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  • Open Access

Moments of Quantum Channel Ensembles

Matthew Duschenes1,2,3,*, Diego García-Martín4, Zoë Holmes5,6, and M. Cerezo4,†

  • *Contact author: mduschen@uwaterloo.ca
  • †Contact author: cerezo@lanl.gov
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PRX Quantum 7, 033050 – Published 9 September, 2026

DOI: https://doi.org/10.1103/grdp-jph9

Abstract

Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate t-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders t, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning and define useful concepts such as that of channel t-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping noise), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest in moment calculations.

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Popular Summary

Randomness plays a central role in modern quantum experiments, from benchmarking quantum processors to understanding noise and chaos. While the statistical properties of random unitary operations are well understood through tools such as unitary designs, far less is known about random quantum channels, which describe realistic, noisy dynamics. In this work, we develop a general framework to compute and compare statistical moments of ensembles of quantum channels. These moments capture how randomness spreads information, independently of any specific input state or measurement. We introduce natural reference ensembles for quantum channels and show that, unlike the unitary case, the maximally depolarizing channel plays a fundamental role. In particular, we establish hierarchies between channel ensembles and define the notion of channel t-designs. Our results reveal that random quantum channels are generically depolarizing, with noise sometimes suppressing fluctuations and sometimes enhancing them. This perspective unifies noise-induced concentration phenomena and provides new insight into noisy quantum circuits and variational algorithms.

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Outline

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PRX Quantum 7, 033050– Published 9 September, 2026
  • Received 22 January 2026
  • Revised 27 April 2026
  • Accepted 6 July 2026
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