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

Abstract

Mutation testing assesses test suites by introducing program changes and checking whether tests detect them. In quantum programs, a surviving mutant may leave program behavior unchanged, produce a change hidden by the chosen input or measurement, or remain undetected because statistical evidence is insufficient. We develop a circuit-aware framework to distinguish these causes and guide execution planning and test changes. The framework covers ideal, noiseless programs and mutations that replace one gate or subcircuit. The method traces how a mutation can affect measured outputs and combines established mathematical checks with statistical estimates. For supported cases, we establish conditions under which a test cannot reveal a mutation and exact formulas for its effect when the required state information is available. Under stated assumptions, we bound the mutation score while retaining surviving and unresolved mutants, and identify a restricted circuit family whose mutation effects admit efficient classical classification. The deliverables include a decision procedure, mathematical guarantees, worked examples of test changes and their implementation requirements, and a small-circuit reference implementation for differential expectation tests. These results help testers decide what to execute, what to change, and what remains unknown. The evidence is limited to proofs and executable examples under the ideal model.

Quantum mutation testing introduces circuit changes to assess whether a test suite detects faults. This paper asks how to distinguish semantic equivalence, changes hidden by the selected input or measurement, and insufficient statistical evidence, and how those distinctions should guide execution and mutation-score reporting. For ideal unitary circuits with one gate or subcircuit replacement, it combines backward light cones, commutation checks, exact response formulas, and statistical intervals into an execution planner. Writing $W$ for the relative replacement, $\tau$ for the mutation-site state, and $O_j$ for the backward-propagated observable, the expectation gap is $|\operatorname{Tr}[(O_j-W^\dagger O_jW)\tau]|$. For involutory-axis rotations, two state expectations determine the response at every angle, including all zero-gap strengths and its maximum. The state-independent bound $\Delta\leq R_O|\sin(\eta/2)|$ permits below-threshold skips and also bounds measurement-distribution changes after normalization. Under pure-stabilizer, Pauli, and Clifford-suffix assumptions, the gap has only three shapes: $0$, $|\sin\eta|$, or $1-\cos\eta$, classifiable in polynomial time. Valid simultaneous certificates yield mutation-score bounds $W_D/W\leq\mathrm{MS}\leq(W_D+W_U)/W$, retaining survivor and unresolved weights. The results use standard quantum response identities, norm duality, stabilizer simulation, and statistical coverage arguments, supported by worked examples and a small-circuit reference implementation. General state-dependent calculations can remain exponential; optimal Helstrom measurements may be inadmissible or costly. Noise, resets, mid-circuit measurements, and classical control are excluded. The prevalence of the survival mechanisms, net execution savings, and augmentation benefits on independent mutants remain empirically unvalidated.