SIMULATING QUANTUM CIRCUITS---UNDERSTANDING THE ROLES OF WIGNER NEGATIVITY, SYMMETRY, AND SYMMETRY BREAKING

这项提议是关于量子计算优势的理论；它提出的中心问题是“我们如何利用量子进行计算？”这种算法的想法可以追溯到4000多年前的巴比伦数学家。相比之下，量子算法的想法，即使用量子力学来解决复杂的数学问题，是非常新的。最初的步骤是由Deutsch、Feynman和Shor在大约30年前迈出的。相应地，我们还不太了解量子算法。像Shor和Grover的算法这样强大的例子是众所周知的，但它们很少。在这个提议中，我们试图更好地理解量子算法的量子力学基础，并扩展量子计算工具集。为此，我们仔细研究了各种已知的核心量子力学性质在量子加速中的作用，如Wigner函数负性和上下文相关性。我们还研究了可能出现量子加速的新起源的现象学领域，例如量子物质的兰姆达多面体和计算相。

This proposal is on the theory of quantum computational advantage; and the central question it asks is ``How can we harness the quantum for computation?''. The idea of the algorithm dates back to the Babylonian mathematicians, more than 4,000 years ago. The idea of quantum algorithms, i.e., using quantum mechanics to solve complex mathematical problems, is very recent in comparison. The first steps were made by Deutsch, Feynman and Shor, only about 30 years ago. Correspondingly, we don't understand quantum algorithms very well yet. Powerful examples such as Shor's and Grover's algorithms are known, but they are few and far between.In this proposal, we seek to better understand the quantum mechanical foundation of quantum algorithms, and to extend the quantum computational tool set. To this end, we scrutinize various known core quantum mechanical properties for their role in quantum speedup, such as Wigner function negativity and contextuality. We also investigate areas of phenomenology where novel origins of quantum speedup may arise, such as the Lambda polytopes and computational phases of quantum matter.