Section: New Results

Determinism and Computational Power of Real Measurement-based Quantum Computation

Participant : Luc Sanselme [contact] .

This is a joint work with Simon Perdrix (CNRS, Carte Team at Loria). This work has begun in 2014.

The starting point for this work was about a problem in «Quantum cloud computing». A person with a classical resource wants to perform a quantum computation. To do so he asks some quantum resources to perform his computation. The difficult part is that he wants to be sure that the quantum resources he asks to perform his computation don't cheat and return him the good results. This kind of «Quantum cloud computing» is called interactive proofs. The quantum resources are called the provers. Real Measurement-based quantum computing (MBQC) has been used for interactive proofs by McKague.

Measurement-based quantum computing (MBQC) is a universal model for quantum computation. The combinatorial characterization of determinism in this model, powered by measurements, and hence, fundamentally probabilistic, is the cornerstone of most of the breakthrough results in this field. To answer our question, we needed to develop some tools in this MBQC field. The most general known sufficient condition for a deterministic MBQC to be driven is that the underlying graph of the computation has a particular kind of flow called Pauli flow. The necessity of the Pauli flow was an open question. We showed that the Pauli flow is necessary for real-MBQC, and not in general providing counter-examples for (complex) MBQC. We explored the consequences of this result for real MBQC and its applications. Real MBQC and more generally real quantum computing is known to be universal for quantum computing. In the interactive proofs developed by McKague, the two-prover case corresponds to real-MBQC on bipartite graphs. While (complex) MBQC on bipartite graphs are universal, the universality of real MBQC on bipartite graphs was an open question. We showed that real bipartite MBQC is not universal: we proved that all measurements of real bipartite MBQC can be parallelized. Therefore, real bipartite MBQC leads to constant depth computations. As a consequence, McKague techniques cannot lead to two-prover interactive proofs.