Randomness Resources for Quantum Technologies

量子信息科学有望从根本上改变我们做事的方式，就像经典信息科学继续改变我们日常生活的方方面面一样。经典的信息科学告诉我们，破解密码是多么困难，或者一台计算机做一次计算需要多长时间；量子信息科学预测了从根本上安全的密码学，以及比任何可以想象的经典机器更快地解决某些问题的计算机。乍一看，认为随机性实际上可以帮助执行信息处理任务是令人惊讶的，然而它确实可以：例如，已知随机加密密钥是隐藏消息的最佳方式；更令人惊讶的是，在某些问题上，与其像经典计算机通常那样执行确定性算法，不如在计算解决方案时进行猜测——也就是说，调用随机性。因此我们说，随机性是经典信息论中的一种资源；手边有一枚可以抛掷的硬币是一种有形资产。当一个人想要测试一个复杂的设备或过程时，这尤其正确：向它发送随机输入，并调查输出的行为。我们也可以有意地将随机性引入量子信息协议，并询问这是否可以使某些任务变得更容易。事实证明，答案也是肯定的，这引发了诸如随机量子电路或随机量子纠错码等研究。在量子力学的形式化中，它们被表示为随机算子，而不是简单的随机数，但它们可以被认为是量子信息科学的资源，就像在经典情况下一样。然而，无论是经典还是量子，生成真正的随机资源都是非常困难的；人们可以想象，通过几十亿次抛硬币来加密tb级的信息。在实践中，我们依赖于所谓的伪随机资源，在给定有限的时间或计算能力的情况下，这些资源永远无法与真正的随机区分开来。如果我们考虑越来越复杂的测试来检查随机性，那么伪随机资源将通过这些测试达到一定程度的复杂性（超过一定程度就会失败）。这样的资源比真正的随机资源更容易创建，而伪随机数生成器是当今信息技术的基石。该研究项目旨在为量子信息技术提供伪随机资源。在量子领域，伪随机性的概念被称为量子“t型设计”。这些是资源——量子算子的集合——它们通过了随机性测试，达到了某种程度的复杂性（更准确地说，t对应于统计矩的程度）。该项目有两个主要组成部分；首先是对t型设计的数学结构进行系统研究，在此过程中发现新的t型设计，然后为特定的量子技术优化这些资源；在布里斯托尔大学，我们关注的一项技术是集成量子光子学，因此这个项目的第二部分将是利用我们的理论工作来提出和执行量子光子实验，以证明量子伪随机性。量子技术还处于起步阶段，这项研究将是理解和解决有效产生对信息科学至关重要的随机性问题的重要早期步骤。在短期内，研究结果将用于解决具有挑战性的问题，比如找到描述日益复杂的量子设备的最佳方法，就像英国量子技术网络（UK quantum Technology Network）数百家合作伙伴正在开发的那些设备一样。从长远来看，它将为任何量子平台提供定制的插件伪随机资源，这将在未来的众多量子信息应用中使用。

Quantum information science promises to fundamentally change the way we do things, not unlike how classical information science continues to change every aspect of our daily lives. Classical information science teaches us how difficult it is to break a cipher, or how long it will take a computer to do a calculation; quantum information science predicts fundamentally secure cryptography, and computers that solve certain problems faster than any conceivable classical machine. At first glance, it is surprising to think that randomness can actually help perform information processing tasks, and yet it can: for example, a random cryptographic key is known to be the best way to hide messages; more surprisingly, there exist problems where, rather than execute a deterministic algorithm as classical computers normally do, it is better to guess -- that is, invoke randomness -- while computing a solution. Thus we say that randomness is a resource in classical information theory; having a coin at hand that one can flip is a tangible asset. This is especially true when one wants to test a complex device or process: send it random inputs, and investigate how the outputs behave.We can also purposely introduce randomness into quantum information protocols and ask if this can make certain tasks easier. It turns out the answer is also yes, giving rise to the study, for example, of random quantum circuits, or random quantum error correcting codes. In the formalism of quantum mechanics these are expressed as random operators, rather than simple random numbers, but they can be thought of as resources for quantum information science in much the same way as in the classical case. However, both classically and quantumly, generating truly random resources is very difficult; one can imagine trying to encrypt terabytes of information by flipping a coin billions of times. In practice we rely on so-called pseudorandom resources that, given a finite amount of time or computing power, can never be distinguished from truly random. If we think of increasingly complex tests one might do to check for randomness, a pseudorandom resource will pass these tests up to a certain level of complexity (and fail beyond that). Such resources are much easier to create than truly random ones, and pseudorandom number generators are a cornerstone of today's information technologies.This research project aims to make pseudorandom resources available to quantum information technologies. In the quantum realm, the notion of pseudorandomness is captured by what are called quantum 't-designs'. These are resources -- ensembles of quantum operators -- that pass randomness tests up to some level of complexity (more precisely, t corresponds to the degree of a statistical moment). The project has two main components; the first will be a systematic study of the mathematical structure of t-designs, finding new ones along the way, and then optimising these resources for specific quantum technologies; at the University of Bristol a technology we focus on is integrated quantum photonics, and so the second part of this project will be to use our theoretical work to propose and perform quantum photonic experiments that demonstrate quantum pseudorandomness.Quantum technology is in its infancy, and this research will be an important early step in understanding and solving the problem of efficiently producing the randomness that is crucial to information science. In the short term, the results will be used to tackle challenging problems such as finding the best way to characterise increasingly complex quantum devices, like the ones being developed by hundreds of partners in the UK Quantum Technology Network. In the longer term, it will enable customised, plug-in pseudorandom resources for any quantum platform, which will be used in a multitude of future quantum information applications.