Combining Viewpoints in Quantum Theory

数学、计算机科学和物理学有着一种美丽而又奇特的共生关系。纯粹的数学推理可以对物理世界产生不可思议的影响。反之亦然，物理直觉和实验可以揭示看似完全抽象和脱离现实的数学真理。一方面，计算机科学可以被看作是模拟物理系统或辅助数学探索的工具。另一方面，它也是物理和数学的一个特例。毕竟，计算机是物理对象，因此受物理定律的支配。然而，驱动计算机科学的经典问题可以独立于计算机的物理构造方式来回答。例如，抽象推理本身就可以决定计算机原则上是否可以被编程来解决给定的问题，如果可以，效率如何。自从我第一次认识自然科学以来，我就对纯粹思维在自然科学中不可思议的有效性感到惊讶。这使我对量子计算机科学产生了兴趣，它是数学、计算机科学和物理学的交汇点。量子计算机本质上是我们可以控制的小型量子力学系统，用于使大自然比任何经典计算机更有效地解决某些问题。充分了解量子计算的细节，使其能够大规模部署，这显然会改变我们的社会。高级量子编程有几个障碍。最基本的问题直指量子力学反直觉性的核心。问题在于，量子力学的体制削弱了通常非常有效的逻辑思维和直觉的力量。例如，如果我给你一块饼干，让你选择茶或咖啡，你会期望得到茶和饼干，或者咖啡和饼干。但在量子力学定律下，这个最基本的逻辑真理不再成立。这是由于人们一次只能从一个经典视点从量子系统中提取数据。为了更多地了解这个系统，我们需要结合多个经典视点的测量结果。同样，量子计算机比经典计算机强大得多，正是因为量子程序员能够在不同的经典观点中工作，并在不同的经典观点之间切换。然而，由于某些原因，经典观点之间的转换没有得到系统的研究，很可能是因为量子系统通常是孤立地研究的。幸运的是，计算机科学、物理学和数学之间的联系比上面所描述的要深入得多。理论计算机科学擅长处理整个系统群落，包括并行的复合系统。因此，计算机科学技术的转移实际上影响了物理学和数学，这些概念还没有得到太多关注。我将把量子系统和经典观点同等地放在一个范畴中，研究它们之间的动态关系，并最终努力恢复抽象思维在这一领域的有效性。这将促进我们对自然的理论认识。同时，它将具有实际的好处，使量子协议和算法的设计更容易被非专业程序员所接受；我的目标是把我的饼干也吃掉。

Mathematics, computer science, and physics enjoy a beautiful but curious symbiotic relationship. Pure mathematical reasoning can uncannily have consequences for the physical world. Vice versa, physical intuition and experiment can uncover mathematical truths that seemed entirely abstract and divorced from reality. Computer science, on the one hand, can be regarded as a tool to simulate physical systems or aid mathematical exploration. On the other hand, it is also a special case of both physics and mathematics. Computers are, after all, physical objects, and are therefore governed by the laws of physics. Nevertheless, the classic questions driving computer science can be answered independently from the physical way computers are built. For example, abstract reasoning alone can decide whether a computer can in principle be programmed to solve a given problem, and if so, how efficiently.Ever since my first acquantaince I have been amazed by this unreasonable effectiveness of pure thought in physical sciences. This has drawn me to quantum computer science, which lies at the interface of mathematics, computer science, and physics. Quantum computers are essentially small quantum-mechanical systems that we can control, used to make nature solve certain problems much more efficiently than any classical computer could. Understanding quantum computing in enough detail to allow its large-scale deployment will clearly transform our society.There are several obstructions to high-level quantum programming. The most fundamental ones run straight to the heart of the counterintuitiveness of quantum mechanics. The problem is that the regime of quantum mechanics diminishes the power of logical thought and intuition that is usually so effective. For example, if I were to offer you a biscuit and a choice of tea or coffee, you would expect to receive either tea and a biscuit, or coffee and a biscuit. But under quantum-mechanical laws, this most basic logical truth no longer holds. This is caused by the fact that one can only extract data from a quantum system from one classical viewpoint at a time. To learn more about the system, we need to combine measurements from multiple classical viewpoints. Similarly, quantum computers are so much more powerful than classical ones precisely because of the ability of a quantum programmer to work in, and switch between, different classical viewpoints. However, the switching between classical viewpoints has escaped systematic study for some reason, most probably because quantum systems are usually studied in isolation. Fortunately, the ties between computer science, physics, and mathematics run even deeper than sketched above. Theoretical computer science excels in handling entire communities of systems, including compound ones, that all live in parallel. Thus, transfer of computer science techniques in fact influences physics and mathematics, where such notions have not received much attention. I will place quantum systems and classical viewpoints on an equal footing in a single category, investigate the dynamical relationships between them, and eventually endeavour to restore the effectiveness of abstract thought in this realm. This will advance our theoretical understanding of nature. At the same time, it will have practical benefits by making the design of quantum protocols and algorithms more accessible to non-specialist programmers; I aim to have my biscuit and eat it too.