Complexity of Quadratic Bosonic Hamiltonian Simulation: $\mathsf{BQP}$-Completeness and $\mathsf{PostBQP}$-Hardness

2026-03-27

AI Summary

This work investigates the computational complexity boundaries of simulating dynamics governed by exponentially large quadratic bosonic Hamiltonians. By leveraging quantum complexity theory, bosonic Hamiltonian modeling, and reductions from quantum circuits, the study establishes that the general quadratic boson simulation problem is $\mathsf{BQP}$-complete. Moreover, it demonstrates that an extended variant of the problem is $\mathsf{PostBQP}$-hard, revealing a complexity phase transition from efficiently quantum-solvable to postselection-hard. These results unify and generalize prior findings on classical oscillator networks and continuous-time quantum walks, precisely delineating the boundary between quantum and classical simulability for large-scale bosonic systems.

Abstract

The computational complexity of simulating the dynamics of physical quantum systems is a central question at the interface of quantum physics and computer science. In this work, we address this question for the simulation of exponentially large bosonic Hamiltonians with quadratic interactions. We present two results: First, we introduce a broad class of quadratic bosonic problems for which we prove that they are $\mathsf{BQP}$-complete. Importantly, this class includes two known $\mathsf{BQP}$-complete problems as special cases: Classical oscillator networks and continuous-time quantum walks. Second, we show that extending the aforementioned class to even more general quadratic Hamiltonians results in a $\mathsf{PostBQP}$-hard problem. This reveals a sharp transition in the complexity of simulating large quantum systems on a quantum computer, as well as in the difference in complexity between their simulation on classical and quantum computers.

Problem

• Bosonic Hamiltonian
• Quantum Simulation
• Computational Complexity
• BQP-completeness
• PostBQP-hardness

Innovation

• quadratic bosonic Hamiltonian
• BQP-completeness
• PostBQP-hardness
• quantum simulation
• computational complexity