DNA as topological quantum computer: IV

In order to have a more concrete view about realization of topological quantum computation (see the previous posting and links from it), one must understand how quantum computation can be reduced to a construction of braidings from fundamental unitary operations. The article Braiding Operators are Universal Quantum Gates by Kaufmann and Lomonaco contains a very lucid summary of how braids can be used in topological quantum computation.

1. The identification of the braiding operator R - a unitary solution of Yang-Baxter equation - as a universal 2-gate is discussed. In the following I sum up only those points which are most relevant for the recent discussion.

2. One can assign to braids both knots and links and the assignment is not unique without additional conditions. The so called braid closure assigns a unique knot to a given braid by connecting n^th incoming strand to n^th outgoing strand without generating additional knotting. All braids related by so called Markov moves yield the same knot. The Markov trace (q-trace actually) of the unitary braiding S-matrix U is a knot invariant characterizing the braid closure.

3. Braid closure can be mimicked by a topological quantum computation for the original n-braid plus trivial n-braid and this leads to a quantum computation of the modulus of the Markov trace of U. The probability for the diagonal transition for one particular element of Bell basis (whose states are maximally entangled) gives the modulus squared of the trace. The closure can be mimicked quantum computationally.

Quantum computer is universal if all unitary transformations of n^th tensor power of a finite-dimensional state space V can be realized. Universality is achieved by using only two kinds of gates. The gates of first type are single particle gates realizing arbitrary unitary transformation of U(2) in case of qubits. Only single 2-particle gate is necessary and universality is guaranteed if the corresponding unitary transformation is entangling for some state pair. The standard choice for the 2-gate is CNOT acting on bit pair (t,c). The value of the control bit c remains of course unchanged and the value of the target bit changes for c=1 and remains unchanged for c=0.

2. The fundamental braiding operation as a universal 2-gate

The realization of CNOT or gate equivalent to it is the key problem in topological quantum computation. For instance, the slow de-coherence of photons makes quantum optics a promising approach but the realization of CNOT requires strongly nonlinear optics. The interaction of control and target photon should be such that for second polarization of the control photon target photon changes its direction but keeps it for the second polarization direction.

For braids CNOT can be be expressed in terms of the fundamental braiding operation e_n representing the exchange of the strands n and n+1 of the braid represented as a unitary matrix R acting on V_n\otimes V_n+1.

The basic condition on R is Yang-Baxter equation expressing the defining condition e_ne_n+1e_n= e_n+1e_ne_n+1 for braid group generators. The solutions of Yang-Baxter equation for spinors are well-known and CNOT can be expressed in the general case as a transformation of form A₁\otimes A₂ R A₃\otimes A₄ in which single particle operators A_i act on incoming and outgoing lines. 3-braid is the simplest possible braid able to perform interesting tqc, which suggests that genetic codons are associated with 3-braids.

The dance of lipids on chessboard defined by the lipid layer would reduce R to an exchange of neighboring lipids. For instance, the matrix R= DS, D =diag(1,1,1,-1) and S=e₁₁+e₂₃+e₃₂+e₄₄ the swap matrix permuting the neighboring spins satisfies Yang-Baxter equation and is entangling.

3. What the replacement of linear braid with planar braid could mean?

Standard braids are essentially linear objects in plane. The possibility to perform the basic braiding operation for the nearest neighbors in two different directions must affect the situation somehow.

1. Classically it would seem that the tensor product defined by a linear array must be replaced by a tensor product defined by the lattice defined by lipids. Braid strands would be labelled by two indices and the relations for braid group would be affected in an obvious manner.

2. The fact that DNA is a linear structure would suggests that the situation is actually effectively one-dimensional, and that the points of the lipid layer inherit the linear ordering of nucleotides of DNA strand. One can however ask whether the genuine 2-dimensionality could provide a mathematical realization for possible long range correlations between distant nucleotides n and n+N for some N. p-Adic effective topology for DNA might become manifest via this kind of correlations and would predict that N is power of some prime p which might depend on organism's evolutionary level.

3. Quantum conformal invariance would suggest effective one-dimensionality in the sense that only the observables associated with a suitably chosen linear braid commute. One might also speak about topological quantum computation in a direction transversal to the braid strands giving a slicing of the cell membrane to parallel braid strands. This might mean an additional computational power.

4. Partonic picture would suggest a generalization of the linear braid to a structure consisting of curves defining the decomposition of membrane surface regions such that conformal invariance applies separately in each region: this would mean breaking of conformal invariance and 2-dimensionality in discrete sense. Each region would define a one parameter set of topological quantum computations. These regions might corresponds to genes. If each lipid defines its own conformal patch one would have a planar braid.

The realization of single particle gates as U(2) transformations leads naturally to the extension of the braid group by assigning to the strands sequences of group elements satisfying the group multiplication rules. The group elements associated with a n^th strand commute with the generators of braid group which do not act on n^th strand. G would be naturally subgroup of the covering group of rotation group acting in spin degrees of spin 1/2 object. Since U(1) transformations generate only an overall phase to the state, one the presence of this factor might not be necessary. A possible candidate for U(1) factor is as a rotation induced by a time-like parallel translation defined by the electromagnetic scalar potential Φ=A_t.

The natural realization for single particle gate s subset SU(2) would be as SU(2) rotation induced by a magnetic pulse. This transformation is fixed by the rotation axis and rotation angle around this axes. This kind of transformation would result by applying to the strand a magnetic pulse with magnetic field in the direction of rotation axes. The duration of the pulse determines the rotation angle. Pulse could be created by bringing a magnetic flux tube to the system, letting it act for the required time, and moving it away. U(1) phase factor could result from the electromagnetic gauge potential as a non-integrable phase factor exp(ie∫ A_tdt/hbar) coming from the presence of scale potential Φ=A_t in the Hamiltonian.

What could then be the simplest realization of the U(2) transformation in the case of cell membrane?

1. There should be a dark spin 1/2 particle associated with each lipid, electron or proton most plausibly. A more complex realization would use J=2 Cooper pairs of electrons.

2. One should a apply the magnetic pulse on the braid strands ending at the lipid layer. The model for the communication of sensory data to the magnetic body requires that magnetic flux tubes go through the cell membrane. This would suggest that the direction of the magnetic flux tube is temporarily altered and that the flux tube then covers part of the lipid for the required period of time.

   The realization of the single particle gates requires electromagnetic interactions. That single particle gates are not purely topological transformations could bring in the problems caused by a de-coherence due to electromagnetic perturbations. The large values of Planck constant playing a key role in the TGD based model of living matter could save the situation. The large value of hbar would be also required by the anyonic character of the system necessary to obtain R-matrix defining a universal 2-gate.