26Interpreting the scientific output

A precise number is the beginning of a review

Suppose a colleague sends the number −108.770051792909 Hartree and says that a quantum calculation reproduced a nitrogen ground-state energy. The first task is not to admire its decimal places. It is to identify the molecule, model, state, operator, method, energy offset, measurement procedure, and reference that give the number meaning.

That number appears in the repository’s completed tutorial record. The record describes the documented QDK Chemistry 2.2.0 workflow, whose version pin is defined in tutorials/_versions.py. We are reading that record at source revision 97a3ff4f; we have not reproduced the computation in this session.

The target is neutral, intended-singlet N₂ at 1.85 Å, with a selected active-space Hamiltonian derived from the 28-spatial-orbital cc-pVDZ model. It is a fixed-nuclear-geometry electronic problem, not the equilibrium molecule’s free energy, a reaction rate, or experimental confirmation of quantum advantage. The separate notebook at 1.270025 Å is a different calculation and cannot silently supply missing evidence.

This lesson turns the entire workflow into an applied interpretation discipline. Each stage owns a different kind of statement. The final energy comparison is useful precisely when those statements remain separate.

Reconstructing the recorded nitrogen chain

The molecular input script fixes the bond distance, charge, and multiplicity, then compares two finite-basis Hartree–Fock calculations. The active-space workflow tests record the subsequent valence and refined CASCI results.

The following are recorded source reference values, not results executed here:

The HF basis change is about 26.582 milliHartree in magnitude. It measures sensitivity between two calculations, not the exact error of either basis relative to a complete-basis result. It also does not measure the missing correlation error of HF. Increasing basis flexibility and improving the many-electron state are different interventions.

The active-space reduction raises the recorded CASCI energy by 0.007317727973 Hartree, about 7.318 milliHartree, relative to the larger valence-space model. That is already larger than the tutorial’s later one-milliHartree algorithmic target. There is no contradiction: the later target asks whether the quantum workflow reproduces the smaller selected Hamiltonian, not whether that model equals the larger-space or real-molecule answer.

The candidate CAS(10e,8o) has fixed active populations (5,5) and 3,136 determinants. The refined CAS(6e,6o) has populations (3,3), 400 determinants, active full-MO indices 4–9, inactive indices 0–3, and eighteen excluded virtual orbitals. Indices are identifiers within a specified orbital representation, not universal chemical labels valid at another geometry.

The tutorial computes natural orbitals from correlated data and solves CASCI again; the recorded energy is unchanged by that representation change before reducing the active space. This is a meaningful consistency check. Merely receiving a localizer’s determinant carrier with rotated orbitals would not establish that the original correlated coefficients were preserved. The fresh correlated solve is part of the evidence chain.

The model is intended to target a singlet, but separate alpha/beta populations constrain spin projection, not generally total $S^2$. An interpretation claiming certified spin purity needs suitable state evidence, not only spin_multiplicity=1 at SCF or a count-derived metadata property. The reported fixed-sector reference should be described as the configured determinant-sector calculation with its stated intended physical state.

Which object owns each constant

The term “total energy” is dangerous without a definition. In the ordinary active-space molecular route,

The scalar includes nuclear repulsion and constant frozen-inactive contributions. Interactions with inactive electrons also modify the active one-body matrix; freezing electrons is not just deleting their rows. A mapped identity Pauli coefficient is another contribution generated by operator algebra and is not automatically the same scalar.

The reference snapshot records

so their sum is the selected CASCI total −108.771051792909 Hartree. These component values and the recorded count of 247 retained Pauli terms are version-pinned snapshots in the mapping tests, not quantities assumed immutable across all software environments or representation choices. The portable check is that the mapped same-sector active eigenvalue plus its stored scalar agrees with the selected reference.

An ECP changes the physical effective model: core-electron effects enter an effective potential and the explicit-electron count changes. It is not generally a single optional scalar to append to an otherwise all-electron energy. Compare consistent ECP/all-electron conventions and appropriate reference quantities; do not invent an energy for omitted core electrons and add it manually.

The lattice PPP background scalar is likewise a model charge-reference term, not automatically molecular nuclear repulsion. A field named core_energy can serve compatible algebraic bookkeeping while its physical origin differs by producer. Record that origin.

Orbital energies are not total energies. Summing occupied HF orbital eigenvalues double counts interaction contributions unless the appropriate energy expression is used, and correlated total energies are not obtained by summing natural occupations times arbitrary orbital energies. The returned energy and valid RDM contraction have specific contracts that must be followed.

Validate the representation before interpreting the algorithm

The nitrogen mapping script validates in the 400-state (3,3) sector, not by taking the lowest eigenvalue of an unrestricted twelve-qubit Fock-space matrix. It adds the core scalar and compares with CASCI for exactly the selected orbitals. This isolates representation correctness from active-space quality.

The tensor convention matters to that check. Current molecular arrays are chemist ordered, with the mixed block alpha-pair on the left and beta-pair on the right. Flattened tensor positions, active-local indices, fermionic mode order, Pauli-string direction, and determinant phases all have explicit meanings. A permutation that changes representation must be applied to states and observables consistently. Equal qubit count and equal encoding strings do not prove equivalence.

A useful verification invariant is Hermiticity, $H=H^\dagger$. Others include state normalization, the expected conserved-number sector, and correct small matrix action. The current classical qubit-solver wrappers can discard imaginary parts and do not generally enforce electron number; they cannot serve as universal references for arbitrary complex operators. The complex-preserving Pauli matrix utilities are useful for small explicit checks but remain exponentially costly.

Likewise, an exact round-trip through an external operator library checks representation under that round-trip’s conventions. It does not certify total-energy shifts, the physical sector, or equivalence to a real experiment. A smoke test that completes without exceptions is weaker still.

Trial quality is not the same as algorithm success

The trial-state script ranks determinant support with deterministic near-tie handling, reoptimizes coefficients using PMC, then aligns trial and reference coefficients by determinant identity before taking a squared overlap. It does not use coefficient-list positions as though they were universal basis labels.

The recorded snapshot has one-, two-, and four-determinant fidelities about 0.48255, 0.58641, and 0.73239 relative to the selected-space CASCI state. The corresponding displayed preparation gate counts are 6, 14, and 30, all on twelve problem qubits. Those are source reference circuit statistics, not physical gate counts measured here. The leading determinant is expressed in the selected orbital gauge; its weight is not a basis-independent property of nitrogen.

Three conclusions are legitimate: this particular compact-state family improves overlap as support grows; the preparation circuits change cost; and the orbital register does not shrink just because fewer amplitudes are loaded. A fourth conclusion is not generally legitimate: that a complete fresh-state-per-bit IQPE run returns the ground energy with probability exactly 0.73239.

The current IQPE protocol reprepares the system for each bit/shot and concatenates majority decisions. Different components can influence different bits. Standard coherent QPE has a different branch-distribution interpretation. If success probability is needed for a resource calculation, specify the actual complete protocol and how success is defined rather than substituting a convenient fidelity number.

Near degeneracy, high overlap with an eigenspace can be more relevant than overlap with one arbitrarily chosen eigenvector inside it. Individual eigenvectors may rotate within a degenerate subspace without changing the physical spectrum. A state or orbital label should therefore be tracked by appropriate character, sector, and subspace, not only by sorted energy position.

The one-milliHartree result and the coarse phase grid

The documented IQPE chapter records twenty complete runs: 010000 appeared nineteen times and 001111 once. The modal active energy was −9.652276065987 Hartree; adding the stored core gave −108.770051792909, exactly one milliHartree above the selected reference to the displayed precision. These are recorded development-run results, not a distribution observed in this session.

The apparent precision conceals a crucial resolution distinction. The recorded evolution time is approximately 0.162738437655 inverse Hartree with six phase bits. Adjacent grid energies are about 0.6033 Hartree apart, not 0.001 Hartree apart. The chosen grid point is close to the known reference because the tutorial used that reference to tune the evolution time deliberately.

That is a legitimate implementation-validation exercise and an illegitimate general strategy for discovering an unknown energy. The reference was not merely compared afterward; it helped select a favorable grid. A report should state this dependence prominently rather than claim general milliHartree resolution from six bits.

The recorded 19/20 modal frequency is an empirical proportion for that configured protocol and seed set. It is not a proved future success probability, a 95% confidence guarantee, or evidence that every source of error is below one milliHartree. The rare neighboring outcome changes energy by a large grid interval. Reporting only the dominant value would hide that behavior.

A QpeResult from the core algorithms does not automatically store the complete histogram, decoder provenance, or all alias candidates. The tutorial’s surrounding record supplies additional context. For ordinary time evolution, principal-angle conversion leaves energies separated by $2\pi/t$ ambiguous unless an appropriate prior interval is justified. For unary LCU/SOSSA protocols, the doubled-phase branch choice has its own signal-relative meaning. More decimal places in raw_energy do not resolve any of these ambiguities.

Accuracy, precision, resolution, and uncertainty

Accuracy is agreement with the relevant target/reference quantity. Precision is repeatability or spread under stated conditions. Resolution is the spacing or distinguishability imposed by a numerical representation. Uncertainty summarizes what values remain plausible given the available evidence and assumptions.

An illustrative estimate with standard error 0.1 milliHartree but a model bias of 5 milliHartree is precise yet inaccurate relative to the broader target. Conversely, a fine phase grid can have poor accuracy because its evolution operator is wrong. Repeating a fixed seeded simulation and obtaining exactly the same output does not establish a narrow uncertainty from independent observations.

A confidence interval concerns the behavior of a statistical procedure under specified sampling assumptions. An empirical standard deviation describes spread among observed values; a standard error concerns uncertainty of an estimated mean. Intrinsic quantum variance describes the state’s outcome distribution for an observable. They are not interchangeable, and none automatically includes chemical-model bias.

Hartree conversion constants are supplied by the repository’s constants.hpp. Approximately, one Hartree is 27.2114 eV or 627.509 kcal/mol, so one milliHartree is about 0.6275 kcal/mol. A named “chemical accuracy” target is a chosen requirement for a quantity, not a universal natural threshold guaranteeing every chemistry prediction is useful.

A layered error budget, beginning with the physical question

The first layer is the physical Hamiltonian and nuclear description. A clamped-nuclei electronic calculation assumes fixed geometry and separates electronic from nuclear motion in the Born–Oppenheimer picture. Temperature, vibrational averaging, solvent, relativistic effects, external fields, or open-system coupling are not automatically present. An accurate solution of the chosen equation cannot repair terms that were deliberately omitted from it.

The second layer is the one-electron representation: finite AO basis, linear-dependence treatment, and any ECP. Compare basis sets while holding other assumptions fixed to assess sensitivity. A smaller basis-change difference is evidence, not proof of a complete-basis error bar; different approximations can cancel accidentally. Increasing basis size does not repair a single-determinant description of strong static correlation.

The third layer is the correlated model: active/inactive/virtual partition, determinant selection, and treatment of outside-space dynamical correlation. Low orbital entropy or occupation near zero/two is a diagnostic, not a rigorous bound on an omitted energy contribution. The nitrogen valence-to-refined energy change quantifies one specific comparison. It is not the full missing-correlation error of the refined model relative to all-electron full-basis chemistry.

MCSCF can adapt orbitals, selected CI can enlarge important determinant support, and coupled cluster or perturbation theory address other correlation regimes under their assumptions. They are not interchangeable corrections to be added without a compatible definition. Some correlated methods, including ordinary truncated coupled cluster, are nonvariational: a lower energy is not automatically a better upper bound to the exact model ground energy. The effective-Hamiltonian factory discussed earlier has no bundled concrete downfolding implementation; it cannot be invoked as an unspecified missing-error correction.

The fourth layer is numerical solution. SCF convergence, residual thresholds, iteration limits, orbital stability, CI eigensolver convergence, and nonlinear coupled-cluster convergence are not the same checks. A converged stationary solution can be unstable or the wrong state. A solver-returned object without checked convergence is not automatically accepted scientific evidence. Record the actual method/backend and relevant status rather than relying only on the final energy’s smooth-looking digits.

The fifth layer is operator approximation: integral screening, Cholesky/double-factorization truncation, Pauli screening, and changed normalization/shift representations. A tolerance on a residual matrix or coefficient is not itself an energy error bound. A representation-aware comparison can estimate a norm difference, or compare the same small model before/after approximation, but must include all changed constants and one-body corrections. Reconstructing a full four-index tensor can also defeat the memory assumptions of a supposedly compressed workflow.

Circuit, state, and statistical error are different layers

The sixth layer is trial state and preparation. Support truncation, projected reoptimization, amplitude rounding, binary/sparse loading, and finite-precision QROM rotations can change the state in different ways. Classical fidelity to an accessible reference helps quantify one relation, but not the unknown true-molecule overlap. A coefficient-loading alias oracle with entangled garbage is not a clean trial state simply because its index probabilities match.

The seventh layer is Hamiltonian simulation and coherent control. A Trotter/Suzuki or Zassenhaus approximation differs from exact evolution; a fixed sampled qDRIFT unitary is not protected by an averaged-channel guarantee for all coherent uses. Power repetition can amplify base error. Controlled operations make previously global phases observable, so phase-dropping rewrites can become material even when a standalone state plot looks unchanged.

A finite-resolution QPE-subspace oracle can leave temporary phase ancillas dirty off the promised exact grid. In the current implementation, that is not merely another reported energy variance; it can invalidate the clean reversible-oracle premise of amplitude amplification. Likewise a requested time-dependent endpoint may fail the current unequal-final-step scale contract. A trustworthy report first establishes that the intended operation/time was realized, before estimating its error.

The eighth layer is readout and post-processing: phase-bin resolution, alias/branch selection, sample counts, grouping, majority votes, noise, and postselection. These can produce biases, discrete failures, or random fluctuations with different statistical models. Choosing the lowest observed QPE energy after many noisy runs is not an unbiased ground-energy estimator. Choosing a mode has its own meaning and must be accompanied by the complete distribution and tie rule.

The final layer, when moving beyond classical simulation, is physical implementation and logical failure under error correction. A requested logical error budget is a probability of faulty computation under a hardware/code model. It is not a bound on basis-set error or the distance between a trial expectation and a chemical ground energy. No physical quantum execution or QEC performance has been demonstrated by the records discussed here.

Combining uncertainties only when the mathematics supports it

For independently justified deterministic energy-error bounds $B_i$, the triangle inequality permits a conservative total bound $\sum_i B_i$. It does not justify adding uncalibrated guesses as if they were certified. Error layers can interact, so the comparison path and scope of each bound must be stated.

For random zero-mean errors, variances add only with the appropriate covariance terms:

Quadrature combination of standard deviations assumes those covariance terms vanish or are otherwise modeled. Deterministic chemical biases should not be treated as independent random noise merely to produce a smaller total uncertainty.

For a difference of two estimated energies,

Shared approximations may cancel in an energy difference, but that cancellation must be investigated, not asserted from similar molecule names. Shared samples or common calibration can create statistical correlations as well.

Failure probabilities are a different unit. If separate bad events have bounded probabilities $\delta_i$, a union bound gives a conservative probability at most $\sum_i\delta_i$ that any occurs, without requiring independence. That does not turn those probabilities into Hartree uncertainties. A complete acceptance condition might specify both an energy-accuracy requirement and a probability of violating it, plus model assumptions that remain outside that statement.

When using operator-norm bounds, convert them to the relevant objective. A Hamiltonian perturbation bounded by $\|\Delta H\|$ bounds a fixed normalized-state expectation change by that norm and can bound eigenvalue shifts under matching Hermitian finite-space assumptions. A unitary approximation bound is dimensionless; its effect on an expectation also involves the observable norm and the evolved state. Do not add such a number directly to an energy tolerance.

Recalculating the uncertainty that grouped measurements need

For a QWC measurement group g, define each shot’s weighted score

where $X_j=\pm1$ is the Pauli outcome inferred from that shot. Then

The current qdk expectation estimator instead returns the diagonal plug-in sum of $|h_j|^2(1-\widehat m_j^2)/n_g$. It omits within-group covariance and uses the plug-in n denominator rather than the unbiased n−1 correction for the mean-variance estimate. Its summary’s plus/minus square root is therefore not automatically a calibrated confidence interval.

A practical report should retain each group’s count dictionary and compute first/second moments of the score Y_g when grouped uncertainty matters. Positive covariance can increase error; negative covariance can reduce it. An eigenstate can have zero intrinsic Var(H) while separate Pauli measurements still have noisy means. These facts explain why a small energy_variance field is not a universal measure of state quality or algorithm correctness.

The shot request also is not necessarily the exact sampled total. The estimator allocates an equal floor share to nonidentity groups and drops the remainder. Identity-only groups are added classically. Native loss records are separate from clean counts, while the estimator uses clean counts and normalizes their sum. A survival-conditioned result can be biased if the discarded events are state-dependent. Report requested, allocated, clean, and lost counts separately when they differ.

Near a rare event, a short run can observe no failures and report zero empirical spread while substantial uncertainty remains. Approximate normal intervals can be poor in such cases. A repeated IQPE histogram with nineteen successes does not justify treating failure probability as exactly one twentieth, nor can a fixed seed set establish all future runtime environments. Statistical methodology should match the output type, rather than reusing one variance formula for expectations, modes, bit votes, and decoded phases.

Regression references versus portable scientific checks

The tutorial regression suite separates many portable invariants from optional exact snapshots. Examples of invariants include active-space counts, valid entropy ranges, normalized trial-state relationships, expected mapping dimensions, fixed-sector mapped/CASCI energy agreement, and consistent phase-grid decoding. Exact Pauli counts, detailed amplitudes, and circuit gate counts are guarded as version-pinned snapshots. Some small-model energy references are still checked numerically with explicit tolerances.

The snapshot mode enforces the tutorial’s public version, ignoring local build metadata. This is valuable provenance, not evidence that tests were executed here. The completed lab notebook’s statement that its required implementations passed belongs to that recorded workflow; it is not a fresh status claim from this lesson.

The repository’s reference_tolerances.py distinguishes CI/SCF, plain-text, RDM, unitarity, and finite-shot comparison tolerances. Those tolerances define particular test acceptance criteria. They are not a universal total chemical error budget, and a stochastic test can occasionally fluctuate even when the underlying algorithm is unchanged.

Always distinguish absolute and relative tolerances. A comparison of the form $|a-b|\le\mathrm{atol}+\mathrm{rtol}|b|$ can allow a surprisingly large absolute difference for a total energy near −108 Hartree. With zero absolute tolerance and relative tolerance $10^{-5}$, the allowed difference is about 1.08 milliHartree. This is a mathematical illustration of why a seemingly small relative tolerance can be comparable to the entire teaching target. Use explicit units and justified scales for each invariant.

Global phase, determinant ordering, degenerate orbital rotations, and low-level compiler decompositions can change snapshots while preserving intended physics. Conversely, a smoke test or visually similar circuit can miss a wrong energy offset, mixed-spin permutation, lost phase, or incorrect sector. The strongest checks compare the intended mathematical action under a fully specified representation, not only a text string or process exit code.

From an electronic energy to chemistry

Many chemical decisions depend on energy differences, not one absolute electronic energy. A reaction electronic energy compares products and reactants in compatible conventions. A barrier compares a transition-region configuration with a reactant minimum. A minimum is a locally stable geometry; a transition state has the appropriate unstable direction along a reaction path. One fixed stretched bond does not establish either an equilibrium structure or a reaction mechanism.

Measured equilibrium and rates often depend on free energy. Zero-point energy is vibrational energy remaining even at zero temperature. Thermal corrections account for thermally populated motions and states. Entropy quantifies the statistical multiplicity of accessible configurations. Solvation free energy accounts for environmental stabilization in a solvent. Pressure, concentration/standard-state conventions, conformer populations, and nuclear quantum effects can also matter.

Schematically, a chemical free-energy model combines electronic energy, zero-point and thermal contributions, entropy, and environment corrections. Those terms are not automatically appended by a fixed-geometry electronic solver or by QPE. A validated Hessian/vibrational workflow and appropriate thermodynamic treatment would be additional work, not information encoded in extra digits of the electronic result.

For an elementary transition-state-theory picture, a rate can have the form

where T is absolute temperature, $k_B$ Boltzmann’s constant, h Planck’s constant, $\Delta G^\ddagger$ the activation free energy per reacting entity, and $\kappa$ a transmission factor correcting the idealized crossing assumption. Molar free energies require the matching molar gas constant instead of $k_B$. This is a physical interpretation framework, not a rate calculator executed or claimed as a new built-in chemistry method here.

The tutorial’s accuracy discussion notes that at 298.15 K a one-milliHartree free-energy error can multiply an equilibrium/rate prediction by about 2.88 if other contributions are exact. It does not say an electronic QPE error automatically produces that exact experimental rate error. The exponential sensitivity explains why the target quantity and omitted contributions matter so much.

An electronic calculation can inform discovery, but it is not by itself a yield, drug efficacy, catalyst lifetime, manufacturing cost, or product value. Such conclusions require additional models and measurements. Comparing with experiment should match state, geometry/environment, temperature, observable, and uncertainty. Comparing with classical methods should use the best relevant methods and the same target/model accuracy, not an artificially weak baseline.

An acceptance record a team can actually use

The repository’s lab-notebook template already separates goal, model, mapping, trial state, and phase estimation. A technical report can extend it into the following acceptance record without claiming that every item has already been demonstrated:

Scientific claim. State the observable and reference: for this record, fixed-geometry selected-space total energy for the specified N₂ model, compared with its same-sector CASCI reference. State separately whether the goal is implementation validation, improved chemistry, or a physical quantum demonstration.

Model identity. Record geometry and units, charge, intended/verified spin information, basis/ECP choice, active/inactive indices and orbital representation, explicit electron populations, and any fitted or omitted interactions. Identify the larger-space and basis comparisons as sensitivity evidence, not automatically calibrated uncertainty.

Energy ledger. Record the object supplying each active, identity, nuclear/frozen-core, and SOS/background shift. Give the exact reconstruction expression. Confirm that the comparison uses the same total-energy convention and no constant is added twice.

Numerical and state evidence. Name solver/backend, convergence and stability evidence, operator/Hermiticity/sector checks, support-selection and coefficient-optimization method, reference overlap definition, normalization, and preparation representation. Distinguish intended trial fidelity from complete-protocol success.

Quantum protocol. Record mapping and mode order, unitary family, time/normalization/shift, powers and approximation settings, controlled-mapper assumptions, phase bits/window/branch rule, and any reference-guided parameter tuning. State which capabilities or cleanup promises were actually verified.

Sampling and outputs. Preserve full available histograms or per-bit records, requested/allocated/usable/lost shots, independent-run/seed policy, estimator formula including covariance handling, tie/postselection rules, and the exact fields used to calculate the reported number. Missing provenance is a limitation, not an invitation to infer it from a plot.

Scope of acceptance. Separate energy bias estimates, statistical uncertainty, probability of failure, and exclusions. For the completed tutorial record, the defensible statement is agreement of a configured, reference-guided classical simulation of IQPE with its compact-model CASCI reference at the one-milliHartree boundary—not experiment-level nitrogen accuracy or practical quantum advantage. This course-writing session performed no new verification run.

This record can be reviewed before any hardware-cost calculation. It makes explicit what a future implementation is being asked to reproduce, how success will be judged, and what remains outside the claim.

The handoff to resource estimation

Four evidence levels must remain distinct: correct simulation of a specified circuit; favorable algorithmic asymptotic scaling; conditional feasibility under a fault-tolerant hardware/QEC model; and demonstrated practical advantage over the best relevant alternative. None automatically proves the next.

The logical workload must include all bits, powers, shots, independent trials, state preparations, predicate/oracle calls, reflections, scratch allocation, and classical preprocessing/post-processing—not just the largest attractive circuit diagram. A failed or insufficiently justified success assumption changes the required repetitions and therefore the resource estimate.

Lessons 27–29 will price that precisely specified workload under logical synthesis, quantum error correction, and hardware models. Those teams need target precision, total success criteria, gate/measurement structure, ancilla ownership, and repeated-use counts. Their resource estimate will answer a conditional cost question. It will not retroactively validate the active space, repair omitted chemistry, or turn a recorded simulator snapshot into a physical experiment.