Thread by @markusheinrich_

We give general guarantees for benchmarking with random #quantum circuits. For fast-scrambling random circuits, linear depth is sufficient for function and sample-efficiency. Our analysis applies to prominent protocols such as linear XEB.

scirate.com/arxiv/2212.061…

We study a variant of RB, dubbed "filtered RB", which applies a random sequence of gates, followed by a basis measurement. The inversion is effectively done in post-processing, making the protocol easier to implement in experiments.

Following the modern approach of "measure first -- ask later", the same data may be used for other randomized protocols from the "toolbox", and gate set tomography.

We discuss adaptions to the protocol which may allow for constant depth. We expect that those are highly relevant for linear XEB, and the discussion of "bias" in this context. However, this part only holds under the "white noise assumption" common in quantum supremacy literature.

In contrast, the main analysis holds for arbitrary Markovian and time-stationary noise which is sufficiently weak. Moreover, the gates can be taken from any compact group and sampled according to an (almost) arbitrary probability measure.

The experimental samples are then used to estimate a function F(m). We show that if m is sufficiently large, then F(m) follows a single (matrix) exponential decay. Standard RB shows as many exponential decays as there are irreps of the group.

Our scheme allows to "filter" onto a specific irrep in post-processing, thereby isolating a single decay. This is particularly significant for benchmarking with smaller groups, where many irreps appear and the identification is otherwise impossible.

By studying the variance of our estimator, we also show that using a random circuit does not substantially change the sampling complexity compared to Haar-random gates. In particular, the protocol is sample-efficient for unitary 3-designs, local unitary 3-designs, the Pauli group

Due to the generality of our approach, it may also be used to study benchmarking of non-universal and analog simulators, as well as CV systems.

Big thanks to my co-authors Martin Kliesch and Ingo Roth, for this fruitful collaboration. From the initial idea to this final draft, it took almost 1.5 years and a lot of work. I gladly take the blame for the length of the paper :)

Get the summary during my talk at #QIP2023