Q-sphere and Bloch spheres

Two pictures of a state: the Q-sphere shows the whole multi-qubit state; Bloch spheres show each qubit alone.

Q-sphere: each basis state with non-zero amplitude is a dot. Latitude = Hamming weight (|0…0⟩ at the north pole, |1…1⟩ at the south pole). Dot size ∝ probability, colour = phase on a 0 → 2π wheel, with the global phase fixed so the largest amplitude is real and positive.

ρ_{q} = Tr_{others} ∣ ψ ⟩ ⟨ ψ ∣, r = (Tr ρ_{q} X, Tr ρ_{q} Y, Tr ρ_{q} Z), purity = Tr ρ_{q}^{2}

Bloch vector of one qubit from its reduced density matrix. |r| = 1 for a pure, unentangled qubit; |r| = 0 and purity ½ for a maximally entangled one (Bell state).

The step slider replays the circuit one DAG layer at a time (step 0 = |0…0⟩), so you can watch entanglement appear when the CNOT is applied.

Five pictures of one state

View	What it shows
Q-sphere	All basis states at once: size = probability, colour = phase, latitude = Hamming weight.
Bars	Probability \|a\|² per basis state; each bar keeps the phase colour, so a Z gate changes colours but not heights.
Amplitudes	Real and imaginary part of every amplitude as grouped bars: the raw statevector, signs included.
Line	Magnitude \|a\| and phase/π as two lines over the basis states; interference shows up as magnitudes moving while phases line up.
Bloch	One sphere per qubit from the reduced density matrix; a qubit is tagged entangled when its purity drops below 0.99.

Debug mode: breakpoints and stepping

The Debug tab in the IDE (and the Playground) runs the whole circuit on the server in one call and lets you walk through it like a program: Debug/Continue (F5), Step over (F10), Step back (Shift+F10), Restart (Ctrl/⌘+Shift+F5), Stop (Shift+F5). Click a gate in the circuit, or the gutter next to a line of OpenQASM 3 or Python, to set a red breakpoint; Continue runs to the next one. The current gate is highlighted, future gates are dimmed, and a timeline under the circuit scrubs to any step.

Every step carries the gate matrix, the statevector before and after (table and kets), the probabilities, the Bloch vectors with purity, and the fidelity. With a fake backend as the noise model, each gate is translated to the device's native gates and run on a density-matrix simulator with that device's noise, so the Fidelity chart shows how far the real state drifts from the ideal one gate by gate. Debug mode covers up to 4 qubits and 80 gates; measurements appear as one final step with sampled counts.

F (ρ, ∣ ψ ⟩) = ⟨ ψ ∣ ρ ∣ ψ ⟩, purity = Tr ρ^{2}

Fidelity of the noisy state ρ against the ideal |ψ⟩, and purity of the whole register (1 = pure).

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Metrics: comparing distributions

Real IBM Quantum hardware

View

What it shows

Q-sphere

All basis states at once: size = probability, colour = phase, latitude = Hamming weight.

Bars

Probability |a|² per basis state; each bar keeps the phase colour, so a Z gate changes colours but not heights.

Amplitudes

Real and imaginary part of every amplitude as grouped bars: the raw statevector, signs included.

Line

Magnitude |a| and phase/π as two lines over the basis states; interference shows up as magnitudes moving while phases line up.

Bloch

One sphere per qubit from the reduced density matrix; a qubit is tagged entangled when its purity drops below 0.99.