QSE Theorem Reference

Convention: Rx(θ)|0⟩ = cos(θ/2)|0⟩ − i·sin(θ/2)|1⟩
Ry convention: Ry(θ)|0⟩ = cos(θ/2)|0⟩ − sin(θ/2)|1⟩
Φ_M(s) = ∏_{i: (M^T s)_i = 1} cos(θᵢ)
M ∈ F₂^{NB×NA}: M[b][a] = 1 iff CNOT(ctrl=a, tgt=b) is present
Qubit layout: A-qubits = bits 0..NA−1, B-qubits = bits NA..NA+NB−1

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Core Theorems

ID	Name	Formula	max_err
T14	Walsh-Hadamard Master	P_b = (1/2^NB) Σ_s (−1)^{b·s} Φ_M(s), VNE = H({P_b})	< 7e-16
T-UNIVERSAL	Diagonal Mixed Input	P_b = Σ_{x: Mx=b} ρ_A[x,x] (works for any diagonal ρ_A)	< 5e-16
T-UNIVERSAL-GEN	Nonlinear Gates	`P_b = Σ_{x: f(x)=b}	ψ[x]
T16	Phase Invariance	VNE depends only on {	cos(θᵢ/2)

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Structural Theorems

ID	Name	Key Result	max_err
T17	Blindness	VNE = 0 ↔ rank_F₂(M) = 0	exact
T-RANK	Rank Upper Bound	VNE(B) ≤ rank_F₂(M)	exact
T-OPT	Rank Saturation	θ = π/2 ⟹ VNE = rank_F₂(M) exactly	exact
T-SUB	Monotone Substructure	Adding CNOT → rank nondecreasing → VNE nondecreasing	exact

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Gate Extensions

ID	Name	Key Result	max_err
T-BINIT	B-Init Independence	T14 formula unchanged for any product-state B init (full-rank M)	< 7e-16
CZ-T14	CZ Equivalence	CZ + |+⟩_A = CNOT + |0⟩_A (same VNE)	< 6e-16
T-fSim	fSim Gate (NA=1)	λ± = (1 ± √D)/2, D = 1 − sin²(2α)sin⁴(θ/2); φ-invariant	< 2e-15
T-FSIM-PRODUCT	fSim Product Gates	NA independent fSim gates (qubit i ↔ B-qubit i): VNE = Σᵢ H(λ±ᵢ)	< 4e-14
iSWAP	Full iSWAP	iSWAP + Rx(θ)|0⟩_B → product state → VNE = 0	< 4e-16
T-SWAP	SWAP via 3 CNOTs	SWAP(A_a, B_b) via CNOT→CNOT→CNOT → VNE(B) = 0	< 7e-16

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Symmetry & Invariance Theorems

ID	Name	Key Result	max_err
T-RY-EQ	Ry = Rx Symmetry	VNE(Ry(θ) circuit) = VNE(Rx(θ) circuit) for same θ and M	< 8e-16
T-MIRROR	Schmidt Symmetry	VNE(A) = VNE(B) for any pure bipartite state |Ψ⟩_AB	< 8e-16
T-COHERENCE-DIAG	Off-diagonal ρ_A Blindness	Off-diagonal ρ_A elements do not affect VNE(B); ρ_B always diagonal; VNE depends only on diag(ρ_A)	0

T-COHERENCE-DIAG note: This is a non-trivial result. Even if A is in a coherent superposition (off-diagonal ρ_A ≠ 0), the partial trace over A produces a strictly diagonal ρ_B, so the CNOT circuit acts as a classical channel at the level of B's entropy.

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Gradient Theorems

ID	Name	Formula	max_err
T-GRAD	Rx VNE Gradient	d(VNE)/dθᵢ = −Σ_b (dP_b/dθᵢ)·log₂(P_b) where dP_b/dθᵢ = (1/2^NB) Σ_{s: (M^T s)_i=1} (−1)^{b·s}·(−tan θᵢ)·Φ_M(s)	< 2e-10 (FD limit)

T-GRAD note: The 2e-10 error is the finite-difference floor, not an analytic error. The closed-form formula is exact; verification uses 4th-order central differences (h=1e-5).

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Information-Theoretic Theorems

ID	Name	Formula	max_err
T-MUTUAL	Mutual Information	I(A;B) = 2·VNE(B) for pure state	exact
T-MI	MI Non-Negativity	I(B₁;B₂) ≥ 0	analytical
T-PAULI	Pauli Representation	Φ_M(s) = ⟨Z^{M^T s}⟩_A (Pauli expectation value form)	exact

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Dynamics

ID	Name	Key Result	max_err
T15A	k-Layer Circuits	M_eff = M₁ ⊕ M₂ ⊕ ... ⊕ Mₖ (mod 2)	< 8e-16
T18	Period-2	Even number of identical CNOT layers → M_eff = 0 → VNE = 0	< 4e-16
T-BIDIR	Bidirectional Circuits	Alternating A→B and B→A CNOT layers reduce to T14(M_eff)	< 2e-15

T-BIDIR details:

For circuits with interleaved A→B layers (CNOTs ctrl=a, tgt=B_b) and B→A layers (CNOTs ctrl=B_b, tgt=a), the effective M matrix is:

M_eff[b, :] = M_raw[b, :] @ T_cum  (mod 2)

where T_cum ∈ F₂^{NA×NA} is the cumulative A-space shear transformation accumulated from all B→A CNOTs seen so far. Update rule after CNOT(ctrl=B_b, tgt=A_a):

T_cum[a, :] ^= M_eff[b_abs, :]   (row XOR, not matrix multiply)

Invertibility condition: T_cum must have full GF(2) rank (= NA). When T_cum is singular, multiple x-values map to the same A-state with different B-states, creating off-diagonal ρ_B elements (quantum coherences) that break the T14 assumption. The theorem applies only to circuits where T_cum remains invertible throughout.

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Noise Theorems

ID	Name	Formula	max_err
T-NOISE	Amplitude Damping	After AD_γ on B: P₀ → P₀+γP₁, P₁ → (1−γ)P₁; VNE_noisy = H(P₀+γP₁, (1−γ)P₁)	< 3e-16
S2	k-Round Transfer Matrix	T = S_AD_A · S_CNOT · S_AD_B · S_CNOT (16×16 superoperator); vec(ρₖ) = Tᵏ·vec(ρ₀)	< 7e-16

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Moment & Rényi Theory

ID	Name	Formula	max_err
T22	Rényi Moments	Tr(ρᵅ) = (1/2^{(α−1)NB}) Σ_{s₁..s_{α−1}} ∏Φ_M(sᵢ)·Φ_M(⊕sᵢ)	< 2e-15
T29	Newton-Girard Spectrum	Power-sum moments (T22) → Newton-Girard identities → all eigenvalues of ρ_B	< 7e-7
T-RENYI-DERIV	Rényi Derivative	dS_α/dα|_{α=1} = −(ln2/2)·Var_P[log₂ P_b] where Var_P = Σ_b P_b(log₂ P_b)² − VNE²	< 4e-10 (4th-order FD)
T-RENYI-CONV	Rényi→VNE Convergence	S_α → VNE as α → 1	0

T-RENYI-DERIV note: Derived by Taylor-expanding log Tr(ρᵅ) around α=1 using T22 moments. The (ln2/2)·Var coefficient follows from the second cumulant of the log-probability distribution. Verified with 4th-order central differences (h=1e-3) to avoid FD floor.

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Random Circuit Statistics

ID	Name	Key Result	max_err
T-RANDOM-VNE	Expected VNE = Expected GF(2) Rank	E[VNE(θ=π/2)] = E[rank_GF₂(M)] over uniform random M ∈ F₂^{NB×NA}	0

Gaussian binomial formula:

E[rank_GF₂(M)] = (1/2^{NA·NB}) · Σ_{r=1}^{min(NA,NB)} r · [NB choose r]₂ · ∏_{j<r}(2^NA − 2^j)

[n choose k]₂ = ∏_{j=0}^{k-1}(2^n − 2^j) / ∏_{j=0}^{k-1}(2^k − 2^j)   (Gaussian binomial)

The formula is exact over integers (no floating-point error).

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Inter-Theorem Connections

ID	Connection	max_err
B1	T22(α=2) = Parseval: Σ P_b² = (1/2^NB) Σ Φ(s)²	< 3e-16
B2	Φ_M(s) = E_{x~P_A}[(−1)^{(M^T s)·x}] — Walsh-Fourier characteristic function of P_A	< 5e-16
B3	T14 = α→1 limit of T22 Rényi hierarchy (T-RENYI-CONV)	analytical
B4	T17+T-RANK+T-OPT+T-SUB = F₂ linear matroid rank structure	analytical
B5	T22 = α-point Pauli correlation function (via T-PAULI)	analytical
B6	VNE(θ) monotone in [0, rank_F₂(M)] for uniform θ ∈ [0, π/2]	500/500
B7	T14 = H(f*(P_A)): VNE is Shannon entropy of the classical pushforward of P_A under M	< 4e-16

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Resolved Open Problems

ID	Problem	Resolution	Theorem
O1	Closed-form for bidirectional circuits	M_eff = M_raw·T_cum (GF(2)), invertible T_cum required	T-BIDIR
O2a	T-fSim eigenvalue formula (NA=1)	D = 1 − sin²(2α)sin⁴(θ/2); φ-invariant	T-fSim
O2b	T-fSim for NA > 1	VNE additivity for independent fSim gates	T-FSIM-PRODUCT
O3	Gradient d(VNE)/dθᵢ for Ry/Rz input	Ry gives same VNE as Rx (T-RY-EQ); Rx gradient in closed form (T-GRAD)	T-RY-EQ, T-GRAD
O4	Rényi approximation error bound	dS_α/dα|₁ = −(ln2/2)·Var_P[log₂ P_b]	T-RENYI-DERIV
O5	Non-diagonal ρ_A extension	Off-diagonal ρ_A does not affect VNE; ρ_B always diagonal	T-COHERENCE-DIAG
O6	Random circuit VNE via F₂ matroid counting	E[VNE] = E[rank_GF₂] via Gaussian binomial	T-RANDOM-VNE

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Scaling & Performance

Tested on NA=12, NB=6 (2^18 = 262,144 amplitudes):

Operation	Complexity	Measured
t14_Pb (WHT formula)	O(2^{2NB})	12 ms
vne_gradient_rx	O(NA · 2^{2NB})	195 ms
exact_rho_B (vectorized)	O(2^{NA+NB} · 2^{NB})	8 ms
exact_rho_B (old O(n²) loop)	O(4^{NA+NB})	~hours

Key insight: T14 complexity depends only on NB (the B register size), not NA. Growing NA to 100+ costs nothing for the WHT formula — only the statevector simulation and gradient scale with NA.

Vectorized partial trace (exact_rho_B):

# Old: O(4^{NA+NB}) double loop — infeasible for NA > 6
# New: O(2^{NA+NB}) reshape + matmul
A = amps.reshape(2**NB, 2**NA)   # B-bits as rows, A-bits as cols
rho_B = A @ A.conj().T           # traces out A exactly

Index layout: amps[a_bits | (b_bits << NA)] — low NA bits are A, high NB bits are B. Reshape naturally groups B as rows and A as columns, making the matmul correct.

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Package API (qse/)

from qse import (
    # Core — T14 machinery
    build_M, phi_dict, t14_Pb, t_universal,
    vne_from_Pb, renyi_entropy, f2_rank,
    t22_moment, newton_girard,

    # States — simulation ground truth
    product_state, product_state_ry,
    exact_rho_B, exact_rho_B_mixed, vne_exact,

    # Gates — statevector operations
    apply_cnot, apply_fsim, apply_swap,

    # Gradients
    vne_gradient_rx,

    # Noise
    amplitude_damping_kraus, superop,
    noise_transfer_Pb, build_cnot_ad_transfer,

    # Random circuits
    gaussian_binom_2, expected_gf2_rank, enumerate_vne,
)

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Test Suite

Run python qse_tests.py — 34/34 tests pass at machine precision (max_err < 1e-8).

Phase	Tests	Notes
Initial commit	23/23	Core + noise + moment theory
Katman 0 (bug fixes)	23/23	T-fSim formula fix, T-BINIT full-rank check, T18 generalization
Katman 1 (open problems)	33/33	O2b–O6 resolved; T-SWAP, T-MIRROR added
Katman 2 (new theorems)	34/34	T-BIDIR (O1) completed