Representations and Numerical Reliability

A trustworthy quantum calculation connects a physical question to a finite representation, a numerical algorithm, and checks of the reported observable’s accuracy. This chapter supplies the common language for that chain: basis and array conventions, conditioning and stability, refinement, independent benchmarks, and the cost of reliable computation.

Enter this chapter

Begin with representing states and operators as arrays. It establishes basis ordering, conjugation, and the difference between a normalized coefficient vector and normalized wavefunction samples. Continue to basis choice and representation for projection error and nonorthogonal overlap matrices, then dense, sparse, and matrix-free operators to choose how an algorithm accesses its Hamiltonian.

Before assigning tolerances, work through numerical units and dimensionless variables. The same tolerance number can mean different physical errors in different units. A complete calculation records the scaled equation and restores the units of each observable correctly.

These pages assume the basic state, operator, and inner-product language of Core Formalism and the linear algebra in the Mathematical Toolkit. They concern numerical representations of that physics, rather than new postulates.

From a residual to an accuracy claim

Read Precision diagnostics before interpreting the last printed digits. Numerical error, conditioning, and stability then explains what a residual measures and why small residuals need not imply accurate eigenvectors or continuum observables.

For example, an approximate normalized eigenpair of a Hermitian matrix satisfies

$$\operatorname{dist}(\theta,\sigma(H))\le\|Hv-\theta v\|.$$

This bounds distance to the spectrum of the finite matrix. It does not bound the error caused by replacing a continuum Hamiltonian with that matrix. Convergence studies varies the approximations that a residual cannot test, and validation against analytic benchmarks supplies independent targets.

Use error budgets to identify the dominant limitation and convergence reports to explain the results. A refined mesh, a large sample count, and a conserved norm answer different questions; the report should say which ones were checked.

Algebra and symmetry as independent checks

Symbolic operator algebra automates ordered products while keeping assumptions and truncation boundaries explicit. Clebsch–Gordan computation develops exact quantum-number bookkeeping, phase conventions, and orthogonality tests. Perturbation theory automation generates coefficients through projected recurrences and verifies them against a small exact problem.

Symmetry and selection-rule checking turns physical structure into tests of commutators, forbidden transitions, and sector leakage. It also explains why an intentionally broken symmetry should not be enforced by the implementation. Use these diagnostics alongside numerical refinement: a wrong Hamiltonian can preserve all the symmetries of the intended one.

Scaling a calculation without changing its claim

High-performance computing basics introduces memory costs, arithmetic intensity, and strong/weak scaling. GPU and accelerator notes examines data residency, asynchronous timing, and precision. Parallelization patterns covers independent tasks, domain decomposition, reductions, and random-stream assignment.

Performance comparisons should preserve the observable and its error criteria. Reducing precision, shortening trajectories, or restricting a symmetry sector may change the scientific problem. Check those choices before interpreting a shorter runtime as an improvement.

A first complete calculation

A productive first route is to choose the infinite square well, represent its kinetic operator with finite differences, and compute its lowest eigenpairs using matrix diagonalization. Compare first with the exact discrete spectrum, then with the continuum spectrum under grid refinement. Report weighted normalization, matrix residuals, observed order, and an observable-specific error estimate.

For time evolution, repeat the same logic with a driven two-level system. For sampling, use Monte Carlo basics and track uncertainty rather than treating random variation as deterministic convergence. Executable investigations and shared run instructions are collected in Labs.

References

• Golub, G. H., and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013 — structured numerical linear algebra.
• Higham, N. J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002 — arithmetic, conditioning, and error analysis.
• LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007 — discretization and refinement.
• Williams, S., A. Waterman, and D. Patterson, “Roofline: An Insightful Visual Performance Model for Multicore Architectures,” Communications of the ACM 52(4), 65–76 (2009), author repository — performance models.