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The limited computational power of constant-depth quantum circuits can beboosted by adapting future gates according to the outcomes of mid-circuitmeasurements. We formulate computation of a variety of Boolean functions in theframework of adaptive measurement-based quantum computation using a clusterstate resource and a classical side-processor that can add bits modulo 2,so-called $l2$-MBQC. Our adaptive approach overcomes a known challenge thatcomputing these functions in the nonadaptive setting requires a resource statethat is exponentially large in the size of the computational input. Inparticular, we construct adaptive $l2$-MBQC algorithms based on the quantumsignal processing technique that compute the mod-$p$ functions with the bestknown scaling in the space-time resources (i.e., qubit count, quantum circuitdepth, classical memory size, and number of calls to the side-processor). Asthe subject is diverse and has a long history, the paper includes reviews ofseveral previously constructed algorithms and recasts them as adaptive$l2$-MBQCs using cluster state resources. Our results constitute an alternativeproof of an old theorem regarding an oracular separation between the power ofconstant-depth quantum circuits and constant-depth classical circuits withunbounded fan-in NAND and mod-$p$ gates for any prime $p$.

Quantum algorithms for classical Boolean functions via adaptive measurements: Exponential reductions in space-time resources