HHL Algorithm 2D: Quantum Linear System Solver

Matrix A — eigenvalues

λ₁1.00

λ₂2.20

λ₃4.00

Input vector b (eigenbasis weights)

β₁1.00

β₂0.70

β₃0.40

Ancilla inversion

Scale C / λ_min 0.60

Clock speed 1.0×

Success P(anc=1)

–

Condition κ = λmax/λmin

–

Shots run

Kept fraction

–

x ∝ A⁻¹b = ( – , – , – )

How it works

|0⟩ → √(1-(C/λᵢ)²)|0⟩ + (C/λᵢ)|1⟩ , θᵢ = 2·asin(C/λᵢ)

Uncomputing the phase register and measuring the ancilla in |1⟩ post-selects the state onto

Σᵢ (βᵢ·C/λᵢ)|uᵢ⟩ / √P  ∝  A⁻¹|b⟩ = |x⟩
P(anc=1) = Σᵢ βᵢ²(C/λᵢ)²

Small eigenvalues get inverted into large amplitudes — the tall purple "1/λ" bars on the small-λ mode — which is exactly why a nearly-singular A (large condition number κ = λmax/λmin) makes HHL's success probability collapse: the run has to be repeated far more often to land the post-selection. The history strip below the dials runs many independent shots (each a fresh Bernoulli draw at the current P) so you can watch the kept-fraction converge to the computed probability.

λ₁, λ₂, λ₃ — the three eigenvalues of A (its spectrum in this eigenbasis).
β₁, β₂, β₃ — how much of |b⟩ lies along each eigenvector.
C scale — the ancilla-rotation constant, expressed as a fraction of λmin (must stay below 1 for a valid rotation on every mode).
Post-select — samples one shot from P(anc=1) and reports whether that run would have kept or discarded the register.
Auto-shots — keeps sampling shots automatically so the running kept-fraction can be compared against the analytic P live.

Eigenmode tower (height = λᵢ) Phase-estimation clock dial Ancilla tilt needle (θᵢ) Output amplitude bar (βᵢ·C/λᵢ)