Quantum Merlin-Arthur with noisy channel

TL;DR

This paper proves that the quantum Merlin-Arthur complexity class remains unchanged even when the quantum channel is noisy, by employing measurement-based quantum computing and error correction techniques.

Contribution

It introduces a relaxed verification test for noisy graph states, maintaining QMA equivalence under noise with only single-qubit measurements.

Findings

QMA remains unchanged with noisy channels under certain error correction conditions.

Fault-tolerant measurement-based quantum computing can verify noisy graph states.

A new relaxed test accepts error-correctable noisy graph states.

Abstract

What happens if in QMA the quantum channel between Merlin and Arthur is noisy? It is not difficult to show that such a modification does not change the computational power as long as the noise is not too strong so that errors are correctable with high probability, since if Merlin encodes the witness state in a quantum error-correction code and sends it to Arthur, Arthur can correct the error caused by the noisy channel. If we further assume that Arthur can do only single-qubit measurements, however, the problem becomes nontrivial, since in this case Arthur cannot do the universal quantum computation by himself. In this paper, we show that such a restricted complexity class is still equivalent to QMA. To show it, we use measurement-based quantum computing: honest Merlin sends the graph state to Arthur, and Arthur does fault-tolerant measurement-based quantum computing on the noisy graph state with only single-qubit measurements. By measuring stabilizer operators, Arthur also checks the correctness of the graph state. Although this idea itself was already used in several previous papers, these results cannot be directly used to the present case, since the test that checks the graph state used in these papers is so strict that even honest Merlin is rejected with high probability if the channel is noisy. We therefore introduce a more relaxed test that can accept not only the ideal graph state but also noisy graph states that are error-correctable.