Computational depth complexity of measurement-based quantum computation

TL;DR

This paper demonstrates that measurement-based quantum computation (one-way model) has equivalent computational power to unbounded fan-out quantum circuits, highlighting its potential for efficient quantum algorithms and its classical-quantum hybrid advantages.

Contribution

It proves the computational equivalence between the one-way quantum model and unbounded fan-out circuits, clarifying the model's power and its implications for quantum algorithms.

Findings

Quantum Fourier transform can be approximated in constant depth in the one-way model.

Factorization can be performed by a polytime classical algorithm with a constant-depth one-way quantum computer.

The extra power of the one-way model stems from unbounded classical parity gates in constant depth.

Abstract

We prove that one-way quantum computations have the same computational power as quantum circuits with unbounded fan-out. It demonstrates that the one-way model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It confirms and completes previous results which have pointed out, for some specific problems, a depth separation between the one-way model and the quantum circuit model. Since one-way model has the same computational power as unbounded quantum fan-out circuits, the quantum Fourier transform can be approximated in constant depth in the one-way model, and thus the factorisation can be done by a polytime probabilistic classical algorithm which has access to a constant-depth one-way quantum computer. The extra power of the one-way model, comparing with the quantum circuit model, comes from its classical-quantum hybrid nature. We show that this extra power is reduced to the capability to perform unbounded classical parity gates in constant depth.