Numerical Benchmarks

Numerical benchmarks test the computation itself: convergence, conservation laws, stability, matrix properties, finite-size behavior, and regression against previously validated outputs. They complement analytic benchmarks. A code can match one exact energy and still fail as a numerical method.

Convergence Tests

A convergence test varies one refinement parameter at a time:

• grid spacing,
• box size,
• basis size,
• time step,
• solver tolerance,
• sample size,
• bond dimension or truncation threshold.

Record the observed change in a physical observable, not only the change in a plot. If two refinement parameters interact, state the refinement path explicitly.

Matrix Assembly Checks

For Hamiltonian matrices, check:

• Hermiticity under the correct inner product,
• dimensions and basis ordering,
• boundary rows,
• units of kinetic and potential terms,
• symmetry blocks when block diagonalization is expected.

For a closed finite-dimensional Hamiltonian, the numerical evolution operator should satisfy

$$U^\dagger U\approx I.$$

The tolerance should reflect the time stepper and floating-point scale.

Normalization and Conservation

Closed-system wavefunction calculations should monitor:

• norm conservation,
• energy conservation for time-independent Hamiltonians,
• expected symmetry quantum numbers,
• probability current conservation in scattering.

Open-system calculations should monitor:

• trace preservation,
• Hermiticity of the density operator,
• positivity or at least absence of unphysical negative eigenvalues beyond tolerance,
• complete-positivity assumptions when a channel representation is used.

Norm conservation is necessary but not sufficient. A method can conserve norm while producing a wrong phase, width, current, or transition probability.

Stochastic Benchmarks

Monte Carlo, quantum trajectories, and randomized algorithms need statistical validation:

• fixed seed for reproducibility,
• multiple seeds for uncertainty,
• sample size,
• estimator definition,
• confidence interval or standard error,
• autocorrelation or effective sample-size check when relevant.

A single seeded run is a regression test, not a statistical uncertainty estimate.

Regression Benchmarks

Regression benchmarks prevent previously validated outputs from drifting. They should store compact reference values, not large opaque outputs. Good regression targets include:

• first few eigenvalues,
• norm drift summary,
• trace-preservation summary,
• convergence slope,
• selected expectation values,
• hash or checksum for exported data when needed.

If a regression changes, investigate before updating the reference value. A changed result can be a bug fix, a dependency change, a convention change, or an actual regression.

Failure Diagnosis

When a benchmark fails, classify the failure:

Symptom	Likely Cause
correct low energies, wrong high energies	grid cutoff or basis truncation
norm conserved, packet width wrong	phase error, dispersion error, or wrong initial convention
$R+T\ne1$	incomplete separation, absorption, current normalization, or time-step error
density matrix trace preserved but negativity appears	integrator, generator, or complete-positivity issue
result changes with box size	finite-domain effect
result changes with package version	dependency or API drift

The diagnosis should be attached to the notebook or benchmark report.

Cross-Links

• Analytic Benchmarks
• Benchmark Problems
• Reproducibility Status
• Convergence Tests
• Error Estimates

References

• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.
• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• Y. Saad, Numerical Methods for Large Eigenvalue Problems, 2nd ed., SIAM, 2011.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.