What is a qubit actually made of?

Strip away the Bloch sphere and the bra-kets: physically, a superconducting qubit is a capacitor wired across one nonlinear component. Here is what that component does, and why everything hinges on it.

You'll be able to name the parts of a transmon qubit and say what each one does.

The problem: oscillators are too polite

Picture a swing. Push it, and it settles into a rhythm: back and forth at one fixed rate, forever, if nothing slows it down. An LC circuit, a capacitor (a bucket that stores charge) wired to a coil (a flywheel that stores current), does the electrical version of the same thing. Charge sloshes into the capacitor, drains out through the coil, sloshes back, and the whole thing rings at one fixed frequency set by how big the bucket and the flywheel are.

Here is the catch. Quantum mechanically, that ringing circuit doesn't just have one energy: it has a whole ladder of allowed energies, evenly spaced, rung after rung, like a musical instrument's overtones. Evenly spaced sounds harmless. It's fatal for a qubit. If you send in a pulse tuned to boost the circuit from the bottom rung to the second rung, that exact same pulse is equally good at boosting it from the second rung to the third, and the third to the fourth. You can never cleanly stop at "just two levels", the two things every classical bit and every qubit need.

So the goal isn't just "build something that oscillates"; a plain LC tank already does that beautifully, and it's useless. The goal is to build something that oscillates unevenly: where the first rung costs a noticeably different amount of energy to climb than the second one. That unevenness is called anharmonicity, and one component supplies almost all of it.

How, in real terms: Working: keep going ↓

The fix: one component that refuses to be linear

The component is the Josephson junction: two superconductors separated by an insulating gap so thin (a couple of nanometers) that paired electrons (Cooper pairs) tunnel straight through it. Below a critical current it carries current with exactly zero resistance, which is already strange. What makes it useful here is stranger still: the relationship between the current flowing through it and the phase difference across it isn't a straight line, the way it is for a normal wire or a normal coil. It's a sine function. Wired into a circuit, it behaves like an inductor, but one whose effective "inductance" depends on how much current is already flowing through it. Push it harder, and it pushes back disproportionately. That disproportion is the nonlinearity the whole field is built on.

Wire an ordinary capacitor straight across a single Josephson junction, and you get the transmon, far and away the most common superconducting qubit design, running under most of Google, IBM and Rigetti's hardware. Electrically it is almost the same object as the plain LC tank above, with one piece swapped: the linear coil is replaced by the junction's nonlinear one. Solve that circuit and the evenly-spaced ladder collapses into an uneven one: each rung costs a little less energy to climb than the rung below it.

The ratio E_J/E_C is the one dial that matters, and it's a real trade-off. A real Cooper-pair box (the transmon's ancestor) ran at low E_J/E_C and paid for it in charge noise: stray, uncontrolled electric fields constantly jittering the qubit frequency. Koch and colleagues' 2007 transmon paper showed that pushing deep into E_J/E_C ≫ 1 (a "fat," well-shielded capacitor) suppresses that charge noise exponentially, while the anharmonicity you need for addressability only shrinks by a slow power law in return. An exponential win paid for with a mild tax is a trade you take every time, which is exactly why the transmon, not its more charge-sensitive ancestor, is the design almost every lab converged on.

With the bottom two levels now sitting at a distinguishable energy gap from the next pair up, a microwave pulse tuned exactly to ω₀₁ drives 0→1 cleanly, without much leaking into level 2. Try it below.

Drive it yourself

Left: the circuit's own voltage, ringing at whatever frequency the controls below set. Right: its actual energy ladder, computed live from the same numbers: evenly spaced with no junction, uneven the moment you add one.

The formalism, with receipts

An LC circuit is quantized the same way a mechanical oscillator is: canonical conjugate variables (flux Φ across the inductor and charge Q on the capacitor) obey [Φ̂, Q̂] = iħ, and the Hamiltonian H = Q̂²/2C + Φ̂²/2L is exactly a quantum harmonic oscillator, with eigenvalues E_n = (n + ½)ħω, ω = 1/√(LC). Evenly spaced, as claimed above: this is not an approximation, it is exact.

A Josephson junction's current-phase relation is I = I_c sin φ, with voltage V = (ħ/2e) dφ/dt across it. Treated as a circuit element this defines a nonlinear inductance L_J(φ) = ħ/(2eI_c cos φ): a real inductor, but one whose value depends on the bias point, which is exactly the sine-versus-straight-line nonlinearity described above. Its energy scale is the Josephson energy E_J = ħI_c/2e.

Shunting a single junction with a capacitor C_Σ (charging energy E_C = e²/2C_Σ) gives the transmon Hamiltonian in the charge basis:

where n̂ is the Cooper-pair number operator and n_g the offset charge. In the transmon regime E_J/E_C ≫ 1, expanding cos φ̂ past its quadratic (harmonic) term to the next, quartic order turns this into a weakly anharmonic (Duffing) oscillator, giving the leading-order level structure quoted above: ω₀₁ ≈ √(8E_JE_C)/ħ − E_C/ħ, and each successive gap ω_n,n+1 smaller than the one before it by α ≈ −E_C/ħ (Koch, Yu, Gambetta, Houck, Schuster, Majer, Blais, Devoret, Girvin & Schoelkopf, arXiv cond-mat/0703002, Phys. Rev. A 76, 042319 (2007)). The paper's headline result is the trade this page keeps returning to: charge dispersion, the residual sensitivity to the offset charge n_g, falls exponentially in √(8E_J/E_C), while the anharmonicity available for gating falls only as a weak power law in E_C/E_J. That asymmetry is the case for operating deep in the E_J/E_C ≫ 1 regime rather than near the Cooper-pair box's original charge-sensitive point.

The drive, and why it is a gate. Coupling a classical microwave tone to the circuit adds a drive term to the Hamiltonian; in the rotating frame, under the rotating-wave approximation, and restricted to the {|0⟩, |1⟩} subspace, this reduces to the textbook two-level Rabi problem. On resonance, the population in |1⟩ evolves as

where Ω is the Rabi frequency set by the drive amplitude. A pulse cut off at Ωt = π (a π-pulse) leaves P₁ = 1: full population transfer from |0⟩ to |1⟩, which is how the X gate is physically implemented. A π/2-pulse instead leaves an equal superposition, the same primitive the homepage's own |0⟩→|+⟩ widget asks you to apply. This is the exact mechanism behind every single-qubit gate on real superconducting hardware, surveyed in full in Krantz, Kjaergaard, Yan, Orlando, Gustavsson & Oliver, arXiv 1904.06560 (Applied Physics Reviews 6, 021318 (2019)); the broader circuit-QED framework this all sits inside is reviewed in Blais, Grimsmo, Girvin & Wallraff, arXiv 2005.12667 (Rev. Mod. Phys. 93, 025005 (2021)).

Off resonance, the same formula generalises. A drive detuned from ω₀₁ by Δ gives P₁(t; Ω, Δ) = (Ω²/(Ω²+Δ²))·sin²(√(Ω²+Δ²)·t/2), with Δ = 0 recovering the on-resonance case above exactly. Detuning does two things at once. The oscillation runs at the generalised Rabi frequency √(Ω²+Δ²), which is faster than Ω, so the peak arrives earlier than a π-pulse's timing would suggest; and the peak itself is capped at Ω²/(Ω²+Δ²), so a badly detuned drive cannot reach full population transfer however long you run it. That ceiling is the whole subject of the section below.

The catch this page's widget deliberately simplifies away. Because the transmon is only weakly anharmonic, a fast drive tuned to ω₀₁ has non-negligible spectral weight at the nearby ω₁₂ gap too, leaking population into level 2, which is an error, not a feature. The standard fix is DRAG (Derivative Removal by Adiabatic Gate): add a second control tone proportional to the time-derivative of the first, which analytically cancels the leading leakage term without slowing the gate down. Motzoi, Gambetta, Rebentrost & Wilhelm, arXiv 0901.0534 (Phys. Rev. Lett. 103, 110501 (2009)) reported roughly an order-of-magnitude error reduction relative to the state of the art at realistic transmon parameters, and some form of DRAG shaping runs on essentially every superconducting qubit gate in production today.

Recalibrating this same gate, blind

Everything above assumed one thing: that the drive is exactly in tune with the qubit. Real hardware will not hold still. Flux noise and stray two-level-system defects nudge a qubit's own frequency hour to hour, so the drive ends up slightly out of tune — detuned — and a detuned drive cannot flip the qubit all the way, however long you run it. Worse, you cannot read the detuning off any dial. All you get back is noisy measurement shots. The game below is what recalibrating against that drift actually looks like. (The Formal section above has the formula that ceiling falls out of.)

References

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