Novel Synthesis Method V · 2026

Quantum Walk†
Synthesis

A walker in quantum superposition traverses a discrete lattice. The coin operator rotates its internal state before each step; the resulting probability amplitude distribution — shaped by quantum interference — becomes the audio waveform. No classical system can produce this spectrum.

0
Step t
1.000
‖ψ‖² (norm)
0.00
σ spread
0.000
Entropy S
0.000
Decoherence
H
Coin
Probability Amplitude · |ψ(x,t)|² — click to place walker
ψ(x) · real / imaginary / |ψ|²
Phase Portrait (Re vs Im per site)
Quantum Parameters
Coin Operator U
Presets
Walk Parameters
Initial State |ψ₀⟩
Audio Output
Source
Sound Shaping
LFO → Quantum Parameter Modulation
Reverb
Delay
Discrete-Time Quantum Walk — Coin + Shift
|ψ(t+1)⟩ = Ŝ · (Û ⊗ Î) · |ψ(t)⟩
Û(θ,φ) = [[cos θ, e^{iφ} sin θ], [−e^{−iφ} sin θ, cos θ]]    coin unitary
Ŝ = Σₓ |x+1⟩⟨x| ⊗ |↑⟩⟨↑| + |x-1⟩⟨x| ⊗ |↓⟩⟨↓|    conditional shift
Decoherence: ρ → (1−ε)ρ + ε·Σₓ |x⟩⟨x| ρ |x⟩⟨x|    position measurement noise

The walker's internal spin |↑⟩, |↓⟩ is rotated by coin Û before each step. Hadamard coin → σ∝√t ballistic spread (quantum speedup over classical √t diffusion). Audio: probability density P(x,t) = |ψ↑(x,t)|² + |ψ↓(x,t)|² read as waveform. Quantum interference produces spectral structure unreachable by classical oscillators.