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.
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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Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Altman, E. et al. Quantum simulators: architectures and opportunities. PRX Quant. 2, 017003 (2021).
Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).
Schäfer, F., Fukuhara, T., Sugawa, S., Takasu, Y. & Takahashi, Y. Tools for quantum simulation with ultracold atoms in optical lattices. Nat. Rev. Phys. 2, 411–425 (2020).
Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).
Morgado, M. & Whitlock, S. Quantum simulation and computing with Rydberg-interacting qubits. AVS Quant. Sci. 3, 023501 (2021).
Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012).
Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021).
Kloeffel, C. & Loss, D. Prospects for spin-based quantum computing in quantum dots. Annu. Rev. Condens. Matter Phys. 4, 51–81 (2013).
Burkard, G., Ladd, T. D., Nichol, J. M., Pan, A. & Petta, J. R. Semiconductor spin qubits. Rev. Mod. Phys. (in the press); preprint available at https://arxiv.org/abs/2112.08863.
Slussarenko, S. & Pryde, G. J. Photonic quantum information processing: a concise review. Appl. Phys. Rev. 6, 041303 (2019).
Pelucchi, E. et al. The potential and global outlook of integrated photonics for quantum technologies. Nat. Rev. Phys. 4, 194–208 (2021).
Kjaergaard, M. et al. Superconducting qubits: current state of play. Annu. Rev. Condens. Matter Phys. 11, 369–395 (2020).
Alexeev, Y. et al. Quantum computer systems for scientific discovery. PRX Quant. 2, 017001 (2021).
Flammia, S. T., Gross, D., Liu, Y.-K. & Eisert, J. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys. 14, 095022 (2012).
Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. In STOC’16 — Proc. 48th Annual ACM SIGACT Symposium on Theory of Computing, 913–925 (ACM, 2016).
O’Donnell, R. & Wright, J. Efficient quantum tomography. In STOC’16 — Proc. 48th Annual ACM SIGACT Symposium on Theory of Computing, 899–912 (ACM, 2016).
Aaronson, S.Shadow tomography of quantum states. In STOC’18 — Proc. 50th Annual ACM SIGACT Symposium on Theory of Computing, 325–338 (ACM, 2018).
Aaronson, S. & Rothblum, G. N. Gentle measurement of quantum states and differential privacy. In STOC’19 — Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing, 322–333 (ACM, 2019).
Bădescu, C. & O’Donnell, R. Improved quantum data analysis. In STOC ’21 — Proc. 53rd Annual ACM SIGACT Symposium on Theory of Computing, 1398–1411 (ACM, 2021).
van Enk, S. J. & Beenakker, C. W. J. Measuring Trρn on single copies of ρ using random measurements. Phys. Rev. Lett. 108, 110503 (2012).
Elben, A., Vermersch, B., Dalmonte, M., Cirac, J. I. & Zoller, P. Rényi entropies from random quenches in atomic Hubbard and spin models. Phys. Rev. Lett. 120, 50406 (2018).
Elben, A., Vermersch, B., Roos, C. F. & Zoller, P. Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states. Phys. Rev. A 99, 1–12 (2019).
Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).
Paini, M. & Kalev, A. An approximate description of quantum states. Preprint at https://arxiv.org/abs/1910.10543 (2019).
Morris, J. & Dakić, B. Selective quantum state tomography. Preprint at https://arxiv.org/abs/1909.05880 (2019).
Knips, L. et al. Multipartite entanglement analysis from random correlations. npj Quant. Inf. 6, 51 (2020).
Ketterer, A., Wyderka, N. & Gühne, O. Characterizing multipartite entanglement with moments of random correlations. Phys. Rev. Lett. 122, 120505 (2019).
Chen, S., Yu, W., Zeng, P. & Flammia, S. T. Robust shadow estimation. PRX Quantum 2, 030348 (2021).
Koh, D. E. & Grewal, S. Classical shadows with noise. Quantum 6, 776 (2022).
Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124, 10504 (2020).
Zhu, D. et al. Cross-platform comparison of arbitrary quantum computations. Preprint at https://arxiv.org/abs/2107.11387 (2021).
Vermersch, B., Elben, A., Sieberer, L. M., Yao, N. Y. & Zoller, P. Probing scrambling using statistical correlations between randomized measurements. Phys. Rev. X 9, 21061 (2019).
Joshi, M. K. et al. Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions. Phys. Rev. Lett. 124, 240505 (2020).
Brydges, T. et al. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).
Elben, A. et al. Mixed-state entanglement from local randomized measurements. Phys. Rev. Lett. 125, 200501 (2020).
Elben, A. et al. Many-body topological invariants from randomized measurements in synthetic quantum matter. Sci. Adv. 6, eaaz3666 (2020).
Gross, D., Audenaert, K. & Eisert, J. Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48, 052104 (2007).
Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).
Vitale, V. et al. Symmetry-resolved dynamical purification in synthetic quantum matter. SciPost Phys. 12, 106 (2022).
Rath, A., Branciard, C., Minguzzi, A. & Vermersch, B. Quantum fisher information from randomized measurements. Phys. Rev. Lett. 127, 260501 (2021).
Evans, T. J., Harper, R. & Flammia, S. T. Scalable Bayesian Hamiltonian learning. Preprint at https://arxiv.org/abs/1912.07636 (2019).
Cotler, J. & Wilczek, F. Quantum overlapping tomography. Phys. Rev. Lett. 124, 100401 (2020).
Rath, A., van Bijnen, R., Elben, A., Zoller, P. & Vermersch, B. Importance sampling of randomized measurements for probing entanglement. Phys. Rev. Lett. 127, 200503 (2021).
Banaszek, K., Cramer, M. & Gross, D. Focus on quantum tomography. New J. Phys. 15, 125020 (2013).
Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. IEEE Trans. Inf. Theory 63, 5628–5641 (2017).
O’Donnell, R. & Wright, J. Efficient quantum tomography II. In STOC’17 — Proc. 49th Annual ACM SIGACT Symposium on Theory of Computing, 962–974 (ACM, 2017).
Chen, S., Huang, B., Li, J., Liu, A. & Sellke, M. Tight bounds for state tomography with incoherent measurements. Preprint at https://arxiv.org/abs/2206.05265 (2022).
Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).
Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).
Ohliger, M., Nesme, V. & Eisert, J. Efficient and feasible state tomography of quantum many-body systems. New J. Phys. 15, 015024 (2013).
Sugiyama, T., Turner, P. S. & Murao, M. Precision-guaranteed quantum tomography. Phys. Rev. Lett. 111, 160406 (2013).
Kueng, R., Rauhut, H. & Terstiege, U. Low rank matrix recovery from rank one measurements. Appl. Comput. Harmon. Anal. 42, 88–116 (2017).
Guta, M., Kahn, J., Kueng, R. & Tropp, J. A. Fast state tomography with optimal error bounds. J. Phys. A 53, 204001 (2020).
Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010).
Vovrosh, J. & Knolle, J. Confinement and entanglement dynamics on a digital quantum computer. Sci. Rep. 11, 11577 (2021).
Neven, A. et al. Symmetry-resolved entanglement detection using partial transpose moments. npj Quantum Inf. 7, 152 (2021).
Zeng, B., Chen, X., Zhou, D.-L. & Wen, X.-G. Quantum Information Meets Quantum Matter. From Quantum Entanglement to Topological Phases of Many-body Systems (Springer, 2019).
Pollmann, F. & Turner, A. M. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B 86, 125441 (2012).
Cian, Z.-P. et al. Many-body Chern number from statistical correlations of randomized measurements. Phys. Rev. Lett. 126, 050501 (2021).
Satzinger, K. J. et al. Realizing topologically ordered states on a quantum processor. Science 374, 1237–1241 (2021).
Kitaev, A. & Preskill, J. Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006).
Levin, M. & Wen, X. G. Detecting topological order in a ground state wave function. Phys. Rev. Lett. 96, 110405 (2006).
Flammia, S. T., Hamma, A., Hughes, T. L. & Wen, X. G. Topological entanglement Rényi entropy and reduced density matrix structure. Phys. Rev. Lett. 103, 261601 (2009).
Huang, H.-Y., Kueng, R., Torlai, G., Albert, V. V. & Preskill, J. Provably efficient machine learning for quantum many-body problems. Science 377, eabk3333 (2022).
Hartigan, J. A. & Wong, M. A. Algorithm AS 136: a K-means clustering algorithm. J. R. Stat. Soc. C 28, 100–108 (1979).
Swingle, B. Unscrambling the physics of out-of-time-order correlators. Nat. Phys. 14, 988–990 (2018).
Lewis-Swan, R. J., Safavi-Naini, A., Kaufman, A. M. & Rey, A. M. Dynamics of quantum information. Nat. Rev. Phys. 1, 627–634 (2019).
Liu, H. & Sonner, J. Quantum many-body physics from a gravitational lens. Nat. Rev. Phys. 2, 615–633 (2020).
Nie, X. et al. Detecting scrambling via statistical correlations between randomized measurements on an NMR quantum simulator. Preprint at https://arxiv.org/abs/1903.12237 (2019).
Qi, X.-L., Davis, E. J., Periwal, A. & Schleier-Smith, M. Measuring operator size growth in quantum quench experiments. Preprint at https://arxiv.org/abs/1906.00524 (2019).
Garcia, R. J., Zhou, Y. & Jaffe, A. Quantum scrambling with classical shadows. Phys. Rev. Research 3, 033155 (2021).
Joshi, L. K. et al. Probing many-body quantum chaos with quantum simulators. Phys. Rev. X 12, 011018 (2022).
Levy, R., Luo, D. & Clark, B. K. Classical shadows for quantum process tomography on near-term quantum computers. Preprint at https://arxiv.org/abs/2110.02965 (2021).
Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).
da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 107, 210404 (2011).
Fedorov, A., Steffen, L., Baur, M., da Silva, M. P. & Wallraff, A. Implementation of a Toffoli gate with superconducting circuits. Nature 481, 170–172 (2011).
Lanyon, B. P. et al. Efficient tomography of a quantum many-body-system. Nat. Phys. 13, 1158–1162 (2017).
Seshadri, A., Ringbauer, M., Monz, T. & Becker, S. Theory of versatile fidelity estimation with confidence. Preprint at https://arxiv.org/abs/2112.07947 (2021).
Seshadri, A., Ringbauer, M., Blatt, R., Monz, T. & Becker, S. Versatile fidelity estimation with confidence. Preprint at https://arxiv.org/abs/2112.07925 (2021).
Kliesch, M. & Roth, I. Theory of quantum system certification. PRX Quantum 2, 010201 (2021).
Carrasco, J., Elben, A., Kokail, C., Kraus, B. & Zoller, P. Theoretical and experimental perspectives of quantum verification. PRX Quantum 2, 010102 (2021).
Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595–600 (2018).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).
Bouland, A., Fefferman, B., Nirkhe, C. & Vazirani, U. On the complexity and verification of quantum random circuit sampling. Nat. Phys. 15, 159–163 (2018).
Liu, Y., Otten, M., Bassirianjahromi, R., Jiang, L. & Fefferman, B. Benchmarking near-term quantum computers via random circuit sampling. Preprint at https://arxiv.org/abs/2105.05232 (2021).
Choi, J. et al. Emergent randomness and benchmarking from many-body quantum chaos. Preprint at https://arxiv.org/abs/2103.03535 (2021).
Cotler, J. S. et al. Emergent quantum state designs from individual many-body wavefunctions. Preprint at https://arxiv.org/abs/2103.03536 (2021).
Garrison, J. R. & Grover, T. Does a single eigenstate encode the full Hamiltonian? Phys. Rev. X 8, 021026 (2018).
Qi, X.-L. & Ranard, D. Determining a local Hamiltonian from a single eigenstate. Quantum 3, 159 (2019).
Bairey, E., Arad, I. & Lindner, N. H. Learning a local Hamiltonian from local measurements. Phys. Rev. Lett. 122, 020504 (2019).
Bairey, E., Guo, C., Poletti, D., Lindner, N. H. & Arad, I. Learning the dynamics of open quantum systems from their steady states. New J. Phys. 22, 032001 (2020).
Li, Z., Zou, L. & Hsieh, T. H. Hamiltonian tomography via quantum quench. Phys. Rev. Lett. 124, 160502 (2020).
Kokail, C., van Bijnen, R., Elben, A., Vermersch, B. & Zoller, P. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936–942 (2021).
Calabrese, P. & Cardy, J. Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009).
Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Sample-efficient learning of interacting quantum systems. Nat. Phys. 17, 931–935 (2021).
Haah, J., Kothari, R. & Tang, E. Optimal learning of quantum Hamiltonians from high-temperature Gibbs states. Preprint at https://arxiv.org/abs/2108.04842 (2021).
Rouzé, C. & França, D. S.Learning quantum many-body systems from a few copies. Preprint at https://arxiv.org/abs/2107.03333 (2021).
Yu, W., Sun, J., Han, Z. & Yuan, X. Practical and efficient Hamiltonian learning. Preprint at https://arxiv.org/abs/2201.00190 (2022).
Seif, A., Hafezi, M. & Liu, Y.-K. Compressed sensing measurement of long-range correlated noise. Preprint at https://arxiv.org/abs/2105.12589 (2021).
Hangleiter, D., Roth, I., Eisert, J. & Roushan, P. Precise Hamiltonian identification of a superconducting quantum processor. Preprint at https://arxiv.org/abs/2108.08319 (2021).
Blume-Kohout, R. et al. Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography. Nat. Commun. 8, 14485 (2016).
Nielsen, E. et al. Gate set tomography. Quantum 5, 557 (2021).
Merkel, S. T. et al. Self-consistent quantum process tomography. Phys. Rev. A 87, 062119 (2013).
Madzik, M. T. et al. Precision tomography of a three-qubit donor quantum processor in silicon. Nature 601, 348–353 (2022).
Samach, G. O. et al. Lindblad tomography of a superconducting quantum processor. Preprint at https://arxiv.org/abs/2105.02338 (2022).
Brieger, R., Roth, I. & Kliesch, M. Compressive gate set tomography. Preprint at https://arxiv.org/abs/2112.05176 (2021).
Evans, T. et al. Fast bayesian tomography of a two-qubit gate set in silicon. Phys. Rev. Appl. 17, 024068 (2022).
van den Berg, E., Minev, Z. K., Kandala, A. & Temme, K. Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors. Preprint at https://arxiv.org/abs/2201.09866 (2022).
Flammia, S. T. Averaged circuit eigenvalue sampling. Preprint at https://arxiv.org/abs/2108.05803 (2021).
Harper, R., Flammia, S. T. & Wallman, J. J. Efficient learning of quantum noise. Nat. Phys. 16, 1184–1188 (2020).
Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).
Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).
Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).
Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).
Hadfield, C., Bravyi, S., Raymond, R. & Mezzacapo, A. Measurements of quantum Hamiltonians with locally-biased classical shadows. Comm. Math. Phys. 391, 951–967 (2022).
Hillmich, S., Hadfield, C., Raymond, R., Mezzacapo, A. & Wille, R. Decision diagrams for quantum measurements with shallow circuits. In International Conference on Quantum Computing and Engineering 24–34 (IEEE, 2021).
Huang, H.-Y., Kueng, R. & Preskill, J. Efficient estimation of Pauli observables by derandomization. Phys. Rev. Lett. 127, 030503 (2021).
Zhang, T. et al. Experimental quantum state measurement with classical shadows. Phys. Rev. Lett. 127, 200501 (2021).
Zhao, A., Rubin, N. C. & Miyake, A. Fermionic partial tomography via classical shadows. Phys. Rev. Lett. 127, 110504 (2021).
Yen, T.-C., Ganeshram, A. & Izmaylov, A. F. Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates. Preprint at https://arxiv.org/abs/2201.01471 (2022).
Shlosberg, A. et al. Adaptive estimation of quantum observables. Preprint at https://arxiv.org/abs/2110.15339 (2021).
Kohda, M. et al. Quantum expectation-value estimation by computational basis sampling. Phys. Rev. Research 4, 033173 (2022).
Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).
Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett. 122, 040504 (2019).
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).
Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021).
Haug, T., Self, C. N. & Kim, M. Large-scale quantum machine learning. Preprint at https://arxiv.org/abs/2108.01039 (2021).
Zhou, Y., Zeng, P. & Liu, Z. Single-copies estimation of entanglement negativity. Phys. Rev. Lett. 125, 200502 (2020).
Horodecki, P. Measuring quantum entanglement without prior state reconstruction. Phys. Rev. Lett. 90, 167901 (2003).
Carteret, H. A. Noiseless quantum circuits for the peres separability criterion. Phys. Rev. Lett. 94, 040502 (2005).
Carteret, H. A. Estimating the entanglement negativity from low-order moments of the partially transposed density matrix. Preprint at https://arxiv.org/abs/1605.08751 (2016).
Yu, X.-D., Imai, S. & Gühne, O. Optimal entanglement certification from moments of the partial transpose. Phys. Rev. Lett. 127, 060504 (2021).
Liu, Z. et al. Detecting entanglement in quantum many-body systems via permutation moments. Preprint at https://arxiv.org/abs/2203.08391 (2022).
Tran, M. C., Dakic, B., Arnault, F., Laskowski, W. & Paterek, T. Quantum entanglement from random measurements. Phys. Rev. A 92, 050301(R) (2015).
Tran, M. C., Dakic, B., Laskowski, W. & Paterek, T. Correlations between outcomes of random measurements. Phys. Rev. A 94, 042302 (2016).
Ketterer, A., Wyderka, N. & Gühne, O. Entanglement characterization using quantum designs. Quantum 4, 325 (2020).
Imai, S., Wyderka, N., Ketterer, A. & Gühne, O. Bound entanglement from randomized measurements. Phys. Rev. Lett. 126, 150501 (2021).
Ketterer, A., Imai, S., Wyderka, N. & Gühne, O. Statistically significant tests of multiparticle quantum correlations based on randomized measurements. Preprint at https://arxiv.org/abs/2012.12176 (2021).
Knips, L. A moment for random measurements. Quantum Views 4, 47 (2020).
Leone, L., Oliviero, S. F. E. & Hamma, A. Stabilizer Rényi entropy. Phys. Rev. Lett. 128, 050402 (2022).
Oliviero, S. F. E., Leone, L., Hamma, A. & Lloyd, S. Measuring magic on a quantum processor. Preprint at https://arxiv.org/abs/2204.00015 (2022).
Feldman, N., Kshetrimayum, A., Eisert, J. & Goldstein, M. Entanglement estimation in tensor network states via sampling. PRX Quantum 3, 030312 (2022).
Vermersch, B., Elben, A., Dalmonte, M., Cirac, J. I. & Zoller, P. Unitary n -designs via random quenches in atomic Hubbard and spin models: application to the measurement of Rényi entropies. Phys. Rev. A 97, 023604 (2018).
van den Berg, E., Minev, Z. K. & Temme, K. Model-free readout-error mitigation for quantum expectation values. Phys. Rev. A 105, 032620 (2022).
Gottesman, D. The Heisenberg representation of quantum computers. Preprint at https://arxiv.org/abs/quant-ph/9807006 (1998).
Hu, H.-Y., Choi, S. & You, Y.-Z. Classical shadow tomography with locally scrambled quantum dynamics. Preprint at https://arxiv.org/abs/2107.04817 (2021).
Bu, K., Koh, D. E., Garcia, R. J. & Jaffe, A. Classical shadows with Pauli-invariant unitary ensembles. Preprint at https://arxiv.org/abs/2202.03272 (2022).
Nakata, Y., Hirche, C., Koashi, M. & Winter, A. Efficient quantum pseudorandomness with nearly time-independent Hamiltonian dynamics. Phys. Rev. X 7, 021006 (2017).
Koenig, R. & Smolin, J. A. How to efficiently select an arbitrary Clifford group element. J. Math. Phys. 55, 122202 (2014).
Notarnicola, S. et al. A randomized measurement toolbox for Rydberg quantum technologies. Preprint at https://arxiv.org/abs/2112.11046 (2021).
Ringbauer, M. et al. A universal qudit quantum processor with trapped ions. Preprint at https://arxiv.org/abs/2109.06903 (2021).
García-Pérez, G. et al. Learning to measure: adaptive informationally complete generalized measurements for quantum algorithms. PRX Quant. 2, 040342 (2021).
Nguyen, H. C., Bönsel, J. L., Steinberg, J. & Gühne, O. Optimising shadow tomography with generalised measurements. Preprint at https://arxiv.org/abs/2205.08990 (2022).
Renes, J. M., Blume-Kohout, R., Scott, A. J. & Caves, C. M. Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004).
Scott, A. J. Tight informationally complete quantum measurements. J. Phys. A 39, 13507–13530 (2006).
Fischer, L. E. et al. Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space. Phys. Rev. Research 4, 033027 (2022).
Stricker, R. et al. Experimental single-setting quantum state tomography. Preprint at https://arxiv.org/abs/2206.00019 (2022).
National Academies of Sciences, Engineering, and Medicine. Manipulating Quantum Systems: An Assessment of Atomic, Molecular, and Optical Physics in the United States (National Academies Press, 2020); https://www.nap.edu/catalog/25613/manipulating-quantum-systems-an-assessment-of-atomic-molecular-and-optical.
Naldesi, P. et al. Fermionic correlation functions from randomized measurements in programmable atomic quantum devices. Preprint at https://arxiv.org/abs/2205.00981 (2022).
Struchalin, G., Zagorovskii, Y. A., Kovlakov, E., Straupe, S. & Kulik, S. Experimental estimation of quantum state properties from classical shadows. PRX Quantum 2, 010307 (2021).
LeCun, Y., Bottou, L., Bengio, Y. & Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE 86, 2278–2324 (1998).
Goodfellow, I., Bengio, Y. & Courville, A. Deep Learning. Adaptive Computation and Machine Learning (MIT Press, 2016); http://www.deeplearningbook.org/.
Silver, D. et al. Mastering the game of go without human knowledge. Nature 550, 354–359 (2017).
Jumper, J. et al. Highly accurate protein structure prediction with alphafold. Nature 596, 583–589 (2021).
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).
The authors declare no competing interests.
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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