Kane Qubit (Phosphorus-in-Silicon)

Description

The Kane qubit, proposed by Bruce Kane in 1998, encodes quantum information in the nuclear spin of individual ${}^{31}\text{P}$ donor atoms embedded in isotopically pure ${}^{28}\text{Si}$. The nuclear spin ($I=1/2$) of phosphorus has extraordinarily long coherence times — $T_2>30\text{s}$ has been demonstrated — because the nuclear spin couples very weakly to the environment.

The architecture uses three types of gate electrodes above each donor:

1. A-gates: control the hyperfine coupling $A$ between the donor electron and nuclear spin, enabling selective NMR addressing of individual nuclei.
2. J-gates: control the exchange interaction $J$ between electrons on neighboring donors, mediating two-qubit gates.
3. Global RF/microwave fields: drive ESR/NMR transitions.

Single-qubit gates are performed by NMR pulses on individual nuclei (made distinguishable by A-gate detuning of the hyperfine coupling). Two-qubit gates use the electron-mediated exchange interaction, controlled by the J-gate voltage.

The silicon host is chosen for its nuclear-spin-free isotope (${}^{28}\text{Si}$), eliminating magnetic noise from the lattice. Single-atom placement with scanning tunneling microscope (STM) lithography has been demonstrated by the Simmons group (UNSW), achieving atomic-precision donor placement.

Figure

Hamiltonian

Two-donor system:

$H={\sum }_{i=1,2}\left[\frac{g_e{\mu }_{B}B}{2}{\sigma }_z^{(e_i)}-\frac{g_n{\mu }_{n}B}{2}{\sigma }_z^{(n_i)}+A_i{\mathbf{S}}_{e_i}\cdot {\mathbf{I}}_{n_i}\right]+J{\mathbf{S}}_{e_1}\cdot {\mathbf{S}}_{e_2}$

where $A_i$ is the gate-tunable contact hyperfine coupling for donor $i$, $J$ is the exchange coupling between donor electrons, $g_e$ ($g_n$) is the electron (nuclear) $g$-factor, and $B$ is the applied magnetic field.

At $B\sim 2\text{T}$, the electron Zeeman splitting ($\sim 56\text{GHz}$) far exceeds the hyperfine coupling ($A\sim 117\text{MHz}$ in bulk), so the electron spin adiabatically follows the nuclear spin state, mediating an effective nuclear-nuclear interaction.

Motivation

Nuclear spins in silicon offer the longest coherence times of any solid-state qubit, and silicon fabrication is the most mature semiconductor technology on Earth. Kane’s proposal connects quantum computing to the existing trillion-dollar silicon fab infrastructure, with qubit densities potentially approaching CMOS transistor scales.

Key Metrics

Metric	Value	Notes	Fidelity reference
Nuclear $T_1$	>30 hours	${}^{31}$P in ${}^{28}$Si at 1.5 K	Muhonen et al. 2014
Nuclear $T_2$ (echo)	>35 s	World record for solid-state qubit	Muhonen 2014
Electron $T_2$	0.5–1 s	In ${}^{28}$Si	—
1Q gate fidelity	99.95%	Nuclear spin, NMR control	Muhonen et al. 2014
2Q gate fidelity	99.4%	Exchange-mediated	Mądzik et al. 2022
Donor spacing	10–20 nm	STM lithography placement	—
Hyperfine coupling $A$	~117 MHz	Bulk value; gate-tunable	Kane 1998
Operating temperature	100 mK – 1 K	Electron spin relaxation limited	—

Linked Papers

• kane-1998-silicon-nuclear-spin

Related Entries

• spin-qubit
• loss-divincenzo-qubit