综合量子态和幺正的交互式证明
Interactive Proofs for Synthesizing Quantum States and Unitaries
摘要：
虽然量子复杂性理论传统上关注经典复杂性理论（如计算布尔函数）产生的问题，但研究固有量子操作的复杂性也是有意义的，如构建量子态或执行幺正变换。基于这个动机，我们定义了用于合成量子态和幺正的交互式证明模型，其中多项式时间量子验证者与不可信量子证明者交互，并且接受的验证者也输出目标态的近似（对于状态合成问题）或应用于输入状态的目标酉式的结果（对于酉合成问题）;此外，应该存在验证者以概率1接受的“诚实”证明者。我们的主要结果是包含$\mathsf{PSPACE} \subseteq \mathsf{IP}$的“状态合成”模拟：任何由多项式空间量子算法（可以运行指数时间）计算的状态序列都允许上述形式的交互协议。利用这个状态合成协议，我们也给出了一个单一的合成协议，多项式空间可计算的酉，只在一个多项式维的子空间上的非平凡的行为。我们在多个纠缠证明者的情况下也得到了类似的结果。
英文摘要：
Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state synthesis problem) or the result of the target unitary applied to the input state (for the unitary synthesis problem); furthermore there should exist an "honest" prover which the verifier accepts with probability 1. Our main result is a "state synthesis" analogue of the inclusion $\mathsf{PSPACE} \subseteq \mathsf{IP}$: any sequence of states computable by a polynomial-space quantum algorithm (which may run for exponential time) admits an interactive protocol of the form described above. Leveraging this state synthesis protocol, we also give a unitary synthesis protocol for polynomial space-computable unitaries that act nontrivially on only a polynomial-dimensional subspace. We obtain analogous results in the setting with multiple entangled provers as well.
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