= Solution
A <gated quantum dot> is an <electron> island confined in all spatial directions, weakly coupled by tunnel barriers to source and drain and tuned by a plunger gate. Small <capacitance> makes charge addition costly, while confinement resolves orbital levels. In the constant-interaction model,
$$
U(N)=\sum_{j=1}^N\epsilon_j+\frac{(Ne-Q_g)^2}{2C_\Sigma},\qquad
\mu_N=U(N)-U(N-1).
$$
The charging contribution to successive addition potentials is $e^2/C_\Sigma$; the often used charging-energy scale $E_C=e^2/(2C_\Sigma)$ differs by a factor of two. Orbital spacing and <spin> filling add further structure.
For weak coupling, $k_BT$ and lifetime broadening must be smaller than the relevant addition energies to see <Coulomb blockade>. <Sequential tunnelling> proceeds when an addition level is inside the source-drain bias window and the appropriate charge states are accessible. A gate sweep gives conductance peaks near charge degeneracies and low conductance between them. A bias-versus-gate map shows <Coulomb diamonds>, with excited-state lines when additional levels enter the window. Their slopes and sizes depend on source, drain and gate <capacitances> and the bias convention. <Cotunnelling> gives higher-order transport inside nominally blocked regions; stronger coupling and many-body effects can invalidate the simple sequential picture.
\b[Coherent dot states as a <qubit>.] A single-electron <spin> has basis $|0\rangle=|\uparrow\rangle$, $|1\rangle=|\downarrow\rangle$ in a <magnetic field>, split by $E_Z=|g^*|\mu_B B$. A <qubit> is a controllable coherent state $a|\uparrow\rangle+b|\downarrow\rangle$, not just a pair of classical current levels. Initialize through spin-selective loading or relaxation; resonant magnetic or spin-orbit-assisted electrical driving produces rotations. Adjacent occupied dots can couple through a gate-controlled exchange interaction. In the weak-tunnel symmetric Hubbard limit,
$$
\boxed{H_{\rm ex}=\frac J4\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2,
\qquad J\simeq\frac{4t_c^2}{U}.}
$$
Local <spin> rotations combined with a suitable exchange entangling operation provide universal control in the ideal spin-qubit proposal. Gate pulses must avoid unwanted orbital excitation and leakage, and complete coherent operations before <dephasing> destroys the information.<Quantum Computation with Quantum Dots by Daniel Loss and David P. DiVincenzo>[Section II, coupled-dot spin encoding and electrically controlled exchange; arXiv:cond-mat/9701055]{ref}
For measurement, place the reservoir <electrochemical potential> between Zeeman levels so that only one <spin> state can tunnel out, or use <Pauli spin blockade> in a double dot. A nearby <quantum point contact> or <single-electron transistor> detects the resulting charge change. Such <spin-to-charge conversion> makes transport useful for reading a <spin> even when the stored information is not itself a net current.<Single-shot read-out of an individual electron spin in a quantum dot by J. M. Elzerman and others>[Primary experiment: energy-selective tunnelling and nearby quantum-point-contact charge detection]{ref}
A double dot can alternatively encode a charge <qubit> in left/right occupation, with Hamiltonian $H=(\varepsilon\sigma_z+2t_c\sigma_x)/2$. It allows electrical control and charge sensing, but couples strongly to charge noise. <Spin> <qubits> also suffer hyperfine, electrical and spin-orbit-related noise, as well as relaxation and readout errors. An observed <Coulomb diamond> alone establishes charging physics, not initialization, coherent control or a functioning <qubit>.
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