Exponentially-improved effective descriptions of physical bosonic systems

TL;DR

This paper demonstrates that under certain energy conditions, the effective dimension needed to describe bosonic quantum systems can be exponentially reduced, enabling more efficient simulation and learning algorithms.

Contribution

It identifies an energy condition that allows exponential improvement in the effective description of bosonic systems and applies this to enhance classical simulation methods.

Findings

Most bosonic states satisfy the energy condition.

The effective dimension scales as log(1/ε) instead of 1/ε².

Classical algorithms for simulating bosonic circuits are significantly improved.

Abstract

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 $\epsilon$, here we identify a natural energy condition which allows us to improve this scaling exponentially to $\log(1/\epsilon)$. We then prove that most bosonic quantum states satisfy this condition, and in particular those produced by combining Gaussian dynamics with generic energy-preserving dynamics, which include the output states of universal bosonic quantum circuits. We apply this finding to enhance learning algorithms for bosonic quantum states and we further 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 classical simulation algorithms for universal bosonic quantum circuits. Our results demonstrate that physical bosonic systems are significantly more well-behaved than previously assumed, allowing for efficient descriptions even at high precision.