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Topological quantum computation using analog gravitational holonomy and time dilation
by Emil Genetay Johansen, Tapio Simula
This Submission thread is now published as
|Authors (as registered SciPost users):||Emil Génetay Johansen|
|Preprint Link:||scipost_202206_00002v2 (pdf)|
|Date submitted:||2022-09-22 11:13|
|Submitted by:||Génetay Johansen, Emil|
|Submitted to:||SciPost Physics Core|
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.
Published as SciPost Phys. Core 6, 005 (2023)
Author comments upon resubmission
List of changes
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).
The gravitational interaction can be described in terms of an ``artificial" gauge field. The gravitational connection 1-forms depend on the velocity in Eq.(10), which arises due to a (probability) density gradient and the source of the gravitational transformation is the quantum pressure term. Transforming to a rotating frame of reference is generically achieved by adding an angular momentum operator multiplied by an angular velocity to the Hamiltonian, which can be expressed in terms of the velocity due to gravity in Eq.(10). The term accounting for the kinetic energy is thereby taking on a similar form to that of an electron minimally coupled to a magnetic field. The gravitational interaction thus looks essentially the same as an electromagnetic interaction on a charged particle. We have added clarifying sentences and expressions in the revised manuscript. See Eq.(11), (12) and the surrounding text.
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?
The referee is correct. The analogy to the Unruh effect is not relevant for the holonomy, only for the Lorentz boost. The time components of the spin connection (first row and first column) belong to the boost part of the Poincaré group while all other components (the spatial ones) belong to the rotational part. Thus, gravity is implementing a boost on the spinor when the time components are non-zero resulting in a transfer in amplitude between the spinor components. The rotational part on the other hand is leaving the distribution of the amplitude invariant. The analogy to the Unruh effect has therefore been described in the manuscript for the boost gate, not for the holonomy gate.
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 Δt which is not a discrete quantity.
We are grateful to the referee for drawing our attention to this point. Indeed, the boost gate is not topologically protected in the conventional sense because its action depends explicitly on the continuous time variable. Prompted by the referee’s observation, we have sharpened our wordings on this in the manuscript. However, it is true that space-time trajectory between two time slices has no bearing on the outcome, which only depends on the distance between the two slices. We have clarified this in section 4.2.
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.
We thank the referee for highlighting this. Indeed, the gate set described is providing a dense cover in SU(2) and as such is only universal on a 1-qubit level (clarification on this is added in section 5). However, given a 4-spinor the gravitational gate should act as a spin-3/2 representation of SU(2) thus implying that the gravitational gates should be applicable to qudit systems as well.
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 px+ipy superfluid that supports MZM?
Our result is extremely general the only assumptions being that the underlying fluid can be characterised by an acoustic metric, and that its excitations carry a spin. As such, we do indeed expect our results to be applicable to both MZM vortices in topological superconductors, e.g. Nature 606, 890 (2022), and to non-abelian vortex anyons in spinor BECs, e.g. Phys. Rev. Lett. 123,140404 (2019).
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?
The referee is correct in that 2-qubit gates are required in order to realize arbitrary quantum algorithms. In order for the gravitational gates to act as a spin-3/2 representation on the kelvon states one would need to find means to entangle two kelvons gravitationally. At this stage it is not clear to us if this can be achieved without developing an analogue model of quantum gravity. In the manuscript, we do not claim that it can, and merely suggest that it would be a natural direction to explore in the future.
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The authors have carefully addressed and clarified the points I made in my first report, and made some adjustments to the manuscript accordingly. While many open ends remain, I reiterate that the proposed scenario is original and interesting, and I'm happy to recommend the manuscript for publication in SciPost Physics Core.