Quantum Dots
A quantum dot is a piece of semiconductor made small enough, in all three directions, that an electron trapped inside it can no longer pretend it lives in a bulk crystal. Where the bulk offers continuous energy bands, the dot offers a discrete ladder of levels — the same quantization a real atom shows, produced here not by the Coulomb pull of a nucleus but by a potential the experimenter builds by hand. In a gate-defined dot that potential is purely electrostatic: metal electrodes patterned on the surface of a heterostructure are biased to deplete a buried two-dimensional electron gas, squeezing what is left into a puddle small enough to hold a single, countable electron. Dial the voltages and you dial the atom — its size, its level spacing, how many electrons it holds, and how strongly it talks to its neighbour. The point of all this control is a qubit, and the qubit is not the artificial atom but one property of one electron inside it: the orientation of its spin. In a magnetic field the two spin states split by the Zeeman energy, and because a spin barely notices the electric fields that rattle every solid-state device, it remembers its phase far longer than a charge would. This is the 1998 Loss–DiVincenzo proposal, which showed that spins in coupled dots can satisfy every DiVincenzo criterion at once: a scalable register, initialization by spin-selective tunneling, single-qubit rotations by resonant drive, a two-qubit gate from the exchange interaction, long coherence relative to gate times, and readout by converting spin into charge. The exchange interaction is the crux — lower the wall between two dots, let the wavefunctions overlap, and Pauli antisymmetry splits singlet from triplet by an energy J you can switch on and off. That same antisymmetry, read the other way, is how you measure the spin at all.
- A quantum dot confines charge carriers in all three dimensions on the scale of tens of nanometres, collapsing the bulk energy bands into a discrete, atom-like ladder — an "artificial atom" whose size, occupation, and couplings are set by gate voltages.
- The qubit is the spin of a single trapped electron: its up and down states split by the Zeeman energy E_Z = gμ_B B, and its weak coupling to electric noise gives long coherence (the Loss–DiVincenzo spin qubit).
- Single-qubit gates are resonant spin rotations — magnetic (ESR) or, via a gradient or spin–orbit field, electric (EDSR); two-qubit gates come from the exchange interaction J, tunable through the inter-dot tunnel barrier.
- Readout is spin-to-charge conversion: energy-selective tunneling or Pauli spin blockade maps the spin onto a charge configuration that a nearby electrometer senses.
- Isotopically enriched ²⁸Si removes the nuclear-spin bath that dephases electrons in GaAs, and silicon dots now reach single- and two-qubit gate fidelities above 99%, in the fault-tolerant regime; the open problems are charge noise, valley splitting, and scaling the control wiring.
- Key idea: confine a single electron in an electrostatically defined semiconductor well so its spin becomes a qubit; drive it with magnetic or electric resonance, couple two qubits through the tunable exchange interaction, and read it out by converting spin into a measurable charge configuration.
Confine a single electron in an electrostatically defined semiconductor potential well so its spin becomes a qubit; drive it with magnetic or electric resonance, couple two qubits through the tunable exchange interaction, and read out by converting spin into a measurable charge configuration.
Where this lives today
Quantum-dot spin qubits are now among the most credible candidates for a large-scale quantum processor, and the reason is written into the silicon. Isotopically enriched ²⁸Si removes the nuclear-spin bath that limited the pioneering GaAs devices, and Si/SiGe and Si-MOS dots have pushed single-qubit gate fidelities past 99.9% and two-qubit CZ fidelities above 99% — into the regime relevant for surface-code fault tolerance. The strategic argument is manufacturing: a spin qubit is, structurally, a transistor operated one electron at a time, so the same CMOS foundries that print billions of transistors could in principle print qubits and their control electronics side by side. Several groups and companies (Intel, Delft's QuTech, UNSW, RIKEN) are now demonstrating small two-dimensional dot arrays with shared control and cryo-CMOS wiring. The live problems are equally concrete: charge noise, which enters through the exponentially barrier-sensitive exchange; valley splitting in silicon, where a near-degenerate second conduction-band minimum must be pushed away to leave a clean two-level system; device-to-device variability from atomic-scale disorder; and the "wiring bottleneck" of routing one or more gate lines to every dot in a dense array. Two research directions attack the last problem: exchange-only and singlet–triplet encodings that trade microwave control for all-electrical exchange pulses, and spin–photon coupling in superconducting resonators (circuit QED for spins) that lets distant dots talk through a microwave photon rather than nearest-neighbour tunneling — borrowing the dispersive-readout machinery developed for superconducting qubits. Germanium hole dots, with strong intrinsic spin–orbit coupling, add fast all-electrical control without micromagnets. The open question is no longer whether a spin qubit works, but whether the artificial atom can be printed by the million.
The physical picture
Picture squeezing a guitar string from both ends and from above and below until only its fundamental tone fits. Confine an electron that tightly in all three directions and its allowed energies stop being a continuum and become a discrete ladder — and because the confinement is built from voltages rather than fixed by a nucleus, the ladder is yours to tune. Two energy scales govern the artificial atom. The first is the confinement (or orbital) spacing, set by how hard you squeeze: it is the energy gap between the ladder's rungs. The second is the charging energy, the electrostatic cost of forcing one more electron onto a small object already holding charge. When both dwarf the thermal energy k_BT, the electron number is sharp and you can load exactly one. The first figure fixes this picture and pins the numbers: for a GaAs dot a few tens of nanometres across, both energies land at a few meV, thousands of times the millikelvin thermal scale.
The artificial atom is the stage; the qubit is one actor on it. Apply a magnetic field and the electron's spin splits into two levels separated by the Zeeman energy — a tiny compass needle that can point up, down, or any quantum superposition, ticking at a frequency of tens of gigahertz. Crucially the Zeeman splitting is a thousand times smaller than the orbital spacing: the qubit lives in the fine structure, well below the atom's gross structure, which is why it is so weakly disturbed by the electric noise that shakes the orbitals. Two-qubit logic then exploits fermion antisymmetry. Lower the wall between two dots so the electron wavefunctions overlap, and the Pauli principle forces the spin-singlet and spin-triplet arrangements to different energies; that difference, the exchange energy J, is the rotation rate of a two-qubit gate. The second figure shows the double dot, the spin ladder, and the exchange handle, with the real numbers: a 14 GHz qubit at half a tesla and a √SWAP gate in about 25 nanoseconds.
The core equations
The starting point is the confinement itself. Model the dot crudely as a hard-wall box of width L; the electron is a standing wave whose allowed wavenumbers are quantized, and the resulting energies set the meV orbital scale that makes the dot an atom. The essential lesson is the scaling, not the exact shape of the well.
where: E_n — confinement energy of the n-th level; ħ — reduced Planck constant; m* — effective mass of the carrier in the host (0.067 mₑ in GaAs, ≈ 0.2 mₑ in silicon); L — confinement length; n — the level index. Reading it: the spacing scales as 1/L², so shrinking the dot pushes the rungs apart — quantum confinement is a size effect, and tens of nanometres is small enough to matter. Adding an electron to a small charged object costs electrostatic energy, and this charging energy is what makes the electron number discrete. It is the second gatekeeper, alongside confinement, of single-electron control.
where: E_C — single-electron charging energy; e — elementary charge; C_Σ — total capacitance of the dot to all nearby electrodes and gates. Reading it: a small dot has a small capacitance and hence a large E_C; when E_C ≫ k_BT the dot is in Coulomb blockade and admits electrons one at a time, which is how you tune down to a single occupant. The qubit is defined the moment a magnetic field lifts the spin degeneracy. The two spin states split by the Zeeman energy, and the qubit's transition frequency is E_Z/h — the frequency your control electronics must hit.
where: E_Z — Zeeman splitting between spin-up and spin-down; g — effective Landé g-factor (≈ 2 in silicon, ≈ −0.44 in bulk GaAs); μ_B — Bohr magneton, 9.274 × 10⁻²⁴ J/T; B — applied field. Reading it: for g = 2 the qubit frequency is 28 GHz per tesla, so a fraction of a tesla puts the qubit in the convenient microwave band — and comfortably below the orbital scale. Two-qubit logic lives in the exchange interaction. When two singly-occupied dots are tunnel-coupled, the low-energy spin physics reduces to a Heisenberg coupling whose strength J is set by virtual hopping between the dots and is exponentially sensitive to the barrier.
where: J(ε) — exchange energy, tunable through the barrier or the inter-dot detuning ε; σ_i — the vector of Pauli operators for spin i; t — inter-dot tunnel coupling; U — on-site Coulomb repulsion. Reading it: J splits singlet from triplet; because t depends exponentially on barrier height, J swings over orders of magnitude with a small voltage change — fast gates, but also a fast route for charge noise to enter. Pulsing the exchange for a controlled time is a two-qubit gate. The canonical exchange gate is the square-root-of-swap, generated when the integrated exchange reaches a quarter-turn of the singlet–triplet phase.
where: τ — the exchange pulse length; the pulse-area condition ∫J dt = πħ/2 sets the gate; two √SWAP in series make a full SWAP. Reading it: combined with single-qubit rotations, √SWAP completes a universal gate set — this is the DiVincenzo two-qubit requirement, delivered by a voltage pulse. Finally, coherence. A slowly fluctuating qubit frequency — from nuclear spins or charge noise — dephases the qubit over a time T₂*, which dynamical decoupling can extend to a longer T₂.
where: T₂ — inhomogeneous (free-induction) dephasing time; T₂ — coherence under Hahn or CPMG decoupling; δf — the low-frequency spread in qubit frequency set by the noise bath; ∂B/∂f — the sensitivity of the qubit frequency to that noise. Reading it: quiet the bath (enrich to ²⁸Si, park at charge-noise sweet spots) and T₂ lengthens; refocus the slow drift with pulses and you recover a much longer T₂.
The goal is to show why two electrons sitting one-per-dot acquire a singlet–triplet energy splitting J, and why that splitting is electrically tunable — because J is the entire resource for two-qubit logic in spin qubits, and its exponential dependence on the barrier is what makes the gates both fast and noise-sensitive. We model the coupled dots as a two-site fermionic Hubbard system, note that Pauli exclusion lets only the singlet hybridize with the doubly-occupied configuration, diagonalize the resulting two-by-two block, and read off J in the strong-repulsion limit.
Step 1. Write the two-site Hamiltonian H = Σ_iσ ε_i n_iσ − t Σ_σ (c†_{1σ}c_{2σ} + h.c.) + U Σ_i n_{i↑}n_{i↓}, with tunnel coupling t and on-site repulsion U. The two triplet states, being symmetric in spin, are antisymmetric in space and so cannot place both electrons in the same orbital; they do not couple to any doubly-occupied state and sit unshifted at energy E_T = 0. Why: Pauli exclusion is doing the sorting here — it is precisely the states that could lower their energy by double occupation that are forbidden from it, so the singlet and triplet fates diverge.
Step 2. The singlet does hybridize. In the basis of the one-per-dot singlet (1,1) and the doubly-occupied singlet (2,0), the Hamiltonian block is H_S = [[0, −√2 t],[−√2 t, U−V]], where V is the inter-dot Coulomb shift and the √2 is the fermionic matrix element for tunneling into an occupied orbital. Why: virtual excursions to the (2,0) configuration are the mechanism — the singlet borrows a little double-occupancy and is pushed down in energy, exactly the superexchange idea.
Step 3. Diagonalize the block. The lower eigenvalue is E_S = (U−V)/2 − √(4t² + ((U−V)/2)²), and since the triplet stays at E_T = 0 the splitting is J = E_T − E_S = √(4t² + ((U−V)/2)²) − (U−V)/2. Why: this exact form is valid at any detuning and shows J is always positive (singlet below triplet) — the ground state of two exchange-coupled electrons is a singlet, as it must be.
Step 4. Take the strong-repulsion limit U − V ≫ t, expanding the square root: J → 4t²/(U−V) → 4t²/U. Because t falls off exponentially with barrier height, so does J, giving orders-of-magnitude electrical tunability. Why: this is the workhorse formula for gate design — it says the coupling is a second-order tunneling process, quadratic in t and inversely proportional to the charging cost, and it is why raising the barrier a little switches the gate fully off. *Verified: standard Hubbard result; reduces to 4t²/U in the symmetric large-U limit, matching Loss and DiVincenzo (1998) and Hanson et al., Rev. Mod. Phys. 79, 1217 (2007).*
Go deeper · Derivation B · 2 minPauli spin blockade as a spin-to-charge readout
The goal is to explain how the same fermion antisymmetry that produces exchange also lets you measure a spin you cannot detect directly, by turning a two-spin state into a charge configuration a nearby electrometer can read. We arrange two electrons across a double dot at a detuning where the charge transition (1,1) → (2,0) is energetically allowed, then show that only the spin-singlet is permitted to make that transition while the triplets are blocked — so the presence or absence of a charge movement reports the spin.
Step 1. Write the spin symmetries: the singlet |S⟩ = (|↑↓⟩ − |↓↑⟩)/√2 is antisymmetric under exchange, while the three triplets (e.g. |T₀⟩ = (|↑↓⟩ + |↓↑⟩)/√2) are symmetric. This distinction is not about energy yet — it is about the exchange symmetry of the two-electron wavefunction. Why: everything downstream follows from this single fact, because the total wavefunction of two electrons must be antisymmetric, tying spin symmetry to spatial symmetry.
Step 2. Apply the antisymmetry constraint to the (2,0) target. Putting both electrons in the same dot's lowest orbital makes the spatial part symmetric, so the spin part must be antisymmetric — a singlet. The (2,0) triplet would require promoting one electron to an excited orbital and sits far higher in energy. Why: this is the blockade — the only accessible (2,0) state is a singlet, so charge can complete the transition only if the incoming (1,1) state is also a singlet.
Step 3. Read the charge. A (1,1) singlet tunnels to (2,0) and moves one electron; a (1,1) triplet cannot and the charge stays put. A capacitively coupled sensor — a quantum point contact or a single-electron transistor — sees two distinct conductance values, so "singlet" and "triplet" become two measurable currents. Why: this completes spin-to-charge conversion, satisfying the DiVincenzo measurement criterion with a mechanism that needs no magnetic detector, only an electrometer.
Step 4. Run it backwards to initialize. Relaxing into the (2,0) charge ground state and then separating the electrons deterministically prepares a (1,1) singlet — a fast, high-fidelity reset. Why: the same blockade that reads the spin also loads a known state, so one physical mechanism covers two of the DiVincenzo criteria at once. *Verified: mechanism established by Ono et al., Science 297, 1313 (2002); reviewed in Hanson et al., Rev. Mod. Phys. 79, 1217 (2007).*
Go deeper · Derivation C · 2 minWhy pulsing the exchange builds a two-qubit gate
The goal is to turn the Heisenberg Hamiltonian into an actual entangling gate and to identify the two useful operating regimes — the exchange-dominated √SWAP and the Zeeman-dominated controlled-phase — because these are the two ways every spin-qubit processor entangles, and knowing which regime you are in tells you what pulse you must apply. We evolve under the exchange coupling, use the identity that relates σ₁·σ₂ to the SWAP operator, and then add a Zeeman difference to see the controlled-phase emerge.
Step 1. Use σ₁·σ₂ = 2·SWAP − 1, so the exchange evolution exp(−i (φ/4) σ₁·σ₂), with accumulated phase φ = ∫J dt/ħ, is (up to a global phase) exp(−i(φ/2)·SWAP). Swapping is generated directly by exchange. Why: this identity is the bridge from a physical spin–spin coupling to a logic operation — the exchange literally rotates the two spins into and through a swap.
Step 2. Choose the pulse area. At ∫J dt = πħ/2 the operation is the square root of SWAP; a second identical pulse completes the full SWAP. Combined with single-qubit rotations, √SWAP generates the whole two-qubit group, so this plus local control is universal. Why: √SWAP is the natural exchange gate because exchange gives you SWAP powers directly — you stop the pulse halfway to get the entangling √SWAP rather than the non-entangling SWAP.
Step 3. Now add a large Zeeman difference ΔE_z between the dots (J ≪ ΔE_z). The transverse parts of σ₁·σ₂ are detuned away and only the σ_z^1 σ_z^2 term survives, leaving H ≈ (J/4) σ_z^1 σ_z^2 on top of the individual Zeeman terms — an Ising coupling that generates a controlled-phase (CZ) gate. Why: with individually resolved qubit frequencies, the exchange acts as a pure conditional phase, which is exactly the CZ that surface-code architectures want, and it is the high-fidelity regime silicon processors run in.
Step 4. Note the idle requirement. Because J ≈ 4t²/U is exponentially barrier-sensitive, any residual J between gates couples idle qubits and converts charge noise into phase error; the barrier must be raised to J ≈ 0 when the qubits are meant to be free. Why: an always-on coupling is not a feature but a leak — the same tunability that makes the gate fast makes a stray exchange a decoherence channel. *Verified: √SWAP from the Heisenberg exchange and the CZ limit match Loss and DiVincenzo (1998); the Ising reduction in the Zeeman-dominated regime is the standard CZ mechanism of Burkard et al., Rev. Mod. Phys. 95, 025003 (2023).*
Limiting cases
The behaviour of a coupled-dot spin qubit is set by the ratio of the exchange J to the Zeeman difference ΔE_z between the two dots — the table reads off the three regimes and the idle condition.
| Case | Result | Meaning |
|---|---|---|
| Exchange-dominated, J ≫ ΔE_z | H ≈ (J/4) σ₁·σ₂ | Singlet–triplet eigenstates; pulsing J generates √SWAP. The natural regime for singlet–triplet and exchange-only encodings. |
| Zeeman-dominated, J ≪ ΔE_z | H ≈ ½(E_{z,1}σ_z¹ + E_{z,2}σ_z²) + (J/4)σ_z¹σ_z² | Effective σ_zσ_z (Ising) coupling with individually addressable spins — the standard regime for high-fidelity CZ gates. |
| Idle, J → 0 | H ≈ ½ Σ_i E_{z,i} σ_z^i | Barrier raised, wavefunction overlap negligible, qubits decoupled. Required between gates to avoid always-on coupling and charge-noise phase errors. |
| Strong confinement, L ≪ λ_dB | E_n large, well separated | Deep quantum-dot limit: a single orbital is occupied and the artificial atom is clean; the opposite limit L ≫ λ_dB recovers a bulk 2DEG with no quantization. |
Worked examples
1 · The two energy scales of a GaAs dot (fully worked)
A gate-defined dot in GaAs (effective mass m = 0.067 mₑ, relative permittivity ε_r = 12.9) has a confinement length L ≈ 30 nm and an effective radius r ≈ 30 nm. Estimate its ground-state confinement energy, its orbital level spacing, and its charging energy, and compare all three with the thermal energy at 100 mK.*
Step 1. Confinement energy from equation (1) with n = 1: E₁ = ħ²π²/(2mL²). With m = 0.067 × 9.109 × 10⁻³¹ = 6.10 × 10⁻³² kg and L = 3.0 × 10⁻⁸ m, E₁ = (1.055 × 10⁻³⁴)² × π² / (2 × 6.10 × 10⁻³² × (3.0 × 10⁻⁸)²) = 9.99 × 10⁻²² J = 6.2 meV.
Step 2. Orbital level spacing. The gap to the next level is E₂ − E₁ = (4 − 1)E₁ = 3E₁ = 18.7 meV. (A real gate-defined dot has a softer, roughly harmonic potential, so its spacing is typically a milder 1–5 meV; the hard-wall box sets the correct scale and the correct 1/L² trend.)
Step 3. Charging energy. Modeling the dot as a conducting disk of radius r in the semiconductor, C_Σ ≈ 8ε₀ε_r r = 8 × 8.854 × 10⁻¹² × 12.9 × 3.0 × 10⁻⁸ = 2.74 × 10⁻¹⁷ F, so E_C = e²/C_Σ = (1.602 × 10⁻¹⁹)²/(2.74 × 10⁻¹⁷) = 9.4 × 10⁻²² J = 5.8 meV.
Step 4. Compare with the thermal scale. At T = 100 mK, k_BT = 1.381 × 10⁻²³ × 0.1 = 1.38 × 10⁻²⁴ J = 0.0086 meV. So E_C/k_BT ≈ 680 and E₁/k_BT ≈ 720 — both energy scales exceed the thermal energy by nearly three orders of magnitude. The electron number is sharply quantized and the orbital ladder is fully resolved: the dot is a clean artificial atom, and one can load exactly one electron.
2 · Qubit frequency and exchange-gate time (fully worked)
A single-electron spin qubit sits in a silicon dot (g ≈ 2) at B = 0.5 T. Find its Larmor frequency and Zeeman energy. Then, tuning up an exchange J with J/h = 10 MHz between two such dots, find the duration of a √SWAP gate, and check that this J is consistent with a tunnel coupling t/h = 5 GHz and charging U = 1 meV.
Step 1. Zeeman energy from equation (3): E_Z = gμ_B B = 2 × 9.274 × 10⁻²⁴ × 0.5 = 9.27 × 10⁻²⁴ J = 57.9 μeV.
Step 2. Larmor (qubit) frequency: f = E_Z/h = 9.27 × 10⁻²⁴ / 6.626 × 10⁻³⁴ = 14.0 GHz — a convenient microwave frequency, and note it is only 57.9 μeV, about 1/100 of the meV orbital scale, confirming the qubit lives well below the artificial-atom structure.
Step 3. √SWAP time from equation (5): the pulse-area condition ∫J dt = πħ/2 with constant J gives τ = πħ/(2J). Writing J = h × (J/h) = 6.626 × 10⁻³⁴ × 10⁷ = 6.63 × 10⁻²⁷ J, τ = π × 1.055 × 10⁻³⁴ / (2 × 6.63 × 10⁻²⁷) = 25 ns (equivalently τ = 1/(4 × J/h) = 1/(4 × 10⁷ s⁻¹)).
Step 4. Consistency check via J ≈ 4t²/U. With t = h × 5 × 10⁹ and U = 1 meV = 1.602 × 10⁻²² J, J = 4t²/U gives J/h = 4h(5 × 10⁹)²/U = 4 × 6.626 × 10⁻³⁴ × 2.5 × 10¹⁹ / 1.602 × 10⁻²² ≈ 414 MHz — comfortably above the 10 MHz we wanted, so operating at J/h = 10 MHz means sitting well inside the tunable range (a modestly raised barrier reduces t and brings J down to the target). Qubit frequency 14 GHz, √SWAP in 25 ns, with J tunable from hundreds of MHz down to the idle point by the barrier gate.
Physics of the portrait
The spin qubit in a quantum dot is one man's answer to a question the whole field was asking in the mid-1990s — where, in a real solid, do you put a qubit that is both controllable and long-lived? The answer, spin, and the machine to run it, coupled dots, came from Daniel Loss.
Origins · 1 minDaniel Loss and how this was discovered
Daniel Loss is a Swiss theoretical physicist, professor at the University of Basel and a founding figure of solid-state quantum information. His 1998 paper with David DiVincenzo, "Quantum computation with quantum dots," is the blueprint on which the entire field of semiconductor spin qubits rests. At a moment when quantum computing had Shor's algorithm and an abstract theory of gates but no agreed physical qubit that could scale, Loss and DiVincenzo made a specific, buildable proposal: encode the qubit in the spin of a single electron confined in a gate-defined quantum dot, rotate it with local magnetic fields, and — the decisive idea — entangle neighbouring qubits by pulsing the Heisenberg exchange interaction on and off through the inter-dot tunnel barrier. Every element of that scheme, from spin-to-charge readout to the exponential tunability of exchange, became a research programme in its own right, and the proposal reads today almost as a checklist of the DiVincenzo criteria it helped crystallize. Loss's later work ranged widely across the theory of decoherence in spin qubits, the hyperfine problem that pushed the field from GaAs toward isotopically enriched silicon, spin–orbit and hole-spin qubits, and topological superconductivity and Majorana modes. But it is the coupled-dot spin qubit — the artificial atom whose spin you compute with — that carries his name into every laboratory now chasing a silicon quantum processor.
Going further
Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, §7 (physical realizations) · Loss and DiVincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120 (1998) · Hanson, Kouwenhoven, Petta, Tarucha and Vandersypen, "Spins in few-electron quantum dots," Rev. Mod. Phys. 79, 1217 (2007) · Zwanenburg et al., "Silicon quantum electronics," Rev. Mod. Phys. 85, 961 (2013) · Burkard, Ladd, Pan, Nichol and Petta, "Semiconductor spin qubits," Rev. Mod. Phys. 95, 025003 (2023).
Provenance. The confinement and charging formulas are standard (particle-in-a-box and the capacitive Coulomb-blockade model); the exchange derivation reproduces the two-site Hubbard result J → 4t²/U, and the Pauli-spin-blockade logic follows from fermion antisymmetry (Ono et al., Science 297, 1313, 2002). A verifier should: (1) reproduce E₁ ≈ 6.2 meV, 3E₁ ≈ 18.7 meV, and E_C ≈ 5.8 meV for the GaAs numbers of Worked Example 1 and confirm all exceed k_BT ≈ 0.0086 meV at 100 mK; (2) confirm E_Z = 57.9 μeV and f = 14.0 GHz for g = 2 at 0.5 T, and τ = 25 ns for the √SWAP at J/h = 10 MHz, in Worked Example 2; (3) reproduce the Hubbard diagonalization J = √(4t² + ((U−V)/2)²) − (U−V)/2 → 4t²/(U−V); (4) check the √SWAP and CZ limits of the exchange gate. Constants: μ_B = 9.274 × 10⁻²⁴ J/T (CODATA); bulk GaAs g ≈ −0.44, silicon g ≈ 2. The above-99% fidelity statements reflect Burkard et al. (2023) and refer to isotopically enriched silicon devices.
Prerequisites. DiVincenzo Criteria · Pauli exclusion and spin-statistics · Zeeman effect. Leads to. Superconducting Qubits · Dispersive Readout · Circuit Quantum Electrodynamics.
Ask the tutor: Ask me to run a specific two-electron input through Pauli spin blockade, or to compare the √SWAP and controlled-phase exchange regimes step by step, and I will take it one line at a time.
Why use the electron's spin as the qubit instead of its charge or position?
Because spin is quiet. The dominant noise in a solid-state device is electrical — fluctuating charges in the oxide and near the gates jiggle every electric potential the electron feels. A charge qubit (which state is defined by which dot the electron sits in) couples directly to that noise and dephases quickly. A spin barely notices electric fields to first order: it responds to magnetic fields, and there is far less magnetic noise around. So the spin remembers its phase orders of magnitude longer. The price is that spins are also hard to talk to and hard to read — which is why the field invented electric-dipole spin resonance to drive them electrically and spin-to-charge conversion to read them out, borrowing the charge channel only when needed.
What is the difference between the orbital energy and the qubit energy?
They are two completely different scales that happen to live in the same object. The orbital (confinement) energy is the spacing of the artificial atom's ladder — a few meV, set by how tightly the dot is squeezed and by the charging energy. The qubit energy is the Zeeman splitting of a single spin — tens of μeV at typical fields, about a thousand times smaller. The orbital structure governs how you load electrons and how strongly EDSR can mix spin with motion, but the computation happens entirely in the tiny Zeeman-split spin manifold. Mixing the two up gives gate frequencies and coherence estimates that are wrong by orders of magnitude, which is why keeping the scales separate is the first thing to get right.
How does lowering a barrier between two dots create a two-qubit gate?
When the barrier is high, the two electron wavefunctions barely overlap and the spins ignore each other. Lower the barrier and the electrons can virtually hop into each other's dot; Pauli exclusion then forces the singlet and triplet spin arrangements to different energies, an exchange splitting J ≈ 4t²/U. That energy difference makes the two-spin state precess — a controlled rotation. Pulse J on for the right time (∫J dt = πħ/2) and you get a √SWAP; in the regime where the two dots have different qubit frequencies, the same exchange acts as a controlled-phase gate instead. Raise the barrier again and J switches off, because t depends exponentially on barrier height. The barrier voltage is, quite literally, the two-qubit gate knob.
Why does isotopic purification to ²⁸Si help so much?
Natural silicon is about 4.7% ²⁹Si, whose nucleus carries spin-1/2. Those nuclear spins produce a small, randomly fluctuating magnetic field through the hyperfine interaction, and that random field shifts the electron's Zeeman frequency from shot to shot, dephasing it. Enriching to nearly pure ²⁸Si (spin-zero) removes essentially the entire bath, extending T₂* from microseconds to often hundreds of microseconds or more. GaAs cannot escape this problem — every gallium and arsenic isotope carries nuclear spin, so there is no spinless version to purify to. That single materials fact is the main reason the field migrated from GaAs, where the physics was pioneered, to silicon, where the coherence lives long enough for fault-tolerant gates.
What is valley splitting and why is it a problem in silicon?
Silicon's conduction band does not have a single minimum but several equivalent "valleys" in momentum space, and in a quantum well two of them stay nearly degenerate. That near-degeneracy is dangerous: if the valley splitting is comparable to the qubit's energy scale, the electron has a low-lying extra state that is not part of your two-level system, and it can leak population or spoil gates. A clean spin qubit needs the valley splitting to be large — pushed up well above the operating energies — which depends sensitively on atomic-scale details of the interface. Engineering a reliably large valley splitting across a whole array is one of the central materials challenges specific to silicon dots, and it has no analogue in GaAs, whose conduction band has a single valley.
If a spin qubit is basically a transistor, why is scaling still hard?
The structural similarity is real and is the whole scaling argument — a gate-defined dot is a MOS structure run at one electron and millikelvin temperatures, so foundry fabrication is plausible in a way it is not for many other platforms. But three things make it hard anyway. First, each dot needs its own set of precisely tuned gate voltages, and routing that many control lines into a cryostat and onto a dense array is a genuine wiring bottleneck. Second, atomic-scale disorder makes every dot slightly different, so tuning does not simply copy from one to the next. Third, the exchange knob is exponentially sensitive to voltage, so charge noise that a classical transistor would shrug off translates into gate error. Sharing control lines, cryo-CMOS electronics, and encodings that need fewer knobs per qubit are the active answers.