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）的量子计算近来引起了广泛关注。然而，针对这些在无限维希尔伯特空间上进行的玻色子计算，尚缺乏一套定义完善的量子复杂性理论。本工作为此研究纲领奠定了基础。我们基于BQP、局域哈密顿量问题及QMA的玻色子推广，引入了自然的复杂性类及相关问题。我们揭示了标准布尔经典计算与离散变量量子复杂性类之间的若干关联及微妙差异，并指明了若干待解决的重要开放性问题。具体而言：1. 我们证明二次（高斯型）量子动力学的计算能力等价于BQL类。更一般地，我们基于高阶门操作定义了具有有界误差概率的连续变量量子多项式时间计算类。由于希尔伯特空间的无限维特性，这些复杂性类是否具有可判定的上界并非显然。我们为此类定义了完全问题，并证明了BQP下界与EXPSPACE上界。进一步表明计算多项式玻色子观测量期望值的问题属于PSPACE。2. 我们证明判定玻色子哈密顿量谱有界性的问题是co-NP困难的。此外，我们发现求解玻色子哈密顿量基态能量的问题关键取决于优化所采用的能量约束态集合的非高斯星秩：对于恒定星秩情形，该问题是NP完全的；对于多项式有界星秩情形，该问题属于QMA；对于无界星秩情形，该问题是不可判定的。