Implementation of single-qubit measurement-based t-designs using IBM processors

TL;DR

This paper explores implementing measurement-based t-designs on IBM superconducting quantum processors, demonstrating the challenges posed by noise and the need for noise reduction to achieve accurate designs.

Contribution

It presents the first implementation of an exact single-qubit 3-design on IBM quantum hardware using graph state measurements, highlighting noise limitations.

Findings

Exact 3-design was not achieved due to depolarising noise.

Approximate 2-design was partially successful but not perfect.

Noise reduction is essential for practical measurement-based t-designs.

Abstract

Random unitary matrices sampled from the uniform Haar ensemble have a number of important applications both in cryptography and in the simulation of a variety of fundamental physical systems. Since the Haar ensemble is very expensive to sample, pseudorandom ensembles in the form of t-designs are frequently used as an efficient substitute, and are sufficient for most applications. We investigate t-designs generated using a measurement-based approach on superconducting quantum computers. In particular, we implemented an exact single-qubit 3-design on IBM quantum processors by performing measurements on a 6-qubit graph state. By analysing channel tomography results, we were able to show that the ensemble of unitaries realised was a 1-design, but not a 2-design or a 3-design under the test conditions set, which we show to be a result of depolarising noise during the measurement-based process. We obtained improved results for the 2-design test by implementing an approximate 2-design, in which measurements were performed on a smaller 5-qubit graph state, but the test still did not pass for all states. This suggests that the practical realisation of measurement-based t-designs on superconducting quantum computers will require further work on the reduction of depolarising noise in these devices.