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REVIEW 2 major objections 6 minor 43 references

Grover search works on any mix of qudits by reducing the problem to two collective states and matching phases for exact or bounded success.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 08:49 UTC pith:AGLKBM36

load-bearing objection Solid methods packaging of known exact and fixed-point Grover variants for arbitrary/heterogeneous qudits; math checks out, depth-advantage claim is the soft spot. the 2 major comments →

arxiv 2607.24658 v1 pith:AGLKBM36 submitted 2026-07-27 quant-ph

Exact and Fixed-Point Grover Search with Qudits

Tanay Roy This is my paper
classification quant-ph
keywords Grover searchquditsphase matchingfixed-point searchdeterministic searchheterogeneous qudit registerscontrolled-phase gatesamplitude amplification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Grover’s search was built for qubits whose total dimension is a power of two. Many real devices instead use qudits—systems with three or more levels—and sometimes mix different dimensions on one chip. This paper shows that the search still collapses to a two-dimensional problem spanned by one collective “target” state and one collective “unmarked” state, no matter how the register is built. On that plane the familiar oracle and diffusion steps can be written with qudit Fourier transforms and controlled-phase gates, with or without a single qubit ancilla. By tuning the phases in those gates the author obtains four deterministic protocols that hit the target with certainty when the target fraction is known, and two fixed-point protocols that keep the success probability above a chosen floor when it is not. The same geometric picture on the Bloch sphere makes the different phase choices easy to compare, giving hardware teams a concrete menu of circuits rather than a purely abstract algorithm.

Core claim

Any homogeneous or heterogeneous qudit register can be reduced to the two-dimensional span of the collective unmarked state |R⟩ and target state |T⟩; on that subspace the standard, deterministic (one-, two-, and three-parameter phase matching plus an ancilla-tuned overlap), and fixed-point (π/3 and YLC) Grover protocols can be realized with explicit qudit Fourier and controlled-phase circuits while preserving their known query scalings.

What carries the argument

The effective two-level reduction to span{|R⟩,|T⟩} together with the generalized Grover iterate G(α,β)=Sr(β)So(α), where the oracle and diffusion phases α and β are chosen either equal (D1p), alternating (D2p/D3p), ancilla-adjusted, or Chebyshev-scheduled (YLC) so that the Bloch vector lands exactly on |T⟩ or stays above a success floor.

Load-bearing premise

The multi-controlled phase gates and qudit Fourier transforms needed for each iterate must be available at high enough fidelity that their cost does not cancel the quadratic query advantage on near-term hardware.

What would settle it

Implement one of the deterministic or YLC circuits on a real multi-qudit processor for a known small λ, measure the final success probability against the predicted unit or Pmin bound, and compare total two-qudit gate count and depth with the equivalent qubit encoding of the same database size.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Heterogeneous qudit chips can run unstructured search without padding the Hilbert space to the next power of two.
  • When the target fraction is known, exact search needs at most one extra oracle query beyond the ordinary optimal count.
  • When the target fraction is unknown, the YLC schedule still scales as O(1/√λ) while guaranteeing a chosen minimum success probability.
  • Native SNAP-style phase gates on bosonic platforms can replace deep recursive decompositions of multi-controlled CPHASE, cutting circuit depth.
  • The same amplitude-amplification toolbox extends directly to qudit-enhanced quantum sensing at the Grover–Heisenberg limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Platforms that already offer high-fidelity selective number-dependent phase gates gain the largest immediate depth reduction, because the paper’s multi-controlled CPHASE is essentially a SNAP operation.
  • The ancilla-tuned-overlap method lets a device keep a fixed, high-fidelity oracle while still reaching certainty, which may be the most practical deterministic option on hardware where the oracle is hard-wired.
  • Comparing measured Bloch-sphere trajectories (via tomography of the two-dimensional subspace) against the paper’s predicted arcs would give a direct diagnostic of phase-control error before full-scale search is attempted.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The manuscript formulates Grover search on homogeneous and heterogeneous qudit registers by projecting the dynamics onto the two-dimensional span of the collective unmarked and target states. It derives the standard oracle, diffusion operator, success probability, and optimal iteration count; gives Fourier-transform and controlled-phase implementations, including ancilla kickback; and presents deterministic D1p, D2p, D3p, and ancilla-tuned variants together with the π/3 and Yoder–Low–Chuang fixed-point protocols. Appendices supply qudit gate identities, phase-parameter equations, Chebyshev definitions, and a recursive elementary-gate count. The claimed contribution is a unified, hardware-oriented toolkit preserving the known query scalings and potentially reducing circuit depth on platforms with native qudit phase operations.

Significance. If the hardware assumptions are stated accurately, this will be a useful reference for implementing amplitude amplification on emerging qudit processors. Its strengths are the parameter-free algebraic derivations once λ is specified, explicit Fourier/CPHASE and ancilla-kickback identities, the complete D2p/D3p phase equations, a reproducible gate-count recurrence and table, and clearly falsifiable O(1/sqrt(lambda)) versus O(1/lambda) query predictions. The principal significance is consolidation and translation to homogeneous and heterogeneous qudit hardware rather than a new asymptotic search result.

major comments (2)
  1. [Abstract; §III after Eq. (19); Fig. 2; Appendix D] The abstract and §III (after Eq. (19)) connect the protocol to reduced circuit depth and “explicit circuit decompositions suitable for diverse hardware platforms,” but the only elementary-gate decomposition supplied is Fig. 2/Appendix D, restricted to n qudits of the same prime dimension. For the heterogeneous and non-prime registers emphasized as the main motivation, the crucial multi-qudit CPHASE remains a primitive in Fig. 1(c), with no decomposition or gate count. Please either provide a mixed-dimension/non-prime decomposition and cost analysis or explicitly restrict the decomposition and depth-reduction claims to native-phase platforms and homogeneous prime registers. This does not affect the query-complexity or phase-matching results.
  2. [§III, gate-complexity paragraph; Appendix D, Eqs. (D.1)-(D.3)] The statement that the Fig. 2 decomposition requires O((d+1)^n) two-qudit gates is inconsistent with the recurrence immediately derived in Eq. (D.1). For fixed d, Eq. (D.3) gives the dominant growth factor x_+=[(d+1)+sqrt((d+1)^2+4)]/2>d+1, so the count is not bounded by O((d+1)^n); it should be stated as O(x_+^n), and likely Θ(x_+^n). The distinction is exponentially large in n and should be propagated to the comparison with native SNAP/CPHASE implementations.
minor comments (6)
  1. [§II.A, Eq. (11); §III, Eq. (19)] The notation CPHASE_{d,m} is ambiguous for a register: Eq. (11) defines |m> as a tensor product over register qudits, while I_d appears to denote a single-qudit identity. For heterogeneous registers, consider CPHASE_{\mathbf d,m} with identity I_D, D=∏_j d_j, and use d consistently for a local dimension.
  2. [§III.D, Eqs. (28)-(30)] The exact-search condition would be clearer if the explicit solution λ'=sin²[π/(4k_certain+2)] were displayed, together with the one-line argument that λ'≤λ follows from the ceiling definition in Eq. (18). As written, Eq. (28) is a condition rather than an obviously constructive choice.
  3. [Appendix D, Table A1] Table A1 reports both N_1 and N_2, but the text gives a recurrence only for N_2. Please add the corresponding N_1 recurrence or explain how the single-qudit entries are generated.
  4. [§IV.B, Eq. (34)] Please state the branch/range used for cot^{-1} and explicitly relate the sign convention in Eq. (34) to the commonly quoted YLC convention α_j=-β_{k-j+1}. This will prevent implementation ambiguities even though phases differing by 2π are equivalent.
  5. [§III.C, Eq. (27); Appendix E, Eqs. (E.2a)-(E.2b)] The D3p protocol allows k_certain-k'≥2 in Eq. (27), but Appendix E supplies equations only for the final-two-step case k'=k_certain-2. Please either provide the general alternating-step equations or state explicitly that the supplied construction is restricted to modifying the last two iterations.
  6. [References [8], [10], and [22]] Reference [8] appears to contain a formatting typo (“PoSLA TTICE2023,” presumably PoS LATTICE2023). References [10] and [22] also use incomplete DOI-style journal information and should be normalized once bibliographic data are available.

Circularity Check

1 steps flagged

No significant circularity: qudit Grover framework is a self-contained 2D reduction plus standard external protocols; self-citations only supply prior D2p phases/errata.

specific steps
  1. self citation load bearing [Sec. III.B, Eqs. (23)–(26); App. E; Refs. [27], [38]]
    "To address this constraint, Roy et al. developed a deterministic two-parameter protocol (D2p) that leaves the oracle So(α) unchanged and adjusts the diffusion operator to achieve exact search [27]. ... The corresponding equations for an odd number of iterations [38] are given in Appendix E."

    D2p phase conditions and the odd-iteration erratum are imported from the present author's prior papers rather than re-derived from scratch here. This is ordinary self-citation of a prior construction being ported to qudits; it is not load-bearing for the paper's main claim (arbitrary qudit registers reduce to the same 2D Grover dynamics with explicit circuits), because D1p, D3p, ancilla, π/3, and YLC routes do not depend on [27]/[38].

full rationale

The load-bearing chain is the standard collective-state reduction (Eqs. 1–2) to span{|R⟩,|T⟩}, the oracle/diffusion algebra (Eqs. 4–5), and explicit qudit Fourier/CPHASE circuit identities (Eqs. 10–17, App. C), all derived in-place from unitary definitions without fitting data or importing a uniqueness theorem. Deterministic and fixed-point variants are generalizations of independently published protocols (Long D1p, Grover π/3, Yoder–Low–Chuang YLC, Mishra ancilla tuning); query scalings and success formulas follow from the same 2D geometry. The only self-citations ([27], [38]) attribute the two-parameter (D2p) phase-matching scheme and its odd-k erratum to prior Roy et al. work and supply numerical phase equations (App. E), but they are not used to forbid alternatives or to force the central qudit-register claim by definition—D1p, D3p, ancilla, and fixed-point routes stand independently. No fitted-input-as-prediction, self-definitional loop, or renaming of an empirical pattern appears. Circularity burden is therefore negligible.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on the standard geometric formulation of amplitude amplification, on the existence of qudit Fourier and selective-phase gates, and on previously published phase-matching schedules. No numerical free parameters are fitted; the only modeling choices are ideal unitaries and the treatment of unused Hilbert-space dimensions as extra unmarked items.

axioms (4)
  • standard math Grover/amplitude-amplification dynamics on any finite database reduce exactly to a two-dimensional rotation in the plane spanned by the normalized collective unmarked state |R⟩ and target state |T⟩.
    Invoked from the outset (Sec. II, Eqs. 1–7) and used for every subsequent protocol; standard since Brassard et al. and the original Grover geometric picture.
  • domain assumption Qudit Hadamard (discrete Fourier transform) gates and selective CPHASE(φ) (or SNAP) gates are available as native or efficiently compilable operations on the target hardware.
    Required for the circuit constructions in Sec. II–III and Fig. 2; treated as a platform primitive rather than derived.
  • domain assumption Unused dimensions D−N of a qudit register may be treated as additional unmarked items, so that λ=M/D without changing the query scaling.
    Stated in Sec. II A; lets the framework cover heterogeneous registers whose product dimension is only an upper bound on N.
  • standard math The published phase schedules of Long (D1p), Roy et al. (D2p), Mishra et al. (ancilla), Grover (π/3), and Yoder–Low–Chuang remain valid when the underlying register is a qudit register.
    Used throughout Secs. III–IV; the paper only re-implements the same abstract iterates with qudit gates.

pith-pipeline@v1.2.0-grok45-kimik3 · 21191 in / 2629 out tokens · 45883 ms · 2026-07-31T08:49:57.368419+00:00 · methodology

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read the original abstract

Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.

Figures

Figures reproduced from arXiv: 2607.24658 by Tanay Roy.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (b) shows an example in which only the final two steps are modified and the state reaches the target with certainty (see Appendix E for the corresponding equa￾tions). D. Method 4 The fourth method, developed for qubits by Mishra et al. [39], uses an ancilla to tune the effective target overlap. The same idea extends naturally to qudit registers, and the ancilla only needs to be a two-level system. The goal… view at source ↗

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Reference graph

Works this paper leans on

43 extracted references · 2 canonical work pages

  1. [1]

    L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett.79, 325 (1997)

    1997

  2. [2]

    G. Long, H. Yan, Y. Li, C. Tu, J. Tao, H. Chen, M. Liu, X. Zhang, J. Luo, L. Xiao, and X. Zeng, Experimental NMR realization of a generalized quantum search algo- rithm, Physics Letters A286, 121 (2001)

    2001

  3. [3]

    Figgatt, D

    C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Debnath, and C. Monroe, Complete 3-qubit Grover search on a programmable quantum computer, Nature Communications8, 1918 (2017)

    1918

  4. [4]

    T. Roy, S. Hazra, S. Kundu, M. Chand, M. P. Patankar, and R. Vijay, Programmable superconducting proces- sor with native three-qubit gates, Phys. Rev. Appl.14, 014072 (2020)

    2020

  5. [5]

    Zhang, K

    K. Zhang, K. Yu, and V. Korepin, Quantum search on noisy intermediate-scale quantum devices, Europhysics Letters140, 18002 (2022)

    2022

  6. [6]

    L. B. Nguyen, N. Goss, K. Siva, Y. Kim, E. Younis, B. Qing, A. Hashim, D. I. Santiago, and I. Siddiqi, Em- powering a qudit-based quantum processor by travers- ing the dual bosonic ladder, Nature Communications15, 7117 (2024)

    2024

  7. [7]

    T. Kim, T. Roy, X. You, A. C. Li, H. Lamm, O. Pronitchev, M. Bal, S. Garattoni, F. Crisa, D. Bafia, et al., Ultracoherent superconducting cavity-based multi- qudit platform with error-resilient control, arXiv preprint arXiv:2506.03286 (2025)

    Pith/arXiv arXiv 2025

  8. [8]

    T. Roy, T. Kim, A. Romanenko, and A. Grassellino, Qudit-based quantum computing with SRF cavities at Fermilab, PoSLA TTICE2023, 127 (2024)

    2024

  9. [9]

    Z. Wang, R. W. Parker, E. Champion, and M. S. Blok, High-EJ /EC transmon qudits with up to 12 levels, Phys. Rev. Appl.23, 034046 (2025)

    2025

  10. [10]

    P. J. Low, N. C. F. Zutt, G. A. Tathed, and C. Senko, Quantum logic operations and algorithms in a single 25-level atomic qudit, Nature Communications 10.1038/s41467-026-72662-8 (2026)

    work page doi:10.1038/s41467-026-72662-8 2026

  11. [11]

    B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher-dimensional Hilbert spaces, Nature Physics 5, 134 (2009)

    2009

  12. [12]

    Gedik, I

    Z. Gedik, I. A. Silva, B. C ¸ akmak, G. Karpat, E. L. G. Vidoto, D. O. Soares-Pinto, E. R. deAzevedo, and F. F. Fanchini, Computational speed-up with a single qudit, Scientific reports5, 1 (2015)

    2015

  13. [13]

    A. S. Nikolaeva, E. O. Kiktenko, and A. K. Fedorov, Effi- cient realization of quantum algorithms with qudits, EPJ Quantum Technology11, 43 (2024)

    2024

  14. [14]

    A. S. Nikolaeva, E. O. Kiktenko, and A. K. Fedorov, Gen- eralized Toffoli gate decomposition using ququints: To- wards realizing Grover’s algorithm with qudits, Entropy 25, 10.3390/e25020387 (2023)

    work page doi:10.3390/e25020387 2023

  15. [15]

    Godwood, D

    S. Godwood, D. M. K¨ urk¸ c¨ uo˘ glu, G. N. Perdue, M. Maneyro, and A. Roggero, Fault-tolerant resource comparison of qudit and qubit encodings for diagonal quadratic operators (2026), arXiv:2604.26792 [quant-ph]

    Pith/arXiv arXiv 2026

  16. [16]

    Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Frontiers in Physics8, 589504 (2020)

    2020

  17. [17]

    S. S. Ivanov, H. S. Tonchev, and N. V. Vitanov, Time- efficient implementation of quantum search with qudits, Phys. Rev. A85, 062321 (2012)

    2012

  18. [18]

    Godfrin, A

    C. Godfrin, A. Ferhat, R. Ballou, S. Klyatskaya, M. Ruben, W. Wernsdorfer, and F. Balestro, Operating quantum states in single magnetic molecules: Implemen- tation of Grover’s quantum algorithm, Phys. Rev. Lett. 12 119, 187702 (2017)

    2017

  19. [19]

    Perez-Garcia, R

    B. Perez-Garcia, R. Hernandez-Aranda, A. Forbes, and T. Konrad, The first iteration of Grover’s algorithm using classical light with orbital angular momentum, Journal of Modern Optics65, 1 (2018)

    2018

  20. [20]

    T. Roy, Z. Li, E. Kapit, and D. Schuster, Two-qutrit quantum algorithms on a programmable superconducting processor, Phys. Rev. Appl.19, 064024 (2023)

    2023

  21. [21]

    Mohit, J

    F. Mohit, J. Guanzon, J. McKinlay, T. J. Weinhold, C. R. Myers, M. P. Almeida, M. Rambach, and A. G. White, Quantum mechanics can find a needle in a haystack every time (2025), arXiv:2506.06435 [quant-ph]

    Pith/arXiv arXiv 2025

  22. [22]

    X. Shi, J. Sinanan-Singh, T. J. Burke, J. Chiaverini, and I. L. Chuang, Efficient implementation of a quantum al- gorithm with a trapped ion qudit, Nature Communica- tions 10.1038/s41467-026-68746-0 (2026)

    doi:10.1038/s41467-026-68746-0 2026

  23. [23]

    Dogra, A

    S. Dogra, A. Dorai, and K. Dorai, Implementation of the quantum fourier transform on a hybrid qubit–qutrit NMR quantum emulator, International Journal of Quan- tum Information13, 1550059 (2015)

    2015

  24. [24]

    Litteken, L

    A. Litteken, L. M. Seifert, J. D. Chadwick, N. Notting- ham, T. Roy, Z. Li, D. Schuster, F. T. Chong, and J. M. Baker, Dancing the quantum waltz: Compiling three- qubit gates on four level architectures, Proceedings of the 50th Annual International Symposium on Computer Architecture (ISCA ’23), Association for Computing Ma- chinery (2023), article 71

    2023

  25. [25]

    M. Meth, J. Zhang, J. F. Haase, C. Edmunds, L. Postler, A. J. Jena, A. Steiner, L. Dellantonio, R. Blatt, P. Zoller, T. Monz, P. Schindler, C. Muschik, and M. Ringbauer, Simulating two-dimensional lattice gauge theories on a qudit quantum computer, Nature Physics21, 570 (2025)

    2025

  26. [26]

    Brassard, P

    G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Quantum amplitude amplification and estimation, Contemporary Mathematics305, 53 (2002)

    2002

  27. [27]

    T. Roy, L. Jiang, and D. I. Schuster, Deterministic Grover search with a restricted oracle, Phys. Rev. Res. 4, L022013 (2022)

    2022

  28. [28]

    T. J. Yoder, G. H. Low, and I. L. Chuang, Fixed-point quantum search with an optimal number of queries, Phys. Rev. Lett.113, 210501 (2014)

    2014

  29. [29]

    Li and S

    P. Li and S. Li, Phase matching in Grover’s algorithm, Physics Letters A366, 42 (2007)

    2007

  30. [30]

    G. L. Long, Grover algorithm with zero theoretical failure rate, Phys. Rev. A64, 022307 (2001)

    2001

  31. [31]

    G. L. Long, Y. S. Li, W. L. Zhang, and L. Niu, Phase matching in quantum searching, Physics Letters A262, 27 (1999)

    1999

  32. [32]

    F. M. Toyama, W. van Dijk, Y. Nogami, M. Tabuchi, and Y. Kimura, Multiphase matching in the Grover al- gorithm, Phys. Rev. A77, 042324 (2008)

    2008

  33. [33]

    K. Park, P. Marek, and R. Filip, Efficient quantum simulation of nonlinear interactions using SNAP and Rabi gates, Quantum Science and Technology9, 025004 (2024)

    2024

  34. [34]

    R. W. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. V. Albert, L. Frunzio, L. Jiang, and R. J. Schoelkopf, Cavity state manipulation using photon-number selective phase gates, Phys. Rev. Lett.115, 137002 (2015)

    2015

  35. [35]

    Bornman, T

    N. Bornman, T. Roy, J. A. Job, N. Anand, G. N. Perdue, S. Zorzetti, and M. S. Alam, Benchmarking the perfor- mance of a high-q cavity qudit using random unitaries, Quantum Science and Technology10, 025062 (2025)

    2025

  36. [36]

    S. L. Braunstein, B.-S. Choi, S. Ghosh, and S. Maitra, Exact quantum algorithm to distinguish Boolean func- tions of different weights, Journal of Physics A: Mathe- matical and Theoretical40, 8441 (2007)

    2007

  37. [37]

    Toyama, W

    F. Toyama, W. Van Dijk, and Y. Nogami, Quantum search with certainty based on modified Grover algo- rithms: optimum choice of parameters, Quantum infor- mation processing12, 1897 (2013)

    2013

  38. [38]

    T. Roy, L. Jiang, and D. I. Schuster, Erratum: Determin- istic Grover search with a restricted oracle [phys. rev. research 4, l022013 (2022)], Phys. Rev. Res.5, 029002 (2023)

    2022

  39. [39]

    Mishra, A

    H. Mishra, A. Balasubramanyam, and G. N. Raghava, Deterministic quantum search for arbitrary initial success probabilities, Quantum Information Processing25, 222 (2026)

    2026

  40. [40]

    L. K. Grover, Fixed-point quantum search, Phys. Rev. Lett.95, 150501 (2005)

    2005

  41. [41]

    Ringbauer, M

    M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, A universal qudit quantum processor with trapped ions, Nature Physics18, 1053 (2022)

    2022

  42. [42]

    Huang, E

    J. Huang, E. Kasaba, T. J. DiNapoli, T. Roy, and S. Chakram, Universal Jaynes-Cummings control of an oscillator (2026), arXiv:2605.18658 [quant-ph]

    Pith/arXiv arXiv 2026

  43. [43]

    R. R. Allen, F. Machado, I. L. Chuang, H.-Y. Huang, and S. Choi, Quantum computing enhanced sensing (2025), arXiv:2501.07625 [quant-ph]

    Pith/arXiv arXiv 2025