Normalized colloidal probability density $ρ^(H)$ profiles without interaction (U = 0) on a 2ⁿ = 16 grid using time-step Δτ = 10⁻⁴ plotted in increments of (a) 2 × 10⁻³ and (b) 2 × 10⁻² up to T = 10⁻¹. Lines show VQS solutions based on full circular ansatz with six to eight layers, and circles show classical solutions based on the same discretization. Mean (c) trace and (d) norm errors plotted on log scale versus the total number of circuits N_q for run-time up to T = 10⁻¹ (insets show peak values). Data are grouped by the number of ansatz layers, each set using time-step Δτ ∈{10, 5, 2, 1} × 10⁻⁴. Closed symbols represent Runge–Kutta solutions using time-step Δτ ∈{10, 5, 2} × 10⁻⁴. Horizontal reference line: 0.05