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Speed limit for entangling gates in diamond color centers

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2026-07-09 07:08 UTC pith:5CKQMBXJ

load-bearing objection Systematic comparison of four entangling-gate protocols for GeV color centers, with a clean parameter-free QSL derivation via Cartan decomposition. the 2 major comments →

arxiv 2607.07549 v1 pith:5CKQMBXJ submitted 2026-07-08 quant-ph

Control Protocols for Entangling Gates for Group-IV Color-Centers in Diamond

Jurek Frey , Frank K. Wilhelm , Matthias M. M\"uller This is my paper
classification quant-ph PACS 03.67.Lx03.67.Ac76.30.Mi61.72.Bb
keywords gatesquantumcontrolentanglinggroup-ivspinaccuratelyalgebraic
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.

This paper identifies three distinct classes of entangling two-qubit gates for group-IV color centers in diamond (exemplified by the germanium-vacancy center coupled to a 13C nuclear spin), each mediated by a different combination of the parallel hyperfine coupling component A_zz and the orthogonal component A_zx. The authors derive a quantum speed limit for any CNOT-equivalent gate of tau_CNOT = pi / sqrt(A_zz^2 + A_zx^2), which for experimentally measured parameters yields approximately 171 nanoseconds. They then demonstrate four practical routes to approach this limit: dynamical decoupling pulse sequences, resonant driving of double-quantum transitions, quantum optimal control pulse shaping, and algebraic decomposition using the Cartan (KAK) decomposition of two-qubit unitaries. The analysis is performed under coherent (noise-free) Hamiltonian evolution to isolate the energy-level structure and the fundamental speed constraints it imposes. The central finding is that the relative magnitudes of A_zz and A_zx, together with the tunable nuclear Larmor frequency, determine which gate protocol is fastest and most experimentally feasible, and that quantum optimal control subject to realistic microwave amplitude constraints emerges as the most practical route for strongly-coupled group-IV systems where dynamical decoupling requires impractically fast pi-pulses.

Core claim

The paper's central result is the derivation of a unified quantum speed limit tau_CNOT = pi / sqrt(A_zz^2 + A_zx^2) for entangling gates in group-IV color-center systems, obtained by recognizing that free evolution under the static hyperfine Hamiltonian generates a non-local gate content whose entangling rate is set by the combined magnitude of both hyperfine components. Using the Cartan decomposition, the authors show that CNOT, iSWAP, and SWAP gates can be synthesized by interleaving free evolution (at multiples of tau_CNOT) with instantaneous single-qubit rotations, with iSWAP requiring 2*tau_CNOT and SWAP requiring 3*tau_CNOT. The three gate types correspond to different interaction medi

What carries the argument

The Cartan (KAK) decomposition of two-qubit unitaries, which separates any two-qubit gate into local single-qubit rotations and a non-local entangling core characterized by three Weyl coordinates (c1, c2, c3). Free evolution under the hyperfine Hamiltonian builds up one Weyl coordinate linearly in time at a rate determined by sqrt(A_zz^2 + A_zx^2)/4, and interleaving this evolution with single-qubit rotations at specific time intervals converts the accumulated non-local content into CNOT, iSWAP, or SWAP gates. The quantum speed limit follows directly from the time needed to accumulate the required Weyl coordinate.

Load-bearing premise

The quantum speed limit derivation for the algebraic decomposition route assumes infinitely fast single-qubit operations, meaning the local rotations interleaved with free evolution take zero time. In practice, microwave control is amplitude-limited (to roughly 2*pi x 15 MHz) and nuclear-spin RF control is slow, so these local gates are not instantaneous and the true achievable gate time will exceed the derived limit.

What would settle it

If the combined hyperfine interaction sqrt(A_zz^2 + A_zx^2) does not correctly set the entangling rate for all three gate types, or if the Cartan-decomposition-based speed limit does not hold under the actual driven dynamics (where single-qubit operations have finite duration and the system is not purely in free evolution), then the derived tau_CNOT bound would not be a tight or achievable speed limit for these gates.

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

If this is right

  • The speed limit tau_CNOT = pi / sqrt(A_zz^2 + A_zx^2) provides a benchmark against which any entangling-gate protocol for group-IV color centers can be assessed: gates operating far above ~171 ns (for typical parameters) are not approaching the fundamental bound and may have room for improvement.
  • The identification of three gate types (A_zz-mediated, A_zx-mediated, and combined) maps directly onto magnetic-field design: the choice of nuclear Larmor frequency determines which hyperfine channel dominates and thus which protocol family is available, giving experimentalists a concrete parameter-space guide.
  • The finding that quantum optimal control achieves infidelities below 10^-8 under realistic amplitude constraints suggests that the practical bottleneck for group-IV color-center gates is decoherence and pulse imperfections, not the coherent control landscape itself.
  • The algebraic decomposition approach, if extended beyond single-nucleus systems, could provide speed limits and synthesis recipes for multi-nuclear-spin registers, where the Weyl-coordinate analysis generalizes to higher-dimensional entangling dynamics.
  • The SWAP gate construction via three sequential CNOT-equivalent evolutions (~1.65 microseconds total) demonstrates that full state transfer between electron and nuclear spin is feasible within coherence times reported for GeV centers, enabling quantum memory protocols.

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 / 9 minor

Summary. This manuscript studies entangling two-qubit gate protocols for a group-IV color center (specifically the GeV center) coupled to a strongly-coupled 13C nuclear spin. The authors derive the system Hamiltonian, perform an exact diagonalization and Schrieffer-Wolff transformation, and identify three types of entangling gates mediated by the parallel (Azz), orthogonal (Azx), or both hyperfine components. They derive a quantum speed limit (QSL) of τ_CNOT = π/√(A²zz + A²zx) ≈ 171 ns using a Cartan decomposition under the assumption of instantaneous single-qubit operations. Four practical gate protocols are compared: dynamical decoupling (DD), double-quantum transition (DQT) driving, quantum optimal control (QOC), and algebraic decomposition. All simulations use experimentally extracted hyperfine parameters and are restricted to coherent Hamiltonian evolution. The QOC-optimized gates achieve infidelities below 10⁻⁸ at gate times of ~2 μs under realistic amplitude constraints.

Significance. The paper provides a systematic and useful comparison of gate protocols for group-IV color centers, a platform of growing importance for quantum networking. The Hamiltonian derivations (Sec. II) are clean, the Schrieffer-Wolff approximation is properly checked against exact diagonalization, and the QSL derivation (Sec. III.C) follows standard Cartan decomposition results. The QOC optimizations achieve high fidelities (>99.8%) with realistic amplitude constraints (Ωmax = 2π×15 MHz), and the data is publicly available (Zenodo). The identification of distinct operating regimes as a function of Azz, Azx, and ωI provides practical guidance for experimental gate-set design.

major comments (2)
  1. Sec. III.C, Eq. (39) and surrounding text: The QSL τ_CNOT = π/√(A²zz + A²zx) ≈ 171 ns is derived under the assumption of 'infinitely fast single qubit operations' (stated explicitly in Sec. III.C). The paper itself acknowledges in Sec. III.B that fast nuclear-spin manipulation via RF pulses is 'experimentally unfeasible.' The practical protocols demonstrated (DD ~1.3 μs, DQT ~2–3 μs, QOC ~2–2.2 μs) are all 8–17× slower than this QSL, even under purely coherent evolution. This gap is not discussed in terms of its implications for the QSL's predictive value. The authors should either (a) explicitly state that the QSL serves as a fundamental bound that is not achievable with current hardware and discuss what fraction of the gap is due to the instantaneous-local-gate assumption versus MW amplitude constraints, or (b) provide an estimate of a practically achievable speed limit given the realΩ
  2. Sec. I and throughout: The analysis is restricted to coherent Hamiltonian evolution, and all reported fidelities (e.g., 99.6% for DQT in Sec. III.B, <10⁻⁸ for QOC in Sec. III.B and III.C) are therefore upper bounds. The practical utility of the proposed gates depends critically on whether these fidelities survive realistic noise, particularly given that the best gate times (~2 μs) are comparable to cited GeV coherence times ('a few microseconds' scale, Refs. [68,69]). The paper defers the noise analysis to Ref. [27], but does not provide even an order-of-magnitude estimate of decoherence-limited fidelity for the protocols studied here. A brief quantitative discussion or back-of-the-envelope estimate of the expected decoherence-limited fidelity for the QOC gates (the paper's recommended protocol) would substantially strengthen the practical claims.
minor comments (9)
  1. Sec. II.B, Eq. (23): The figure of merit is defined as 1−F = 1 − (1/d²)|Tr[W U†_T]|², but in the text this is referred to as 'gate infidelity.' The standard gate infidelity for a d-dimensional system is 1 − |Tr[W U†_T]|²/d², which differs by a factor related to the dimension normalization. Please clarify whether Eq. (23) is the standard average gate infidelity or a different metric, and ensure consistency with the d=4 claim.
  2. Sec. III.A, Eq. (26): The ZZ/2 gate condition requires ωI = 2Azz = 2π×5.72468 MHz, but later in the same section (Eq. 31) the ZX/2 gate uses ωI = 2nAzx/π. The text should clarify that these are different operating points (different magnetic field values) and that one cannot simultaneously optimize for both gate types at the same field setting.
  3. Fig. 3 caption: The text states 'gate fidelity of 1−F<10⁻⁸%' but the y-axis label reads 'FoM.' It is unclear whether the y-axis is the FoM (1−F) or the fidelity F. Please clarify the axis label.
  4. Sec. III.B: The DQT gate fidelity of F=99.6% is reported for ωI = ±2π×13.65 MHz, but the corresponding gate time is not explicitly stated in the text (only visible in Fig. 4, bottom panel). Please state the gate time in the main text for completeness.
  5. Sec. III.C, Eq. (38): The entangling frequency is defined as ω_ent = √(A²zx + A²zz)/4. The factor of 1/4 should be explicitly traced to the spin-1/2 operator normalization (Sz = σz/2, Iz = σz/2) to avoid confusion for readers accustomed to Pauli-operator conventions.
  6. Appendix C, Eq. (C4): The local gate ansatz assumes a specific form K1 = diag(1, e^{iφ}) ⊗ C1. The justification for this particular form is not stated. Please briefly explain why this ansatz is sufficient (e.g., by symmetry or gauge freedom).
  7. Sec. V (Conclusion): The claim that 'these approaches bring gate fidelities closer to the threshold required for distributed quantum information processing' should specify which threshold is being referenced (e.g., fault-tolerance thresholds for specific error correction codes) and note that this claim applies only to the coherent-evolution upper bounds.
  8. Typographical: 'Lamor frequency' appears multiple times (e.g., Sec. III.A, Sec. III.B, Fig. 4 caption) and should be 'Larmor frequency.' Also, 'Wie thank' in the Acknowledgements should be 'We thank.'
  9. Fig. 1: The y-axis label 'Coefficient (2π×MHz)' and the legend entries (XI, XX, XZ, YY, ZX) are not fully explained in the caption. Please clarify what these coefficients represent (presumably the Pauli decomposition coefficients of the transformed Hamiltonian, Eq. 13) and how they relate to the eigenvalues shown in gray.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Both major points are well-taken and address genuine gaps in the manuscript's discussion of practical implications. We address each below.

read point-by-point responses

  1. Referee: Sec. III.C, Eq. (39): The QSL τ_CNOT ≈ 171 ns is derived under the assumption of infinitely fast single-qubit operations, but the paper acknowledges that fast nuclear-spin manipulation via RF is experimentally unfeasible. The practical protocols are 8–17× slower. The gap is not discussed. The referee asks to either (a) explicitly state the QSL is a fundamental bound not achievable with current hardware and discuss what fraction of the gap is due to the instantaneous-local-gate assumption versus MW amplitude constraints, or (b) provide an estimate of a practically achievable speed limit.

    Authors: We agree with the referee that the gap between the fundamental QSL (~171 ns) and the practical gate times (~1.3–2.2 μs) is insufficiently discussed in the current manuscript. We will add an explicit paragraph in Sec. III.C (and cross-reference it in Sec. IV) addressing this point along the lines of option (a), with a semi-quantitative decomposition of the gap. Specifically, the gap arises from two distinct sources: (1) the assumption of instantaneous local gates, which in practice requires RF control of the nuclear spin—experimentally unfeasible at the required speeds—and (2) the MW amplitude constraint Ω_max = 2π×15 MHz, which limits the rate at which the electron spin can be driven. Regarding source (1): the algebraic decomposition approach (Sec. III.C) requires local rotations on the nuclear spin interleaved with free evolution. Without fast RF, these must be synthesized indirectly via MW-driven gates, adding substantial overhead. The SWAP gate synthesis via ZX+YY+ZZ gates (Sec. III.C) illustrates this: the sum of individual entangling gate times is ~1.6 μs before accounting for the local gate overhead, already ~9× the QSL. Regarding source (2): the QOC results in Figs. 3 and 8 show that convergence to high fidelity occurs around T ≈ 2 μs under the amplitude constraint. We can estimate a practical speed limit by noting that the QOC convergence region for the ZX/2 gate (Fig. 3) sharpens dramatically between 1.5 and 2 μs, suggesting that the amplitude-constrained practical limit is on the order of ~1.5–2 μs for entangling gates in this system. This is consistent with the DQT gate times of ~2–3 μs. We will add this discussion explicitly, stating that the QSL serves as a fundamental bound that is not achievable with current hardware, and that the practically achievable速度 revision: no

Circularity Check

0 steps flagged

No significant circularity; the QSL derivation is a parameter-free mathematical result and the hyperfine inputs are externally measured.

full rationale

The paper's central theoretical result, the quantum speed limit τ_CNOT = π/√(A²zz + A²zx) (Eq. 39), is derived from the Cartan decomposition of the free-evolution Hamiltonian (Eq. 4). The non-local content of the static Hamiltonian reduces to a single σz⊗σz term with strength √(A²zz + A²zx)/4 (Eq. 38), and the QSL follows from the requirement that the Weyl coordinate c3 reach π/2 for a CNOT-equivalent gate. This derivation is self-contained and parameter-free given the Hamiltonian. The hyperfine parameters (Azz = 2π×2.86234 MHz, Azx = 2π×0.60281 MHz) are taken from experimental measurements by other groups [12, 27, 38], not fitted to gate performance data in this paper. The QOC results are numerical optimizations against external target unitaries (CNOT, SWAP) using the dCRAB algorithm, and the achieved fidelities and gate times are outputs of the optimization, not inputs. The self-citations to Refs. [26, 27] (co-authored by M.M. Müller) are used for context on prior QOC work and the Weyl figure of merit (Eq. 24), but neither is load-bearing for the QSL derivation itself. Ref. [27] is invoked for the noise analysis context, which the paper explicitly states it does not reproduce here. The QSL derivation cites Refs. [66, 67] (external authors) for the bound under full single-qubit control. The gap between the QSL (~171 ns) and practical gate times (~2 μs) is a limitation of the practical protocols relative to the theoretical bound, not a circularity. The derivation chain is self-contained against external benchmarks, so the circularity score is 1, reflecting only minor self-citations that are not load-bearing for the central theoretical claim.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

No new physical entities, particles, or forces are introduced. The paper works entirely within the standard spin-1/2 Hamiltonian framework with established hyperfine coupling terms. All parameters are either experimentally measured or standard algorithmic choices.

free parameters (5)
  • Azz = 2π×2.86234 MHz
    Longitudinal hyperfine coupling, fixed by experimental geometry from [12, 27, 38], not fitted in this paper
  • Azx = 2π×0.60281 MHz
    Transverse hyperfine coupling, fixed by experimental geometry from [12, 27, 38], not fitted in this paper
  • ωI = varied over 2π×[-15,15] MHz
    Nuclear Larmor frequency, tunable experimental parameter used as control knob
  • Ωmax = 2π×15 MHz
    Maximum Rabi frequency, experimental constraint
  • Nc (dCRAB basis size) = 20
    Number of basis functions in dCRAB optimization, algorithmic choice
axioms (5)
  • domain assumption Secular approximation for hyperfine interaction
    Standard in high-field ESR, drops non-secular terms in Eq. (1)
  • standard math Rotating wave approximation (RWA)
    Applied to control Hamiltonian Eq. (2) to obtain Eq. (3), standard for near-resonant driving
  • domain assumption Coherent Hamiltonian evolution (no noise)
    Stated in Sec. I: 'intentionally performed under coherent Hamiltonian evolution focusing on theoretical insights'
  • ad hoc to paper Infinitely fast single-qubit operations for QSL
    Sec. III.C: QSL derivation assumes instantaneous local gates, which is not experimentally realistic
  • domain assumption Full controllability of the electron-nuclear system via MW control
    Invoked in Sec. II.B, supported by prior work [23-27]

pith-pipeline@v1.1.0-glm · 19400 in / 2632 out tokens · 401795 ms · 2026-07-09T07:08:41.505371+00:00 · methodology

0 comments
read the original abstract

Accurately controlling entangling gates remains a major challenge for quantum technology applications with solid-state spin qubits. Here, we study a group-IV color-center with a strongly-coupled nuclear spin and approach the problem from a quantum control perspective. We show that there are three different types of entangling gates where the entanglement is mediated by the parallel hyperfine-coupling component, the orthogonal one or both. We derive the respective quantum speed limits (QSL) and show by means of dynamical decoupling, resonant driving of single- and double-quantum transitions, quantum optimal control and algebraic gate decomposition how these gates can be realized. We finally discuss the experimental applicability.

Figures

Figures reproduced from arXiv: 2607.07549 by Frank K. Wilhelm, Jurek Frey, Matthias M. M\"uller.

Figure 1
Figure 1. Figure 1: FIG. 1. Shown in color is the Pauli decomposition of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Population dynamics during a dynamical decoupling [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Optimization of a ZX/2 gate with the dCRAB [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Performance of the double-quantum transition [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Shown is a dCRAB Optimization of a YY/2 gate [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Time evolution of the Weyl coordinates. The system [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The upper panel shows the optimization procedure of a SWAP gate. The control of the system is optimized once with [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

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Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · 8 internal anchors

  1. [1]

    Wei, P.-J

    Y.-C. Wei, P.-J. Stas, A. Suleymanzade, G. Baranes, F. Machado, Y. Q. Huan, C. M. Knaut, S. W. Ding, M. Merz, E. N. Knall,et al., Science388, 509 (2025)

    work page 2025

  2. [2]

    Oleynik, J

    L. Oleynik, J. u. Rehman, S. Koudia, and S. Chatzinotas, Scientific Reports15, 29778 (2025)

    work page 2025

  3. [3]

    Stas, Y.-C

    P.-J. Stas, Y.-C. Wei, M. Sirotin, Y. Huan, U. Yazlar, F. A. Arias, E. Knyazev, G. Baranes, B. Machielse, S. Grandi,et al., arXiv preprint arXiv:2509.09464 (2025)

    work page arXiv 2025

  4. [4]

    Halder, A

    J. Halder, A. Rajabov, R. Bassoli, F. H. Fitzek, and G. P. Fettweis, Scientific Reports14, 19262 (2024)

    work page 2024

  5. [5]

    X. Guo, C. R. Breum, J. Borregaard, S. Izumi, M. V. Larsen, T. Gehring, M. Christandl, J. S. Neergaard- Nielsen, and U. L. Andersen, Nature Physics16, 281 (2020)

    work page 2020

  6. [6]

    Quantum Networks Using Color Defects in Diamond: Principles, Progress, and Perspectives

    A. Majumder, C. G. Torun, T. Schr¨ oder, G. Pieplow, P. Kumar, and K. Saha, Quantum networks using color defects in diamond: Principles, progress, and perspec- tives (2026), arXiv:2605.30005 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  7. [7]

    Schirhagl, K

    R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Annual review of physical chemistry65, 83 (2014)

    work page 2014

  8. [8]

    Neumann,Towards a room temperature solid state quantum processor-the nitrogen-vacancy center in dia- mond, Ph.D

    P. Neumann,Towards a room temperature solid state quantum processor-the nitrogen-vacancy center in dia- mond, Ph.D. thesis, Stuttgart, Universit¨ at Stuttgart, Diss., 2011 (2012)

    work page 2011

  9. [9]

    I. B. Harris, C. P. Michaels, K. C. Chen, R. A. Parker, M. Titze, J. Arjona Mart´ ınez, M. Sutula, I. R. Chris- ten, A. M. Stramma, W. Roth,et al., PRX Quantum4, 040301 (2023)

    work page 2023

  10. [10]

    H. K. Beukers, C. Waas, M. Pasini, H. B. Van Ommen, Z. Ademi, M. Iuliano, N. Codreanu, J. M. Brevoord, T. Turan, T. H. Taminiau,et al., Physical Review X15, 021011 (2025)

    work page 2025

  11. [11]

    I. B. Harris and D. Englund, Physical Review B109, 085414 (2024)

    work page 2024

  12. [12]

    Grimm, K

    N. Grimm, K. Senkalla, P. J. Vetter, J. Frey, P. Gund- lapalli, T. Calarco, G. Genov, M. M. M¨ uller, and F. Jelezko, Physical Review Letters134, 043603 (2025)

    work page 2025

  13. [13]

    Patel, Z

    A. Patel, Z. Chowdhry, A. Prabhakar, A. Rathi, and V. P. Bhallamudi, Scientific reports14, 22421 (2024)

    work page 2024

  14. [14]

    C. M. Knaut, A. Suleymanzade, Y.-C. Wei, D. R. As- sumpcao, P.-J. Stas, Y. Q. Huan, B. Machielse, E. N. Knall, M. Sutula, G. Baranes,et al., Nature629, 573 (2024)

    work page 2024

  15. [15]

    M. Ruf, N. H. Wan, H. Choi, D. Englund, and R. Hanson, Journal of Applied Physics130(2021)

    work page 2021

  16. [16]

    T. H. Taminiau, J. Cramer, T. van der Sar, V. Do- brovitski, and R. Hanson, Nature Nanotechnology9, 171 (2014)

    work page 2014

  17. [17]

    M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, Na- ture Communications9, 2552 (2018)

    work page 2018

  18. [18]

    Pompili, S

    M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson, Science372, 259 (2021), https://www.science.org/doi/pdf/10.1126/science.abg1919

    work page doi:10.1126/science.abg1919 2021

  19. [19]

    Childress and R

    L. Childress and R. Hanson, MRS bulletin38, 134 (2013)

    work page 2013

  20. [20]

    N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. W. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, Science356, 928 (2017), https://www.science.org/doi/pdf/10.1126/science.aan0070

    work page doi:10.1126/science.aan0070 2017

  21. [21]

    C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, Physical Review X 9, 031045 (2019)

    work page 2019

  22. [22]

    Takou, E

    E. Takou, E. Barnes, and S. E. Economou, Quantum8, 1304 (2024)

    work page 2024

  23. [23]

    Rembold, N

    P. Rembold, N. Oshnik, M. M. M¨ uller, S. Montangero, T. Calarco, and E. Neu, AVS Quantum Science2, 024701 (2020), https://doi.org/10.1116/5.0006785

    work page doi:10.1116/5.0006785 2020

  24. [24]

    Chou, S.-Y

    Y. Chou, S.-Y. Huang, and H.-S. Goan, Physical Review A91, 052315 (2015)

    work page 2015

  25. [25]

    H. Yuan, R. Zeier, N. Pomplun, S. J. Glaser, and N. Khaneja, Physical Review A - Atomic, Molecular, and Optical Physics92, 10.1103/PhysRevA.92.053414 (2015)

    work page doi:10.1103/physreva.92.053414 2015

  26. [26]

    Y. X. Wang, T. Calarco, F. Motzoi, and M. M. M¨ uller, Circuit design for a star-shaped spin-qubit processor via algebraic decomposition and optimal control (2025), arXiv:2506.16900 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  27. [27]

    J. Frey, K. Senkalla, P. J. Vetter, F. Jelezko, F. K. Wilhelm, and M. M. M¨ uller, Optimal two-qubit gates for group-iv color-centers in diamond (2026), arXiv:2604.11340 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  28. [28]

    Hanson, V

    R. Hanson, V. Dobrovitski, A. Feiguin, O. Gywat, and D. Awschalom, Science320, 352 (2008)

    work page 2008

  29. [29]

    J. Maze, J. Taylor, and M. Lukin, Physical Review B—Condensed Matter and Materials Physics78, 094303 (2008)

    work page 2008

  30. [30]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge university press, 2010)

    work page 2010

  31. [31]

    Travesedo, J

    J. Travesedo, J. O’Sullivan, L. Pallegoix, Z. W. Huang, P. Hogan, P. Goldner, T. Chaneliere, S. Bertaina, D. Est` eve, P. Abgrall,et al., Science Advances11, eadu0581 (2025)

    work page 2025

  32. [32]

    Zhao, S.-W

    N. Zhao, S.-W. Ho, and R.-B. Liu, Physical Review B—Condensed Matter and Materials Physics85, 115303 (2012)

    work page 2012

  33. [33]

    Kraus and J

    B. Kraus and J. I. Cirac, Physical Review A63, 062309 (2001). 11

    work page 2001

  34. [34]

    Khaneja and S

    N. Khaneja and S. J. Glaser, Chemical Physics267, 11 (2001)

    work page 2001

  35. [35]

    Makhlin, Quantum Information Processing1, 243 (2002)

    Y. Makhlin, Quantum Information Processing1, 243 (2002)

    work page 2002

  36. [36]

    Watts, J

    P. Watts, J. Vala, M. M. M¨ uller, T. Calarco, K. B. Wha- ley, D. M. Reich, M. H. Goerz, and C. P. Koch, Physical Review A91, 062306 (2015)

    work page 2015

  37. [37]

    Interaction-resolved decomposition of multi-qubit unitaries via computational-basis phases

    B. Baran, T. Calarco, M. M. Mueller, and F. Mot- zoi, Towards scalable multi-qubit optimal control via in- teraction decomposition in the diagonal frame (2026), arXiv:2602.18375 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  38. [38]

    Gundlapalli, P

    P. Gundlapalli, P. J. Vetter, G. Genov, M. Olney-Fraser, P. Wang, M. M. M¨ uller, K. Senkalla, and F. Jelezko, High-fidelity single-shot readout and selective nuclear spin control for a spin-1/2 quantum register in diamond (2025), arXiv:2510.09164 [quant-ph]

    work page arXiv 2025

  39. [39]

    Margolus and L

    N. Margolus and L. B. Levitin, Physica D: Nonlinear Phenomena120, 188 (1998), proceedings of the Fourth Workshop on Physics and Consumption

    work page 1998

  40. [40]

    Caneva, M

    T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Mon- tangero, V. Giovannetti, and G. E. Santoro, Phys. Rev. Lett.103, 240501 (2009)

    work page 2009

  41. [41]

    Deffner and E

    S. Deffner and E. Lutz, Phys. Rev. Lett.111, 010402 (2013)

    work page 2013

  42. [42]

    Deffner and S

    S. Deffner and S. Campbell, Journal of Physics A: Math- ematical and Theoretical50, 453001 (2017)

    work page 2017

  43. [43]

    All-electron $\mathrm{\textit{ab-initio}}$ hyperfine coupling of Si-, Ge- and Sn-vacancy defects in diamond

    A. Karim, H. H. Vallabhapurapu, C. Adambukulam, A. Laucht, S. P. Russo, and A. Peruzzo, arXiv preprint arXiv:2309.13913 (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  44. [44]

    Schweiger and G

    A. Schweiger and G. Jeschke,Principles of pulse electron paramagnetic resonance(Oxford university press, 2001)

    work page 2001

  45. [45]

    O’sullivan, J

    J. O’sullivan, J. Travesedo, L. Pallegoix, Z. W. Huang, P. Hogan, A. S. May, B. Yavkin, S. Lin, R.-B. Liu, T. Chaneliere,et al., Nature Physics , 1 (2025)

    work page 2025

  46. [46]

    Entanglement Generation on the Double Quantum Transition of NV Ground State Via Globally Addressing Microwave Pulse

    M. Morillas-Rozas, A. L´ opez-Garc´ ıa, and J. Cerrillo, En- tanglement generation on the double quantum transition of nv ground state via globally addressing microwave pulse (2025), arXiv:2501.18244 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  47. [47]

    J. R. Schrieffer and P. A. Wolff, Physical Review149, 491 (1966)

    work page 1966

  48. [48]

    C. Brif, R. Chakrabarti, and H. Rabitz, New Journal of Physics12, 075008 (2010)

    work page 2010

  49. [49]

    J. P. Palao, D. M. Reich, and C. P. Koch, Physical Re- view A—Atomic, Molecular, and Optical Physics88, 053409 (2013)

    work page 2013

  50. [50]

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K¨ ockenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbr¨ uggen,et al., The Euro- pean Physical Journal D69, 279 (2015)

    work page 2015

  51. [51]

    Barnes, C

    E. Barnes, C. Arenz, A. Pitchford, and S. E. Economou, Physical Review B96, 024504 (2017)

    work page 2017

  52. [52]

    Preti, T

    F. Preti, T. Calarco, and F. Motzoi, PRX Quantum3, 040311 (2022)

    work page 2022

  53. [53]

    M. M. M¨ uller, R. S. Said, F. Jelezko, T. Calarco, and S. Montangero, Reports on progress in physics85, 076001 (2022)

    work page 2022

  54. [54]

    Calzavara, T

    M. Calzavara, T. Calarco, and F. Motzoi, arXiv preprint arXiv:2509.25930 (2025)

    work page arXiv 2025

  55. [55]

    Caneva, T

    T. Caneva, T. Calarco, and S. Montangero, Physical Re- view A84, 022326 (2011)

    work page 2011

  56. [56]

    Doria, T

    P. Doria, T. Calarco, and S. Montangero, Physical review letters106, 190501 (2011)

    work page 2011

  57. [57]

    N. Rach, M. M. M¨ uller, T. Calarco, and S. Montangero, Physical Review A92, 062343 (2015)

    work page 2015

  58. [58]

    M. H. Goerz, G. Gualdi, D. M. Reich, C. P. Koch, F. Mot- zoi, K. B. Whaley, J. Vala, M. M. M¨ uller, S. Montangero, and T. Calarco, Phys. Rev. A91, 062307 (2015)

    work page 2015

  59. [59]

    Meiboom and D

    S. Meiboom and D. Gill, Review of Scientific Instruments 29, 688 (1958)

    work page 1958

  60. [60]

    Haeberlen and J

    U. Haeberlen and J. S. Waugh, Physical Review175, 453 (1968)

    work page 1968

  61. [61]

    L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. 76, 1037 (2005)

    work page 2005

  62. [62]

    H. Y. Carr and E. M. Purcell, Phys. Rev.94, 630 (1954)

    work page 1954

  63. [63]

    Casanova, Z.-Y

    J. Casanova, Z.-Y. Wang, J. F. Haase, and M. B. Plenio, Phys. Rev. A92, 042304 (2015)

    work page 2015

  64. [64]

    Resch, I

    J. Resch, I. Karapatzakis, M. Elshorbagy, M. Schrodin, P. Fuchs, P. Graßhoff, L. Kussi, C. S¨ urgers, C. Popov, C. Becher,et al., arXiv preprint arXiv:2509.03354 (2025)

    work page arXiv 2025

  65. [65]

    Klotz, K

    M. Klotz, K. G. Fehler, R. Waltrich, E. Steiger, S. H¨ außler, P. Reddy, L. F. Kulikova, V. A. Davydov, V. N. Agafonov, M. W. Doherty,et al., Physical Review Letters128, 153602 (2022)

    work page 2022

  66. [66]

    Speed limits of two-qubit gates with qudits

    B. Basyildiz, C. Jameson, and Z. Gong, arXiv preprint arXiv:2312.09218 (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  67. [67]

    Impens and D

    F. Impens and D. Gu´ ery-Odelin, Physical Review A112, 042614 (2025)

    work page 2025

  68. [68]

    Senkalla, G

    K. Senkalla, G. Genov, M. H. Metsch, P. Siyushev, and F. Jelezko, Phys. Rev. Lett.132, 026901 (2024)

    work page 2024

  69. [69]

    Banasiak, M

    M. Banasiak, M. Pierpaoli, and R. Bogdanowicz, Mate- rials Today Physics , 102088 (2026)

    work page 2026

  70. [70]

    Khaneja and S

    N. Khaneja and S. Glaser, Cartan decomposition of su(2n), constructive controllability of spin systems and universal quantum computing (2000), arXiv:quant- ph/0010100 [quant-ph]

    work page arXiv 2000

  71. [71]

    Zhang, J

    J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Phys. Rev. Lett.91, 027903 (2003)

    work page 2003

  72. [72]

    Zhang, J

    J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Phys. Rev. A69, 042309 (2004)

    work page 2004

  73. [73]

    A new algorithm for producing quantum circuits using KAK decompositions

    Y. Nakajima, Y. Kawano, and H. Sekigawa, A new algo- rithm for producing quantum circuits using kak decom- positions (2005), arXiv:quant-ph/0509196 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  74. [74]

    D. J. Egger and F. K. Wilhelm, Physical Review Letters 112, 240503 (2014)

    work page 2014