Choose one function class and prove a uniform rank/degree certificate.
Acceptance: A source-faithful class predicate and formula-to-core action theorem, including phases and supports.
planned; no claimed closure
Structure before circuit tricks
High-dimensional structured function states
Multivariate PDE data, localized scientific functions and conditional distributions need more than generic amplitude loading.
research target setting:structured-tt-supplied-cores
\[|f\rangle=\frac{1}{\|f\|_{2,N}}\sum_{\mathbf j\in[N]^D}f(\mathbf x_{\mathbf j})|\mathbf j\rangle,\quad N=2^n\]
Frozen target and input model
Prove a constructive cost and state-error theorem for a precisely specified low-rank, sparse-frequency, mixed-smoothness or compositional class; do not equate these assumptions.
Access model: An explicit formula-to-core supplier or a charged sparse coefficient oracle; grid, complex phase and a nonzero norm are part of the input.
- D and n specified; rank r and degree q are certified, not numerical fit labels
- Core/coefficients and normalizer are computably supplied
- Approximation error includes model compression, grid, arithmetic and primitive synthesis
Desired result, not an achieved bound
A candidate target is poly(D,r,q,n,log(1/epsilon)) cost under stated bit-length and stability promises; this bound is not claimed here.
Generic tensor-product degree-q expansions have (q+1)^D coefficients; smoothness alone does not remove dimensionality. Hermite's one-dimensional exact-real compiler is a substrate, not this multivariate theorem.
Dependency-ready execution route
Bound normalized-state error from core approximation and arithmetic.
Acceptance: An explicit norm lower bound and a theorem for the exact requested vector norm; no uncharged condition number.
planned; no claimed closure
Compose the class supplier with the existing clean TT compiler.
Acceptance: Primitive-list semantics, workspace cleanup, and every cost coordinate under the same model.
planned; no claimed closure
Next bounded advance: Start from the Hermite corridor; add a tensor-product or shallow coupled class with a proved rank bound before claiming general high dimension.
Reusable mathematics
Named Lean substrates below have their own exact signatures. They do not certify every sentence or proposed generalization on this page.
QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec
Lower-bound comparison contract
model-definition-pending
Separate arbitrary-vector counting barriers from lower bounds for the selected structured class. Specify which class parameters enter the hard family.
Same-model key: setting:structured-tt-supplied-cores
Diagnostic examples, not proofs
- Product Gaussian with explicitly bounded widths
- Weakly coupled Gaussian with a proved rank estimate
- Hermite-smoothed PDE auxiliary profile
Primary-source ledger
primary-metadata-checked Abstract: explicit function to MPS to circuit; requested v1 HTML was unavailable in this audit
Prior art for function-to-MPS preparation. The ASPBE Hermite specialization must not be credited with inventing polynomial-to-MPS or sequential MPS preparation.
primary-metadata-checked Abstract and article record
Multivariate preparation baseline; a tensor-product coefficient count is not automatically polynomial in dimension.
primary-text-checked Abstract, Introduction and Discussion: shallow compositional state preparation
Composition-dependent bounds, not an efficient preparation theorem for every continuous multivariate function.
Download bounded agent / contributor packet
python3 website/scripts/research_atlas.py context --route spw-structured