Schreiber Quantum Computation with Defect Branes

The idea of topological \,quantum computation with anyons is traditionally thought of as physically implemented within solid state physics, where anyons are realized, in one way or other, as codimension-2 defects in some quantum material, governed by an effective 3d Chern-Simons theory.

On the other hand, it has become widely understood that relevant solid state quantum systems (potentially) supporting quantum computation tend to be analogs of intersecting brane models in string theory/M-theory, a statement sometimes known as AdS/CMT correspondence (closely related to the AdS-QCD correspondence, both of which being cousins of the more famous AdS/CFT correspondence, i.e. the “holographic principle” of string/M-theory).

Now, while the realization of anyons in solid state physics still remains somewhat elusive, we may observe that the analogous anyonic defects in string theory/M-theory are ubiquituous: These are are the defect branes with worldvolume of dimension 2 less than that of the ambient spacetime. Examples include the D7-branes in 10d type IIB string theory, but also the 3-branes in 6d.

In fact, the entire zoo of exotic branes in string theory/M-theory has been argued (de Boer & Shigemori 2012, p. 12) to be constituted by defect branes with U-duality monodromy, which essentially translates to: braid representations with values in U-duality groups.

This suggest that topological \;quantum computation may have its more natural implementation not in “mesoscopiccondensed matter physics, but in the truly microscopic high energy physics according to string theory/M-theory, with the former only being the approximate image of the latter under the AdS/CMT correspondence.

More concretely, under Hypothesis H the quantum states of flat M-branes intersecting\;codimension-1 branes are given by the cohomology of configuration spaces of points in the relative transversal space, according to Sati Schreiber 2019.

As discussed there, for D6-branes intersecting D8-branes the relative transversal space is 3-dimensional and the ordinary cohomology of its configuration space of points is given by weight systems on horizontal chord diagrams.

But this discussion transfers from branes inside the full ambient spacetime to branes inside an M5-brane: Here the charge quantization law changes from twisted\;4-Cohomotopy to twisted\;3-Cohomotopy (Fiorenza, Sati Schreiber 2020) and the analogous situation now concerns 3-branes inside the M5-brane, whose configurations are points in the transversal 2-dimensional space.

If we understand (as we should) the cohomology of this configuration space of points in 2d – which is now the space of quantum states of the 3-branes in 6d, according to Hypothesis H – in the generality of non-abelian and/or twisted\,generalized cohomology, hence as generalized cohomology with local coefficients, then every such cocycle now subsumes the datum of a braid representation (since the fundamental group of the configuration space of points in 2d is the braid group). This means, inder Hypothesis H, that every quantum state of 3-branes in 6d encodes, among further information, an anyonic\,quantum system (hence a set of topological\; quantum gates), just as expected from a system of quantum defect branes.

It is noteworthy that this theoretical state of affairs may not be all that remote from realistic high energy physics: In many models of string phenomenology (notably in the Witten-Sakai-Sugimoto model and more generally in M-theory-lifts of Randall-Sundrum-type intersecting brane models) the worldvolume of the 3-branes in 6d is identified with our observed 3+1d spacetime. Under this identification, the above model of topological quantum computation on defect branes should amount to quantum computation with the hypothetical axio-dilaton-field. One could go on to quote speculations that axion(-dilatons) are secretly already observed, in the form of fuzzy dark matter (e.g. Alexander & McDonough 2019).

Last revised on March 15, 2022 at 05:13:09. See the history of this page for a list of all contributions to it.