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Enhanced measurements on quantum computers via the simultaneous probing of non-commuting Pauli operators

Rick P. A. Simon, Zheng Shi, Charlie Nation, Andrew Jena, and Luca Dellantonio

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AbstractMeasuring the state of quantum computers is a highly non-trivial task, with implications for virtually all quantum algorithms. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. Based on Bayesian statistics, it accurately estimates not only the average of the desired observable but also the error en route. This enables an adaptive shot-allocation algorithm that preferentially samples the most uncertain Pauli terms. In regimes with many non-commuting Pauli operators, this “double'' scheme can outperform the state-of-the-art measurement protocol in minimizing total shots for a given precision. We also numerically confirm the finding in previous theoretical works that the two-copy scheme incurs an overhead due to the square-root relationship between the variance of measured quantities and the number of measurement shots.Featured image: Scheme of our algorithm for the toy example $\hat{O}=\hat{IX} + \hat{XI} + \hat{XX} + \hat{YY} + \hat{ZZ}$ (weights equal one for all $i=1,\dots,5$). These Pauli strings are depicted as vertices of a graph, connected when they commute. All-to-all connected groups (pink, green, and blue) can be simultaneously measured. Alternatively, one can employ the more expensive double scheme to assess the magnitude of all $\hat{P}_{i}$. To choose which group to probe, we assign a virtual measurement that predicts the group that minimizes the estimation variance $(\Delta \widetilde{O})^2$ (based on previous measurements, green in the figure as an example). After the real measurement, $\widetilde{O}$ and $(\Delta \widetilde{O})^2$ are updated and are either outputted by the algorithm (if the budget $M$ is depleted) or employed to assign the next measurement.Popular summaryThe simulation of physical systems is one of the most promising applications for quantum computers, both in the near and long term. To carry out such simulations, the system is often broken down into smaller mathematical pieces, known as Pauli strings, which in general can only be measured a few at a time. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. A challenge of this approach is that the joint measurements do not reveal the signs of the Pauli strings. We address this with an advanced statistical framework that combines information from both the joint and standard measurements, while also giving real-time feedback on which terms are most important to reduce the error. Additionally, the same framework allows us to estimate the simulation error with only a single experiment run, rather than requiring multiple repeated experiments. Looking ahead, we aim to extend this scheme to recover sign information more directly, potentially with the help of ancillary systems, and to test its performance on realistic noisy hardware. Our approach opens new possibilities for improving measurement efficiency in quantum simulations, with potential applications in fields such as chemistry and materials science.► BibTeX data@article{Simon2026enhanced, doi = {10.22331/q-2026-10-07-2229}, url = {https://doi.org/10.22331/q-2026-10-07-2229}, title = {Enhanced measurements on quantum computers via the simultaneous probing of non-commuting {P}auli operators}, author = {Simon, Rick P. A. and Shi, Zheng and Nation, Charlie and Jena, Andrew and Dellantonio, Luca}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2229}, month = oct, year = {2026} }► References [1] Seth Lloyd.

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Blanchard, T. F. Sjolander, J. P. King, M. P. Ledbetter, E. H. Levine, V. S. Bajaj, D. Budker, and A. Pines. Measurement of untruncated nuclear spin interactions via zero- to ultralow-field nuclear magnetic resonance. Phys. Rev. B, 92: 220202, Dec 2015. 10.1103/PhysRevB.92.220202. https://doi.org/10.1103/PhysRevB.92.220202 [54] Ariel Shlosberg et al. In preparation. [55] Alexander Terenin, Shawfeng Dong, and David Draper. GPU-accelerated Gibbs sampling: A case study of the Horseshoe Probit model. Statistics and Computing, 29 (2): 301–310, March 2019. ISSN 1573-1375. 10.1007/s11222-018-9809-3. https://doi.org/10.1007/s11222-018-9809-3Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2026-10-11 10:00:59). Could not fetch ADS cited-by data during last attempt 2026-10-11 10:00:59: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractMeasuring the state of quantum computers is a highly non-trivial task, with implications for virtually all quantum algorithms. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. Based on Bayesian statistics, it accurately estimates not only the average of the desired observable but also the error en route. This enables an adaptive shot-allocation algorithm that preferentially samples the most uncertain Pauli terms. In regimes with many non-commuting Pauli operators, this “double'' scheme can outperform the state-of-the-art measurement protocol in minimizing total shots for a given precision. We also numerically confirm the finding in previous theoretical works that the two-copy scheme incurs an overhead due to the square-root relationship between the variance of measured quantities and the number of measurement shots.Featured image: Scheme of our algorithm for the toy example $\hat{O}=\hat{IX} + \hat{XI} + \hat{XX} + \hat{YY} + \hat{ZZ}$ (weights equal one for all $i=1,\dots,5$). These Pauli strings are depicted as vertices of a graph, connected when they commute. All-to-all connected groups (pink, green, and blue) can be simultaneously measured. Alternatively, one can employ the more expensive double scheme to assess the magnitude of all $\hat{P}_{i}$. To choose which group to probe, we assign a virtual measurement that predicts the group that minimizes the estimation variance $(\Delta \widetilde{O})^2$ (based on previous measurements, green in the figure as an example). After the real measurement, $\widetilde{O}$ and $(\Delta \widetilde{O})^2$ are updated and are either outputted by the algorithm (if the budget $M$ is depleted) or employed to assign the next measurement.Popular summaryThe simulation of physical systems is one of the most promising applications for quantum computers, both in the near and long term. To carry out such simulations, the system is often broken down into smaller mathematical pieces, known as Pauli strings, which in general can only be measured a few at a time. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. A challenge of this approach is that the joint measurements do not reveal the signs of the Pauli strings. We address this with an advanced statistical framework that combines information from both the joint and standard measurements, while also giving real-time feedback on which terms are most important to reduce the error. Additionally, the same framework allows us to estimate the simulation error with only a single experiment run, rather than requiring multiple repeated experiments. Looking ahead, we aim to extend this scheme to recover sign information more directly, potentially with the help of ancillary systems, and to test its performance on realistic noisy hardware. Our approach opens new possibilities for improving measurement efficiency in quantum simulations, with potential applications in fields such as chemistry and materials science.► BibTeX data@article{Simon2026enhanced, doi = {10.22331/q-2026-10-07-2229}, url = {https://doi.org/10.22331/q-2026-10-07-2229}, title = {Enhanced measurements on quantum computers via the simultaneous probing of non-commuting {P}auli operators}, author = {Simon, Rick P. A. and Shi, Zheng and Nation, Charlie and Jena, Andrew and Dellantonio, Luca}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2229}, month = oct, year = {2026} }► References [1] Seth Lloyd.

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Enhanced measurements on quantum computers via the simultaneous probing of non-commuting Pauli operators

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