A raised cosine distribution is a bounded continuous distribution whose density follows a shifted, scaled cosine curve on [μ−s, μ+s] and vanishes outside that interval. It approximates the normal distribution when long tails can be ignored. section 8.4.1
Quantum state preparation turns the distribution into a reusable Fourier-based pattern:
- Encode two Fourier coefficients into the amplitudes associated with outcomes 0 and 2ⁿ−1. section 8.4.1
- Apply the quantum Fourier transform to the qubits in reverse order without swaps. section 8.4.1
- Measure probabilities matching the discretized raised cosine; for s = 2ⁿ⁻¹, the discrete probabilities can be checked against equivalent cosine-squared expressions. section 8.4.1
Three-qubit encoding
For n = 3, N = 8, and μ = 3.25, construct a circuit with a Hadamard gate, a phase rotation of −2πμ/N, and a reversed, swap-free QFT. Running it produces amplitudes proportional to cos((k−μ)π/N) with a phase factor; their squared magnitudes match the raised cosine probabilities. section 8.4.1
Figure 8.29 The raised cosine probability density function
Use the density plot to see the finite support and cosine-shaped profile before examining its quantum encoding. section 8.4.1
