Semi-device-dependent blind quantum tomography

Ingo Roth1,2, Jadwiga Wilkens1,2, Dominik Hangleiter3,2, and Jens Eisert2,4

1Quantum Research Center, Technology Innovation Institute (TII), Abu Dhabi, UAE
2Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
3Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland/NIST, USA
4Helmholtz-Zentrum Berlin für Materialien und Energie, Germany

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Extracting tomographic information about quantum states is a crucial task in the quest towards devising high-precision quantum devices. Current schemes typically require measurement devices for tomography that are a priori calibrated to high precision. Ironically, the accuracy of the measurement calibration is fundamentally limited by the accuracy of state preparation, establishing a vicious cycle. Here, we prove that this cycle can be broken and the dependence on the measurement device's calibration significantly relaxed. We show that exploiting the natural low-rank structure of quantum states of interest suffices to arrive at a highly scalable `blind' tomography scheme with a classically efficient post-processing algorithm. We further improve the efficiency of our scheme by making use of the sparse structure of the calibrations. This is achieved by relaxing the blind quantum tomography problem to the de-mixing of a sparse sum of low-rank matrices. We prove that the proposed algorithm recovers a low-rank quantum state and the calibration provided that the measurement model exhibits a restricted isometry property. For generic measurements, we show that it requires a close-to-optimal number of measurement settings. Complementing these conceptual and mathematical insights, we numerically demonstrate that robust blind quantum tomography is possible in a practical setting inspired by an implementation of trapped ions.

Quantum technologies, in particular, quantum computers require highly accurately functioning components. To test and improve these components one needs flexible diagnostic tools that are able to characterise them precisely. A basic such characterisation task is quantum state tomography, the task of determining the quantum state of a device from measurements. The precision of the tomography is limited by the accuracy of the calibration of the employed measurement device. Calibrating the measurement device in turn requires the highly accurate preparation of quantum states by such a device as we set out to characterise in the first place. We encounter a vicious cycle. A way to break this vicious cycle is to simultaneously determine the quantum state and the measurement device's calibration.

In our work, we give a mathematical proof that this is possible if the quantum states are sufficiently pure. We assume that measured data depend linearly on the state and a set of calibration parameters that model small deviations from a calibration baseline. We develop an algorithm that can be efficiently run on a classical computer and prove that given the data as an input, the algorithm converges to the correct state and calibration parameters provided the measurement is suitably unstructured. The number of required measurement scales close to optimally in the degrees of freedom of the problem. Since a pure state is described by fewer parameters than an arbitrary quantum state, our structure assumption allows us to additionally estimate calibration parameters without increasing the number of measurements. Furthermore, we numerically demonstrate that our method works in a setting that is motivated by ion traps experiments. Altogether, we develop a method that we expect to be useful in practice for high-precision state tomography based on the theoretical and numerical evidence.

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Cited by

[1] Shin-Liang Chen and Jens Eisert, "Semi-Device-Independently Characterizing Quantum Temporal Correlations", Physical Review Letters 132 22, 220201 (2024).

[2] Raphael Brieger, Ingo Roth, and Martin Kliesch, "Compressive Gate Set Tomography", PRX Quantum 4 1, 010325 (2023).

[3] Fernando G. S. L. Brandão, Richard Kueng, and Daniel Stilck França, "Fast and robust quantum state tomography from few basis measurements", arXiv:2009.08216, (2020).

[4] Jens Eisert, Axel Flinth, Benedikt Groß, Ingo Roth, and Gerhard Wunder, "Hierarchical compressed sensing", arXiv:2104.02721, (2021).

[5] Axel Flinth, Benedikt Groß, Ingo Roth, Jens Eisert, and Gerhard Wunder, "Hierarchical Isometry Properties of Hierarchical Measurements", arXiv:2005.10379, (2020).

[6] Burhan Gulbahar, "K-sparse Pure State Tomography with Phase Estimation", arXiv:2111.04359, (2021).

[7] Axel Flinth, Ingo Roth, Benedikt Groß, Jens Eisert, and Gerhard Wunder, "Guaranteed blind deconvolution and demixing via hierarchically sparse reconstruction", arXiv:2111.03486, (2021).

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