perhaps say that a dark electron is exchanged
between the ends of the flux tube associated with the dark HB.

2.3.5 Connection with time-like character of music experience and cognition

A connection with the model of DNA based on bioharmony is suggestive.

1. DNA and RNA codons are identified as points at the orbits of icosahedral and tetrahedral subgroups
of quaternion automorphisms. Amino acids (AAs) have been identified as orbits of the icosahedral
and tetrahedral groups, which are discrete subgroups of quaternionic automorphisms, which is
completely analogous to Galois groups.

2. Harmony is the basic element of music and music involves time in an essential way. Same is true
of cognition. Perhaps the time-like braid strands could give a concrete content to the proposal.
Codons would correspond to 3-chords and gene would correspond to a piece of music in a much
more concrete way than originally proposed. Genes would also represent primitive cognitions.

2.4 Cognitive measurement cascades as counterparts of measurements of
anyon charges

The measurements of topological charges reduce the tensor products for the representations of the braid
group to irreducible representations. What would the counterpart for this process be at the level of the
NQC?

1. I have discussed cognitive measurements [34, 47] as a cascade of ”small” state function reductions
(SSFRs) for the irreducible representations of the Galois group of extensions of extensions of.... .
The full Galois group has a representation as a product of relative Galois groups Rn = Galn/Galn−1.
The SSFR cascade means a reduction of the representation to a product of representations of the
relative Galois groups Rn.

2. This measurement cascade would be the opposite for the measurement of anyonic topological charges
involving an analogous decomposition of the tensor product of representations to irreducible repre-
sentations of the full braid group.

In ZEO, the counterpart for the measurement of topological charges would correspond to the time
reversal of this process starting with BSFR, which creates a completely entangled state as the
representation of the full Galois group, and is followed by SSFR cascade proceeding in an opposite
time direction. The formation and decomposition of tensor products would occur in different time
directions.

2.5 Comparison of standard view about TQC with the TGD view

It is useful to compare the standard view about TQC with its TGD counterpart.

1. Qubits as states are replaced by representations of the braid group characterized by the value of
the topological charge and of the quantum group G assignable to the Chern-Simons action.

Quantum groups [3, 1, 2] are discussed from the TGD point of view in [21] and in chapters about
possible role of von Neumann algebras known as hyperfinite factors of type II1 in TGD [32, 23].

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Quantum group SU(2)q quantum group characterized by quantum phase q = exp(iπ/k), k = 5, is
the simplest option. One can say that anyons correspond to electrons assignable to the orbits of
2-D systems, whose time evolution could be described by Chern-Simons action.

In TGD, these 3-surfaces would correspond to the light-like orbits of partonic 2-surfaces which for
larger values of heff can have rather large size. For hgr = GMm/v0 the gravitational Compton
length for a particle with mass m is GM/v0 = rs/2v0 independent of the mass of the particle and
for Earth this gives .45 cm for v0 = c, one half of the Schwartschild radius.

2. Topological qubits correspond to topological charges such as the already mentioned parity for con-
densed matter Majorana electrons, which would have degenerate energies because they correspond
to momentum vectors k and −k differing by lattice momentum.

3. Quite generally, quantum measurements are Hilbert space projections. Measurement of qubit cor-
responds to a measurement of a topological charge. The qubit can be measured by a fusion process
for the representations of the gauge group G. Fusion means a formation of a tensor product of
representations and could result as a final state of TQC. Measurement means a projection to a
particular representation characterized by a topological charge.

One can also consider the opposite operation in which one decomposes a given representation to
a direct sum of product representations and projects out one particular product representation by
measuring topological charges for the composites.

4. Fibonacci TQC with quantum group SU(2)q for quantum phase q = exp(iπ/5), serves as the
simplest candidate for an interesting TQC. Condensed matter Majorana fermions could correspond
to Fibonacci anyons with q = exp(iπ/5)
(https://phys.org/news/2014-12-fibonacci-quasiparticle-basis-future-quantum.html).
The fusion for Fibonacci anyons is non-commutative and non-associative. These properties are
coded by a non-commutative R matrix and non-trivial F matrix (see Appendix). For a fusion of N
representations the number of degenerate ground states is N :th Fibonacci number.

This has a counterpart in TGD.

1. In the TGD framework, Galois group elements in general change the value of cosmic time as a real
part of the root of the polynomial defining the mass shell in M8 and its image in H. Therefore the
associated virtual quark states are not energy degenerate.

That mass squared values for anyons are different conforms with the idea of time-like braiding as a
genuine quantum state rather than time evolution of quantum state, which is natural in ZEO. One
can of course challenge this assumption. For states containing N particles with the same polynomial
P and represented as an iterate P ◦ .... ◦ P mass squared values as roots approach to Julia set for
P , and this could give rise to approximate degeneracy of mass squared values and corresponding
values of light-cone proper time a.

One can also consider a situation in which one has several roots with the same real part (say roots
of a second order polynomial). One can ask whether the analogs of condensed matter Majorana
fermions correspond to these kinds of states.

2. The topological structure in question would be realized in terms of the space-time topology as a
monopole flux tube not possible in Maxwellian electrodynamics. Also the strings assignable to the
flux tubes and corresponding string world sheets as representation of time-like braiding inducing
space-like braiding would play a key role. Chern-Simons action would be assigned to the light-like
3-surfaces defining the orbits of partonic 2-surfaces and string world sheets would connect these
orbits.

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3. The quaternionic automorphism group, defining the analog of the Galois group and having SU(2)
or its quantum variant as a covering group, serves as the analog of the gauge group G and acts in
the normal space of the space-time surface. Discrete and finite subgroups assignable to the Platonic
solids and regular polygons define the natural finite discretizations of this group.

The braid group could be replaced with a subgroup identifiable as the Galois group for an extension
of rationals or for extension of extension of rationals. Also this group can be non-Abelian and would
be naturally represented as a subgroup of the braid group.

4. Time reversed fusion corresponds to a cognitive measurement cascade consisting of unitary evolu-
tions followed by SSFRs as counterparts of ”weak” measurements. Cognitive measurement cascade
and its reversal are initiated by a BSFR changing the arrow of time. Two subsequent BSFRs would
correspond to fusion and its reversal and the time evolution between them would correspond to the
braiding as a unitary evolution. In TGD inspired theory of conscious experience, the sequence of
SSFRs gives rise to the flow of consciousness.

5. Quantum group SU(2)q for Fibonacci TQC has an interpretation as quantum automorphism. What
makes this biologically highly interesting is that the twist exp(iπ/5) is realized geometrically in the
structure of the DNA. This suggests that DNA and dark DNA could involve TQC. One can wonder
whether genes with N codons correspond to a fusion of N Fibonacci representations.

2.6 Could the MB of DNA perform intentional TQC?

In TQC and also in AI as human endeavours, human intention plays a key role. This fact has been often
forgotten by AI extremists. The braiding defining the TQC would be constructed using technological
tools developed by humans. What about the situation at the level of DNA based TQC? Could the MB
of DNA play the role of humans to some degree? What kind of quantum computations could the MB of
DNA perform?

1. When the braid ends can participate in the flow defined by cellular water or by 2-D liquid defined
by the lipids of the cell membrane in liquid crystal phase, one can consider the possibility that the
MB induces this flow and in this way builds time-like TQC program, which is also stored as spatial
braiding to memory.

As will be found in the next section, this situation would be true for braids possibly defined by the
gravitational flux tubes connecting the oxygens of phosphates of DNA with the lipid ends of nuclear
or cell membrane containing also phosphates. Also the GTPs and GDPs of microtubules contain
phosphates and their oxygens could be connected with those of lipid phosphates.

The braiding would serve a memory storage purpose. If MB can induce the flow of water or of
lipids, one can say that it can build TQC programs. For instance, a representation of function
involving two registers could be constructed by starting from entangled register and using the flow
of water or lipids to induce the needed braiding for the second register implying the entangled
state

∑
|n〉〈f(n)| . The TQC ending with cognitive state function reduction cascade would define

a conscious cognitive representation of the flow.

2. It will also be found that A-G base pairs by the N...H-N ↔ N-H...N symmetry of gravitational flux
tubes define candidates for HBs assignable to TQC. In this case the braid ends cannot move but
the reconnections of braid strands could produce braiding and TQC. Similar situation is true for
the sequence of identical DNA codons of, say, genes. They could define an orbit of the Galois group
and give rise to its representation. There would be 63 types of orbits which could decompose to
separate representations corresponding to various codons. Besides single electron states also many
electron states would be possible.

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In this kind of situation, the cognitive measurement cascade would give rise to a conscious cog-
nition at DNA level. In ZEO, reconnections would be forced by the preferred extremal property
and unavoidable by the 4-D character of the space-time surface. Therefore they would reflect the
underlying physics. The failure of the strict determinism could be interpreted as a selection between
a finite number of alternatives at the frames defining the space-time surface as a 4-D analog of soap
film. The analog of TQC would give rise to a sensory perception accompanied by cognition.

Factorization of integers into primes is one of the most interesting applications of QC. At first, it looks
unlikely that the MB of DNA could be able to do something like this. However, finite groups have a prime
decomposition to a product of finite groups and in the same way Galois groups have a decomposition to
a product of relative Galois groups, which do not have a similar decomposition.

Group theoretical prime decomposition is analogous but more general than the prime decomposition
of integers and more general composition of algebraic numbers to algebraic primes. Since groups with a
prime number of elements are certainly prime groups, prime factorization would follow as a consequence
and would be a side product of any cognitive SSFR cascade. This conforms with the paradoxical finding
that idiot savants, who do not have any idea about the notion of prime, can factorize large integers [22].

Could Quantum Fourier Transform (QFT) have any analog at the level of DNA? The states in the
irreps of the Galois group serve as candidates for the plane waves defining Fourier components. Could
cognitive measurements naturally involve a measurement of these quantum numbers as eigen values for
maximal set of commuting Galois group elements acting as a minimal Galois transformation. For instance,
a rotation by exp(i2π/n) would be analogous to this kind of transformation in Zn. These measurements
would induce a localization to a single Fourier component and repeated measurements of the same state
would give the probabilities of various Fourier components. These states are superpositions of states at
mass shells with varying mass squared and involve time delocalization making sense by the finite non-
determinism. A repeated measurement of Galois momenta would make it possible to find the factors of
an integer as in the ordinary QC.

3 DNA as quantum gravitational TQCer?

In this section a detailed model for DNA as a TQCer will be developed. The attribute ”quantum
gravitational” is not necessary since also smaller values of heff than hgr can be considered.

3.1 Concrete questions concerning DNA TQC

Before representing a concrete model for TQC using Galois anyons as qubits, the basic questions are
discussed.

3.1.1 How could DNA qubits be realized physically?

For TQC temperature topological charge identifiable replaces spin as qubit. In the TGD framework
Galois charges replace topological charges and one can talk about Galois anyons.

The basic question is how DNA makes it possible to realize anyonic qubits.

1. Dark nucleons associated with dark DNA codons, that is with O...H-O type HBs cannot realize
dynamical qubits in terms of spin because the codons must be fixed if they are to represent ge-
netic code. Only in the communications based on resonant cyclotron transitions their states can
temporarily change but should return back to the original state as a state of minimum (free) energy.

One can assign to A-T, G-C pairs 1+1 asymmetric HBs, which do not allow electronic anyons. This
gives rise to 3 +3 dark electrons, which could give rise to dark representation of the genetic code.

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The tentative interpretation is that the dark codons define the analog of computer hardware with
a fixed ROM. The dark codons would serve as addresses in the resonance mechanism: the analogy
with LISP is obvious.

2. The dynamical working memory should correspond to an anyonic realization of qubits. A dark
electron associated with the quantum HB of type X...H-X +X-H...X can give rise to two bi-localized
states with odd and even Z2 parity where Z2 exchanges the ends of HB. These two dark electron
states could serve as anyons.

This could work for electrons of O...H-O bonds between the oxygens of phosphate and water
molecules. This could be also the case for the N...H-N bond of C-G base pair, which is sym-
metric. The HB can be assigned with C codon. In this case, the notion of Z2 anyon makes sense
and could make possible TQC using gravitational variants of symmetric HBs of C-G base pairs
(hhttps://cutt.ly/WGNddJ3).

3.1.2 How could the unitary time evolution be realized?

Superpositions of HBs of type X...H-X + X-H...X could give rise to electronic anyons with bi-localized
dark electrons. Depending on the situation, braiding or reconnections having, at least apparently, the
same effect would define the unitary gates.

1. If the molecules containing X can move, braiding is possible. This is the case if the HBs are
associated with the phosphates of lipids of the cell membrane forming a liquid crystal and connect
them to the molecules of the cellular water. In the sol phase for intracellular water, the flow of
water molecules could define braiding.

The original proposal [18, 31] was that the flux tubes connecting the oxygens of the phosphates
associated with the DNA strand with the phosphates of the lipid ends would define TQCer. The
flow of the lipids of the lipid layer forming a 2-D liquid could define a braiding and thus TQC
program. For gravitational flux tubes this option could make sense. The oxygens of the phosphates
of DNA could be also connected with the molecules of the water surrounding the DNA if they can
move.

In this case, the dance metaphor makes sense: the TQC as time-like braiding produces a log file as
a spatial braiding.

2. For N..-H-H + N-H...N HBs of C-G base pairs the nitrogen atoms cannot move. The reconnections
of dark braid strands could produce the same effect as braiding and induce flux tube connections
between C:s and G:s belonging to distinct C-G pairs. For gravitational flux tubes these connections
could be very long.

String word sheets are fundamental objects in TGD and by the 4-dimensionality of the space-time
surface, 2-D string world sheets at flux tubes representing the orbits of space-like braids intersect at
a discrete set of points and for preferred extremals the reconnections are forced by topology. The
non-determinism is associated with the choice whether the time-like strand pair AC+ BD transforms
to AC+BD or AD+BC.

3.1.3 What about ordinary QC or TQC using electron spin of HB as qubit?

I do not understand TQC enough to say whether electron spin could also appear as a qubit when braidings
and reconnections define the gates. In any case, this option meets the same objections as the QC option
since a very low temperature would be needed in the standard physics framework.

1. The hyperfine splitting (https://cutt.ly/oGNdeA3), causing the 21-cm line of hydrogen, corre-
sponds to the magnetic interaction energy of nuclear dipole moment with electron’s magnetic field

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and is proportional to heff . The energy of hydrogen hyperfine splitting is ∆E = 5.89 × 10−6 eV.
This corresponds to a temperature of 5.89× 10−2 K. If the electrons are dark, the energy of hyper-
fine splitting is proportional to heff . The energy is above thermal energy at room temperature for
heff/h > 5× 103.

Note that the temperature T at the MB of DNA is assumed to be very low but during aging
identified as an approach to thermal equilibrium with the biological body T is assumed to increase
and approaches the Hagedorn temperature assignable to the flux tubes of MB [62].

2. If spin serves as a qubit, the manipulation of electronic qubits by changing their spin direction
using photons or braiding or reconnection, which at least apparently seems to have the same effect
as braiding, would be needed. Both braiding and reconnection involve the replacement A → C +
B → D with A → D + B → C but reconnection involves temporary touch of the braid strands
which might have some effect.

3.2 Number theoretical generalization of Kitaev’s proposal

Kitaev [5, 4] has proposed an elegant model for TQC using as qubits the two states of condensed matter
Majorana fermion [6] with two bi-localized states, which have parities +1 and -1 under Z2 symmetry.

3.2.1 Galois group as subgroup of braid group and Galois anyons

In the TGD framework, the representations of the Galois group would naturally replace these repre-
sentations and one could speak of TQC which is also number theoretic as far as anyon-like states are
considered.

Topological robustness would be replaced by number theoretical robustness due to the fact that the
extension of rationals depends only weakly on the polynomial: this is obvious from the fact, the number
of extensions is finite for a polynomial of given degree. M8 − H duality [42, 43] indeed implies that a
given space-time region is determined by a polynomial. In QFT approximation one is forced to replace
many-sheeted space-time with ordinary space-time and the nice picture is lost. One might however hope
that in TQC this loss is fatal.

1. Galois group replaces Z2. Instead of topological charges, one can speak of number theoretical
charges. Representations of the Galois group would correspond to number theoretical qubits. Num-
ber theoretical anyon would be identified as a superposition of states localized at points of orbit of
Galois group Z2 associated with DNA double strand.

As already found, the Galois ground state corresponding to heff = h = n0h0 is not completely
unique but would naturally correspond to a polynomial Pg = Qg ◦ P2 where P2 is second order
polynomial, all roots of P = P1 ◦ Pg appear in pairs x ± y and Z2 permutes the members of the
pairs. Fibonacci polynomial PF = X2 − x− 1 is highly attractive candidate for P2 and would give
the roots (1 ±

√
5)/2 as roots of all polynomials P . Also the twisting geometry of DNA favors

Fibonacci TQC, which is also the minimal option.

2. Hydrogen bonds X...H-X and X-H...X are symmetric and their possibly gravitationally dark variants,
could give rise to states with opposite parity. The electron of the hydrogen could define the number
theoretic anyon.

3. The gravitational flux tubes as counterparts of H-bonds could define the braid strands but alsos
smaller values heff ≥ h assignable to electromagnetic flux tubes could work. Braiding would take
place for these strands.

4. What about the protonic option for X...H-X type HBs based on the identification of anyons as
delocalized states of the dark proton with opposite parity? Also now one can consider a superposition

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of N-H...N and N...H-N gravitational bonds and two different parity states with respect to Z2. The
quantum gravitational model for the metabolic energy quanta however suggests that the dark proton
is localized mostly in the interior of the gravitational flux tube so that the dark proton should not
have a large amplitude at the ends of the flux tube.

Hydrogen bonded structures of type X...H-X populate living matter. Water and DNA and the first
examples that come into mind.

1. The hydrogen bonds between water molecules are of type O..H-O. Hydrogen bonded water molecule
clusters could give rise to multiply localized anyonic states of electrons and serve as TQCers.

2. The HBs of the oxygens of phosphate atoms with oxygens of water molecules allow poly-localized
electrons if the HB is superposition of O-...H-= and O-H...H. This would allow to associate electronic
anyons and TQC also with the dark nucleon triplet codons, which cannot have dynamical spin.

3. G-C base pair has one N..H-N type HBs (hhttps://cutt.ly/WGNddJ3). N-H...N ↔ N...H-N are
could be possible for h−eff > h HBs, and could lead to the delocalization so that one could assign
anyonic state with Galois Z2 symmetry with it. The G-C base pairs of the DNA double strand
could define a sequence of topological qubits. Note that the splitting of the N-H...N bond in the
G-C base pair leading to N + H-N is known to occur during DNA transcription and replication and
also in the temporary splitting of the HB [60].

4. Benzene allows delocalized states of electron pairs, which could be poly-localized and be analogous
to Z6 anyons. Also Z2 and Z3 anyons can be considered. The atoms of the aromatic ring could be
connected by flux tubes with heff > h and perhaps even heff = hgr.

In DNA , the sequences of the aromatic 5- and 6-rings, possibly defining Z5 and Z6 anyons, could
give rise to a delocalization of the anyonic states along DNA strands possibly involving gravitational
analogs of valence bonds.

5. In DNA strand nucleotides A and G contain aromatic 5- and 6- rings glued together whereas T and
C contain aromatic 6-ring (hhttps://cutt.ly/WGNddJ3). The members of base pairs contain fused
5- and 6-ring and 6-ring respectively. One can wonder whether the Galois representations associated
with these structures in the double DNA strand structure could make possible TQC. Also the side
chains of amino acids Phe, Tyr, and Trp contain aromatic rings and HBs between oxygens of water
molecules might be relevant for information processing at, say, microtubular level.

3.2.2 The non-symmetric HBs of base pairs and possible new dark realizations of the
genetic code

The symmetric HBs of C-G base pairs (hhttps://cutt.ly/WGNddJ3) would be in a very special role.
What about the remaining non-symmetric HBs associated with codons?

1. Besides N..H-N HB there are 3+3 electrons per codon with asymmetric HB of form X..H-Y, with
X,Y= O,N or N,O. The proposal that an electronic variant of metabolism is realized, leads to the
question of whether the spins of these 6 electrons could realize genetic code as a 6-bit code. Now
only the analogs of DNA codons would be realized.

2. For asymmetric HBs, anyonic dynamics for electrons is not possible but the electronic dark codons
could serve as addresses in the resonance mechanism of communication based on the transformation
of Josephson radiation to pulse sequences by cyclotron resonance [61, 60]. This is possible if the
electrons are dark so that the energy of the hyper-fine splitting is scaled so that it is higher than
thermal energy. This would require heff ≥ 50.

One can also imagine resonance-based communications between dark electron 6-plets and dark
nucleon triplets using dark photons.

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3. The dark proton at flux tube and dark electron at the hydrogen end could define an analog of dark
H atom. Dark H would have 4=3+1 spin states with spins 1 and 0 and these states could define the
analogs of nucleotides in 1-1 correspondence with A,T,C,G. C as a special codon would naturally
correspond to the spin singlet. Hyper-fine splitting for this dark atom would distinguish between
triplet and singlet. For large heff the energy this splitting would be above thermal energy so that
the spin configurations would be stable.

These observations challenge the details of the earlier view [61] about the genetic code.

1. The dark nucleon realization of the genetic code [61] predicts both DNDA, DRNA, DtRNA, and
DAAs. One can criticize the realization since also neutrons are required.

The model of the code has several variants but the most recent model [60] requires dark variants of
both neutron and proton residing at the gravitational flux tube defining gravitational HB connecting
the oxygens of phosphate and water. The charge of the delocalized dark proton would not be visible
in the scale of DNA so that its replacement with dark neutron would not affect the situation in this
scale.

Dark protons would be generated from ordinary protons in Pollack effect [9, 8, 11, 10]. They could
transform to dark neutrons by the dark variant of strong interactions or of weak interactions at the
gravitational flux tubes. Dark weak interactions could be realized in even cellular scales and imply
that dark variants of weak bosons are massless in the scales below the dark Compton length of weak
bosons. This would explain chiral selection of biomolecules difficult to understand in the standard
model.

The conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis
(PCAC) [30] relate the descriptions of hadrons in terms of strong and weak interactions, which
suggests that these views might provide dual descriptions. The duality might in fact reduce to
M8 −H duality. The interpretation of anyonic electron as a color octet electro-pion [30] involving
color octet meson-like state associated with the gravitational flux tube was already discussed.

If HB is associated with oxygen of phosphate (water molecule), the hydrogen of phosphate (water
molecule) would look negatively charged. For anyonic states the electron of H would spend half of
the time near the two oxygens involved implying that negative charge would be delocalized in a
longer scale.

2. Could the standard genetic code be associated with the electron triplets at HB associated with base
pairs rather than with the phosphate water HBs? One can imagine two realizations.

(a) For both dark DNA strands, both dark proton triplet and dark electron triplet would have 23

dark entangled states and together they would combine to form 64 states. Could they provide
a dark realization of the genetic code consistent with the chemical genetic code?

(b) Could the dark protons at the HBs associated with base-pairs pair with dark electrons at their
ends give rise to analogs of dark H atoms? This could give 64 states perhaps allowing an
interpretation as a dark realization of genetic code.

There are objections against both proposals. The counterparts of RNA,tRNA, and AAs are not
predicted so that the correspondence with the chemical realization of the genetic code is not plau-
sible. Dark codons would have integer spin varying from 0 to 3 and the code table does not show
any grouping of codons to these multiplets containing an odd number of states.

To sum up, it would seem that several realizations of the genetic code are possible as indeed suggested
by the proposed universality of the genetic code [52, 61].

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3.2.3 Could protonic and electronic anyons define a pair of registers?

Two registers are needed to represent a Boolean function x → y = f(x) in terms of entanglement (see
Appendix). n qubits represent the values of x and y. The simplest representation of f is as a maximally
entangled state

∑
|n〉〈f(n)| . In this representation quantum Fourier transform (QFT) is exponentially

faster than the ordinary fast Fourier transform. Also the quantum counterparts of number theoretic
algorithms such as finding prime factors and greatest common divisor are faster than their classical
counterparts.

How could one realize these registers in the recent case? There should be a natural interaction inducing
the entanglement between qubits. The realization of the genetic code fixes the states of dark proton and
electron triplets completely for a given codon so that these qubits are non-dynamical.

In the case of HBs of type X...H-X, this however leaves the anyonic degrees of freedom assignable to
the dark electron as Z2 degeneracy and perhaps also with dark protons as a similar degeneracy. The
entanglement between electronic and protonic anyons would commute with the spin degrees of freedom.
Could the two registers correspond to electronic and protonic anyons? Could the braidings of the flux
tube, possibly induced by reconnections, generate entanglement between these anyons? The objection is
that the anyonic dark protons would not be delocalized in long scales as the model for metabolic energy
quantum requires. The metabolic dark proton states would correspond to different states concentrated
near the top of the gravitational flux tube.

4 Appendix: Basic concept and ideas of quantum computation

I am not a specialist in quantum computation and since some readers might also have the same problem,
I have added some remarks about QC, which I believe to be relevant for this article. I have discussed
the TGD view about TQC for about 13 years ago [19, 18, 31]. These chapters reflect my views at that
time and a lot has happened in the TGD based view of quantum biology after that. Perhaps I also have
a little bit deeper understanding of TQC now.

4.1 About key ideas of QC

In the following the basic ideas QC and TQC are briefly described.

4.1.1 Gates as unitary transformations

Quantum computation can be seen as circuits consisting of gates, which realize unitary transformations
assigning to n incoming qubits n = m outgoing qubits: unitary forces m = n. For qubits, which reduce
to ordinary bits one obtains as a special case Boolean functions from n to n bits.

Unitarity forces m = n but by using control qubits for which nothing happens in the the gate but the
outcome from the remaining qubits depends on the value of the control qubit, one can realize also gates
which for bits reduce to Boolean maps from n bits to a smaller number of bits so that ordinary logic
circuits can be realized as a special case.

n-gates with n = 1, 2, 3 are enough for obtaining a universal set of gates. The interested reader can
learn details from the slides of Viterbi: for instance the slides at https://cutt.ly/EGNsmcR describe
Quantum Fourier Transform.

1. 1- port represents a unitary transformation of a single qubit.

(a) Phase gate, Hadamard gate and rotations by Pauli spin matrices are basic gates of this kind.
Discrete rotation as SU(2) transformation represents the general unitary transformation. Ro-
tation is specified by two orthogonal rotation axes and by 3 rotation angles.

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(b) Discrete subgroups of rotation group assignable to Platonic solids and regular polygons define
especially interesting selections for the set of possible quantization axes and for the possible
directions of spin representable as a point of Bloch sphere. For Platonic solids the subgroup
of SU(2) is discrete. These subgroups can produce unitary transformation in a finite accuracy
only but one can consider the possibility of transformations obtained as products of elements
of these subgroups.

(c) Quantum variant of SU(2) emerges in TQC and also the braid group defines a quantum variant
of the permutation group as a finite covering of the braid group. The gates in topological
computation correspond to the elements of the braid group. In the TGD framework, SU(2)
has a representation as the covering of the automorphism group of quaternions (analogous to
Galois group) acting in the normal space of the space-time surface.

2. Arbitrary N ×N -D unitary transformation can be constructed as a product of 2-D unitary trans-
formations. In the N = 2n case, the transformation can be represented at qubit level and using
control gates one can represent unitary transformations by using qubit representation with N < 2n.

The representation of a general unitary transformation in dimension n requires of order n2n gates.
The subset needed as unitary transformations is however believed to be much smaller than all
possible transformations.

3. Swap, which permutes subsequent incoming qubits and CNOT are examples of 2-gates.

4. The notion of controlled gate generalizes to n qubits. Toffoli gate as CCNOT defines a 3-gate and
together with 1- and 2-gates it defines a universal set of gates.

4.1.2 Bloch sphere and Platonic solids

Block sphere gives a parameterization for the directions of the spin quantization axis and spin has two
directions for a given quantization axis. In the twistorialization of TGD at the level of M8

c this interpre-
tation of the twistor sphere is natural [42, 43].

1. In the number theoretic vision these directions correspond to sines and cosines and in the number
theoretic vision these must belong to the extension of rationals considered assignable to a given
space-time region. This discretization can be interpreted in terms of finite measurement resolution.

2. The allowed quantization directions are obtained from each other by the transformations of the
rotation group SU(2). If these rotations form a finite group, only the symmetry groups of Platonic
solids and regular polygons are possible. For Platonic solids there are 4, 6, 8, 12, and 20 quantization
axes corresponding to tetrahedron, octahedron, cube, icosahedron and cube.

4.1.3 Some applications of QC

Examples of the applications of QC working faster than their classical counterparts are discussed in the
Wikipedia article (https://cutt.ly/8Hs5qdG). For instance, the following examples are discussed.

1. A very simple application is the finding of the inverse image of function by measurement the of
value of function f = f(n) for

∑
|n〉〈f(n)| giving the superposition

∑
|n〉〈f(n) = y|.

In a more general case this localization gives the inverse image of a map f of m-D discrete space to
n-D discrete space. The repeated application of this algorithm can be used to find the boundary of
a region of the inverse image of f .

2. Quantum Fourier transformation calculates a discrete Fourier transformation exponentially faster
than ordinary fast Fourier transform. Other related applications find a prime factor of integer,

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Pitkänen, M., Quantum Gravitation & Topological Quantum Computation

period of a periodic function represented as an entangled state
∑
|n〉〈f(n)| of two registers as, and

number theoretic logarithm.

Quantum Fourier transform (QFT) is discussed (https://cutt.ly/EGNsmcR) takes place exponen-
tially faster than the classical fast Fourier transform. For N = 2n qubits the number of computation
steps is O(n) whereas classically it is O(n2n). The discrete Fourier transform has a huge number
of both physical and number-theoretical applications.

QFT can be represented in terms of n qubit registers as an un-entangled product of states of n
qubits and this state can be constructed using only gates inducing phase rotations Rk = ep(i2π/k)
of qubits, Hadamard gates producing the superposition of 0 and 1, and control gates.

3. There is an algorithm calculating the phase produced by a unitary transformation: this algorithm
involves one additional qubit, whose phase is opposite.

4. There is a search algorithm, which increases the probability of the searched integer before localization
in discrete space defined by integers. The number of trials is O(

√
N) whereas classically it is O(N).

5. Error correction algorithms localizing the logical qubits relevant for the computation to a subspace
of logical qubits. These algorithms detect the error by using parity qubits and correct the error by
action of a unitary gate in the case that the number of errors is below a given number.

4.1.4 Finding a period of a periodic function

One assumes that the function f(n) is periodic but the period is not known. The entangled state of the
registers is

∑
|n〉〈f(n)|.

1. One assumes that one has measured y = f(x) and has obtained
∑
|n〉〈f(n) = y|. If f is periodic,

one obtains a superposition of points n0 + nr, where n0 is the offset and r is the period, which
should be measured.

2. A QFT is performed for the input register. One obtains a superposition for states with momenta
mN/r.

3. The measurement of momentum this state gives momentum state with momentum pm = mN/r for
some m, which is however unknown.

4. The operation is repeated. This gives a series of outcomes m1,m2,m3, .... Eventually the minimum
value of momentum corresponds to m = 1.

4.2 About key ideas and notions of TQC

It is appropriate to briefly recall the basic ideas and concepts of TQC [19].

4.2.1 Topological gates and qubits

The topological stability of braiding guarantees that the TQC program coded by the braiding is robust
against perturbations. If qubits were spins, there would still be the instability of qubits and entanglement
caused by the interaction of spins with the environment, in particular thermal instability.

1. Qubits as spins are replaced by representations of the braid group characterized by the value of the
topological charge and the quantum group Gq assignable to the Chern-Simons action. The quantum
group SU(2)q is the simplest option. Topological charge replaces spin as qubit. One can say that
anyons correspond to electrons assignable to 2-D topological structures, whose time evolution as
3-surfaces could be described by Chern-Simons action.

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The mathematics of quantum groups [3, 1, 2] is discussed from the TGD point of view in [21] and
in the chapters about the possible role of von Neumann algebras known as hyperfinite factors of
type II1 (HFFs) in TGD [32, 23]. Quantum groups would be assigned to the inclusions of HFFs
characterizing the finite measurement resolution. Cognitive representations are an alternative way
to describe the finite measurement resolution.

2. Topological qubits correspond to topological charges such as the already mentioned parity for the
condensed matter Majorana electrons, which would have degenerate energies because they corre-
spond to momentum vectors k and −k differing by lattice momentum.

The idea of Kitaev [5, 4] [6] is to use anyons as topological qubits instead of spin. The condensed
matter Majorana electrons bi-localized at the ends of superconducting wire have two states with
opposite parities associated with the exchange of the ends of the wire. These states with degenerate
energies would serve as qubits, which would be much more stable than spins.

3. Topological approach allows to realize gates in terms of braiding operation. Braid group BN as a
covering of the permutation group of N braid strands would define the allowed unitary transforma-
tions induced by braidings. This implies finite accuracy but the increase of the covering improves
the accuracy.

This allows to overcome the problem of the Hamiltonian approach in which the gate Hamiltonian
defining the unitary transformation must be ”on” for a very precise time ∆T . It is not easy to
arrange this by external interaction. A possible way to avoid this altogether is to assume a permanent
Hamiltonian but allow the qubit system to move with a fixed velocity past the Hamiltonian system
with a velocity, which gives the desired ∆T .

4. Non-abelianity is required since the manifold of the energy degenerate states in which the braid
group would act, is determined by states and must be a higher-dimensional representation of the
braid group in order to give rise to a large enough number of logical qubits. There exist no well-
established candidate for the needed non-abelian anyon yet.

4.2.2 R and F matrices

R- and F matrices are central notions in TQC (https://arxiv.org/pdf/2005.03236.pdf) and charac-
terize what happens in the fusion of the representations of quantum groups. These matrices are believed
to characterize quantum phases as topological orders and were discovered in 2-D fractional quantum Hall
systems.

1. Fusion corresponds to a tensor product, which is commutative and associative for ordinary group
representations. For quantum groups and braid groups, the discrete group elements are replaced
by flows in plane so that the situation changes. The commutativity of the product ab of the
representations is lost and associativity for the product a(bc) of three representations is only modulo
unitary transformation: a(bc) is equal to (ab)c only modulo unitary transformation.

2. R matrix characterizes the braid operation, swap, in which the two braid strands are permuted by
flow-like continuous transformation. Braiding as an element of BN replaces the discrete permutation
of adjacent braid strands as an element of SN . The R-matrix characterizes the effect of the braid
operation and reduces to a phase in the abelian case but is a genuine matrix in the physically more
interesting non-Abelian situation.

3. F matrix characterizes the associativity modular unitary transformation for fusion operations. The
F matrix is trivial for the ordinary tensor product . This means that the fusions a(bc) and (ab)c
produce different states but do not change the state-space. F-matrix F (a, b, c) relates these two
states as a unitary transformation in the tensor product of the 3 state spaces.

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4. Fibonacci quantum computation with quantum group SU(2)q for quantum phase q = exp(iπ/5)
represents the simplest example of a non-commutative situation
(https://phys.org/news/2014-12-fibonacci-quasiparticle-basis-future-quantum.html).
For a fusion of N representations the number of energy degenerate ground states is N :th Fibonacci
number.

Received May 25; Accepted July 23, 2022

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https://tgdtheory.fi/public_html/articles/twisttgd2. pdf
https://tgdtheory.fi/public_html/articles/precns.pdf
https://tgdtheory.fi/public_html/articles/precns.pdf
https://tgdtheory.fi/public_html/articles/darkcode.pdf
https://tgdtheory.fi/public_html/articles/aging.pdf
https://tgdtheory.fi/public_html/articles/aging.pdf

	Introduction
	Two visions about physics in TGD framework
	Physics as geometry
	Physics as a generalized number theory and M8-H duality

	Zero energy ontology (ZEO) and QC
	Finite field approximation and QC
	TQC and the new view about space-time
	Galois anyons
	Braidings and reconnections as basic elements of TQC
	Different TGD based views of TQC


	What could the replacement of the braid group with the Galois group mean?
	Functional composition of the polynomials and many-particle states
	Ground state polynomial
	Mass squared formula and inheritance hypothesis
	Decomposition of Galois group to a product of relative Galois groups

	M8-H duality at the level of M4
	The orbits of the Galois group in H transform hyperboloids to each other
	Orbits of the Galois group as braidings?
	Do Galois anyons require tachyonic states?
	How could dark DNA correspond to time-like braids strands for dark DNA?
	String world sheet interpretation of time-like braidings at the level of H
	Connection with time-like character of music experience and cognition

	Cognitive measurement cascades as counterparts of measurements of anyon charges
	Comparison of standard view about TQC with the TGD view
	Could the MB of DNA perform intentional TQC?

	DNA as quantum gravitational TQCer?
	Concrete questions concerning DNA TQC
	How could DNA qubits be realized physically?
	How could the unitary time evolution be realized?
	What about ordinary QC or TQC using electron spin of HB as qubit?

	Number theoretical generalization of Kitaev's proposal
	Galois group as subgroup of braid group and Galois anyons
	The non-symmetric HBs of base pairs and possible new dark realizations of the genetic code
	Could protonic and electronic anyons define a pair of registers?


	Appendix: Basic concept and ideas of quantum computation
	About key ideas of QC
	Gates as unitary transformations
	Bloch sphere and Platonic solids
	Some applications of QC
	Finding a period of a periodic function

	About key ideas and notions of TQC
	Topological gates and qubits
	R and F matrices



