Exercise PLAN-214

A Hermitian clean block does not force a Hermitian unitary completion. Definition9.23 adds global self-adjointness; the exact scaled qubitization proof is Eq10.30-10.40, with2D interior sectors and1D endpoint sectors.

1. Only the clean compression A/alpha is prescribed by an exact block encoding; the remaining coupled unitary-completion conditions do not force U_A=U_A^dagger.
2. An admissible rotation completion for A=(1/2)I_2 has the required amplitude-oracle first column but is not Hermitian; other completions of that first column may be Hermitian.
3. Section9.8/Definition9.23 adds global U_A=U_A^dagger, strictly stronger than Hermiticity of the encoded block.
4. For exact HBE and eigenvalues of C=A/alpha strictly inside[-1,1], U_A^2=I closes the2D clean/leakage sector as in Eq10.30-10.35; at ±1 the clean sector is1D.
5. Combining U_A with Z_P yields the rotation and exact Chebyshev clean block T_k(C) in Eq10.38-10.41; no unproved approximate or finite-bit guarantee is inferred.

(i) Write P=|0^m><0^m| tensor I. For positive normalizer alpha, an exact block encoding constrains only the clean compression: (<0^m| tensor I) U_A (|0^m> tensor I)=A/alpha. The other blocks are not independently arbitrary: they must jointly complete a unitary. But that unitarity constraint does not require the whole completion to be self-adjoint. Thus a Hermitian encoded block need not imply U_A=U_A^dagger.

(ii) A concrete completion consistent with Example9.17 is the following. Take A=(1/2)I_2 and one ancilla qubit, set b=sqrt(3)/2, and choose R=[[1/2,-b],[b,1/2]], U_A=R tensor I_2, with the ancilla first. Then R^T R=I_2, so U_A is unitary, and its clean block is A. On input |0>|i> it produces ((1/2)|0>+b|1>)|i>, exactly the source amplitude-oracle action for this diagonal A. Yet R^dagger=R^T has the opposite off-diagonal signs and differs from R, so U_A is not Hermitian. More generally use a real a in(0,1) and b=sqrt(1-a^2). The specified |0> input action alone does not uniquely determine the completion on |1>: [[a,b],[b,-a]] is a different, Hermitian-unitary completion with the same first column. The point is that a non-Hermitian completion is allowed, not that every amplitude oracle must be non-Hermitian.

(iii) Definition9.23 in Section9.8, printed159, calls a block encoding Hermitian when it also satisfies the GLOBAL condition U_A=U_A^dagger. This is strictly stronger than merely encoding Hermitian A, as the preceding example proves. The nontrivial qubitization closure is in Section10.2.1, printed167-168, Eqs10.30-10.40; the old references9.30/9.34 (and9.30-9.40 in historical primary metadata) are incorrect equation locators, not alternative versions of this proof.

Here is the precise exact-encoding motivation. If U_A is an exact Hermitian block encoding with scale alpha, put C=A/alpha. Compression of a unitary implies ||C||<=1. For an eigenvector v of C with eigenvalue lambda, let a_vec=|0^m>v. Then U_A a_vec=lambda a_vec+e_perp, with P e_perp=0 and ||e_perp||^2=1-lambda^2, as in Eq10.30-10.33. If |lambda|<1, define b_vec=e_perp/sqrt(1-lambda^2). Hermitian unitarity gives U_A^2=I; applying U_A a second time gives U_A b_vec=sqrt(1-lambda^2)a_vec-lambda b_vec (Eq10.34). Consequently span{a_vec,b_vec} is invariant and U_A has matrix [[lambda,sqrt(1-lambda^2)],[sqrt(1-lambda^2),-lambda]] there (Eq10.35), a reflection. Z_P=2P-I has matrix diag(1,-1) on this basis, so the iterate Q=U_A Z_P has matrix [[lambda,-sqrt(1-lambda^2)],[sqrt(1-lambda^2),lambda]], a rotation (Eq10.38-10.39). Its kth power has clean diagonal entry T_k(lambda), giving the exact clean block T_k(C), with source Eq10.40-10.41. When alpha=1 this is T_k(A).

If lambda=+1 or-1, e_perp=0 and the clean vector already spans a ONE-dimensional invariant subspace. No normalized leakage vector is needed or divided by zero; the same clean-block identity follows from lambda^k=T_k(lambda). In particular one must not claim every eigenvector necessarily yields a nontrivial2D sector. For an approximate block encoding, self-adjointness still defines HBE, but the preceding exact formula applies to its actual compression; it is not automatically an exact T_k(A/alpha) identity or an error bound for the requested A. These distinctions explain the structural usefulness of Hermiticity without claiming it is necessary for every more general qubitization construction. This is source/NL algebra, not a freshly executed circuit, finite-bit bound or local Lean certificate.

1. States the exact clean compression A/alpha and distinguishes coupled unitary-completion constraints from global self-adjointness; does not infer whole-unitary Hermiticity from the encoded block. (1 weight)
2. Gives a unitary non-Hermitian rotation completion matching the amplitude-oracle |0> action and a Hermitian diagonal clean block, without claiming the first-column specification uniquely forces that completion. (1 weight)
3. Correctly states Section9.8/Definition9.23's extra global requirement U_A=U_A^dagger, and identifies Section10.2.1/Eq10.30-10.40 as the actual qubitization proof locators. (1 weight)
4. Explains exact normalized closure through U_A^2=I and two reflections for |lambda|<1; handles lambda=±1 as1D, gives T_k(A/alpha) at general scale, and does not extend an exact identity to arbitrary approximate HBE without an error argument. (1 weight)

No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.

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