Guaranteed recovery of quantum processes from few measurements

Martin Kliesch1,2, Richard Kueng3,4,5, Jens Eisert4,6,7, and David Gross3,8

1Institute for Theoretical Physics, Heinrich Heine University Düsseldorf, Germany
2Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland
3Institute for Theoretical Physics, University of Cologne, Germany
4Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
5Institute for Quantum Information and Matter, California Institute of Technology, USA
6Department of Mathematics and Computer Science, Freie Universität Berlin, Germany
7Helmholtz-Zentrum Berlin für Materialien und Energie, Germany
8Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Australia

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Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.
Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and $\ell_2$-norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.

Characterizing the time evolution of a quantum system is a fundamental task both in theory and practice. Prospective quantum technologies, in particular, require precise control of both the systems and their evolutions (e.g., ``quantum gates’’). The task of inferring the state of a system from measurements is well-understood (``state tomography’’). In contrast, much less is known about characterizing evolutions from observations alone (``process tomography’’). In this work, we provide a practical procedure for this task.

Process tomography demands a huge number of measurements. However, many relevant time evolutions exhibit additional structure (approximately low Kraus rank). A central result of our paper states that significantly fewer measurements suffice for a full characterization of such evolutions. This includes unitary time evolution -- as given by the Schrödinger equation -- where our method achieves a quadratic reduction of the measurement effort over conventional characterization methods. Our reconstruction technique is conceptually simple and does not require any prior assumptions. Moreover, it is supported by analytic convergence guarantees, as well as extensive numerical demonstrations.

An important application concerns the implementation of quantum gates which form the basic building blocks of quantum circuits. Realistic implementations inevitably suffer from errors. The resulting (noisy) time evolution can be measured. In this way, one can learn what errors were made in order to correct them. Our proposed method is geared towards achieving this goal with substantially fewer measurements than traditional estimation techniques. In this way, it complements single-parameter performance certificates such as randomized benchmarking.

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