Exercise SP-117

All ordered cut ranks support compatible TT/MPS and existential short arbitrary-real circuits via additional mathematics; they do not by themselves return a uniform finite-bit compiler. All-cut rank1 is a product state with a short circuit, yet may lack effective amplitude input. Current separate Hermite/TT compilers are acknowledged rather than erased.

1. The ordered Hermite cut factorization bounds width/rank; it is algebraic structural evidence, not an emitted gate list or finite-bit work certificate, while all-cut bounds have positive TT existence consequences through additional mathematics.
2. Scalar normalization preserves factor width but does not evaluate its normalizer or produce compatible canonical cores and finite-precision gates; those are separate interfaces whose actual current suppliers must be distinguished.
3. All-ordered-cut rank1 implies product structure and short arbitrary-real circuit existence; a generic non-effective product amplitude instead isolates the missing effective-input premise for a uniform finite-bit claim, not Hermite hardness or a gate-count lower bound.

The rank statement concerns a FIXED ordered prefix/suffix flattening with integer j=2^sWidth*x+y, not an arbitrary permutation of physical sites. FactorsThrough retains witnesses A,B with F=AB. Matrix rank is bounded by their common inner dimension. Scalar normalization changes A to A/Z, so it preserves this width. That algebraic scale operation works as a formula with a supplied scalar; it does not compute Z, provide an effective representation of its value, make different cut witnesses compatible, or return a gate list. For the actual target the table is nonempty and samples are strictly positive, so Z>0 and the normalized state is well defined. For N=0 there is one positive sample: a_0/Z=1, and the empty reference circuit suffices. It is not a generic zero-vector normalization case.

For contract I, all ordered cut ranks at most D have a substantive positive consequence. Successive exact Schmidt decompositions (or real SVDs for this real tensor) give a compatible MPS/TT with each internal bond dimension at most D and scalar endpoint bonds. This is ADDITIONAL mathematics beyond reading off the independent two-factor certificates. Choosing canonical gauges gives isometries from each incoming bond into the emitted qubit and next bond. Pad the active bond into dimension B=2^ceil(log2(max(1,D))); B<2D for D>1. Extend the active isometries to local unitaries on a fresh physical qubit plus that bond, preserving the required clean input columns. Sequentially emit the qubits and end with the bond clean. Standard finite-dimensional unitary/real-orthogonal decomposition then compiles each local unitary with polynomially many one/two-qubit gates in B (hence in D), giving N times a polynomial in D gates and O(log D) bond wires. Register reversal/reindexing between the MPS site order and physical little-endian order must be handled explicitly, not dropped. Gate parameters may be arbitrary real numbers under contract I; computing their finite descriptions and numerical values was NOT charged. Thus the rank information plus these mathematical existence/compilation results supports existential polynomial gate size. It is wrong to turn the certificate file's lack of an emitted gate list into a theorem that such circuits cannot exist.

Here is the elementary all-cut rank-one check. For N=1 every normalized state is already a one-qubit factor. For N>1, rank one at the first site/rest flattening writes the nonzero tensor as u_1 tensor v, with u_1 and v normalized after moving a scalar/sign between them. For each later cut, its flattening is the residual flattening of v tensored with the nonzero vector u_1; multiplication by that nonzero vector is injective and leaves the rank unchanged. Hence all internal ordered flattenings of v also have rank one. Induction gives u_1 tensor ... tensor u_N. A real normalized factor u_j=(alpha_j,beta_j) is the first column of a real planar rotation [[alpha_j,-beta_j],[beta_j,alpha_j]], realized by an arbitrary-real RY angle; any overall sign can be absorbed into one factor for N>=1. One such gate per qubit gives exact O(N) gate existence without a bond register. In contrast, rank one at one chosen cut only gives a product of TWO possibly highly entangled multi-qubit factors. For example on four sites, Bell_{12} tensor Bell_{34} has rank one across12|34 but rank two across1|234 and123|4. The old arbitrary-factor argument therefore does not address rank one at every ordered cut.

A genuine algorithmic gap can coexist with the positive existence result. Choose a real q in(1/4,3/4) that is not computable, and consider for N>=1 the normalized product state (sqrt(q)|0>+sqrt(1-q)|1>) tensor |+>^(tensor(N-1)). Every ordered cut has rank one; an arbitrary-real RY parameter for the first factor and N-1 fixed RY(pi/2) gates prepare it in O(N) gates. But suppose there were a finite program with no oracle/effective representation for q that output certified fixed-computable-gate approximations to this state for every requested b. The finite output circuit has computable probabilities, and its first-qubit-zero probability approximates q. For normalized vectors, probability error for a projector is at most twice the vector2-norm error, so requesting sufficiently many extra precision bits would compute q to arbitrary accuracy, a contradiction. This is not a claim that q is an allowed finite input encoding: it demonstrates why a rank predicate that permits arbitrary reals does not itself supply the EFFECTIVE INPUT PROMISE needed by contract II. Even when entries are computable, computability alone supplies no polynomial evaluation bound. This illustration is a generic product-state discriminator, not a claim about the actual Hermite family, and not a quantum gate-size lower bound.

For contract II one must specify at least: an effective access/encoding and cost for L and the coefficient/core data; a method for producing compatible normalized TT cores and their canonical/completed stage data with a charged preprocessing bound; an evaluated nonzero normalizer with enough certified relative precision; computable gate-angle values and a finite-gate synthesis procedure; and a uniform propagated error/conditioning analysis that turns their local inaccuracies into the requested global target error. These are construction/cost promises, not additional hypotheses needed for the exact algebraic rank lemma. Normalization sensitivity matters: absolute raw-table error alone does not give a normalized-state error without control of the nonzero norm. One must not silently assign unit finite-bit cost to arbitrary exact-real comparisons, transcendental calls, or expanding an exponentially sized dense input. No assertion that every one of these obligations is absent from the entire current library follows.

Indeed the current library has separate constructions. HermiteStatePreparation.hermiteCircuit is a full-table exact-real reference loader with2^N-1 RY and2(2^N-1-N) CX gates. ConstructiveTensorTrain takes actual cores and produces a canonical chain/residual. ConstructiveTensorTrainCompiler.compile takes a supplied normalized scalar-boundary real chain plus a bond cap and returns an actual primitive circuit with all-sector clean-bond column theorem and polynomial bound in its bond dimension. ConstructiveHermitePreparation.prepare supplies a literal Hermite chain normalized through a local Gram norm and has an existing gate/depth bound48(n+1)(2k+6)^3 for N=n+1, with explicit clean bond wires and no opaque oracle. Its sharper2k+6 source-chain bond is not obtained just by identifying it with the8k+12 cut certificate. HermiteFiniteNorm supplies the same true norm through local environments; stored raw-source modules also provide actual unnormalized source data and separately scoped exact-real/selected-integer ledgers. These are additional interfaces and proofs, compatible with contract I and with genuine constructive progress, not objects returned by sampled_cut_rank_le. Their inspected signatures do not by themselves establish contract II's complete uniform finite-bit algorithm/runtime/error guarantee. A noncomputable annotation alone proves neither inefficiency nor incomputability.

The corrected verdict is therefore model-specific: the cut certificate is useful all-cut structure supporting existential short exact-real preparation via extra mathematics, while it alone is not a certified effective, finite-bit polynomial-time compiler. It is not a necessity theorem for every efficient circuit architecture, an impossibility result for Hermite samples, or a new execution/formal-view certificate.

1. States the exact ordered prefix/suffix factorization and normalization-scaling scope, actual normalized target/nonzero norm and N=0 positive singleton. Separates independent cut witnesses from already compatible TT cores. (1 weight)
2. Explains the positive all-cut TT/MPS/isometry/completion route to polynomial arbitrary-real gate-size existence with charged-as-data real angles and logarithmic bond, without confusing that additional mathematics with output of the rank declaration or claiming rank is necessary for arbitrary architectures. (1 weight)
3. Proves all-internal-ordered-cut rank1 implies a product by induction, gives O(N) arbitrary-real one-qubit preparation, and distinguishes one-cut rank1 (e.g. Bell12 tensor Bell34). Gives a valid effective-input/precision obstruction consistent with all-cut rank1, not an unsupported short-circuit lower bound or a hardness claim about Hermite. (1 weight)
4. Identifies at least two concrete effective-input/normalization/compatible-core/angle-synthesis/error/cost obligations for a uniform finite-bit compiler, acknowledges current separate constructive and norm/raw-source interfaces, and makes no absent-global-compiler, noncomputable-means-hard, or new kernel/closure claim. (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.

Read the Lean track and existing diagnostic warmups

• QuantumBlockEncoding/HermiteSampleStructure.lean

  lines 1-58: 'Neither a rank bound nor normalization supplies a primitive circuit compiler.' sampled_cut_rank_le : rank <= 8k+12; normalized_cut_factorization : the real amplitudes retain the same factor width after exact normalization (a factorization, not a compiler)

  Source SHA-256: eda04790bd59d39a49da0e98bc9602261ef7c5b0cdb531d88a97c3f6d1036d1f · MIT (repository LICENSE)

Contract: e5c98cae0eaae7d1e5cfc5785dbffaf7f537f9d0cdebae3fb688e7faf21672ea

Item digest: 4fbcba06bb019f260453d215171cfda9b196b17062d17eef6da10ced46d160c2

Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a

Authored support steps: 3. This is not measured graph distance or model difficulty.

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