Exponential concentration of the tensor-product RY quantum kernel

The global-measurement example of Thanasilp et al. (2022): the fidelity kernel of the embedding U(x) = ⊗ₖ R_Y(xₖ) is κ(x,x') = ∏ₖ cos²((xₖ-x'ₖ)/2). By translation invariance each coordinate reduces to a uniform variable on [-π,π], so we study ryKernel n θ = ∏ₖ cos²(θₖ). Its moments are elementary (𝔼[κ]=(1/2)ⁿ, Var[κ]=(3/8)ⁿ-(1/4)ⁿ), giving genuine exponential concentration with NO Haar assumption.

noncomputable def QuantumAlg.unifAngle : MeasureTheory.Measure ℝ

Uniform probability measure on [-π, π].

Equations

• QuantumAlg.unifAngle = (ENNReal.ofReal (2 * Real.pi))⁻¹ • MeasureTheory.volume.restrict (Set.Icc (-Real.pi) Real.pi)

noncomputable def QuantumAlg.ryKernel (n : ℕ) (θ : Fin n → ℝ) : ℝ

The reduced tensor-product RY fidelity kernel ∏ₖ cos²(θₖ).

Equations

• QuantumAlg.ryKernel n θ = ∏ k : Fin n, Real.cos (θ k) ^ 2

noncomputable def QuantumAlg.ryMeasure (n : ℕ) : MeasureTheory.Measure (Fin n → ℝ)

The data distribution: n independent uniform angles.

Equations

• QuantumAlg.ryMeasure n = MeasureTheory.Measure.pi fun (x : Fin n) => QuantumAlg.unifAngle

instance QuantumAlg.instIsProbabilityMeasureRealUnifAngle : MeasureTheory.IsProbabilityMeasure unifAngle

theorem QuantumAlg.ryKernel_nonneg (n : ℕ) (θ : Fin n → ℝ) : 0 ≤ ryKernel n θ

theorem QuantumAlg.ryKernel_le_one (n : ℕ) (θ : Fin n → ℝ) : ryKernel n θ ≤ 1

theorem QuantumAlg.continuous_ryKernel (n : ℕ) : Continuous (ryKernel n)

theorem QuantumAlg.integral_cos_sq_unifAngle : ∫ (x : ℝ), Real.cos x ^ 2 ∂unifAngle = 1 / 2

∫ cos²θ over the uniform measure on [-π,π] equals 1/2.

theorem QuantumAlg.integral_cos_pow4_unifAngle : ∫ (x : ℝ), Real.cos x ^ 4 ∂unifAngle = 3 / 8

∫ cos⁴θ over the uniform measure on [-π,π] equals 3/8.

instance QuantumAlg.instIsProbabilityMeasureRyMeasure (n : ℕ) : MeasureTheory.IsProbabilityMeasure (ryMeasure n)

theorem QuantumAlg.ryKernel_memLp (n : ℕ) : MeasureTheory.MemLp (ryKernel n) 2 (ryMeasure n)

theorem QuantumAlg.mean_ryKernel (n : ℕ) : ∫ (θ : Fin n → ℝ), ryKernel n θ ∂ryMeasure n = (1 / 2) ^ n

First moment: 𝔼[κ_n] = (1/2)ⁿ.

theorem QuantumAlg.mean_ryKernel_sq (n : ℕ) : ∫ (θ : Fin n → ℝ), ryKernel n θ ^ 2 ∂ryMeasure n = (3 / 8) ^ n

Second moment: 𝔼[κ_n²] = (3/8)ⁿ.

theorem QuantumAlg.variance_ryKernel (n : ℕ) : ProbabilityTheory.variance (ryKernel n) (ryMeasure n) = (3 / 8) ^ n - (1 / 4) ^ n

Exact variance: Var[κ_n] = (3/8)ⁿ - (1/4)ⁿ.

theorem QuantumAlg.variance_ryKernel_le (n : ℕ) : ProbabilityTheory.variance (ryKernel n) (ryMeasure n) ≤ (3 / 8) ^ n

theorem QuantumAlg.ryKernel_concentrates : ExpConcentratedProb (fun (n : ℕ) => ryMeasure n) fun (n : ℕ) => ryKernel n

The tensor-product RY quantum kernel concentrates exponentially (Thanasilp 2022), with NO Haar assumption.

theorem QuantumAlg.ryKernel_tendsto_zero {δ : ℝ} (hδ : 0 < δ) : Filter.Tendsto (fun (n : ℕ) => (ryMeasure n) {θ : Fin n → ℝ | δ ≤ |ryKernel n θ - ∫ (x : Fin n → ℝ), ryKernel n x ∂ryMeasure n|}) Filter.atTop (nhds 0)

The RY-kernel deviation probability vanishes as the qubit count grows: the kernel landscape becomes exponentially flat, so a polynomial shot budget cannot resolve it.