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REVIEW 3 major objections 4 minor 80 references

Resonating Kagome Dimer coverings in Rydberg atom arrays

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Rokhsar-Kivelson state of kagome dimer coverings can be prepared on a cylinder by a sequence of local Rydberg blockade gates that grows the state strip by strip, in a time proportional to the cylinder length and…

desk verdict New and solid MPS for kagome RK cylinders; the gate-growth protocol is plausible but its arbitrary-width correctness rests on an unproved reachability invariant, and the paper still deserves a real referee. read the letter →

arxiv 2506.21255 v2 pith:SUG6IAGO submitted 2025-06-26 quant-ph cond-mat.quant-gas

Xicheng Wang , Erich J Mueller This is my paper · ORCID
classification quant-phcond-mat.quant-gas MSC 81P6882B20 PACS 03.67.-a03.67.Lx
keywords RydbergatomarrayskagomelatticedimermodelRokhsar-KivelsonstatematrixproducttopologicalorderZ2spinliquidquantumpreparation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes an efficient experimental protocol for preparing the kagome-lattice Rokhsar-Kivelson (RK) state, the uniform quantum superposition of all valid dimer coverings, on a cylinder or torus. It shows that this topologically ordered Z2 spin-liquid state has a matrix product state (MPS) representation on cylinders of any width, and that this representation directly dictates a sequence of local Rydberg blockade gates that grows the state one annular strip at a time. If correct, the protocol prepares the state in a time proportional to the cylinder length and independent of its circumference, and the same construction adapts to other hardware such as transmon arrays. The paper also maps the thinnest cylinders to simpler phases, a resonating-bond crystal and a spin-1 valence-bond chain analog, bridging one-dimensional physics and two-dimensional topological order.

What carries the argument

The machinery is a matrix product state (MPS) representation of the dimer coverings, used as the blueprint for a sequence of local gates. Each annular strip of a YC-2N cylinder is labelled by binary strings $L$ and $R$ (where a 1 means a dimer touches the external vertex of a triangle) plus a bit $u$; the parity of $L$ must equal the parity of $R$, and neighboring strips are connected by $\bar{L}_{j+1}=R_j$. Writing the equal-weight superposition in this form gives Eq. (7), a translationally invariant MPS whose bond dimension and entanglement entropy are set by the circumference. The gates, such as the controlled-Hadamard $U^H_{1c1t}$, controlled-not $U^X_{1c1t}$, $U_{1c2t}$, $U^X_{2c2t}$, $U^H_{2c2t}$, and $U_{2c4t}$, are Rydberg-blockade operations whose branching on target atoms implements exactly the matrix-multiplication steps of the MPS; the branching diagrams are the same object as the matrix product diagram of the MPS. This is what makes the growth time linear in length and independent of width: all gates within one strip act in parallel, and each strip adds a fixed layer of the MPS.

What would settle it

An exact numerical simulation, or a small experimental run, of the full gate sequence for a YC-6 or YC-8 cylinder, checking after every gate that the control atoms are in one of the configurations for which the gate is defined and that the resulting state matches the MPS wavefunction of Eq. (7) in the appropriate topological sector, would settle the claim. A single unexpected control configuration, or any nonzero amplitude on a configuration that is not a valid dimer covering consistent with the boundary, falsifies the protocol.

Watch

Extended reading notes

Core claim

The central claim is that the uniform superposition of all dimer coverings of the kagome lattice on a cylinder, the Rokhsar-Kivelson state, is not only a convenient theoretical object but a state that can be built dynamically. On a YC-2N or XC-2N cylinder, every strip configuration is labelled by binary strings $(L,R,u)$ obeying a parity condition and a connection condition, so the equal-weight superposition is exactly a matrix product state with bond dimension $2^{N-1}$ per topological sector. The authors design a sequence of six controlled-blockade gates, each acting on a small set of atoms, whose branching structure reproduces the matrix multiplication of the MPS: applying the gates strip by strip from one end of the cylinder to the other yields the RK state. On thin cylinders the state simplifies to a product of resonating plaquettes (the eye model) or an entangled state equivalent to a spin-1 valence-bond chain (the hourglass model), and on wider cylinders it carries Z2 topological order and $e$/$m$ anyonic excitations with mutual semionic statistics. The same construction extends to tori by gluing cylinder ends, where the four topological sectors are isomorphic to the logical states of the toric code.

Load-bearing premise

The whole construction rests on the assumption that, as gates are applied one after another, the atoms that serve as controls are only ever found in the configurations the protocol has defined, and every atom being manipulated starts in its ground state; if that assumption fails, some gate has no defined action and the final state is not the target superposition.

Editorial extensions

If this is right

  • For a cylinder of length $L_x$ and circumference $L_y$, the strip-by-strip protocol prepares the Rokhsar-Kivelson state in time $O(L_x)$, independent of $L_y$; on an $L\times L$ patch this is $O(\sqrt{N})$ for $N\sim L^2$ atoms, saturating the known bound on entanglement production by local circuits.
  • The thinnest cylinders become experimentally accessible toy models: the YC-2 state is a product of resonating plaquettes with no long-range entanglement, while the XC-4 state has $\ln 2$ entanglement entropy across cuts and effective spin-1/2 edge degrees of freedom.
  • On wider cylinders the prepared state has $\mathbb{Z}_2$ topological order: closed loop operators have fixed eigenvalues, non-contractable loops label two (cylinder) or four (torus) topological sectors, and the excitations are $e$ and $m$ anyons that are mutual semions.
  • The state can be probed by projective measurements of Z-strings and basis-mapped X-strings, and an ancilla-controlled string loop gives a direct measurement of the mutual statistics phase.
  • Because the required gates reduce to standard single- and two-qubit operations, the protocol transfers from Rydberg arrays to other quantum computing hardware such as transmon arrays.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: The MPS-as-blueprint strategy is not tied to the kagome dimer constraint; any constrained superposition that can be labelled strip-wise by binary strings satisfying local parity and connection conditions could in principle be grown by the same kind of blockade-gate sequence.
  • Editorial inference: The paper gives a length-scaling algorithm and a dual circumference-scaling algorithm; an open question it does not address is whether intermediate space-time tradeoffs exist that interpolate between the two while keeping the gates local.
  • Editorial inference: The 'configurations that never appear' assumption could be tested cheaply by exact enumeration or classical simulation of the growth paths for widths up to about $N=6$; this would either verify the inductive claim or expose a width at which the gate definitions become incomplete.
  • Editorial inference: On a torus the construction yields a superposition of the two $m_x$ sectors, i.e. an eigenstate of a non-contractable X-loop; applying or measuring a Z-loop would project this onto a definite logical state, suggesting a direct route to preparing and manipulating logical qubits of the toric code.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the Rokhsar-Kivelson (RK) state on the kagome lattice, defined as the uniform superposition of all valid dimer coverings. It develops a strip parametrization of dimer coverings on YC-2N and XC-2N cylinders, leading to matrix product state (MPS) representations, and analyzes the thin-cylinder limits YC-2 and XC-4, where the state reduces to a resonating-bond crystal and an AKLT-like state, respectively. The central proposal is a sequential local-gate protocol in Rydberg atom arrays that grows the RK state annulus by annulus, with a claimed time scaling linear in cylinder length and independent of circumference, together with a dual width-proportional scheme and a torus-gluing construction. The paper also discusses seeding the initial edge, experimental probes of topological order and anyons, and implementations on digital quantum hardware.

Significance. If the proposed protocol is correct, it is a significant contribution: it offers a concrete route to preparing a topologically ordered RK state with circuit depth saturating the entanglement lower bound, and it provides an MPS representation that clarifies the connection between 2D topological order and 1D precursors. The strip parametrization with parity and connection conditions is carefully constructed and combinatorially well motivated, and the target state is defined independently of the protocol, so there is no circular fitting. The main weakness is that the arbitrary-width gate sequence is justified by diagrammatic reasoning rather than by a formal proof or numerical verification, and the protocol's gates are defined only on a subset of control configurations whose reachability is asserted rather than demonstrated.

major comments (3)
  1. [Sec. V A, Eq. (12)] The correctness of the growth protocol rests on a reachability invariant that is asserted but not proved. Equation (12) defines U^X_{2c2t} only on the control states |1c0c>, |0c1c>, and |1c1c>, and the text immediately after Eqs. (9)-(14) states "Since the gate operations are applied sequentially, some configurations will never appear." In Step 4 of Sec. V B, U^X_{2c2t} acts on (m,n,1) and (m,n,2), with the two control qubits formed from (m,n-1,2), (m,n-1,3), (m,n,3) and (m,n+1,1), (m,n+1,3), (m,n,3), respectively. If both control qubits were in |0c>, the gate would be undefined, and the physical pulse would not leave the target pair in |00>. The thin-cylinder illustrations do not establish the invariant for arbitrary cylinder width N, and Appendix B.2 repeats the assertion ("This gate is never applied to a state where the control bits are set to |0c0c>") without proof. The manuscript needs either a formal induction over m and n verifying that every gate application lies in the defined domain, or an explicit enumeration of all reachable control states; the same requirement applies to the XC-2N gates in Eqs. (13)-(14).
  2. [Sec. V B/V C, Eqs. (17)-(18)] The central claim that the gate sequence "produces a uniform superposition of all valid dimer configurations" is supported only by diagrammatic reasoning. The MPS decomposition in Eqs. (17)-(18) and the statement that the gates were designed to follow the branching diagram are suggestive, but they do not by themselves prove that the sequential application of the conditional gates in Eqs. (9)-(14) implements the tensor contraction of Eq. (7) for every strip and every width. In particular, the action of each gate depends on control states that are themselves entangled with the already-grown portion of the cylinder, so a diagrammatic correspondence to matrix multiplication is not self-evidently a proof of the final amplitudes. Please provide a formal inductive proof that the circuit maps the MPS state on m strips to the MPS state on m+1 strips, or a numerical check (for example, exact statevector fidelity for N=2,3,4 and several lengths) demonstrating that the output is exactly |Ψ>.
  3. [Sec. V D, Appendix C] The seeding step for a uniform superposition also needs a quantitative check. Applying U^{√ψ} to every left-facing triangle produces a product state in which the amplitude of a specific edge configuration with k selected vertices is (1/√2)^{N-k}(1/2)^k, and it is not explained how the subsequent unitary growth gates convert this into equal amplitudes for every full dimer covering. The manuscript should specify the nonzero entries, including any normalization factors, of the MPS tensors A in Eq. (7) and show that the seed state plus the growth sequence produces exactly the MPS coefficients; without this, a reader cannot verify that the protocol prepares the RK state rather than a weighted superposition.
minor comments (4)
  1. [Appendix B 1] The word "extapolate" should be "extrapolate."
  2. [Appendix C] The definition of U^{Ψ} writes U^{Ψ}|00⟩ = (|01⟩+|01⟩)/√2; the second term should evidently be |10⟩, and the same duplication appears in the definition of U^{√Ψ}.
  3. [Throughout] There are several typos: "converings" in Appendix D should be "coverings," "arbitary" in Sec. V C should be "arbitrary," "manor" in the Introduction should be "manner," and "asociated" in the Introduction should be "associated."
  4. [References] The same paper by Giudici, Lukin, and Pichler appears as Refs. [10] and [46]; one duplicate should be removed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the protocol is a constructive implementation of an independently derived MPS; the only issues are a non-load-bearing self-citation and an omitted reachability proof.

full rationale

The paper's central claim is a state-preparation protocol, not an empirically predicted quantity. The target state |Ψ⟩ is defined independently as the uniform superposition of all valid kagome dimer coverings (Rokhsar-Kivelson [1]). The MPS representation in Eq. (7) is built from a bijective labeling (L,R,u) of dimer coverings with parity and connection constraints (Sec. IV and Appendix A), so the MPS is not defined by the gates that later realize it. The gate sequence of Sec. V is presented as a blueprint derived from the MPS branching diagrams (Sec. V C: 'We designed our gates so that at each step we produce the superposition of states prescribed by the diagram'), which is a constructive synthesis, not a fitted input renamed as a prediction. No parameter is fitted, and the protocol's output is not used to define the RK state. The only self-citation, [52] (Harrington, Mueller, Murch) in Appendix B, is a peripheral remark on dissipative gates and is not load-bearing. The main caveat is a correctness gap: the assertion in Sec. V A that 'some configurations will never appear' and the claim in Sec. V B that the sequence 'produces a uniform superposition of all valid dimer configurations' are supported only by thin-cylinder illustrations, not by a formal induction for arbitrary cylinder width. The U^X_{2c2t} gate in Eq. (12) is undefined on the |0c0c⟩ control state, so the reachability invariant is load-bearing. This is an omitted proof rather than a circular reduction: the invariant is not identical to the conclusion, and an independent combinatorial induction would close the gap. Accordingly, the paper shows no significant circularity; score 2 reflects only the minor self-reliance on its own newly introduced MPS and the peripheral self-citation, while the central construction has independent content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The protocol is a construction, not a fit: no free parameters are tuned to match data. The main unproved inputs are the physical Rydberg blockade model and the imported topological properties of the RK state; the combinatorial strip encoding is proved in Appendix A, and no new physical entities are postulated.

assumptions (5)
  • domain assumption Hard-core Rydberg blockade: V_alpha,beta can be treated as 0 or infinity, forbidding simultaneous excitation of neighboring atoms.
    Used throughout Sec. V and App. B to define control-qubit gate actions and to argue that only valid dimer coverings are produced; appears in Eq. (8) and the following paragraph.
  • domain assumption Kagome dimer coverings of a YC-2N or XC-2N strip are uniquely encoded by (L,R,u) with the parity condition and the connection condition Lbar_{j+1}=R_j.
    Central to writing the wavefunction as an MPS in Eq. (7); proved in Appendix A for the YC strips and asserted by analogy for the XC strips.
  • domain assumption The RK state has Z2 gauge structure: it is an eigenstate of contractible loop operators and has topological sectors distinguished by non-contractible Z-loops.
    Imported from Refs. [6,11] and used in Secs. II and III to identify thin-cylinder states and sectors; not re-derived in this paper.
  • standard math Adiabatic theorem and Landau-Zener approximation apply to the slow sweeps used for the X and H gates in App. B 1.
    Used to justify the physical implementation of the gates; the authors explicitly invoke the gap and the Landau-Zener model.
  • standard math The Crosswhite-Bacon Matrix Product Diagram (Ref. [49]) faithfully represents MPS matrix multiplication and can be used as a gate blueprint.
    Used in Sec. V C to justify designing the growth gates from the MPS diagram; relies on an external result.

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Cite this review

Pith. "Pith review of Resonating Kagome Dimer coverings in Rydberg atom arrays." pith.science (2026). https://pith.science/paper/SUG6IAGO

@misc{pith2026250621255,
  author       = {Pith},
  title        = {Pith review of: Resonating Kagome Dimer coverings in Rydberg atom arrays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SUG6IAGO}},
  note         = {Machine review of arXiv:2506.21255}
}
read the original abstract

Motivated by experiments on Rydberg atom arrays, we explore the properties of uniform quantum superpositions of kagome dimer configurations and construct an efficient algorithm for experimentally producing them. We begin by considering the thin cylinder limit, where these states have simple descriptions. We then develop a matrix product representation of the states on arbitrary cylinders, which leads to a natural protocol to efficiently grow them. We explain how our approach can be adapted to other quantum computing hardware.

Figures

Figures reproduced from arXiv: 2506.21255 by the authors.

Figure 2
Figure 2. FIG. 2. Constructing kagome lattice cylinders. (a) The blue [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Utilizing periodic boundary conditions, the XC [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. A schematic illustration of transitions between topo [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (18 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Dimer coverings on the YC-8 cylinder. (a) One an [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Dimer coverings on the XC-8 cylinder. (a) One an [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Topological Sectors and String Operators on YC [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Blockaded configurations during YC-2 [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Gates for XC-2 [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Blockaded configurations during XC-2 [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Gate sequence for growing the eye model on a YC-2 [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Gate sequence for growing the Rokhsar-Kivelson [PITH_FULL_IMAGE:figures/full_fig_p009_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Gate sequence for growing the Rokhsar-Kivelson [PITH_FULL_IMAGE:figures/full_fig_p010_16.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Gate sequence for growing the hourglass model on a [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Seeding the dimer covering. (a) For a seed with [PITH_FULL_IMAGE:figures/full_fig_p012_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Illustration of how the unique dimer covering config [PITH_FULL_IMAGE:figures/full_fig_p014_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. (a) Illustration for an X-type sweeping pulse. Here [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]
Figure 21
Figure 21. Figure 21: FIG. 21. (a) Arrangement of target and control atoms for adi [PITH_FULL_IMAGE:figures/full_fig_p016_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Realizing [PITH_FULL_IMAGE:figures/full_fig_p016_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Realizing [PITH_FULL_IMAGE:figures/full_fig_p017_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Gate sequence to glue together two previously [PITH_FULL_IMAGE:figures/full_fig_p018_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. (a) Gluing two ends of a cylinder to make a torus. [PITH_FULL_IMAGE:figures/full_fig_p019_25.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 62 canonical work pages

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    U1c2t gate: The atoms at positions ( m, 1) and (m, 2) are designated as target atoms , while the control bit is composed of atoms at ( m−1, 5) and (m−1, 6)

    work page

  2. [2]

    UH 1c1t gate: The atom at ( m, 3) serves as the target, with the control bit consisting of atoms at (m, 1) and ( m, 2)

    work page

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    UX 1c1t gate: The atom at (m, 4) is set as the target, with the control bit composed of (m, 1), (m, 2), and (m, 3)

    work page

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    For ( m, 5), the control bit consists of atoms ( m, 1), ( m, 3), and ( m, 4); for (m, 6), the control bit consists of (m, 2), (m, 3), and (m, 4)

    UX 2c2t gate: The atoms at ( m, 5) and ( m, 6) are designated as targets. For ( m, 5), the control bit consists of atoms ( m, 1), ( m, 3), and ( m, 4); for (m, 6), the control bit consists of (m, 2), (m, 3), and (m, 4). This process is schematically depicted in Fig 13(b) and (d). Each arrow shows the state of subsequent tar- get atoms, after the listed ga...

    work page

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    For each odd n, the atoms at positions (m, n,1) and (m, n,2) are designated as target atoms, colored in green in Fig. 14. The control bit is composed of the atoms located at ( m−1, n,1) and ( m−1, n,2), which touch the target atoms. The gate operation U1c2t is applied

    work page

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    The control bit is composed of the atoms ( m, n,1) and ( m, n,2), colored in green

    Again for odd n, the atom at ( m, n,3) serves as the target atom , colored in blue. The control bit is composed of the atoms ( m, n,1) and ( m, n,2), colored in green. The gate U H 1c1t is then applied

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    The control bit consists of atoms at ( m, n−1, 2), ( m, n−1, 3), (m, n+1, 3), and ( m, n+1, 1)

    For even n, the atom at ( m, n,3) is selected as the target atom , colored in purple. The control bit consists of atoms at ( m, n−1, 2), ( m, n−1, 3), (m, n+1, 3), and ( m, n+1, 1). These are the blue atoms adjacent to the target, as well as the closest green atom on each side. The gate operation U X 1c1t is applied. 10

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    For (m, n,1), the control bit is composed of (m, n−1, 2), (m, n−1, 3), and ( m, n,3); for ( m, n,2), the con- trol bit consists of ( m, n+1, 1), ( m, n+1, 3), and (m, n,3)

    For even n, the atoms at ( m, n,1) and ( m, n,2) are treated as targets, colored in yellow. For (m, n,1), the control bit is composed of (m, n−1, 2), (m, n−1, 3), and ( m, n,3); for ( m, n,2), the con- trol bit consists of ( m, n+1, 1), ( m, n+1, 3), and (m, n,3). These are the blue and purple atoms ad- jacent to the targets, as well as the closest green ...

    work page

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    The resulting quantum state is a superposition of all paths through this diagram

    (b) Each gate results in a superposition of excitations, which are contingent on the existing dimer configurations. The resulting quantum state is a superposition of all paths through this diagram. C. State Creation for XC cylinders Similar to Sec. V B, we illustrate state cre...

  2. [10]

    Under this operation the resonating dimer state grows: → +√ 2 → +√ 2 (15) → +√ 2 → +√ 2 (16) These superpositions are illustrated in Fig

    U2c4t gate: The atoms at positions (m, 1), (m, 2), (m, 3), (m, 4) are designated as target atoms, while the control bit is composed of atoms at ( m−1, 5) and (m−1, 6). Under this operation the resonating dimer state grows: → +√ 2 → +√ 2 (15) → +√ 2 → +√ 2 (16) These superposit...

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    After performing these sequential gate operations, the hourglass unit, initially in its ground state, is transformed into a matrix product state which is one unit cell larger

    UH 2c2t gate: The atoms at positions ( m, 5), (m, 6) are designated as target atoms , while the control bit is composed of atoms at ( m, 1), ( m, 2), ( m, 3), (m, 4) . After performing these sequential gate operations, the hourglass unit, initially in its ground state, is tran...

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    For each even n, the atoms at positions ( m, n,1), (m, n,2), (m, n,3), and ( m, n,4) are designated as target atoms , marked in purple in Fig. 16. There are two control bits: the first consists of atoms at (m, n− 1, 4) and ( m, n− 1, 6), controlling the tar- gets at ( m, n,1) ...

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    For each even n, the atoms at (m, n,5) and (m, n,6) are designated as target atoms , marked in purple in Fig. 16. There are again two control bits: the first consists of atoms at ( m, n,1), ( m, n,2), and (m, n,3), controlling the target at (m, n,5); the sec- ond consists of a...

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    For each odd n, the atoms at positions ( m, n,1), (m, n,2), (m, n,3), and ( m, n,4) are designated as target atoms , marked in purple in Fig. 16. There are two sets of control bits: the first consists of atoms at (m, n− 1, 4) and (m, n− 1, 6), controlling the targets at ( m, n...

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    For each odd n, the atoms at (m, n,5) and (m, n,6) are designated as target atoms , marked in orange in Fig. 16. There are again two control bits: the first consists of atoms at ( m, n,1), ( m, n,2), and (m, n,3), controlling the target at (m, n,5); the sec- ond consists of at...

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    No gates are applied to the atoms where lj = 0

    One applies a U Ψ gate to any pair of atoms in a left-facing triangle for which we want lj = 1. No gates are applied to the atoms where lj = 0. One pro- ceeds with the same gate set that was previously used to 12 (a) (b) (c) (d) FIG. 17. Seeding the dimer covering. (a) For a s...

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    The ancillae act as the control bits, and the atoms in the triangles act as the targets

    One applies U1c2t gates, shown in green, which entangle the ancillae with the dimers. The ancillae act as the control bits, and the atoms in the triangles act as the targets. At this step there will be an excited dimer on each left-facing triangle if and only if the correspond...

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    Adiabatic Gates For a time-dependent Hamiltonian H(t), the system evolves under the unitary operator U (t) = T exp − i ℏ Z t 0 H(t′) dt′ . If the system is initially in an eigenstate of H(0), and H varies slowly enough, it will evolve into the corresponding eigenstate of H(T )...

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    These non-adiabatic gates are typically much faster

    Nonadiabatic Gates As an alternative to the adiabatic approach, our quan- tum gates can be implemented via non-adiabatic proto- cols where the pulses ∆( t) and Ω( t) are carefully timed so that the system makes a Rabi transition from the ini- tial to final state. These non-adi...

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    Hence to the Ψ gate is implemented by a pulse with R √ 2Ω(t) dt = π, followed again by a correc- tive phase pulse R ∆(t) dt = π 2

    |00⟩. Hence to the Ψ gate is implemented by a pulse with R √ 2Ω(t) dt = π, followed again by a correc- tive phase pulse R ∆(t) dt = π 2 . With these fundamental gates as building blocks, we construct the six gate operations: U X 1c1t: We arrange the atoms as shown in Fig 20 (a...

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    The gate sequences are simpler for the YC- 2N geometry, making it more suitable for implementing on a digital quantum computer

    Implementation with digital quantum circuits The gates in our protocol can also be implemented in digital quantum circuits, enabling the production of the Rokhsar-Kivelson state in other platforms, such as trans- mon arrays. The gate sequences are simpler for the YC- 2N geomet...

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