Quantum Process Tomography: The Bloch Ellipsoid

Ideal Gate

Noise Channel

Depolarizing p 0%

Dephasing γ (T2-type) 0%

Amplitude damping γ (T1-type) 0%

Reconstructed From 4 Probes

R = [ 1.00 0.00 0.00 ] [ 0.00 1.00 0.00 ] [ 0.00 0.00 1.00 ] t = ( 0.00, 0.00, 0.00 )

Avg. gate fidelity

100.0%

Bloch volume ratio

1.000

Mean output purity ⟨|r|⟩

1.000

Probes used

How it works

A single-qubit state is a point r = (x,y,z) in the Bloch ball, ρ = (I + r·σ)/2. Any physical process (gate + noise) acts as an affine map r → R·r + t. Standard quantum process tomography reconstructs R and t by sending four known probe states through the channel — the maximally mixed state (r=0) gives t directly, and the +x, +y, +z probes give the three columns of R once t is subtracted:

t      = Λ(0)
R·x̂  = Λ(x̂) − t
R·ŷ  = Λ(ŷ) − t
R·ẑ  = Λ(ẑ) − t

The reconstructed map is exactly what the point cloud on the right shows: every point started on the unit Bloch sphere (a pure state) and is redrawn at R·r + t. Since |det R| is the volume-scale factor of a linear map, it directly reports how much of the state space the channel destroys — a value near 1 means the sphere survives intact; near 0 means almost every state collapses toward a single mixed point.

Depolarizing p — Bloch vector shrinks isotropically: r → (1−p)r.
Dephasing γ — kills coherence (x,y) while leaving the measured z-population alone: (x,y) → √(1−γ)(x,y), z unchanged. Models T2 decay.
Amplitude damping γ — models energy relaxation toward |0⟩: (x,y) → √(1−γ)(x,y), z → (1−γ)z + γ. This is the only non-unital channel here, so it is the one that shifts t away from the origin.

Average gate fidelity is estimated by Monte-Carlo: many random pure probe states r are sent through the ideal gate (r_ideal = R_gate·r, still on the unit sphere) and through the full reconstructed channel (r_out = R·r + t), then averaged using F(ψ) = (1 + r_ideal·r_out)/2 — the standard overlap between a pure state and a general mixed state on the Bloch ball.

Ideal Bloch sphere Reachable states (image of Λ) Probe: after channel

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