Dispersive Qubit Readout (2D) — Circuit QED Measurement

Qubit State

Excited-state population p₁ 50%

Readout Chain

Coupling g / 2π 60 MHz

Drive amplitude ε 2.5

Integration time τ 500 ns

Live Readout

Dispersive shift χ/2π

3.6 MHz

Pointer separation D

0.94

Signal-to-noise (SNR)

2.2

Single-shot fidelity

87%

Assignment error (shots fired)

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How it works (2D model)

Same dispersive-readout model as the 3D version, flattened onto the IQ plane you actually read a real chip from. A transmon qubit dispersively coupled (detuning Δ ≫ g) to a readout resonator shifts the resonator's frequency by ±χ depending on its own state — the Jaynes–Cummings Hamiltonian in the dispersive limit reduces to:

H/ħ ≈ ω_r a†a + (ω_q/2)σ_z + χ σ_z a†a,   χ = g²/Δ

Driving the resonator at its bare frequency populates it with a coherent state whose steady-state amplitude depends on the qubit state:

α_g = ε / (κ/2 − iχ)
α_e = ε / (κ/2 + iχ)
D  = |α_e − α_g| = 2εχ / [(κ/2)² + χ²]

Those two pointer states land at different points on the IQ (in-phase/quadrature) plane. Averaging the homodyne signal for a time τ suppresses noise as 1/√τ, so:

SNR(τ) = D·√τ / σ₀
F = ½[1 + erf(SNR / 2√2)]

This is the interface every superconducting quantum processor uses to talk to the classical world — a threshold on this exact IQ plane turns the measurement into a 0/1 bit.

p₁ slider / Prepare buttons — sets the qubit's population before the readout pulse; a QND dispersive readout measures only population, not superposition phase.
g, ε, τ — coupling strength, drive amplitude and integration time; watch how each reshapes the pointer separation and shrinks the noise clouds on the plane.
Fire 400 single shots — samples individual measurement outcomes from the two Gaussian clouds and reports the fraction an optimal threshold would assign to the wrong state.

Drag the IQ plane to pan and scroll to zoom in on the pointer-state clouds.

drag to pan · scroll to zoom

Qubit

Resonator

|g⟩ pointer / shots |e⟩ pointer / shots I / Q axes

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