Flip-Flop Qubit

Figure

Description

The flip-flop qubit encodes quantum information in the combined electron-nuclear spin states of a phosphorus (${}^{31}\text{P}$) donor atom in silicon. Using the convention adopted in the proposal and experimental demonstration, the computational basis states are:

$|0⟩=|{\downarrow }_e{\Uparrow }_n⟩,|1⟩=|{\uparrow }_e{\Downarrow }_n⟩$

where arrows denote electron spin ($\uparrow /\downarrow$) and nuclear spin ($\Uparrow /\Downarrow$) orientations. In each state, the electron and nuclear spins point in opposite directions — hence “flip-flop.” These states are connected by a simultaneous electron-nuclear spin flip driven by transverse hyperfine interaction, which can be electrically tuned by displacing the electron wavefunction relative to the nucleus using gate electrodes.

The 2017 processor proposal adds a deliberately hybridized orbital degree of freedom: a donor-bound orbital $|d⟩$ and an interface orbital $|i⟩$. Their vertical separation $d$ creates a tunable electric dipole $ed$, and hyperfine coupling transfers that electric interaction to the flip-flop spin subspace. In the proposal, a $1\text{MHz}$ effective coupling can be maintained for donor separations from 180 to 500 nm by retuning the donor-interface tunnel coupling. That is far less placement-sensitive than the exchange interaction used by conventional donor-spin architectures.

The first experiment, reported in 2023, demonstrated coherent electric control of the same flip-flop spin encoding in a single implanted phosphorus donor. It modulated the hyperfine coupling by electrically distorting the donor electron wavefunction, but did not reach the donor-interface large-dipole regime or demonstrate the proposed long-range two-qubit coupling. The logical flip-flop rotation itself uses a microwave electric field; the experiment still used auxiliary ESR/NMR operations for preparation and readout.

Hamiltonian

The proposal requires the orbital degree of freedom explicitly. In frequency units, its two-level Hamiltonian is

$${\mathcal{H}}_{\mathrm{orb}}=\frac{1}{2}\left[V_t{\sigma }_x-\frac{ed(E_z-E_z^0)}{h}{\sigma }_z\right],$$

where $V_t$ is the donor-interface tunnel coupling and $E_z^0$ is the ionization field. Let $P_i=|i⟩⟨i|=(1+{\sigma }_z)/2$ and $P_d=|d⟩⟨d|=(1-{\sigma }_z)/2$. A representative spin-orbital Hamiltonian is

$$\frac{H}{h}={\mathcal{H}}_{\mathrm{orb}}+{\gamma }_e{B}_0(1+\Delta \gamma P_i)S_z-{\gamma }_n{B}_0{I}_z+A{P}_d\mathbf{S}\cdot \mathbf{I},$$

where $\Delta \gamma$ allows for the interface-dependent electron gyromagnetic ratio. The bare flip-flop splitting is

$${ϵ}_{\mathrm{ff}}=\sqrt{[({\gamma }_e+{\gamma }_n)B_0{]}^2+A(E_z{)}^2}.$$

with ${\gamma }_{+}={\gamma }_e+{\gamma }_n$. The orbital splitting is

${ϵ}_o=\sqrt{V_t^2+[ed(E_z-E_z^0)/h{]}^2}.$

The transverse hyperfine term couples the flip-flop and charge sectors with $g_{\mathrm{so}}=(A/4)(V_t/{ϵ}_o)$. In the donor-interface regime, an ac electric field drives the spin transition through the second-order charge-mediated rate

$$g_E^{\mathrm{ff}}=\frac{g_{\mathrm{so}}{g}_E}{2}\left(\frac{1}{{\delta }_{\mathrm{so}}}+\frac{1}{{\delta }_E}\right),g_E=\frac{e{E}_{\mathrm{ac}}d}{4h}\frac{V_t}{{ϵ}_o}.$$

The 2023 single-donor experiment is described directly in the flip-flop subspace, after dropping a common energy shift, by

$$\frac{H_{\mathrm{ff}}}{h}=\frac{1}{2}\left[{\gamma }_{+}{B}_0{\sigma }_z+A(E){\sigma }_x\right],$$

so a small hyperfine modulation gives $f_{\mathrm{Rabi}}^{\mathrm{ff}}\simeq [\partial A(E)/(2\partial E)]E_{\mathrm{ac}}$ in the rotating-wave limit.

For two vertical donor-interface dipoles, the direct charge-sector interaction is

$$\frac{H_{\mathrm{dip}}}{h}=V_{dd}({\sigma }_{z,1}{\sigma }_{z,2}+{\sigma }_{z,1}+{\sigma }_{z,2}),V_{dd}=\frac{e^2{d}_1{d}_2}{16\pi {ϵ}_0{ϵ}_{r}h{r}^3}.$$

The logical flip-flop coupling is not this bare interaction: it appears at second order through the two charge orbitals and can be tuned with $V_t$ and $E_z$.

Motivation

The flip-flop qubit addresses the central scaling bottleneck of silicon donor qubits: exchange coupling demands extremely precise placement. Replacing it with tunable donor-interface dipole coupling over hundreds of nanometres relaxes placement tolerances and leaves room for interconnects. The all-electric control via gate electrodes (no oscillating magnetic field for logical operations) further simplifies the proposed control architecture.

Experimental Status

Donor-interface processor proposal — Tosi et al. (2017):

• Proposed the flip-flop qubit encoding and electric dipole coupling mechanism.
• Calculated one- and two-qubit gate errors near $10^{-3}$ under the paper’s realistic-noise assumptions.
• Showed that coupling to a superconducting microwave resonator could extend entanglement to macroscopic distances.
• Predicted a tunable $1\text{MHz}$ effective coupling across 180–500 nm donor separations; the illustrated $\sqrt{i\mathrm{SWAP}}$ gate used $r=180\text{nm}$.

Single-donor demonstration — Savytskyy et al. (2023):

• Demonstrated coherent microwave-electric control of the $|{\downarrow }_e{\Uparrow }_n⟩\leftrightarrow |{\uparrow }_e{\Downarrow }_n⟩$ transition in one implanted ${}^{31}\text{P}$ donor.
• Reached a Rabi frequency of $118.5(2.5)\text{kHz}$ and an average native one-qubit gate fidelity of $98.4(2)\%$.
• Measured $T_1^{\mathrm{ff}}=173(12)\text{s}$, Ramsey $T_2^{\ast }=4.09(88)\mu \text{s}$, and Hahn-echo $T_2^{\mathrm{H}}=184(24)\mu \text{s}$.
• The device used hyperfine Stark modulation of a distorted donor orbital; it did not transfer the electron to an interface dot or test dipole-mediated two-qubit gates.

2024–2026 updates:

• De Michielis and Ferraro (2024/2025) simulated parallel gates in four-qubit arrays and found spectator/parallel-operation effects to be especially severe for simultaneous two-qubit gates; this is a theoretical scaling result, not a hardware benchmark.
• D’Onofrio, Ferraro, and De Michielis (2026) developed a spin-orbital simulator and predicted composite $\sqrt{i\mathrm{SWAP}}$ and $i\mathrm{SWAP}$ infidelities of $4.72\times 10^{-4}$ and $6.14\times 10^{-4}$ after local phase compensation in an isolated pair. They also found that spectator qubits require geometry-aware recalibration; the work is an arXiv preprint.
• A 2026 APS conference abstract reported MHz-rate electric driving of flip-flop states in an antimony-donor/MOS-dot device. This is a conference-level extension to a high-spin donor, not yet a peer-reviewed phosphorus flip-flop processor result.

Status as of September 2026: the single-${}^{31}\text{P}$ flip-flop encoding and electric control are experimentally demonstrated. The donor-interface large-dipole operating point, the predicted 180–500 nm logical coupling, and any flip-flop two-qubit gate remain undemonstrated.

Key Metrics

Metric	Value	Notes	Fidelity reference
1Q native gate fidelity	$98.4(2)\%$	Randomized benchmarking; demonstrated single donor	Savytskyy et al. 2023
Electric-drive Rabi frequency	$118.5(2.5)\text{kHz}$	Highest applied microwave power in the demonstrated device	Savytskyy et al. 2023
$T_1^{\mathrm{ff}}$	$173(12)\text{s}$	Demonstrated single donor	Savytskyy et al. 2023
Ramsey $T_2^{\ast }$	$4.09(88)\mu \text{s}$	Demonstrated single donor	Savytskyy et al. 2023
Hahn-echo $T_2^{\mathrm{H}}$	$184(24)\mu \text{s}$	Demonstrated single donor	Savytskyy et al. 2023
Coupling range	180–500 nm	Range over which $g_{2q}^{\mathrm{ff}}=1\text{MHz}$ was maintained by retuning $V_t$; predicted	Tosi et al. 2017
Coupling strength	1 MHz	Tunable second-order logical coupling; predicted	Tosi et al. 2017
$\sqrt{i\mathrm{SWAP}}$ error	$\sim 3\times 10^{-3}$	Optimized model near $r=180\text{nm}$; not demonstrated	Tosi et al. 2017

Scaling Considerations

• Fabrication tolerance: tunability maintains the proposed $1\text{MHz}$ coupling across 180–500 nm spacing, but donor depth and interface quality still control $d$, $V_t$, and charge noise.
• Charge noise sensitivity: operating near the charge-spin hybridization point exposes the qubit to charge noise from the Si/SiO₂ interface, which could limit coherence. Optimal operating points that balance dipole strength against charge noise sensitivity need experimental validation.
• Magnetic field requirements: a static magnetic field of ~0.4 T is needed to define the qubit, plus precise tuning to the anticrossing regime — adding complexity.
• Experimental gap: single-qubit electric control is demonstrated, but the intended donor-interface ionization point, long-range dipole coupling, and two-qubit gate have not been demonstrated.
• Resonator coupling: the proposal includes coupling to superconducting resonators for long-range (mm-scale) interactions, but this adds the complexity of integrating superconducting and semiconductor technologies.

References

Original proposal

• G. Tosi et al., “Silicon quantum processor with robust long-distance qubit couplings,” Nat. Commun. 8, 450 (2017) | arXiv:1509.08538

Experimental demonstration

• R. Savytskyy et al., “An electrically driven single-atom ‘flip-flop’ qubit,” Sci. Adv. 9, eadd9408 (2023) | arXiv:2202.04438

Scaling and control studies

• M. De Michielis and E. Ferraro, “Impact of Parallel Gating on Gate Fidelities in Linear, Square, and Star Arrays of Noisy Flip-Flop Qubits,” Adv. Quantum Technol. 8, 2400341 (2025) | arXiv:2407.20166
• L. D’Onofrio, E. Ferraro, and M. De Michielis, “Numerical Optimization of Two-Qubit Gates in Silicon Flip-Flop Qubit Arrays under Electrical Control,” arXiv:2607.29123 (2026 preprint)

Recent conference report

• S. Kruskic et al., “Integrating high-spin antimony donors with MOS quantum dots in silicon, Part 2: Flip-Flop qubit operation,” APS Global Physics Summit 2026 abstract

Linked Papers

• tosi-2017-flip-flop-qubit
• savytskyy-2023-electrically-driven-flip-flop-qubit
• de-michielis-2025-parallel-gating-flip-flop-arrays
• donofrio-2026-flip-flop-two-qubit-optimization

Evergreen context

• charge-noise-sweet-spot — the donor-interface orbital creates both the useful electric dipole and the main electric-noise exposure.
• resonator-as-quantum-bus — covers the proposal’s route from hundreds-of-nanometres dipole coupling to longer-range microwave links.
• sqrt-swap-as-universal-gate — use only for its explicit SWAP-versus-iSWAP boundary: the predicted flip-flop entangler comes from projected dipolar XY coupling and is $\sqrt{i\mathrm{SWAP}}$, not the isotropic-exchange $\sqrt{\mathrm{SWAP}}$ primitive.

Related Entries

• kane-qubit — original phosphorus-in-silicon qubit proposal; requires atomic-precision placement
• silicon-spin-qubit — broader silicon spin qubit family
• loss-divincenzo-qubit — quantum dot spin qubit with short-range exchange coupling
• spin-qubit — general spin qubit concept
• circuit-qed — proposed resonator bus for millimeter-scale coupling