[论文解读] The Computational Complexity of Linear Optics

Scott Aaronson|arXiv (Cornell University)|Jan 25, 2010

Quantum Computing Algorithms and Architecture参考文献 58被引用 12

一句话总结

本文提出了玻色采样（BosonSampling）模型，这是一种基于线性光学网络中非相互作用光子的量子计算框架，用于解决经典计算机难以处理的采样问题。论文证明，若经典计算机能够高效模拟该模型，则多项式层级将坍塌，从而为仅使用线性光学和永久值计算实现量子优越性提供了强有力证据。

ABSTRACT

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

研究动机与目标

• 为基于线性光学的量子计算机能够解决经典计算机无法处理的问题提供证据。
• 将玻色采样模型形式化为涉及线性光学网络中非相互作用玻色子的采样问题。
• 确立若该模型能被高效经典模拟，则在合理的复杂性理论假设下，多项式层级将坍塌。
• 尽管该模型不具备通用性，但仍激励利用当前光子技术实现其实验实现。

提出的方法

• 该模型生成相同光子，使其通过线性光学酉变换网络，并测量每个输出模式的光子数分布。
• 特定测量结果的概率与酉变换矩阵子矩阵的平方永久值成正比。
• 论文将模拟玻色采样的困难性归约为近似随机复高斯矩阵永久值的计算难度。
• 提出两个关键猜想：高斯矩阵永久值猜想（近似计算永久值为#P难）与永久值反浓度猜想（永久值模长通常较大）。
• 分析使用复杂性理论约化方法表明，若经典模拟是高效的，则 P^#P = BPP^NP，导致多项式层级坍塌。
• 该框架设计为可实验实现，利用当前光子技术，避免对量子比特纠缠或通用量子门的需求。

实验结果

研究问题

• RQ1 仅基于线性光学的量子计算机能否解决经典计算机无法处理的采样问题？
• RQ2 若经典计算机能高效模拟线性光学量子系统，将产生何种复杂性理论后果？
• RQ3 近似计算随机高斯矩阵的永久值是否如论文所猜想的那样为#P难？
• RQ4 能否证明随机矩阵的永久值通常较大，从而确保输出分布不会集中在少数结果上？
• RQ5 实现通过玻色采样证明量子优越性的最小实验设置是什么？

主要发现

• 若经典计算机能精确模拟玻色采样模型，则 P^#P = BPP^NP，意味着多项式层级坍塌至第三层。
• 在高斯矩阵永久值猜想与永久值反浓度猜想成立的前提下，即使是对玻色采样的近似经典模拟，也将导致多项式层级的相同坍塌。
• 论文提供了数值与理论证据支持这两个猜想，特别是表明高斯矩阵的永久值通常为 √n! / poly(n) 量级，从而确保采样分布具有非平凡性。
• 该模型虽不具备通用量子计算能力，但仍能解决被认为对经典计算机困难的采样问题。
• 该框架可利用当前光子技术实现，为在无需全尺寸量子计算机的情况下展示量子优越性提供了潜在路径。
• 结果表明，线性光学量子系统或能提供迄今最强有力的证据，证明扩展的邱奇-图灵论题被违反。