210 lines
7.7 KiB
Python
210 lines
7.7 KiB
Python
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"""
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ghz3_perturbation_symbolic.py
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Exact symbolic (sympy) first-order degenerate perturbation theory for the
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GHZ3 shadow map under single-qubit dephasing noise on party B.
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Background / formalism
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-----------------------
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At rho0 = GHZ3, the normalized combined shadow map M_A(rho0) has an exactly
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threefold-degenerate singular value sigma = sqrt(2/3), with orthonormal
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bases U0 (15x3, target side) and V0 = I_3 (3x3, source side) -- see
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ghz3_shadow_map_symbolic.py.
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For a perturbation rho(eps) = rho0 + eps * delta_rho, the map itself is
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exactly linear: M_A(rho(eps)) = M_A(rho0) + eps * M_A(delta_rho).
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To first order in eps, the perturbed singular values within the degenerate
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block are
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sigma_i(eps) = sigma + eps * lambda_i(K) + O(eps^2),
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where K is the symmetrized projection of the perturbing map onto the
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degenerate subspace:
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K = (1/2) * ( U0^T M_A(delta_rho) V0
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+ V0^T M_A(delta_rho)^T U0 ).
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This is the direct analogue, for singular values, of ordinary degenerate
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perturbation theory for Hermitian eigenvalues. K is automatically real
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symmetric here because U0, V0 are real orthonormal bases.
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Physical perturbation studied here: single-qubit dephasing on party B,
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i.e. the (unnormalized, direction-only) Lindbladian jump direction
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delta_rho^(P) = P_B rho0 P_B - rho0, P in {X, Y, Z}.
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P = Z models T2-type dephasing in the computational (stabilizer) basis --
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the dominant error channel on most physical qubit platforms. P = X, Y model
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dephasing along an axis that does not commute with the GHZ3 stabilizer
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group.
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We show, exactly:
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- Z-dephasing on B: eigenvalues of K are {-2*sqrt(6)/3 (x2), 0 (x1)}
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-> the "z" target-response channel is exactly protected to first order,
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while the "x","y" channels decay at twice the generic rate.
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- X- or Y-dephasing on B: eigenvalues of K are {-sqrt(6)/3 (x3)}
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-> fully isotropic decay, no protected direction.
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The physical reason: Z_A Z_B is a stabilizer generator of GHZ3 and commutes
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with Z_B, so the z-channel survives Z_B-dephasing unchanged to first order;
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X_AX_B, Y_AY_B do not commute with Z_B and decay.
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A numeric finite-difference cross-check (against the singular values of the
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exactly perturbed matrix, not just the leading-order K prediction) is
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included at the end.
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Run: python3 ghz3_perturbation_symbolic.py
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"""
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import sympy as sp
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from sympy import sqrt, I, simplify, Matrix, eye, zeros, re, Rational, N
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# ---------------------------------------------------------------------
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# 1. Rebuild the same primitives as in ghz3_shadow_map_symbolic.py
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# (kept self-contained so this script can be run standalone)
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# ---------------------------------------------------------------------
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X = Matrix([[0, 1], [1, 0]])
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Y = Matrix([[0, -I], [I, 0]])
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Z = Matrix([[1, 0], [0, -1]])
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I2 = eye(2)
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PAULIS = {'x': X, 'y': Y, 'z': Z}
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def kron(A, B):
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mA, nA = A.shape
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mB, nB = B.shape
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out = zeros(mA * mB, nA * nB)
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for i in range(mA):
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for j in range(nA):
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out[i * mB:(i + 1) * mB, j * nB:(j + 1) * nB] = A[i, j] * B
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return out
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def kron3(a, b, c): return kron(kron(a, b), c)
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def op_A(P): return kron3(P, I2, I2)
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def op_B(P): return kron3(I2, P, I2)
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def op_C(P): return kron3(I2, I2, P)
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def ghz3_state():
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psi = zeros(8, 1)
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psi[0, 0] = 1 / sqrt(2)
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psi[7, 0] = 1 / sqrt(2)
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return simplify(psi * psi.H)
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def entry(rho, ops):
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M = None
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for op in ops:
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M = op if M is None else M * op
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return simplify(re(simplify((rho * M).trace())))
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def build_M(rho):
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"""Same 15x3 unnormalized shadow-map matrix as in the companion script."""
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rows = []
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for pb in ['x', 'y', 'z']:
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rows.append([entry(rho, [op_A(PAULIS[pa]), op_B(PAULIS[pb])])
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for pa in ['x', 'y', 'z']])
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for pc in ['x', 'y', 'z']:
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rows.append([entry(rho, [op_A(PAULIS[pa]), op_C(PAULIS[pc])])
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for pa in ['x', 'y', 'z']])
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for pb in ['x', 'y', 'z']:
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for pc in ['x', 'y', 'z']:
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rows.append([entry(rho, [op_A(PAULIS[pa]), op_B(PAULIS[pb]), op_C(PAULIS[pc])])
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for pa in ['x', 'y', 'z']])
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return Matrix(rows)
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NORM_CONST = 1 / sqrt(3) # combined-map normalization for n=3 qubits
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# ---------------------------------------------------------------------
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# 2. Degenerate-perturbation-theory machinery
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# ---------------------------------------------------------------------
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def Kmatrix(delta_rho, U0, V0):
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"""Symmetrized first-order splitting matrix for the degenerate block
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spanned by (U0, V0), given a perturbation direction delta_rho."""
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Md = simplify(NORM_CONST * build_M(delta_rho))
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A = simplify(U0.T * Md * V0)
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return simplify(Rational(1, 2) * (A + A.T))
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def main():
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rho0 = ghz3_state()
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M0 = build_M(rho0)
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Mn0 = simplify(NORM_CONST * M0)
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# Degenerate subspace bases (see companion script for derivation):
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# Gram matrix Mn0^T Mn0 = (2/3) I_3 exactly, so V0 = I_3 and
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# U0 = Mn0 rescaled to unit-norm columns.
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sigma = sqrt(Rational(2, 3))
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U0 = simplify(Mn0 / sigma)
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V0 = eye(3)
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print(f"Unperturbed degenerate singular value: sigma = {sigma} "
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f"= {float(sigma):.6f} (should be sqrt(6)/3, threefold)\n")
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ops_B = {
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'Z (T2-type, computational-basis dephasing)': op_B(Z),
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'X': op_B(X),
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'Y': op_B(Y),
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}
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K_store = {}
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for label, OB in ops_B.items():
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delta_rho = simplify(OB * rho0 * OB - rho0)
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K = Kmatrix(delta_rho, U0, V0)
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K_store[label] = (K, delta_rho)
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print("=" * 70)
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print(f"Dephasing on qubit B along {label}")
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print("K =")
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sp.pprint(K)
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eigs = K.eigenvals()
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print("Exact eigenvalues of K (first-order singular-value shifts):")
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for ev, mult in eigs.items():
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print(f" {sp.nsimplify(ev)} (multiplicity {mult}) "
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f"= {float(ev):.6f}")
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print(f"trace(K) = {simplify(sp.trace(K))} "
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f"= {float(sp.trace(K)):.6f} "
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f"(this is d/d(eps) ||M_A(rho(eps))||_* at eps=0)\n")
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# -------------------------------------------------------------
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# 3. Numeric finite-difference cross-check (independent of the
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# symbolic K-matrix machinery): compute the *exact* singular
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# values of M_A(rho0 + eps*delta_rho) for small eps and compare
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# to sigma + eps*lambda_i(K).
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# -------------------------------------------------------------
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print("=" * 70)
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print("Finite-difference cross-check for Z-dephasing on B")
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print("(exact singular values of the perturbed matrix vs. first-order "
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"prediction from K)\n")
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K_Z, delta_rho_Z = K_store['Z (T2-type, computational-basis dephasing)']
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eig_list = sorted(K_Z.eigenvals().keys(), reverse=True) # e.g. [0, -2sqrt6/3, -2sqrt6/3]
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# build the multiset of 3 eigenvalues (respecting multiplicity)
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eig_multiset = []
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for ev, mult in K_Z.eigenvals().items():
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eig_multiset += [ev] * mult
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eig_multiset = sorted(eig_multiset, reverse=True)
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for eps_val in [sp.Rational(1, 100), sp.Rational(1, 1000)]:
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rho_eps = rho0 + eps_val * delta_rho_Z
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M_eps = simplify(NORM_CONST * build_M(rho_eps))
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G_eps = simplify(M_eps.T * M_eps)
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sv_exact = sorted([sp.sqrt(ev) for ev in G_eps.eigenvals().keys()
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for _ in range(G_eps.eigenvals()[ev])],
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key=lambda v: float(v), reverse=True)
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sv_predicted = sorted([sigma + eps_val * ev for ev in eig_multiset],
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key=lambda v: float(v), reverse=True)
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print(f"eps = {eps_val} :")
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print(" exact singular values:", [f"{float(v):.6f}" for v in sv_exact])
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print(" 1st-order prediction :", [f"{float(v):.6f}" for v in sv_predicted])
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print()
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if __name__ == "__main__":
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main()
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