Define and prove an isometric finite-to-continuous embedding.
Acceptance: Inner products and normalization match the stated quadrature/Fock convention.
planned; no claimed closure
Structure before circuit tricks
CV–DV function preparation and non-Gaussian resources
Schrödingerisation uses auxiliary profiles in both continuous-variable and qubit implementations, but their resources are not interchangeable.
research target setting:finite-energy-cv-dv-embedding
\[\|V_N|f_N\rangle-|f\rangle_{L^2}\|\le\epsilon_{\rm trunc}+\epsilon_{\rm grid}+\epsilon_{\rm prep}\]
Frozen target and input model
Relate a declared finite-energy oscillator encoding to qubit-grid preparation with a rigorous embedding and separate resource budgets.
Access model: An explicit isometric embedding V_N, cutoff, grid or Fock truncation, energy/squeezing and non-Gaussian gate model.
- Finite-energy/domain conditions and target normalization
- The embedding and quadrature weights are fixed
- CV noise, squeezing and non-Gaussian resources are not counted as free qubits
Desired result, not an achieved bound
A model-specific conversion, approximation and preparation theorem, not a universal equivalence of CV and DV costs.
The finite Hermite state is a useful first example. No CV–DV equivalence or GKP preparation theorem is currently supplied by this route.
Dependency-ready execution route
Prove energy-tail and discretization error for a fixed profile family.
Acceptance: All cutoff, domain and regularity hypotheses explicit.
planned; no claimed closure
Compose preparation and conversion costs.
Acceptance: Separate DV gates and CV physical resources plus total state error.
planned; no claimed closure
Next bounded advance: Choose a finite-energy Hermite-smoothed profile and one embedding before comparing grid, oscillator or GKP implementations.
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
source-survey-pending
First specify the energy and non-Gaussian resource model; no cross-model lower-bound transfer is asserted.
Same-model key: setting:finite-energy-cv-dv-embedding
Diagnostic examples, not proofs
- Hermite-smoothed auxiliary profiles
- Gaussian plus controlled non-Gaussian perturbation
Primary-source ledger
No external source is attached to this local mechanism note. It remains authored exposition, not a literature-priority claim.
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python3 website/scripts/research_atlas.py context --route spw-cvdv