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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 
Submission information  

Preprint Link:  scipost_202206_00002v1 (pdf) 
Date submitted:  20220603 03:29 
Submitted by:  Génetay Johansen, Emil 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Nonuniversal topological quantum computation models, such as the Majorana fermionbased Ising anyon model, have to be supplemented with an additional nontopological noisy gate in order to achieve universality. Here we endeavour to remedy this using an EinsteinCartan 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 spacetime holonomy and (ii) as gravitational time dilation. The nonabelian geometric phases are enabled by gravitational interactions, which are mediated by the spinconnection. 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
Anonymous Report 1 on 202281 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202206_00002v1, delivered 20220801, doi: 10.21468/SciPost.Report.5471
Strengths
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 nonabelian energymomentum flux tubes.
Weaknesses
Mechanism for carrying quasiparticle excitations around flux tube is left open; proposed scheme is for 1qubit gates only and needs to be supplemented by 2qubit gates.
Report
This paper proposes to exploit the ``analog gravity” arising in nonuniform superfluid condensates for a mechanism to realize quantum gates on quasiparticle excitations. Key ingredient is an AharanovBohm effect for a carefully designed nonabelian energymomentum 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 quasiparticle excitations around a flux tube are left open.
Requested changes
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 PauliiX 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 1qubit Bloch sphere, leaving aside the 2qubit 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 gravityonly platform for TQC can be developed, for example by employing the kelvon quasiparticles. It is unclear to me how 2qubit gates can be realized in such a scenario?
Author: Emil Génetay Johansen on 20220922 [id 2842]
(in reply to Report 1 on 20220801)Dear referee,
Please see the file attached containing our detailed response to the report.
Best regards,
Emil Génetay Johansen and Tapio Simula
Attachment:
Response_letter.pdf