Semiconductor Spin Qubits

A semiconductor spin qubit is the modern realisation of the 1998 Loss–DiVincenzo proposal: trap a single electron in an electrostatically defined quantum dot and store the qubit in its spin. Metal gates patterned on a silicon or gallium-arsenide heterostructure sculpt a potential landscape in a buried two-dimensional electron gas; adjusting the gate voltages loads a controlled number of electrons — ideally exactly one per dot — and sets the tunnel barriers between neighbours. The trick that makes the platform work is a separation of scales. The electron's orbital wavefunction sits in a deep, charged well that couples strongly to the noisy electric fields rattling the gates, but its spin carries no electric dipole at leading order, so it floats almost untouched above that electrical storm. That is why a spin can stay coherent for microseconds to milliseconds while the charge around it fluctuates on nanosecond scales. Everything else is engineering built on that idea. A static field $B_0$ Zeeman-splits the two spin states into a qubit; a resonant microwave or electric drive rotates it; lowering the wall between two dots switches on the Heisenberg exchange $J$ that entangles neighbours; and a charge sensor, watching a single electron hop on or off, converts the un-measurable spin into a measurable current. The great promise is that all of this is fabricated with the same lithography and materials as the transistors in a laptop — so a mature qubit could, in principle, ride the entire semiconductor industry to million-qubit scale. This article assembles the physics of that platform, from the Zeeman ladder to the exchange sweet spot to the isotopic purification that gives silicon its quiet magnetic vacuum.

Key takeaways

A semiconductor spin qubit stores quantum information in the spin-1/2 of a single electron (or hole) confined in a gate-defined quantum dot, in silicon (Si/SiGe, SiMOS) or GaAs.
A static field Zeeman-splits the spin, $\hbar\omega_0=g\mu_B B_0$ ; single-qubit gates are driven by electron spin resonance (a microwave $B_1$ ) or electric-dipole spin resonance (an electric drive plus a field gradient or spin–orbit coupling).
Two-qubit gates use the Heisenberg exchange $J(\varepsilon)=4t^2U/(U^2-\varepsilon^2)$ set by tunnel coupling $t$ and detuning $\varepsilon$ ; a pulse of $J$ gives $\sqrt{\mathrm{SWAP}}$ or, with a Zeeman gradient, a CPHASE/CNOT.
Readout is spin-to-charge conversion: energy-selective (Elzerman) tunnelling or Pauli spin blockade maps spin onto a charge state sensed by a nearby SET or quantum point contact.
Isotopically enriched $^{28}$ Si removes the nuclear Overhauser field, pushing $T_2^*$ from $\sim 10$ ns in GaAs to tens of microseconds, with one- and two-qubit fidelities now above 99%.
Key idea: store the qubit in a single confined spin (small, electrically quiet), split it by the Zeeman effect, rotate it with ESR/EDSR, entangle neighbours with the tunable Heisenberg exchange $J$ , and read it out by converting spin into charge.

The core insight

Store the qubit in a single confined spin (small, electrically quiet), split it by the Zeeman effect, rotate it with ESR or EDSR, entangle neighbours through the electrically tunable Heisenberg exchange $J$ , and read it out by converting spin into charge.

Where this lives today

Spin qubits have graduated from proof-of-principle to small integrated processors. Industrial fabrication is now central: Intel has produced 300 mm-wafer silicon spin-qubit chips (the "Tunnel Falls" test vehicle) on the same lines that make classical CMOS, exploiting the platform's headline advantage — a dot pitch of tens of nanometres and full lithographic compatibility. Academic groups in Delft, Munich, Sydney and elsewhere have run six-qubit silicon arrays with all-electrical control, demonstrated germanium hole-spin qubits (which need no micromagnet because spin–orbit coupling is intrinsic), and shown "hot" operation above 1 K, which eases the punishing cryogenic power budget for control wiring. Two frontier ideas address scaling directly: coherent electron shuttling, in which the electron itself is conveyed across a chain of dots by a travelling potential while keeping its spin, and spin–photon coupling, in which a spin is hybridised with charge and coupled to a superconducting microwave resonator to import circuit-QED dispersive readout and long-range buses. The open problems are unglamorous but decisive: device-to-device uniformity, the interconnect bottleneck of routing one or more gates to every dot in a dense two-dimensional array, and fast high-fidelity readout — the practical obstacles between today's tens of qubits and the millions a fault-tolerant machine will need.

The physical picture

Think of each quantum dot as an artificial atom holding exactly one electron. The orbital sits in a deep electrostatic well built by the gates, but the spin is almost free — a tiny compass needle that a magnetic field aligns. Because the spin carries no electric dipole, the fluctuating electric fields that plague every charge-based qubit barely couple to it; this is the whole reason a spin can stay coherent while the charges around it jitter. A static field $B_0$ sets the qubit frequency through the Zeeman effect, and since the electron gyromagnetic ratio is $g\mu_B/h\approx 28$ GHz per tesla, a modest half-tesla field already puts the qubit in the convenient 10–20 GHz microwave band. The first figure lays out the device and that Zeeman ladder together.

Figure

A gate-defined double quantum dot: two electrostatic wells each hold one electron whose spin is the qubit. A static field B0 Zeeman-splits the levels by E_Z = gμ_B B0, and since gμ_B/h ≈ 28 GHz per tesla, B0 = 0.5 T gives a qubit frequency f0 ≈ 14 GHz. Lowering the barrier turns on the tunnel coupling t and hence the exchange J.

To make two spins talk, you lower the wall between their wells so the wavefunctions overlap. Pauli exclusion then forces the antisymmetric (singlet) and symmetric (triplet) spin combinations to have different energies, and that splitting is the exchange $J$ . Turning $J$ on and off with a voltage pulse is like turning an entangling knob: a calibrated pulse performs a $\sqrt{\mathrm{SWAP}}$ or a controlled-phase gate. But this same electrical control is the platform's Achilles' heel — $J$ depends steeply on the inter-dot detuning $\varepsilon$ , so the charge noise that a lone spin ignores comes roaring back during two-qubit gates. The saving grace is a sweet spot: at the symmetric point $\varepsilon=0$ the exchange curve is flat, $\partial J/\partial\varepsilon=0$ , so $J$ is first-order immune to detuning noise. The second figure is the quantity a device physicist actually tunes.

Figure

Exchange versus detuning, J(ε) = 4t²U ⁄ (U²−ε²), for t = 10 μeV and U = 1 meV. At the symmetric point ε = 0 the curve is flat (∂J ⁄ ∂ε = 0) with J = 0.4 μeV ≈ 97 MHz — the charge-noise sweet spot. Toward the charge transition ε → U it rises steeply and grows sensitive to detuning noise.

Reading a spin is a trick of conversion. Spin direction is nearly impossible to measure directly, but a single electron hopping on or off a dot is easy to sense with a nearby electrometer. By arranging that only one spin orientation is energetically allowed to tunnel — energy-selective Elzerman readout for a single spin, or Pauli spin blockade for two electrons — the un-measurable spin is transcribed into a measurable charge. Finally, silicon's superpower is isotopic purification: natural silicon carries $4.7\%$ $^{29}$ Si nuclei, each a tiny magnet that fluctuates the electron's Zeeman frequency, but enrichment to nearly spin-free $^{28}$ Si removes them, leaving the spin in a magnetically quiet vacuum and extending coherence by orders of magnitude over GaAs.

The core equations

The qubit itself is defined by the Zeeman interaction: a static field $B_0$ along the quantisation axis splits spin-up from spin-down, and the splitting sets the qubit frequency. This is the anchor from which every gate frequency and pulse time is measured.

H_Z=\tfrac{1}{2}\,g\,\mu_B\,B_0\,\sigma_z,\qquad \hbar\omega_0=g\,\mu_B\,B_0 \qquad (1)

where: $g$ — Landé $g$ -factor, $\approx 2$ in Si and $\approx -0.44$ in bulk GaAs; $\mu_B$ — Bohr magneton, $9.274\times10^{-24}\ \mathrm{J\,T^{-1}}$ ; $B_0$ — static field defining the quantisation axis; $\omega_0$ — Larmor (qubit) angular frequency; $\sigma_z$ — Pauli operator on the spin. Reading it: the qubit is a two-level system whose gap is directly tunable by the magnet, with $g\mu_B/h\approx 28$ GHz/T for a free-electron $g$ . Single-qubit rotations come from a transverse oscillating drive at the Larmor frequency. In electron spin resonance a real microwave magnetic field $B_1$ does the driving; the rotating-wave approximation strips it down to a static transverse field in the rotating frame, producing Rabi oscillations.

H_{\mathrm{drive}}(t)=\tfrac12 g\mu_B B_1\cos(\omega t)\,\sigma_x \ \xrightarrow{\ \text{RWA},\ \omega=\omega_0\ }\ \tfrac12\hbar\Omega_R\,\sigma_x,\qquad \Omega_R=\frac{g\mu_B B_1}{2\hbar} \qquad (2)

where: $B_1$ — amplitude of the oscillating drive field perpendicular to $B_0$ ; $\Omega_R$ — Rabi angular frequency, setting the single-qubit gate speed; $\omega$ — drive frequency, resonant at $\omega=\omega_0$ . Reading it: the gate rate scales linearly with drive amplitude, and a $\pi$ rotation ( $|\!\uparrow\rangle\to|\!\downarrow\rangle$ ) takes a time $t_\pi=\pi/\Omega_R$ . At scale, delivering a microwave magnetic field to each of a million dots is impractical, so electric-dipole spin resonance is preferred: an AC gate voltage shakes the electron back and forth, and a spatially varying magnetic field (from an on-chip micromagnet) makes that motion look like an oscillating field in the electron's own frame.

\hbar\Omega_{\mathrm{EDSR}}=\tfrac12\,g\mu_B\,\Big(\frac{\partial B_\perp}{\partial x}\Big)\,x_0,\qquad x_0=\frac{eE_{\mathrm{ac}}}{m^*\omega_0^{2}} \qquad (3)

where: $\partial B_\perp/\partial x$ — transverse field gradient from the micromagnet; $x_0$ — AC displacement of the electron driven by field $E_{\mathrm{ac}}$ ; $m^*$ — effective mass of the confined electron; $E_{\mathrm{ac}}$ — oscillating electric field from a gate. Reading it: the drive is now electrical (easy to route and to focus), and the gradient is what converts electric motion into a magnetic spin drive. Two-qubit logic uses the Heisenberg exchange, switched on by lowering the inter-dot barrier so the wavefunctions overlap. Its magnitude is set by the tunnel coupling $t$ , the on-site charging energy $U$ , and the detuning $\varepsilon$ between the dots — all electrically tunable.

H_{\mathrm{exch}}=J(\varepsilon)\,\mathbf{S}_1\!\cdot\!\mathbf{S}_2,\qquad J(\varepsilon)=\frac{4t^2 U}{U^2-\varepsilon^2}\ \xrightarrow{\ \varepsilon\to0\ }\ \frac{4t^2}{U} \qquad (4)

where: $J$ — exchange energy (the singlet–triplet splitting at the symmetric point); $t$ — inter-dot tunnel coupling; $U$ — on-site Coulomb (charging) energy; $\varepsilon$ — detuning; $\mathbf{S}_i$ — spin operator of electron $i$ . Reading it: $J$ is controlled either through the barrier ( $t$ ) or the detuning ( $\varepsilon$ ), and its flatness at $\varepsilon=0$ is the sweet spot of the previous figure. A pulse of exchange rotates the two-spin state; a swing of exchange phase $\pi/2$ produces the entangling $\sqrt{\mathrm{SWAP}}$ , and two of these interleaved with single-qubit rotations build a CNOT.

U_{\sqrt{\mathrm{SWAP}}}=\exp\!\Big(\!-\frac{i}{\hbar}\!\int_0^\tau\! J(t)\,\mathbf{S}_1\!\cdot\!\mathbf{S}_2\,\mathrm{d}t\Big),\qquad \int_0^\tau\! J\,\mathrm{d}t=\frac{\pi\hbar}{2} \qquad (5)

where: $\tau$ — exchange-pulse duration; the accumulated exchange phase $\theta=\int_0^\tau J\,\mathrm{d}t/\hbar=\pi/2$ gives $\sqrt{\mathrm{SWAP}}$ up to single-qubit phases. Reading it: the gate time trades against exchange strength, $\tau=\pi\hbar/(2J)$ for a square pulse — larger $J$ means faster, but noisier, gates. Coherence is the whole ballgame, and it splits into relaxation and dephasing. In GaAs the free-induction (Ramsey) time is set by the fluctuating nuclear Overhauser field; in $^{28}$ Si that field is largely gone.

\frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_\phi},\qquad \frac{1}{T_2^{*}}\Big|_{\mathrm{GaAs}}\sim\frac{g\mu_B\sqrt{\langle B_{\mathrm{nuc}}^2\rangle}}{\hbar} \qquad (6)

where: $T_1$ — spin relaxation (energy-decay) time, often ms–s in Si; $T_\phi$ — pure dephasing time; $T_2^{*}$ — free-induction dephasing time; $B_{\mathrm{nuc}}$ — fluctuating Overhauser field from host nuclear spins. Reading it: relaxation and dephasing add in the $T_2$ budget, and magnetic noise controls $T_2^{*}$ — which is why removing the nuclei matters so much.

The core derivations

Derivation AExchange from the two-electron double dot

The goal is to derive the exchange law $J(\varepsilon)=4t^2U/(U^2-\varepsilon^2)$ from a minimal two-site Hubbard model, so that the electrical tunability of two-qubit gates — and the existence of a charge-noise sweet spot at $\varepsilon=0$ — falls out as a theorem rather than an assertion. We model two dots (L and R) each holding one electron, allow an inter-dot tunnel coupling $t$ and an on-site Coulomb repulsion $U$ , and ask for the energy difference between the singlet and triplet spin states. Because only the singlet can virtually hop into the doubly occupied orbital, that virtual excursion lowers the singlet alone, and the resulting splitting is the exchange.

Step 1. Set up the Hilbert space. In the $(1,1)$ charge configuration (one electron per dot) the spin states are the singlet $|S\rangle$ and the three triplets $|T_0\rangle,|T_\pm\rangle$ . Tunnelling can move an electron across to make the doubly occupied $(0,2)$ singlet $|S(0,2)\rangle$ , which costs the charging energy $U$ and is shifted by the detuning $\varepsilon$ . Why: the $(0,2)$ excursion is the only door through which spin correlations acquire an energy cost, so the whole exchange lives in this small subspace.

Step 2. Restrict to the singlet subspace. In the basis $\{|S(1,1)\rangle,|S(0,2)\rangle\}$ the Hamiltonian is

H_S=\begin{pmatrix}0 & \sqrt{2}\,t\\ \sqrt{2}\,t & U-\varepsilon\end{pmatrix}.

Only the singlet couples to $(0,2)$ , because Pauli exclusion forbids two same-spin electrons from sharing the lowest orbital — the triplets cannot tunnel there at all. Why: this asymmetry is the physical origin of exchange; the triplets are inert, so any energy shift of the singlet immediately becomes a singlet–triplet gap.

Step 3. Lower the singlet by second-order perturbation theory in $t$ . For $t\ll U-\varepsilon$ the ground singlet energy shifts to

E_S\approx-\frac{2t^2}{U-\varepsilon},

while the triplet $|T_0\rangle$ is unshifted at this order. Why: the electron makes a virtual round trip to the far dot and back, and second-order perturbation theory is exactly the accounting of such virtual excursions.

Step 4. Symmetrise over both detuning directions. Virtual hops are possible into the $(0,2)$ and the $(2,0)$ configurations, so adding the $\varepsilon\to-\varepsilon$ partner gives

J=E_{T_0}-E_S=\frac{2t^2}{U-\varepsilon}+\frac{2t^2}{U+\varepsilon}=\frac{4t^2U}{U^2-\varepsilon^2}.

At $\varepsilon=0$ this reduces to $J=4t^2/U$ and, crucially, $\partial J/\partial\varepsilon=0$ : the symmetric operating point is first-order insensitive to detuning (charge) noise. Why: the even function of $\varepsilon$ has a flat minimum at the origin, which is precisely the sweet spot two-qubit gates are parked at. *The result reduces to the Heisenberg–Hubbard limit $J=4t^2/U$ at $\varepsilon=0$ and is an even function of $\varepsilon$ with vanishing first derivative at the origin — confirming the sweet-spot property.*

Go deeper · Derivation B · 1 minRabi flopping from a resonant transverse drive

Derivation BRabi flopping from a resonant transverse drive

The goal is to show that a single classical microwave field, tuned to the Zeeman frequency, drives coherent $|\!\uparrow\rangle\leftrightarrow|\!\downarrow\rangle$ oscillations at the Rabi rate $\Omega_R=g\mu_B B_1/2\hbar$ , which is the mechanism behind every ESR single-qubit gate. We start from the full time-dependent Hamiltonian of a spin in a static field plus a transverse oscillating field, move to the frame rotating at the drive frequency, and drop the fast counter-rotating term to expose a static effective field about which the spin precesses.

Step 1. Write the Hamiltonian as the Zeeman term plus a transverse drive, $H=\tfrac12\hbar\omega_0\sigma_z+\tfrac12 g\mu_B B_1\cos(\omega t)\,\sigma_x$ . Transform into the frame rotating at $\omega$ about $\hat z$ ; the static term becomes $\tfrac12\hbar(\omega_0-\omega)\sigma_z$ , which vanishes on resonance. Why: the rotating frame removes the fast Larmor precession, leaving only the slow dynamics the gate actually cares about.

Step 2. Decompose the drive into co- and counter-rotating parts, $\cos(\omega t)\,\sigma_x=\tfrac12(e^{i\omega t}+e^{-i\omega t})\,\sigma_x$ . In the rotating frame one component is static and the other oscillates at $2\omega$ ; discard the latter (the rotating-wave approximation). Why: the $2\omega$ term averages to nothing over a gate for any realistic drive, so keeping it would only add a tiny Bloch–Siegert correction.

Step 3. Read off the surviving static term $\tfrac12\hbar\Omega_R\sigma_x$ with $\Omega_R=g\mu_B B_1/2\hbar$ . It generates the rotation $e^{-i\Omega_R t\,\sigma_x/2}$ about $\hat x$ — coherent Rabi flopping between spin-up and spin-down at frequency $\Omega_R$ . Why: a static transverse field in the rotating frame is exactly a rotation generator, so on resonance the qubit simply precesses about $\hat x$ at the Rabi rate, and a pulse area $\Omega_R t_\pi=\pi$ inverts the spin. *This is the standard rotating-wave approximation result; the $\pi$ -pulse time $t_\pi=\pi/\Omega_R$ is used directly in the worked example below.*

Go deeper · Derivation C · 2 minOverhauser nuclear dephasing and the Ramsey time

Derivation COverhauser nuclear dephasing and the Ramsey time

The goal is to explain why the free-induction decay of a spin qubit is Gaussian rather than exponential and to pin the Ramsey time $T_2^{*}$ to the width of the nuclear field distribution — the calculation that quantifies why $^{28}$ Si is so much quieter than GaAs. The key physical input is that the host nuclear spins evolve slowly compared with a single experimental shot, so each run of a Ramsey sequence sees a fixed but random extra magnetic field, and the coherence we measure is an average over that static randomness.

Step 1. Treat the Overhauser field as quasi-static. Each shot adds a fixed detuning $\delta\omega=g\mu_B B_{\mathrm{nuc}}/\hbar$ , drawn from a Gaussian $P(\delta\omega)\propto e^{-\delta\omega^2/2\sigma_\omega^2}$ with width $\sigma_\omega=g\mu_B\sigma_B/\hbar$ , since the field is the sum of many independent nuclear spins. Why: the central-limit theorem makes the many-nucleus field Gaussian, and its slow evolution justifies treating it as frozen during one measurement.

Step 2. Average the accumulated Ramsey phase over the distribution:

\langle e^{i\delta\omega t}\rangle=\int \mathrm{d}(\delta\omega)\,P(\delta\omega)\,e^{i\delta\omega t}=e^{-\sigma_\omega^2 t^2/2}.

The Gaussian integral of the random phase gives a Gaussian — not exponential — coherence decay. Why: a static frequency spread dephases the ensemble quadratically in time, the hallmark of inhomogeneous (as opposed to Markovian) broadening.

Step 3. Identify the Ramsey time by matching to $\langle\sigma_x(t)\rangle=\exp[-(t/T_2^{*})^2]$ , giving

T_2^{*}=\frac{\sqrt{2}\,\hbar}{g\mu_B\,\sigma_B}.

In GaAs a field width $\sigma_B$ of a few millitesla yields $T_2^{*}\sim 10$ ns; in enriched $^{28}$ Si the residual $^{29}$ Si makes $\sigma_B$ smaller by roughly three orders of magnitude, so $T_2^{*}$ reaches tens of microseconds. Why: $T_2^{*}$ is inversely proportional to the field width, so removing nuclei by isotopic purification directly buys coherence.

Step 4. Note that the noise being quasi-static is what makes echo work. A Hahn-echo $\pi$ pulse refocuses a frozen detuning, so $T_2\gg T_2^{*}$ ; isotopic purification (removing the noise) and dynamical decoupling (refocusing it) are therefore complementary, not redundant. Why: only slowly varying noise can be refocused, and the Overhauser field's slowness is precisely why echo sequences recover so much coherence. *The Gaussian free-induction decay and the value $T_2^{*}\sim10$ ns in GaAs are consistent with the landmark two-spin experiments in GaAs quantum dots.*

Limiting cases

Spin-qubit physics lives between competing regimes set by the magnetic field, the host material, and the exchange–detuning relation.

Case	Result	Meaning
Symmetric exchange (low detuning)	$J\approx 4t^2U/(U^2-\varepsilon^2)\ \xrightarrow{\varepsilon\to0}\ 4t^2/U$	The sweet spot: $J$ set by barrier ( $t$ ) and charging ( $U$ ) with $\partial J/\partial\varepsilon=0$ — first-order immune to charge noise.
Detuned (charge-like) exchange	$J\approx 2t^2/(U-\varepsilon)\quad(\varepsilon\to U^-)$	Near the $(1,1)$ – $(0,2)$ transition $J$ grows steeply and becomes very sensitive to detuning noise; fast but noisy.
Singlet–triplet vs single-spin encoding	$\hbar\omega_{ST_0}=\sqrt{J^2+(g\mu_B\Delta B_z)^2}$	With a Zeeman gradient $\Delta B_z$ two electrons form an $S$ – $T_0$ qubit driven by $J$ ; without it one uses single-spin (Loss–DiVincenzo) encoding split by $g\mu_B B_0$ .
GaAs vs enriched $^{28}$ Si	$T_2^{}\sim10\ \mathrm{ns}\ (\mathrm{GaAs})\ \longrightarrow\ T_2^{}\gtrsim 10\text{–}100\ \mu\mathrm{s}\ (^{28}\mathrm{Si})$	Nuclear Overhauser dephasing dominates in GaAs; isotopic enrichment removes it, vastly extending coherence.

Common trapA long idle T2 does not mean high two-qubit fidelity

It is tempting to quote a spin qubit's beautiful idle $T_2^{*}$ or echo $T_2$ as the coherence figure and assume gates inherit it. They do not. The exchange $J$ used for two-qubit gates is steeply — near the charge transition, nearly exponentially — sensitive to the inter-dot detuning and barrier voltages, so the very charge noise that a dipole-free idle spin shrugs off becomes the leading error during the gate. That is why two-qubit operations, not idle memory, are almost always the fidelity bottleneck, and why devices are parked at the $\partial J/\partial\varepsilon=0$ sweet spot. A companion trap lives in GaAs: the headline "long $T_2$ " numbers presuppose dynamical decoupling that refocuses the quasi-static Overhauser field — the bare $T_2^{*}$ is a hundred times shorter. Always ask which coherence time, under which pulse sequence, and during which operation.

Worked examples

1 · A silicon qubit: frequency, Rabi rate, and gate time (fully worked)

A silicon spin qubit ( $g=2$ ) sits in a field $B_0=0.5$ T and is driven by an ESR field of amplitude $B_1=1$ mT. Find the qubit frequency, the Rabi frequency, and the time for a $\pi$ (bit-flip) gate.

Step 1. Qubit frequency from equation (1). The electron gyromagnetic ratio is $g\mu_B/h=2\times(9.274\times10^{-24})/(6.626\times10^{-34})=2.80\times10^{10}$ Hz/T, i.e. $\approx 28$ GHz/T. So $f_0=(28.0\ \mathrm{GHz/T})\times0.5\ \mathrm{T}=\mathbf{14.0\ GHz}$ — squarely in the microwave band, as intended.

Step 2. Rabi frequency from equation (2): $f_R=\Omega_R/2\pi=g\mu_B B_1/(2h)$ . Numerically $f_R=(9.274\times10^{-24}\times2\times10^{-3})/(2\times6.626\times10^{-34})$ . The drive is a thousand times weaker than $B_0$ , so this comes out a thousand times smaller than $f_0$ up to the factor of two: $f_R=\mathbf{14.0\ MHz}$ .

Step 3. A $\pi$ gate needs pulse area $\Omega_R t_\pi=\pi$ , so $t_\pi=1/(2f_R)=1/(2\times1.40\times10^7\ \mathrm{Hz})=\mathbf{35.7\ ns}$ . The qubit runs at 14 GHz, flops at 14 MHz, and flips in about 36 ns — a clean separation of drive from carrier that keeps the rotating-wave approximation excellent.

2 · An exchange two-qubit gate at the sweet spot (fully worked)

Two dots have tunnel coupling $t=10$ μeV and charging energy $U=1$ meV. Working at the symmetric point $\varepsilon=0$ , find the exchange $J$ , its frequency, and the $\sqrt{\mathrm{SWAP}}$ gate time. Confirm the sweet-spot condition.

Step 1. Exchange from equation (4) at $\varepsilon=0$ : $J=4t^2/U=4\times(10\ \mu\mathrm{eV})^2/(1000\ \mu\mathrm{eV})=4\times100/1000\ \mu\mathrm{eV}=\mathbf{0.40\ \mu eV}$ . As a frequency, $J/h=(0.40\times10^{-6}\times1.602\times10^{-19})/(6.626\times10^{-34})=\mathbf{97\ MHz}$ .

Step 2. $\sqrt{\mathrm{SWAP}}$ time from equation (5): the required exchange phase is $\theta=\pi/2$ , so for a square pulse $\tau=\pi\hbar/(2J)=1/(4\,f_J)=1/(4\times9.7\times10^7\ \mathrm{Hz})=\mathbf{2.6\ ns}$ — comparably fast to the single-qubit gate above.

Step 3. Check the sweet spot. Differentiating $J(\varepsilon)=4t^2U/(U^2-\varepsilon^2)$ gives $\partial J/\partial\varepsilon=8t^2U\varepsilon/(U^2-\varepsilon^2)^2$ , which is $\mathbf{0}$ at $\varepsilon=0$ . So a small detuning fluctuation $\delta\varepsilon$ shifts $J$ only at second order, $\delta J\propto(\delta\varepsilon)^2$ — the gate is first-order immune to charge noise. A 97 MHz exchange gives a 2.6 ns entangling gate that, run at $\varepsilon=0$ , is protected against the dominant detuning noise.

Physics of the portrait

Semiconductor spin qubits were born as a theory paper — a blueprint for taking a single confined electron and turning its spin into a quantum bit, complete with a prescription for one- and two-qubit gates and readout, years before any device met the bar.

Origins · 1 minDaniel Loss and how this was discovered

Daniel Loss

born 1958 · spin qubits and the Loss–DiVincenzo proposal

Daniel Loss, a Swiss theoretical physicist at the University of Basel, together with David P. DiVincenzo at IBM, published Quantum computation with quantum dots in Physical Review A in 1998 — the paper that founded the field. Their proposal was audacious in its concreteness: encode each qubit in the spin of one electron trapped in a gate-defined quantum dot, drive single-qubit rotations with local magnetic fields, and — the decisive idea — entangle neighbouring qubits by pulsing the Heisenberg exchange between their dots, an interaction that could be switched purely with gate voltages. The exchange gate they analysed, $\sqrt{\mathrm{SWAP}}$ , remains the archetype of two-qubit logic in the platform. Loss's broader work spans spintronics, spin–orbit physics, Majorana bound states, and the decoherence of solid-state qubits, and DiVincenzo is equally famous for the five-plus-two criteria that any quantum-computing hardware must satisfy — criteria the spin-qubit platform was designed to meet. For two decades the proposal was a target the experimentalists chased; the arrival of isotopically enriched silicon, micromagnet-driven EDSR, and above-99% gate fidelities has finally made the 1998 blueprint a working machine.

History

1998

— Loss and DiVincenzo propose spin qubits in quantum dots and the exchange gate

2005

— Petta et al. coherently manipulate two coupled electron spins and observe singlet–triplet oscillations in GaAs

2007

— Hanson, Kouwenhoven, Petta, Tarucha and Vandersypen review spins in few-electron quantum dots

2014

— a single-electron spin qubit is demonstrated in isotopically enriched silicon, pushing coherence into microseconds

2022

— three groups independently report silicon two-qubit gate fidelities above 99%, crossing the fault-tolerance threshold

Going further

Texts and reviews. Loss and DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57, 120 (1998) · Hanson, Kouwenhoven, Petta, Tarucha, Vandersypen, Spins in few-electron quantum dots, Rev. Mod. Phys. 79, 1217 (2007) · Petta et al., Coherent manipulation of coupled electron spins in semiconductor quantum dots, Science 309, 2180 (2005) · Burkard, Ladd, Pan, Nichol, Petta, Semiconductor spin qubits, Rev. Mod. Phys. 95, 025003 (2023) · Nielsen and Chuang, Quantum Computation and Quantum Information, §7 (physical realisations). Key numerical values used throughout: $\mu_B=9.274\times10^{-24}$ J/T; $g\approx2$ (Si), $g\approx-0.44$ (GaAs); gyromagnetic ratio $g\mu_B/h\approx28$ GHz/T; $^{29}$ Si natural abundance $4.685\%$ ; worked-example results $f_0=14.0$ GHz, $f_R=14.0$ MHz, $t_\pi=35.7$ ns; exchange results $J=0.40$ μeV $=97$ MHz, $\tau_{\sqrt{\mathrm{SWAP}}}=2.6$ ns; and the Gaussian-average integral giving $e^{-(\sigma_\omega t)^2/2}$ . No device-specific fidelity numbers are asserted beyond the standard above-99% threshold statement.

Prerequisites. The DiVincenzo Criteria · Spin and the Zeeman effect · Pauli exclusion and two-electron states. Leads to. Superconducting Qubits · Dispersive Readout · Trapped-Ion Qubits.

Ask the tutor: Ask me to walk through why lowering a single gate voltage entangles two spins, or to compare the coherence and gate-fidelity budgets of silicon spin qubits against superconducting and trapped-ion qubits.

Why put the qubit in the spin rather than the electron's charge or position?

Because charge is loud and spin is quiet. A charge qubit — say, an electron in the left versus right dot — has a large electric dipole, so it couples strongly to exactly the noise source that dominates in a gated semiconductor: fluctuating electric fields from trapped charges and gate voltage noise. Its coherence times are correspondingly short, often nanoseconds. A spin has no electric dipole at leading order, so those same fields barely see it, and coherence stretches to microseconds or beyond. The price is that spin is harder to control and to read out, which is why the platform spends so much of its engineering on EDSR drives, micromagnets, and spin-to-charge conversion. It is a deliberate trade: give up easy control to buy a quiet qubit.

What is the difference between ESR and EDSR, and why prefer EDSR at scale?

Electron spin resonance drives the spin with a real oscillating magnetic field $B_1$ at the Larmor frequency — conceptually clean, but delivering a strong, localised AC magnetic field to each of a million dots is an engineering nightmare, since magnetic fields are hard to confine and to switch quickly. Electric-dipole spin resonance sidesteps this: an AC gate voltage physically shakes the electron, and a magnetic-field gradient (from an on-chip micromagnet) or intrinsic spin–orbit coupling turns that motion into an effective oscillating field in the electron's frame. The drive is now purely electrical — easy to route, focus, and multiplex with the same gate lines used for everything else. That is why nearly every scalable silicon and germanium architecture uses EDSR rather than ESR.

Why is the exchange interaction electrically tunable at all?

Because exchange is a consequence of wavefunction overlap, and overlap is set by voltages. When two electrons in adjacent dots can virtually hop onto each other's site, Pauli exclusion allows the singlet to make that excursion but forbids the triplet, so the singlet is pushed down in energy — that singlet–triplet gap is the exchange $J$ . How much the electrons overlap depends on the height of the tunnel barrier (a gate voltage) and on the detuning between the two dots (another gate voltage), so $J$ is a knob you turn with the same electrodes that define the dots. Equation (4) makes this explicit, and it is the reason two-qubit gates can be pulsed on nanosecond timescales — but also why charge noise on those same gates limits their fidelity.

What exactly is the sweet spot, and does it eliminate charge noise?

The sweet spot is the symmetric operating point $\varepsilon=0$ where the exchange curve $J(\varepsilon)$ is flat, $\partial J/\partial\varepsilon=0$ . There a small detuning fluctuation $\delta\varepsilon$ from charge noise changes $J$ only at second order, $\delta J\propto(\delta\varepsilon)^2$ , rather than first order — so the gate is far more robust than at a detuned bias. It does not eliminate charge noise: barrier-voltage noise still modulates $t$ and hence $J$ , and second-order detuning sensitivity is nonzero. But operating symmetrically and controlling $J$ through the barrier ("symmetric" or "baseband" exchange control) was the key advance that pushed two-qubit fidelities across 99%. The sweet spot is a suppression, not a cure.

How does reading a single electron's charge tell you its spin?

Through spin-to-charge conversion, which turns an un-measurable quantity into a measurable one. In energy-selective (Elzerman) readout you tune the dot so the reservoir's Fermi level sits between the spin-up and spin-down electrochemical potentials; then only a spin-up electron has enough energy to tunnel out to the reservoir (with a fresh spin-down tunnelling back in), while spin-down stays put. A nearby charge sensor — a single-electron transistor or quantum point contact — sees the transient change in dot occupation as a blip in current. In Pauli spin blockade, used for two-electron qubits, tunnelling into a doubly occupied state is allowed for a singlet but Pauli-forbidden for a triplet, so the spin state gates a charge movement. Either way, the sensor never measures spin directly; it measures a charge that spin has been made to control.

Silicon versus GaAs — is silicon simply better?

For coherence, yes, and decisively so: GaAs has no spin-free isotope (every gallium and arsenic nucleus carries spin), so the Overhauser field gives $T_2^{*}\sim10$ ns and caps memory times, whereas isotopically enriched $^{28}$ Si removes almost all nuclear spins and stretches $T_2^{*}$ into the tens of microseconds. Silicon also has the priceless advantage of CMOS compatibility. GaAs is not obsolete, though — its clean, well-controlled heterostructures and strong, uniform confinement made it the workhorse for the foundational physics (the 2005 two-spin experiments were GaAs), and it remains valuable for fast prototyping and for studies where the nuclear bath is itself the object of interest. But for a scalable, coherent processor, enriched silicon (and increasingly germanium) is where the field has moved.