Theory and Exactness Scope

This chapter defines what SchroSIM means by "exact" and where that claim stops.

1) Mathematical Conventions

SchroSIM’s CV Gaussian model uses:

  1. quadrature ordering R = (q1, p1, q2, p2, ...),
  2. units hbar = 1,
  3. Gaussian state parameterization by mean d and covariance V,
  4. symplectic form Omega = direct_sum_k [[0,1],[-1,0]].

Formal derivations are in 11-mathematical-proofs.md.

2) Exactness Taxonomy in SchroSIM

2.1 Model-Exact (Gaussian Core)

Within the Gaussian model assumptions, updates are exact identities:

  1. affine symplectic evolution:
  2. d' = S d + u
  3. V' = S V S^T
  4. Gaussian channels (loss, thermal-loss) in closed form,
  5. linear-Gaussian conditioning for homodyne and heterodyne updates.

This is the exactness regime referenced in the README for tractable workflows.

Terminology used across docs:

  1. model-exact: exact with respect to the implemented Gaussian model equations,
  2. controlled approximation: bounded/explicit approximation regimes (cutoff, truncation, routed non-Gaussian paths).

2.2 Deterministic Runtime Reproducibility

Given the same:

  1. circuit payload,
  2. seed,
  3. backend selection and runtime configuration,
  4. foundry profile/policy inputs,

SchroSIM targets deterministic replay at the result/trace level.

2.3 Approximation Regimes

Model-exactness does not imply exact infinite-dimensional quantum dynamics in every mode of operation. Controlled approximation enters when:

  1. Fock simulations use finite cutoff dimension,
  2. matrix exponential operators are series-truncated numerically,
  3. effective/derived non-Gaussian handling paths are used for practical workflows,
  4. backend routing selects an alternate compatible execution path.

3) Physicality and Validity Conditions

Core validity conditions include:

  1. uncertainty compatibility:
  2. V + i Omega/2 >= 0,
  3. channel parameter domains:
  4. eta in [0,1], n_th >= 0,
  5. IR parameter finiteness and mode-index validity,
  6. foundry policy constraints (max_modes, squeezing limits, measurement/non-Gaussian allowances).

4) Claim Boundary

SchroSIM claims:

  1. exact implementation of the stated Gaussian model equations,
  2. explicit policy-aware execution and diagnostics,
  3. deterministic, reproducible runtime behavior under fixed inputs/configuration.

SchroSIM does not claim:

  1. exact simulation of arbitrary infinite-dimensional non-Gaussian dynamics without truncation,
  2. hardware-faithful behavior without calibrated model assumptions,
  3. approximation-free Fock evolution at finite cutoff.

5) Practical Interpretation for Users

Use this decision rule:

  1. For Gaussian and tractable circuits: treat SchroSIM results as model-exact.
  2. For Fock/non-Gaussian-heavy or very large circuits: treat results as controlled approximations and report cutoff/backend settings with outputs.
  3. For pre-hardware studies: always bind runs to foundry profile/policy identifiers and seeds.

6) Cross-References