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# The randomized measurement toolbox

## Abstract

Programmable quantum simulators and quantum computers are opening unprecedented opportunities for exploring and exploiting the properties of highly entangled complex quantum systems. The complexity of large quantum systems is the source of computational power but also makes them difficult to control precisely or characterize accurately using measured classical data. We review protocols for probing the properties of complex many-qubit systems using measurement schemes that are practical using today’s quantum platforms. In these protocols, a quantum state is repeatedly prepared and measured in a randomly chosen basis; then a classical computer processes the measurement outcomes to estimate the desired property. The randomization of the measurement procedure has distinct advantages. For example, a single data set can be used multiple times to pursue a variety of applications, and imperfections in the measurements are mapped to a simplified noise model that can more easily be mitigated. We discuss a range of cases that have already been realized in quantum devices, including Hamiltonian simulation tasks, probes of quantum chaos, measurements of non-local order parameters, and comparison of quantum states produced in distantly separated laboratories. By providing a workable method for translating a complex quantum state into a succinct classical representation that preserves a rich variety of relevant physical properties, the randomized measurement toolbox strengthens our ability to grasp and control the quantum world.

## Key points

• Increasingly sophisticated quantum simulators and quantum computers are becoming available, but they are difficult to characterize accurately using classically measured data.

• Randomized measurements provide a feasible procedure for converting a many-qubit quantum state to succinct classical data that can later be processed to estimate many properties of interest with rigorous guarantees.

• Randomized measurements are readily implemented in noisy intermediate-scale quantum devices by repeatedly preparing and measuring a quantum state in a randomly selected basis.

• Many applications of randomized measurements have been conceived and experimentally demonstrated, including Hamiltonian simulation tasks, probes of quantum chaos, measurements of non-local order parameters, and comparison of quantum states produced in distantly separated laboratories.

• Experimental imperfections in performing randomized measurements can often be easily mitigated; a wide range of different physical platforms realizing qubits, bosonic and fermionic quantum many-body systems is accessible.

• Viewed as a powerful quantum-to-classical converter, randomized measurements enable the use of classical algorithms to learn and predict properties of quantum systems that may never have been realized before.

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## Acknowledgements

A.E. acknowledges funding by the German National Academy of Sciences Leopoldina under grant no. LPDS 2021-02 and by the Walter Burke Institute for Theoretical Physics at Caltech. J.P. acknowledges funding from the US Department of Energy Office of Science, Office of Advanced Scientific Computing Research (DE-NA0003525, DE-SC0020290), and the National Science Foundation (NSF) (PHY-1733907). The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center. B.V. acknowledges funding from the French National Research Agency (ANR-20-CE47-0005, JCJC project QRand) and from the Austrian Science Foundation (FWF, P 32597 N). P.Z. acknowledges support by the US Air Force Office of Scientific Research (AFOSR) via IOE grant no. FA9550-19-1-7044 LASCEM, by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS), and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440).

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Elben, A., Flammia, S.T., Huang, HY. et al. The randomized measurement toolbox. Nat Rev Phys 5, 9–24 (2023). https://doi.org/10.1038/s42254-022-00535-2

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