The effective description of a bosonic quantum system identifies the minimum finite dimension required to capture its essential dynamics. This effective dimension plays an important role in the complexity of classical and quantum algorithms for learning and simulating bosonic systems. While generic bosonic states require a dimension scaling as 1/\epsilon^2 for a precision of approximation \epsilon, we identify a natural energy condition that improves this scaling exponentially to log(1/\epsilon). We then prove that a broad class of physically relevant bosonic states satisfies this condition, including states produced by combining Gaussian dynamics with arbitrary energy-preserving dynamics, which encompass the output states of universal bosonic quantum circuits. We apply this finding to improve learning algorithms for bosonic quantum states and obtain new classical simulation algorithms for a large class of bosonic systems. Finally, using efficient decompositions of Kerr gates as sums of Gaussian gates, we significantly refine these simulation algorithms for universal bosonic quantum circuits.

I will begin the talk with a comprehensive introduction to the necessary technical background in bosonic quantum information. Familiarity with the basic language of finite-dimensional quantum information will be assumed, but no prior background in bosonic quantum information will be required. The talk is based on the paper: https://arxiv.org/pdf/2604.18720.

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