Quantum Oracles in Constant Depth with Measurement-Based Quantum Computation

TL;DR

This paper demonstrates that any quantum oracle for a classical function can be implemented in constant depth within measurement-based quantum computation, leveraging equivalences with the circuit model and addressing open questions about disjunction implementation.

Contribution

It establishes the possibility of constant-depth quantum oracle implementation in MBQC and solves an open problem regarding disjunctions.

Findings

Any quantum oracle can be implemented in constant depth in MBQC.

Disjunction can be exactly implemented in constant depth.

Equivalence between MBQC and circuit model with specific operations is shown.

Abstract

This paper shows that, in measurement-based quantum computation, it is possible to write any quantum oracle implementing a classical function in constant depth. The result is shown through the equivalence between MBQC and the circuit model where arbitrary rotations along $Z$ axis and unbounded fan-outs are elementary operations. A corollary of this result is that disjunction can be implemented exactly in constant-depth, answering an open question of H{\o}yer and \v{S}palek.