#!/usr/bin/env python3 """ Computation 55 -- Closed-form sigma_2 = D - 2 of the Walsh-tail off-diagonal block =================================================================================== This completes Lemma 2-prime by exhibiting an explicit pair of vectors u_1 := |e_0> - (1 / D) sum_{a = 0}^{D-1} |e_a> in [D-1, 1] cap H_1, u_2 := - sum_{b = 1}^{D-1} |{0, b}> in [D-1, 1] cap H_2, (both S_{D-1}-fixed under the fermionic-signed S_D representation rho_F, which is the representation that commutes with D_sub) and verifying analytically that M := matrix of T_pm = P_+ T_tail P_- on span(u_1, u_2)/normed reduces to a rank-1 (D - 2)/2 . [[-1, +1], [-1, +1]] matrix whose singular values are (D - 2, 0). Combined with Comp 52's closed form sigma_1 = 2^(D-1) - D on the trivial S_D component, this proves || P_+ T_tail P_- ||_op = 2^(D-1) - D exactly for all D >= 4, because the exponential gap dominates: for D >= 4, 2^(D-1) - D >> D - 2. Method ------ Verify the algebraic identities used by the closed form, to machine precision, for D in {4, 5, 6, 7, 8}: (a) D_sub u_1 = u_2 (b) D_sub u_2 = D * u_1 (weight-3 leakage cancels exactly) (c) T_tail u_1 = beta * u_1 with beta = - binom(D-1, 2) (d) T_tail u_2 = gamma * u_2 with gamma = (D-2)(5-D)/2 Then check (e) M_analytic = (D-2)/2 * [[-1, +1], [-1, +1]] matches M_numerical (machine precision), and that the saturating right singular vector v_star = (-e_1 + e_2)/sqrt(2) gives || T_pm v_star || = (D - 2) || v_star || exactly. """ import math import numpy as np import numpy.linalg as la def popcount(x): c = 0 while x: c += x & 1 x >>= 1 return c def build_T_tail_diagonal(D, kD): """T_tail = sum_{|S| > kD} chi_S is diagonal in the computational basis.""" c_by_weight = [] for w in range(D + 1): total = 0 for j in range(kD + 1, D + 1): kraw = 0 for i in range(0, min(w, j) + 1): if j - i <= D - w: kraw += ((-1) ** i) * math.comb(w, i) * math.comb(D - w, j - i) total += kraw c_by_weight.append(total) dim = 1 << D return np.array([c_by_weight[popcount(x)] for x in range(dim)], dtype=complex) def build_D_sub(D): """D_sub = sum_a chi_a^Cliff via Jordan-Wigner.""" n = 1 << D M = np.zeros((n, n), dtype=complex) for a in range(D): for x in range(n): xp = x ^ (1 << a) sign = 1 for b in range(a): if (x >> b) & 1: sign *= -1 M[xp, x] += sign return M def spectral_projectors(D_sub_mat, D): n = D_sub_mat.shape[0] sqrtD = math.sqrt(D) Pp = (np.eye(n, dtype=complex) + D_sub_mat / sqrtD) / 2.0 Pm = (np.eye(n, dtype=complex) - D_sub_mat / sqrtD) / 2.0 return Pp, Pm def main(): print("=" * 90) print(" Computation 55 -- Closed-form sigma_2 = D - 2 of the Walsh-tail block") print("=" * 90) print() print(" Explicit S_{D-1}-fixed input vectors:") print(" u_1 = |e_0> - (1/D) sum_a |e_a> in [D-1, 1] cap H_1") print(" u_2 = -sum_{b > 0} |{0, b}> in [D-1, 1] cap H_2") print() print(" Verify algebraic identities, then verify || M_analytic - M_numerical ||") print(" and the closed-form sigma_max = D - 2.") print() kD = 2 print(f" {'D':>3} {'D_sub u1=u2':>15} {'D_sub u2=D u1':>15} " f"{'T u1=beta u1':>14} {'T u2=gamma u2':>15} " f"{'||M_a - M_n||':>14} {'sigma_max':>12} {'target':>8}") for D in range(4, 9): n = 1 << D D_mat = build_D_sub(D) T_tail_diag = build_T_tail_diagonal(D, kD) T_tail = np.diag(T_tail_diag) Pp, Pm = spectral_projectors(D_mat, D) T_pm = Pp @ T_tail @ Pm # Build u_1, u_2 u1 = np.zeros(n, dtype=complex) for a in range(D): u1[1 << a] = -1.0 / D u1[1] += 1.0 u2 = np.zeros(n, dtype=complex) for b in range(1, D): u2[(1 << 0) | (1 << b)] = -1.0 # Identities (a)-(d): err_a = la.norm(D_mat @ u1 - u2) err_b = la.norm(D_mat @ u2 - D * u1) beta = -math.comb(D - 1, 2) gamma = (D - 2) * (5 - D) / 2 err_c = la.norm(T_tail @ u1 - beta * u1) err_d = la.norm(T_tail @ u2 - gamma * u2) # Normalize: e1 = u1 / la.norm(u1) e2 = u2 / la.norm(u2) # Numerical M: basis = np.stack([e1, e2], axis=1) M_num = basis.conj().T @ T_pm @ basis # Analytical M: M_ana = (D - 2) / 2.0 * np.array([[-1.0, 1.0], [-1.0, 1.0]], dtype=complex) err_M = la.norm(M_num - M_ana) # Saturating right singular vector: v_star = (e2 - e1) / math.sqrt(2) sigma_max = la.norm(T_pm @ v_star) / la.norm(v_star) print(f" {D:>3} {err_a:>15.2e} {err_b:>15.2e} " f"{err_c:>14.2e} {err_d:>15.2e} " f"{err_M:>14.2e} {sigma_max:>12.6f} {D-2:>8}") print() print("=" * 90) print(" Findings") print("=" * 90) print() print(" (a) D_sub u_1 = u_2 and D_sub u_2 = D u_1 are exact identities (no") print(" weight-3 leakage in D_sub u_2 because each triple {0, b, c}") print(" receives contributions +|{0,b,c}> and -|{0,b,c}> from the (b, c)") print(" and (c, b) JW orderings that cancel).") print() print(" (b) T_tail u_1 = beta u_1 (weight-1) and T_tail u_2 = gamma u_2") print(" (weight-2) follow from T_tail being diagonal in the computational") print(" basis with eigenvalues depending only on Hamming weight.") print() print(" (c) The 2-x-2 multiplicity-space matrix") print() print(" M = (D-2)/2 . [[-1, +1], [-1, +1]]") print() print(" is RANK 1, with sigma values (D - 2, 0). Combined with the") print(" closed form sigma_1 = 2^(D-1) - D on the trivial S_D-component") print(" (Comp 52) and the vanishing of all higher 2-row irreps under") print(" the fermionic rho_F (Comp 54), the operator norm of the") print(" Walsh-tail off-diagonal block is") print() print(" || P_+ T_tail P_- ||_op = 2^(D-1) - D exactly,") print() print(" for every D >= 4 at k_D = 2.") if __name__ == "__main__": main()