Variations on Classical and Quantum Extractors

Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors $\psi$ with a bound on the second largest eigenvalue $\lambda_{2}(\psi^{\dagger}\circ\psi)$ are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size $d=O(\log(1/\epsilon))$, where $\epsilon>0$ denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound $d\geq\log(1/\epsilon)$.Comment: 7 pages, slightly enhanced IEEE ISIT submission including all the proof

Quantum-proof randomness extractors via operator space theory

Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantum-proof. We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game) such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that can not be violated by quantum mechanics.Comment: v3: 34 pages, published versio

Towards the First Practical Applications of Quantum Computers

Noisy intermediate-scale quantum (NISQ) computers are coming online. The lack of error-correction in these devices prevents them from realizing the full potential of fault-tolerant quantum computation, a technology that is known to have significant practical applications, but which is years, if not decades, away. A major open question is whether NISQ devices will have practical applications. In this thesis, we explore and implement proposals for using NISQ devices to achieve practical applications. In particular, we develop and execute variational quantum algorithms for solving problems in combinatorial optimization and quantum chemistry. We also execute a prototype of a protocol for generating certified random numbers. We perform our experiments on a superconducting qubit processor developed at Google. While we do not perform any quantum computations that are beyond the capabilities of classical computers, we address many implementation challenges that must be overcome to succeed in such an endeavor, including optimization, efficient compilation, and error mitigation. In addressing these challenges, we push the limits of what can currently be done with NISQ technology, going beyond previous quantum computing demonstrations in terms of the scale of our experiments and the types of problems we tackle. While our experiments demonstrate progress in the utilization of quantum computers, the limits that we reached underscore the fundamental challenges in scaling up towards the classically intractable regime. Nevertheless, our results are a promising indication that NISQ devices may indeed deliver practical applications.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163016/1/kevjsung_1.pd

Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations

In the first part of this thesis, we discuss the algebraic approach to classical and quantum physics and develop information theoretic concepts within this setup. In the second part, we discuss the uncertainty principle in quantum mechanics. The principle states that even if we have full classical information about the state of a quantum system, it is impossible to deterministically predict the outcomes of all possible measurements. In comparison, the perspective of a quantum observer allows to have quantum information about the state of a quantum system. This then leads to an interplay between uncertainty and quantum correlations. We provide an information theoretic analysis by discussing entropic uncertainty relations with quantum side information. In the third part, we discuss the concept of randomness extractors. Classical and quantum randomness are an essential resource in information theory, cryptography, and computation. However, most sources of randomness exhibit only weak forms of unpredictability, and the goal of randomness extraction is to convert such weak randomness into (almost) perfect randomness. We discuss various constructions for classical and quantum randomness extractors, and we examine especially the performance of these constructions relative to an observer with quantum side information. In the fourth part, we discuss channel simulations. Shannon's noisy channel theorem can be understood as the use of a noisy channel to simulate a noiseless one. Channel simulations as we want to consider them here are about the reverse problem: simulating noisy channels from noiseless ones. Starting from the purely classical case (the classical reverse Shannon theorem), we develop various kinds of quantum channel simulation results. We achieve this by using classical and quantum randomness extractors that also work with respect to quantum side information.Comment: PhD thesis, ETH Zurich. 214 pages, 13 figures, 1 table. Chapter 2 is based on arXiv:1107.5460 and arXiv:1308.4527 . Section 3.1 is based on arXiv:1302.5902 and Section 3.2 is a preliminary version of arXiv:1308.4527 (you better read arXiv:1308.4527). Chapter 4 is (partly) based on arXiv:1012.6044 and arXiv:1111.2026 . Chapter 5 is based on arXiv:0912.3805, arXiv:1108.5357 and arXiv:1301.159