We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states can be sensitive to arbitrarily small changes in a coin's bias. This contrasts with classical probabilistic finite automata, whose sensitivity to changes in a coin's bias is bounded by a classic 1970 result of Hellman and Cover. Despite this finding, we are able to bound the power of advice coins for space-bounded classical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin, of languages decidable by classical and quantum polynomial-space machines with advice coins. Our main theorem is that both classes coincide with PSPACE/poly. Proving this result turns out to require substantial machinery. We use an algorithm due to Neff for finding roots of polynomials in NC; a result from algebraic geometry that lower-bounds the separation of a polynomial's roots; and a result on fixed-points of superoperators due to Aaronson and Watrous, originally proved in the context of quantum computing with closed timelike curves.
No Size-Preserving Amplification with Quantum Advice
National Taiwan University（台湾大学）；National Tsing Hua University. National Center for Excellence in Quantum Information Science and Engineering.（国立清华大学。量子资讯科学与工程卓越中心）
The Information Content of Systems in General Physical Theories
什么样的物体是量子态？它是一个编码成指数增长的信息量（在系统的大小中）的对象，还是更类似于概率分布？事实证明，这些问题对我们如何处理这些信息很敏感。例如，Holevo的界限告诉我们，n个量子比特只编码n比特的经典信息，但对于某些通信复杂性任务，量子和经典资源之间存在指数分离。我们不只是对比量子物理和经典物理，我们可以把两者放在一个广阔的物理理论景观中，并询问非量子（和非经典）理论与量子理论有何不同，或者比量子理论更强大。例如，在通信复杂性中，某些（非量子）理论可以使所有通信复杂性任务变得微不足道。在最近的工作[C。M. Lee和M. J. Hoban，Proc. Royal Soc. A 472（2190），2016]，我们证明了一般（非量子）物理理论中状态信息内容的巨大力量并不限于通信复杂性。我们表明，在一般的物理理论中，状态可以被视为计算机在这些理论中的“建议”，并且这个建议允许计算机轻松地解决任何决策问题。Aaronson强调了量子通信复杂性和接受量子建议的量子计算之间的密切联系，我们的工作进一步表明这是一个非常普遍的联系。在这项工作中，我们回顾了我们以前的工作中的结果，并讨论了通信复杂性和计算机之间的错综复杂的关系，采取一般理论的意见。
CQST, IIIT Hyderabad（印度理工学院海得拉巴分校量子科学与系统中心）；QuICS, UMIACS, University of Maryland, College Park（马里兰大学学院公园校区量子信息与计算科学研究所）；IQIM, CMS, Caltech（加州理工学院理论物理与量子信息研究所）；Faculty of Electrical Engineering, Czech Technical University in Prague（布拉格捷克理工大学电气工程学院）
