A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (low-degree) multivariate identity testing are weaker. To obtain our lower bound it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the Shub-Smale tau-conjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a good enough bound on the number of real roots of sums of products of sparse polynomials (Descartes' rule of signs gives such a bound for sparse polynomials and products thereof). In this third version of our paper we show that the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent.
The Power of Shallow-depth Toffoli and Qudit Quantum Circuits
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；International Iberian Nanotechnology Laboratory（伊比利亚纳米技术国际实验室）
Worst-case depth hierarchy for shallow quantum circuits
Hon Hai (Foxconn) Quantum Computing Research Center（鸿海（富士康）量子计算研究中心）；University of Oxford（牛津大学）；The University of Hong Kong（香港大学）；University of Technology Sydney（悉尼科技大学）
Experimentally Testable Quantum Advantage in Shallow Circuits
IonQ Inc.（IonQ公司）；Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算机科学联合中心，美国国家标准与技术研究院/马里兰大学）；University of Maryland Institute for Advanced Computer Studies, University of Maryland（马里兰大学高级计算机研究所，马里兰大学）；Universidad de Sevilla（塞维利亚大学）；Instituto Carlos I de Física Teórica y Computacional, Universidad de Sevilla（塞维利亚大学卡洛斯一世理论与计算物理研究所）
Unconditional Quantum Advantage for Sampling with Shallow Circuits
Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Computer Science, Columbia University（哥伦比亚大学计算机科学系）；Perimeter Institute for Theoretical Physics（理论物理珀蒂默研究所）
Quantum hardness of learning shallow classical circuits
Center for Theoretical Physics, MIT（麻省理工学院理论物理中心）；CWI（CWI数学与计算机科学中心）；QuSoft（QuSoft量子软件研究中心）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）
本文研究了均匀分布和分布无关的PAC学习框架下等深度经典电路的量子可学性。为了实现我们的结果，我们建立了量子学习和量子安全密码系统之间的联系。然后，我们实现以下结果。1)在均匀分布下学习AC $^0$和TC $^0$的难度。我们的第一个结果涉及概念类TC $^0$（resp. AC $^0$），具有无界扇入多数门（分别为）的恒定深度和多项式大小的电路类。AND、OR、NOT门）。我们证明，如果不存在量子多项式时间（分别）。强次指数时间）算法来解决带误差的环学习（RLWE）问题，那么TC $^0$（分别为. AC $^0$）（即使访问量子成员资格查询）。该结果中的主要技术使用被认为是量子安全的显式伪随机函数来构造在均匀分布下难以量子学习的概念类。2)在PAC设置中学习TC $^0_2$的难度。我们的第二个结果表明，如果不存在LWE问题的量子多项式时间算法，则不存在TC类$^0_2$的多项式时间量子PAC学习算法，即，深度2 TC $^0$电路。在这个结果中的主要技术是建立一个连接的公钥密码系统的量子安全性和可学习的概念类，包括解密功能的密码系统。这对Aaronson [Aar 05]提出的“量子计算理论的十大半重大挑战”之一给出了强（条件）否定的答案，Aaronson询问AC $^0$和TC $^0$是否可以在量子多项式时间内进行PAC学习。
Circuit complexity lower bounds for quantum spin glasses
Broad Institute of MIT and Harvard（麻省理工学院和哈佛学院博德研究所）；Operations Research Center and Center for Statistics and Data Sciences, IDSS, Massachusetts Institute of Technology（麻省理工学院运筹学中心与统计与数据科学中心IDSS）
