The Time-Space Complexity of Checking Multiple Assertions in Quantum Programs

Runtime assertions are a promising mechanism for testing and debugging quantum programs. But unlike the classical world, checking a quantum program that contains multiple assertions often requires using additional space or running the program additional times. For example, on current quantum hardware where mid-circuit measurement is restricted or costly, an assertion's pass/fail outcome cannot be revealed immediately. Instead, it is routed into an ancilla qubit during execution and read out by a terminal measurement. For a program with $n$ assertions, a naive strategy uses $n$ ancillas to learn all $n$ outcomes, while an alternative uses one ancilla but repeats program execution over $n$ rounds, checking one assertion per round. Both satisfy $S \cdot T = O(n)$, where $S$ is the number of ancillas and $T$ the number of executions: a fundamental time-space trade-off. Can one do asymptotically better? We reveal that the answer depends sharply on the information to be learned. Reporting the outcomes of all assertions requires linear complexity, but two partial-information tasks of detecting whether any assertion fails, and of identifying the first failing assertion, require only logarithmic complexity -- an asymptotic improvement. Moreover, the checking strategies for these tasks can trade time for space in useful ways. In this work, we formalize the complexity of checking multiple assertions in a quantum program. Using this definition, we establish its landscape of asymptotic lower bounds and constructive upper bounds. We confirm via a case study on Grover's algorithm that the resource costs of constructed strategies match theoretical predictions, illustrating the practical design space for quantum programmers.