39. Superconducting Qubits
39.1Josephson junctions
Two superconductors separated by a 1–2 nm insulator: Cooper pairs tunnel through, producing the current–phase relation I = Ic·sin(φ) — an inductor with a nonlinearity. The nonlinearity is the entire point: an ordinary LC oscillator has equally spaced energy levels, so a drive addressing |0⟩ → |1⟩ also drives |1⟩ → |2⟩, and no qubit can be isolated. The Josephson element makes the spacing unequal, so one transition can be addressed without the next. Junctions are patterned by electron-beam lithography with aluminum oxidation, and they are the most fabrication-sensitive part of the chip — the two-level defects of 29.10 live in their oxide layer.
39.2Transmons
The dominant design: a Josephson junction shunted by a large capacitance. The large shunt makes the energy nearly insensitive to offset charge — killing the charge noise that limited earlier charge qubits — and carried coherence from nanoseconds (1999) to ~100 µs and beyond. The price is weak anharmonicity, only ~−200 to −300 MHz between adjacent transitions, so pulses must be carefully shaped or they leak (29.6). Variants: fixed-frequency transmons (better coherence, slower gates — IBM's choice) versus flux-tunable transmons (faster gates, more noise-sensitive — Google, Rigetti). When someone says "superconducting qubit" today, they almost always mean a transmon; IBM, Google, and Rigetti all build them.
39.3Microwave control
Drive the |0⟩ ↔ |1⟩ transition with microwave pulses at 4–6 GHz, delivered down coaxial lines to a capacitive antenna on chip. Rabi oscillations calibrate pulse amplitude; pulse phase is controlled digitally, which makes rotations about the Z axis nearly free in software (virtual-Z gates). Shaping matters: Gaussian-derivative (DRAG) pulses suppress leakage into |2⟩ (29.6). Gate durations run 10–30 ns. Count the wiring: per qubit, 2–3 control lines plus a shared or dedicated readout line — multiply by a million qubits and the scaling wall of 38.6 becomes concrete. This per-qubit line count is why cryogenic control electronics is a research field.
39.4Readout resonators
Dispersive readout: the qubit couples to a microwave resonator whose frequency shifts by ±χ depending on the qubit state; send a probe tone through and the outgoing phase or amplitude tells you the state. Signals are at the single-photon level, so quantum-limited amplifiers (Josephson parametric amplifiers, traveling-wave parametric amplifiers) are mandatory before conventional electronics can see anything. Readout takes ~100 ns to 1 µs at 99–99.9% fidelity — and during that window, T1 decay corrupts the answer (29.5). Resonators also couple qubits to each other unintentionally, making them a crosstalk channel as well as a measurement instrument (39.8).
39.5Cryogenics
Why 10–20 mK: thermal excited-state population n = 1/(exp(hf/kT) − 1) must be far below 1, and at 5 GHz that demands temperatures well under 240 mK. A dilution refrigerator provides microwatts of cooling at the base stage; every coax line enters through attenuators (keeping blackbody radiation out) and exits through amplifiers. The refrigerator defines the machine's size, cost, and wiring budget, and its cooling power defines how much control electronics can live inside — which is why moving digital control to the 4 K or millikelvin stage (cryo-CMOS) is the central scaling battleground for this platform (38.6). You cannot shortcut this: warm transmons are simply not qubits.
39.6Calibration
The daily pipeline (38.7, 29.8): spectroscopy to find each qubit's frequency, Rabi calibrations for pulse amplitudes, DRAG tuning, two-qubit gate calibration across every coupled pair, readout discriminator fitting. On a 100+ qubit device this is thousands of interacting parameters under frequency crowding — qubits sit only 100–500 MHz apart, so one qubit's calibration can invalidate a neighbor's. Deciding which calibrations to run, in what order, and when to re-run them is a genuine operations-research problem, made perpetual by drift. This is where superconducting-hardware companies employ their most software engineers — a classical control-and-optimization pipeline wrapped around physics.
39.7Two-qubit gates
Three main mechanisms. Cross-resonance (IBM, fixed-frequency qubits): drive the control qubit at the target's frequency through their shared coupling; ~100–300 ns, errors 10⁻³–10⁻². Tunable couplers (Google): a dedicated coupler element is flux-pulsed to switch an effective interaction on and off, achieving faster gates (~30–60 ns) and the platform's best reported two-qubit errors. Flux-tunable qubits (Rigetti) gate by pulsing the qubits themselves. All are nearest-neighbor on a lattice — heavy-hexagonal for IBM — so every long-range interaction costs SWAP gates, and compiler overhead (Part XII) is part of the platform's true cost of ownership. Compare platforms at the compiled-circuit level, never the bare gate level.
39.8Crosstalk
The platform's characteristic engineering disease. Qubits share control lines, readout chains, and couplers, so a pulse on qubit i drives qubit j at the 10⁻³ level; flux pulses leak through shared circuitry; and gates run simultaneously measurably degrade versus isolated calibration. The structural damage exceeds the per-gate number: crosstalk creates correlated errors between physical neighbors, straining the independence assumptions that error-correction thresholds are proven under (29.7, Part X). Countermeasures are classical engineering: careful frequency allocation, pulse shaping, dynamical decoupling on idle qubits, and simultaneous randomized benchmarking to quantify the damage. When comparing vendors, prefer the ones who publish crosstalk measurements — the ones who do not are not measuring it.