Toward Circuit-Aware Mutation Testing for Quantum Programs: Exact Visibility and Execution Planning

1 Introduction

• A surviving mutant may be equivalent, hidden by the input or measurement, or undetected because finite-shot evidence is insufficient.A one-qubit sign-error mutation has the same computational-basis output probabilities as the original for the fixed input, but differs under an X-basis measurement.Source: §1 · pp. 1–2

1.1 Scope and Claims

• The formal analysis uses ideal noiseless reversible circuits and one gate or subcircuit replacement; hardware effects and higher-order mutants are outside its direct scope.A source-level edit that expands into multiple gate changes must be represented as a subcircuit replacement or analyzed at the source-level semantic boundary.Source: §1.1 · pp. 2–3
• Mutant realism, non-equivalence, detectability, and non-redundancy are distinct, and adequacy evaluation must fix mutant sampling independently of the suite.Test-conditional information may guide execution planning, but resulting detection yield must not be reported as an unbiased adequacy score.Source: §1.1 · pp. 2–3

1.2 Objectives and Contributions

• The procedure distinguishes semantic equivalence, input- or measurement-dependent invisibility, finite-strength cancellation, and insufficient evidence.For fixed-axis rotations, two state expectations determine when every strength has zero gap and the maximum gap for the selected test.Source: §1.2 · p. 3
• The execution planner combines valid certificates and uncertainty bounds into a mutation-score interval that retains survivors and unresolved mutants.It also uses a constant-cost angle check to certify some rotations below a differential-expectation or total-variation threshold before state calculation.Source: §1.2 · p. 3
• State-side and observable-side certificates identify when the input or measurement must change, with the Helstrom observable giving an unrestricted fixed-input optimum.The reference implementation and worked examples make decisions reproducible.Source: §1.2 · p. 3

2.1 Reading the Quantum Model as a Software Test

• A differential expectation-value test compares original and mutant expectations on the same input and observable, with a prespecified threshold determining which population-level gaps count as detection.Changing the measurement can reveal a mutant that survives another measurement on the same input, provided the declared interface permits that change.Source: §2.1 · pp. 4–5

2.2 Program Semantics and Permitted Observations

• Equivalence is relative to allowed inputs and observations: state-transforming semantics differs from interface-relative outcome distributions, so the domain and interface must be fixed independently of the test suite.With restricted measurements, distinct output states may be observationally equivalent; changing the test within the interface differs from extending the interface. Matching averages may also conceal differing distributions for observables with more than two outcomes.Source: §2.2 · pp. 5–6

2.3 Mutation Signal and Measurement Cost

• The expectation gap is specific to an input and observable; finite-shot estimates have range-dependent uncertainty, and execution planning must weigh paired-shot cost, threshold distance, and measurement implementation.A constant observable has zero range and cannot distinguish unitary mutants by expectation. More generally, statistical bounds provide sufficient sampling costs, but gaps near the threshold can remain unresolved.Source: §2.3 · p. 6

2.4 Test Collections, Detection, and Adequacy

• The mutation score counts each detected mutant once in a population fixed independently of the suite; non-equivalent survivors remain in its denominator, while unresolved outcomes are not treated as detections.Native assertions retain their own acceptance rules: gap bounds can certify survival or a kill only when the mutant’s expectation enclosure is respectively inside or disjoint from the acceptance region. Statistical decisions require their prespecified procedure and error control.Source: §2.4 · pp. 7–8

2.5 From Analysis to Execution Decisions

• The planner first checks equivalence under declared semantics, then evaluates each remaining mutant–test pair and records whether to skip, predict detection, execute, or leave unresolved.A certified zero must be proved using exact values; a value merely near zero numerically is not a zero certificate. Survivors and unresolved mutants remain in the declared population, and a zero expectation gap on one test does not establish equal output distributions.Source: §2.5 · pp. 9–10

2.6 Computational-Basis Distribution Assertions

• Computational-basis distribution assertions fit the expectation framework through bit-string projectors, but skipping for zero effect requires proof covering every probability that could change the assertion’s decision.The distribution assertion keeps its own statistical procedure and pass/fail rule.Source: §2.6 · pp. 10–11
• For full computational-basis distributions, the structural backward light cone cannot exclude register mutations, though exact-state invariance and an angle-only total-variation skip rule remain available.That exact-distance certificate does not decide sampled histogram or chi-square assertions, which retain their own decision procedures.Source: §2.6 · p. 11

2.7 Analyzing a Mutant–Test Pair

• At the mutation site, the analysis compares the original and mutant expectations by propagating the final observable backward and isolating the gate change as a relative unitary.Conditions that quantify over every input or observable require proofs over those full domains; equality for one test does not establish equality for other tests.Source: §2.7 · p. 11

2.7.1 Sound Structural Filtering

• A mutation entirely outside the structural backward light cone of the measured observable has exactly zero effect and can be skipped, while cone overlap alone does not prove an effect.The cone is computed from gate connectivity and supports before propagating the full observable or estimating the mutation-site state; it conservatively contains the exact algebraic support.Source: §2.7.1 · p. 12

2.7.2 Exact Local Mutation Effect

• For a fixed observable, a mutation is invisible for every input exactly when its relative unitary commutes with the propagated observable; for a fixed input, invisibility to every observable is equivalent to commutation with the mutation-site state.Neither condition rules out a zero gap for a particular input–observable pair when both commutators are nonzero; evaluating the exact gap requires the propagated observable and mutation-site state, which can be expensive to obtain.Source: §2.7.2 · pp. 12–14
• A fixed input and observable can still yield zero effect even when neither the observable nor state commutes with the relative unitary, so the exact gap must be evaluated to decide that pair.For a fixed input, the maximum gap over Hermitian propagated observables with operator norm at most one is determined by the trace norm of the state difference.Source: §2.7.2 · pp. 13–14

2.7.3 Exact Effects for Common Gate Changes

• For involutory-axis rotations, two state expectations determine the exact effect at any angle, and the angle and observable range can sometimes certify a below-threshold gap.For fixed-axis rotations with nonzero response coefficients, the invisible strengths modulo 2π are 0 and 2 atan2(−b_m, a_m).Source: §2.7.3 · pp. 14–16
• Fixed one-qubit replacements reduce to an effective axis and angle; a relative unitary that is only a global phase is channel-equivalent to the original.The global phase cancels from conjugation, so the rotation formulas apply to the reduced replacement.Source: §2.7.3 · pp. 17–18
• Clifford replacements update propagated Pauli terms exactly, and Pauli replacements retain only anticommuting terms; evaluating the realized gap still requires state expectations [13].Exact spectral formulas also cover generators with two distinct eigenvalues, though computing projectors and state expectations may be expensive.Source: §2.7.3 · p. 18

2.7.4 Bounded Approximation for Other Small Mutations

• A double-commutator remainder bounds the finite-strength gap in a deterministic interval that can certify below- or above-threshold behavior; commutation gives an exact zero effect.The guarantee requires exact inputs or validated numerical error bounds; the fallback is not established as scalable or beneficial in measured use.Source: §2.7.4 · pp. 18–19

2.7.5 Evidence Supporting the Decisions

• Execution status is recorded separately from evidence type: an exact observable-side update alone cannot determine the realized gap or a threshold decision without state expectations.The first-order perturbative coefficient and remainder bound can instead provide a deterministic interval for threshold decisions.Source: §2.7.5 · pp. 19–20
• Pilot estimates may support compatibility with a visibility tolerance but cannot prove zero, while the operator proxy is only a worst-case bound and unavailable evidence leaves effect size unresolved.A large operator proxy does not establish a large response for the selected test or a large finite-strength gap.Source: §2.7.5 · p. 20

3 Circuit-Aware Execution Planning

• The planner records each pair’s decision, evidence, and unresolved status, while its reference filter order is reproducible but not cost-optimal.Analysis should be compared with direct statevector evaluation using propagation, equivalence checking, state evaluation, and memory costs.Source: §3 · p. 20

3.1 Preserving Classifications and Mutation Scores

• Under stated soundness and coverage assumptions, resolved classifications are correct and detected, surviving, and unresolved weights bound the population mutation score without dropping unresolved mutants.For example, 60 detected, 25 surviving, and 15 unresolved mutants in a population of 100 give a score interval of 60/100 to 75/100.Source: §3.1 · p. 24

3.2 GHZ Example: Decisions and Population Accounting

• For other fixed oracle rules, valid oracle-specific classifiers can use the same accounting, while unresolved weight remains in the score bounds.The GHZ example holds the input and eight-mutant population fixed while changing the test; all eight mutants remain non-equivalent under the declared state-transforming semantics.Source: §3.1 · p. 24, §3.2 · pp. 24–25

3.2.1 Single-Qubit Expectation Test

• For the Z1 expectation test, exact gaps yield two detected mutants, six survivors, and no unresolved mutants, so the score is 2/8 before execution.The planner schedules two candidates; removing the six skipped survivors would produce a misleading 2/2 score.Source: §3.2.1 · p. 25

3.2.2 Full Output-Distribution Test

• Changing the GHZ test from the Z1 expectation to the full output distribution raises the population mutation score from 2/8 to 5/8 for the same mutants.The additional detections are m1, m2, and m4; the outcomes are analytical, not finite-shot rejections.Source: §3.2.2 · pp. 25–26, Table 6 · p. 26

4.1 One-Qubit Pauli-Axis Rotation

• For Ry(θ)→Ry(−θ), the (|0⟩, Z) test is invisible, while changing the input or observable can expose the mutation; fixed nonzero identification advantage requires Ω(1/η^2) executions as η→0.A zero first-order coefficient does not establish a zero exact gap; evaluate the exact expression at the declared mutation strength.Source: §4.1 · pp. 26–27

4.2 Fixed Cross-Axis One-Qubit Replacement

• For Rx(π/2)→Rz(π/2), the relative replacement is a mixed-axis rotation, and the example changes the Z expectation from 0 to 1; global-phase-only replacements remain invisible.Adding separate X- and Z-axis effects would not describe the relative operation.Source: §4.2 · pp. 27–28

4.3 Mutation-Guided Test Augmentation

• For a fixed input, O⋆ = sgn(∆τj) attains the largest gap among unit-norm observables, ∥∆τj∥1; if ∆τj = 0, no observable change can expose the mutant.The optimum can require the full mutation-site state and an added eigenbasis measurement; screening a restricted output family gives a lower bound that can be exponentially smaller.Source: §4.3 · p. 29
• The rotation examples show that a sign-reversal mutant can be exposed by changing the observable, while an entangled two-qubit example requires a constructed measurement to attain a non-Pauli optimum.The two-qubit construction adds one CNOT and two one-qubit gates per measured circuit, plus readout and classical relabeling; it is admissible only if the contract permits the basis change. Because tests are built from the mutations they expose, adequacy should be evaluated on held-out mutants or another independent population.Source: §4.3 · pp. 29–30

4.4 A Local Mutation Exposed Through Correlation

• A CNOT suffix lets a final Z measurement on one qubit read a two-qubit correlation: for input |00⟩, the gap grows as 1 − cos η and reaches the full outcome range at |η| = π.For input |++⟩, the exact gap is zero at every strength despite the absence of either zero-effect certificate, showing that structural visibility alone does not guarantee a signal.Source: Figure 5 · p. 31, §4.4 · p. 31

4.5 Efficient Analysis for Stabilizer Circuits

• With a supplied pure stabilizer tableau, Clifford suffix, and Pauli output and mutation axis, gaps have only three forms: zero, |sin η|, or 1 − cos η; M candidates cost O(Ln + Mn^3) binary operations without shots.The classification excludes arbitrary convex mixtures and requires the stated state representation; even one non-Clifford preparation gate can invalidate the three-shape result.Source: §4.5 · pp. 32–33

4.6 Computational Scope and Scalability

• State-dependent rules can scale through supplied product or stabilizer descriptions, small causal cones with reduced inputs, or quantum execution of relevant prefixes; the planner does not automatically select tableau or causal-cone methods.Small mutation support alone does not guarantee a small causal cone or an available reduced input, and local reduced-state commutation cannot replace the global state-invariance condition for entangled registers.Source: §4.6 · p. 33
• Coefficient sums can replace a dense norm calculation, but still require the state-dependent first-order expectation, may yield a loose bound, and need omitted-term allowances when truncated.The general-generator fallback uses bounded dense products; the coefficient bound is an available extension, not an implemented performance result.Source: §4.6 · p. 36
• Population-wide redundancy and subsumption add costs: identical compiled forms can be grouped by hashing and equality checks, while unrestricted equivalence checks may require O(M2) proofs and all-pairs subsumption costs O(M2T) once indicators are known.Obtaining exact detection indicators remains a separate analysis or execution cost, and effect-based grouping requires profiles without providing a semantic certificate.Source: §4.6 · p. 36
• Scalability depends on circuit and state structure, precision, candidate and test counts; resource-limited or unresolved mutants retain population weight rather than being treated as equivalent or surviving.A Clifford suffix can make classical Pauli propagation cheap while the stated shadow bound is exponential; wide circuits with small relevant cones or tractable states can still admit informative exact decisions.Source: §4.6 · p. 36

4.7 Practitioner Checklist: Do, Do Not, and Report

• Report each decision with its test, evidence, uncertainty, costs, semantics, mutant population, threshold, operations, and measurement implementation, including unresolved-mutant weights and independent evaluation of proposed test improvements.For reusable fixed-axis analyses, retain (am, bm), signed response, and zero set; distinguish family angle offsets from fixed-replacement axes and angles, and record preprocessing, pilot, and execution costs.Source: §4.7 · p. 36
• For RO = 2 and γ = 0.1, canonical angles below 2 arcsin(0.05) ≈0.1000 radians are undetectable by that test; compare pilot, direct-execution, and reconstruction costs at matched precision and confidence.Pilot error allocations and retained terms must satisfy the sampling and truncation inequalities in Appendix B.9.Source: §4.7 · p. 37

5.1 Evidence and Implementation Scope

• The probes provide proofs, hand-checkable examples, executable demonstrations, and bounded synthetic checks, but no corpus-effect, filter hit-rate, measured-savings, or held-out adequacy results.The authors identify population-level effects and execution-planning or test-augmentation benefits as requiring separate empirical evaluation.Source: §5.1 · p. 37

5.2 Bounded Synthetic Feasibility Probes

• Stored-cone computation completed through 4,096 qubits at 676.0 MiB, while full-register symbolic comparisons completed at six qubits and timed out at eight and ten.The memory watchdog threshold was 768 MiB; sampled maxima for stopped cases are lower bounds on worker peaks, not evidence of operating-system out-of-memory failures.Source: §5.2 · pp. 38, 40
• 33 of 39 attempted synthetic cases completed; four crossed the RSS threshold and two timed out.The table groups the declared feasibility cases by computation and outcome; stopped-case memory values are sampled RSS, not completed-worker peaks.Source: §5.2 · p. 37, Table 10 · p. 39

5.3 Choosing Rules for a Circuit

• Rule applicability depends on the oracle, circuit and mutation structure, observable and state representations, and resource budget; the implementation does not automatically choose settings.Omitted or exhausted checks provide no evidence of equivalence or survival; unresolved candidates retain their population weight.Source: §5.3 · p. 40
• For non-Clifford suffixes, planners should track discarded Pauli-term mass, and state-dependent rules require affordable access to needed expectations.A Clifford suffix alone does not make an arbitrary input state tractable; an error allowance too large to resolve a threshold cannot justify a decision.Source: §5.3 · p. 40

5.4 Empirical Validation Still Required

• Censuses must fix populations and classification rules beforehand, while savings and adequacy claims require separate evaluations under common budgets and on independent mutants.Results should be separated by mutant population and test suite, and distinguish algebraic zeros from numerical and tolerance-based zero classifications.Source: §5.4 · pp. 40–41

5.5 Model and Interpretation Limits

• The formal results assume finite-dimensional noiseless unitary circuits and single gate or subcircuit replacements; equal observable means need not imply equal distributions.Ideal-equivalence certificates do not establish hardware indistinguishability, which requires noise and drift models and measured device costs.Source: §5.5 · p. 41

6.1 Mutation Generation and Observed Survival

• Quantum mutation tools generate gate and parameter changes, but the fraction of generated mutants covered by this local-unitary analysis remains unmeasured [25] [26] [27].Source: §6.1 · p. 41

6.2 Equivalence and Test-Relative Detection

• Circuit-level analysis explains some surviving mutants through input- or measurement-hidden changes and strength-specific cancellations, beyond proof of semantic equivalence or coverage-based indications.These mechanisms need not be mutually exclusive.Source: §6.2 · p. 42
• The planner uses exact certificates or bounds with stated simultaneous coverage and retains unresolved outcomes, tying score accounting to evidence for the declared oracle rather than prediction accuracy.Source: §6.2 · p. 42
• Quantum software testing also uses mutation-based test generation, varied inputs, statistical assertions, and differential or metamorphic oracles; a reference-program difference alone does not establish native assertion failure.Source: §6.2 · p. 42

6.3 Established Mathematical Methods

• Established trace-duality, propagation, stabilizer, and statistical tools support the analysis; response identities and reconstruction methods identify zero or maximal gaps for a declared test [13].Three-evaluation sinusoidal reconstruction also supports response reuse within fixed-axis families.Source: §6.3 · p. 42

7 Conclusion

• Semantic checks, circuit-dependency analysis, and exact local-effect formulas distinguish several causes of mutant survival, including certified zero effects and insufficient evidence.For supported fixed-axis rotations, the response classification identifies every zero-gap strength; under stated stabilizer assumptions, response curves can be classified in polynomial time.Source: §7 · pp. 43–44
• Under stated assumptions, the planning guarantee bounds the mutation score while retaining survivors and unresolved mutants in adequacy accounting.Skipping execution does not remove a non-equivalent mutant from the denominator.Source: §7 · p. 44
• State differences guide whether an observable can expose a mutant, but test changes must respect the declared interface and detection rule.The guarantee distinguishes state-transforming semantics from a fixed computational-basis interface.Source: §7 · p. 44
• The results provide a basis for deciding what to execute, change, and leave unknown, but their practical frequency, cost savings, and adequacy effects remain for empirical validation.The evidence described comprises proofs, worked examples, and a reference implementation under ideal unitary semantics.Source: §7 · p. 44

A Detailed Statistical and Local Framework

• The detailed framework appendix supplies statistical bounds, local derivations, and implementation algorithms supporting the main text.Source: A Detailed Statistical and Local… · p. 51

A.1 Planning Soundness: Assumptions and Proof

• The guarantee assumes sound semantic exclusions, valid zero-gap certificates and intervals, prespecified error allocations, and certified representative-class indicators with retained multiplicities.The target population consists of non-equivalent mutants, and ordinary floating-point evaluation alone does not certify a threshold comparison.Source: §A.1 · p. 51
• The planning proof combines pilot and observed-test coverage to classify detected, surviving, and unresolved mutants and bound the population mutation score.Detected classes contribute their weights, survivors contribute zero, and unresolved classes contribute between zero and their total weight.Source: §A.1 · pp. 51–52
• The guarantee needs neither complete execution nor optimal scheduling, but an unresolved semantic-equivalence status cannot be silently treated as the fully non-equivalent population.Source: §A.1 · p. 52

A.2 Hoeffding-Based Estimation Efficiency

• For fixed-separation testing, the Hoeffding protocol gives a distribution-independent sufficient shot count; variance-aware estimators or direct tests may need fewer samples.The cited bound is sufficient, while lower bounds for particular distributions require a separate argument.Source: §A.2 · p. 52

A.2.1 Classification at a Practical Threshold

• A fixed-shot interval classifies a mutant as below threshold, detected, or unresolved, but cannot uniformly resolve every gap arbitrarily close to the threshold.The threshold decision, detecting a prespecified difference from zero, and ranking by estimation efficiency are distinct tasks.Source: §A.2.1 · pp. 52–53
• The estimation-efficiency score ranks signal relative to sufficient mean-estimation cost, but does not decide a practical threshold or measure suite adequacy.Its cost accounting must adjust for unequal shot allocation or reused reference estimates.Source: §A.2.1 · p. 53
• Per-decision error control does not automatically control a collection of tests and mutants; family-wise control uses an allocation across all decisions or models shared-sample correlation.Source: §A.2.1 · p. 53

A.3 Exact Local Replacement Form

• For a local unitary replacement, the exact gap depends on the mutation-site state and propagated observable; every state has zero gap exactly when the relative replacement commutes with that observable.A fixed state's gap can also vanish by cancellation even when the operator difference is nonzero.Source: §A.3 · p. 54

A.4 State-Side Dual and Trace-Distance Interpretation

• The trace-norm variational characterization makes the mutation-site state difference determine the largest achievable observable gap, with an attaining observable described in the proof.Source: §A.4 · p. 54
• Observable-gap and fidelity-based tests quantify different aspects of state differences, and their cost bounds do not imply an unconditional asymptotic ordering outside the structured class.The small-angle comparison depends on visibility and nonvanishing coefficients; direct or variance-aware detection may be faster.Source: §A.4 · p. 54
• For the supported involutory-axis rotation, the state and Helstrom-observable construction attains the normalized-gap ceiling and yields a tight optional score ceiling when comparison cost is test independent.Source: §A.4 · p. 55
• For predetermined independent copies, the discrimination bound is test independent; entangled multi-use inputs fall outside that product-input argument.The text reports Θ(1/|η|) scaling for an entangled construction versus Θ(1/η^2) for independent executions in the stated involutory-axis setting.Source: §A.4 · p. 56

A.4.1 Optional Unknown-Source Discrimination Policy

• The optional unknown-source policy gives a necessary angle condition, but it neither decides whether a differential gap exceeds the practical threshold nor rules out classical analysis.For Ncopy = 104 and δdisc = 0.05, the condition is approximately 0.0258 radians.Source: §A.4.1 · p. 56

A.5 Exact Structured Rotation and Clifford Forms

• The exact finite-angle Pauli-rotation gap retains trigonometric dependence and a state expectation needed when the sine contribution cancels.The expression uses the same propagated Pauli strings as the first-order coefficient.Source: §A.5 · p. 58
• For an axis rotation, the maximum absolute response is |am| + sqrt(am^2 + bm^2), and the gap vanishes at every strength only when am = bm = 0.For nonzero coefficients, the finite-strength zero set includes η ≡ 0 and η ≡ 2 atan2(−bm, am) modulo 2π.Source: §A.5 · p. 58
• A Pauli relative mutation changes exactly the anticommuting component, while a general Clifford mutation can also permute Pauli strings; evaluating either expectation still requires state information.Like terms should be combined before exact expectation evaluation or pilot scoring.Source: §A.5 · p. 59
• Every two-eigenvalue generator reduces to involutory-axis rotation algebra, and every fixed one-qubit replacement has a canonical such representation without a perturbative approximation.Evaluating projector blocks or state expectations may nevertheless be computationally expensive.Source: §A.5 · pp. 59–60

A.6 Fallback General-Generator Local Perturbation Form

• For general small generators, operator sensitivity bounds first-order response across states, but the realized gap depends on the selected state and finite-strength effects; zero operator sensitivity alone does not prove invisibility.Certified commutation with the propagated observable does prove invisibility at every strength.Source: §A.6 · p. 60

A.7 Backward Light Cone Criterion

• A mutation outside the structural backward light cone has exactly zero effect on the observable for every input, but membership in the cone is necessary, not sufficient, for visibility.The structural cone can overapproximate exact support because of commutation or cancellation.Source: §A.7 · p. 60

A.8 Circuit Characteristics That Predict Efficient Mutants

• Visibility and tractability depend on light-cone membership, noncommutation, state support, distance to measurement, and observable locality; none of these heuristics alone establishes a nonzero detectable effect.A single Pauli propagated through a Clifford suffix makes the noncommutation proxy binary; distance to measurement alone does not predict effect or estimation cost.Source: §A.8 · p. 61
• Execution planning distinguishes certified-zero skips from below-threshold skips and above-threshold or unresolved executions, retaining provenance and unresolved cases rather than treating them as zeros.A statistical skip requires a simultaneous interval wholly below the threshold at declared confidence; unavailable gap evidence leaves the outcome unresolved.Source: §A.8 · p. 61
• For fixed one-qubit replacements outside the rotation and Clifford branches, axis–angle reduction constructs the exact observable difference, including for non-Pauli axes; scalar relative unitaries yield zero difference without an axis.For a simultaneous Euler-parameter path, the construction is applied separately to each fixed candidate and assumes no common axis along the path.Source: §A.8 · p. 64
• A separate symbolic path for small circuits with exact gates and inputs independently proves observable invariance or zero expectation, but unsupported data or resource limits leave the certificate unavailable.It accepts recognized discrete Clifford gates, explicitly exact one- or multiqubit unitaries and Pauli rotations, and explicitly exact pure or mixed states; angles must be symbolic expressions such as rational multiples of π.Source: §A.8 · p. 64
• The semantic filter checks pinned transpiler matches independently of input and observable, using exact common-prefix and suffix semantics before testing whether the changed-support relative unitary is scalar.The local proof can cover circuits wider than four qubits; if unavailable, full-register symbolic comparison is attempted only within symbolic and dense width limits.Source: §A.8 · p. 64
• Truncated observable-side updates support pilot intervals only with bounded discarded coefficient mass; exhausted resources leave candidates unresolved, and truncated observables alone cannot establish complete-observable evidence.The built-in pilot samples a simulated mutation-site state separately for each candidate and allocates αpilot/|M| before filtering without recycling unused allocations.Source: §A.8 · p. 64

A.9 Interpretation and Consequence

• Promising structured candidates affect the measured observable, have state support on that exact difference, yield a detectable shift, and avoid costly state-preparation or measurement changes.The operator score is a state-independent first-order upper-bound proxy, whereas realized efficiency requires the test’s state-dependent gap; proxy, pilot, and realized scores must remain separately labeled.Source: §A.8 · p. 64, §A.9 · p. 65
• Observable-gap analysis can reduce execution cost for a fixed test suite and stratify mutants by expected difficulty, but suite-specific gaps must not define the mutant population.When exact expectation evaluation is infeasible, truncated structured differences can be paired with pilot measurements; otherwise first-order sensitivities provide an approximation, while operator norms bound worst-case first-order response across states.Source: §A.9 · p. 65

B Extended Examples and Pilot-Estimator Details

• The examples instantiate the observable gap and optional estimation-efficiency score, while later sections develop pilot estimators for wider circuits.Source: B Extended Examples and… · p. 65

• At θ=π/4 and η=0.05, Δm≈0.036224 and E_m∝1.312 × 10−3, versus first-order estimates of ≈0.035355 and 1.25 × 10−3.The examples use pure inputs and Pauli observables, treating C_m as constant.Source: §B.1 · p. 65
• Removing Ry(θ) gives Δm=|cosθ−1|: at θ=π/4, Δm≈0.293, but at θ=0.05, Δm≈0.00125.Source: §B.1 · p. 65
• For X on |0⟩ with Z readout, X→I gives Δm=2, while X→Y gives Δm=0 and identical output density operators for that input; global-phase mutations are equivalent for every observable.Source: §B.1 · p. 66
• Removing CNOT under |+⟩c|0⟩t with Zt gives Δm=1, but input |1⟩c|0⟩t makes Δm=0.Source: §B.1 · p. 66
• Flipping CNOT’s control gives Δm=0 for Zt but Δm=2 and E_m∝4 for ZcZt.Source: §B.1 · p. 66
• A swap gives Δm=2 for Z1 but Δm=0 for the symmetric observable Z1+Z2.Source: §B.1 · p. 66
• With X, η=0.05 gives Δm≈0.00125 and rare-outcome detection takes O(1/Δm) shots; switching to Y gives Δm≈|η|.Source: §B.1 · p. 67
• Entangling-gate removal and subcircuit replacement each give Δm=2 for a joint Z observable but Δm=0 for a local marginal.Source: §B.1 · pp. 66–67

B.2 End-to-End GHZ Planning Walk-Through

• The eight candidates are author-defined, with γ=0.1 illustrative; representativeness of the 0.01-radian fault offset remains unestablished.Source: §B.2 · p. 67
• The fixed-test planner schedules only m7 and m8 for execution, pruning six of eight candidates without treating test-relative zeros as program equivalence.Source: Table 12 · p. 68, §B.2 · p. 68

B.3 Summary of Example Scores

• Table 13 collects representative gaps whose interpretations depend on each example’s displayed input and observable.Source: Table 13 · p. 68, §B.3 · p. 68

B.4 Simultaneous Parameter Mutations of a General One-Qubit Gate

• For a fixed-axis one-parameter sweep, one coefficient pair determines the complete zero set and maximum along that sweep.The signed coordinate offset is used directly as the effective angle for each generated replacement.Source: §B.4 · p. 69
• For simultaneous offsets, construct the complete relative unitary before evaluating its effective rotation and exact gap; component effects cannot be scored and added independently.The effective axis varies along a general simultaneous-parameter path, so one coefficient pair does not generally determine its full zero set or maximum.Source: §B.4 · pp. 69–70
• At θ = 0, δϕ = −δλ and at θ = π, δϕ = δλ are null directions modulo 2π.Offsets by integer multiples of 2π in any Euler coordinate likewise change the gate by at most a global phase.Source: §B.4 · pp. 69–70
• Near θ = π/2 the leading local effect is the squared Euclidean norm of the three offsets, while at θ = 0 and θ = π it exposes the respective phase-offset cancellations.The Euler-coordinate norm depends on representation; the relative unitary up to global phase and its effective rotation angle are invariant measures of physical mutation.Source: §B.4 · p. 70

B.5 Pauli-Expansion View

• In the Pauli expansion, only terms anticommuting with the mutation axis contribute, and each contributes a paired-string term to the exact gap.The exact expression includes both a sine contribution and a (1 − cos η) contribution, which can remain nonzero when the first-order coefficient vanishes.Source: §B.5 · pp. 70–71

B.6 How Surrounding Gates Amplify or Wipe Out the Mutation

• A mutation is exactly invisible when the backward-propagated observable commutes with its generator; a Z-rotation followed by direct Z-measurement is one example.A later Hadamard can make the same mutation potentially visible, with response depending on state expectations of X and Y.Source: §B.6 · p. 71
• After an entangling gate, a local mutation can depend on multiqubit correlations, and missing correlations can remove both linear and quadratic response.In the stated CNOT example, both Tr(XqZrτj) and Tr(ZqZrτj) must vanish for the expectation test to be exactly insensitive at every strength.Source: §B.6 · pp. 71–72

5. Cancellation among Pauli components; B.7 Bounded Amplification

• Contributions from multiple propagated Pauli components can add coherently or cancel destructively, wiping out an effect even inside the light cone.The effect of a suffix circuit is bounded; for a unit-norm Pauli observable, the gap is at most 2| sin(η/2)|.Source: §5 · p. 72, §B.7 · p. 72

B.8 Restricted Pauli Test-Augmentation Screen

• A restricted Pauli screen can underestimate distinguishability: a small retained-Pauli score need not mean the mutant is intrinsically hard to detect.For the described Haar-random orthogonal state pair, the trace-norm difference is 2 while the largest Pauli expectation difference is bounded by O(√n 2^-n/2) with high probability.Source: §B.8 · pp. 72–73

B.9 Pilot Estimation of Mutation Effect in Wide Circuits; B.9.1 Computational Growth and Bounded Fallbacks; B.9.2 Backward Pauli Propagation

• Wide-circuit exact state and observable evaluation can require exponential resources, so the proposed pilot extension truncates backward-propagated Pauli terms and estimates signed state expectations with an explicit error budget.A term cap bounds the retained representation but does not itself guarantee a useful error bound; the practical benefit remains unvalidated.Source: §B.9 · p. 73, §B.9.1 · p. 74, §B.9.2 · p. 75

B.9.3 Pilot Measurements of Signed State Support

• For fixed retained Pauli weight, pilot sample complexity is independent of total register size but grows exponentially with weight, and one site-level pilot can be reused across candidate mutations.Analytically tractable states such as known stabilizer or product states can supply expectations exactly instead.Source: §B.9.3 · pp. 75–76

B.9.4 Pauli-Rotation Coefficient Estimates

• For Pauli rotations, the pilot estimates two strength-independent coefficients to score candidate strengths, but finite data cannot certify exact all-strength invisibility.The joint confidence region instead indicates whether invisibility is compatible with a selected tolerance.Source: §B.9.4 · pp. 76–77

B.9.5 Relative Clifford Replacements

• Relative Clifford replacements use the same site-level pilot data after retained observable-difference terms are combined, preserving observable-side cancellation.The signed-versus-magnitude-only difference serves as a Clifford cancellation diagnostic.Source: §B.9.5 · p. 77

B.9.6 Cost, Amortization, and Interpretation

• Shared-shadow cost comparisons must include pilot preparation, execution, measurement, post-processing, truncation, and calibrated costs; strength reuse alone establishes no advantage over reconstruction baselines.A forecast also needs calibrated constants and a pre-execution estimate or conservative bound for the avoided-comparison fraction.Source: §B.9.6 · pp. 78–79
• Both truncation and pilot-sampling error shrink linearly near zero mutation strength, so weaker mutations do not by themselves improve certified relative precision.A target estimation error and a population-boundary decision are distinct; classifications can remain unresolved near the detection threshold.Source: §B.9.6 · p. 78

B.9.7 Precision-Matched Analytical Baselines

• For many strengths on a few axes, interpolation uses 2Naxis + 1 configurations; targeted prefix measurements can suit modest target lists, while shared shadows require enough low-weight reuse to offset setup and sampling costs.Direct execution is competitive when candidate counts are low or setup and auxiliary-configuration costs exceed sampling savings; all methods can leave threshold cases unresolved.Source: §B.9.7 · pp. 79–81

B.9.9 Algorithm

• The algorithm propagates and truncates Pauli terms, gathers site-level pilot estimates, scores rotation and Clifford candidates, and charges approximation and pilot costs to the planning budget.It reports truncation and sampling uncertainties separately and distinguishes evidence type from execution-decision status.Source: §B.9.9 · p. 81

B.9.10 Interpretation

• Pilot estimates retain signs needed to represent state orthogonality and cancellation, while high-weight strings require exponentially more local-Pauli shadow samples.For a single Pauli observable with a Clifford suffix, a finite-strength rotation generally requires two signed expectations unless a coefficient is known or its strength multiplier vanishes.Source: §B.9.10 · pp. 81–82

B.9.8 Light-Cone Filtering; C.1 Backward-Light-Cone Filter

• The light-cone rule can filter test-relative zero effects using exact circuit topology and gate supports, independently of input state or a truncated observable approximation.A retained Pauli approximation cannot prove that the full mutation effect is exactly zero.Source: §B.9.8 · p. 81, §C.1 · p. 82

C Detailed Pre-Execution Filters; C.2 Transpiler-Based Equivalence Filter

• Matching deterministic, semantics-preserving transpiler outputs is a sufficient equivalence filter only under stated pipeline and channel-preservation assumptions, followed by independent exact proof in the reference implementation.A failure to prove equivalence does not prove non-equivalence.Source: §C.2 · p. 82

C.3 Circuit-Equivalence Filter

• Circuit-equivalence checks must match the declared semantics; exact methods can filter tractable families, while unrestricted unitary non-equivalence has worst-case hardness.Comparing state vectors for one input establishes ground truth only for that input, not full circuit equivalence.Source: §C.3 · pp. 82–83

C.4 Mutant–Mutant Redundancy Filter; C.5 Subsumption and Mutation Buckets

• Mutant redundancy is test-relative and requires equality of signed expectations or relevant distributions, while finite-sample near-equivalence is only a heuristic.Subsumption likewise depends on exact detection sets; buckets offer no semantic guarantee and should preserve the declared population when assessing adequacy.Source: §C.4 · p. 83, §C.5 · p. 84

D Local Perturbation Remainder

• The local perturbation remainder bounds the signed expectation change using a first-order generator term and a nonasymptotic Taylor remainder.The bound uses unitary invariance, the unit trace norm of the state, and the reverse triangle inequality.Source: D Local Perturbation Remainder · p. 84

E Mutation Operators, Test Oracles, and Applicability

• Table 14 maps published operators and tests to available analyses, but the inventory does not measure the fraction of generated mutants covered.QuanForge also applies gate and parameter mutations to quantum neural networks [26].Source: E Mutation Operators, Test Oracles,… · pp. 84–85, Table 14 · p. 86
• Classify edits on the union of original and replacement supports; overlapping rotation, one-qubit, Clifford, and spectral cases cover different local replacements.Deletion and insertion are represented by setting the removed or original operation to identity; changed controls or wires can enlarge the support.Source: E Mutation Operators, Test Oracles,… · p. 85
• A passing test is not a semantic-equivalence certificate: syntactic grouping and input-specific statevector filtering do not establish channel equivalence.The cited tools differ in handling equivalent mutants, and absent certification does not show that every generated mutant is non-equivalent.Source: E Mutation Operators, Test Oracles,… · p. 85
• The published RGI insertion is analyzable as a fixed involutory-axis rotation, but evaluating the planner and validating the illustrative 0.01-radian offset need further evidence.For fixed ϕ, the axis A = cos ϕ X + sin ϕ Y satisfies A^2 = I [27, Sec. IV-A].Source: E Mutation Operators, Test Oracles,… · p. 85