How much can we learn about a quantum device from a limited number of experiments?

How can one figure out how well a quantum computer works and what its dominant errors are? These devices are hard to check for the same reason they are interesting: in the regime that matters, no classical computer can predict the correct output beforehand. Quantum system characterization aim to closes this gap.
We design protocols that turn measurement data into a (partially very partial) description of a device, and aim to support every protocol by rigorous guarantee.
Our methods cover a wide range of measurement and classical postprocessing costs that by the desired information gain: 

• randomized benchmarking and similar cheap benchmarks, which return an average error rate,
• classical shadows, which estimate many properties of a state from few randomized measurements, and
• gate set tomography, which reconstructs states, gates and readout self-consistently.

We aim to answer the questions of 

• How many experimental runs does a given guarantee cost?
• Do the guarantees survives realistic laboratory settings? 

What we work on

• Quantum state and process tomography, including rigorous recovery guarantees and error bounds.
• Compressive tomography: exploiting low-rank and sparsity structure so that far fewer measurement settings suffice.
• Gate set tomography, which characterises gates, state preparation and measurement together and without assuming any of them to be perfect.
• Randomized benchmarking in its many variants, and the mathematics of the random gate ensembles it relies on.
• Classical shadows and shadow estimation: predicting many observables at once from randomized measurements.
• Certification of quantum states, gates and devices, and the question of when a certification test is actually sound.

Selected results

• A tutorial on quantum system certification.
  Our tutorial collects the scattered techniques for verifying that a quantum device works accuratly into one framework with explicit sample-complexity statements.
  • Kliesch, Roth, Theory of quantum system certification: a tutorial, [arXiv:2010.05925], PRX Quantum 2, 010201 (2021).
• Gate set tomography (GST) made cheaper and more flexible.
  GST is the arguable most informative characterization method but also the most expensive one. We develop a new GST data processing pipeline for constructing an entire gate set from fewer experiments in a more flexible way by exploiting tensor network techniques and the low-rank structures.
  • Brieger, Roth, Kliesch, Compressive gate set tomography, [arXiv:2112.05176], PRX Quantum 4, 010325 (2023).
• Guarantees for compressed-sensing process tomography.
  • Kliesch, Kueng, Eisert, Gross, Guaranteed recovery of quantum processes from few measurements, [arXiv:1701.03135], Quantum 3, 171 (2019).
• Analytical classical shadow estimators without group structure.
  • Arienzo, Heinrich, Roth, Kliesch, Closed-form analytic expressions for shadow estimation with brickwork circuits, Quantum Inf. Comput. 23, 961–993 (2023), [doi:10.26421/QIC23.11-12-5]; Presented as a talk at TQC 2023, Aveiro.
• Do classical shadows still work under gate-independent noise?
  Short answer: yes, for Pauli-type observables and no for observables with large stabilizer rank.
  • Brieger, Heinrich, Roth, Kliesch, Stability of classical shadows under gate-dependent noise, [arXiv:2310.19947], Phys. Rev. Lett. 134, 090801 (2025).
• Randomized benchmarking with random circuits.
  Rigorous randomized benchmarking (RB) theory assumed exact sampling from a group. We extend RB to the shallow random circuits used in practice, addressing scalability issues of RB to many qubits while keeping the error bars meaningful.
  • Heinrich, Kliesch, Roth, Randomized benchmarking with random quantum circuits, [arXiv:2212.06181], preprint (2022). Presented at QIP 2023, Ghent.
• Tomography with average gate fidelities.
  • Roth, Kueng, Kimmel, Liu, Gross, Eisert, Kliesch, Recovering quantum gates from few average gate fidelities,  Phys. Rev. Lett. 121, 170502 (2018), [doi:10.1103/PhysRevLett.121.170502], presented as a talk at QIP 2019, Boulder.
• Sound gate set certification.
  Common techniques like randomized benchmarking are guaranteed to give good scores if the gate implementations are good. Here, we achieve the coverse: If our certification test is passed then it is also guaranteed that the gate set is good -- that's the direction one usually wants.
  • Nöller, Miklin, Kliesch, Gachechiladze, Classical certification of quantum gates under the dimension assumption, [arXiv:2401.17006], Quantum 9, 1825 (2025);
  • Nöller, Miklin, Kliesch, Gachechiladze, Sound certification of memory-bounded quantum computers, [arXiv:2411.04215], preprint (2024). Presented as SPAM-free sound certification of quantum gates via quantum system quizzing at TQC 2025, Bengaluru, and as Self-testing memory-bounded quantum computers at QCTiP 2025, Berlin.

Funding

This work is supported by the DFG through the Emmy Noether Programme which funds early-career researchers to lead an independent junior research group.
Our group is Verification and characterization of quantum technologies, DFG project [441423094].