Parallel Implementation of CNOT^N and C₂NOT² Gates via Homonuclear and Heteronuclear Förster Interactions of Rydberg Atoms

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the manuscript by Farouk et al, the authors perform careful analysis of the implementation of multi-qubit gates on neutral atom arrays, considering in detail challenges around suppressing target-target interactions to enable high fidelity gate operations and propose a novel C_2NOT^2 gate protocol that exploits the benefits of utilising heteronuclear interactions.

Neutral atoms have emerged as a promising candidate for scalable quantum information processing with recent demonstrations of many of the core building blocks for high fidelity two-qubit gates, first steps towards logical qubit encodings, and mid-circuit measurement as required to enable quantum error correction and future fault-tolerant hardware. A major advantage of the neutral atom platform is the ability to perform parallel multi-qubit gates, however when using a single species the strong interactions being used between control and target qubits also lead to parasitic control-control and target-target couplings that rapidly degrade gate performance. 

This paper explores dual species operations using Cs and Rb as a route to overcoming this limitation, enabling strong control-target interactions whilst suppressing target-target couplings. Two gate protocols are considered here, the first an EIT based CNOT^N gate previously proposed in 2009 and demonstrated for two qubits in 2022, and the authors demonstrate significant improvement in scaling to 4 or 5 target qubits when using a Cs-Rb Forster resonance. Additionally, they provide a novel protocol for extending the EIT approach to performing C_2NOT^2 protocols that are far simpler than previous multi-step pulse proposals and leverage the benefits of the EIT gate mechanism.

These gates are essential for implementation of surface code stabilisers and the work provides a route to overcome previous limitations. This is highly relevant to a number of on-going research activities both in academia and emerging quantum industry, where a number of dual-species Cs/Rb platforms are under development and thus represents high impact research worthy of publication in Photonics.

The paper is well written and easy to follow, with commendable details on exactly how the pair states were chosen and calculated for easy reproduction (even down to settings used in the open source code libraries), and the results are placed in context.

However, prior to publication, I would like the authors to address a few minor comments/questions:

(1) All fidelities in the paper are quoted assuming only a single intermediate state. As explicitly detailed in Ref. 46, the intermediate state has two hyperfine levels leading to a time-dependent differential shift that means when performing the Raman pi-pulse, there is a dynamical AC shift which can introduce phase errors and either needs a dynamic or static shift applied during the pulse. Additionally, to accurately calculate the probability of scattering from the Rydberg state it is necessary to couple the target Rydberg states to the full intermediate excited state manifold as even though the Raman pathway is exclusively through the |mf=1> levels, the other mf levels can couple to the Rydberg states. Both of these effects will mean the quoted numbers underestimate spontaneous decay and gate fidelities, and the authors should provide more detailed discussion around this (or even perform calculations using the accurate intermediate excited state to quantify this error for smaller system sizes (recognising that for 5 target qubits it may not be possible to calculate exactly).

(2) Figure 3 shows a 3D projection of 2D configurations - this adds nothing to the paper and would be better drawn as 2D graphs.

(3) On page 14 line 401 the authors say quote a time Tp and give it a value in µm (perhaps it should be µs?) but this needs clarifying.

(4) Hopefully this would be fixed by the proof stage but avoid floating figures in the middle of paragraphs as it breaks the text up and becomes harder to follow. 

 

Author Response

"Please see the attachment."

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript entitled “Parallel implementation of CNOT^N and C₂NOT² gates via homonuclear and heteronuclear Förster interactions of Rydberg atoms” (corresponding authors A.M. Farouk and I.I. Beterov) submitted to the Photonics review addresses high-fidelity multiqubit CNOT gates using EIT and Rydberg blockade with Rb and Cs atoms following an idea proposed by M. Müller et al. [PRL 102, 170502 (2009)].

Improvement of the fidelity of such quantum gates is demonstrated by exploiting strong dipole-dipole interaction between target and control atoms while the van de Waals interactions between the target atoms are kept small. The manuscript presents interaction with the radiation of a 3-level control atom and of a 4-level target atom in the inverted Y configuration and models the interactions (dipole-dipole and vdW) between the control atom and N target atoms. Fidelity of multiqubit CNOT quantum entangled gates is simulated for homonuclear and heteronuclear architectures and a scheme for a Rydberg EIT C₂NOT² gate is discussed. Some error sources for various CNOT gates are addressed.  

I recommend publication of the manuscript in the Photonics review upon taking into account some small typographical corrections and add some explanations.

1.       Line 142: Indicate and explain which are the dimensions of the identity matrices that you use. Use the same notation in the body of the manuscript as in Eq. (7).

2.       The intermediate state 7P_3/2 of Cs has a hyperfine structure. Explain clearly at all levels of your manuscript if that impact your analysis.

3.       Starting from line 256 : Give physical reasons on the fact that there is no fluctuation on the blockade radius dependence for homonuclear species while there are some variations on the dependence of the blockade radius for the heteronuclear species. Which are the physical parameters that define the positions of the fluctuations that are observed? It may be some fluctuations at higher quantum numbers for the blockade radius dependence for the homonuclear species?

Comments on the Quality of English Language

In Figure 4 you plot probability while in the manuscript you talk on fidelity (starting from line 157). Please use the same notion.

 

Author Response

"Please see the attachment."

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

 

In the manuscript "Parallel implementation of CNOT N and C 2 NOT 2 gates via

homonuclear and heteronuclear Förster interactions of Rydberg atoms" by

Ahmed M. Farouk et al., schemes of high-fidelity multiqubit CNOT N and

C 2 NOT 2 gates for alkali-metal neutral atoms used as qubits, have been analyzed.

This article is useful and well written, conclusions are supported by the results,

the literature is complete and the figures are well done. I found only a couple of misprints.

I recommend the paper for publication after their correction.

L 159: in not --> is not

L 259: a local maxima --> a local maximum

or: local maxima

L 260: the principal quantum corresponding --> the principal quantum number corresponding ?

L 377. an non --> a non

L 480: it the same --> is the same

Comments on the Quality of English Language

 Minor editing of English language required

Author Response

"Please see the attachment."

Author Response File: Author Response.pdf