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

TL;DR

This paper demonstrates that adaptive measurement-based quantum computation significantly reduces the resource requirements for computing Boolean functions, especially mod-$p$, by leveraging adaptive strategies and quantum signal processing.

Contribution

It introduces adaptive $l2$-MBQC algorithms that compute mod-$p$ functions with optimal space-time resource scaling, surpassing nonadaptive methods.

Findings

Adaptive $l2$-MBQC reduces exponential resource needs of nonadaptive approaches.

Constructs algorithms based on quantum signal processing for mod-$p$ functions.

Provides an alternative proof of a separation between quantum and classical circuit powers.

Abstract

The limited computational power of constant-depth quantum circuits can be boosted by adapting future gates according to the outcomes of mid-circuit measurements. We formulate computation of a variety of Boolean functions in the framework of adaptive measurement-based quantum computation using a cluster state resource and a classical side-processor that can add bits modulo 2, so-called $l2$-MBQC. Our adaptive approach overcomes a known challenge that computing these functions in the nonadaptive setting requires a resource state that is exponentially large in the size of the computational input. In particular, we construct adaptive $l2$-MBQC algorithms based on the quantum signal processing technique that compute the mod-$p$ functions with the best known scaling in the space-time resources (i.e., qubit count, quantum circuit depth, classical memory size, and number of calls to the side-processor). As the subject is diverse and has a long history, the paper includes reviews of several previously constructed algorithms and recasts them as adaptive $l2$-MBQCs using cluster state resources. Our results constitute an alternative proof of an old theorem regarding an oracular separation between the power of constant-depth quantum circuits and constant-depth classical circuits with unbounded fan-in NAND and mod-$p$ gates for any prime $p$.