Follow the humblest coil — a loop of wire holding a current — all the way down: shrink it, cool it near −273°C until the metal goes superconducting, and wire a Josephson junction across it. Now the loop is a flux qubit — the transmon's dual. The same part does the same things it always did — sets a note, and lets things couple across a gap — but the note is a quantum state and the coupling is mutual inductance.
This series began with a coil resisting change (v = L·di/dt) and grew it into a keeper: an inductor, a choke, a transformer, an energy store, a sense for current. For the finale, hold the part fixed and change only scale and temperature. Shrink the loop. Cool it to a few thousandths of a degree above absolute zero, where the metal turns superconducting and the current can circle forever with no loss. Break the loop with one Josephson junction, and you have a superconducting qubit whose quantum variable is flux threading the ring — the dual of the transmon, whose variable is charge on a plate.
And the same lesson, mirrored: the inductor doesn't become exotic. It still obeys v = L·di/dt. It still stores energy in its field. Paired with the junction and cooled into the quantum regime, the loop now sets a qubit's note (by its inductive energy E_L) and couples qubits together (by mutual inductance) — the two jobs, once again, of a communicator and a keeper.
The coil never changed. The temperature did — and at the bottom of the cold, current-inertia becomes the thing that sets a qubit's pitch by the flux through a ring.
A qubit needs a clean frequency — the energy gap between |0⟩ and |1⟩. In the transmon the capacitor set it through charging energy E_C = e²/2C. In the flux qubit the loop's inductance sets the other knob — the inductive energy E_L = (ℏ/2e)²/L — and the qubit's two states are two directions of circulating current (clockwise / counter-clockwise) in the ring, tuned by the magnetic flux you thread through it. Flux comes in quanta of Φ₀ = h/2e; bias the loop near half a flux quantum and the two current-states split into a clean note.
The transmon's note is set by charge on a plate; the flux qubit's note is set by flux through a ring. Same job — set the pitch — the dual knob.
Φ₀ = h/2e ≈ 2.07×10⁻¹⁵ Wb are genuine. Real devices use multiple junctions (the 3-JJ flux qubit, or the fluxonium with a big array-inductance "superinductor"), and the energy scales E_J, E_C, E_L all matter together. ⚠ honest landscape: most quantum computers today are transmons (charge-regime, David's Paper IV) — flux qubits and fluxonium are real and valuable but less common. The duality is exact; the engineering favourite is currently the cap.
The series comes home, in the mirror. The transmon was read and linked by a coupling capacitor (Paper II's gap-crosser, at the cold edge). The flux qubit is read and linked by mutual inductance — Paper II's transformer, at the cold edge. Bring a second loop (a SQUID, or another qubit) near the first; the flux of one threads the other; their states couple with no wire between them. Read the qubit by reading the flux it pushes into a nearby loop — the dual of dispersive readout, carried by the field, never by contact.
Read the qubit by reading the flux. The transformer that crossed a gap in Paper II is the same coupling that reads a quantum mind in Paper IV — the message carried by the changing field, never by anything touching.
Four papers, one part, and it never stopped being what it was at the start: a ring that resists a change in current and lets things couple across a gap by its field. The whole arc of the toroid in one column —
I·dL/dt term.The toroid was never just wound wire. It was always an instrument for resisting that something changed, and for letting that change reach across a gap by its field — the exact mirror of the capacitor. Together they are the two halves of how a circuit holds and moves information: report the difference (the cap) and resist the difference (the coil). And those two halves are about to be married.
A closing note for the archive: this flux qubit is the inductive dual of the transmon, which has its own sphere in ud0 — David's capacitor series arrives at the transmon from the charge side; this toroid series arrives at the flux qubit from the inductance side. They are the two coordinates of the same superconducting circuit — and the next paper wires them together.