We introduce a framework that allows for the exact analytic treatment of quantum dynamics subject to noise, captured by unitary evolution under a Hamiltonian drawn from a random-matrix ensemble. The unitary acts on a Hilbert space of arbitrary dimension $N$, which may include ancillary (e.g., “bath”) degrees of freedom, and the “strength” of the noise is tunable. While the methods we develop apply to generic random-matrix ensembles with a notion of rotation invariance, we explicitly consider the Gaussian unitary ensemble (GUE). Averaging over the ensemble of “noisy” Hamiltonians produces an effective quantum channel, the properties of which are analytically calculable and determined by the spectral form factors of the relevant ensemble. We compute spectral form factors of the GUE exactly for any finite $N$, along with the corresponding GUE quantum channel and its variance. Importantly, the GUE channel captures the effects of generic unitary sources of noise. Key advantages of our approach include the ability to access exact analytic results for any $N$, the ability to tune to the noise-free limit (in contrast, e.g., to the Haar ensemble), and analytic access to moments beyond the variance. We also highlight some unusual features of the GUE channel, including the nonmonotonicity of the coefficients of various operators as a function of noise strength and the failure to saturate the Haar-random limit, even with infinite noise strength.

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• Open quantum systems & decoherence
• Quantum chaos
• Quantum circuits
• Quantum error correction
• Quantum fluctuations & noise
• Quantum information theory
• Quantum spin models
• Random matrix theory
• SU(N) symmetries