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# Quantum certification and benchmarking

## Abstract

With the rapid development of quantum technologies, a pressing need has emerged for a wide array of tools for the certification and characterization of quantum devices. Such tools are critical because the powerful applications of quantum information science will only be realized if stringent levels of precision of components can be reached and their functioning guaranteed. This Technical Review provides a brief overview of the known characterization methods for certification, benchmarking and tomographic reconstruction of quantum states and processes, and outlines their applications in quantum computing, simulation and communication.

## Key points

• To ensure the correct functioning of a quantum device, its components must be certified and benchmarked.

• Certification, benchmarking and characterization tasks are particularly demanding in quantum simulation and computing applications.

• The most common tools for certification and benchmarking are surveyed and assessed according to the information that may be extracted from the protocol, the assumptions underlying the protocol, and its complexity in terms of samples, measurements and post-processing.

• We highlight particularly important concepts, protocols and applications and list key figures of merit — information gain, complexity and underlying assumptions — for several protocols.

## Relevant articles

• ### Scalable estimation of pure multi-qubit states

npj Quantum Information Open Access 13 May 2022

• ### Learning the noise fingerprint of quantum devices

Quantum Machine Intelligence Open Access 01 April 2022

• ### Optimal verification of the Bell state and Greenberger–Horne–Zeilinger states in untrusted quantum networks

npj Quantum Information Open Access 26 November 2021

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## Acknowledgements

We gratefully acknowledge discussions with D. Gross and J. Helsen, in addition to many other members of the scientific community. J.E. acknowledges funding from the DFG (CRC 183 project B01, EI 519/9-1, EI 519/14-1, EI 519/15-1, MATH+ project EF1-7, Deadalus, CRC 1114 project B06, FOR 2724), the BMWF (Q.Link.X), the BMWi (PlanQK), FQXi (2019-207756) and the Templeton Foundation. This work also received funding from the European Union (EU) Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS). N.W. acknowledges funding support from the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 750905. E.K. and D.M. acknowledge funding from the ANR project ANR-13-BS04-0014 COMB, E.K. by the EPSRC (EP/N003829/1).

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Correspondence to Jens Eisert or Dominik Hangleiter.

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## Glossary

Noisy intermediate-scale quantum (NISQ) devices

A first generation of quantum computational devices, partially already realized, that consist of up to a few hundred qubits (intermediate scale), which are only partially controlled and not protected by quantum error correction (and hence are noisy).

Tomographic reconstruction

The process of reconstructing an unknown quantum state or process from the observed statistics of a well-chosen collection of measurements carried out on an ensemble of identically prepared systems.

Resampling techniques

Refers to a body of methods in statistics aimed at estimating the precision of sample statistics, of validating models by using random subsets or of estimating confidence regions by statistical means.

Bayesian prior

In Bayesian statistical inference, an initial estimate of the probability of a hypothesis (the Bayesian prior) is updated according to Bayes’ theorem as more information becomes available.

Compressed sensing

A field of applied mathematics that studies rigorous guarantees for algorithmic solutions to linear inverse problems under structure assumptions, such as sparsity and low-rankness.

PAC

‘Probably approximately correct’ (PAC) learning is an algorithm that, given sample inputs and outputs of an unknown function f, returns a candidate function (hypothesis) that with high probability closely approximates f when applied to unseen data.

Unitary 2-designs

A finite set of unitary gates with the property that averages over any quadratic polynomial with respect to this set are equal to the Haar measure average over the full unitary group, useful in particular when estimating average fidelities.

Sub-universal models of quantum computing

Computational tasks that are insufficient for an arbitrary or universal quantum computation but remain provably hard for classical computers. They are potentially feasible using NISQ devices.

Clifford gates

A group of quantum gates that map Pauli operators onto Pauli operators and play a key role specifically in fault-tolerant quantum computing.

Quantum twirling lemma

A lemma that reduces arbitrary quantum maps to convex combinations of local Pauli maps by using averages over the Pauli or the Clifford group.

Post-quantum-secure collision-resistant hash functions

Functions scrambling a larger space into a smaller space (hash) which rarely map distinct inputs to the same output (collision-resistant) and cannot be efficiently inverted by quantum computers (post-quantum-secure).

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Eisert, J., Hangleiter, D., Walk, N. et al. Quantum certification and benchmarking. Nat Rev Phys 2, 382–390 (2020). https://doi.org/10.1038/s42254-020-0186-4

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