Pith. sign in

REVIEW 4 major objections 4 minor 70 references

Quantum Walks on Arbitrary Spatial Networks with Rydberg Atoms

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proposes a Rydberg-atom implementation of staggered quantum walks on arbitrary spatial networks and reports a quadratic speedup for spatial search on random geometric graphs: $\sqrt{N}$ oracle calls and $O(\sqrt{N}\log N)$ total…

desk verdict Concrete Rydberg implementation scheme for staggered walks, but the quadratic speedup hangs on a fitted angle the paper admits it can't yet compute. read the letter →

arxiv 2507.21011 v1 pith:W4L6P35D submitted 2025-07-28 quant-ph cond-mat.quant-gasphysics.app-phphysics.atom-ph

Gabriel Almeida , Raul Santos , Lara Janiurek , Yasser Omar This is my paper · ORCID
classification quant-phcond-mat.quant-gasphysics.app-phphysics.atom-ph PACS 03.67.Ac03.67.Lx
keywords staggeredquantumwalkRydbergatomarraysspatialsearchtessellationcoverrandomgeometricgraphsalgorithmsmulti-qubitgatesquadraticspeedup
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

Quantum walks are the natural quantum analogue of random walks on graphs, and this paper shows how to run them on Rydberg-atom arrays for graphs whose vertices can be placed in the plane. The central move is to decompose each step of a staggered walk—the coinless version defined by reflecting about cliques of a graph—into products of W-state preparations and multi-controlled Z gates, both of which the Rydberg blockade makes native, while reconfiguring the atom array between tessellations. A classical companion algorithm constructs the required tessellation cover in $O(md^2)$ time, and on random geometric graphs the number of tessellations grows only as $(r/r_c)\log N$. Running the search with the generalized staggered walk at an optimally scanned angle yields $\sqrt{N}$ oracle calls, the Grover quadratic speedup, with $O(\sqrt{N}\log N)$ total steps. If correct, this gives a concrete route from spatial graph problems to a reconfigurable neutral-atom platform.

What carries the argument

The machinery has two halves. The graph-theoretic half is the tessellation cover: a set of partitions of vertices into cliques (the tessellations) such that every edge lies in at least one clique; the paper's greedy algorithm builds one in $O(md^2)$ time, and the count of tessellations $T \approx (r/r_c)\log N$ on random geometric graphs sets the circuit depth. The quantum half is the diagonalization of each clique reflection $W_\alpha = \mathbb{1} - 2\sum_k |\alpha_k\rangle\langle\alpha_k|$ as $\bigotimes_k U_{\alpha_k} C^{s-1}Z\, U_{\alpha_k}^\dagger$: $U_{\alpha_k}$ prepares the clique's W-state with a known $O(s)$-gate circuit, and $C^{s-1}Z$ is a native multi-qubit Rydberg gate, so no two-qubit decomposition overhead appears and the whole walk is implemented by interleaving these operators with dynamic rearrangement of the atom array.

What would settle it

Simulate the generalized staggered walk on the same random geometric graphs with $\theta$ chosen from graph data alone (for example, from the degree distribution or spectral gap) instead of from the scan over the marked vertex; if the first-peak search time then fails to scale as $\sqrt{N}$, or if the success probability stays bounded away from one, the quadratic speedup is tied to the fitted parameter rather than to the Rydberg implementation. A direct formula for $\theta_{\mathrm{op}}$ in terms of the graph would settle the question either way.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is a gate-level prescription for the staggered quantum walk in the single-excitation subspace of $N$ atoms, together with numerical evidence that the prescription inherits the optimal search scaling. Each clique $\alpha_k$ of a tessellation is assigned the W-state $|\alpha_k\rangle = (|10\ldots0\rangle + |01\ldots0\rangle + \cdots + |0\ldots01\rangle)/\sqrt{s}$, and the walk operator for tessellation $\alpha$ is diagonalized as $W_\alpha = \bigotimes_k U_{\alpha_k} C^{s-1}Z\, U_{\alpha_k}^\dagger$, with $U_{\alpha_k}$ the W-state preparation circuit of depth $O(s)$ and $C^{s-1}Z$ a native Rydberg multi-controlled gate. The generalized search walk $W_\theta = e^{-i\theta W_1}\cdots e^{-i\theta W_T}$ with a phase oracle reproduces, in simulation on random geometric graphs at $r/r_c=2$, a first-peak search time of $\sqrt{N}$ oracle calls and $O(\sqrt{N}\log N)$ steps; the paper reports this as a quadratic speedup, with the caveat that the optimal angle $\theta_{\mathrm{op}}$ is located by a numerical scan and its direct derivation from the graph is left open.

Load-bearing premise

The speedup depends on knowing, in advance, the rotation angle that maximizes amplitude amplification: the paper obtains this angle by scanning all values numerically and admits that computing it directly from the graph remains an open question.

Editorial extensions

If this is right

  • On random geometric graphs with $r/r_c=2$, the number of oracle calls to find a marked vertex scales as $\sqrt{N}$, and the total number of walk steps as $\sqrt{N}\log N$.
  • A single walk step costs $O(N)$ gates per tessellation, so implementing the walk on an $N$-atom array requires circuit depth proportional to the tessellation count $T = O((r/r_c)\log N)$.
  • The classical preprocessing finds a valid tessellation cover in $O(md^2)$ time; on random geometric graphs this evaluates to $O((r/r_c)^6 N \log^3 N)$.
  • Because $C^{s-1}Z$ is native to the Rydberg blockade, the protocol avoids the overhead of decomposing multi-qubit gates into two-qubit gates.
  • Setting the generalized walk angle to $\theta = \pi/2$ recovers the standard staggered walk, so the search construction contains the ordinary walk as a special case.

Reading between the lines

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

  • The open problem of computing $\theta_{\mathrm{op}}$ from the graph is the single step separating this from a fully specified algorithm; a spectral or degree-based estimator for $\theta_{\mathrm{op}}$ on random geometric graphs would make the $\sqrt{N}$ query bound an end-to-end claim.
  • Since the tessellation preprocessing cost grows as $(r/r_c)^6$, the practical benefit of the proposal is largest for sparse spatial networks near the connectivity threshold; dense graphs may spend more classical time on the cover than the quantum walk saves.
  • The same W-state diagonalization applies to any staggered walk whose generators are clique reflections, so the construction could serve as a general Rydberg subroutine for optimization and simulation, not only for search.
  • A natural experimental follow-up is to measure the success probability at the predicted stopping time over many graph realizations, since a useful spatial search requires high probability of finding the marked vertex, not just a favourable mean time.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a Rydberg-atom implementation of staggered quantum walks on arbitrary spatial networks. The walker is encoded in a single excitation among N atoms; each tessellation's reflection operator is diagonalized via a W-state preparation and a native multi-qubit C^{s-1}Z gate. The authors also present a greedy algorithm for constructing a tessellation cover of a graph, benchmark it on random geometric graphs, and report numerical evidence that a generalized staggered walk with an optimized rotation angle θ achieves a quadratic speedup for spatial search. The central claim is that the proposal achieves quadratic speedup in spatial search, with oracle calls O(√N) and total steps O(√N log N).

Significance. The gate-level construction is a useful step toward programmable quantum walks on Rydberg arrays, and the tessellation algorithm addresses a necessary preprocessing problem. The paper explicitly describes how the walk operators are decomposed into native Rydberg gates and provides a classical algorithm with benchmarks, which is reproducible and clearly presented. However, the central speedup claim is not yet fully demonstrated because the optimal rotation angle θ_op is obtained by a scan that assumes knowledge of the marked vertex, and the paper concedes that computing θ from the graph is open. The complexity of the tessellation algorithm is also overstated by a factor, and the scaling evidence in Fig. 4 lacks a quantitative fit. If the θ_op issue is resolved, the contribution would be significant for the quantum-walk and Rydberg-platform communities.

major comments (4)
  1. [Section V, Eq. (8) and Fig. 4] The claimed quadratic speedup rests on θ_op being chosen by a linear scan that maximizes the success probability for a known marked vertex. In an actual spatial search the marked vertex is unknown, so this scan cannot be part of the quantum procedure unless θ_op is independent of the marked vertex or can be computed classically from the graph structure without knowing the solution. The paper's own statement in Section VI that "determining this value directly from the graph remains an open question" confirms that the algorithm is not fully specified. This is load-bearing because the abstract and Section V claim a quadratic speedup as a property of the proposal. Please either provide a method to compute θ_op from the graph in time polynomial in log N without knowledge of the marked vertex, or restrict the claim to a numerical demonstration for an optimized θ and clearly state that the end-to-end query complexity is not yet established.
  2. [Section III, Algorithm 1] The stated worst-case complexity O(md²) omits the loop over "each color c in used_colors" in the main routine. For each edge (u,v), is_colorable is called for each of the T colors, and each call examines neighbors of u and v, which can be O(d) in the worst case. The per-edge cost is therefore O(T d), not O(d²), giving a total of O(m T d) = O((r/r_c)^5 N log^3 N) under the paper's scalings T = O((r/r_c) log N) and d = O((r/r_c)^2 log N), rather than the claimed O((r/r_c)^6 N log^3 N). Please correct the derivation or justify why T is not a multiplicative factor.
  3. [Section III, Algorithm 1] No correctness proof is provided for the greedy tessellation algorithm. The statement that the routine "ensures the graph remains properly tessellated" is not demonstrated. A formal induction showing that every color class is a union of cliques and that every edge is assigned to at least one tessellation is needed, since the algorithm is a central contribution and its output defines the walk operators used in the rest of the paper.
  4. [Section V, Fig. 4] The power-law scaling claim is not supported quantitatively. No fitted slope, confidence interval, or goodness-of-fit is reported; the visual impression of √N scaling from a log-log plot with five data points is insufficient, especially given that no error bars are shown for the search times. Please provide a fit of the data to a function a N^b with uncertainties, and report the number of random graph realizations used for each point.
minor comments (4)
  1. [Section II, Fig. 2] In the W-state preparation description, the condition sin(θ_m/2) = -1/√(s+1-m) for m > 1 is missing a definition of the index m; please specify the qubit ordering and the correspondence between the angles and the circuit in Fig. 2.
  2. [Section III, Fig. 3 caption] The caption states that error bars indicate the standard error of the mean (σ/√N), but N is also used for the graph size; this is confusing because the number of realizations is 6000/N. Please use a distinct symbol for the number of realizations.
  3. [Section IV] The Trotter error statement is somewhat imprecise: the first-order product formula has a per-step error O((Δt)^2), leading to a total error O(t^2/K) for K steps. The text's phrasing "proportional to t^2/ϵ" is consistent with this, but the derivation would be clearer if the K dependence were written explicitly.
  4. [Section V, Eq. (8)] The statement that θ = π/2 recovers the original staggered walk "up to a global phase" is correct, since e^{-i(π/2)W} = -i W, but the text should explicitly note the factor -i to avoid confusion.

Circularity Check

1 steps flagged · score 6.0 of 10

Quadratic speedup claim rests on a numerically fitted rotation angle θ_op, which the paper admits cannot yet be derived from the graph.

  1. fitted input called prediction [Section V (Spatial search), surrounding Eq. (8) and Fig. 4; Section VI (Conclusions)]
    "First, we determine the value of θ that produces the highest amplitude amplification of the marked vertex with a linear scan. Then, using this optimal value θ = θop, we measured the search time, defined as the number of oracle calls needed to reach the first maximum in probability. As shown in Fig. 4, the search time scales with √N ... Regarding the quantum search algorithm, although we showed the existence of an optimal angle that enables the quadratic speedup, determining this value directly from the graph remains an open question."

    The claimed quadratic speedup is demonstrated by first choosing θ_op through a linear scan that maximizes the success probability of the known marked vertex, then measuring the search time with that fitted angle. In an actual search the marked vertex is unknown, so the scan cannot be part of the quantum procedure unless θ_op is computable from the graph alone — which the paper explicitly concedes is an open question. The speedup therefore reduces to an existence result conditioned on a data-dependent fit rather than a fully specified, parameter-free algorithm. The √N scaling is observed after optimizing θ against the target outcome, not derived or predicted independently of the fit.

full rationale

The paper's contributions decompose into three parts: the Rydberg implementation of staggered quantum walks, a classical tessellation-cover algorithm, and a spatial search speedup demonstration. The first two parts are built on established independent results (Portugal's staggered walk framework, W-state preparation circuits, native Rydberg multiqubit gates) and are supported by numerical benchmarks; no circularity is present there. The central claim of 'quadratic speedup in spatial search algorithms' (Abstract, Section V) relies on a rotation angle θ_op selected by a linear scan to maximize the probability of the marked vertex in the simulated graphs. Since the marked vertex is unknown during a real search, this fitting procedure cannot be part of the algorithm as specified, and the paper itself states that determining θ from the graph remains open. Thus the speedup 'prediction' reduces to a fitted parameter that is renamed as an optimal angle, which is the fitted_input_called_prediction pattern. There are no load-bearing self-citations: the cited staggered-walk and Rydberg-gate works are external and independent. The circularity is localized but affects the paper's headline positive result, so a score of 6 is appropriate: one central prediction reduces by construction to a numerical fit.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the staggered walk formalism (standard), the assumed availability of native Rydberg multiqubit gates and reconfigurable atom arrays (domain assumptions from cited experiments), the single-excitation subspace fidelity, and a correctness/complexity claim for the new tessellation algorithm that is only empirically validated. The speedup benchmark additionally depends on a fitted rotation angle θ_op.

free parameters (2)
  • θ_op (walk rotation angle) = Not reported explicitly; scanned numerically to maximize P_max
    Section V: 'First, we determine the value of θ that produces the highest amplitude amplification ... with a linear scan.' The quadratic speedup result is obtained after this optimization.
  • Tessellation scaling constant (coefficient of (r/rc) log N) = Not reported; inferred from Fig. 3
    Section III: the relation T = O((r/rc) log N) is obtained empirically from random geometric graph benchmarks, not derived.
assumptions (5)
  • standard math Staggered quantum walk evolution defined by reflection operators W_α = 1 - 2 Σ_k |α_k⟩⟨α_k| (Eq. 1)
    Background formalism from Portugal (Ref. [37]); the paper builds on it.
  • domain assumption Rydberg arrays can natively implement multi-controlled Z gates (C^{s-1}Z and C^{s-1}Z^θ) and dynamic atom rearrangement
    Sections II and V rely on these platform capabilities, citing Refs. [6,52,53]; no error model or reconfiguration cost is included.
  • domain assumption The single-excitation subspace is preserved throughout the walk
    The walker is encoded as exactly one hyperfine excitation; the implementation assumes no leakage into multi-excitation states.
  • standard math Connectivity threshold r_c = sqrt(log N / (π N)) for random geometric graphs
    Used to set the natural radius scaling; cited from Refs. [44,45].
  • ad hoc to paper The greedy tessellation algorithm (Algorithm 1) always produces a valid tessellation cover
    No correctness proof is provided; the paper only benchmarks it numerically on random geometric graphs (Section III).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Walks on Arbitrary Spatial Networks with Rydberg Atoms." pith.science (2026). https://pith.science/paper/W4L6P35D

@misc{pith2026250721011,
  author       = {Pith},
  title        = {Pith review of: Quantum Walks on Arbitrary Spatial Networks with Rydberg Atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4L6P35D}},
  note         = {Machine review of arXiv:2507.21011}
}
read the original abstract

Rydberg atoms provide a highly promising platform for quantum computation, leveraging their strong tunable interactions to encode and manipulate information in the electronic states of individual atoms. Key advantages of Rydberg atoms include scalability, reconfigurable connectivity, and native multi-qubit gates, making them particularly well-suited for addressing complex network problems. These problems can often be framed as graph-based tasks, which can be efficiently addressed using quantum walks. In this work, we propose a general implementation of staggered quantum walks with Rydberg atoms, with a particular focus on spatial networks. We also present an efficient algorithm for constructing the tessellations required for the staggered quantum walk. Finally, we demonstrate that our proposal achieves quadratic speedup in spatial search algorithms.

Figures

Figures reproduced from arXiv: 2507.21011 by the authors.

Figure 1
Figure 1. FIG. 1. To perform a staggered walk, a tessellation cover [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Proposed implementation of a staggered quantum walk. Each vertex of the spatial network is encoded in a single [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scaling of the number of tessellations in random [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Spatial search with staggered quantum walk on random geometric graphs with [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 63 canonical work pages

  1. [1]

    F or eachneighbor v of u:

    work page

  2. [2]

    F or eachcolor c in used_colors:

    work page

  3. [3]

    If is_colorable(u, v, c):

    work page

  4. [4]

    color_edges(u, v, c)

    work page

  5. [5]

    Continue to next neighbor

    work page

  6. [6]

    If no color c worked:

    work page

  7. [7]

    F or eachy neighbor ofu with color (u, y) = c:

    work page

  8. [8]

    color_edges(u, v, cnew) F unctionis_colorable(u, v, c)

    work page

Show all 70 references
  1. [10]

    If (u, x) not an edge: Return False

  2. [12]

    If (v, y) not an edge: Return False

  3. [13]

    Return True F unctioncolor_edges(u, v, c)

  4. [15]

    Set color (u, x) ← c

  5. [16]

    F or each y neighbor of u with color (u, y) = c:

  6. [17]

    Set color (v, y) ← c

  7. [18]

    F or each x neighbor of v with color (v, x) = c:

  8. [19]

    colors”, and “coloring an edge

    Set color (x, y) ← c III. TESSELLA TION ALGORITHM Since the staggered quantum walk relies on the tessellation structure, efficiently computing the tessellation cover is essential for scaling the model to large graphs. We developed an algorithm to find 102 103 N 2 3 4 5 6 7T rc...

    2020

  9. [20]

    Pósfai and A.-L

    M. Pósfai and A.-L. Barabási,Network science, Vol. 3 (Citeseer, 2016)

    2016

  10. [21]

    Fortunato, Community detection in graphs, Physics reports 486, 75 (2010)

    S. Fortunato, Community detection in graphs, Physics reports 486, 75 (2010)

    2010

  11. [22]

    Danila, Y

    B. Danila, Y. Yu, J. A. Marsh, and K. E. Bassler, Optimaltransportoncomplexnetworks,PhysicalReview E—Statistical, Nonlinear, and Soft Matter Physics 74, 046106 (2006)

    2006

  12. [23]

    G. Yan, T. Zhou, B. Hu, Z.-Q. Fu, and B.-H. Wang, Efficient routing on complex networks, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 73, 046108 (2006)

    2006

  13. [24]

    Moreno, M

    Y. Moreno, M. Nekovee, and A. F. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 69, 066130 (2004). 6

    2004

  14. [25]

    Morgado and S

    M. Morgado and S. Whitlock, Quantum simulation and computing with Rydberg-interacting qubits, AVS Quantum Science 3 (2021)

    2021

  15. [26]

    McInroy, N

    K. McInroy, N. Pearson, and J. Pritchard, Benchmarking the algorithmic performance of near-term neutral atom processors, arXiv preprint arXiv:2402.02127 (2024)

    2024 arXiv

  16. [27]

    Maskara, S

    N. Maskara, S. Ostermann, J. Shee, M. Kalinowski, A. McClain Gomez, R. Araiza Bravo, D. S. Wang, A. I. Krylov, N. Y. Yao, M. Head-Gordon, et al., Programmable simulations of molecules and materials with reconfigurable quantum processors, Nature Physics , 1 (2025)

    2025

  17. [28]

    Ebadi, A

    S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar,et al., Quantum optimization of maximum independent set using Rydberg atom arrays, Science376, 1209 (2022)

    2022

  18. [29]

    M. Kim, K. Kim, J. Hwang, E.-G. Moon, and J. Ahn, Rydberg quantum wires for maximum independent set problems, Nature Physics18, 755 (2022)

    2022

  19. [30]

    Jeong, M

    S. Jeong, M. Kim, M. Hhan, J. Park, and J. Ahn, Quantum programming of the satisfiability problem with Rydberg atom graphs, Physical Review Research 5, 043037 (2023)

    2023

  20. [31]

    A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman, Tweezer-programmable 2D quantum walks in a Hubbard-regime lattice, Science 377, 885 (2022)

    2022

  21. [32]

    M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo, et al., Quantum approximate optimization of non-planar graph problems on a planar superconducting processor, Nature Physics17, 332 (2021)

    2021

  22. [33]

    Y. Zhu, Z. Zhang, B. Sundar, A. M. Green, C. H. Alderete, N. H. Nguyen, K. R. Hazzard, and N. M. Linke, Multi-round QAOA and advanced mixers on a trapped- ion quantum computer, Quantum science and technology 8, 015007 (2022)

    2022

  23. [34]

    Deng, S.-Q

    Y.-H. Deng, S.-Q. Gong, Y.-C. Gu, Z.-J. Zhang, H.- L. Liu, H. Su, H.-Y. Tang, J.-M. Xu, M.-H. Jia, M.- C. Chen, et al., Solving graph problems using Gaussian boson sampling, Physical Review Letters 130, 190601 (2023)

    2023

  24. [35]

    A. M. Childs, D. Gosset, and Z. Webb, Universal computation by multiparticle quantum walk, Science 339, 791 (2013)

    2013

  25. [36]

    A. M. Childs and J. Goldstone, Spatial search by quantum walk, Physical Review A—Atomic, Molecular, and Optical Physics70, 022314 (2004)

    2004

  26. [37]

    Magniez, M

    F. Magniez, M. Santha, and M. Szegedy, Quantum algorithms for the triangle problem, SIAM Journal on Computing 37, 413 (2007)

    2007

  27. [38]

    Ambainis, Quantum walk algorithm for element distinctness, SIAM Journal on Computing 37, 210 (2007)

    A. Ambainis, Quantum walk algorithm for element distinctness, SIAM Journal on Computing 37, 210 (2007)

    2007

  28. [39]

    Marsh and J

    S. Marsh and J. B. Wang, Combinatorial optimization via highly efficient quantum walks, Physical Review Research 2, 023302 (2020)

    2020

  29. [40]

    K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi, and P. Xue, Simulating dynamic quantum phase transitions inphotonicquantumwalks,PhysicalReviewLetters 122, 020501 (2019)

    2019

  30. [41]

    Di Molfetta and A

    G. Di Molfetta and A. Pérez, Quantum walks as simulators of neutrino oscillations in a vacuum and matter, New Journal of Physics18, 103038 (2016)

    2016

  31. [42]

    Yan, Y.-R

    Z. Yan, Y.-R. Zhang, M. Gong, Y. Wu, Y. Zheng, S. Li, C. Wang, F. Liang, J. Lin, Y. Xu, et al., Strongly correlated quantum walks with a 12-qubit superconducting processor, Science364, 753 (2019)

    2019

  32. [43]

    M. Gong, S. Wang, C. Zha, M.-C. Chen, H.-L. Huang, Y. Wu, Q. Zhu, Y. Zhao, S. Li, S. Guo, et al., Quantum walks on a programmable two-dimensional 62-qubit superconducting processor, Science 372, 948 (2021)

    2021

  33. [44]

    Tang, X.-F

    H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, et al., Experimental two-dimensional quantum walk on a photonic chip, Science advances4, eaat3174 (2018)

    2018

  34. [45]

    C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.- C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, et al., Observation of topologically protected edge states in a photonic two-dimensional quantum walk, Physical Review Letters 121, 100502 (2018)

    2018

  35. [46]

    Peruzzo, M

    A. Peruzzo, M. Lobino, J. C. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff,et al., Quantum walks of correlated photons, Science 329, 1500 (2010)

    2010

  36. [47]

    Zähringer, G

    F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, Realization of a quantum walk with one and two trapped ions, Physical Review Letters 104, 100503 (2010)

    2010

  37. [48]

    Schmitz, R

    H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, Quantum walk of a trapped ion in phase space, Physical Review Letters 103, 090504 (2009)

    2009

  38. [49]

    Tamura, T

    M. Tamura, T. Mukaiyama, and K. Toyoda, Quantum walks of a phonon in trapped ions, Physical Review Letters 124, 200501 (2020)

    2020

  39. [50]

    Matjeschk, C

    R. Matjeschk, C. Schneider, M. Enderlein, T. Huber, H. Schmitz, J. Glueckert, and T. Schaetz, Experimental simulation and limitations of quantum walks with trapped ions, New Journal of Physics14, 035012 (2012)

    2012

  40. [51]

    P. Xue, B. C. Sanders, and D. Leibfried, Quantum walk on a line for a trapped ion, Physical Review Letters103, 183602 (2009)

    2009

  41. [52]

    Karski, L

    M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Quantum walk in position space with single optically trapped atoms, Science325, 174 (2009)

    2009

  42. [53]

    Dadras, A

    S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G. S. Summy, Experimental realization of a momentum- space quantum walk, Physical Review A 99, 043617 (2019)

    2019

  43. [54]

    P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, and M. Greiner, Strongly correlated quantum walks in optical lattices, Science 347, 1229 (2015)

    2015

  44. [55]

    Clark, C

    J. Clark, C. Groiseau, Z. Shaw, S. Dadras, C. Binegar, S. Wimberger, G. Summy, and Y. Liu, Quantum to classical walk transitions tuned by spontaneous emissions, Physical Review Research3, 043062 (2021)

    2021

  45. [56]

    Portugal, Staggered quantum walks on graphs, Physical Review A93, 062335 (2016)

    R. Portugal, Staggered quantum walks on graphs, Physical Review A93, 062335 (2016)

    2016

  46. [57]

    Szegedy, Quantum speed-up of markov chain based algorithms, in 45th Annual IEEE symposium on foundations of computer science (IEEE, 2004) pp

    M. Szegedy, Quantum speed-up of markov chain based algorithms, in 45th Annual IEEE symposium on foundations of computer science (IEEE, 2004) pp. 32– 41. 7

    2004

  47. [58]

    Portugal, M

    R. Portugal, M. C. de Oliveira, and J. K. Moqadam, Staggered quantum walks with Hamiltonians, Physical Review A 95, 012328 (2017)

    2017

  48. [59]

    Khatibi Moqadam, M

    J. Khatibi Moqadam, M. C. de Oliveira, and R. Portugal, Staggered quantum walks with superconducting microwave resonators, Physical Review B 95, 144506 (2017)

    2017

  49. [60]

    Abreu, L

    A. Abreu, L. Cunha, T. Fernandes, C. de Figueiredo, L. Kowada, F. Marquezino, D. Posner, and R. Portugal, Boundsandcomplexityforthetessellationproblem,Mat. Contemp 45, 22 (2017)

    2017

  50. [61]

    Abreu, L

    A. Abreu, L. Cunha, T. Fernandes, C. de Figueiredo, L. Kowada, F. Marquezino, D. Posner, and R. Portugal, The graph tessellation cover number: extremal bounds, efficient algorithms and hardness, in LATIN 2018: Theoretical Informatics: 13th Latin American Symposium, Buenos Aire...

    2018

  51. [62]

    Abreu, L

    A. Abreu, L. Cunha, C. de Figueiredo, L. Kowada, F. Marquezino, R. Portugal, and D. Posner, A computational complexity comparative study of graph tessellation problems, Theoretical Computer Science 858, 81 (2021)

    2021

  52. [63]

    M. D. Penrose, The longest edge of the random minimal spanning tree, The annals of applied probability7, 340 (1997)

    1997

  53. [64]

    M. J. Appel and R. P. Russo, The connectivity of a graph on uniform points on [0, 1]d, Statistics & Probability Letters 60, 351 (2002)

    2002

  54. [65]

    Chakrabarti, C

    A. Chakrabarti, C. Lin, and N. K. Jha, Design of quantum circuits for random walk algorithms, in2012 IEEE Computer Society Annual Symposium on VLSI (IEEE, 2012) pp. 135–140

    2012

  55. [66]

    Möttönen and J

    M. Möttönen and J. J. Vartiainen, Decompositions of general quantum gates, Trends in quantum computing research , 149 (2006)

    2006

  56. [67]

    J. J. Vartiainen, M. Möttönen, and M. M. Salomaa, Efficient decomposition of quantum gates, Physical Review Letters 92, 177902 (2004)

    2004

  57. [68]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information, Vol. 2 (Cambridge university press Cambridge, 2001)

    2001

  58. [69]

    Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    1996

  59. [70]

    Portugal and T

    R. Portugal and T. D. Fernandes, Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians, Physical Review A95, 042341 (2017)

    2017

  60. [71]

    D. Yu, H. Wang, J.-M. Liu, S.-L. Su, J. Qian, and W. Zhang, Multiqubit Toffoli gates and optimal geometry with Rydberg atoms, Physical Review Applied 18, 034072 (2022)

    2022

  61. [72]

    He, J.-X

    Y. He, J.-X. Liu, F.-Q. Guo, L.-L. Yan, R. Luo, E. Liang, S.-L. Su, and M. Feng, Multiple-qubit Rydberg quantum logic gate via dressed-state scheme, Optics Communications 505, 127500 (2022)

    2022

  62. [73]

    Marsh and J

    S. Marsh and J. Wang, A quantum walk-assisted approximate algorithm for bounded np optimisation problems, Quantum Information Processing 18, 61 (2019)

    2019

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.