Operational Quantum Average-Case Distances

Filip B. Maciejewski1,2, Zbigniew Puchała3,4, and Michał Oszmaniec1

1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
2Research Institute for Advanced Computer Science (RIACS), USRA, Moffett Field, CA
3Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 44-100 Gliwice, Poland
4Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland

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We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) between output statistics of protocols in which quantum objects are intertwined with random circuits and measured in standard basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects – the average-case distances (ACDs). We apply them to analyze the effects of noise in quantum advantage experiments and for efficient discrimination of high-dimensional states and channels without quantum memory. We argue that ACDs are better suited for assessing the quality of NISQ devices than common distance measures such as trace distance or the diamond norm.

In the world of Noisy Intermediate Scale Quantum (NISQ) devices, traditional metrics like trace distance or diamond norm provide a worst-case view of how different two quantum protocols are. These metrics may be impractical, often requiring complex circuits and potentially exaggerating the impact of noise or errors. Our study introduces quantum Average-Case (AC) distances, which consider the total variation between the outputs of two protocols using random circuits. We show that these AC distances can be approximated by simple second-degree polynomials in objects of interest (states, measurements, or general channels), and we argue that they are more practical for assessing real-world quantum devices than standard measures.

We apply AC distances to evaluate the impact of noise on quantum advantage protocols based on random circuit sampling. AC distances provide both lower and upper bounds for average total variation, helping to quantify how far noisy outputs stray from ideal or become trivially uniform. For instance, we show that even in the absence of gate and state-preparation noise, the local, symmetric bitflip error in measurements causes noisy distribution to approach trivial one exponentially quickly in system size.

Our findings also extend to randomized quantum algorithms. AC distances can efficiently quantify the distinguishability of quantum objects using simple, local random circuits. We study two scenarios related to those recently analyzed in the context of so-called Quantum Algorithmic Measurement and complexity growth of quantum circuits: (i) distinguishing Haar random N qubit pure state from maximally mixed state and (ii) distinguishing N qubit Haar random unitary from maximally depolarizing channel. Our findings imply that protocols employing random circuits can be used to efficiently discriminate quantum objects. Since they do not depend on the objects to be distinguished, randomized measurement schemes can be interpreted as "universal discriminators", analogous to the SWAP test but not requiring the usage of entanglement or coherent access to copies of quantum systems.

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Cited by

[1] Giacomo De Palma and Dario Trevisan, "The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices", Annales Henri Poincaré 24 12, 4237 (2023).

[2] Michał Oszmaniec, Michał Horodecki, and Nicholas Hunter-Jones, "Saturation and recurrence of quantum complexity in random quantum circuits", arXiv:2205.09734, (2022).

[3] Filip B. Maciejewski, Zbigniew Puchała, and Michał Oszmaniec, "Exploring Quantum Average-Case Distances: proofs, properties, and examples", arXiv:2112.14284, (2021).

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