Section 6 4 October 2026

Protected coherent transport in the same inventory

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131069

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6 Protected coherent transport in the same inventory

Actual archive storage and protection of unknown logical amplitudes are different tasks. The monograph's bounded internal gap construction can be implemented here without a stochastic interface. To display its nonempty domain, encode two qubits into four by

C\ket{a,b}=\frac{\ket{0,a,b,a\oplus b}+\ket{1,1\oplus a,1\oplus b,1\oplus a\oplus b}}{\sqrt2}.

Let $S_X=X_1X_2X_3X_4$ , $S_Z=Z_1Z_2Z_3Z_4$ , $P=(I+S_X)(I+S_Z)/4$ and $Q=I-P$ , with identities on the retained internal nuisance systems and inaccessible reference understood. Set

H_{\rm pen}=\tfrac\Delta2(I-S_X)+\tfrac\Delta2(I-S_Z),\quad H_\Delta=H_{\rm pen}+H_0+V.

The penalty satisfies $H_{\rm pen}P=0$ and $H_{\rm pen}\ge\Delta Q$ . Assume the bounded self-adjoint operators are stationary on the protected exposure, with $[H_0,P]=0$ , $\norm{H_0}\le b$ , and

V=\sum_{i=1}^4\sum_{\alpha=x,y,z}\sigma_i^\alpha\otimes B_{i\alpha}, \quad B_{i\alpha}=B_{i\alpha}^\dagger,\quad\norm V\le v.

The $B$ 's act on retained finite internal nuisance systems. During the protected exposure, the spatial holding Hamiltonian is a commuting spectator; it is factored out. We do not assert a bounded-norm theorem for arbitrary unbounded coordinate couplings. One-site Paulis anticommute with a stabilizer, so $PVP=0$ . All encoding and decoding gates are finite internal unitaries.

Proposition 6.1 (Retained-bank gap bound)

For $\Delta-2b-v=\gamma>0$ ,

\norm{\bigl(e^{-itH_\Delta/\hbar}-e^{-itH_0/\hbar}\bigr)P} \le\min\left\{2,\frac{2v+tv^2/\hbar}{\gamma}\right\}.

(26)

The same bound holds with every inaccessible reference and retained internal nuisance system included.

Proof

Decompose the complete internal bank as $\operatorname{ran}P\oplus\operatorname{ran}Q$ and define its compressed blocks

A=PH_0P,\qquad B=QVP,\qquad D=QH_\Delta Q,

where $A$ and $D$ act on their respective subspaces and $B:\operatorname{ran}P\to\operatorname{ran}Q$ . Then

H_\Delta=H_d+W,\qquad H_d=\begin{pmatrix}A&0\\0&D\end{pmatrix},\qquad W=\begin{pmatrix}0&B^\dagger\\B&0\end{pmatrix}.

The stated bounds imply

D\ge(\Delta-b-v)Q=(b+\gamma)Q, \qquad A\le bP,\qquad \norm B\le v.

Thus the norm-convergent integral

X=\int_0^\infty e^{-rD}Be^{rA}\dd r

satisfies $DX-XA=B$ and $\norm X\le v/\gamma$ . Indeed the integrand has norm at most $v e^{-r\gamma}$ ; differentiating it and integrating its vanishing boundary term gives the identity. Here $r$ has inverse-energy units; it is not physical time. The skew-adjoint block operator

S=\begin{pmatrix}0&-X^\dagger\\X&0\end{pmatrix} \quad\text{satisfies}\quad [S,H_d]=-W,\qquad \norm S=\norm X.

Put $f(u)=e^{uS}We^{-uS}$ . Differentiation and integration yield

\begin{aligned}e^SH_\Delta e^{-S} &=H_d+f(1)-\int_0^1f(u)\dd u=H_d+R,\\ R&=\int_0^1u e^{uS}[S,W]e^{-uS}\dd u. \end{aligned}

These conjugations are unitary, so

\norm R\le\tfrac12\norm{[S,W]}\le v^2/\gamma, \qquad \norm{e^{\pm S}-I}\le\norm S\le v/\gamma.

Two changes of frame and Duhamel in physical time therefore give

\norm{e^{-S}e^{-it(H_d+R)/\hbar}e^S-e^{-itH_d/\hbar}} \le\frac{2v}{\gamma}+\frac{tv^2}{\hbar\gamma}.

On $\operatorname{ran}P$ , the last unperturbed propagator agrees with $e^{-itH_0/\hbar}$ . The trivial norm bound two completes (26). Tensoring an identity preserves each operator norm, so the same estimate retains the inaccessible reference and nuisance bank.

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This is the monograph's coherent protection estimate, with its assumptions preserved; Hamiltonian error suppression has independent primary precedent [11]. Its role here is compatibility with actual material writes and records, not selection of a noise generator.