Numerical Optimization of Two-Qubit Gates in Silicon Flip-Flop Qubit Arrays under Electrical Control

Elena Ferraro; Lorenzo D'Onofrio; Marco De Michielis

REVIEW 3 major objections 5 minor 26 references

Numerical Optimization of Two-Qubit Gates in Silicon Flip-Flop Qubit Arrays under Electrical Control

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The paper claims that high-fidelity entangling gates in silicon flip-flop qubit arrays require co-optimizing control pulses, local phase corrections, and device geometry rather than tuning the interaction in isolation.

desk verdict Solid, self-contained numerical study of flip-flop qubit gates; the co-design conclusion is plausible, but the multi-qubit fidelity numbers need an explicit statement of whether spectator leakage is included. read the letter →

arxiv 2607.29123 v1 pith:P647KTS3 submitted 2026-07-31 quant-ph

Lorenzo D'Onofrio , Elena Ferraro , Marco De Michielis This is my paper · ORCID

classification quant-ph MSC 81P6881V70 PACS 03.67.Lx

keywords flip-flopqubitssiliconquantumcomputingtwo-qubitgateoptimizationMakhlininvariantsdipole-dipolecouplingcontrolRzrotationcalibrationspectator

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that two-qubit entangling gates in silicon donor flip-flop qubit arrays cannot be made high-fidelity by optimizing the interaction alone; the control pulse, the local single-qubit phase corrections, and the surrounding qubit geometry must be tuned together. Using a full spin-orbital simulator, it maps the electrically induced dipole-dipole interaction onto canonical gates such as sqrt(iSWAP) and iSWAP via Makhlin invariants, then shows that electrically driven Rz rotations can compensate accumulated local phases. With this co-design, composite-gate infidelities reach 4.72e-4 for sqrt(iSWAP) and 6.14e-4 for iSWAP, below the 1e-3 fault-tolerance threshold. Extending to multi-qubit arrays, spectator qubits shift the interaction landscape and the optimal correction angle, and symmetric environments consistently outperform asymmetric ones. A careful reader would care because this outlines a concrete optimization route toward scalable, all-electrical silicon quantum processors.

What carries the argument

The argument is carried by three interlocking tools: FlipFlopQSim, a simulator that propagates the coupled electron-nuclear spin-orbital Hamiltonian under realistic time-dependent electric fields; Makhlin invariants, which classify the two-qubit entangling content up to local operations and identify hold-time windows where the dipole-dipole interaction is locally equivalent to sqrt(iSWAP) or iSWAP; and the electrically driven Rz rotation calibrated at the clock-transition bias point, whose hold time maps linearly onto rotation angle with a tunable fine structure. The Makhlin invariants do the work of separating the nonlocal gate from local phases, while the calibrated Rz dataset supplies the

What would settle it

Recompute the optimized sqrt(iSWAP) and iSWAP pulse sequences in the full 8^N basis, or experimentally measure population in parallel-spin states at the optimal operating points. If the resulting infidelity exceeds 1e-3, or if observed leakage is non-negligible, the quantitative support for the central co-design claim fails.

Watch

Extended reading notes

Core claim

The central claim is that isolated two-qubit optimization is insufficient: residual dipolar couplings from spectator qubits alter both the nonlocal entangling evolution and the local phases, so the correction angle calibrated for an isolated pair is no longer optimal in larger arrays. The paper demonstrates this by simulating the full spin-orbital dynamics, classifying the entangling landscape with Makhlin invariants, calibrating electrically driven Rz rotations over a full 2π phase range, and reconstructing complete pulse sequences. The resulting sqrt(iSWAP) and iSWAP gate infidelities, 4.72e-4 and 6.14e-4, fall below the 1e-3 threshold, and symmetric active-idle configurations in linear, s

Load-bearing premise

The quoted fidelities assume a reduced four-dimensional-per-qubit subspace that discards parallel electron-nuclear spin configurations; if those discarded states participate in the driven dynamics at the optimized operating points, the infidelity numbers are optimistic.

Editorial extensions

If this is right

Two-qubit gate calibration must be performed in the actual array environment, not inferred from isolated-pair simulations.
Symmetric qubit layouts are preferable; they reduce spectator-induced distortion and can outperform asymmetric layouts even when the latter assume ideal local corrections.
Electrically driven Rz rotations calibrated over a full 2π range are sufficient to implement the necessary local corrections at the required accuracy.
iSWAP gates are far more sensitive to geometry and system size than sqrt(iSWAP), making sqrt(iSWAP) the more practical target for early multi-qubit demonstrations.
The absence of any CNOT-equivalent region confirms that the XY-type dipole-dipole interaction cannot produce CNOT in a single entangling step, guiding gate set selection.

Reading between the lines

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

The reduced 4^N subspace used throughout could be checked against the full 8^N basis at the optimized operating points; if leakage into parallel-spin states is non-negligible, the quoted infidelities would rise and possibly cross the threshold.
The strong sensitivity of the correction angle (deviations of 1e-4 rad degrade fidelity) suggests that closed-loop experimental calibration will be needed, not just open-loop simulations.
The co-design principle likely transfers to other dipolarly coupled spin-orbital qubit platforms, where spectator-induced phase shifts are similarly layout-dependent.
Dynamically tunable metallic couplers, mentioned as future work, would make the geometry dependence controllable and could turn asymmetric layouts from liabilities into assets.

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.

Desk Editor's Note

read the letter

Take on arXiv:2607.29123: this is a solid numerical study. The new thing is a full simulation pipeline for flip-flop qubit gates that combines Makhlin classification, electrically driven Rz calibration, and a multi-qubit spectator analysis. The authors show convincingly that the entangling interaction's local phases and the spectator environment must be handled together; the symmetric-vs-asymmetric layout result is the most useful design principle here. The code is openly available, and the Hamiltonian truncation to the 4^N subspace is actually safe because every term conserves S_z+I_z, so the excluded parallel-spin states are never populated under closed evolution. That concern can be set aside.

The real soft spot is the multi-qubit section. The paper explicitly projects spectator qubits onto |0⟩ and says this restricts the analysis to conditional trajectories, but it never reports whether spectator excitations are negligible. If the quoted fidelities in Figs. 5–6 come from the projected two-qubit operator rather than the full N-qubit propagator, they are conditional success probabilities, not unconditional gate infidelities. The qualitative conclusion that symmetric layouts are better may survive even with leakage, but the reader cannot tell. That is a gap; the fix is to report the spectator excitation probability or the full-register infidelity. It is not fatal, but it needs to be addressed.

The headline infidelities (4.72e-4 and 6.14e-4) are minima from optimization, so they are not independent predictions. The paper is transparent about this, but it would be good to see a validation with a different pulse shape or an out-of-sample point. The Makhlin distance threshold is also somewhat arbitrary; minor.

Who should read this: anyone working on flip-flop or donor-based silicon qubits, and people doing pulse-level optimization for spin qubits. It does not break new theoretical ground, but it is a useful design tool and a reasonable template for co-optimization studies.

Recommendation: send it out. The method is sound, the code is available, and the open questions (spectator leakage, out-of-sample fidelities) are addressable in revision. A good referee should look at it; I'd probably take it after moderate revision.

Referee Report

3 major / 5 minor

Summary. This paper presents a numerical study of two-qubit entangling gates in silicon flip-flop qubit arrays. Using a MATLAB framework (FlipFlopQSim), the authors simulate coherent spin-orbital dynamics under electrical control, classify two-qubit operations via Makhlin invariants, and identify hold-time windows locally equivalent to sqrt(iSWAP) and iSWAP. They then design electrically driven Rz rotations to compensate local phases, reporting full physical infidelities of 4.72e-4 for sqrt(iSWAP) and 6.14e-4 for iSWAP. The multi-qubit section analyzes linear, square, and star arrays and concludes that spectator couplings shift the entangling landscape, so gate optimization must co-optimize pulses, local phases, and geometry; symmetric layouts are consistently better than asymmetric ones. All simulations use the reduced 4^N anti-parallel spin basis.

Significance. The paper's contribution is a concrete computational pipeline and a qualitative design principle: isolated two-qubit optimization is insufficient in multi-qubit flip-flop arrays, and symmetric qubit environments mitigate spectator-induced distortions. Strengths include the openly available code, the use of Makhlin invariants to separate local from nonlocal content, and the comparison of ideal versus physically simulated Rz corrections, which goes beyond bare optimization. If the spectator-excitation and basis-truncation concerns are resolved, the framework will be useful for device design. The spin-conservation structure of the Hamiltonian suggests the 4^N reduction may be exact, but this is not demonstrated in the manuscript; and the multi-qubit fidelity numbers are not yet shown to be unconditional.

major comments (3)

[Sec. 6, Figs. 5-6] The analysis projects idle spectators onto |0> to define a reduced two-qubit operator and notes that this 'implicitly restricts our analysis to conditional trajectories where the spectator qubits remain unexcited.' The manuscript never states whether the reported infidelities are computed from the full N-qubit propagator or from this projected operator. If the latter, spectator excitations are discarded by construction, and the quoted values are conditional success probabilities rather than unconditional gate fidelities. Please report spectator excitation probabilities or full-register infidelity for the optimal operating points.
[Appendix A; Sec. 5] All quantitative results, including the headline values 4.72e-4 and 6.14e-4, are obtained in the reduced 4^N basis with no 8^N cross-check. The excluded parallel-spin configurations are asserted to be 'energetically detuned and weakly coupled,' but no truncation error is quantified. Given the spin-projection conservation of the Hamiltonian in Eqs. (5)-(12), the reduction may be exactly decoupled; this should be stated and ideally verified numerically at the two-qubit level. Without this, the sub-threshold claims are not fully justified.
[Sec. 5, Eq. (14)] The reported infidelities are minima obtained by optimizing correction angle and hold time with the same fidelity metric used to evaluate performance. This does not invalidate the optimization, but reporting only the minimum is fragile; the paper itself notes strong sensitivity to theta. To support the fault-tolerance claim, the authors should report the infidelity over a finite operating range (e.g., a small neighborhood of the optimal theta and thold) or provide an out-of-sample check for the optimized parameters.

minor comments (5)

[Sec. 2, Eq. (5)] The expression 'ˆI+' appears garbled; please clarify the identity and formatting throughout the Hamiltonian definitions.
[Eq. (16)] The relative threshold 0.1 is arbitrary; please state how the classification changes with the threshold or justify the choice.
[Fig. 4] The left panel's axis label 'piSWAP' should be '√iSWAP'; also consider using consistent notation for the square-root gate in the text and figures.
[Sec. 6] The statement that the two local-correction forms give a 'negligible difference' should be quantified (e.g., maximum infidelity difference) rather than left qualitative.
[Appendix A] 'Energetically detuned and weakly coupled' is imprecise; if the excluded states are decoupled by spin conservation, say so explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity found; the reported fidelities are explicitly optimized quantities and the stated approximations are limitations, not circular reductions.

full rationale

Walking the derivation chain: Sec. 2 defines the physical Hamiltonian explicitly (Eqs. 5-13); Sec. 4 classifies gates with Makhlin invariants from [22]; Sec. 5 optimizes Rz correction pulses and reports the infidelity at the optimized operating point; Sec. 6 extends the same procedure to multi-qubit arrays, noting that spectator projection restricts results to conditional trajectories. No step reduces an output to an input by construction: the reported infidelities (4.72e-4, 6.14e-4) are minima of an explicit cost function, not out-of-sample predictions, and the paper is transparent that they are optimized quantities. The Makhlin invariants and the XY-interaction limitation on CNOT are independent external results [22,26]. Self-citations [21,23,24] supply the simulation code and prior geometry studies, but the central co-design claim is supported by the new full-physics simulations reported here, not by an appeal to those citations. The stated limitations are genuine but non-circular: the 4^N subspace truncation (Appendix A) and the spectator-ground projection (Sec. 6, 'implicitly restricts our analysis to conditional trajectories') are unquantified approximation choices, and a full-register unconditional infidelity is not reported; these are correctness-risk items, not circular reasoning. No fitted parameter is renamed as a prediction; no uniqueness theorem from the authors' prior work is invoked to force a choice.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central results rest on a specific device model taken from Tosi et al., a truncated Hilbert-space approximation, and hand-chosen pulse/layout parameters. No new physical entities are introduced; FlipFlopQSim is software, not a physical postulate.

free parameters (4)

Hyperfine profile steepness c = 5.174e-4 m V^-1
Shape parameter in A(ΔEz)=A0/(1+exp(c·ΔEz)), Eq. (9); adopted from Ref. [15], controls all gate dynamics and is not re-derived here.
Baseline device parameters (B0, Δγ, d, Vt) = 0.4 T, -0.002, 15 nm, 11.29 GHz
Section 2: 'Following the operational baseline established in Ref. [15]'. The entangling landscape and optimized angles are specific to these values.
Control and layout parameters = ΔE_idle=1e4 V/m, ΔE_int=1300 V/m, τ1=2 ns, τ2=20 ns, r=360 nm, ΔE_ct=290 V/m
Hand-fixed pulse-sequence and geometry values used in Fig. 2-6; results depend on these choices.
Makhlin equivalence threshold = 0.1 ||g_ideal||
Eq. (16): the relative threshold used to assign local-equivalence classes. This hand-chosen criterion determines which t_hold windows are labeled sqrt(iSWAP) or iSWAP.

assumptions (6)

domain assumption Parallel electron-nuclear spin configurations are energetically detuned and can be neglected, so the reduced 4^N flip-flop subspace is sufficient.
Appendix A states all results use the reduced basis; if this fails, the quoted leakage numbers are incomplete.
standard math Makhlin invariants uniquely classify two-qubit gates up to local unitary equivalence.
Sec. 4, Ref. [22]; used to call the generated gates sqrt(iSWAP) or iSWAP.
standard math The dipole-dipole Hamiltonian reduces to an XY-type interaction in the orbital eigenbasis, so CNOT cannot be reached in a single entangling step.
Sec. 4, Ref. [26]; used to explain the absence of CNOT in the landscape.
domain assumption Spectator qubits can be projected onto their |0> ground states, yielding a unitary reduced description for conditional trajectories.
Sec. 6 explicitly restricts the multi-qubit analysis to trajectories where spectators remain unexcited; this conditions the fidelity results.
domain assumption Time evolution is noiseless and computed by piecewise-constant matrix exponentiation.
Appendix A, Eq. (21); classical noise is optional but not included in the reported infidelities.
domain assumption The hyperfine coupling profile A(ΔEz)=A0/(1+exp(c·ΔEz)) and the adopted device parameters from Ref. [15] describe the physical system.
Eq. (9) and Sec. 2; the entire simulation rests on this prior model.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Numerical Optimization of Two-Qubit Gates in Silicon Flip-Flop Qubit Arrays under Electrical Control." pith.science (2026). https://pith.science/paper/P647KTS3

@misc{pith2026260729123,
  author       = {Pith},
  title        = {Pith review of: Numerical Optimization of Two-Qubit Gates in Silicon Flip-Flop Qubit Arrays under Electrical Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P647KTS3}},
  note         = {Machine review of arXiv:2607.29123}
}

abstract

Silicon-based donor flip-flop qubits offer a promising path toward scalable, fault-tolerant quantum computing by combining the long coherence times of nuclear spins with fast, fully electrical control and long-range dipole-dipole coupling between qubits. However, realizing high-fidelity entangling operations in this platform remains challenging. The entangling interaction is intrinsically coupled to electron orbital dynamics, which can lead to leakage into non-computational states and unwanted phase accumulation. Furthermore, in multi-qubit architectures, residual dipolar couplings from spectator qubits distort the effective interaction landscape. In this work, we employ a numerical simulation framework, FlipFlopQSim, that models the spin-orbital dynamics of interacting flip-flop qubits to extract effective logical operations from realistic electrical control pulses. Using Makhlin invariants, we map the entangling landscape generated by electrically controlled dipole-dipole interactions and identify operating regions locally equivalent to canonical two-qubit gates, such as $\sqrt{iSWAP}$ and $iSWAP$. We then optimize the control parameters of physically realizable, electrically driven $R_z$ rotations to implement the necessary local corrections and maximize composite gate fidelity. Finally, we scale our analysis to multi-qubit registers with various geometries and connectivity patterns to evaluate spectator-induced distortions. Our results demonstrate that high-fidelity entangling operations cannot be optimized in isolation; rather, they require a co-design approach that simultaneously optimizes pulse control, local phase compensation, and physical device geometry. This work provides a robust numerical framework for assessing the scalability of electrically controlled silicon quantum processors and outlines key design principles for robust multi-qubit gate implementation.

Figures

Figures reproduced from arXiv: 2607.29123 by the authors.

**Figure 1.** Schematic of a flip-flop qubit. A 31P donor atom is embedded in bulk 28Si at a distance zd from the Si/SiO2 interface. An electric field Ez, applied through a metal gate, enables control of the electron position between the donor-bound state (|d⟩) and the interface-bound state (|i⟩). The parameter d denotes the separation between the mean positions of the donor- and interface-bound electron wavefunctions. A static m… view at source ↗

**Figure 2.** Classification of two-qubit operations via Makhlin invariants as a function of the hold time [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Phase calibration landscape for an Rz(θ) gate in a single-qubit system. The rotation angle θ (y-axis) is extracted via bounded optimization of the effective propagator as a function of the hold time thold at the clock-transition point (x-axis). Color mapping indicates gate infidelity (1−F), accounting for both coherent errors and leakage to higher energy levels. The inset shows the control sequence, displaying the v… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Optimization landscape for composite two-qubit entangling gates. Gate infidelity is shown as a [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Lowest gate infidelity for √ iSWAP gate in linear arrays with varying qubit numbers and activeidle layouts. Symmetric and asymmetric configurations are compared using both ideal Rz corrections (instantaneous and error-free) and numerically simulated electrically drive…

**Figure 6.** Figure 6: Lowest gate infidelity for √ iSWAP gate across different four-qubit geometries and activeidle configurations. Symmetric and asymmetric layouts are compared using both ideal Rz corrections (instantaneous and error-free) and realistic electrically driven Rz implementati…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 12 canonical work pages

[1]

Semiconductor Spin Qubits

Guido Burkard et al. “Semiconductor Spin Qubits”. In:Rev. Mod. Phys.95 (2023), p. 025003.doi: 10.1103/RevModPhys.95.025003

doi:10.1103/revmodphys.95.025003 2023
[2]

Silicon Spin Qubits from Laboratory to Industry

M. De Michielis et al. “Silicon Spin Qubits from Laboratory to Industry”. In:Journal of Physics D: Applied Physics56 (2023), p. 363001.doi:10.1088/1361-6463/acd8c7

doi:10.1088/1361-6463/acd8c7 2023
[3]

Donor-based qubits for quantum computing in silicon

J. C. McCallum, B. C. Johnson, and T. Botzem. “Donor-based qubits for quantum computing in silicon”. In:Applied Physics Reviews8 (2021), p. 031314.doi:10.1063/5.0060957

work page doi:10.1063/5.0060957 2021
[4]

A silicon-based nuclear spin quantum computer

B. E. Kane. “A silicon-based nuclear spin quantum computer”. In:Nature393 (1998), pp. 133–137. doi:10.1038/30156

doi:10.1038/30156 1998
[5]

Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent

L. M. K. Vandersypen et al. “Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent”. In:Npj Quantum Information3 (2017), p. 34.doi:10.1038/s41534-017-0038-y

doi:10.1038/s41534-017-0038-y 2017
[6]

Donor spins in silicon for quantum technologies

A. Morello et al. “Donor spins in silicon for quantum technologies”. In:Advanced Quantum Tech- nologies3 (2020), p. 2000005.doi:10.1002/qute.202000005

doi:10.1002/qute.202000005 2020
[7]

A single-atom electron spin qubit in silicon

J. J. Pla et al. “A single-atom electron spin qubit in silicon”. In:Nature489.7417 (2012), pp. 541– 545.doi:10.1038/nature11449

work page doi:10.1038/nature11449 2012
[8]

High-fidelity readout and control of a nuclear spin qubit in silicon

Jarryd J. Pla et al. “High-fidelity readout and control of a nuclear spin qubit in silicon”. In:Nature 496.7445 (2013), pp. 334–338.doi:10.1038/nature12011

doi:10.1038/nature12011 2013

Show all 26 references

[9]

Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots

J. R. Petta et al. “Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots”. In:Science309.5744 (2005), pp. 2180–2184.doi:10.1126/science.1116955

2005 doi
[10]

Electrically controlling single-spin qubits in a continuous microwave field

Arne Laucht et al. “Electrically controlling single-spin qubits in a continuous microwave field”. In: Science Advances1 (2015), e1500022.doi:10.1126/sciadv.1500022

2015 doi
[11]

Storing quantum information for 30 seconds in a nanoelectronic device

Juha T. Muhonen et al. “Storing quantum information for 30 seconds in a nanoelectronic device”. In:Nature Nanotechnology9 (2014), pp. 986–991.doi:10.1038/nnano.2014.211

2014 doi
[12]

Quantum Information Storage for over 180 s Using Donor Spins in a 28Si

M. Steger et al. “Quantum Information Storage for over 180 s Using Donor Spins in a 28Si ”Semicon- ductor Vacuum””. In:Science336.6086 (2012), pp. 1280–1283.doi:10.1126/science.1217635

2012 doi
[13]

Electron spin relaxation of single phosphorus donors in metal-oxide- semiconductor nanoscale devices

Stefanie B. Tenberg et al. “Electron spin relaxation of single phosphorus donors in metal-oxide- semiconductor nanoscale devices”. In:Physical Review B99 (2019), p. 205306.doi:10 . 1103 / PhysRevB.99.205306

2019
[14]

Electron spin coherence exceeding seconds in high-purity silicon

Alexei M. Tyryshkin et al. “Electron spin coherence exceeding seconds in high-purity silicon”. In: Nature Materials11 (2012), pp. 143–147.doi:10.1038/nmat3182

2012 doi
[15]

Silicon quantum processor with robust long-distance qubit couplings

Guilherme Tosi et al. “Silicon quantum processor with robust long-distance qubit couplings”. In: Nature Communications8 (2017), p. 450.doi:10.1038/s41467-017-00378-x

2017 doi
[16]

Robust electric dipole transition at microwave frequencies for nuclear spin qubits in silicon

Guilherme Tosi et al. “Robust electric dipole transition at microwave frequencies for nuclear spin qubits in silicon”. In:Physical Review B98 (2018), p. 075313.doi:10.1103/PhysRevB.98.075313

2018 doi
[17]

An electrically driven single-atom flip-flop qubit

Rostyslav Savytskyy et al. “An electrically driven single-atom flip-flop qubit”. In:Science Advances 9.6 (2023), eadd9408.doi:10.1126/sciadv.add9408

2023 doi
[18]

Fast high-fidelity single-qubit gates for flip-flop qubits in silicon

F. A. Calderon-Vargas, Edwin Barnes, and Sophia E. Economou. “Fast high-fidelity single-qubit gates for flip-flop qubits in silicon”. In:Phys. Rev. B106 (2022), p. 165302.doi:10 . 1103 / PhysRevB.106.165302. 12

2022
[19]

Universal set of quantum gates for the flip-flop qubit with 1/f noise

E. Ferraro et al. “Universal set of quantum gates for the flip-flop qubit with 1/f noise”. In:EPJ Quantum Technology9.2 (2022).doi:10.1140/epjqt/s40507-022-00120-7

2022 doi
[20]

Fast noise-resistant control of donor nuclear spin qubits in silicon

James Simon et al. “Fast noise-resistant control of donor nuclear spin qubits in silicon”. In:Phys. Rev. B101 (20 May 2020), p. 205307.doi:10.1103/PhysRevB.101.205307

2020 doi
[21]

Version 1.0.0

Lorenzo D’Onofrio, Elena Ferraro, and Marco De Michielis.FlipFlopQSim. Version 1.0.0. 2026. doi:10.5281/zenodo.21669875

2026 doi
[22]

Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations

Yuriy Makhlin. “Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations”. In:Quantum Information Processing1.4 (2002), pp. 243–252.doi: 10.1023/A:1022144002391

2002 doi
[23]

Parallel Gate Fidelity of Flip-Flop Qubits in Small 1D- and 2D-Arrays in a Noisy Environment

M. De Michielis, D. Rei, and E. Ferraro. “Parallel Gate Fidelity of Flip-Flop Qubits in Small 1D- and 2D-Arrays in a Noisy Environment”. In:Advanced Quantum Technologies7 (2024), p. 2300455. doi:10.1002/qute.202300455

2024 doi
[24]

Impact of Parallel Gating on Gate Fidelities in Linear, Square, and Star Arrays of Noisy Flip-Flop Qubits

M. De Michielis and E. Ferraro. “Impact of Parallel Gating on Gate Fidelities in Linear, Square, and Star Arrays of Noisy Flip-Flop Qubits”. In:Advanced Quantum Technologies8.1 (2025), p. 2400341. doi:10.1002/qute.202400341

2025 doi
[25]

Fidelity of quantum operations

Line Hjortshøj Pedersen, Niels Martin Møller, and Klaus Mølmer. “Fidelity of quantum operations”. In:Physics Letters A367.1–2 (2007), pp. 47–51.doi:10.1016/j.physleta.2007.02.069

2007 doi
[26]

Natural two-qubit gate for quantum computation using the XY interaction

Norbert Schuch and Jens Siewert. “Natural two-qubit gate for quantum computation using the XY interaction”. In:Physical Review A67 (2003), p. 032301.doi:10.1103/PhysRevA.67.032301. 13

2003 doi

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Semiconductor Spin Qubits

Guido Burkard et al. “Semiconductor Spin Qubits”. In:Rev. Mod. Phys.95 (2023), p. 025003.doi: 10.1103/RevModPhys.95.025003

doi:10.1103/revmodphys.95.025003 2023

[2] [2]

Silicon Spin Qubits from Laboratory to Industry

M. De Michielis et al. “Silicon Spin Qubits from Laboratory to Industry”. In:Journal of Physics D: Applied Physics56 (2023), p. 363001.doi:10.1088/1361-6463/acd8c7

doi:10.1088/1361-6463/acd8c7 2023

[3] [3]

Donor-based qubits for quantum computing in silicon

J. C. McCallum, B. C. Johnson, and T. Botzem. “Donor-based qubits for quantum computing in silicon”. In:Applied Physics Reviews8 (2021), p. 031314.doi:10.1063/5.0060957

work page doi:10.1063/5.0060957 2021

[4] [4]

A silicon-based nuclear spin quantum computer

B. E. Kane. “A silicon-based nuclear spin quantum computer”. In:Nature393 (1998), pp. 133–137. doi:10.1038/30156

doi:10.1038/30156 1998

[5] [5]

Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent

L. M. K. Vandersypen et al. “Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent”. In:Npj Quantum Information3 (2017), p. 34.doi:10.1038/s41534-017-0038-y

doi:10.1038/s41534-017-0038-y 2017

[6] [6]

Donor spins in silicon for quantum technologies

A. Morello et al. “Donor spins in silicon for quantum technologies”. In:Advanced Quantum Tech- nologies3 (2020), p. 2000005.doi:10.1002/qute.202000005

doi:10.1002/qute.202000005 2020

[7] [7]

A single-atom electron spin qubit in silicon

J. J. Pla et al. “A single-atom electron spin qubit in silicon”. In:Nature489.7417 (2012), pp. 541– 545.doi:10.1038/nature11449

work page doi:10.1038/nature11449 2012

[8] [8]

High-fidelity readout and control of a nuclear spin qubit in silicon

Jarryd J. Pla et al. “High-fidelity readout and control of a nuclear spin qubit in silicon”. In:Nature 496.7445 (2013), pp. 334–338.doi:10.1038/nature12011

doi:10.1038/nature12011 2013