Quantum computing involving physical systems with continuous degrees of freedom, such as the quantum states of light, has recently attracted significant interest. However, a well-defined quantum complexity theory for these bosonic computations over infinite-dimensional Hilbert spaces is missing. In this work, we lay foundations for such a research program. We introduce natural complexity classes and problems based on bosonic generalizations of BQP, the local Hamiltonian problem, and QMA. We uncover several relationships and subtle differences between standard Boolean classical and discrete variable quantum complexity classes and identify outstanding open problems. In particular: 1. We show that the power of quadratic (Gaussian) quantum dynamics is equivalent to the class BQL. More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error based on higher-degree gates. Due to the infinite dimensional Hilbert space, it is not a priori clear whether a decidable upper bound can be obtained for these classes. We identify complete problems for these classes and demonstrate a BQP lower and EXPSPACE upper bound. We further show that the problem of computing expectation values of polynomial bosonic observables is in PSPACE. 2. We prove that the problem of deciding the boundedness of the spectrum of a bosonic Hamiltonian is co-NP-hard. Furthermore, we show that the problem of finding the minimum energy of a bosonic Hamiltonian critically depends on the non-Gaussian stellar rank of the family of energy-constrained states one optimizes over: for constant stellar rank, it is NP-complete; for polynomially-bounded rank, it is in QMA; for unbounded rank, it is undecidable.

翻译：涉及连续自由度物理系统（如光的量子态）的量子计算最近引起了显著关注。然而，对于这些在无限维希尔伯特空间上进行的玻色子计算，尚缺乏明确定义的量子复杂性理论。在本工作中，我们为这一研究计划奠定了基础。我们基于BQP、局域哈密顿量问题和QMA的玻色子推广，引入了自然的复杂性类和相关问题。我们揭示了标准布尔经典复杂性类与离散变量量子复杂性类之间的若干关系与微妙差异，并指出了突出的开放问题。特别地：1. 我们证明了二次（高斯）量子动力学的能力等价于BQL类。更一般地，我们基于高次门定义了具有有界错误概率的连续变量量子多项式时间计算类。由于无限维希尔伯特空间的存在，这些类是否具有可判定的上界并非先验明确。我们确定了这些类的完全问题，并证明了BQP下界与EXPSPACE上界。进一步，我们证明了计算多项式玻色子可观测量期望值的问题属于PSPACE。2. 我们证明了判定玻色子哈密顿量谱有界性的问题是co-NP难的。此外，我们展示了寻找玻色子哈密顿量最小能量问题关键依赖于优化过程中能量受限态族的非高斯恒星秩：对于常数恒星秩，该问题为NP完全的；对于多项式有界秩，属于QMA；对于无界秩，则为不可判定的。