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Topological quantum computation using analog gravitational holonomy and time dilation
by Emil Genetay Johansen, Tapio Simula
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|Authors (as registered SciPost users):||Emil Génetay Johansen|
|Preprint Link:||scipost_202206_00002v1 (pdf)|
|Date submitted:||2022-06-03 03:29|
|Submitted by:||Génetay Johansen, Emil|
|Submitted to:||SciPost Physics|
Non-universal topological quantum computation models, such as the Majorana fermion-based Ising anyon model, have to be supplemented with an additional non-topological noisy gate in order to achieve universality. Here we endeavour to remedy this using an Einstein--Cartan analog gravity picture of scalar fields. Specifically, we show that the analog gravity picture enables unitary transformations to be realized in two distinct ways: (i) via space-time holonomy and (ii) as gravitational time dilation. The non-abelian geometric phases are enabled by gravitational interactions, which are mediated by the spin-connection. We analytically compute its matrix elements as a function of the scalar field density distribution. This density can be regarded as the gravitating distribution of matter in an analog universe. We show via explicit calculations that there exists an infinite set of asymptotically flat analog gravitational fields, each of which implements a unique unitary transformation, that render the interactions topological. We emphasise the generality of this result by asserting that such gravitational gates could potentially be implemented in a broad range of real systems modeled by scalar field with an acoustic metric.
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:scipost_202206_00002v1, delivered 2022-08-01, doi: 10.21468/SciPost.Report.5471
Original and interesting idea for mechanism to realize quantum gates exploiting analog gravity as it arises in superfluid condensates. Explicit construction of field configuration for the required non-abelian energy-momentum flux tubes.
Mechanism for carrying quasi-particle excitations around flux tube is left open; proposed scheme is for 1-qubit gates only and needs to be supplemented by 2-qubit gates.
This paper proposes to exploit the ``analog gravity” arising in non-uniform superfluid condensates for a mechanism to realize quantum gates on quasi-particle excitations. Key ingredient is an Aharanov-Bohm effect for a carefully designed non-abelian energy-momentum flux tube. The proposed scenario is original and interesting but, at this stage, a bit unbalanced. While the gravitational field configurations are worked out in mathematical detail, the mechanisms for creating such field configurations and carrying quasi-particle excitations around a flux tube are left open.
I would like to invite the authors to address and clarify a number of points.
1. It would be helpful to this reader to give a bit more detail on how the quasiparticles (arising as spinors in a BdG formalism) transform under gravitational transformations, backing up expressions such as eq. (21), (22).
2. The ``pertinent remarks” on the Unruh effect (end of section 4) seem unrelated to the gravitational holonomy gates. Is there a connection, or a physical effect that’s relevant in this context?
3. In section 4.2, it is unclear to this reader in what sense the mechanism for the Pauli-iX boost is topological, as the effect seems to depend on a time $\Delta t$ which is not a discrete quantity.
4. I find it confusing that section 5 uses the term ``universality” in the sense of being dense on the 1-qubit Bloch sphere, leaving aside the 2-qubit gates.
5. In section 6 it is suggested that gravitational holonomy gates supplement a quantum computational scheme based on MZM or quantum doubles. Can this be made more concrete – for example could the analog gravity scenario be realized in a $p_x+ip_y$ superfluid that supports MZM?
6. In section 6 it is suggested that a gravity-only platform for TQC can be developed, for example by employing the kelvon quasi-particles. It is unclear to me how 2-qubit gates can be realized in such a scenario?